I

NATL INST OF STANDARDS & TECH R.I.C.

A1 11 01 725742

/Mechanics of pneumatic tires QC100 .U556 V1 22:1971 C.2 NBS-PUB-R 1<

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I UNITED STATES DEPARTMENT OF COMMERCE • Maurice H. Stans, Secretar^r

,j ^ O S NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

Mechanics of Pneumatic Tires

Samuel K. Cl.ark, Editor

Department of Engineering Mechanics

University of Michigan Ann Arbor, Michigan 48104

Prepared for the

Office of Vehicle Systems Research

Institute for Applied Technology

National Bureau of Standards

Washington, D.C. 20234

This Monograph was prepared in fulfillment of the National Bureau of Standards Contract CST-386

funded by the National Highway Traffic Safety Administration,

Department of Transportation,

through the NBS Interagency Agreement FH-1 1-6090.

National Bureau of Standards Monograph 122

Nat. Bur. Stand. (U.S.), Monogr. 122, 853 pages (Nov. 1971) CODEN: NBSMA

Issued November 1971

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD

Catalog No. C13.44:122), Price $4.75
Abstract

The pneumatic tire has been an integral part of automotive transportation almost

since its inception, yet it remains a product whose characteristics are not easily predictable

or comprehensible by conventional engineering techniques. This treatise is an attempt

to provide a rational descriptive and analytical basis for tire mechanics. Chapters of

this book are contributed by active research workers in the fields of rubber and textile properties, friction, material properties, tire stress problems, tire design and construction,

vehicle skid and handling, and tire mechanical properties.

Key words: Friction; rubber; skid; tires; tire cord; tire contact; tire stress; tire structure;

vehicles.

Library of Congress Catalog Card Number: 70-611535

II
Foreword

Without the pneumatic tire, the development of motor vehicle trans- port would not have been possible. The tire carries the load, absorbs

road shocks, transmits engine power to the road, and steers and stops

the vehicle.

The first patent describing a pneumatic tire was issued to R. W.

Thomson in 1845, and the first practical pneumatic (bicycle) tire was made by J. B. Dunlop in 1888. Since that time, tires have been improved

so that they can operate safely and efficiently at high speeds for sus- tained periods under large loads on wet and dry roads, in hot weather or

cold, on unimproved dirt or gravel roads, or on the modern high speed

turnpikes of the Federal highway system. Tires have been designed for dependable service on a wide variety of vehicles including aircraft and

earthmovers.

The evolution of this complex product has been the result of much

successful experimental and theoretical research carried out largely by

the major tire manufacturers. Partly because much of this work has been

considered proprietary, no previous book has given an overview of the

state of knowledge of the mechanics of pneumatic tires. Because of its responsibility to the Department of Transportation for the development

of a technical basis for tire safety performance standards, NBS stimulated the compilation of this book, and is pleased to make it available. We hope that this volume wiU provide a comprehensive introduction

to a subject central to the tire safety problem, and that it wiU help to establish reasonable, practical, and effective safety standards for use by

those who are responsible for improving tire performance.

Lewis M. Branscomb, Director

III

Preface

This volume was conceived as an aid to the practicing engineer or research worker who has need of information concerning the mechanics

of pneumatic tires. While a number of smaller and less comprehensive

reviews of particular aspects of tire mechanics or technology have ap- peared in the last few years, particularly the excellent review by Hofferberth and Frank in 1968, there has none the less been no comprehensive

evaluation of the field since Hadekel's work in 1952, nearly 20 years ago. Even in the case of Hadekel's report, attention was directed primarily

to aircraft tires and the emphasis was somewhat more on the presentation of pertinent data rather than on a description of the theoretical side

of the various problems.

A further reason for compiling such a volume as this lies in the chaotic

state of the literature on tire mechanics. There is no single accepted

source in the English speaking world which prints papers primarily con- cerned with tire mechanics. None of the technical societies are exclusively directed to this goal, so that worthwhile research contributions

are to be found in a number of different journals, and in some cases papers are given orally at various meetings and then not published in permanent form. In addition, a great deal of the worthwhile literature appears in the German and Russian languages so that a research worker

wishing to keep abreast of the field finds it necessary to scan a wide

range of publications. For this reason comprehensive summary reviews

are probably worthwhile and valuable if for no other reason than to bring together pertinent recent research references.

This book attempts to assess the current state of tire mechanics both

theoretically and experimentally. It attempts to do so without avoiding

or emphasizing either the mathematics or the physics which can be

helpful and informative, or the reality of fact and experiment upon which

all engineering is ultimately based. It does not claim to be without

shortcomings, since there are undoubtedly valuable research papers

which have missed the attention of the various authors. This is particularly true in the Russian and Japanese languages, where so few re- search workers have a reading familiarity. Nevertheless, the editor would

appreciate notification of errors or omissions and will make every effort

to accommodate them in subsequent editions. The idea for this book was suggested to the editor by Dr. F. Cecil

Brenner, Office of Vehicle Systems Research, National Bureau of Standards. Financial support for it was furnished by funds from the

Department of Transportation, and the editor would like to acknowledge

his debt to those two sources as being primarily responsible for making

this volume possible.

V
Whatever success or failure this book achieves will be due almost

entirely to two groups of people. The first is made up of the authors of the various individual chapters, who are identified publicly with their

efforts. The second group, who will receive no such public recognition,

have also contributed greatly and should be recognized here for their

efforts in reviewing parts of the texts during its preparation. I would

like to acknowledge such assistance from Ralph Paterson of Textile

Fibers Laboratory, E. I. Dupont Company, Seymour Lippman of Uniroyal, Inc., Dr. Robert Harrington of B. F. Goodrich Research Laboratories, and Dr. F. S. Conant of Firestone Central Research Laboratories.

Finally, special thanks should be given to Milton Piatt and G. A. M.

Butterworth of Fabric Research Laboratories, and to Donald Nordeen

of the General Motors Corporation, for substantial efforts in the review

of two of the chapters.

A great deal of effort was needed to bring the manuscripts from the

various authors into a common format. This is not a simple matter in a book of this size, and I would like to express my thanks to Miss Elaine

Beale and Mrs. Bertha Darruw of the National Bureau of Standards and

to Dr. J. T. Tielking, consultant, for their diligence and care in this task. Samuel K. Clark

Ann Arbor, Michigan

September, 1970

VI
Contents

Page Foreword Ill

I Lewis M. Branscomb, Director, National Bureau of Standards, WashI ington, D.C. 20234

Preface V

Samuel K. Clark, University of Michigan, Ann Arbor, Michigan 48104

I Conversion of U.S. Customary Units to SI Units X

Chai>ter 1. Material Characteristics 1 1.1. Rubber Structure and Properties 1 S. D. Gehman, formerly with The Goodyear Tire & Rubber Company, Akron, Ohio 44316 (now retired) Introduction 2

j Composition of tire compounds 3 Technical evaluations of physical properties of tire compounds... 5 Rubber elasticity 14 Rubber viscoelasticity 23 Reinforcement of rubber with carbon black 32

1.2. Friction of Rubber 41

K. C. Ludema, Department of Mechanical Engineering, University

of Michigan, Ann Arbor, Michigan 48104

Introduction 42

Early phenomenogical laws of friction vs. new information 43 Mechanical classes of tire and rubber friction 44 Rubber friction 45 Combined components of friction 54

1.3. Tire Cord Structure and Properties 63

Stanley Backer, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Introduction 65 Cord geometry 65 Mechanics of tensile behavior of tire cords Ill Bending mechanics of twisted structures 191

1.4. Recent Developments with Tire Cords and Cord-to-Rubber Bonding 219

T. Takeyama and j. Matsui, Industrial Products Laboratory,

Fibers and Textiles Research Laboratories, Toray Industries, Inc. Otsu, Japan

Physical properties of tire cords 220

Rubber-to-cord bonding 265 Chapter 2. Properties of Cord-Rubber Laminates 307 Samuel K. Clark, Department of Engineering Mechanics, University

of Michigan, Ann Arbor, Michigan 48104

Elastic properties of cord reinforced rubber 308

Strength and failure mechanisms for cord reinforced rubber 341 VII
CONTENTS- Continued

Page Chapter 3. Structure of the Tire 355

V. E. Gough, formerly with the Dunlop Company, Birmingham, England (now retired) General considerations ; 356 Pneumatic tire structure — general features 360

Flexible filament and soft matrix constructions 369

Tire construction methods 379 Cord path— practical factors determining selection 385

Calculation of cord length 386

Analysis of manufacturing methods by R vs cos (f) charts 390 Mechanism of load carrying— infinitely flexible membrane 392 Mechanism of load carrying— tire structure 398 Chapter 4. Tire Stress and Deformation 405 Joseph D. Walter, The Firestone Tire & Rubber Company, Akron,

Ohio 44317

Introduction 406

Analytical techniques 407 Experimental techniques 430

Principal notation 439 Chapter 5. The Contact Between Tire and Roadway 445 Samuel K. Clark, Department of Engineering Mechanics, University

of Michigan, Ann Arbor, Michigan 48104

Introduction 446

Contact area 447

Slip between tire and roadway 463 Contact pressures.. 471 Chapter 6. Skid Resistance and Directional Control 501

A. Schallamach, The Natural Rubber Producers' Research Association, Welwyn Garden City, Hertfordshire, England

Introduction 502 Theory 502 Cornering and side force 508 Braking and circumferential slip 515 The speed dependence of rubber friction 518 Hydrodynamic effects on wet roads 520 The velocity dependence of the braking coefficient on wet roads 523 The effect of tread pattern and carcass construction on tire friction. . . 526 Tread compound effects on wet friction 531 Aquaplaning (hydroplaning) 539 Winter tires 544 Chapter 7. The Tire as a Vehicle Component 545

H. C. A. van Eldik Thieme and H. B. Pacejka, Technological University of Delft, Delft, The Netherlands

Introduction 548

Straight hne rolling experiments (H. C. A. van Eldik Thieme) 553

VIII
CONTENTS - Continued

Chapter 7. — Continued Page Cornering and camber experiments (H. C. A. van Eldik Thieme) 631

Tire in-plane dynamics (H. B. Pacejka) 695 Yaw and camber analysis (H. B. Pacejka) 763 Index 841

IX
Conversion of U.S. Customary Units to SI Units

The International System of Units (SI) was adopted by the Thirteenth

General Conference on Weights and Measures, Sevres, France, October

15-17, 1968. Conversion factors for the units used here are given in the following table:

Physical TTU.o.^ onversion quantity Customary factor SI Unit Unit (*)

inch 0.0254 meters (m) Temperature (T+459.67) 5/9 degrees Kelvin (K) Force Ibf 4.448 newtons (N) Density Ibm/ff^ 16.02 kilograms per cubic meter (kg/m^)

Stress, Pressure.. psi=lbf/in- 6895 new^tons per square meter (N/m-)

Torque in-lbf 0.1130 meter-newtons (m • N)

^Multiply value given in U.S. Customary Unit by conversion factor to obtain equivalent value in SI Unit. Prefixes to indicate multiple of units are as foUows:

Prefix Multiple

kilo (k) 103 mega (M) 10«

giga (G) 10«

centi (c) 10-2 micro (/x) 10-6

In addition, certain engineering units which have come into common

usage in Europe are rarely seen in English or American literature. Their

conversion is summarized by the relations given below:

1 kp. = l kgf. =2.205 Ibf.

1 bar =10-^ N/m2

1 atmos. (Standard) = 1 kgf./cm^

The standard acceleration due to gravity is taken to be 9.807 m/s".

X
CHAPTER 1

Material Characteristics

1.1. Rubber Structure and Properties

S. D. Gehman ^ Page 1.1.1. Introduction 2

1.1.2. Composition of tire compounds 3

1.1.3. Technical evaluations of physical properties of tire com- pounds 5

Stress-strain properties and evaluation of cure 5 Aging tests 9

1 ear tests 10

Hardness 11 Dynamic tests 12

Flex cracking 12

Rubber abrasion tests 12

1.1.4. Rubber elasticity 14

Thermodynamic aspects 14

Molecular picture: elasticity of a rubber molecule 16

Elasticity of the molecular network 19

Strain energy representation of rubber elasticity 21

1.1.5. Rubber viscoelasticity 23

Molecular and model concepts of rubber viscoelasticity... 23

Viscoelastic relations between creep, stress relaxation, and

complex dynamic modulus 26

Superposition principle 26

Time-temperature superposition principle 26

Dynamic properties 27

Energy losses in tires 31

1.1.6. Reinforcement of rubber with carbon black 32

References 37

' Formerly with The Goodyear Tire & Rubber Company. Akron. Ohio 44316.

(Retired. Present address: 214 Kenilworth Drive. Akron, Ohio 44313.)

1
2 RUBBER STRUCTURE AND PROPERTIES

1.1.1. Introduction

Pneumatic tires usually contain a variety of rubber compositions,

each designed to contribute some particular factor to overall performance. Rubber compounds designed for a specific function will usually

be similar but not identical in composition and properties, although in some cases there can be significant differences between compounds in

tires of various types. The guiding principle in development of rubber

compositions for tires is to achieve the best balance of properties for a particular type of tire service. Since a tire is a mechanical structure, a rubber component should be judged on how it functions in the system

rather than on its individual properties or performance capabilities.

Thus a rubber compound which did not adhere well to other tire components, or which required vastly different vulcanization conditions

than other parts of the tire, could be useless in the tire even though it had excellent strength and other mechanical properties. Tire perform- ance is the result of skill and experience in producing a mechanically

harmonious structure of rubber compounds, fabric and adhesive, beads

and other components which work together to give optimum service.

The principal functions of the rubber compositions in a tire are fairly

obvious. The tread compound must provide wear resistance and be tough

and resilient to minimize cuts, tears, and cracks, as well as to protect

the tire body from bruising impacts. Low mechanical hysteresis loss in the tread is desirable since lower tire operating temperatures are ad- vantageous. Good friction properties of the tire tread for all driving conditions are, of course, very important. In some cases optimum tread

properties are obtained by using a cushioning compound between tread

and tire body as additional protection against fabric bruises, thus making

a "double layer" tread. This cushion can also serve, especially in re- treading, as a bonding or transition layer between tread and body

compounds. Intermediate hardness properties between those of tread

and body are usually used in the cushion or breaker under the tread. Tire body or carcass rubber compounds must form strong bonds to the adhesive-coated fabric. Their strength and durability should be adequate to insulate the cords and hold them in their paths. The rubber

must, however, be soft enough to permit a slight change of cord angles

when the tire is flexed. The body rubber serves as insulation between

the fabric plies. Outstanding fatigue resistance is required of body com- pounds in order to withstand cyclic deformation. It is essential that they

retain adequate physical properties and durability at the internal tire temperatures generated in service. Hence, low mechanical energy losses are needed for body compounds. There may be gradations in the properties of body compounds, with hardness usually diminishing some- what from tread to cushion to top plies to inner plies. In tubeless tires, a liner or coating on the inside ply retards diffusion
COMPOSITION OF TIRE COMPOUNDS 3

of inflating air into the fabric, and protects against ensuing ply

separations.

Rubber compositions around the wire bead are called bead insulation,

and give it geometric stability, shape it to fit the rim, and provide firm

anchorage for the cords.

Finally, sidewall compounds must be especially durable in flexing

and weathering, and be scuff and impact resistant to protect the body

from curb impacts. The sidewall may include a decorative compound

as a surface layer.

Processing requirements impose additional restraints on rubber

compositions suitable for tires, so that the rubber technologist must

consider many factors in compounding a rubber for a specific use.

1.1.2. Composition of Tire Compounds

Ingredients [1]- in tire compounds can be classified as: (1) the rubber

which may be a single polymer or a blend of polymers [2, 3] and, with

high molecular weight polymers, may include an extending oil [4-6];

(2) fillers, principally various types of highly-developed carbon blacks

[7-9]; (3) relatively small additions of softeners, plasticizers, or reclaim

rubber which serve principally as processing aids; (4) the chemical

vulcanization system [10-12], which is likely to include two accelerators

[13], sulfur, and a small amount of zinc oxide; (5) chemical protective

agents, known as antioxidants and antiozonants [14, 15]. With such a wide variety of ingredients, the important mechanical

properties for a given tire compound can usually be obtained from a number of different compositions. For example, modulus and hardness can be

controUed by varying either the amount of carbon black, the amount of extender-oil or softener, the fineness and structure of the black, or the

number of molecular crosslinks introduced during vulcanization. Thus

even small advantages in cost, performance, and processing, which may

only become apparent with extensive testing or service experience,

become important factors in compound selection. No tire compound is ever final but is always subject to changes as test results and experience

accumulate.

In the United States, in 1968, about 75 percent of the rubber being used

in tires and tire products was synthetic [16], and of this about 73 percent was SBR (styrene-butadiene copolymers) [17], 21 percent was stereo- elastomer [17] and about 5 percent was butyl [18] or chlorobutyl rubber

[19]. The latter is used for liners and inner tubes because of low air permeability and resistance to oxidative deterioration. The stereo rubber,

ci5-l,4-poly-butadiene, has come into wide use as a componding in- gredient which can improve durability of tire body compounds, can

reduce groove cracking and can increase abrasion resistance of tire treads.

^ Figures in brackets indicate the literature references at the end of this chapter.
4 RUBBER STRUCTURE AND PROPERTIES

Airplane tires have been the last stronghold of completely natural

rubber tires, but synthetic rubber is now beginning to appear in them

also. Of the 30 or so standard types of carbon black, the HAF (High Abrasion Furnace) and ISAF (Intermediate Super Abrasion Furnace) types

with their HS (High Structure) variants are currently used most ex- tensively in tire treads. There is now a trend toward high structure blacks in treads, i.e., from ISAF black to HAF-HS. Relative use of

HAF and ISAF in 1967 was about 68 percent and 32 percent respectively

[20]. The high structure blacks are reported to give somewhat better wear and resistance to cracking and cut growth, along with improving

processing of polymers used in treads. A physical description of these

blacks is given in table 1.1.3. of this chapter.

Larger particle size carbon blacks, such as SRF, GPF, and FEF types

are usually preferred for carcass compounds.

Representative tread and carcass compound recipes can be found in the rubber technical literature [2, 3] as can cushion or undertread

formulations [21], wire bead insulation [22], and liner recipes [19]. For illustration a few recipes are given which are representative of

tire compounds.

Table 1.1.1. Tread-type formulations Example

A [23] Example B [24] OE-SBR" 89.38 96.00 BR^ 35.00 30.00 Carbon black ^70.00 ^70.00 Zinc oxide 3.00 4.00

Stearic acid 2.00 2.00 Antiozonant n.oo n.50

19.63 24.00

" 1.05 " 1.20 Secondary accelerator '0.02 Sulfur 1.86 2.00 222.94 230.70

Ciirp - 40 min/

293° F

° Oil-extended styrene-butadiene copolymer; 100 parts polymer, 37.5 parts oil

^ ci5-polybutadiene

' HAF or ISAF types HAF types

^ N-Cyclohexyl-N-phenyl-p-phenylenediamine

^ Aromatic secondary amine type

^ N-Oxidiethylene-2-benzothiazylsulfenamide

^ Sulfenamide type

' Diephenyl guanidine
EVALUATION OF PHYSICAL PROPERTIES 5

Table LL2. Carcass-type formulations [2]

1 ruck tire compound

Passenger

tire body

compound

Smoked sheet 25.00 25.00 Pale crepe 25.00 25.00 ciVBR 50.00 50.00 FEE Carbon black 40.00 35.00 Medium processing oil 8.00 8.00 Rosin oil 2.00 2.00 Wing-Stay 100 1.00 1.00 Stearic acid 2.00 2.00 Amax #1 .80 .80 Zinc oxide 5.00 5.00 Sulfur 2.50 2.50 Optimum cure 40 min/ 8 min/

275° F 310° F

Simpler standard formulations are available for test and material

specification purposes [25], and these, or similar ones, are often pre- ferred for research studies on physical properties of rubber compounds.

1.1.3. Technical Evaluations Of Physical Properties

Of Tire Compounds

Evaluation of the physical properties of rubber compounds for technical purposes proceeds by subjecting them to a battery of standard

laboratory tests [26, 27]. Occasionally individual laboratories may

deviate from these standard procedures, and may also use special tests and testing devices which they have found to be advantageous. However, the standard tests for rubber in the United States represent many

years of testing experience [4]. International standardization of rubber

tests is also well advanced and under constant development [28]. It should be realized that these standard tests, although they fulfill practical purposes in rubber technology, often involve combinations of several basic properties of rubber and hence are of limited general use. They usually do not furnish a very fundamental description of the

physical properties, such as might be desired for research purposes.

Here we discuss very briefly the most important of these test pro- cedures, and examine a few typical results when they are applied to

tire compounds.

Stress-strain properties and evaluation of cure. The physical properties of any rubber compound depend upon the state of cure [10], that

is, upon how far the chemical vulcanization reactions have been carried. Vulcanization introduces chemical crosslinks or bonds between the

long chain polymer molecules. This crosslinked network is decisive for
6 RUBBER STRUCTURE AND PROPERTIES

the physical properties and is determined, for a given rubber compound, by vulcanization time and temperature. The traditional rule of thumb is that the vulcanizing time to reach a given level of a property,

such as static modulus, is halved if vulcanization temperature is raised

18° F, and vice versa. Although this rule is still often adequate, a more

precise description of the time-temperature dependence of vulcanization requires determination of the activation energy of vulcanization

for each rubber compound [10, 29, 30]. The effect of cure on the physical properties of a rubber compound

is usually determined by vulcanizing a series of test sheets for different

times at the same temperature [25]. This may, of course, also be done at several temperatures if more thorough tests are desired. Dumbbellshaped test specimens are cut from the sheets and static stress-strain curves taken [31]. Rubber stress-strain curves under static conditions are concave to- ward the load axis, i.e., strain hardening, except for a short portion near

the origin. The concavity is accentuated by the occurrence of stress- induced crystallization in natural rubber at higher elongations [32, 33]. There is no yield point before failure, as is usual with metals. The curves

in figure 1.1.1. represent approximately the effect of cure time on static stress-strain properties for a range of cures of a tread-type compound.

Stress and tensile strength are calculated using the original cross-section

area. In rubber technology, this stress is called modulus and is designated for a specific elongation, so that the 300 percent modulus for a rubber compound is the stress required to extend a strip to four times

its original length. Volume changes are negligible [34], when rubber is strained, so that Poisson's ratio is assumed to be one-half. Static modulus provides a convenient parameter to assess the temperature range in which elastomeric properties are exhibited. While

this varies somewhat with individual polymers, in general the modulus

of an elastomer varies with temperature as shown in figure 1.1.2. At

low temperatures a hard or glassy character is evident. As temperature

is raised the rubber passes through a transition region in which properties

change rapidly. Rubber properties prevail over a range of temperatures

above the transition temperature, and finally at yet higher temperatures

viscoelastic of flow properties become important and predominate. It should be noted that properties other than modulus generally also show

significant changes in passing from the glassy to the rubbery state. At high temperatures and for long times, the flow properties of rubber

are marked. Stress relaxation in this region can often be attributed to oxidative degradation. In the elastomeric region there is relatively little stress relaxation, and in this state the molecular network comes into

equilibrium with applied stress, so that here the concept of modulus is valuable.

As shown in figure 1.1.1., modulus or stress at a given elongation in- creases, and breaking elongation decreases, as cure advances. This is almost invariably the case. Tensile strength usually goes through a maximum, although this is not always observed in a range of test cures. In some compounds a phenomenon known as reversion occurs and mod-
EVALUATION OF PHYSICAL PROPERTIES

200 400

STRAIN . %

Figure Ll.l. Variation of stress-strain curves with cure; data from reference [108].

GLASSY region

ELASTOMERIC

REGION

VISCOELASTIC OR FLOW REGION

Figure L1.2. TEMPERATUREGeneral variation of rubber modulus with temperature.
8 RUBBER STRUCTURE AND PROPERTIES

ulus, tensile strength, and breaking elongation all decrease with overcuring. Natural rubber is especially prone to this reversion. Stress-strain curves such as those shown in figure 1.1.1. are widely used in tire com- pound development to adjust cure rates and to insure that modulus,

tensile strength, and breaking elongation of a compound fall inside a desired range. For tire tread compounds the usual tensile strength will be in the range 2500 to 4000 psi, 300 percent modulus in the range of 1000 to 1700 psi, and breaking elongation in the range of 400 to 600

percent.

Although such discrete test cures are traditional in the rubber industry,

and are useful for evaluation of physical properties in relation to cure, various instruments are also used which furnish a continuous record of modulus or stiffness as cure progresses [30, 35]. Such data are very useful in showing how modulus develops, whether or not it reaches a flat plateau and whether or not there is reversion or decrease in modulus

with overcure. Figure 1.1.3 illustrates these possibilities diagrammatically. The character of such a curve is determined by the polymer

and the vulcanizing system [10]. The start of the curve is also significant,

because it gives a measure of processing safety, that is, an indication

of time-temperature conditions which the compound can endure

in mixing, extrusion, etc., without excessive prevulcanization or "scorching".

Optimum or "best" cure for a rubber compound cannot in general be

uniquely defined, since it depends upon the type of service and the par0 20 40 60 80

CURE TIME, MIN.

Figure 1.1.3. Diagram of continuous cure curves.

Curve A is representative for an SBR tread compound cured at 280° F; the term "marching modulus" indicates a slowly increasing modulus for long cure times.
EVALUATION OF PHYSICAL PROPERTIES 9

ticular property, or properties, which should be optimized [36]. The

cure designated "best" in figure 1.1.1 gives a balance of properties best

suited by experience for tire tread service. There is usually a range of cures for which a vulcanizate would be expected to give about the same

service performance, but large deviations from the best cure can be

disastrous. Optimum cure time is usually found to be within the time

required to obtain 90 to 95 percent of the rise in modulus, as shown by

a curve such as in figure 1.1.3. When a tire is cured, the time-temperature history wiU vary throughout the tire cross section since heat must flow into it from the mold and

through the internal bladder used to form the tire. The thermal dif- fusivity of rubber is relatively low [37], so that the equivalent cure, i.e., minutes of cure expressed through calculation at a given reference

temperature, may vary considerably between the outer surface and the

interior point of least cure, even for a passenger tire. It is desirable to formulate tire compounds to compensate for this as much as possible,

and to use vulcanization systems which provide "flat" cures, that is, cures such that physical properties tend to reach a plateau as vulcanization progresses. It is important to have a good, adequate cure at every

point in the tire and yet to avoid excessive cure at any point.

Tire cures are originally determined by incorporating thermocouples

at strategic points in the tire. Temperature is measured during vul- canization as a function of time. The production curing cycle is gotten

by analysis of these time-temperature curves in relation to the vulcanizing characteristics of the compounds. Cure times for passenger car

tires may be on the order of 20 minutes [38]. The state of cure of the rubber in a tire can probably be most con- veniently examined or studied by equilibrium swelling measurements in benzene using small pieces of rubber cut from the tire [10]. The molecular crosslinks formed during vulcanization limit the swelling.

Aging tests. Physical properties of tire compounds, especially stress- strain properties, are also routinely examined in rubber laboratories

after the rubber has been exposed to one or more accelerated aging

tests [39]. These are usually run at elevated temperatures in order to simulate deterioration in service over a long period of time. Several of the tests are oven-aging tests under carefully controlled conditions,

while others use a bomb fiUed with air or oxygen under pressure to further accelerate degradation.

Figure 1.1.4 gives an example of the effect of oven aging on the stress- strain curve of a tread compound. Although such results do not correlate

perfectly with aging deterioration in service, they can be very helpful,

especially with a background of experience, in anticipating whether or not a compound will be satisfactory in this respect. These aging effects are quite compHcated [40, 41] as they depend on oxidative chemical

reactions with the polymer [42, 43]. Hence they are very dependent both on the chemical nature of the polymer and on antioxidants in the com- pound recipes. In general, SBR is less sensitive to aging than natural

rubber. The basic mechanism of degradation in these two rubbers ap- pears to be quite different, SBR tending to harden on aging and natural
10 RUBBER STRUCTURE AND PROPERTIES

rubber to soften, reflecting, respectively, predominance of additional

molecular crosslinks and chain scission, i.e., cutting of chain molecules

into smaller molecules.

Effectiveness of antiozonants in rubber compounds is usually eval- uated by outdoor exposure tests [44] and accelerated ozone tests [45], to promote the characteristic cracks which result from action of ozone

on stretched rubber [46, 47]. Tear tests. Rubber tear tests [48] are designed to cause a high stress gradient at the end of a cut or notch in an angle or crescent shaped

testpiece, which is pulled in a testing machine. Figure 1.1.5 illustrates various types of tear test specimens. Although tearing phenomena with

rubber are most important and revealing in regard to mechanisms of rubber failure [49, 50], technical tear tests have very limited practical

significance, probably because the notch effect is so complicated and

difficult to control for rubber. Tear test values are reported as load per

unit of specimen thickness. An SBR tread compound with tensile
EVALUATION OF PHYSICAL PROPERTIES 11

STRIP TEST SPECIMEN TROUSERS TEST SPECIMEN

1

CRESCENT TEST SPECIMEN PURE SHEAR TEST SPECIMEN

ANGLE TEST SPECIMEN RING TEST SPECIMEN Figure 1.1.5. Tear test specimens.

Strength about 3000 psi might reasonably show a crescent tear strength

of about 250 lb/in. In general, tear values tend to increase with greater

tensile strength and breaking elongation but to decrease with higher

modulus.

Hardness. Rubber hardness is an important quality control parameter.

It is conveniently measured with a Shore A Durometer,^ a pocket instru- ment which has been standardized [51] but use of which often leaves

much to be desired in the way of precision. This may be improved by

mounting it in a rigid stand. The durometer uses a small, spring loaded

indenter with a truncated conical point protruding from a flat base. When

indenter and base are pressed against the rubber, the resulting spring

deflection, which depends upon rubber hardness, is indicated by a pointer

with a scale graduated from 0 (no hardness) to 100 (no indentation). Shore

A hardness for rubber tread compounds is typically in the range of 50

to 65 units and for unfilled vulcanizates about 25 to 30.

^ Certain commercial products and instruments are identified in this book in order to specify adequately technical procedures. In no case does such identification imply recommendation or endorsement by the National Bureau of Standards, nor does it imply that the products or equipment identified are necessarily the best available for the purpose.
12 RUBBER STRUCTURE AND PROPERTIES

There are a variety of other hardness test instruments for rubber and

a well-developed International Rubber Hardness Degree (IRHD) scale

which agrees approximately with the Shore A Durometer scale [52, 53]. Indentation of thick rubber obeys the classical elasticity analysis of

Hertz very well [54], and this gives a mechanism to relate elastic modulus

to hardness measurements.

Dynamic tests. Many different test procedures are available to measure

rubber stiffness and energy loss for relatively small cyclic deformations,

often over ranges of temperature and frequency. These evaluations are

especially pertinent for tire compounds because heat generation and

temperature rise from rubber hysteresis losses are important factors

in tire durability [55]. One of the oldest and still most widely used types of test for this purpose is a pendulum rebound test, in which a pendulum is released

from a fixed height to strike a rubber block and then rebound [56]. Superiority or natural rubber or synthetic ci5-l,4-polyisoprene in this test is pronounced. Percent rebound for SBR tread compounds will usually be

in the range of 52 to 62 percent while that for comparable natural rubber

compounds may be more than 70 percent. A falling ball instead of a pendulum is often used in a rebound test. Free vibration tests [57] and forced nonresonant and resonant vibration

tests are also found in great variety [58, 59]. These are used to measure dynamic modulus,^ internal friction,^ and resilience ^ of rubber compounds. Reference [60] includes a useful table of storage ^ and loss moduli ^ for a wide variety of rubbers and other materials.

Energy losses in rubber may be evaluated by measuring the temperature rise when a block specimen is cyclically deformed in shear or compression or both. These rubber testers are called flexometers [61]. There are many limitations, however, on the usefulness and interpretation

of such temperature rise data. Compression set of the rubber specimen

after a flexometer test is usually also reported and is used in compound

evaluations, especially for state of cure. Occasionally the tests are run

to destruction.

Flex cracking. Initiation and growth of small cuts or tears in tire treads,

especially in the pattern grooves and in tire sidewalls, are so significant

for tire performance that much effort has been expended in developing

laboratory tests to simulate and clarify these phenomena. Several test- ing machines and procedures have been standardized for such tests

[62]. The DeMattia machine, where a specimen with a transverse groove

is cyclically bent as shown in figure 1.1.6, is probably the most familiar

of these. Results of such tests are sensitive to compounding factors such

as type of elastomer, state of cure, and protective agents in the recipe

[63].Rubber abrasion tests. Laboratory abrasion tests are used in tread

compound development because they provide inexpensive, rapid

screening of experimental polymers and rubbers, in spite of the fact that correlations with roadwear have limited success. Two methods are

^ These terms are defined in the section on dynamic properties.
EVALUATION OF PHYSICAL PROPERTIES 13

^

25mm^ ^

din)

^'

140- 155mm

6.35*0.13

mm

(0.25*0.005

in) ° R 2.38* 0.03 mm X"[R0.094* 0.001 in) J

t (a)

Figure 1.1.6.

(a) DeMattia testpiece. (b) Mounting of DeMattia testpiece. given in ASTM Standards [64], the Dupont abrader where two rubber

specimens are abraded against a revolving disk, and the Pico abrader in which the surface of the specimen is rubbed by revolving tungsten

carbide knives under carefully controlled conditions. Much abrasion test- ing is also done with the Goodyear angle abrader [65], in which a specimen in the shape of a small rubber wheel is mounted at a slip angle and

driven by a revolving abrasive wheel. A variation of this is the Lambourn

[66] abrader. in which slip of the rubber wheel specimen is controlled

by an electromagnetic brake so that tests can be run either at constant

slip or constant transmitted power. Developments with Lambourn-type

abraders have become quite sophisticated in efforts to secure correlations with road tests on different road surfaces and under different

driving conditions [67].

It is widely recognized that the severity of abrasion, or rate of wear,

is an important factor in abrasion testing. Rankings of a series of comI
14 RUBBER STRUCTURE AND PROPERTIES

pounds may change if rate of wear is changed. This introduces obvious

complications in trying to correlate laboratory data with road wear,

since severity of wear on the road depends on many factors including

road surface, frequency of curves, amount of rain, braking and many

others. The severity level of a laboratory abrasion test, that is, the amount

of rubber removed per unit distance traveled, should presumably be

comparable to that in road tests for best correlations. This may be very

low, by American standards, in the order of 100 miles/mil*^ of wear

normally and 50 miles/mil or less for severe service. To reduce testing

time, laboratory tests are usually accelerated and have much larger wear rates than road tests. 1.1.4. Rubber Elasticity

Thermodynamic aspects. Distinctive features of rubber elasticity are easy deformability or low modulus, enormous deformations, and rapid

recovery when deforming forces are released. There is also more sensitivity to temperature than for many elastic materials. Figure 1.1.7 shows the dramatic effect of low temperatures on relative modulus of

unfilled vulcanizates of several polymers. For any elastomer there is a range of temperatures over which transition occurs from a rubbery to

-100 -80 -60 -40 -20

TEMPERATURE, •€

Figure 1.1.7. Effect of low temperatures on rubber modulus; data by ASTM method

1053-65, reference [4].

' One mil equals .001 inch equals .0254 mm.
RUBBER ELASTICITY 15

a glassy state, as shown in figure 1.1.2. This transition temperature

range for SBR is from — 60°C to — 40°C, which is about as high as can

be tolerated for a general purpose tire rubber. The curve for c/s-polybutadiene in figure 1.1.7 is complicated by crystallization, which starts

to affect the warming curve at about —95° C. In the transition range from the glassy to the rubber state modulus

falls rapidly with increasing temperature, but further temperature increase

results in a slowly rising modulus. This rise was first observed in ex- periments by Gough, published in 1805, as a contraction when a rubber

specimen stretched by a weight was heated. Joule, about 50 years later,

studied thermoelastic phenomena exhibited by rubber and interpreted

them in terms of the new science of thermodynamics then being de- veloped by Kelvin. Treloar [72] gives a very good review of the thermodynamic fundamentals of rubber elasticity. For a reversible process, the

first and second laws of thermodynamics provide,

dE=TdS + dW (1.1.1) in which E is internal energy of a system, T is absolute temperature,

S is entropy, and W is work done on the system. At once there is a

difficulty here because ordinary rubber deformations are not completely

reversible. It is necessary to take special measures with any test specimens, such as solvent vapor treatments or prestretching them at an ele- vated temperature, in order to secure reversible deformations.

If the tensile force on a rubber strip is /, then the work done during an isothermal displacement dl is, neglecting small volume changes,

dW = fdl (1.1.2) and, with eq (1.1.1)

(1.1.3) Equation (1.1.3) resolves the force into two terms. The first arises from

changes in internal energy and the second from entropy changes with

changes in length.

By differentiation of eq. (1.1.3), it follows that

(1.1.4)

so that eq (1.1.3) can be written

(1.1.5) Equation (1.1.5), in conjunction with eq (1.1.4), has been very important

for understanding rubber elasticity because it allows experimental
16 RUBBER STRUCTURE AND PROPERTIES

evaluation of internal energy and entropy changes upon deformation. If the equilibrium force exerted by a stretched rubber strip held at constant

length is measured and plotted as a function of temperature, the slope

at any value of T is ( t4 I , which eq (1.1.4) shoves to be the entropy

change per unit change in length for isothermal extension at T. The

corresponding internal energy change I -rj ) is given by the intercept

of the tangent with the zero T axis. Careful experimental work of this type by Meyer and Ferri [68],

Anthony, Caston, and Guth [69], Wood and Roth [70], Gee [71] and

many others showed that over a considerable temperature range stress was closely proportional to absolute temperature, and this led to the

conclusion that rubber elasticity resides principally in the entropy

term of eq (1.1.5). There is an entropy decrease on extension and an

increase on retraction, except:

a. at very low elongations, below about 10 percent where a so-called

thermoelastic inversion is observed due to thermal expansion obscuring

the entropy effect

b. at large elongations, where high orientation and crystallization may

occur. Volume changes and internal energy effects, however, appear never to be entirely absent. This entropy basis indicates that rubber elasticity

must have an entirely different molecular origin or mechanism than

ordinary elasticity, where stresses increase the internal energy by

increasing molecular or atomic spacings.

Molecular picture: elasticity of a rubber molecule. The unique

thermoelastic behavior of rubber is related to molecular structure by

the kinetic or statistical theory of rubber elasticity. The theory provides

a very satisfactory explanation of what might be called the mainspring

of rubber elasticity, but it involves idealizations which have restricted

its quantitative application to very carefully controlled equilibrium ex- periments with suitable rubber compounds and limited elongations and

temperatures. It can only be regarded as semiquantitative for rubber

in real applications. Reasons for these deviations from the theory, however, are quite comprehensible in light of what is known from many

sources concerning the molecular geometry and forces. There is extensive evidence that a rubber is composed of long chain

molecules, as shown in figure 1.1.8. The monomer repeating unit in the

chain molecule of cis-polyisoprene or natural rubber is

T

CH3

-^H2—C=CH^H2—
RUBBER ELASTICITY 17

while in ciVpolybutadiene it is —CH2—CH=CH—CH2. In SBR,

styrene units amounting to about 23 percent by weight occur at ran- dom in the polybutadiene chain. Such molecules are flexible by

virtue of rotation around the single bonds, except at low temperatures

where packing becomes too close or crystallization may occur for the

first two rubbers. They tend to assume haphazard or chance configurations because of thermal agitation of their segments. Most of these

configurations will be very crumpled, so that a chain molecule can be

extended by an external tensile force provided that interaction with

its neighbors is not too strong. As temperature is lowered, this inter- action increases until the typical low modulus and rubber elasticity are no longer present. This temperature influence is shown in figure 1.1.7. Chemical composition of the molecules may vary widely provided

that chain length, flexibility, and interactions all lie within ranges ap- propriate for rubber elasticity. In order to secure mechanical stability in such a molecular structure it is necessary to connect the chain molecules into a network by introducing chemical bonds or crosslinks

between them during vulcanization. The chemical nature of the crosslinks is not relevant for the theory because they are idealized simply

as network connections. The nature of the crosslinks is, however, technically important. The crosslinks cannot be too numerous, or flexibility of the network wiU suffer, but their number must be adequate to suppress

plastic flow. To be more definite, rubber chain molecules having 1000 to 2000 monomer units per chain may very well have about 10 to 40 of these

units crosslinked at random along a chain in a vulcanizate.

History of the kinetic theory of rubber elasticity, which dates back to about 1932, has been recounted by Treloar [72] and by Flory [73]. The

theory has been thoroughly examined both experimentally and analytically and has been refined in many respects. Treloar gives an excellent

exposition of developments up to about 1958, and a review article by

Figure 1.1.8. Diagram to illustrate concept of molecular structure of rubber.
18 RUBBER STRUCTURE AND PROPERTIES

Krigbaum and Roe [74] brings the subject up to recent years. Only a

brief recapitulation of a few of the main features of the theory will be

given here. A single, long chain rubber molecule assumes random statistical configurations to the extent permitted by hindered bond rotations, fixed

valence-bond angles,^ excluded volumes (since no two atoms can occupy

the same space simultaneously), and intermolecular forces. Some, of these effects can be accommodated in the theory, but in the first mathematical model the molecule is assumed to undergo random thermal

fluctuations among all possible configurations of its n links, each of

length /, just as if they were freely orienting. Thus the problem of describing the configurations is the random walk problem, and the configurations have analogies to the Brownian motion of a particle.

The distance between the ends of a chain, r, is called the displacement

length, end-to-end distance, or simply the chain length. The distance

measured along the chain is the chain contour length.

The distribution of chain lengths is Gaussian provided they are not extended more than about one-third of their fully extend length, nl. This distribution is given by

P(r)dr= (463/7ri/2)r2 exp {-b^r^)dr. (1.1.6) In eq (1.1.6) P{r) is the probability function for r and

b^= (3/2) lint'). (1.1.7) The root mean square value of r is

ir-) 'l^ = ln^l\ (1.1.8) This shows at once that n must be large to account for rubber elongations,

since the ratio of the average unstretched chain end separation to the

fully extended chain length is n~^l'^. Higher molecular weights thus favor

greater extensibility.

Equation (1.1.6) is for a freely orienting chain. For a real molecule,

where valence angles are retained and rotation may be hindered, it is necessary to introduce the idea of an equivalent random chain and an

equivalent random link. Length of the equivalent random link depends

upon the chemical structure. Usually the equivalent link contains up to ten main chain bonds. For polyisoprene, Treloar estimated that there were 0.76 monomer units per random link. If a rubber network chain

molecule, i.e., a chain between crosslinks, contains about 80 monomer

units, or 105 freely orienting links, the ratio of fully stretched length

to average unstretched end-to-end distance will be (lOS)^/^ ~ 10.2. This

is adequate to account for rubber elongations.

According to the familiar Boltzmann relation the entropy of a system

is proportional to the logarithm of the number of possible configurations.

^ For a paraffin chain molecule the angle between two adjacent bonds in the -C-C structure is 109.5°.
RUBBER ELASTICITY 19 Hence from eq (1.1.6) the entropy S of a single chain molecule is S = c-k-b^-f' (1.1.9) in which c is an arbitrary constant and k is Boltzmann's constant. It is apparent from eq (1.1.9) that the entropy decreases as r becomes larger, that is, as the molecule is stretched. The work required to increase r to r+ dr is fdr=-Tf (1.1.10)

in which / is the stretching force on the molecule. From eq (1.1.9) dSldr = -2kb'r so that f=2kmr. (1.1.11)

This is the average fluctuating statistical force exerted at the end of a stretched molecule. It is proportional to absolute temperature and to the value of r, the end-to-end distance, and it acts along the Hne of r. Elasticity of the molecular network. A chain molecule reaching from

one crosslink to another is called a network chain. The Gaussian dis- tribution of eq (1.1.6) is assumed to apply to each network chain so that the entropy change for deformation of the network can be calculated by

summing the entropy changes for all the chains in the network. In doing

this the assumption is made that the deformation in affine, that is, the

vector components of length of each chain are changed in the same ratio as the corresponding dimensions of the rubber specimen. Treloar [72] gives the entropy change A5 of the network due to deformation as A5 = -iA^A:(X2 + \| + \2_3) (1.1.12)

in which N is the number of network chains per unit volume and \i,

k-z, and A.3 are the principal extension ratios along the three mutually

perpendicular axes of strain for a pure homogeneous strain. The ex- tension ratio is defined as the ratio of the deformed to the undeformed

length.

If the deformation is not accompanied by any change in internal energy

the work of deformation, is — TAS, so that W=^ NkTikj^ kl^ kl-3) =^ Gikj + Xl^ kl-3) (1.1.13)

W is the elasticaUy stored free energy per unit volume and is known as the stored energy function. In eq (1.1.13) G is the modulus of rigidity which in this simpler version

of the theory depends uniquely on chain molecular weight through the

relation

G = NkT=pRT/Mc. (1.1.14)
20 RUBBER STRUCTURE AND PROPERTIES

Here p is the density of the rubber, R is the gas constant and Mc is the

number-average network chain molecular weight. The theory thus pro- vides a single elastic constant which is proportional to the degree of

crosslinking.

To calculate the principal stresses from eq (1.1.13) it is assumed that volume changes can be neglected so that Xi-\2-X3 = l (1.1.15)

Work done by the applied forces is dW=fidXi +f2d\2 -^fsdXs (1.1.16)

in which /i, /2, and fs are forces per unit initial unstrained area and act along the principal axes. Comparing dW obtained by differentiating

eq (1.1.13) with dW in eq (1.1.16), after eliminating X3 from both by use

of eq (1.1.15), and equating the coefficients of dki and dk-z gives the

general stress-strain relations:

kiA-\sf3-G{\'i-kl), (1.1.17)

k2F2-ksfs = G{kl-kl). (1.1.18)

Equations (1.1.17) and (1.1.18) can be written in terms of principal stresses ti, ?2, ^3, defined as forces per unit area after straining, by use

of relations such as /i = ^1X2^3, in the form

h-ts = G{k'i-kl), (1.1.19)

t2-t3 = G{kl-kl). (1.1.20)

These equations give only the difference between two principal stresses,

since eq (1.1.15) has introduced the indeterminancy of an arbitrary

hydrostatic stress. This may be recognized by writing the principal

stresses [72] as h = Gk'l -\-p; t2 = Gkl +p\h = GKl + p. (1.1.21)

However, if one or more of the principal stresses is given, a unique

solution can be obtained for the other two stresses provided that the

extension ratios A.i, X2, A.3 are known. For uniaxial extension fz and /3 both vanish, and X2 = A-3 = Xr^^^ so that from eqs (1.1.17) and (1.1.18),

the rubber stress-strain relation is /=G(X-X-2). (1.1.22)

Treloar carried out experiments to verify eqs (1.1.19) and (1.1.20)

using simple extension, uniaxial compression, uniform two dimensional

extension and shear deformation on natural rubber gum vulcanizates. He
RUBBER ELASTICITY 21

found that these equations using the single physical constant G provided

a fairly satisfactory first approximation to experimental results. However,

deviations were observed both at moderate strains, where measured

stresses tended to be lower than predicted, and at very high strains

where they were larger than predicted. The effects at large strains are caused by failure of the Gaussian distribution, eq (1.1.6) to apply for large extensions of the chain molecules. This can be explained by non- Gaussian statistics [72]. Deviations at moderate strains have been attributed to inadequacy

of the stored energy function using only a single constant. Krigbaum

and Roe [74] analyze such deviations from the kinetic theory in detail,

summarize the evidence that there are, in fact, appreciable energy

changes in rubber deformations, and describe more recent attempts

to test and refine this theory.

Correction terms have been introduced into the theory to account for free ends of chains which are not tied into the network and for chain

entanglements which act as effective crosslinks [72, 73, 75, 76]. Fillers and crystallites seem to provide fixed attachment points for network

chains and thus simulate to some extent the effects of additional

crosslinks.

Strain-energy representation of rubber elasticity. The preceding

sections have shown how thermoelastic phenomena, along with rubber

and thermodynamic analysis, led to the kinetic or statistical theory of rubber elasticity. In turn, this leads to a description of rubber elasticity

in terms of a stored energy function using a single elastic constant, called Gin eq (1.1.13). The stored-energy approach has been developed for rubber in an

entirely phenomenological way, independent of any molecular theory,

both by Mooney [77] and by Rivlin [78]. In applying the general elasticity

theory for large deformations of incompressible, isotropic elastic materials, Mooney assumed a linear shear stress law consistent with

kinetic theory. Rivlin showed that it was unnecessary, for many purposes,

to assume any particular elastic law and that the elastic law could be

determined experimentally through relations derived from the theory.

The strain energy per unit unstressed volume stored in a material

is a function of the general components of strain at any point [79]. The

stored-energy function is unaffected by coordinate transformations. The

form of this function is as characteristic of the material as the stress- strain relation, with which, of course, it is closely connected. The

nature of the stored energy function for a pure homogeneous strain completely determines the elastic properties of the material. The

assumption of incompressibility simplifies the function, and is justified

for rubber in practical terms because stresses required for changing the

volume are so much larger than those required to change the shape.

It follows from this general theory for large elastic deformations of rubber [72, 79-81] that the stored energy W is a symmetric function of the three principal extension ratios Xi, A.2, and A.3, and can be expressed
22 RUBBER STRUCTURE AND PROPERTIES

in terms of the three following strain invariants: = + + (1.1.23)

/2 = X2 . Xi + Xi • Xl+Xi • (1.1.24)

h=k\'kl- X|. (1.1.25)

Assuming incompressibility, h= I, and ks can be eliminated from

eqs (1.1.23) and (1.1.24) so that W can be expressed in terms of two

independent variables h and which in turn contain only A.i and A.2. This means, of course, that only two of the extension ratios can be varied

independently.

The most general form of this stored energy function can be written

[72]

^=tt Cy(^i - - 3)^ (1.1.26)

(/i — 3) and (/2 — 3) are used in eq (1.1.26) instead of /i and h so that W will be zero for zero strain. The first term of the series, i= 1,7 = 0, gives for W the form derived

from the kinetic theory, eq (1.1.13) which gives a reasonably good first approximation to the rubber stress-strain relations. Retention of the first two terms, i=l, j=0 and i = 0, j=l gives

r=Ci(/i-3) +C2(/2-3) (1.1.27)

This was the form derived by Mooney which, having two constants,

provides better agreement with experimental data than eq (1.1.13). From eq (1.1.27) for simple extension, the force /per unit initial cross

section is derived as /=2(^-l)(c. + §). (1.1.28)

Equation (1.1.28) is known as the Mooney-Rivlin equation. For simple

t^=2{C,^C2)(T (1.1.29) in which to- is shear stress, cr shear strain and hence 2(Ci + C2) is the

modulus of rigidity.

Equation (1.1.28) has been exhaustively tested and debated [72, 74, 79]. There is little question that it gives better agreement with experiments

at moderate elongations than does eq (1.1.22) but the physical origin or

significance of C2 is obscure. Ciferri and Flory [82] showed that it de- creased in value when better techniques were used to reach equilibrium

strains, but it evidently does not vanish entirely and sometimes is quite

appreciable [74].
RUBBER VISCOELASTICITY 23 Energy effects have been shown to make a substantial contribution

to C2, but there are probably also entropy effects related to the orienting

properties of particular network structures as affected by chain length

distribution and chain packing [74]. The exact nature of this is still nebulous.

It would be necessary to retain more terms in eq (1.1.26) in order to account for the upward curvature in rubber stress-strain curves at high

elongation. These occur because of limited chain extensibility.

A convenient way to test eq (1.1.28) experimentally is to plot

I //2 i\ — l/K-) against 1/X. If the equation holds, a straight Hne will be

obtained, the slope of which is C2 while the intercept is Ci.

1.1.5. Rubber Viscoelasticity

' Viscoelasticity, or delayed response to stress change, of rubber com- pounds is especially important for tire performance because it is greatly

enhanced by reinforcing carbon blacks. Equilibrium or reversible

stress-strain relations from the kinetic theory become submerged in viscoelastic effects for such compounds. Tire deflections occur so rapidly that equilibrium conditions are not approached for the rubber

deformations in tires. In general, rubber viscoelasticity becomes evi- dent in a dependence of properties on rate of deformation, time-deformation history, and temperature dependence in excess of the relatively

small kinetic theory effect, according to which the modulus should be

proportional to absolute temperature. In recent years it has been found

that tensile and tear strength mechanisms [83] and even abrasion proc- esses for rubber compounds [84] are dominated by viscoelastic effects. Figure 1.1.9 gives a diagram illustrating the delayed response of rubber in a creep test where a test specimen is loaded with a constant

weight. There is a rapid deformation at the start followed by slow, gradual approach to an equilibrium deformation. When the weight is removed,

there is a rapid retraction followed by a slow recovery which may never

be complete, so that there is permanent set as well. Figure 1.1.10 illustrates compressive stress-relaxation curves for an SBR tread-type compound, when a test specimen was subjected to constant compression. Rate of stress decay, i.e., time rate of change of

stress at constant strain, for this compound evidently went through a minimum at about 78° F. Stress relaxation at moderate temperature is largely a physical phenomenon, but at elevated temperatures it is usually

associated with chemical changes such as oxidation and degradation.

Molecular and model concepts of rubber viscoelasticity. The molecular basis for rubber viscoelasticity Hes in viscous forces acting on the

segments of a chain molecule as they move in response to an applied

stress. Each segment is essentially drawn through a very viscous me- dium consisting of its neighbors, so that its motion is retarded and an

appreciable time is required to adjust to a stress. Semiquantitative cal- culations based on such a molecular picture show that it is essentially correct [75].
24 RUBBER STRUCTURE AND PROPERTIES

1 ^

Lo J

/ CONSTANT TIME ^ LOAD APPLIED

Figure LL9. Typical deformation in creep, recovery, and permanent set.

The mathematical theory of linear viscoelasticity can be presented

by an analysis of ideal spring-dashpot models to represent, respectively,

the elastic and viscous components of the response of the material to stress. These models have been widely described [75, 85-87]. For stress relaxation, that is, decay of stress at constant strain, it is convenient for the model to consist of a large number of Maxwell elements in parallel

Ld

tr 0.7

0 6 ' ' '—'—I Mill 1 1—I

I I I 1 1 1 1—I—i_L_

'O.l 1.0 10.0 TIME. HR. Figure LLIO. Compressive stress relaxation for an SBR tread compound under 30%

compression.
RUBBER VISCOELASTICITY 25

such as shown in figure 1.1. 11(a). For descriptions of creep Voigt-Kelvin

elements in series are advantageous as shown in figure 1.1.11(b). Al- though any rubber viscoelastic curve, such as a creep or stress relaxation curve, can be matched by a suitable model with sufficient elements

[88], this is entirely formal since the model elements cannot be identified

with specific features of molecular or network structure. If the material

does not truly display linear viscoelasticity, such compHcated models

cannot be expected to cover a wide range of behavior. For a VoigtKelvin element, stress on the unit at any time is the sum of the force

arising from spring elongation and that from dashpot velocity, so that one may write

de E • €-\-7) dt' (1.1.30)

Response of a Voigt-Kelvin element to a constant load is [75]

Z) ( t) = €/cr = ( l/E ) [1 - exp (- f/t ) ]. (1.1.31)

Using the element to represent a unit cube of material, e in eqs (1.1.30)

and (1.1.31) is the strain, the strain rate, cr the stress, E is Young's

modulus and r is the retardation time of strain. This latter quantity is defined as 'r)IE, where j) is the internal friction of the material, that is, the stress per unit strain rate. D{t), or e/cr, is called the creep

compliance.

If there are a number of elements in a series model, as in figure

1.1.11(b), there will be a discrete spectrum of retardation times, and

Hi

(a)

1

T

(b)

Figure 1.1.11. Models for viscoelasticity.

(a) Maxwell elements in parallel, (b) Voigt-Kelvin elements in series.
26 RUBBER STRUCTURE AND PROPERTIES

creep compliance will be given by a summation of the contributions

to e from the individual elements.

Viscoelastic relations between creep, stress relaxation, and complex

dynamic modulus. When the mathematical summation described above

is carried to the limit for a continuous distribution of retardation times

of strain, the distribution function is known as the retardation spec- trum of the material. The creep compliance is obtained from this dis- tribution as an integral expression [88, 89]. Similarly, the relaxation

modulus can be derived as an integral expression involving the distribu- tion function of the relaxation times of stress, conveniently derived

through a generalized Maxwell model, figure 1.1.11(a). The relaxation

and retardation spectra are alternative ways of specifying the viscoelastic behavior of a vulcanizate. They are related by the mathematical

theory of linear viscoelasticity [88, 90]. In theory, these spectra can be

determined for a material which conforms to linear viscoelasticity from

suitable stress relaxation or creep curves. This is usually done by ap- proximation methods. Similarly, when the models are subjected to a sinusoidal driving force, the response can be calculated by properly

formulated integrations of either of the distribution functions. Thus, in principle, any of these experimental methods, suitably applied, can be

used to completely determine the viscoelastic response of a material

to forces. For instance, data from a relaxation text can be used to cal- culate creep [91, 92]. Superposition principle. A basic assumption of linear viscoelasticity

theory is that the material responds to stresses in the same way re- gardless of its past stress history. The effect of a change in stress can be

superimposed on effects remaining from previous stresses. This principle permits the response to be calculated for variable applied forces

[88, 90]. As with the kinetic theory, the linear theory of viscoelasticity provides

a useful framework of reference from which to describe rubber properties. Unfortunately it is not truly descriptive of tire compounds which

are filled with reinforcing carbon black. Reasons for this are apparent

in light of the restrictions of the superposition principle. In the first place,

aside from small shear stresses, rubber has a nonlinear stress-strain

relation. In addition there is an irreversible deformation component, or permanent set, which can be described in particular cases by model

elements but which cannot be well described for all deformations.

Finally, in vulcanizates fiUed with reinforcing carbon black there appear

to be stress-induced changes in internal structure associated with

breaking or slipping of polymer bonds, polymer-filler bonds or filler- fiUer bonds. These tend to invalidate application of the superposition

principle and hence the linear theory of viscoelasticity.

Time-temperature superposition principle. A very useful viscoelastic

concept is that time and temperature are equivalent for describing

viscoelastic behavior [75, 86, 89]. Thus a creep curve observed for short

times at a given temperature is identical with one observed for longer

times at a lower temperature, except that the curves are shifted on a logarithmic time axis. They can be superimposed once more by proper
RUBBER VISCOELASTICITY 27

scale changes on this axis. Similarly, portions of a creep curve or stress relaxation curve can be observed at different temperature, and these curve segments can then be shifted along the log-time axis to construct

a composite curve or master curve, applicable for a given temperature,

extending over many decades of time. Figures 1.1.12 and 1.1.13 from

Bueche [75] illustrate this procedure for a plot of creep compliance

against log-time. The shift factor for a curve segment is designated

ar, log gt being the horizontal displacement necessary to allow it to join smoothly into the master curve. This is the factor by which the time

scale is altered due to the difference in temperature, and is, of course,

a function of temperature.

For exact work, there is also a small vertical shift, modulus values

being multiplied by Topo/Tp (or compliance values by the reciprocal ratio)

to take account of the entropy effect of temperature on stress. To and po

are absolute temperature and density, respectively, for standard conditions or for the master curve, and T and p apply for the curve segment

which is to be shifted [86, 89].

It has been found [89] that for all linear viscoelastic materials over a limited temperature range horizontal shift factors are given by the

empirical Williams-Landel-Ferry (WLF) equation:

-ll.U{T-T,)

log

51.6+ (r-r.;)

Tf, is the glass transition temperature of the material. Equation (1.1.32)

provides quite satisfactory shift factors in the range T<j< T < T,,-\-\20

for all types of viscoelastic phenomena. Either Kelvin or Centigrade

temperatures may be used in it. The applicability of eq (1.1.32) is often

used as a criterion for whether a process, such as abrasion or tearing of rubber is viscoelastic in nature. For tire deformations, this time-temperature equivalence principle implies that response of the rubber at high speeds wiU not be correctly evalu- ated with conventional laboratory stress-strain testing speeds at the same

temperature, and that tire temperature and speeds should be taken into consideration. To estimate the order of magnitude of the effect, assuming

tire temperature to be 25° C and T(, = — 10°C, eq (1.1.32) gives log aT = -11.3 for r=25°C and for r=0°C, log ar = -10.0. Thus lowering

the test temperature from 25° C to 0°C is roughly equivalent to increasing testing speed by a factor of 10^ or 20, say from 20 in/min. to 400

in/min. In the same way, high frequency vibration response of the rubber at 25° C could be simulated by testing at 0°C with frequencies lowered by

a factor of about 20. AU this assumes that eq (1.1.32) is valid for tire compounds, which in general is not strictly so. Dynamic properties. When a sinusoidal force is applied to a viscoelastic

material, the viscous reaction causes a lag of strain behind stress, just

as in a creep test, as illustrated in figure 1.1.14. This phase lag is indicative of mechanical energy loss in each cycle.
28 RUBBER STRUCTURE AND PROPERTIES

LOG TIME , SEC.

Figure 1.1.12. Plot of creep compliance vs. log time for a natural rubber vulcanizate at a series of temperatures [75].

LOG TIME , SEC.

Figure 1.1.13. Master curve, constructed by horizontal shifting of curve segments of

figure 1.1.12, giving creep compliance at —56° C,from reference [75].
RUBBER VISCOELASTICITY

STRESS

STRAIN

29

Figure 1.1.14. Diagram to illustrate phase lag of strain behind stress when a sinusoidal

force acts on a rubber specimen.

E* = E' + iE"; TAnS =-|7-

AREA OF ELLIPSE = 7rE"€o^

Figure 1.1.15. Hysteresis loop showing its relations to storage and loss moduli and the phase angel, 8.

this energy appearing as equivalent heat generation in the rubber. The

amount of energy loss is given by the area of the hysteresis loop when

stress in the material is plotted against strain, as shown in figure 1.1.15. The dynamic modulus of a material is defined as a complex number

whose real part is the ratio of the stress component in phase with the

strain to the strain itself, while the imaginary part is the ratio of the stress component 90° out of phase with the strain to the strain itself. The latter
30 RUBBER STRUCTURE AND PROPERTIES

component, of course, is responsible for the energy losses. These

relations are expressed by

in which G"^ is the total complex shear modulus. The real component G'

is called the storage modulus or the dynamic modulus, and G" is the

loss modulus. The complex Young's modulus is defined similarly and,

assuming incompressibility, it is three times the complex shear modulus.

Tan 6, the tangent of the angular phase lag of strain behind stress is given by

Tan 6 is a basic parameter for expressing the energy losses relative to the energy stored. Losses in various dynamic test methods, such as rebound experiments or decay of free vibrations, can all be expressed

in terms of tan 6 [93]. All of these dynamic properties such as G\ G" , and tan S, being

viscoelastic in nature, show marked dependence on temperature and

frequency. However, in a limited range of mechanical frequencies, from

about five or ten to several hundred Hz and at temperatures well above

the glass transition temperature Tg, the frequency dependence is rather

flat. Storage and loss moduli decrease with increase in temperature,

showing that viscoelastic effects dominate over the increase predicted

by the kinetic theory. Tan 8 also decreases with increase in temperature.

This is a saving feature for tire compounds, since otherwise they would

tend to run hotter and hotter in service rather than to approach an

equilibrium temperature for a given operating condition.

To give some idea of numerical values, tan 8 for tread compounds will usually be found in the range 0.1 to 0.2 while dynamic modulus E' will range from 1500 to 2500 psi for low amplitude vibrations of about 60 Hz

at room temperature.

Dynamic modulus for tread compounds exceeds the static modulus,

sometimes by a factor of two or more, depending on the vibration

conditions. It should be noted that the apparent Young's modulus for rubber determined by either dynamic or static tests in compression

depends upon the shape of the test specimen unless the bearing surfaces

are well lubricated. Thus the tread "buttons" in a tread pattern will be

somewhat stiffer in compression on a road which is dry than when it is wet. Empirical shape factors [53, 94] such as the ratio of load area to free area are used to take this shape effect into account. The effect is somewhat larger for dynamic than for static tests [95]. Dynamic properties of tire tread compounds are further complicated

by a marked dependence on amplitude [96], giving rise to nonlinear

effects. Except for extremely small amplitudes [97], dynamic modulus

decreases with increasing amplitude, slowly approaching an asymptotic

value at high amplitudes. This gives rise to a skewed resonance curve as

G'' = G'-i-iG" (1.1.33)

tan o = -—r (1.1.34)
RUBBER VISCOELASTICITY 31 shown in figure 1.1.16 obtained in this case by varying the mass of the

system while holding excitation frequency constant. These effects are

structural or thixotropic in character. However, for a steady forced

vibration amplitude an equilibrium structure is soon reached so that there is a sinusoidal response without harmonics.

Energy losses in tires. Mechanical hysteresis losses in the rubber

compounds and in the fabric of a tire result in a drag component manifesting itself in internal heat generation which is readily observed as an increase in tire temperature above ambient. Internal heat generation

is especially severe for thick, heavy-duty tires. It limits speeds and

loads at which such tires can operate, and it delayed the use of synthetic rubber in compounds for airplane, bus, truck, and off-the-road

tires. In comparing energy losses for rubber compounds it is necessary

to specify the deformation circumstances. Hysteresis loss H per cycle

per unit volume under sinusoidal displacement is [98]

//= TTCToeo sin S. (1.1.35)

Here (To and eo are stress and strain amplitude, respectively, and sin

If rubber compounds are compared at the same strain amplitude,

0-0= €o •

I and

He=7r€lE". (1.1.36)
32 RUBBER STRUCTURE AND PROPERTIES

If they are compared at the same stress amplitude, €0 = (ToI\E* \ and

H^=i7TalE")/\E''\\ (1.1.37) In both cases, energy loss is proportional to loss modulus. At constant

stress cycle amplitude stiffer compounds have lower losses because the

loss per cycle is inversely proportional to the square of the complex

modulus.

Deformation conditions in a tire are so complicated that it is difficult to consider them to be purely constant stress amplitude or constant

strain amplitude cycles. Amplitude of bending cycles in the tire tread

and body do not depend much on compound modulus and so are approximately constant strain amplitude cycles. Compressive stress cycles of

the tread itself are more like cycles at constant stress amplitude.

Detailed experimental analysis to determine contributions of indi- vidual tire components of a 9.00-20 truck tire to drag [98] showed that tread compression contributed 32 percent of the drag, tread bending

27 percent, body rubber 12 percent and cord system 29 percent.

The coefficient of rolling resistance of a tire, defined as the drag force

to pull the free roUing tire divided by the vertical tire load, is related to power loss by the equation

(1.1.38)

where R is the dimensionless coefficient of rolling resistance, P is power

loss, ft-lb/min.; S is speed, ft/sec; and L is tire load, lb. In dynamometer

tests on 9.00-20 truck tires [99], power loss for a tire made with SBR

compounds was about 1.5 times as large as that for a comparable tire made with natural rubber compounds. This is symptomatic of the problem which is encountered with heat generation in using synthetic rubber

for heavy duty tires.

1.1.6. Reinforcement of Rubber With Carbon

Black

Carbon black is unique in tire technology. The benefication which

ensues from dispersing it in rubber is described by the ad hoc term

"reinforcement." The carbon black/rubber system is very complex.

It has been studied exhaustively [100, 101] and many factors have

been shown to contribute to its effectiveness in tires. These include:

a. chemical aspects, such as effects on vulcanization reactions through

adsorption of vulcanizing ingredients or through functional chemical

groups or elements on the carbon black surface

b. physical or chemical adsorption of segments of the chain molecules
REINFORCEMENT WITH CARBON BLACK 33

c. physical bonding between carbon black particles resulting in carbon

black chains and a reticulated structure

d. microstress fields at the particles.

The complex interaction of these chemical, physical chemical, and

physical effects has precluded any simple, definitive explanation of

all of the phenomena. The fine carbon black particles are dispersed

by high shearing forces in the thermoplastic rubber, where they tend

to form a sort of loose, flocculated or reticulated structure because of

their surface activity and mutual attractions. This structure is then

interlaced by a network of rubber chain molecules crosslinked during

vulcanization. The carbon black particles are fine enough and have suffi- cient surface activity to influence the rubber network structure pro- foundly, perhaps furnishing more or less stable junction sites which

act like additional crosslinks to produce effects not observed with an

ordinary, diluent type of filler [101, 102]. There is also a theory that the

particles act to distribute the rubber network stresses [75]. One effect

is to greatly increase the viscous forces on the chain segments of the

rubber molecules as they move to accomodate to stresses. Carbon black in rubber increases the modulus or hardness, as do

all fillers. However, distinctive improvements are observed in strength

properties such as tensile strength, tear strength and above all, in road- wear. On the other hand, mechanical energy losses, hysteresis loss and

viscoelastic responses to forces are greatly augmented. It is now realized

that these two broad characterizations of the effects of carbon black on physical properties are closely connected. All types of strength fail- ures in rubber probably originate at small flaws and proceed by essentiaUy a tearing process [50]. Viscoelastic mechanisms have been

clearly demonstrated in tearing and abrasion of unfilled vulcanizates

[103, 104], and are incorporated in current theories of rubber tensile strength [83]. Stress relaxation and creep reduce stress concentrations

at the tip of a growing flaw or cut. This appears as increased strength.

There is an additional strengthening mechanism with carbon black in that enhanced stresses at the particles produce molecular orientation

or alignment, a sort of fibering, which blunts the tear tip and tends

to divert the tear from a straight line course. Carbon blacks are characterized by particle size, surface area, and

structure. Particle size is measured from electron micrographs. Surface

area is determined by iodine [105] or nitrogen adsorption. "Structure" measures the proclivity of a carbon black to form reticulated structures.

It is evaluated by oil absorption tests [106] or measurement of packing

volume under pressure. Manufacturing methods for carbon blacks are

so advanced that they can now be produced commercially with practically any desired combination of these three characteristics. Table

1.1.3 summarizes ranges of these properties for carbon blacks used fre- quentlv in tires. Figures 1.1.17 and 1.1.18 display the effect of carbon black loadings on tensile strength, pendulum rebound, breaking elongation, and hard- ness [108].
34 RUBBER STRUCTURE AND PROPERTIES

Table 1.1.3. Description of several carbon blacks [107] Av. particle Suriace area. Apparent sp. lype diam, A (Elec- m7g (btil IN

vol. at /34 psi, tron microscope) adsorption) cc/g

HMF (High Modulus Furnace 460-660 30-45 1.20-1.46 FEE (Fine Extrustion Furnace) 310-580 36-48 1.40-1.70 HAF (High Abrasion Furnace) 260-350 62-88 1.45-1.60 ISAF (Intermediate Super

Abrasion Furnace) 175-275 95-135 1.45-1.75 SAF (Super Abrasion Furnace) 140-270 120-145 1.55-1.75

Tensile strength usually goes through a broad maximum as carbon

black loading increases, as does abrasion resistance. The level of 45-55

parts of black per hundred of rubber in tire treads, if not the optimum

for wear in a particular compound, will generally represent the best

balance between wear, resistance to tread cracking, heat generation,

traction, etc., and give best overall performance.

4000

0 25 50 75 100 CARBON BLACK LOADING, PTS./lOO PTS. POLYMER Figure 1.1.17. Effect of loadings of ISAF carbon black in SBR-1500C on tensile strength

and pendulum rebound.
REINFORCEMENT WITH CARBON BLACK 35

800

0 25 50 75 100 CARBON BLACK LOADING, PTS./lOO PTS. POLYMER Figure 1.1.18. Effect of loadings of ISAF carbon black in SBR-1500C on breaking

elongation and hardness.

Stress-Strain curves for carbon black filled vulcanizates show pro- nounced stress softening [109, 110], that is, the rubber compound has

a much lower modulus on the second extension than on the first, as shown in figure 1.1.19. This effect was widely studied and for a long

time was thought to be an important characteristic of the reinforcement

process. It is now known to be a much more general phenomenon

[111, 112] and its significance for reinforcement has been obscured.

This effect is obviously related to the fact that stable reproducible mechanical properties for tires are observed only after a "break-in" run. Abrasion of rubber involves very complicated failure processes

[50, 103], including softening and fatigue of a thin surface layer, probably associated also with oxidative deterioration and actual smearing

under severe conditions, localized cutting and chipping from road as- perities, and shearing off and rolling up of thin flakes of rubber from the

surface. Carbon blacks have evolved largely from requirements to improve treadwear, and have been developed through tire experience and

carefully controlled road tests. No one type of black can be optimum

for the wide range of rubber compounds, tires, and service conditions.
36 RUBBER STRUCTURE AND PROPERTIES

ELONGATION, %

Figure 1.1.19. Stress-softening for a tread rulcanizate illustrating approach to an e. 'ibrium hysteresis loop after many cycles.

SURFACE AREA , m^/g

Figure 1.1.20. Correlation of treadwear with specific surface area of the carbon black.

Data assembled by Studebaker [113] from several sources.
REFERENCES 37 RELATIVE WEAR RESISTANCE

120-

HIGH

STRUCTURE < BLACK

NORMAL

STRUCTURE

BLACK

SMALL

PARTICLE

SIZE LARGER

PARTICLE

SIZE

VERY

SEVERE

I I I I L_ 20 40

TEST SEVERITY.

60 80 100 MILES/MIL OF WEAR

Figure 1.1.21. Relations between relative treadwear, particle size and "structure" of carbon black and severity of service [773]. A broad, general correlation exists between specific surface area of the

carbon black and treadwear [113] as shown in figure 1.1.20. This emphasizes the importance of surface interactions between rubber and

filler for good wear resistance. Figure 1.1.21, however, reflects experi- ence showing that high structure in carbon black is especially advantageous for very severe service [113]. This is an indication of the

present sophistication in development and use of carbon blacks in tire compounds.

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38 RUBBER STRUCTURE AND PROPERTIES

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.

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40 RUBBER STRUCTURE AND PROPERTIES

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Cie, Paris, France, 1953); J. Appl. Phys. 18, 212 (1947); 19, 257 (1948); 22,

1035 (1951).

[91] Hopkins, I. L., and Hamming, R. W., J. Appl. Phys. 28(8), 906 (1957). [92] Aklonis, J. J., and Tobolsky, A. V., J. Appl. Phys. 36(11), 3483 (1965). [93] Gehman, S. D., Rubber Chem. Tech. 30(5), 1202 (1957). [94] Kimmich, E. G., India Rubber World 103(3), 45 (1940). [95] Gehman, S. D., J. Appl. Phys. 11(6), 402 (1943). [96] Gui, K. E., Wilkinson, Jr., C. S., and Gehman, S. D., Ind. Eng. Chem. 44,720 (1952). [97] Payne, A. R., Dynamic properties of filler-loaded rubbers, chap. 3, p. 69, in Ref. [50]. [98] CoUins, J. M., Jackson, W. L., and Oubridge, P. S., Trans. Inst. Rubber Ind. 40,

239 (1964); Rubber Chem. Tech. 38(2), 400 (1965). [99] Stiehler, R. D., et al, Proc. Intern. Rubber Conf., p. 73, Washington, D.C., 1959. [lOO] Kraus, G., ed.. Reinforcement of Elastomers (Interscience Publishers, Inc., New

York, N.Y., 1965); Rubber Chem. Tech. 38(5), 1070 (1965). [101] Parkinson, D., Reinforcement of Rubbers, Monograph of the Inst. Rubber Ind. (London, 1957). [102] Bueche, A. M., J. Polymer Sci. 25, 139 (1957). [103] Schallamach, A., Rubber Chem. Tech. 41(1), 209 (1968). [104] Greensmith, H. W., Mullins, L., and Thomas, A. G., Strength of rubbers, chap. 10,

p. 249, in Ref. [12]. [105] Ref. [4], Method D1510-65, Iodine adsorption number of carbon black, p. 710. [106] Ref. [4], Method D2414, Dibutyl phthalate absorption number of carbon black,

p. 1044. [107] Studebaker, M. L., Compounding with carbon black, chap. 12, p. 319, in Ref. [50]. [l08] Data from Goodyear Tech Book for the Chemical Process Industry, The Goodyear

Tire & Rubber Co., Chemical Div., Akron, Ohio 44316.

[109] Dannenberg, E. M., and Brennan, J. J., Rubber Chem. Tech. 39(3), 597 (1966). [110] Kraus, G., Childers, C. W., and Rollmann. K. W., J. Appl. Polymer Sci. 10, 229

(1966); Rubber Chem. Tech. 39(5), 1530 (1966). [Ill] Trick, G. S., J. Appl. Polymer Sci. 3(8), 252 (1960). [112] Harwood, J. A. C, Mullins, L., and Payne, A. R., J. Appl. Polymer Sci. 9, 3011

(1965); Rubber Chem. Tech. 39(4), 814 (1966). [113] Studebaker, M. L., Rubber Chem. Tech. 41(2), 373 (1968).
1.2. Friction of Rubber

K. C. Ludema ^ Page 1.2.1. Introduction 42

1.2.2. Early phenomenological laws of friction vs. new

information 43

1.2.3. Mechanical classes of tire and rubber friction 44

1.2.4. Rubber friction 45

1.2.5. Combined components of friction 54

Dry friction 54

Wet friction 56

Rubber properties 57

Longitudinal and lateral tire slip and slide 57

References 61

' Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48104. 41
42 FRICTION OF RUBBER

1.2.1. Introduction

The efficiency and convenience of the automobile is to a great extent due to the use of rubber tires. The frictional properties of rubber make

relatively high forces available for accurately controlling road vehicles

w^ithin the reaction capability of most drivers. As with, most consumer materials, rubber was not developed from

first principles to fulfill a defined need. Rather, it existed for other

purposes, and it was adapted and developed for use as a tire tread material. Understanding of rubber properties lags behind its public use. The existence of its high friction is known to almost everyone, but the

method of controlling the friction is known to only a few. Again, the

performance of tires is known in detail by many, but the origin or fundamental cause of rubber friction is not completely known by anyone.

A major impediment to the understanding of tire frictional perform- ance is the lack of adequate, broad, and clearly expressed laws of rubber

friction. Total tire performance is acknowledged in a number of publications [1, 2]^ to be the summation of tire carcass properties plus the

frictional behavior of the tread elements at and near the tire road inter- face. Considerable study is being directed toward general tire behavior

related to the "comfort" dynamics of vehicles. However, basic work in friction mechanisms is lagging far behind. There is no lack of conceptual

models of the basic friction process or of tire behavior. However, re- search and testing in these topics is done under such restricted conditions as to seriously compromise the applicability of the results to broader practical conditions. As a result there are many gaps in our knowledge, and there are several research conclusions that are not yet

well recognized as being relevant to tire friction. Another problem is the ambiguity in the use of terminology in describing tire friction processes. This probably arises from the general tendency

to define all ratios of traction force to normal force as the coefficient of

friction, whether sliding occurs or not. This paper is addressed to these two problems as well as to the task

of bringing research conclusions to bear upon practical tire behavior.

The starting point is a section on the classical laws of friction, followed

by a section on terminology of the mechanical classes of friction. Then

follows a new delineation of the basic mechanisms or components of rubber friction. The emphasis will be upon what is well estabHshed

without a detailed analysis of various points of view. The references

cited are not necessarily supportive of the statements of this paper but

rather should be used as a source where more can be found on the

particular topic.

2 Figures in brackets indicate the literature references at the end of this chapter.
PHENOMENOLOGICAL LAWS VS. NEW INFORMATION 43 1.2.2. Early Phenomenological Laws of Friction

Versus New Information

To the practicing automotive or tire engineer a knowledge of the

fundamentals of rubber friction is important for a general understanding

of observed data. The main interest has been to make preliminary

estimates of the kinds of maneuvers and stopping distances which can

be accomplished under given conditions. In this general area, the litera- ture has been prone to rely somewhat heavily on classical descriptions

of friction.^ The most popular source of statements on frictional behavior is the

French engineer C. A. Coulomb (1736-1806). His popularity is a curious

state of affairs since Coulomb was neither the first to publish, nor were

his explanations of the cause of friction correct. The French architect

G. A. Amontons (1663-1705) published many of the same conclusions,

and the Englishman J. D. Desaguliers (1683-1744) was the first to connect

dry friction with interface "cohesion". The latter view has been repeatedly confirmed since the time of Desaguliers, although it is now referred

to as the adhesion theory of friction. However, Amontons and Coulomb were both careful observers and

both came to essentially the same conclusions. Both used the common

engineering materials of their day — building stone, metals, wood, and

earth. The simplest of their conclusions are the best known, and may be

summarized in the statements:

"1. A higher force is required to begin sliding than to perpetuate

sliding (i.e., the static coefficient of friction is greater than the

kinetic value).

Furthermore, the coefficient of friction of a sliding pair is: 2. independent of applied load,

3. independent of sliding speed,

4. independent of nominal area of contact, and

5. dependent on the nature of the materials in contact." These statements are incomplete for describing frictional phenomena

in general and particularly misleading when applied to the friction of

tire rubber on road. Coulomb himself had a broader view than many

moderns who quote him. However in the last few years many new and

definitive conclusions concerning rubber friction have been published.

It is now possible to see that the simple categories listed in these classi- cal laws are not entirely separable. For example, the applied load and

the nominal area of contact interact to produce a nominal contact stress. Since surfaces are rarely smooth one would expect that there might be

varying local contact pressures, different from the average, and these

would be controlled by the geometric features of the interface roughness,

and by the material properties. Thus a change in any of these variables

'For a ver\ interesting historic account of the early work on friction see chapter 24 of reference [3].
44 FRICTION OF RUBBER

may (an usually does) influence sliding friction. New information also suggests a connection between the stick-slip phenomenon and the

effect of sliding speed on rubber friction. All of these effects vary as a function of other factors such as the damping loss in the rubber and

the quantity of contaminant, including moisture, in the contact region. In view of the complex interaction of the variables mentioned above we will not proceed with a point by point modification of Coulomb's

(or Amontons') law. Rather we will begin with more fundamental observations and attempt to synthesize real tire behavior.

1.2.3. Mechanical Classes of Tire and

Rubber Friction

The following list of five friction processes and a definition of terms

is offered in the interest of reducing the ambiguity surrounding the use

of the words "coefficient of friction."

a. The classical coefficient of friction, /x, is defined [3] as being equal

to Fs/W, where Fs is the force, tangent to the contact surface, applied

to a solid slider to initiate or maintain sliding and W is the force, normal

to the contact surface, holding the sliding elements in contact. This

latter is the wheel load in the case of tires. Thus

b. Where no sliding occurs at any point, a force tangent to the contact

surface, F, may be appHed to a perfectly rigid slider producing a uniform

tractive stress r at the interface. We define FIW=9, and obviously the

maximum values of F and 6 are given by F^ax^Fs and Omax— f^-

c. A force Fp, tangent to the contact surface, may be applied to a

flexible slider such as a non-rotating tire, producing a nonuniform dis- tribution of tractive stress r at the interface. This may occur where

the nominal contact stress is nonuniform and/or where elastic constraints on the interfaces are nonuniform. Using the findings of Mason [4], we

can state that a very small value of tangential force will produce slip over at least a small part of the interface. Fp thus always produces

partial slip. As the value of Fp increases, a greater fraction of the inter- face slips. We define FplW= cj) and obviously Fp and at full slip

equals Fs and ^t, respectively.

d. A force Fr is required to roll a loaded tire on a level surface. When

control forces are not applied, this mode of operation is referred to as

free rolling. Fr is usually considered to be due to all of the hysteresis

losses in a deflecting tire, which results in moving the center of pres- sure toward the leading edge of the contact patch. However, from the

work Reynolds and others [5], it is apparent that even in free rolling
MECHANICAL CL.^ SES OF RUBBER FRICTION 45

of a cylinder tractive stresses and sometimes slip may be induced in the contact region. In addition, for tires, tractive stresses are developed

in the contact path because distortion is required to change the curved

surface of the tire to conform to the flat road surface. For any complete

model of the tire-road interface friction, these tractive stresses must be

superimposed upon the tractive stresses caused by forces other than F,-.

e. A control force Fc, either braking, traction, or lateral, may be

applied to a rotating tire. In this case, the tractive stress and slip dis- tributions in the contact region are more complex than described in paragraph (c) above. This case is the subject of a number of papers

[6. 7]. The point to be made here is that (Fc)max/^ is often referred to as the coefficient of friction, confusing it with /it. During braking, as the rotating wheel is slowed toward complete

slip, an increasing fraction of the contact patch slips. In this case, Fc

approaches Fs. and this is a result of increasing relative speed between

the tire carcass and road surface. On the other hand, Fp approaches

Fs as a result of the distance of movement of the center of mass of the

slider, a very different mechanism.

It is apparent that the operation of a tire at high slip will be directly

influenced by the frictional properties of the tire rubber. It has been

found that [8] various frictional conditions can influence tire elastic because behavior at small values of slip as well, where carcass deflections

usually control tire behavior. On the other hand, tire carcass properties strongly influence the frictional behavior of tread elements, particularly by way of controlling tire-road contact pressure and local

slip velocities. Because of these interactions, it is imperative that tire performance studies include detailed considerations of friction as well

as of tire deformation and normal pressure distribution. One common method of acknowledging the combination of the several events in the contact patch is to describe tire friction in terms of three functional categories. They are dry friction, wet friction, and

hydroplaning [6. 7. 9]. The role of the road surface is often included

under the description, smooth, polished, abrasive, rough, well-drained,

etc. The tire may be characterized as being with or without tread and

in some instances the tread material is given. Unfortunately, none of these descriptions is adequate for a frictional characterization of a particular tire on a particular road in a particular environment. A more

fundamental approach follows.

1.2.4. Rubber Friction

For rubbing systems in general, there are several factors that contribute to sliding resistance. The friction of tire rubber on practical sur- faces can be divided into at least four components, or causes. The

naming of these four components is arbitrary to some extent, and they

are here separated in terms of friction force, F, rather than the coefficient
46 FRICTION OF RUBBER

of friction /x:

^total ^ad(hesive) ^~ ^deformation) ~^ ^vis(cous) ~^ ^tear(ing) and each component is defined below. Each component is distinctive as shown by simple experiments. Each component varies in a different

way, and affects tire performance in a different way, with variation in contact stress, sliding speed, temperature, tire material, road surface

profile, etc. The several components of friction will first be described

individually, for clarity. In later paragraphs, the likely balance of the

components will be discussed in relation to a real tire. The existence of the adhesive, or dry, component of rubber friction

is verified, and values are measured, by sliding a rubber specimen on a carefully cleaned, smooth surface, such as glass^. Such experiments

show that a thin smooth film of rubber about 100 thick is deposited

and remains attached to the mating surface. The sliding retardation

force varies considerably with contact pressure, sliding speed, and

temperature and is consistent with the view that the value of Fad is dependent on the viscoelastic mechanical properties of the polymer

[10, 11].

When a sliding body leaves a thin film or track, a friction test may be

merely a shear test. If this is the case, then Fad depends on bulk mechanical properties of the polymer and not on adhesion kinetics. Other

authors [12-14] are of the opinion that adhesion kinetics is responsible

for either a part of, or the entire, adhesive frictional behavior. In their

view, the rate of making and breaking bonds controls the magnitude of

Fad- Apparently they assume that the bonds that form also break at the original interface.

One observation at least may be explained by surface kinetics, and

that is the need to slide rubber a few inches before a steady state value

of friction is measured. On the other hand, this may also be due to the

existence of surface films. Whether Fad is controlled by adhesion kinetics or bulk mechanical

properties may in the end be a moot point. There is a possibility that these two properties may be derived from the same source. In any case, there is general agreement that Fad varies with sliding

speed and temperature as shown in figure 1.2.1. The plotted master

curve (a) is from reference [10] for an acrylonitrile-butadiene tire material with a tg (glass transition temperature) of —20° C (—4° F). The

curves are shifted using the WLF transform to show friction at various

temperatures.

The location of the peak of the friction curves can be predicted

from the glass transition temperature of the rubber. For SBR rubber

with — 45° C (—49° F), for example, the friction maxima at the

temperatures designated in figure 1.2.1 would be shifted nearly two

^ A — Angstrom units, 10"* cm.
RUBBER FRICTION 47

2.5

2.0

1.5

-20°C(-4°F)

~~ >^ / N - / ^\

/ \

/ \

\

\ >

20°

/(a)

C (68°F)

/

/ \

/ \

/ \

80''(I76°F) \

\

\

\

\\\

r- (b)

1.0

.5

1 I 1 1 ^

1 1 1 1 1 -8 -6 -4

Loq Sliding Speed (Cm/sec) -2 0 2 4 6 8

Sliding Speed (Miles/hr)

1 0.00022

1 1 0022 2.2

1 1 220

1

Figure 1.2.1. Coefficient of sliding friction plotted against sliding speed in cm.jsec. and

miles per hour.

Curve (a) is the master curve for acrylonitrile-butadiene (unfilled) at 20° C (68° F). Positions of curve (a) shifted to — 20° C (—4° F) and 80° C (176° F) are shown. Curve (b) is for acrylonitrile-butadiene containing 50 parts of high abrasion furnace carbon particles [11].

orders of 10 to the right on the velocity scale. In the same manner, nat- ural rubber with = — 60° C (— 76° F) would have the peaks of the curves

shifted nearly three orders of 10 to the left. The main point to be gotten from figure 1.2.1 is that it appears that in practice friction most often decreases as sliding speed increases.

However, tire-road slip causes surface heating which was not accounted

for in figure 1.2.1. Heating wiU modify the effect of sliding speed, by an amount unknown at this time.

These curves "imply" that friction should increase as the ambient

temperature increases. In practice, this is not found to the extent shown

in figure 1.2.1 and the cause of this discrepancy is not known. Perhaps

the reason lies in the fact that other mechanisms besides the adhesive

component of friction are important in most real conditions. It should

be noted that practical values of /u, are not as high as shown in figure 1.2.1. The reason for this is also not known.

The values plotted in figure 1.2.1 are for kinetic or dynamic coeffi- cient of friction fx^. It is tempting to extrapolate the curves to zero

velocity to find the static value, ^Xg. It would appear that /Xs is smaller

than /xa-, which is opposite to general expectations. The problem most

likely lies in identifying fjig with absolute zero veloctiy, which may be

reasonable for some sliding systems involving more rigid materials.

However, it is easy to see that in order to achieve a practical sliding
48 FRICTION OF RUBBER

speed from zero velocity one passes through low velocities and possibly

through the velocity range where high values of fJi are measured. Thus

the value of /x which must be supplied to start sliding can be higher

than the value of fx to maintain sliding. The literature is not always

clear on where the transition or peak value occurs on the velocity

scale. Reference [15] reports a few test results showing a nearly linear

decrease in fJL with increasing sliding speed, with emphasis on common

automobile speeds. Hurry and Prock [16] report an increase in /x with increasing sliding speed, but only up to speeds of 10 ft/min. (— 0.1 mph). There is no conflict between these two sets of data. Friction induced vibrations are often caused by materials exhibiting

reduced friction with an increase in speed. Such vibrations are often

attributed to the "stick-slip phenonenon," which occurs in cases where

/Xs is greater than juLk. These vibrations are initiated by random disturb- ances either at sliding interface or in the machinery driving the friction experiment, and the frequency is determined by the dynamics of the

system.

Standing vibrations are never found to occur at sliding speeds less than that at peak values of fju (fig. L2.1). Vibrations occur to the right

of the peak and the vibrations are the more severe where the slope of the curve is steepest. Careful analysis of vibration data shows that in many cases the vibrations produce an oscillation in sliding speed between

two finite values, and both values are to the right of the peak of the

curve. This cannot be called the "stick-slip phenomenon"!

The entire explanation for friction induced vibration is not to be

found in the adhesion component of friction. More of the topic will be

found in the later discussion on the viscous component of friction. The adhesion component of friction has been shown repeatedly [3]

to result from strong interface bonding at the small local sites of contact between two surfaces. If bond strength is the same wherever bonds

exist, the force that resists sliding is proportional to the total of all of the minute areas of contact. A number of factors control this true area of contact. We could expect that the true area of contact of two

atomically smooth surfaces would be equal to the nominal area of contact and jjL would be very high. However, for the usual surface which is very

rough on the atomic scale, contact would be limited to the highest pro- turbances on the surfaces. The resulting true area of contact will de- pend upon details of the surface profile, the magnitude of the average

contact pressure and certain properties of the materials. Estimates of true contact area are available for two classes of material behavior.

For materials that yield plastically the total true area of contact A is A = (1.2.1) where W is the load per nominal unit area, Y is the yield strength of the

material, and Ki is a factor of proportionality. For elastomers on rough

surfaces represented by an array of hemispheres we obtain the area of
RUBBER FRICTION 49

contact from the Hertz contact stress equations;

t

11.2.2)

where r is the radius of the hemispheres. E is the modulus of elasticity of the rubber, and Ki is a factor of proportionahty. The values of /vi and Ki vary ^sith a number of conditions including surface roughness.

For example, if surface roughness is represented by an array of spheres, an eight-fold decrease in the population density of spheres will decrease

A by one-half. From eq < 1.2.2 1 above it can be seen that the adhesive is proportional

to laverage contact pressurei- \ This is observed in rubber and other

elastic materials, as shown in figure 1.2.2 [17]. Again, it is known that

yi is proportional to (1/rubber hardness i- on dry surfaces. More general

conclusions on this subject can be reached after discussion of the

remaining components of rubber friction.

2 5 o

ICT 20

ir

Li. u_ 1.5 o

h-z

UJ 1.0

u.

Ll_ UJ 0.5 oo

5 10 15 NORMAL APPARENT PRESSURE (KG/CM^

20

Figure 1.2.2. Coefficient offriction is. normal apparent pressure [17].

Ffief is a retardation force available when sliding a rubber specimen on

perfectly lubricated surfaces with smooth bumps or proturberances [3].

The retardation force is due to the partial irreversibility of deformation, i.e.. damping loss of the rubber caused by passage of the bumps or protuberances. This component of friction is not significant until there

is sliding. The need to attribute F^^^ to the damping loss of tread rub- ber is not often disputed. Attempts have been made to connect dry {hetional behavior ^dth the damping properties of rubber. More work is required to resolve this point.

Damping loss varies with frequency of deformation as shown schematically in figure 1.2.3. It is reasonable to expect that sliding over a regular array of bumps produces a vibration in the rubber, the frequency
50 FRICTION OF RUBBER

of which would be related to sliding speed. Where the bumps are 1/4

inch (0.6 cm.) diameter spheres and the rubber is acrylonitrile-butadiene

with t(, — — 20° C (— 4°F), the retardation force due to damping alone

varies with sliding speed as shown in figure 1.2.4. If topographical features of a real road are included, it is possible

to synthesize a more general deformation component loss curve. The

large texture of a road surface is approximated by the 1/4 inch (0.6 cm.)

diameter spheres just mentioned [18]. A second finer level of texture

exists which has a radius of about two orders of 10 less than the course

texture. Finally, a third very fine level of texture exists at about three to four orders of 10 less than the course texture. The friction performance

of the three textures is shown in figure 1.2.5 together with the approximate theoretical maximum value of Fdef — 0.16 for a very high loss rubber.

If in the place of discrete steps in texture size we assume a con- tinuous distribution, a very broad curve could be plotted as shown by

curve a in figure 1.2.6. On the other hand, if one range of texture size is missing from the road surface, the retardation force curve could be

altered as shown by curve b in figure 1.2.6. This curve represents an

alteration of the fine texture of figure 1.2.5, as occurs on polished roads.

This effect is more fuUy described below.

Curve b of figure 1.2.6 may also serve as a basis for explaining the de- crease in wet friction as temperature increases. The curves in figure 1.2.6

shift to the right with an increase in temperature, which has the effect of lowering F^ef at moderate to slow sliding speeds. The effect is found

by measurement to be the greatest on well-polished roads, which also fits the curves of figure 1.2.6. An interesting finding is that the maximum friction coefficient avail- able from the deformation component of friction depends upon the

population density of the fine protuberances on the road surface. For

example, for most tire rubber, with an average tire-road contact pressure

of 30 psi, Ffjef — 0.07 W when sliding over a tight-packed array of spheres.

Q-<

-3-2-1 0 I 2 3 4 5 6 7 LOG FREQUENCY OF VIBRATION Figure 1.2.3. Typical curve for damping loss as a function of log frequency of vibration from a vibrating reed test.
RUBBER FRICTION 51

LOG SLIDING SPEED (M.P.H.) Figure 1.2.4. Sliding retardation force due to damping loss in rubber when sliding over an array of sphere V4 in. (0.6 cm.) diameter.

LOG SLIDING SPEED

Figure 1.2.5. Schematic representation of the sliding retardation force due to damping

loss in a high loss rubber when sliding over a three order array of spheres, b = V4 in. (0.6 cm.), bx70-^ bx 70-^-5. The three textures are not known to produce equal effects.

while Fdef — 0,09 W when sliding over the same array with alternate spheres removed. The theoretical maximum F^ef — 0.16 W, as previously

indicated.

The deformation component of friction is difficult to separate from

other components on complex surfaces. However, in spite of the low

theoretical maximum value of /x = 0.16, this could be the primary

friction mechanism at high sliding speeds on films of water. It would

be expected that F^ef would be primarily influenced by the damping

loss (rebound) properties of the rubber. This has been found to be true
52 FRICTION OF RUBBER

II 0.I6W Q

li.

0) D

-4 - 3 -2 - 10 12 3 4 LOG SLIDING SPEED

5 6 7

Figure 1.2.6. Sliding retardation force due to damping loss in a high loss rubber sliding on "a", a surface of a continuous distribution i.e., continuous effectiveness of sphere

sizes within the range offigure 1.2.4, and "6" with size 6 X 10~'- missing. for tires on wet roads although the effect of rubber hardness confused

the picture to some extent [19]

.

Increased rubber hardness would have the effect of increasing contact pressure and when contact pressure is increased by increasing the

load, Fdef increased [18]. On the other hand, harder rubber often has

lower damping loss (higher rebound) which could more than offset the

effect of contact pressure. The viscous component of friction force Fyis is defined to take account

of the existence of a layer of either adsorbed or liquid species between

the tire and the road surfaces. It is presumed that this layer is thick

enough to significantly reduce direct bonding, or adhesion, of the tire rubber to the road material. The uniqueness of the viscous component

can be demonstrated by rubbing a rubber slider on a glass that has been

carefully cleaned and exposed to water vapor. A high friction force may

be measured on this glass surface, with F approaching WI2. In this ex- periment, there is probably no F^ef to confuse the picture because of the

smoothness of glass, and no thin film of rubber remains attached to the

glass.

It would appear that the experiment described above, producing

F — W/2, is one extreme example of a fluid film. However, this may, in

fact, be the most common condition of tire-road contract, due to the

adsorption of water vapor on tire and road surfaces.

The thickness of these adsorbed water films is not known. The closest approximation can be made from the work on water adsorption on

lime-soda glass, a material of the same class as some road stones [20]. A surface water film, apparently of 105A (0.4 X 10~^^ in.) thick, may

persist in a dry atmosphere at 23° C (41° F), and 55 A thick at 215° C

(390° F). Very likely much thicker films would exist at lower temperatures and higher relative humidities. In addition, when two mating

surfaces each have an adsorbed film, the total separating film is still thicker. More careful work is needed in this area.
RUBBER FRICTION 53 When water films exceed a few tens of molecules in thickness we may

expect the following competing events to occur. The normal load on the

rubber will eject or squeeze out the liquid film allowing the rubber to approach the mating sohd. The Stefan equation describes this situation

[21],

where / is the force appHed to a circular disc of radius R. The initial film thickness hi of a fluid of viscosity iq is reduced to a thickness /i2 in time t. Where viscosity, velocity, and separation hi are large and

contact pressure is small, conditions are favorable for maintaining

complete separation of the rubber slider from the mating surface. On the

other hand, increasing rubber hardness would be expected to produce

thin water films and therefore higher viscous drag force. The condition from which the Stefan equation was derived is that of nonsliding contact. Where sliding occurs and where the rubber slider has a rounded leading edge, the rubber will tend to slide over the top of the liquid film. There is as yet no rigorous mathematical expression for

this behavior. However we may approximate it by slipper bearing action. Such hydrodynamic lift occurs over a larger fraction of tire surface as

sliding speed increases. Hydrodynamic lift must be distinguished from

aquaplaning or hydroplaning of a tire. The latter is due to insufficient squeezing out of water film in the time of contact of each tread element.

This may explain why a rolling tire can hydroplane at a high linear

speed, spin down to a stop, and restart only after a considerable decrease

in linear speed.

It should be noted that if the leading edge of the rubber slider is sharp,

hydrodynamic lift may be partially averted, although there will be no

"wiping" or dr\ing of the road surface. A water film persists in approximate accord with the Stefan equation. In the real case, the leading

edges are not sharp, and, in fact, the contact stress near the edge is less than the average stress over the contact area. Furthermore, the road

surface geometry is as important to the problem of local hydrodynamic

Hft as is the tire surface geometry.

A complete and formal description of the viscous effects in rubber

friction is not possible at this time. The difficulty does not lie with the

classical basic equations that govern film thickness. Rather, difficulties arise in finding meaningful quantities to put into the equations. For

example, a satisfactory mathematical description of mating surface

geometries has not yet been developed [22]. Another important unknown quantity is the description of the appropriate properties of the film separating the rubbing surfaces. A thin

film of lubricant may behave as a fluid of greater apparent viscosity than

the bulk viscosity [21]. This effect is thought to be due to the nature of the bond between a liquid and solid, which is manifested by the tenacity

with which liquids remain adsorbed to a solid. There is some criticism

of this view, but it appears to be a criticism of form rather than substance.

hi

(1.2.3)
54 FRICTION OF RUBBER

Other recent work shows an opposite effect [23]. Electrical double

layers in an electrolytic liquid film between rubber and glass were ob- served to maintain a larger equilibrium separation between the sHders

than can be accounted for by viscous or inertia effects. Thus, a value of

Fvis was measured for a fluid film thicker than normal for the contact

stress, producing a very low value of yu- = 0.04 without hydrodynamic

lift.In principle, at least, Fyis can be sliding speed dependent in such a way as to encourage frictional vibrations. The proper conditions are achieved by rounding the edges of rubber elements so that hydrodynamic lift increases with velocity. These conditions are apparently met

when a tire slides on a wet polished concrete surface. A high frequency

vibration (squeal) can be heard, particularly at speeds below 15 mph,

and it originates in the tread region of the tire.

Ftear IS a Component of friction which takes account of the observation that some solid surfaces tear particles from the rubber. These

particles usually do not remain attached to the mating surface, and this mechanism can occur on contaminated surfaces where values of /x may

be low. In some cases, where values of [x are high, it is easy to explain

the tearing of particles from a rubber surface by high traction stress, in combination with contact stresses, which cause fracture in the rubber.

Likewise, it could be argued that i'tear is already accounted for under

one of the other three components of friction. On the other hand, the

deformation and thick film viscous components of friction would not

ordinarily be expected to produce large wear fragments.

1.2.5. Combined Components of Friction

The operation of tires on roads involves some combination of the

several mechanical classes and components of friction. Research has

not yet reached the stage where it is possible to mathematically express

either the distribution of slip velocities in the contact patch, or the

traction stress distribution due to the components of friction on tire and

road surfaces of undefined surface geometry. It will therefore be nec- essary to qualitatively describe some of the events in the contact patch

under various operating conditions, and to further define some terms.

In these descriptions, very many details are omitted, in the interest of maintaining the broader and more unifying view.

Dry Friction

Dry friction is one phenomenological category of tire operation that

is thought to be simple, and it usually produces high friction. In fact, it is a wide range of conditions. A common but probably inaccurate as- sumption is that dry friction involves primarily the adhesive component

of friction. Several interesting coincidences seem to bear this out, but

the situation is far from resolved. For example, begin with curve a of

figure 1.2.1 for dry friction of an unfilled rubber where a film of rubber
COMBINED COMPONENTS OF FRICTION 55 was not left on the glass surface. Reference [11] shows the peak value

of friction for the same rubber at nearly the same sliding speed as that

of reference [10], in an experiment that produced a thin film of rubber

attached to the glass. Reference [10] also reports data for carbon fiUed

tire rubber, which is a curve that overlaps the curves of figure 1.2.1 except that the peak of the curve is replaced by a plateau showing a maximum sliding coefficient of friction near 1.2, a value that has been

reported for tires on dry roads. On the other hand, a of 1.2 is not proof

of the adhesion component of friction alone. Reference [11] measures a maximum of 3.2. but reference [23] shows a value in excess of 10. It is likely that these differences are due to complex surface films. A "dry" surface may be^ covered with adsorbed water up to the limit of invisibility at about 500 A (2 X 10"^ in.) or V2 micron thick. A reduction

in braking friction is possible on surfaces exposed to cool moist air with- out a visible water film [24]. This could be true with a 500 A thick film. The tenacity of thin water films is surprisingly high. Water is usually

not completely removed by the local pressures in the tire contact patch.

From eq (1.2.3), for a local contact pressure of 250 psi, a vehicle speed

of 30 mph and a water viscosity of one centipoise (1.45 X 10~^ Reyn.-^),

and R = 10-^ cm. (0.4 X 10-^ in.), a water film of 5 X 10"^ cm. (2 X 10"^

in.) is squeezed to 1.2 X 10~^ cm. thick in the time of local contact of the tire on the road. Thin films or more viscous films would thin pro- portionately less in the same time.

It can be estimated, using the simple Newtonian viscosity equation

for drag force per unit wetted area, f= r)v/ ho, that the viscous drag of

a water film of about 10 ~^ cm. thick between two flat surfaces may pro- duce Fvis in excess of 0.25 W at a sliding speed in the order of 1 mph.

For thinner films or higher velocities, Fyis would be higher. This line of reasoning ends with the comment that (Fc) max for tires does not often

exceed in which case /x=l. Conversely, it has not been proven that some penetration of the water film [21] does not occur on very sharp

asperities, thus producing some adhesion friction. However, it is not necessary to invoke adhesive friction to account for the high values of

tire traction measured on so-called dry surfaces.

At larger slip velocities, such as during braking or severe cornering,

the thin water film would probably be heated and possibly boiled away

to produce more adhesive friction. A simple calculation shows that where

F — 0.5W. average contact pressure is 30 psi, and due account is taken

of the heat of adsorption of water to glass, adiabatic heating, and boiling of a water film 10~^ cm. (0.4 X 10~^ in.) thick takes place by rubbing a distance of from one to two inches. Thus, for a typical automobile tire a slip of about 15 percent would boil such a water film away. For larger

slip ratios or for thinner water films, the water would be removed in the

leading part of the slip region in the contact patch, producing dry or adhesive friction over the remainder of the contact patch. Traction

stresses could, therefore, decrease as slip velocities increase (see

"^Reyn. — Re>-nolds units. Ibf • sec/in.-.
56 FRICTION OF RUBBER

fig. 1.2.1), the opposite of what would be expected if a viscous film were

to persist. Under still more severe conditions, the surface temperature

of the rubber could reach the softening or reversion temperature where

rubber is easily removed from the tire surface [26]. Wet Friction

Wet friction covers a range of conditions from visible wetness to standing water. In this case, the balance between the various friction components is strongly influenced by tire rolling speed.

A rolling tire can be viewed as a device that places tread elements on

top of a water film, applies a load for a period of time and then reverses

the process [27]. During the time of loading the tread element sinks

toward the condition of static dry contact, without achieving this end.

Therefore, through the conta t excursion there is an increasing viscous

drag force potential, which is not realized until there is a slip. In principle,

separating the tire-road surfaces should require a force, limited by cavi- tation and peeling. This force would have the effect of an additional small

force holding the tire near the road.

When slip occurs, there may be sufficient hydrodynamic lift to de- crease viscous drag. It is reasonable to expect that this effect should be

greatest as the slip ratio of a tire increases toward complete sliding.

For instantaneous water film thickness of 10~^ cm. (0.4 X 10~^ in.) between flat surfaces, viscous drag forces are negligibly small even at 100 mph sliding speed. This type of statement begs the question of how

much total water on a road surface produces an effective water film 10~^ cm. thick, because of several factors. If each sinking element can freely

expel water from the pressure surfaces, it will sink in a viscous manner,

ignoring inertia effects. If it cannot do so, it, together with neighboring

elements in the same situation, becomes a single operating element with an increased effective value of R in eq (1.2.3). In the extreme, all elements

in the contact region might operate together, with a value of R that properly represents contact between an entire contact patch and a rough road surface. This situation is often described as the cause of hydroplaning [28]. When hydroplaning occurs, the viscous drag forces

are very small and appropriate directional control forces are not achieved.

In practice, a water film much deeper than 10~^ cm. is required to achieve very small friction forces [7]. Most road surfaces consist of pro- tuberances or projections rising above the average surface plane. A

sinking element will sink more slowly when approaching one of these

projections than will a nearby element located over a valley, if water can be expelled. Thus, the road surface contour is to some extent impressed upon the tire surface. This is called draping [29]. For a sliding

tire the impressions move parallel with the surface with some loss of energy. Thus, there can be a useful Fdef even though Fvis may be very

small. When the water film over the larger projections becomes less than

10~^ cm., the second order or fine projections [30, 31] will begin to be

impressed upon the tire surface, which will produce a higher traction

potential. For thinner water films, —10-^ cm. (0.4 X 10"^ in.) the full Fdef potential is equal to the viscous drag at 10 mph producing Ftotai
COMBINED COMPONENTS OF FRICTION 57 — 0.4 W. Thinner water films would expose the very fine projections but

their exposure does not enhance tire traction. It is apparent that a simple

tire friction test does not reveal the fundamental causes of friction. Rubber Properties

There is no tire rubber that produces high friction on all surfaces,

under all conditions. The effect of base polymer and of compounding

ingredients on frictional properties appears to depend upon a number of other conditions describing the sliding process, including the nature of the opposing surface, the lubrication, and the load. For example, in- creasing carbon black content has been reported to increase friction coefficient on an abrasive surface but to decrease it on a smooth surface

at low loads. Soft compounds usually give higher coefficients of friction than do

harder compounds on smooth surfaces under low loads. In line with

this observation the effect of rubber plasticizer is to produce softer rubber, while increased state of cure produces harder rubber and lower

friction. The effect of rubber hardness on its frictional properties is complicated by a reduction in hardness of the rubber layer in immediate contact with the opposing surface as a result of severe mechanical flexure during sliding [32]. The hardness of a layer of filled rubber in immediate

contact with an abrasive surface is very close to that of an unfilled rubber. For tires, any effect of hardness increase, caused by addition

of fillers, on frictional force must therefore be attributed to the effect of the properties of the rubber below the surface layer. It is well known

that these substrate or bulk properties can be important, since in comparing the friction of thick and thin rubber membranes on ice the thin

specimens exhibited consistently lower coefficients of friction than did

the thicker ones. The influence of rubber properties is complex. However, in summary,

it can be said that the choice of base polymer and of the ingredients used

in compounding can affect frictional properties not only through their

influence on such physical characteristics as hardness, damping loss

[19]. and surface roughness, but also by changes in chemical adhesion

and the ease of surface contaminations through bleeding of pigments or adherence of extraneous contaminants. Furthermore, the compounding

necessary to achieve maximum friction may need to be specific to the

opposing surface. In view of the several constraints on the practical

range of tire rubber properties the available range of friction is not

large. By far the most important variables are the nature of the road

surface and the tread pattern.

Longitudinal and Lateral Tire Slip and Slide

The frictional forces between a tire and road vary with the amount of

deviation from straight line free rolling operation. For the classical rigid

wheel rolling on a flat plane any slip whatsoever is complete sliding.

However, the flexible tire structure can deviate from straight line free roUing without complete sliding at the tire-road interface. Although the
58 FRICTION OF RUBBER

1.2 ^20 VIPH (a)

SLIDE

/ —

•

/60MPH , OMPH^

/ PEAK/SLIDE RATIO ^ '

/ 20MPH-I.35

40MPH

60MPH

• 1.50 •1.45

0 20 40 60 80 100 PERCENT SLIP

Figure 1.2.7a. Brake torque coefficient vs. slip — wet asphalt [33].

I.2i h-y 1.0 o

^ ft o

o

UJ .6

=)o

g .4 20 MPH- 1.30 40MPH-I.20

60 MPH - 1. 10

20 40 60 80

PERCENT SLIP

100

Figure L2.7b. Brake torque coefficient vs. slip — dry asphalt [33].

exact mechanics of this behavior have not been worked out the results are well known. Some are summarized below.

The behavior of the tire in braking is shown in figure 1.2.7 [33]. The

data are presented in terms of brake torque coefficient and percent slip.

The brake force coefficient is a value which is, to within a few percent,

the ratio of braking force to normal load. Percent slip is defined as Wr-Wt

(100) = percent slip

where Wr = rotational speed of a free rolling reference wheel and

Wt = rotational speed of the test wheel.
COMBINED COMPONENTS OF FRICTION 59

0 20 40 60 80 100 PERCENT SLIP

Figure 1.2.7c. Brake torque coefficient vs. slip — wet portland concrete [33].

0,

20 MPH

^--4C MPH (d)

//

// "^eol/ff SLIDE

1/ PEAK/SLIDE F AT 10 20MPH-I

40MPH- 1 60MPH- 1

25

15 05

0 20 40 60 80 100 PERCENT SLIP

Figure L2.7d. Brake torque coefficient vs. slip — dry portland concrete [33].

The behavior of the tire in side sHp can be seen in figure 1.2.8, giving

data for a number of tires, road surfaces, and vehicle speeds [34], Data

for combined braking and side slip are found in the same paper. However,

a clearer picture of combined effects are found in figure 1.2.9 [35]. Several attempts have been made to mathematically express the forces

available in all possible directions and degrees of slip. This usually

takes the form of a "friction circle" or "friction ellipse." In the case of an assumed friction circle, one hopes that it would be possible to predict

total behavior from a simple braking test. Unfortunately, available data

[34] do not show this to be practical. An expression of the form

R- = B- -\- S- is often used, where R is the resultant horizontal force, B is the braking force, and S is the side force. There is not a detailed dis-
60 FRICTION OF RUBBER

1.2 y i.o

o

o A

PEAKy'SLIDE F PH DRY

PH DRY

PH WET

RATIO (e) 20 M

40 M

20 M

-1.55

-1.30 -1.30

Ol-lUC

,20 V1PH DF Y .40 MPH DF P^20MPH WET 1 0 20 40 60 80 100

PERCENT SLIP

Figure L2.7e. Brake torque coefficient vs. slip — ice [33].

IQo 150 20" SLIP ANGLE oc

10*' IS* 20« SLIP ANGLEoC

Figure 1.2.8. Effect ofslip angle on sidewayforce coefficient [34\

cussion of this topic in hand. It would be surprising if a single simple

expression of the above form could be applied to all cases. For example,

the effect pf speed on the above expression is known to depend on tread

design. Furthermore, it makes some difference whether the data used

are peak values of braking force and side force, or sliding values. Speculation on this subject may be near an end since the advent of high quality

test devices [35] to measure force values at all combinations of braking

and side slip.
REFERENCES 61

20 mph

0 20 40 60 80 100 WHEEL SLIP, PERCENT

Figure 1.2.9. Friction coefficient vs. wheel slip [35].

References

[1] Julian, M. A., Revue Generale du Caoutchouc 36 (10), (1959).

[2] Gough, v., and French, T., Proc, First International Skid Prevention Conf., Charlottes- viUe,Va.,1959,p. 189.

[3] Bowden, F., and Tabor. D., Friction and Lubrication of SoHds, Vol. 2, (Oxford Univ.

Press, New York, 1964).

[4] Mason. W. P., ASLE Trans. 2 (1), 39 (1959).

[5] Ref. [3], p. 288.

[6] Maycock, G., Proc. IMF 180-2A (4), 122 (1956-66).

[7] AUbert, B., and Walker, J., Proc. IMF 180-2A (4), 105 (1956-66).

[8] Daube, J., La Technique Routiere 4 (1), 4 (1959).

[9] Leland, T., and Taylor, R., J. Aircraft 2 (2), 72 (1965).

[10] Grosch, K., Proc. Roy. Soc. A274, 21 (1963).

[11] Ludema, K., and Tabor, D., Wear 9, 329 (1966).

[12] SchaUamach. A., Wear 6,375 (1963).

[13] Savkoor, A., Wear 9,66 (1966).

[14] Bartenev, G., and Elkin, A., Wear 8,8 (1965).

[15] Krempel, G., A. T. Z. 69 (1), 128 (Jan. 1967), A. T. Z. 69 (8), 1334 (Aug. 1967).

[16] Hurry, J. A., and Prock, J. D., India Rubber World 128,619(1953).

[17] Atach, D., and Tabor, D., Proc. Roy. Soc. 246, 539 (1958).

[18] Ludema, K., and Lee, C. S.. ASME Paper No. 69-LUB-20 (1969).

[19] Bevilacqua, E., and Percarpio, E., Rubber Chem. Tech. 41 (4), 832-894 (Sept. 1968).

[29] Holland, L., Properties of Glass Surfaces (John Wiley & Sons, Inc., New York, 1964).

[21] Cameron, A., Principles of Lubrication (John Wiley & Sons, Inc., New York, 1966).

[22] Moore, D., Wear 13,381 (1969).

[23] Roberts, A., and Tabor, D., Nature 219, 1122 (1968).

[24] Leland, T. J. W., Proc. 9th Stapp Car Crash Conf., p. 383, University of Minnesota,

1966.
62 FRICTION OF RUBBER

[25] Sabey, B. E., Brit. Granite and Whetstone Fed. J., 7 (Autumn, 1965).

[26] Nybakken, G., Staples, R., and Clark, S., Laboratory Experiments on reverted rubber

friction, Tech. Report 7 (Office of Research Administration, The University of Michigan, Ann Arbor, Mich., Oct. 1968).

[27] Moore, D. F., Research Trends, p. 8 (Spring-Summer 1966).

[28] Home, W., and Dreher, R., NASA TN D-2056 (National Aeronautics and Space

Administration, Washington, D.C. (1963).

[29] Moore, D. F. , Wear 8, 245 (1965).

[30] Wehner, B., Der Bauingenieur 10 (1) (1965).

[31] Schulze, K. H. , Strasse und Autobahn 1 0 (10), 379 (1959).

[32] Schallamach, A., Rubber Chem. Tech. 27,439(1954).

[33] Goodenow, G., Kolhoff, T., and Smithson, F., SAE Paper No. 680137, 13 (Jan. 1968).

[34] Holmes, K. E., and Stone, R. D., Symposium on Handling of Vehicles Under Emergency Conditions, Inst. Mech. Engrs., Jan. 8, 1969, p. 81.

[35] Dugoff. H., and Brown. B. J., Instrumentation for measuring tire shear forces, SAE

Paper No. 700093 (Jan. 1970).

[36] Davisson, J. A., SAE Paper No. 700093 (Jan. 1970). Annual Meeting, Detroit, Mich.,

Jan. 1968).

[37] Dillard, J., SAE Paper No. 970A (SAE 1968 Annual Meeting, Detroit, Mich., Jan. 1968).

[38] Kummer, H. W., and Meyer, W. E., Tentative skid-resistance requirements for main

rural highways, Nat. Coop. Hwy. Res. Program Report 37, Highway Research Board,

NAS, Washington, D.C. 20418, 1967.
1.3. Tire Cord Structure and Properties

Stanley Backer ^ Page 1.3.1. Introduction 65

1.3.2. Cord geometry 65

1.3.2.1. Definitions 65

1.3.2.2. Geometric models: singles yarns 67

1.3.2.3. Geometric models: plied yarns 73

1.3.2.4. Experimental observations of plied yarn

structures 87

Singles yarn retraction 88

Plied yarn retraction 93

Helix angles 96

Cross sections 100

Migration 101

1.3.3. Mechanics of tensile behavior of tire cords Ill

1.3.3.1. An introduction to the mechanics of simple yarn

structure Ill

1.3.3.2. Analyses of tensile behavior of continuous

filament singles yarn 117

The case of pure fiber tension and idealized

geometry 117

The case with transverse forces 121

The case with large extensions 125

Yarn analysis using an energy method 127

Analyzing the role of fiber uniformity 129

1.3.3.3. Experimental observations of singles yarn

tensile behavior 139

1.3.3.4. Analysis of tensile behavior of continuous fila- ment plied yarns and cords 153

Simple fiber tension and idealized geometry 153

Plied yarn analysis using an energy method 162

Analysis of the role of singles yarn variability in plied yarn mechanics 166

Cord behavior in compression 172

1.3.3.5. Observations of plied yarn behavior in tension

and compression 175

' Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. 63
64 TIRE CORD STRUCTURE AND PROPERTIES

Page Tensile behavior 175

Effect of temperature on tensile mechanical

properties 182

Effect of strain rate on tensile mechanical

properties... 183

Compression and buckling 186

Spectrum of stress strain properties * 186

Effect of lubricants 190

1.3.4. Bending mechanics of twisted structures 191

1.3.4.1. Geometry of bent yarns 191

1.3.4.2. Bending rigidity of twisted yarns (with no

friction) 196

1.3.4.3. Bending rigidity of twisted yarns (with friction)... 205

References 213
DEFINITIONS 65

1.3.1. Introduction

The textile cord reinforced rubber system which we call a tire differs

in many respects from the classical composite material discussed so intensively in the current literature [1, 2].- For the tire is, in fact, a struc- ture whose form and composition vary from point to point according to design. Likewise, its reinforcing cords are placed in specific regions

and in different orientations. The levels of strength and stiffness of the

reinforced rubber tire fall well below those of the usual composite. On

the other hand, the ratio of breaking stress of the reinforcing textile fiber to that of the rubber matrix is higher than for most fiber strengthened composites. This ratio is approximately 50 for the rein- forced rubber tire [3]. The reinforcing element in a tire is, in itself, a structure with nu- merous filament components organized in a geometrical array with a view towards enhancing the translation of filament properties into

optimum in situ tire cord performance. And, finally, the filament itself has been shown in recent studies to possess an internal structure at the microscopic as well as at the molecular level [4, 5, 6]. The uniqueness of tire reinforcement design is covered in chapter 3

of this monograph. Likewise, the analytical treatment of those portions

of the tire which can be approximately considered as uniform composite material will be found in chapter 2. This chapter will emphasize

stiU another of the cited differences between the tire and the more usual

composite. It will consider the structural geometry of textile 'tire cords

and will illustrate the interaction of this geometry with the mechanical

properties of the component filaments to influence cord behavior. This

treatment is but a part of the general subject of structural mechanics

of fibers, yarns and fabrics which has received considerable coverage

in the textile literature of the last 25 years. The consolidation and re- view of such material by Hearle [7] is recommended for more detailed

study.

1.3.2. Cord Geometry

1.3.2.1. Definitions

Engineers attempting for the first time to use textile components

as structural elements are frequently confused by the uniqueness and

ambiguity of terms used to describe textile materials and processes. The presence of such ambiguity can frequently be traced to the develop-

- Figures in brackets indicate the literature references at the end of this chapter.
66 TIRE CORD STRUCTURE AND PROPERTIES

ment of textile language during the early part of the industrial revolution

when communication and transport between communities were limited.

Thus the developing industries in towns or local regions not distant in a modern sense, often coined entirely different terms for the same concept. This trend was accentuated by the complete separation of different

parts of the industry, say that dealing with cotton and that dealing with

wool.

The confusion experienced by outside users of textiles has likewise

plagued the professionals working in the industry, particularly in their

efforts to develop information storage and retrieval systems. And of

late, considerable progress has been made in cooperative efforts to foster a universal textile language [8, 9]. The selected group of definitions which follows is listed according to U.S.A. terminology and is limited to the structures and processes as- sociated with tire cords [9, 10, 11]. Cord: two or more plied yarns twisted together in one operation

(also called a cable yarn).

Count (yarn): a number to indicate linear density of yarn systems,

either in mass per unit length or length per unit mass.

Denier: a specific count system often used for filament (as contrasted

to staple) yarns. It is the weight in grams of 9000 meters of the

yarn. Drawing (filament): extending a polymeric filament beyond its yield

point thereby attenuating its cross section, and increasing its modulus and its internal orientation. Generally, filament as- semblies are drawn prior to twisting.

Fiber: a unit of matter characterized by flexibility, fineness, and high

ratio of length to thickness.

Filament: a fiber of infinite length (as compared to short staple fibers).

Filament Yarn: a yarn comprised of filaments of indefinite length.

Helix Angle: angle between a fiber segment as it lies in a yarn and the

corresponding segment of the yarn axis, or the angle between a

single yarn segment as it lies in a plied yarn and the corresponding

segment of the plied yarn axis. Packing Factor (Yarn): the relative packing of fibers or filaments in a yarn structure expressed as the ratio of fiber cross sectional area

to corresponding yarn cross section.

Plied Yarn: a yarn in which two or more single yarns are twisted

together in one operation.

Staple Yarn: a yarn comprised of relatively short (1 to 5 inches) staple

fibers. Stretching (Cord): extending a cord structure under controlled conditions to increase its mechanical properties, its dimensional

stability, and its uniformity from cord to cord. Tenacity: the maximum specific stress developed in a tensile test taken to rupture, often expressed in g.p.d. or grams per denier.

(Notice the dimension of tenacity is that of length, and g.p.d.

is a measure of how many 9000 meter lengths of itself which the

yarn will support.)
GEOMETRIC MODELS: SINGLES YARNS 67

Tex: a specific decimal count system. It is the weight in grams of

1000 meters of a yarn. Twist: the spiral disposition of the component(s) of a yarn which is usually the result of relative rotation of the extremities of the

yarn(s), expressed as turns per unit length of the twisted structure. Yarn: a continuous length of assembled staple fibers or filaments,

often twisted together in one operation. In the tire industry the element of textile reinforcement is called a

tire cord. And in the early 1900's the twisted cotton reinforcement

structure then used was a true cord according to the above definitions,

for it was made by cabling together several pHed yarns. And each plied

yarn was made by twisting together several singles staple yarns. With

the introduction of filament rayon cords in 1938, then nylon in 1947, and

polyester in 1962, the structure was modified and it became in effect a two or a three ply yarn. This reminder that today's tire cord is a plied

yarn in textile terms is intended as a guide to information retrieval in the

textile literature as should be clear from the appended reference titles. 1.3.2.2. Geometric Models: Singles Yarns

The single filament yarn has been assigned the simplest geometrical

model based on the assumptions that (1) the yarn and its component fila- ments are uniform in dimensions along their length, (2) each filament

of the yarn lies in a helical path, rotating around the yarn axis at the same rate (expressed in turns per unit length of that axis), (3) each

filament helix has a constant radius. Thus the yarn is composed of

rings of filaments and all the filaments of a single ring lie in equal helices.

Figure L3.L Geometric model of singles yarn.
68 TIRE CORD STRUCTURE AND PROPERTIES

The form of the model is illustrated in figure 1.3.1 where, according to Hearle [7], singles yarn radius.

r = radius of helix containing a particular fiber. T = yarn twist, in turns per unit length,

h = l/r or the length of one turn of twist. Qy = surface angle of twist, i.e., the helix angle of the outermost rin of filament helices. q = corresponding helical angle at radius r,

1 = length of fiber in one turn of twist at radius r, L = length of fiber in one turn of twist at radius Ry

Clearly, the length of the central fiber in the model is h. And from the

developed surfaces of the cylindrical rings, it follows that,

/2 = A2 + 477^2 (1.3.1) tan q = 27rrT (1.3.2)

X2

Figure 1.3.2. Single helix.

r Now, redrawing the single helix in Cartesian coordinates in figure 1.3.2, we locate the point P at an arbitrary point on the helix and note that,

B = the vertically projected base point from P to the horizontal

plane Xioxz,
GEOMETRIC MODELS: SINGLES YARNS 69

6 = the rotation of OB from the starting point along oxi,

s = the distance along the helix path from Bo to P,

q = the helix angle formed between f and the vertical,

i = the unit tangent vector along the helix path at P, n = the principal normal of the helix path at P,

b = the binormal of the heHx path at P,

X = position vector from 0 to P. The coordinates of point P are seen to be:

xi = rcos 9 (1.3.3) X2 = r sm 6 (1.3.4) xs = rO cot q (1.3.5) and

From Frenet:

s = rO cosec q. (1.3.6)

(1.3.7) dx

ds

^=kn (1.3.8) ds

dh

ds

(1.3.9) where k is the curvature of the helix path and r is the geometric torsion

of the helix path. The classical texts in differential geometry carry through from equations (1.3.3) to (1.3.9) to expressions for curvature and

torsion, i.e.:

k = '^;r = '^

(1.3.10)

two important relationships in the structural mechanics of yarns, which

indicate that the individual filament paths in a twisted yarn possess

both curvature and torsion. Now, substituting in (1.3.10) for the value

of rtaken from (1.3.2),

A; = 27rr sin g

• cos (1.3.11)

T= 27rrcos2 q. (1.3.12)

For the inside central fiber whose ^ 0,

ki = Q,

Ti = 27tT radians/unit length,
70 TIRE CORD STRUCTURE AND PROPERTIES

while for the outermost fiber with maximum heUx angle Qy,

ko = 2ttT sin Qy • cos Qy, (1.3.13)

To = 2777 cos 2 <?y. (1.3.14)

Thus, the central fiber possesses a path-torsion equivalent to the yarn

twist, and this geometric torsion is reduced in the outermost fiber by

the factor cos^ Q, while the central fiber path is straight at the center and maximally curved at the outer layer. It turns out that the geometric

torsion calculated above is equal to the mechanical twist felt by each

fiber in the yarn (about its own axis) if it is assumed that the process of

twisting starts with a cylindrical array of parallel untwisted fibers and all fibers are grasped at one end and rotated together around the yarn axis. Likewise, the path curvature determined from (1.3.10), (1.3.11), and

(1.3.13) all reflect the mechanical curvature of the fibers themselves.

This is one caution to be observed in converting from the fiber-path

parameters A" and r to the actual curvature ka and twist Tq of the fibers.

If the parallel components arrayed for simultaneous twisting into a higher order structure are already pretwisted, then the quantities deter- mined from (1.3.10), (1.3.12) and (1.3.14) represent only the differences

in local twist from the intial state of the component to the in situ configuration in the higher order structure.

It is unlikely that single filaments will have a pretwist prior to their

combination into a twisted yarn. Singles yarn being ply twisted are usually

pretwisted, and hence their changes in local twist will be reflected by

(1.3.10), (1.3.12) and (1.3.14). The assumptions lying behind the model illustrated in figure 1.3.1 lead to the conclusion that the central filament in a given turn is shorter

than the outer filaments. Observations of short segments of yarns show

that this indeed is the case. But this means that the filament path which

is short in one yarn segment is likely to be longer in the next segment

along the yarn axis. For the condition of equal filament velocity input

to the twisting operation requires that over a long length of yarn each

filament has equal path length. Which is to say, if a filament path is relatively short at one point along the yarn, it will be compensatingly

longer at another point. Of course, the local path lengths of a filament

will vary if the filament migrates from helix radius to helix radius, and

this is precisely what occurs in the conventional twisting process. This

kind of migration will be considered briefly at a later point.

Another phenomenon can be anticipated from knowledge of the

varying path lengths of filaments at a given yarn segment. And that is the contraction which takes place in the parallel assembly of filaments

as they are twisted. Since in figure 1.3.1, only the center filament hes

in a straight line, most of the filaments occupy longer paths than the

axis of the twisted yarn. Hearle [7] suggests that the pretwisted, pre- contracted length of a given yarn segment is equal to the average length

of the filaments in that segment, and indicates a direct procedure of calculating this average.
GEOMETRIC MODELS: SINGLES YARNS 71 Viewing figure 1.3.1, it can be shown that the number of fibers dn

which he in the indicated cyhndrical ring of radius r, thickness dr, and

hehx angle q is dn= {Nlrrr) cos qdr (1.3.15)

where is the fiber packing number per unit area perpendicular to the

fiber axes. From eq (1.3.1) it can be shown that dr I Now, from figure 1.3.1, cos q = h/l (1.3.17)

whence from eqs (1.3.15-17)

dn_ dn dr_ Nh

irrirrjr2^'

Since dnldl is constarit, it follows that the variation of n with / is linear

and that the average / is simply the average of the maximum and minimum fiber path lengths in the given segment, i.e..

I (I)

(l + secC>) (1.3.19)

and, as Hearle poinis out, the retraction incurred during twisting is Re={l-h)/l=tsin'

(^1^ (1.3.20)

which shows the numerical relationship of table 1.3.1.

Table 1.3.1. Yarn contraction (after Hearle) Outside twist angle Q Retraction Re

0 0

10 .008 20 .031 30 .072 40 .132 50 .217

Calculations according to eq (1.3.20) predict well the twisting behavior

of continuous filament yarns, according to Hearle, who warns, however,

that deviations can occur if the yarn twists irregularly or if there is
72 TIRE CORD STRUCTURE AND PROPERTIES

permanent straining of the outer filaments or if buckling of the center

filaments occurs. (See also Kilby, 1959 [12].) The twist multiplier, T.M., is technologically a more useful quantity

for calculating the retraction, Re. Twist multiple is the product of the

yarn twist and the square root of the direct yarn count (say denier, D).

For a yarn denier of Z), the weight of a 9000 meter length is D = —^ • 9000 • 100 (1.3.21)

Vy

where Vy is the specific volume of the yarn expressed in cubic cm. per

gram weight. Thus, as stated above

T.M. = Ts VD (1.3.22)

From eq (1.3.2) and eqs (1.3.21-22) one can express the relationship

between twist multiplier (in the denier system) and outside helix angle

of the yarn, as: for denier system: tan Qy = 0.00373 V^(T.M.)d (1.3.23)

and for the tex system [7]: tan Qy = 0.0112 V^(T.M.)x (1.3.24)

The determination of Vy is best made by an optical method, either

examining the yarn profile, or its cross section. The cross sectional

method, although requiring painstaking techniques, gives more information for it permits measurement of specific volume, cross sectional shape,

packing factor, and variation in these parameters.

The specific volume of the yarn is dependent on the specific volume

of the fiber and on the degree of packing of the fibers. The packing

fraction i// is defined by Hearle [7] as

x\) = ^

(1.3.25)

and is shown to vary from 0.3 for certain staple fiber yarns to as high

as 0.9 for a highly twisted nylon filament yarn. The theoretical value

for hexagonal close packing is 0.91. Specific volumes of fibers vary from

1.1 for polypropylene to 0.4 for glass, with rayon at 0.66, polyester at 0.72, and nylon at 0.88. Specific volumes of filament yarns are shown by

Hearle [7] to decrease with twist, starting at levels of 2.65 for polyester

and 3.12 for nylon at very low twist levels and dropping just below 1.00

at very high twists. Conversely, the packing factor of the low twist yarns

in question was about 0.25 and this increased to 0.75 for the highly

twisted polyester and 0.89 for the corresponding nylon.
I

I GEOMETRIC MODELS: PLIED YARNS 73 Under extreme conditions of temperature and tension it may be ex- pected that the hexagonal packing factor of 0.91 (for round fibers) can

be exceeded where the lateral pressures between fibers caused lateral distortion of their cross sections and the yarn cross section approaches

a solid polymeric section.

1.3.2.3. Geometric Models: Plied Yarns

The twisting of two or more singles yarns together to form a plied

yarn is similar in many geometric aspects to the twisting of two or more

filaments to form a single yarn. Instead of the fiber following the helical

path BqP in figure 1.3.2, it is now the singles yarn axis which lies along

BoPy. And the curvature, and torsion, r, calculated according to eq (1.3.10) now refer to the bending and torsion of the singles yarn. What happens to the original fiber helix in the singles as it is bent and

twisted into the plied structure can be followed in terms of differential

Figure L3.3. Plied yarn geometry.
74 TIRE CORD STRUCTURE AND PROPERTIES

geometry. In figure 1.3.3 the singles yarn length, 5, is now shown lying

along the helic BoPy and the yarn section through Py is shown to contain

the principal normal, fi, and the binormal, b. The helix tangent t lies perpendicular to the singles yarn section at Py. The position vector x connects the arbitrary point Py of the helix with the coordinate system

center. A point Pj marks where the fiber "S" lying in the yarn "5", in- tersects the yarn cross section containing to point Py on the singles yarn

axis. The Frenet relations necessary for further analysis are: dy^f

dS ' dT

dS'

KN

(1.3.26)

(1.3.27)

where y — the position vector of point Pf (not shown),

S = the distance along the fiber path, which twists around the

singles yarn helix,

T = the unit tangent vector to the fiber at P/,

K = the curvature of the fiber at point P/,

N = the principal normal of the fiber path at Pf (not shown),

d = the vector joining the points Py and P/,

(f) = the angle between n and d and it locates any given fiber in the yarn cross section at Py.

Now (f) is taken to be proportional to 6, for as the yarn turns around the

ply axis by 6, the fibers rotate around the singles axis by cf). The constant

of proportionality is X, i.e.,

(1.3.28)

The objective of the following summary analysis after Schwarz [13]

and Treloar [17] is to provide expressions for specific parameters to define the interaction between filament geometry and singles geometry,

i.e..

(1) the angle, a, between the filament axis and the singles yarn axis

at any point, i.e., between i and T. This is the local helix angle of the fiber as it lies in the singles yarn (see eq (1.3.44) below);

(2) the length of a single turn of fiber around the plied singles yarn

(see eq (1.3.47) below);

(3) the contraction of the plied yarn as it is twisted from the singles

(see eqs (1.372-76) below);

(4) the curvature K of the fiber as it lies in the plied singles yarn (see

eq (1.3.92) below).

The value of these expressions wiU become obvious in the sections
GEOMETRIC MODELS: PLIED YARNS 75 which follow and which treat the subject of the mechanics of twisted

structures. Now, back to the Frenet equations for the tangent vectors t and T, we

can write

dx dx do ,^ „

=, dy dy dO

In this form lies the suggestion that if x and y can be expressed as functions of 6, they can be differentiated and simply multiplied by dO/ds

and dO/dS respectively to provide expressions for i and T. Then from

?-f=cosa (1.3.31) we have a. Now X is expressed as a function of 0 in eqs (1.3.3-5). Similar treatment

of y is a bit more complicated fory^x-^d (1.3.32)

and, assuming very smaU fiber diameters,

d— a cos 4) • n -\- a sin (f)

• b (1.3.33) n = — cos ^ • Jci — sin ^ • jC2 (1.3.34) 6 = sin ^ cos Q Xi — cos 6 cos Q xz -\- sin Q • x^ (1.3.35)

whence

yi = r cos 0 — a cos 8 cos \d -\- a cos Q sin 6 sin A.^ (1.3.36)

y-z = r sin 6 — a sin 6 cos KB — a cos Q cos 6 sin \6 (1.3.37)

ys = cot + G sin Q sin (1.3.38)

The Xi. x-i, and jcs coordinates of x and the yi, y2, ya coordinates of y

are then differentiated with respect to 0 and multiplied by dOjds and

dO/dS respectively. Now the value of dO/ds is obtained from eq (1.3.6): ? = ^

(1.3.39) ds r however, the value dO/dS must be determined from the differential geometric expression
76 TIRE CORD STRUCTURE AND PROPERTIES

The 6 derivatives of eqs (1.3.36-38) supply the terms under the square

root and lead to the expression — = [r2 cosec2 Q a^X' ^ cos^ + cos^ Q sin^ Xd

do - 2ra cos KS -f 2a^\ cos Q] (1.3.41)

Now the Xi, X2 and xs components of T are: f [ [ ]- (1.3.42)

and cosa = f-r= [^i-r, + ^2- 7^2 + ^3-^3] (1.3.43)

whence from eqs (1.3.3-5), eq (1.3.29), eqs (1.3.36-39), and eqs

(1.3.41^3) we can derive after Chow and Schwarz [13]

ak -\- a cos Q ......

tan a = — 1.3.44a

r cosec (J — a sm (J cos which leads to determination of «, the local helix angle of the fiber as it lies in the singles yarn as it lies in the ply yarn. On the other hand, the

angle between the local filament axis and the ply yarn axis, ^, is ^ + a

and there is merit in developing an expression for the angle, jSo, between

the outermost filament of the singles yarn and the ply yarn axis, in terms

of other ply yarn parameters. This was done by Treloar in a relationship

consistent with eq (1.3.44a):

or, in other terms. r+ a(l + \ cos Q)

tan fio = 7^ , . 1.3.44b)

r cot Q — ak sm a

. p

(l + «/r) tan Q + aT.(Ci) n o aa ^ tan /3o = TTTT- 7^ (1.3.44c)

1 — aTs(Ci) tan Q

where Ci is the ratio of the singles yarn length per unit filament length

(prior to plying) to the ply yarn length per unit filament length (after

plying). Ts is the twist (expressed in radians) of the singles yarn prior to plying. Discussion of Ci follows later, after eq (1.3.71) below.

The general view of the state of torsional balance in a plied yarn (i.e. dimensional stability with respect to untwisting where all external

restraints are removed) has in the past been correlated with the oc- currence of parallel alignment of the outside filament of the singles yarn

with the ply yarn axis, i.e., when /3o equals zero. Schwarz [13] and

Treloar [17] have both treated this special case. But Freeston et al. [14^15] emphasize that recovery from torsional strains is an important

factor to be included in consideration of balance in low-singles low-ply

twist yarns. While for high-singles high-ply twist yarns bending and
GEOMETRIC MODELS: PLIED YARNS 77 bending recovery, as well as friction, become important variables in consideration of ply yarn balance.

By integrating eq (1.3.41) from 6 values of 0 to 277, we obtain the fiber length as it lies in the singles, which, in turn, lies in the ply, for one turn

of ply twist. From (28) we note that (f) will have rotated through A.277 while 6 rotates 277. Thus the fiber path length for a single turn of (/) will be

1/ A. that for a single turn of 6. The integration of eq (1.3.41) was shown by Chow [16] for ^ = 0 to <9- 277, to be:S0 = 277 cosec Qlr'^a' sin' <3(X + cos Q)''yi'- (1.3.45)

when the following condition is met, i.e.

sin^ ^ + (0 < (^)'+ + cos Qy sin' Q (1.3.46)

and thus for (/> = 0 to (/) = 277, the length of a single turn of fiber around

the plied singles yarn is:

Sc^, cosec Q^r' + a' sin' Q{X + cos QY'yi'' (1.3.47) A which is in agreement with Treloar's [17] eq (20).

Still another useful expression given by Chow is the length of fiber

(in a singles lying in a pHed yarn) per unit length of the plied axis. Taking

Tp as the turns of ply twist per unit length of ply axis, the fiber length

is simply

Sp = 277rp cosec (?[r2 + a2 sin'Q {X-^ cos Q)']''^ (1.3.48)

From these expressions (1.3.45 and 1.3.48) Chow [16] determines the

"crimp'' of the fiber as it lies in the higher order structure — first, the

"crimp" of the fiber related to the singles yarn axis Cfs; second, the

fiber "crimp" related to the ply yarn axis C/pi ^

_Sd — 27Tr cosec Q

277r cosec Q

Cfs =

^

{[r2 + a2 sin2 (^(X+cos Q)^yi''-r} (1.3.49)

and then

^fp —

J Cfp = 27TTp cosec Q[r'^a' sin^ Q(\+cos Q)'yi'-l

(1.3.50)
78 TIRE CORD STRUCTURE AND PROPERTIES

as later reported by Schwartz [13, 18].

Using eq (1.3.2), i.e., tan Q = 2TrrTp, the form of eq (1.3.50) can be so altered as to show

seq(3(C/.+ l)=C/p+l. (1.3.51)

As Chow suggests, eqs (1.3.49-50) can be used in the design of plied yarns, cords, and cables, for they indicate the required length of the

primary components per unit length of structures. Equation (1.3.49), for example, indicates how much filament is needed per unit length

of singles yarn (as it lies in the plied yarn). Equation (1.3.50) indicates

how much filament is needed per unit length of the plied yarn. And it follows that the crimp of the singles as it lies in the ply is C,p=sec Q-l (1.3.52)

As Schwarz [18] points out, these equations for crimp reflect directly

the extension which will take place upon untwisting of the various

structures. And the contraction, or retraction, expected upon twisting

is Re = -^^ (1.3.53)

where C may be C/s, Cfp or Csp. Clearly, insertion of Cjp in eq (1.3.53)

is a measure of contraction of filament length to the final ply length and

of Csp is a measure of contraction of singles length to the final ply length.

The use of Cfs in eq (1.3.53) has less technological significance than

does Cfp or Csp. The principal drawback of these determinations of contraction and

extension is their dependence on a, the radius of the secondary helix. The quantity a signifies, in terms of our technological example, the radius

of the singles yarn as it lies in the ply. For the case of a singles yarn

whose fiber diameter is small compared to the singles yarn diameter,

the quantity a approximates the radius of the secondary helical path

of the fibers lying in the outer ring of the singles yarn cross section,

as shown in figure 1.3.3. Clearly, a fiber lying in the singles yarn of figure 1.3.3 at a radius of

(l/2a) or (l/3a) will have a different crimp and retraction calculated

according to eq (1.3.50) or eqs (1.3.52-53) than the fiber at the radius

a. It is only when a single value of a is used that the indicated equation

(1.3.53) for retraction can be used in a technological sense, and figure

1.3.4 illustrates such cases, particularly figures 1.3.4a, 6, d, e, g, h. With

the exception of the central singles yarns of figures 1.3.4^^, h (marked x),

all the other singles yarn of these figures exhibit the same helix radius, r, and within these singles, the filaments all exhibit the same secondary

helix angle a. The same is true for figure 1.3.4i, except for the core

singles and the core filament in the singles. In contrast, the secondary

helical radii a of figures 1.3.4c, /, j all vary from fiber ring to fiber ring,

within each singles yarn.
GEOMETRIC MODELS: PLIED YARNS 79

Figure L3.4. Plied yarns constructions.

In short, direct application of eq (1.3.53) can be made in the cases of

figures 1.3.4a, b, d, e, and possibly to cases 1.3.4g, h, i, but under no

circumstances to 1.3.4c, f, j. Unfortunately, cases 1.3.4c, and f are of

great technological importance both in apparel textiles and in industrial

textiles, such as tire cords; and it is necessary to apply the path averaging techniques for calculation of retraction in multifilament singles yarns. This procedure was reported by Treloar [17], who assumed that the

number of filaments crossing unit area normal to the yarn (singles) axis, m, is constant over the cross section. Treloar also showed that the results which follow from this assumption are similar to those following from the

assumption of constant packing density of the filaments in the section

normal to the filament axes as shown in eqs (1.3.15-20) above.

Treloar started with the component of length of the filament along the

singles yarn axis (first considered to be straight) and showed that for a

unit length of filament at a radius r, the axial component Ir is: /.= [l+(277rr,)2]-i/^ (1.3.54)

It follows that the total axial component of the unit length filaments in the cylindric shell of thickness dr and radius r, is: 27Tmr [1 + (277rn)2]-i/2 dr. (1.3.55)

Then the mean axial length of the twisted singles yarn, /«, formed from

unit length of the (straight, untwisted) filaments is shown to be: la=(^, ^27TT,y2 )

li^ + Ri (27rn)2)'/2-l] (1.3.56)

where Ry is the singles yarn radius.

The singles yarn retraction Re is, by definition. Re—l — la (1.3.57)
80 TIRE CORD STRUCTURE AND PROPERTIES

and it can be shown that

Re=tain^(^^ (1.3.20) as stated above.

Treloar determines the ply yarn contraction by calculating the mean

length of the singles axis per unit filament length, then resolves this quantity along the ply axis through multiplication by cos Qp {Qp is the

ply helix angle). But to calculate the singles yarn axis length, it is useful

to obtain a more tractable expression for A.. Thus from eq (1.3.28)

X =f (1.3.58)

and , 1,7

do ds dcf) From figure 1.3.3, it follows that

(1.3.59)

\ = ^ • r cosec q (1.3.60) or

^

(t> = (1.3.61)

r cosec q Now going back to eq (1.3.47) and integrating from (/>— 0 to (f)>27T we

have

Sc^> =f- cosec Q rr2 + af sin^ q (X+cos (?)2]i/2 (1.3.62) A =

^

[r2 + a2 sin2 ^ (x+cos (?)2]i/2

where ai is the radius of the secondary helix. Now for the length of

singles yarn axis (as it lies in the ply) per unit of filament length, /r, one

can show

^^[l + af(A±^)T"^ (1.3.63)

<^ L V cosec QJ j

^

^

It turns out that the quantity in the parentheses has special meaning as

will be seen shortly. The total twist of the singles yarn as it lies in the

ply yarn is: r;,,= r;+r;, (1.3.64)

where Tfsp is the twist per unit length along the singles yarn axis as it lies in the ply yarn. T's is the twist of the singles prior to plying (calculated per unit length along the singles axis corrected for the change in singles axis length which occurs during plying and Tp is the twist of the
GEOMETRIC MODELS: PLIED YARNS 81

singles yarn axis around the ply axis per unit length along the singles

yarn axis as it lies in the ply. Thus,

277r>,p=^ + T.p (1.3.65)

where Tsp is the torsion of the singles axis as it lies in the primary helix. Stated another way, the total twist (in radians) r/sp of the filament constituting the secondary helix in figure 1.3.3 is T/sp==^+Tsp (1.3.66)

Since from eqs (1.3.61) and (1.3.10)

d(f) A. , sin q cos q -r = and Tsp =

as r cosec q r _ A sm q cos q

Tfsp — \ r cosec q r _X+cos_g__ , T^fsp— r cosec q ~ "^^^fsp

(1.3.67)

where eq (1.3.67) is the quantity in the parentheses of eq (1.3.63). In other words,

lrp= [1 + G? (27rr;,p)2]-i/2 (1.2.68) an expression similar to eq (1.3.54), which following the sequence of eqs (1.3.55-56) provides an expression for the mean singles yarn axial length (as it lies in the ply yarn) formed by unit length of the (straight,

untwisted filaments)

'^""^ O-.rnr V

[(l+(2^a7},p)2)-'/2-l] (1.3.69)

where a is the radius of the singles yarn as it lies in the ply yarn. Now Treloar shows that from eq (1.3.69) one may deduce that 2{i-j_apy

^ap Now eq (1.3.64) can be rewritten as

27TaT}sp = ^'^

r'^"^^ =aTfsp (1.3.70)

If

Tfsp — 27rTs 'k'

Jap.

~^ Tsp Ts ^Tsp (1.3.71)

where la/lnp is the correction factor applied to the original singles twist
82 TIRE CORD STRUCTURE AND PROPERTIES

to account for the additional contraction or extension in the singles axial length due to plying. Note that this correction factor appears in eq

(1.3.44c) as the term Ci. Tg is the singles yarn twist based along its straight (preplied) axis and Ts is Tg expressed in radians rather than

turns. Now in eq (1.3.56) /« is seen as a function of Ts, i.e.,F(a, r^), using a in lieu of Rs. This leads to 2(l-/ap)^/^ arsF (a, r.) = ^ J --^ajsp (1.3.72)

i^ap Lap which can be solved for lap, as reported by Riding [19].

/ap={-(2 + ci/)+2(l + cu+i^2)i/2}/^2 (1.3.73)

where

c=aTsF{a, Ts) (1.3.74) see eq (1.3.56) and

u = ^

sin Q cos Q.

Using eq (1.3.73) one now determines the average length, L, of the ply

yarn axis per unit length of filament.

L = lap cos Q (1.3.75)

and by definition the retraction of the ply yarn is Rep=l-L (1.3.76)

Treloar treated the case of the two ply yarn where r=a presenting a somewhat simplified reduction of eq (1.3.73). Stansfield [20] also developed expressions for singles yarn contraction

upon twisting, for the plied yarn contraction in twisting, and for the plied

yarn helix angle developed in plying. However, he used a geometrical

model different than that of Chow, Schwarz and Treloar.

The earlier models were based on the concept of a singles yarn whose

axis followed a right circular helix. This helix is generated by a radial

vector, r, advancing along the straight ply yarn axis and rotating per- pendicular to it. A secondary helix comprising the individual filament of the singles yarn, twists around the singles yarn axis as it lies in the ply

and its path is generated by a radial vector, a, advancing along the

singles yarn axis and rotating perpendicular to it. Thus, the singles yarn

section shown in figure 1.3.3 at Py is perpendicular to the singles axis. Stansfield's model, figure 1.3.5, differs from the above in that the

secondary vector rotates in a plane perpendicular to the ply yarn axis. And the path of the filament is described as an epi-helix with coordinates
GEOMETRIC MODELS: PLIED YARNS 83

differing from those of eqs (1.3.36-38) as follows:

yi = r cos 6 — a cos (f) cos 6 -\- a sin (ft sin $ (1.3.77)

y- = r sin 0 — a cos (/) sin 6 — a sin 0 cos 6 (1.3.78)

y3 = recoiQ (1.3.79)

Note that the r and a and the sign of the angle 4> in Treloar's and Stansfield's expressions are altered to agree with the nomenclature of models

of Chow as in figure 1.3.3. And from eqs (1.3.77-79), Riding [19] de- rived the following expression for the singles yarn axis as it lies in the

pHed yarn, hp = [-(2 + cw) + 2(1 -\-cw-\- w'^yl^W (1.3.80)

where w = a/r sin Q. Riding [19] points out that eq (1.3.80) is equivalent

to eq (1.3.73), except that sin Q is substituted for sin Q cos Q. And thus

Figure L3.5. Geometrical model of Stansfield for a plied yarn construction.
84 TIRE CORD STRUCTURE AND PROPERTIES

at low ply twists, the two expressions give numerically close predictions.

Gracie [21] objects to the treatment of yarn contraction upon twisting

according to the method of Treloar. For as he points out, the selection

of a unit length of a filament at a given radius from the yarn center and

the averaging of its projected length along the yarn axis with the pro- jected lengths of other unit filament lengths at other radii is based on

limited model conditions. It "implies that in a length Lq of untwisted

yarn, the filaments acquire on twisting a range of axial lengths from

Lo (at the center) downwards, i.e., the yarn acquires a profiled end or ends. This can then be related to the case of a real yarn by imagining

filament migration as a practical mechanism which over a fairly long

sample, equalizes the axial lengths, but is of sufficient long period that

it may be neglected in retraction and volume considerations." Gracie

feek this approach is in error "because it uniquely fixes the retraction

ratio without including the condition that the volume of the material

is the same before and after twisting."

Gracie [21] further points out that the derivation of eq (1.3.19) by

Hearle, is based on the constancy of in eq (1.3.15), i.e., the constancy

across the yarn section of the number of fibers per unit area perpendicular to the fiber axes. This, in turn, implies a constancy of packing

factor (/) across the yarn section and therefore a constancy of yarn vol- ume before and after twisting. In contrast, Treloar's assumption of a constant number of fibers per unit area perpendicular to the yarn axis, over the yarn section, implies a low packing factor at the yarn center and a high packing factor at the outside of the yarn. (It should be noted

that if packing factors are not constant across the twisted yarn section,

the higher factors would be intuitively expected at the yarn center and

lower packing at the periphery.)

Employing the Treloar assumption regarding packing factor, Gracie

shows that in a unit length of twisted yarn, the number of filaments dM

in an arbitrary annulus (where M is the total number of filaments in the

yarn) is, dM=27Trdrl'TTRl (1.3.81)

and the total length of the filaments in the unit length of singles

(twisted) yarn is,

1^ Vl+(277nr)W3^^J^^,^, { [1 + {27TTsRyYYI' - 1}

(1.3.82)

The average filament length, L, is then the expression (1.3.82) divided

by M, which differs somewhat from the expression for la of eq (1.3.56)

derived on the assumption of constant packing factor across the yarn

and before and after twisting. Gracie points out that values of U based on

these two expressions (1.3.82)/M and (1.3.56) differ very little at low

twist levels, e.g. by 0.98 percent at a twist of 2TTRyTs=l.O and 3.7 percent at 27TRyTs = 3.0.
GEOMETRIC MODELS: PLIED YARNS 85 However, he points out, there is a much larger error in twisted radius

Ry because the packing factor (/) is highest at the outside with Treloar's

assumption. At most, (f) = 0.90 at the outside, hence for 27rRyTs=1.0,

(/) at the center is 29.3 percent lower than at the outside and the twisted

radius is 17.4 percent higher than for constant packing factor case (as

per Hearle). Gracie gives the example of low levels of twist as i^irRyT^^ = 1.0, 3.0, 8.0, which actually represent the cases of outside helix angles

Q > 45°, hardly a low twist level. But he concludes that eq (1.3.82)

is more awkward to deal with than eq (1.3.56) and therefore utilizes eq (1.3.56) in his further analysis, leading to the following expressions

of value in yarn structural mechanics. First, the retraction ratio follows

from substitution of Ts = LTb in eq (1.3.56), j-=[l-{n27TRytYV^ (1.3.83)

where L is the average length of filament in a unit length of a singles

twisted yarn. It corresponds to Ijh and can be easily related to retraction

Re of eq (1.3.20). Tt, is the basic twist, defined by Gracie as the twist per

unit of untwisted length. Ryt is the radius of the twisted yarn as contrasted to Ryu, the radius of the yarn before twisting.

Substituting in eq (1.3.56) the requirement for constancy of yarn volume

during twisting

Rlr = LPlu (1.3.84)

L=l+(277rs/?y„)'/4 (1.3.85)

and then substituting in eq (1.3.83) he writes:

^= f9 Ti, ^2 {l±[l-i27TT,Ryuryi^ (1.3.86)

which can be rewritten as,

1 1 {1 ± Vl^ {27TnRyur} (1.3.87)

Now from eq (1.3.83) Gracie points out that 27TTb cannot exceed 2IRyt

and from eq (1.3.86) 27rTb cannot exceed l/Ryt. Stated somewhat differ- ently, l/Tb must exceed (27ri?y?)/2 and also it must exceed (27r7?y„), i.e., the average length filament in one turn of the twisted yarn must exceed

half the circumference of the twisted yarn. It must also exceed the cir- cumference of the untwisted yarn. If Tb is increased beyond these

limits, then snarHng of the singles yarn is induced. Snarling due to lateral jamming of the singles in plied yarn is also considered by Gracie,

who shows that for an m-ply yarn, the maximum ply twist Nm must be

1 1 a2^ m2 ^2^2

1/2 (1.3.88)
86 TIRE CORD STRUCTURE AND PROPERTIES

where a is the radius of the singles yarn and r is the radius of the heUx

formed by the singles yarn as it lies in the ply yarn. For practical purposes we note eq (1.3.85) as the expression useful

for calculating retraction which occurs during twisting when the singles

twist (nominal) and the original (pretwisted) yarn radius is known,

while eq (1.3.87) serves for prediction of retraction if the basic twist and

the original (untwisted) yarn radius are known. Finally, one may predict

the yarn radial enlargement from the requirement of constancy of yarn

volume during twisting, i.e., from eq (1.3.84). {^J=[1 + 7t'R„uT,] (1.3.89)

which may also be derived from Piatt's [24] eq (3) and shown equivalent

to his eq (7). The fourth objective in the above summary analysis of cord geometry was to provide an expression for the curvature K of the fiber as it lies in the plied singles yarn. To derive K one may start with eq (1.3.27), then

differentiate the components of T in eq (1.3.42) with respect to 6 and

multiply by the dd/dS of eq (1.3.41). The dot product of the results should

give the expression for K^. But this expression turns out to be unmanageable and an approximate expression for K can be obtained by considering

fiber curvature in the singles yarn as this yarn is bent in an imagined

plane to a curvature corresponding to sin^ Qplrn where Qp is the heHx

angle and rn the helix radius which the singles yarn axis follows in the

actual plied yarn. For the case of bending a singles yarn in a plane. Backer [80] has

shown that eq (1.3.41) reduces to

--T^=[(a COS kO-ry^a^k^yi^ (1.3.90)

where r is the radius of curvature into which the yarn is bent, and eq

(1.3.27) becomes

KN-- dT do d

do dS do Ids

^

dy 1 de_

dS (1.3,91)

where is the principal normal of the fiber path and u is the quantity

in brackets in eq (1.3.90). By taking the product KN • KN one obtains

with g= r/a:

/ (cos \e-gY^\* + 2\'' cos d (cos kd-g)+^\'' sin^ d-K' sin^X(9 (cos Q-gY

(1.3.92) which simplifies greatly for the inside, outside, and middle filaments, to:
GEOMETRIC MODELS: PLIED YARNS 87

(/) = 0 (inside)

(1.3.93) a[(l-^)2+X^]

0 = 77 (outside)

[(l + ^)+X^]

(1.3.94) a[(l+g)2+X2]

and

7T

(/) = — (middle)

(1.3.95)

Now the radius of curvature r in eq (1.3.90) is set equal to r/i/sin^ Qp, the curvature realized in the ply helix, and g in eqs (1.3.93-95) becomes

rh/a sin^ Qp, where a is the singles yarn axis. In the two ply yarn rh = a

(with no flattening) and g then becomes 1/sin^ Qp. On the other hand, \

becomes 27TrhT'fspl sin Qp where T}sp is the twist per unit length of the

singles yarn axis as it lies in the ply yarn (see eq 1.3.64).

1.3.2.4. Experimental Observations of Plied Yarn Structures

In the decade of the 50's a fair number of papers appeared in the textile literature on the subject of plied yarn structures and properties.

Many of these papers reported experimental results to establish the

validity of various geometrical relationships, such as those discussed

above. In general, it was argued that the theoretical relationships which

provided predictions of geometric parameters closest to those observed

in experiment were the most valid. And it was suggested that close agree- ment between theory and experiment followed from selection of the

most realistic model and from suitable analytical logic.

As it turned out, the early experimental results reported contained

little information concerning experimental methods of studying yarn

geometry. Later papers included such experimental details, and showed

that test data tended to vary with conditions of experiment as much as did predictions based on analysis rooted in different geometrical models.

It was not surprising therefore to note the conclusions of several writers

that when experimental factors closely match the model conditions which

form the basis of a given analysis, then experiment and theory are generally in close agreement. And wide variations between theory and

experiment follow from observed experimental violations of model

assumptions.
88 TIRE CORD STRUCTURE AND PROPERTIES

20

. 12 O 10

^ 6

J

1—

-°/

c/1 l

Jk

0 0.2 0.4 0.6 0.8 1.0 1.2 27rn, b Figure 1.3.6. Yam retraction, calculated from eqs (1.3.56) and (1.3.57) after Treloar [77]. RyTs = n, b.

0 =Courtauld's data +=Dunlop data Singles Yarn Retraction

Th^ retraction, Re, was, in a derivation by Treloar, shown to equal

(1 — /a) in eq (1.3.57), where

(1.3.56)

where x is 2TrRyTs. And Treloar reported singles yarn contraction data

furnished by Courtaulds and by Dunlop to check the prediction of eq

(1.3.56) and eq (1.3.57) as shown in figure 1.3.6. In plotting the experimental results it was noted that the yarn radius Ry varied somewhat with

Ts and so an average value of Ry was used for graphing. The theoretical curve had no adjustable parameter, and so the agreement was con- sidered very good. On the other hancL, Hearle shows some variation from

fiber to fiber contraction ratios Cy = ///i, where by dividing eq (1.3.19) by

h we have:

C,4= (i±^ . (1.3.19)

Figure 1.3.7 shows the experimental values of contraction ratio, Cy, for viscose, Tenasco nylon and acetate plotted against twist angle, Qy. The

theoretical curve, V2 (sec ^y+1) vs. Qy falls amidst the experimental

data, with the high denier Tenasco yarn showing an excessively high

contraction ratio and two of the acetate yarns significantly low values

compared to eq (1.3.19). The remaining yarns contracted at levels close

to those predicted by eq (1.3.19). Tattersall [22] reports the results of twisting contraction tests run under "static" twisting conditions where the entire yarn segment is twisted at once by one revolving clamp, while the other clamp is free

to slide longitudinally, but not to rotate. He shows that the agreement
SINGLES YARN RETRACTION 89

0* 10° 20° 30° 40° 50° TWIST ANGLE, a Figure 1.3.7. Comparison of contraction factors, Hearle et al. [7J. between experimentally measured retraction Re and predicted values

is not as good as for the Dunlop and Courtaulds data reported by Treloar,

but shows that the test data can be made to fit the theoretical curve by

multiplying the radius by a suitable factor, or by selecting a proper

tension at which to conduct the tests (see fig. 1.3.8a). He also noted that surface treatments such as prewashing the yarn in petroleum ether or applying a spinning oil did not influence the results. Finally, he pointed

to the occurrence of kinks at intervals of 1 cm. which became progressively worse as the twist was increased.

Tattersall [22] recognized the difference in conditions for "static"

twisting a yarn and simultaneously measuring its contraction at pro- gressively higher twists as against dynamically twisting a yarn in a commercial uptwister to a given twist level, then untwisting it in the

ZttDi b Figure 1.3.8a. Retraction as function of twist in 1650 denier Tenasco yarn under various

tensions, after Tattersall [22]. RyTg = h, b.
90 TIRE CORD STRUCTURE AND PROPERTIES

uj 10 o

z9 6

I-o

<

LU

a:

I Load (g

(untwisting) / 1 100

• 250

X 250

500

- Theoretical / X /•

X

(b)

i

1

1 0 O.i 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 2 TT n, b Figure 1.3.8b. Retraction determined from untwisting of uptwisted 1650 denier Tenasco

yarn, after Tattersall [22].

laboratory tester while measuring its elongation at progressively higher

levels of untwist. He reasoned that in the commercial twister, the re- distribution of filaments required by the theory could occur, in contrast

to the restrictions of the static twisting method. In his untwisting vs retraction measurements on a commercially twisted 1650 denier Tenasco

yarn, Tattersall showed excellent agreement with theory (according to eqs (1.3.56-57)) at several levels of untwisting tension (see fig. 1.3.8b).

Riding [23] studied further the variables discussed by Tattersall, and

used the same instrument to twist, then untwist the same yarn at various

tension levels, while measuring twist retraction. He found that the twist- untwist process was perfectly reversible at low tensions, but at high

tensions, permanent straining of the outer filaments occurred as evi- denced in microscopic examination of the yarn and in the dropping of the untwist retractions to negative values, during the final stages of untwisting. Riding then designed a laboratory twister which was comprised of a rotating head which continuously fed new portions of un- twisted yarn into the twist zone and a rotationally fixed head which fed

the twisted yarn to a down stream windup. He then ran a series of twist- retraction experiments with "twisting zones" of 1, 4, 8, and 40 inches,

and found that all twist-retraction curves were very close and in close

agreement with Treloar's eqs (1.3.56-57). Riding used experimentally

observed values of Ry to calculate the retractions, reading the appropriate value of Ry from a mean value curve of Ry vs. twist.
SINGLES YARN RETRACTION 91

Stansfield [20J showed that eq (1.3.56) can be written simply as: 7TRyTs=^ Vl-L (1.3.96)

and he plotted experimental values of the right hand side of eq (1.3.96) versus Ts in figure 1.3.9a. His data were based on static twist tests at a

fixed tension. The resulting straight line had a slope equal to irRy, and

this value of Ry was used to calculate la on the basis of eq (1.3.56). The

agreement between theoretical and experimental contraction was shown

to be excellent in Stansfield's figure 1.3.6. But, as Riding points out, one may obtain different predictions oi Re = (1 — la) from eq (1.3.56) de- pending on what values oi Ry are used. Stansfield's values of /?y obtained

from the slope of figure 1.3.9a were, according to Riding [23], much

larger than those observed in actual experiment. And Riding in figure 1.3.9b shows the variation in Re vs. 27TRyTs when: (a) individual measured values of Ry are used (giving excessive retraction over the the- oretical value), (b) the mean of the measured value oi Ry is used, giving

a small improvement, and (c) the hypothetical value of Ry, in excess of measured values, but giving good agreement with the prediction of eqs (1.3.56-57). Clearly, both Riding and Stansfield have used experimentally derived values of yarn radius to obtain improved predictions

0 5 10 15

YARN TWIST n (TURNS/IN.)

Figure 1.3.9a. Graph of (l/pjVl — against yarn twist n, after Stansfield [20]. Pn = h.
92 TIRE CORD STRUCTURE AND PROPERTIES

20.0

18.0

16.0

14.0

12.0

10.0

8.0

6.0

4.0

2.00

A Measured value of b {0.0268-0.0301 cm) ff

II

- u iviec (0.0 Hyp

" (O.C

m vaiu 277 cm

othetic 307 cm

e OT u

)al valu

)

3 of b

\ 1 u J

/ /

II 1

/ii

ii //

(b) 0 0.2 0.4 0.6 0.8 1.0 2Trnb [CURE 1.3.9b. Variation of retraction with different values of hypothetical radius, after Riding [231 Ry = b.

Figure 1.3.9c. Effect of tension on retraction of 1650 denier Tenasco, after Riding [23].
PLIED YARN RETRACTION 93 from eq (1.3.56), but Stansfield has used that average value of Ry which

satisfies eq (1.3.96) which is simply another form of eq (1.3.56) in order

to show good agreement with eq (1.3.56). Stansfield shows good agree- ment of predicted values of Re for statistically twisted yarns. Riding

shows poor agreement of predicted values of Re (using actually measured values of Ry) with experimental values of statically twisted yarns

(his fig. la), but very good agreement with experimental values of con- tinuously (dynamically) twisted yarns (fig. 1.3.9c). This agreement is seen to hold for a wide range of twisting zone lengths (his fig. 6) over a wide range of twisting tensions (fig. 1.3.9c) and yarn deniers (his fig. 9). We prefer Riding's approach to the subject of retraction. Plied Yarn Retraction

The calculation of plied yarn length per unit of initially straight

filament has been based on the determination of singles yarn contracted

length la at initial twist, followed by a modification of singles yarn length

lap as it is somewhat backtwisted during plying according to eq (1.3.73). And the projection of this hehcally formed singles on the plied yarn axis provides the desired quantity L = lap cos Q as in eq (1.3.75). The re- duced form of eq (1.3.73) (shown by Treloar in his equation (40) for a two ply

yarn) has been calculated in conjunction with eq (1.3.75) and the results plotted by Treloar in figure 1.3.10, while in figures 1.3.11a and 1.3.11b

are plotted theoretical and experimental contraction ratios for the

Dunlop yarns (at various singles yarn twist). It is clear that while the

general features of the observed retraction curves are predicted by the

theory, the quantitative values are considerably different.

50

-20*- 10

0 No or nib h-.02 Figure 1.3.10. Theoretical retraction data for two-ply cord, after Treloar.
94 TIRE CORD STRUCTURE AND PROPERTIES

The theoretical curves of figure 1.3.11 were calculated using the ob- served average value of a, the singles yarn radius, over the range of

twists, and the radius of the ply helix, r, was taken equal to a. Treloar

felt that changes in ply yarn radius with ply twist could have accounted

for much of this discrepancy. Accordingly, he recalculated the retrac- tion for one set of yarns using measured values of ply yarn diameter

and assuming a = r = observed diameter/4. The result showed con- siderable improvement in the theory vs. experimental agreement, sug- gesting that variations in actual cord diameter played an important

role in experimental retractions during plying.

Tattersall, figure 1.3.12, conducted retraction tests on two ply yarn

(2 X 1650 den. Tenasco) and compared experimental with theory. His

experiments included (a) "static" twisting of the singles followed by

"static" plying, [b) continuously twisted (on an uptwister) singles yarns

which were then statically twisted in the laboratory to form a plied yarn

and (c) uptwisted singles which were then uptwisted to form a two-ply

yarn. The poor agreement he obtains for case (a) is similar to that re- ported by Treloar for the Dunlop results. The cases (6) and (c) still show

considerable divergence from the theoretical, but are much closer to

it than is case (a). Stansfield [20], as in the case of singles yarn retraction, runs pre- liminary cord static twisting tests and plots the results of tan Q vs. Tp

over a range of 0 to 15 turns/inch. The relationship is linear over a large

part of the test range. He then uses the average (hypothetical) singles

radius, a, obtained as before from preliminary singles static twisting

tests, and the measured value of Q versus Tp to calculate expected ply

yarn retracted lengths for different ply twists according to his ply re- traction equations which Riding [19] has rederived as in eq (1.3.80). The

agreement between Stansfield's theory and experimental results (see
PLIED YARN RETRACTION 95

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 27rn, b

l-IGURE 1.3.12. Retraction on twisting of two-ply cord (2X1650 den. Tenasco). After Tattersall [22].

his fig. 8) is excellent, but his method is still one of preliminary fitting of the experimental data to derive data for final calculation and may be

expected to give improved agreements so long as the general shape of the theoretical curves are correct. As was pointed out by Riding [19], the retraction equations derived from Treloar's geometric model and

from Stansfield's models are very similar and give numerically close

predictions at low ply twists. Riding [19] measured the ply yarn axial length, L, per unit filament

length at different ply twists, by several techniques and found significant

differences in the L vs. Tp curves for the same ply yarns, according to the test method used. He made up individual ply yarns at several ply

twist levels on a commercial uptwister and then unplied the yarn and

untwisted one of the singles to obtain readings at the beginning and end

only— considered to be the most reliable method. He then took commercially (continuously) twisted singles and statically twisted (plied)

them on the Tattersall trolley twister, measuring plied yarn length vs. ply twist on the same specimen. He also took a commercially produced

plied yarn, twisted to the level of Tp = 5 turns/inch and statically un- twisted it, measuring ply yarn length (on a single specimen) vs. residual

ply twist. The results of the three methods were significantly different

as shown in figure 1.3.13. The deviation of the "static plying" data was

attributed to the restriction on de-migration of the filaments in the singles

yarn due to the relatively high pressure of the singles against each other

in the plied structure and causation of spurious ply yarn lengths at each

ply twist. The serious deviation of the "static unplying" test was at-
96 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.13. Contracted cord length for different experimental techniques. Two-fold 1650

denier Tenasco ( — 4.88 t.p.cm. = — 12.40 t.p.i.) after Riding [19].

tributed to the static retwisting of the singles during the unplying op- eration—which, in effect, created unreal singles-in-ply yarn lengths and

spuriously effected ply yarn and lengths at each ply yarn twist. Riding

also observed the effect of tight vs. loose singles readings during his untwisting tests.

Helix Angles

The simple relationship between the tangent of the helix angle and

the turns of twist in a simple helix, as expressed in eq (1.3.2) should

apply equally for the case of a filament helix in a twisted singles and for a singles yarn helix as it lies in a plied yarn. Treloar [17], however,

shows that the angle of the singles in the plied yarn (cord heUx angle in his terminology) varies as a function of the original singles twist, as well

as of the ply twist. The data shown in figure 1.3.14 indicate lower ply

helix angles at any given ply twist when the singles twists are larger.

This dependence is not consistent with the theory, and it is suggested

by Treloar that the deviation might be connected with variations in ply

diameter with ply twist. On the other hand, it might be argued that the

ply diameters at two or three turns per inch of ply should be larger for the case of the 10 TPI and 12 TPI (prior) singles twist than for the

0 TPI prior singles twist and thus eq (1.3.2) suggests that angle q will increase with the increase in the radius, r, of the singles yarn heUx. The

data, however, show it to decrease for the pretwisted singles specimens.

Actually, the simplicity of the geometric expression for the primary

helix angle is misleading when applied to a plied yarn, for while the ply

twist can be accurately measured, the determination of the hehx radius

for the singles of the ply is virtually impossible. For, according to Riding

[19], the singles axis is actually a mathematical line within the singles

and it is possible to measure only the angles which the edges of the
HELIX ANGLES 97

singles, as seen in profile, make with the ply axis. One must know more

about the actual shape of the singles cross section as it lies in the ply.

This need is conveyed in figure 1.3.15 in which Riding shows the sig- nificant flattening which takes place in a two-ply yarn and in a three- ply yarn. It is seen that flattening in a seven-ply yarn is negligible, but

in terms of tire cords it is the lower ply structures which are most

widely used. Riding, accordingly, determines the helix radius by two

methods: (1) by assuming the singles axis to he midway across the section, and (2) by assuming it lies at the centroid of the flattened circles of

figure 1.3.15 (one side flattening for the two-ply yarn, two side flattening

for the three-ply). And with these two values he calculates the singles

(in ply) helix angle and compares it with observed angles at various

twists for two-, three-, and seven-ply yarns. In figure 1.3.16a, the two-ply

yarn shows some departures at the higher ply twists. Considerably more

departure of observed from calculated values occur in his three-ply

sample. However, Riding's data for the seven-ply yarn lies close on to his predicted values as shown in figure 1.3.16b, as one might expect

for the more controlled geometry of the seven-ply structure. The local angle, a, between a filament following a secondary helix in a singles of a ply and the axis of that singles is predicted by eq (1.3.44a).

But measurement of this quantity is more difficult than one would initially

expect. The angle in question represents the difference in local direction

of the filament axis and of the corresponding local segment of the singles

yarn axis. Chow [13] has calculated the error introduced into measure- ment of the outer filament direction by misalignment of the observer's

microscope. Riding has underscored the difficulty in accurately locating

the local singles yarn axis, particularly with regard to its effective heHx

I

i HELIX ANGLES 99

<£l 10

X Experimentol

• Theoretical , centroid axis o Theoretical, mid -point oxis

(0) 0 1.0 2.0 3.0 4.0 5.0 CORD TWIST, N (t.p.cm)

X — At

o 1 Apci line heoretic;al X /

/

/ (b) -2.0 -1.0 0 1.0 2.0 3.0 4.0 5.0 CORD TWIST, N (t.p.cm) Figure L3.16.

(a) Cord helix angle, after Riding [19\. (b) Cord helix angle, after Riding [19]. Two-fold 1650 denier Tenasco. Seven-fold 550 denier Fortisan H. n,=4.88 t.p.c.m.=-12.40 t.p.i. n,=-7.f^^ » - =-20.09 t.p.i.

radius, r. Thus with uncertainties relating to measured values on both

sides of eq (1.3.44a) it is not surprising to find differences in direct

measurements vs. calculated values of a. Treloar has calculated [17] the angle a for a particular sized plied

yarn and has measured corresponding a's over a range of ply twists. His

data presented in figure 1.3.17 evidences serious experimental departures from the predicted relationship.

Gracie [21] who has developed a modified expression for the outside

filament angle, compares its calculated prediction to the Dunlop data

shown in figure 1.3.17. He reports that the agreement is not very good,

though somewhat better than for Treloar's calculations.

Riding, on the other hand, has used eq (1.3.44b) to calculate values of

(3o (angle between the outside filament axis and the plied yarn axis) for two-, three- and seven-ply structures. His experimental values for the

two-ply yarn depart significantly from the predicted values, but the dif- ference between theory and experiment lessen considerably for the
100 TIRE CORD STRUCTURE AND PROPERTIES

35"

UJz

<

K 20 z

UJ

_J^ 10

UJQ

5

IDO ^

\ Co

1 cum lea Nk o \

- Observed

1 1

0 2 4 6 8 10 12 14 CORD TWISK turns per Inch) Figure 1.3.17. Calculated and observed filament angles, after Treloar [77].

ni = — 10 t.p.i.

30' o

uj" 20

_i<

I/) 10

X Experimental

• Theoreticol, centroid axis o Theoretical.

b = 0.01 10 in.

^ 20

i 10 X

< 0 Q

O

o

10

20

1.0 2.0 3.0 4.0 5.0 CORD TWlST,N(t.p.cm)

5 30 o

40

X Experin^entol

• Theoretical,

centroid oxis o Th(joretical mid-poi nt axis

X /

x/

X P

(b)

1.0 2.0 3.0 4.0 CORD TWIST, N (t.p.cm) 5.0

Figure 1.3.18.

(a) Outer filament angle, after Riding [19]. (6) Outer filament angle, after Riding [19]. Two-fold 1650 denier Tenasco ni=-4.88 t.p.cm.=-12.40 t.p.i Three-fold 1650 denier Tenasco. n,=—4.62 t.p.cm.=— 11.73 t.p.i.

three- and seven-ply structures. In view of the experimental difficulties which he so carefully evaluates. Riding considers the overall agreement

between curves in figure 1.3.18a, b as satisfactory — in other words,

eq (1.3.44b) is reasonably well backed by experiment.

Cross Sections

Riding, in figure 1.3.15, shows the cross sections of two-, three- and

seven-ply yarns before and after plying ar»d the degree of flattening of
CROSS SECTIONS 101 each singles component, as it lies in the plied yarns, is quite evident.

Extreme flattening is seen in the two-ply structure and negligible flat- tening in the seven-ply structure. Hearle [7] shows a case of extreme

flattening (his fig. 2.16) of a two-ply hot stretched yarn in which hexagonal distortion of individual fiber cross sections is also evident. But it is Hamilton [25] who provides the most comprehensive picture of

changes in yarn diameters and bulk densities under conditions of thread

flattening. Although his work related to singles yarns, it does include

both filament and staple yarn. And in figure 1.3.19, Hamilton shows the

change in flattened cross sectional width and depth of yarns wound under

tension around a slender mandrel with varying degrees of twist. A logi- cal extension of Hamilton's data can be considered for the cross sectional distortion of the singles lying in a ply, provided that the radius of curvature of the singles helix is substituted for the radius of the mandrel

of Hamilton's test equipment. The analytical prediction of singles dis- tortion in the ply is a problem of considerable complexity.

Migration

The ideal geometric models discussed earlier in this chapter are suit- able for calculation of incremental structure and local mechanical

behavior. But the requirement of filament continuity and the practicaHty of equivalent filament feed rates into the twisting zone necessitate the interchange of filaments between helical rings, so that the

filament which occupies a central position at one point along the yarn

must move to an outer position at a subsequent point. And filaments in outer rings at one point move inward further down the yarn. Such

lateral movement, as contrasted with the circumferential movement

associated with conventional twist, has been termed migration, and a

great deal has been written about it. Hearle [7], for example, devotes a

RN TWIST, turns per inch y^^^ TWIST, Cotton twist factor ^'^ (ii) Figure 1.3.19. Effect of twist on yarn cross section, after Hamilton [25].

(i) Filament yarn

(ii) Staple yarn
102 TIRE CORD STRUCTURE AND PROPERTIES

full chapter to the subject, and El Behery [26] has written a useful

review article on the topic.

Because of the availability of Hearle's consolidated treatment of the

subject of migration, we will give only limited space to its consideration.

But, at the same time, it must be stressed that lateral migration is a recognized, analyzed and experimentally verified aspect of twisted yarn

structure, and it must be kept in mind in treating special fiber assemblies such as the singles and plied filament yarns as are in use today

for tire reinforcement. The subject can be treated as a phenomenon

inherent in twisted structures and studied on a geometric basis, both in theory and experiment. It can also be studied on a mechanistic basis,

relating fiber properties, yarn structure, and processing conditions to fiber movement within the yarn cross section during twisting.

Hearle discusses the geometry of ideal migration in which the filament

"migrates regularly and uniformly from the outside to the center of the

yarn and then back to the outside, in such a way that the density of packing of fibers in the yarn is constant throughout the yarn." He divides the

yarn into cylindrical zones with increments of radius dr and from the

requirement that the length of fiber crossing each zone is proportional

to the volume of the zone, demonstrates a linear relationship between

the square of the relative radial position of a filament (r//?y) and the

distance along the filament S. A corresponding relationship is shown to

exist, for ideal migration, between (r//?y)^ and the length, z, along the

yarn axis, and is seen in figure 1.3.20 to approximate linearity.

Treloar [27] reports a more rigorous analysis of ideal migration,

which shows an approximately linear relationship between and

z for low twists and low rates of migration, but also indicates the imaginary form of the relationship at small values of r/i?, i.e., near the yarn

axis. Observation of migration behavior starting with Morton [28] and pro- ceeding with Hearle [7], Gupta [29], Treloar [30], Riding [31], El- Shiekh [32], and MarzoH [33] has shown that migration behavior occurs,

although, as Hearle points out, in a somewhat irregular manner.

0 Z/2 Z LENGTH ALONG YARN Z Figure 1.3.20. Migration envelopes for yarn with twist angle of 40°, plotted against

length along yarn, after Hearle's figure 34 [7].

(a) Variation of with Z.

(b) Variation of r with Z.
MIGRATION 103 Much has been written about experimental techniques for deter- mining fiber migration in twisted structures and about computational

methods for presenting and analyzing the observations of migration.

Morton and Yen [28] used a dyed tracer fiber against the background

of undyed fibers and made their observations in a medium of matching

refractive index. A mirror placed at 45° below the yarn being observed

provided simultaneous images corresponding to yarn views at right

angles. Chaikin [34, 35] et al, prepared radioactive wool fibers and employed the techniques of autoradiography and shadowgraphy in meas- urement of fiber configuration and longitudinal strain. Block [36, 37] et al. embedded yarn in a soft polymer, microtomed it and examined the

placement of blended fibers within the cross section of the yarn as a

basis for observing and measuring longitudinal, radial and rotational

distribution of fibers. Hamilton [38] similarly cross sectioned blended

yarns as a basis for characterizing the radial distribution of fibers by a migration index. It remained for Bunting [39] to automate the microtoming and photographing of successive yarn sections and playing

back the results in motion picture form to permit visual conception of the migrating and interlacing movement of individual filaments along

the yarn length. Finally, the simplest method of measuring migration

behavior is the direct observation of a seven (or lesser element) ply yarn

with one color distinguishable ply. For here, as Hearle [7] points out, the absence of the distinguishable ply from the surface of the yarn means it is occupying the central position of the plied yarn. El-Shiekh

[32] and Marzoli [33] have used this direct observation as their primary

technique.

The results of these migration studies can be illustrated in innumerable

examples. Hearle's [7] diagram best illustrates the phenomenon in figure 1.3.21 where comparison is made between the projection of a fiber path of an ideal structure versus one of a migrating fiber path. One may

take measurements of the x coordinates of the fiber path from figure

Yorn Surface

Dimensions compressed in this direction Yarn Surfoce

Figure 1.3.21. (a) Projection of the path of a fiber forming part of an idealized yarn

structure, [b) Projection of the path of a fiber migrating over the yarn cross section. (From Gupta, 1963.)
104 TIRE CORD STRUCTURE AND PROPERTIES

1.3.21 and simultaneously measure the y coordinates (point for point)

from a projection at right angles (to that of the fig. 1.3.21). Squaring and

summing the x and y values to give the local helix radius r permits the

experimenter to plot r\R versus the length along the yarn, as done by

Morton in examples given by Hearle [7] in figure 1.3.22a. Data of this sort can be replotted as in figure 1.3.226 for an ideal fiber, two typical

normal fibers and a wild fiber. The differences are striking and can be

used to study the effect of processing and fiber variables such as the

position of fibers upon entering the twist zone, the fiber modulus and

the effect of trailing fiber ends. Such graphs can also show up varia- tions in density of fiber packing across the yarn section.

Hearle [7] has studied fiber migration in spun rayon yarn with emphasis on mean fiber position, RMS deviation of fiber position, mean

migration intensity, equivalent migration frequency, and the coefficient

of variation of each of these parameters. And he reports little effect of

1.0 ^s^—

0.8 —

0.6

0.4

0.20

0.8 —

l_ 0.6 R

0.4

0.2 Length along Yarn Figure 1.3.22a. Examples of helix envelope showing arbitrary zone boundaries, after Morton, 1956. 50

25

(i) thZL

(ii (ui: 100

50 -

(Iv) Figure 1.3.22b. Zonal distributions for individual fibers.

(i) ideal.

'

(ii) , (iii) typical normal fibers.

| (iv) wild fiber. ,
MIGRATION 105

twisting tension on migration during ring spinning of staple rayon yarns. But migration intensity and frequency increase significantly with twist —

in fact, he reports "the ratio of migration frequency to twist frequency

is almost constant, with a migration cycle corresponding to about four

turns of twist." The reduction of C.V. values for the above parameters

as twist increases is taken to indicate a more regular migration pattern

at higher twists. Finally, Hearle points out the presence of a long term

drift (or migration) corresponding in period to the twist in the roving

from which the yarn was spun — corrected for the draft employed during

spinning. This long term drift is superposed on the short term migration

discussed above.

The migration of filaments during the twisting of multifilament yarns

is, of course, more pertinent to the subject of tire cords, and considerable effort has been expended on this topic. Riding [40], Hearle and

Gupta [41], Treloar [27], El-Shiekh [32], and Marzoli [33] have studied

filament migration in multifilament or multi-ply yarns, and their results are pertinent to tire cord problems. Riding [40] analyzed the migration

of radial position of individual filaments in a twisted yarn by calculating

a serial correlation coefficient, Ts, for pairs of r/R values separated by 5 intervals along the yarn length. A correlogram is then plotted (r^ versus

5) for a given yarn. Hearle [7] summarizes this work and illustrates the

different correlograms obtained for: (1) constant radial position, (2) regular migration, (3) random variation, and (4) a combination of regular and

irregular variation. Riding studied migration in different types of twist- ing devices and found that the correlogram of uptwisted yarn generally

gave the appearance of a damped oscillation with a well-defined periodicity. Ring twisted yarn correlograms generally reflected irregular

variation of migration patterns. Controlled changes in yarn denier, fila- ment denier and twisting tension did not appear to have consistent ef- fects on the migration pattern and frequency (obtained from the correlo- gram). However, the producers' twist present in the yarn before the

experimental twisting (to a fixed twist level) was observed to correlate

directly and significantly with the correlogram frequency. Further work

on continuous filament yarns, reported by Hearle and Gupta [41], also showed the correlogram frequency was very close to the frequency of producers' twist. But the equivalent migration frequency calculated by

Hearle [7] was greater than that determined by the correlogram method.

Both "equivalent migration frequency" and "mean migration intensity"

increase with higher twists, but decrease with higher tensions of twist- ing according to Hearle [7]. Some confusion may arise from the fact that the producers' twist (or for the purpose of the experiment, the pretwist of the yarn) appears to govern migration frequency in some instances and the conditions of

twisting, i.e., twist level and twisting tension, significantly aff"ect migration frequency in other instances. Part of the confusion may lie in the use of different measures of migration frequency, but a large part of the

difference in yarn structure may have real mechanical and structural

origins. As Hearle [7] points out, there are two different mechanisms of twisting and twist migration, each of which may dominate in a given case — or both may be active at one time.
106 TIRE CORD STRUCTURE AND PROPERTIES

There is the "Tension Mechanism", first proposed by Morton, and

quantified by Hearle and Merchant [42] for the case of the seven-ply

yarn. The mechanism is based on the assumption that during twisting

the external six singles yarns will bear the major portion of the twisting

tension. The central yarn, on the other hand, will tend towards slackness

since it is being fed into the twist zone at the same velocity as the other

six yarns, but its in situ path length is less than that of the six helically

disposed singles in the outer ring. The assumption is made that when the

central yarn tension falls to zero (and even into compression by an amount X, representing the degree of buckling necessary to initiate migration) then one yarn in the external ring, carrying 1/6 of the twisting

tension will move to the central position, displacing the yarn originally

in that place. At this instant, one of the six outer yarns is at zero tension,

and the others are at 1/6 of twisting tension. The central yarn is also, for an instant, at 1/6 of twisting tension. Then as twisting proceeds, the

central yarn accumulates increments of excess length relative to the

central path length of the plied yarn, and its tension starts to fall, eventually reaching zero tension (and beyond) at which time buckling is initiated and a second migration is underway. While this was taking place, the

slack yarn which had moved to the outer ring, then starts to build up in tension, since it now follows a longer helical path. Its rate of tension

buildup is assumed to equal the rate of tension fall off of the central

yarn. Quantitization of this descriptive mechanism is provided by Hearle [7]

who shows that the length along the ply yarn for which one singles re- mains in the central position Zc is

Zc = L (cos q + Tyl6y) \ In 1 +

Ty sec q

67(1 - cos q)-Ty

yx

7(1 — cos q) —Tyl6

(1.3.97)

where L is the free length of yarn in the twisting zone,

q is the helix angle of the singles in the plied yarn, y is the tensile spring constant of the singles,

Ty is the twisting tension of the plied yarn, X is the fractional excess of yarn in the free zone necessary! to

initiate buckhng.

There are, Hearle points out, two migrations for each occasion that a yarn occupies the central position and each yarn should be in the central

position 1/7 of the total time. Thus, the average migration per 100 cm.

for one of the seven singles involved is 200/7Zc. And it is seen that migration frequency will vary in a prescribed manner with twist angle,

q, tension Ty, free length, L, in the twist zone and the buckling parameter, X. Only X has to be determined experimentally by trial calculations

of the average effective migrations per singles yarn per 100 cm., and comparing the results with experimental observations as in figure 1.3.23. El-Shiekh [32] examined the geometric configuration of a ribbon of seven equispaced singles moving into the twist zone and calculated
MIGRATION 107

Figure 1.3.23. Experimental and theoretical results obtained by Hearle and Merchant

{1 962) for twisting of seven-ply structure. X is the degree of buckling needed for migration. the amount of central element slack, jc, necessary to initiate migration

from center to outer ring. He assumed that the slack central yarn would

be caught up in the external yarns, thus initiating the interchange, when

the axis of the nearest external yarn intersected the surface of revolution of the revolving slack yarn. This calculation removed the need for experimental determinations of x and led to quite reasonable agreement

between theory and experiment as shown in figure 1.3.24. Its applicability is, of course, limited to the geometry of an open ribbon of sepa0 .005 .010 .015

Figure 1.3.24. Variation offrequency of migration with twist angle, a, and yarn tensile strain. Eg.
108 TIRE CORD STRUCTURE AND PROPERTIES

rated elements. Contact and crowding of the filaments as occurs in real twisting of multifilament yarns makes such a determination most diffi- cult—in fact, the entire procedure of calculating migration frequency

becomes extremely complex when the simple geometry of the limitedply yarn gives way to the real conditions met in twisting of commercial

yarns. Yet, in viewing the successive cross sections of the twisting zone

in a seven monofil ply as in figures 1.3.25a, 6, one learns a great deal

which can apply to the real twisting process. In figure 1.3.25, El-Shiekh

shows seven monofils S twisted at five TPI under a constant tension of 200g. In figure 1.3.25a, the twist triangle was embedded when monofil

#1 was slack, while #6 was slack in figure 1,3.256. The slack element

can be readily detected in both cases, and it is seen to end up as the

central element of the plied yarn. The different location of the twist

point is easily observed, generally in a position near the center of gravity

of the tensioned yarn, i.e., it moves from the slack yarn. Note that subtwisting tak'cs place between groups of components before formation

of the final structure — in fact, one monofil in each case is seen to join

the plied yarn only at the last moment.

The lasting effect on migration frequency of producers' twist in the case of Riding [40] and of roving twist in the case of Hearle [7] is seen by Hearle to be the result of a second mechanism of twisting quite unlike

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MIGRATION 109

that dictated by the tensile motivation of migration during cylindrical

twisting. Hearle [7] terms the second twist form as ribbon twisting.

The geometry of simple twisting of a ribbon is shown in figure 1.3.26a

and the formation of a wrapped ribbon by bending on a skew is shown

in figure 1.3.266. According to Hearle [7] both theory and experiment

indicate that a flat rubber strip wiU seek the twisted form of figure 1.3.26a

at lower twists, but the wrapped form of figure 1.3.266 at higher twists.

(I) (U)

Figure 1.3.26a. Geometry of simple twisting of ribbon of width b and thickness a to a twist angle a, and period h, after Hearle and Base, 1966.

(i) Flat ribbon. (ii) Twisted ribbon.

Figure 1.3.26b. Formation of a wrapped ribbon by bending on skew, due to the action of

the torque Tw
110 TIRE CORD STRUCTURE AND PROPERTIES

(a) (b) (c) Figure 1.3.27. (a) Cylindrical twisting, (b) Twisted ribbon form, (c) Wrapped ribbon form

(Hearle and Bose, 1966).

(a) (b) Figure 1.3.28. (a) Schematic representation of ribbon-form of yarn, with filament twist- ing round due to producer's twist, (b) The same yarn twisted into a wrapped form, and

hence showing migration.
MECHANICS OF SIMPLE YARN STRUCTURE 111

Tension

of no

Migration

Figure 1.3.29. Variation offrequency of migration with twisting tension resulting from a combination of mechanisms. {Hearle, Gupta, and Goswami, 1965). There is an intermediate region in which both forms exist together. If, however, the twisting ribbon is made of individual separated filaments,

the form of figure 1.3.26a will not be a stable one, and the structure will degenerate to a form similar to that obtained in cylindrical (conventional)

twisting as discussed above. Hearle provides an excellent summary of the structural form which will occur in twisting, as shown in figure 1.3.27,

which reflects the influence of twist form on expected filament migrations of the diff'erent kinds of twisting and migration patterns and pinpoints the relationship between migration period and producers' twist period (corrected for contraction) on the basis of the wrapped ribbon

structure as shown in figure 1.3.28. Finally, Hearle concludes that the

combination of the two mechanisms will show the relationship of migration frequency and twist tension as in figure 1.3.29.

1.3.3. Mechanics of Tensile Behavior of Tire Cords

1.3.3.1. An Introduction to the Mechanics of Simple

Yarn Structure

The prediction and observation of the tensile behavior of singles

twisted yarn has been covered so eff"ectively by Hearle [7 J that any at- tempt at developing a comprehensive treatment of this topic in the current chapter would be highly repetitive. It is our purpose to focus on

the mechanics of tensile behavior of higher order structures, i.e., tire cords and plied yarns. However, to dive into this topic without a summarization of the analysis and experimental observations on singles

yarns would leave the reader at considerable disadvantage, and so some

repetition must be endured.
112 TIRE CORD STRUCTURE AND PROPERTIES

As a start we use the introductory treatment to simple yarn mechanics

recently prepared by Backer [7]. In approaching the subject we have

the choice of following papers published on a chronological basis, or we can consider the literature in the order of logical development of

subject matter. We take the latter course. Consider a series of fibers infinitely long, uniform in cross section,

and iniform in mechanical properties. The fibers are twisted together

so as to form a series of uniform helices, with each fiber lying on a helix

of fixed radius (and staying at that radius). Each fiber has properties constant along its length; each fiber is similar to its neighbor in geometry

and in mechanical properties. We assume that there is no interaction

between the fibers and we consider each fiber element to be so slender

that it can withstand only tensile forces along its axis. This is the background of Piatt's [43] initial development in yarn mechanics. GegaufF

[44] attempted a similar approach early in the century, but his work was not brought to Ught until some years later by Hearle [45]. Piatt's approach was as follows. Assume that when a yarn is stretched,

each fiber in that yarn has a fixed geometric location identifiable by its inclination angle relative to the yarn axis. A given stretch for the yarn causes fiber strains which differ from position to position. The fiber strain at any location equals the product of the yarn extension and the

cosine squared of the helix angle of the fiber. From the stress-strain curve given for the fiber we can determine the tensile force which wil] develop along its axis as a result of the strain imposed. The contribution of this force to the normal stress acting on the cross section of the

yarn at that location is equal to the product of the axial stress in the fiber and the cosine squared of its helix angle. One now considers this stress acting on a small \ increment of area and integrates the resulting incremental force over the cross section of the yarn for all fibers. Once this integration process is undertaken, we have an expression

for the tensile load or tensile resistance which the yarn puts up to axial stretch at a given level of strain. But the prediction of when the yarn will break depends on the criteria for failure. One may decide that the yarn

will break when the fiber in the position of maximum strain reaches its own breaking strain. This fiber lies at the yarn center with a zero helix

angle. If we know the fiber elongation at rupture, we can then assume

that the yarn breakdown will initiate at this same strain and from the

expression developed by Piatt for the tensile resistance of the yarn to strain, we can calculate the breaking strength of the yarn. An analytical

expression for this breaking strength will be a function of the properties

of the fiber and the goemetry of the yarn, exemplified by its twist and

radius. Figure 1.3.30 shows a plot of yarn strength calculated from Piatt's expression and also determined experimentally at various twist levels For this rcalculation Piatt assumed that the twist angle did not change

significantly with yarn extension, nor did the radius of the yarn decrease

during the test. Later, Piatt [46] dealt with the questions of helix angle change and

lateral yarn contraction. He concluded that lateral contraction plays a secondary part in the overall efficiency of strength translation of fibers
MECHANICS OF SIMPLE YARN STRUCTURE 113

900-

850 - X

UJ

f 800

UJf 750 \ 700 z X

UJ 650 X Experimental Values 600/208 Acetate 600

0 2 4 6 8 10 12 14 16 182022 YARN TWIST, TURNS PER INCH Figure 1.3.30. Yarn strength vs. twist. into yarns for helix angles less than thirty degrees. He also concluded

that his first assumption — that of negligible effect of helix angle

changes — was valid for most textile yarns. Up to this point both Gegauff and Piatt had neglected transverse

forces. And in considering spun yarns they treated only the cases where

the twist level was high enough to prevent slippage of the fibers. So the

analysis was essentially that for continuous filament yarns, even though

applied to both filament and spun yarns. Hearle [45] treated the case of both tensile and transverse or lateral forces. His first report was restricted

to small strains and provided for no lateral contraction. Later Hearle

et al. [47] took lateral contraction and transverse forces into account in< arriving at expressions for the tensile behavior of twisted filament yarns. For a full treatment of this subject, see Hearle [7].

It is interesting to note that Galileo [48] showed some interest in the transverse forces between fibers in a yarn as he attempted to explain to a friend how "fibers, each not more than two or three cubits in length, so

tightly bound together in the case of a rope one hundred cubits long"

would require a great force to break them. "Can you not hold a hempen

fiber," he writes, "so tightly between your fingers that I, pulling by the

other end, would break it before drawing it away from you? Certainly

you can. And now when the fibers of hemp are held not only at the ends,

but are grasped by the surrounding medium throughout their entire

length, is it not manifestly more difficult to tear them loose from what

holds them than to break them? But in the case of rope, the very act of twisting causes the threads to bind one another in such a way that when

the rope is stretched with a great force, the fibers break rather than separate from each other. At the point where a rope parts the fibers are, as everyone knows, very short, nothing like a cubit long, as they would be

if the parting of the rope occurred, not by the breaking of the filament,

but by their slipping one over the other."

Galileo also concerned himself with the nonuniformity of filament

strength along its axis. However, by faulty reasoning, he arrived at the
114 TIRE CORD STRUCTURE AND PROPERTIES

wrong conclusion concerning the effect of weak spots in a long rope

and, as he said, "How can you therefore maintain that the long rope is weaker than the short one. Give up then this erroneous view which you

share with very many intelligent people." The treatment of nonuniformity in yarns was taken up by a highly intelligent individual, F. T. Peirce

[49] whose pioneering work on the weak link theory for textile yarns

serves as reference in numerous current researches on the failure of engineering structures.

Piatt [50] found that it was necessary, in many practical applications,

to deal with fibers which varied considerably in their stress-strain properties. His assumption of uniformity of mechanical properties had to be

set aside and the nonuniformity or variation of stress-strain characteristics of the fiber became another parameter to be included in the

analysis. Piatt's reasoning was roughly as follows. If one takes ten nat- ural fibers and compares their individual stress-strain curves, it is evi- dent that their rupture does not occur all at one point of extension. A

frequency distribution of the rupture extension of the individual fibers can be determined and is shown in figre 1.3.31 in normalized form. The

objective of the analysis is to treat the idealized single yarn geometry

without the assumption of uniformity. If now the normal frequency dis- tribution is converted into a cumulative distribution and inverted, that

is, subtracted from the quantity 1, we have in effect a measure of the

relative number of fibers which will not have broken at any given extension. Note that we are referring here to fiber extension. For the most

simple case, that of parallel untwisted fibers, fiber extension and bundle

extension are the same.

Fiber

Axis

LONGITUDINAL VIEW CROSS- SECTIONAL

(a) ^'^^

Figure 1.3.31. (a) Idealized singles yarn geometry. (6) Standardized normal distribu- tion curve of rupture elongation.
MECHANICS OF SIMPLE YARN STRUCTURE 115

Treating this simple case of parallel fibers, Piatt sets the bundle

strength equal to the product of the average fiber resistance to a given

strain (taken from the average stress-strain curve) times the number

of fibers active at that moment. The fiber stress-strain curve can be

considered linear or taken in any form that can be expressed mathematically. The plot of percentage of unbroken fibers in figure 1.3.32a can also be expressed mathematically. And thus an expression is avail- able for the tensile resistance of the bundle at any given strain. The fact that fibers start to rupture part way along in a bundle test and continue to rupture until the entire assemblage is broken is ac- counted for by the ever-reducing percentage of unbroken fibers shown

in figure 1.3.32a. If this bundle resistance is calculated on a point by

point basis, it does not matter whether the stress-strain curve of the

average fiber is Hnear or not. In fact, a more realistic form of the fiber stress-strain curve consists of two Hnear portions first, the so-called

elastic region, and the second with a reduced slope, the so-called plastic

region. The second region is of more importance if we are considering

the rupture of a bundle of parallel or even twisted fibers and this region

(a) -300 -250 -200 -150 -100 -050 000 050 100 150

T-- ELONGATION AS NO OF STANDARD DEVIATIONS FROM MEAN

RUPTURE ELONGATION

100 r

50

V = 0 %

V= 10%

*0ther curve groups in similor

<^ order of decreosing A/B

J I I IJ V = 30 %

0 2 5 10 20 MEAN FIBER ELONGATION . E

,

(b)

Figure 1.3.32. (a) Unbroken fibers vs. strain. (6) Bundle efficiencies, with coefficient of variation V as a parameter, after Piatt.
116 TIRE CORD STRUCTURE AND PROPERTIES

may be characterized in terms of its slope B and its intercept A. Now if one takes the three parameters: (1) average rupture strain for the fiber population, (2) coefficient of variation of the fiber strain (which dictates

the breadth of the frequency distribution pictured in fig. 1.3.316), and

(3) the rheological properties of the fiber (expressed as the ratio A over

B), one can determine the transmission of fiber properties into maximum

bundle resistance to extension.

The function representing tensile resistance of the bundle is simply

differentiated with respect to strain and its derivative is set equal to zero

to establish the maximum resistance. If this maximum, i.e., the breaking

strength, is compared to the breaking strength summation of the indi- vidual fibers, tested separately, one develops a measure of relative bundle efficiency. It is this relative bundle efficiency which is plotted

in figure 1.3.326 by Piatt. Note that the bundle efficiency is a hundred

percent where the coefficient of variation, v, is zero. As the coefficient of variation increases to 10, then 20, then 30 percent, the bundle efficiency

for a given mean fiber elongation and ratio of intercept to slope (A/B)

decreases significantly. The efficiencies are highest for the high A over B ratios, indicating that a stress-strain curve with a high flat plateau

will provide maximum bundle efficiency. It is noted also that as the mean fiber elongation is increased, the bundle efficiency may go up or go down, depending upon the stress-strain properties of the fiber. Physically what these data say is that if the fiber stress-strain curve

is quite flat in the region of rupture, it does not matter whether some of the fibers in the bundle are breaking at the left-hand side of the flat plateau while others are breaking way over to the right-hand side, at much higher elongations, and some are breaking in the middle. The

point is that they are all up on the plateau closely corresponding to their maximum load capacity and therefore at the moment of bundle rupture,

they are very nearly contributing their maximum support to the bundle.

And the bundle as a result will tend to have a high strength efficiency.

Piatt has superimposed the analysis for the varying fiber stress- strain properties on the analysis for the idealized singles yarn geometry

(fig. 1.3.31a) and thus has extended the treatment of twisted yarn mechanics to include the practical case of highly variable fibrous material.

This work represents a significant contribution to the structural theory

of textile materials. Note also that this approach permitted the treat- ment of nonelastic fibers, that is to say, it allowed for a change in the

slope of the stress-strain curve and dealt with the behavior of the fiber

in its region of rupture. To generalize the treatment one need but express

mathematically the resistance of a fiber population to an average strain,

and further express the variation in the rupture strain for the fiber population. Finally, one must convert from average fiber assembly

strain to. the local fiber strain, depending upon its geometric configuration. Then by integrating the force contribution of the unbroken fibers

in each region of the assembly to the axial tension in the assembly, one can determine the total resistance of the structure. Differentiating

this total resistance with respect to strain and equating the derivative

to zero permits calculation of the strength of the structure.
PURE FIBER TENSION AND IDEALIZED GEOMETRY 117

This treatment does not include the effect of lateral compression of individual fibers within the structure, and it does not provide for the

contraction of the structure itself. These two elements were dealt with

in Hearle's later studies (Hearle et al. [47]). Consideration of lateral contraction or the Poisson effect was essential in the development of a

fiber web theory for nonwovens, for it was quite obvious that bonded

fiber webs contracted significantly in the lateral direction as a result of longitudinal stresses. It is interesting to note that the analysis of the

stress-strain behavior of nonwovens (Backer and Petterson [51]) fol- lowed the general pattern established by Piatt in his analysis of twisted

structures, with fiber inclination angles replacing fiber helix angles.

The principal difference between the analytical treatment of the non- woven and that of the twisted structure is the fact that the inclination

angle follows a statistical function in the nonwoven, while in the twisted

structure it is determined simply from the yarn twist and the distance of the fiber from the center of the yarn. The expression for resistance of a nonwoven to extension in a longitudinal direction contains terms relat- ing to the angular distribution of the fibers, to the fiber modulus and to the contraction ratio of the structure. The latter quantity is unknown,

but it may be calculated by equating the stresses developed in the lateral direction during a longitudinal pull to zero. This expression for the

lateral stress similarly consists of the fiber modulus, its angular distribution in the structure, and the contraction ratio. Equating this expression

to zero leads to a solution for the Poisson ratio expressed solely in terms

of the geometry of the structure. The Poisson ratio of a nonwoven is indeed independent of the properties of its fibrous material; it is actually

a structural parameter, but it can be used as a basic property of non- wovens considered as orthotropic sheet materials, as shown by Backer

and Petterson [51] and Petterson and Backer [52]. One can now imagine the wrapping of a web of nonwoven material into a cylindrical roU to form a three-dimensional structure for which calculations of lateral contraction occurring during application of longitudinal axial tensions can be made.

A formal listing of papers on the theory of tensile properties of twisted

continuous filament yarns, furnished in Hearle's [7] table 4.1, provides

the serious student of this subject with the "must" reading of the structural mechanics of tire cord.

1.3.3.2. Analyses of Tensile Behavior of Continuous Filament Singles Yarn

The Case of Pure Fiber Tension and Idealized Geometry

Piatt [43] assumed an ideal geometry for the singles continuous fila- ment yarn as pictured in figure 1.3.31a, with aU component filaments

round in cross section and uniform in properties and geometry along

their length. Each filament maintains a fixed helical path, within the yarn

with constant helix radius, curvature and helix angle. Packing density
118 TIRE CORD STRUCTURE AND PROPERTIES

is also constant along the yarn length. The forces acting on the fiber of

figure 1.3.33a can be characterized as:

a. a tensile force, Pr, along the fiber axis,

b. a shear force, v, acting in the fiber cross section,

c. a bending moment, m,

d. a torsional moment, ^,

Piatt sets V, t, and m = 0 and, neglecting forces acting on the longitudinal surfaces between fibers, treats the components of p,-, the force act- ing axially along the fibers, so as to develop a relationship between F, the

tensile force on the yarn, and the properties of the filaments as well as

their geometric arrangement. From the simple geometry of a helix (with

helix angle q, radius r and twist T)

tanq = 27rrT (1.3.2)

and the force relationship for the ring of Figure 33b

dpr — (Tf27Trdr cos q (1.3.98)

where cr/ is the tensile stress acting along the axis of the fibers of the

ring. Piatt expresses the yarn tension, P, as: P = 27T r afrcos^qdr (1.3.99)

jr=0

r Now, using the relationship between the applied yarn strain \L/L = ey and the axial fiber strain €/ derived from the condition of small strains

from the geometry of figure 1.3.34, i.e., €/= €y cos^ q (1.3.100)
PURE FIBER TENSION AND IDEALIZED GEOMETRY 119

Figure 1.3.34. Deformations in twisted yarns. and the identity:

cos^ q

1

1 + tan^ q

(1.3.101)

one obtains for the fiber whose generahzed stress strain relationship

is of the form

o/= a + bef (1.3.102)

the expression for yarn tensile resistance to an imposed axial yarn

strain of €y

P = 2Tr r (a + 6ey cos^ g)rcos2 g^/r (1.3.103)

jr=0

which Hearle [7] illustrates can be integrated and reduced to the simple

expression when the constant a vanishes

P= {7TR^)Ef€y cos2 Q (1.3.104)

where the packing factor 0 has been introduced to permit a proper substitution of the fiber tensile modulus Ef for b in eq (1.3.102). Actually,

Hearle shows the quantity l/vy (i;y= specific volume of the yarn) in lieu of (f) in eq (1.3.104) and expresses the specific stress and fiber modulus

in units of [g-wt/(g per cm.)] equal to 10~^ • g-wt/tex. We chose to report

eq (1.3.104) with the packing factor c^, the ratio for fiber area A/ to yarn

cross sectional area Ay, or ttRI, and with the fiber modulus Ef as well

as the stresses ay and cry expressed in units of force per unit cross- sectional area. Care must be taken in use of the correct units for whatever equation is used.
120 TIRE CORD STRUCTURE AND PROPERTIES

STRAIN (7o) Figure 1.3.35. Stress-strain characteristics of high tenacity yarns To reach the simple form af eq (1.3.104) the quantity a is taken as zero for simplification, i.e., the stress-strain curve of the fiber is a simple

linear relationship starting at the Q-0 origin. Such a simplification is satisfactory for such tire cord materials as wire, glass fiber, or rayon as shown in figure 1.3.35, but the polyester yarn would no doubt require

the full form of eq (1.3.102) for its post yield region. The form of the nylon curve suggests a more exacting expression than that of eq (1.3.104).

As Hearle [7] points out, the specific yarn stress multiplied by its solid cross-sectional area equals the yarn tensile load, P, i.e., (Ty{TTR^(f))=P (1.3.105)

and, by definition, the yarn tensile modillus Ey

Ey = ~ (1.3.106)

whence, from eqs (1.3.104^106),

Ey = EfCos^ Q

a simple relationship shown in reasonable agreement with experimental

results. Recall that Q is the outside helix angle.

Piatt reasoned that since the fiber strain at the yarn center equals

the axial strain of the yarn, then initial rupture will take place in the

central fiber, and, following on this event, there occurs a rapid progressive rupturing of all remaining filaments and the yarn breaks. While

this chain of events is observed in certain yarns, where catastrophic

failure occurs as a result of almost instantaneous rupture propagation,

it is by no means universal, as is seen, for example, in Piatt's later work [50] on cordage structures. Here variations in the rupture strain levels of the component fibers are shown to slow down the propagation

of the initial fiber rupture in the yarn structure. In other instances,

the propagation of failure is observed to depend on yarn twist levels, on blend ratio (of a two component yarn), on friction and on relative

fiber properties in a blend. More wiU be said about this later.
CASE WITH TRANSVERSE FORCES 121

The Case With Transverse Forces The importance of lateral forces in affecting load transfer from fiber

to fiber of a staple fiber (spun) yarn was recognized as far back as

Gallileo [48] and many researchers in yarn mechanics have incorporated

the effect of transverse forces on fiber and yarn behavior for twisted

structures. For example, Gurney [53] developed a relationship between

yarn strength, twist, lateral contraction ratio and fiber stress strain properties. Sullivan [54] determined the strength of staple fiber yarns

by analyzing the lateral pressure distribution across the yarn and its contribution to axial stress transfer between fibers and to strength

development. Machida [55] analyzed the lateral pressure in blended

yarns under tension as a means of predicting the critical fiber length

during yarn rupture. Grosberg [56] et al. calculated the lateral pressure

distribution in a low twist sliver during extension as a means for pre- dicting sliver strength.

Hearle [45, 47] provided the most complete treatment of the case of yarn strength taking into account transverse forces. His procedure was

to establish first a more complete relationship between filament strain and yarn strain e/=ey(cos- 9 — /Xy sin- Qf) (1.3.108)

where /Xy is the contraction ratio (or Poisson's ratio) of the yarn, i.e., the ratio of transverse, radial contraction eyt resulting from an axial extension, eyn.

,Jiy = -eytl€ya (1.3.109)

It is worth noting that Petterson [57] started with the same expression

in analysis of the load extension properties of a nonwoven web taking

lateral contraction into account. Assuming linearly elastic fiber behavior,

Hearle then established the generalized stress strain relationship for the fiber as:

€,„ = ^-2m„,^ (1.3.110)

where: €fa is the axial fiber strain,

(Tfa is the axial fiber stress, cr/t is the transverse fiber stress assumed by Hearle to be equal in both radial and circumferential directions,

Efa is the fiber modulus in the axial direction, and

Eft is the fiber modulus in the transverse directions,

IJifit is the Poisson's ratio of the fiber, i.e., the ratio of resultant

axial strain to lateral strain. In a later paper, Hearle and El-Behery [58] actually measured the ratio of transverse to applied pressure (where the applied pressure is lateral
122 TIRE CORD STRUCTURE AND PROPERTIES

to the axes of a set of fibers and the transverse pressure is measured in a perpendicular direction, still lateral to the fiber axes). On the basis of these measurements, they suggest that the circumstantial-radial pres- sure ratio be taken as a constant with a value between 0.1 and 0.5. The assumption followed that the fiber had an axis of symmetry coinciding with its geometric axis, therefore:

llatlEft = fltJEfa (1.3.111)

which permits rewriting of eq (1.3.110) as

17 r (1.3.112)

tfa tfa

where the classical sign convention applies for stresses cr/o and (jft. Combining eq (1.3.108) and eq (1.3.110) leads to: (Tfa^ Efa€y{cOS'^ q — /Jiy siu^ q) + 2lXta(Tft (1.3.113)

Hearle normalized o/a and dft by dividing each by £/aey, the stress which

would be present in a fiber subjected to axial strain of ey. Thus he took

X=(rfal{Efaey) (1.3.114)

g=aftl{Efaey) (1.3.115)

then establishing the elements of force acting on the yarn element pictured in figure 1.3.36 (where X^^fa and g=aft), he established the dif- ferential equation governing radial equilibrium in the yarn. Its solution

combined with eqs (1.3.113-115) for the case of /x<a = 0.5 led to the

following expressions for stresses in a yarn

Figure 1.3.36. Exaggerated view of element.
CASE WITH TRANSVERSE FORCES 123 X=^U cos2 9[l+(//L)2]- [i + in (//!)] (I.3.II6)

^==^/4Cos^ 9[l-(//L)2]+ V2 In (//L) (1.3.117)

plotted in figure 1.3.37, where /, L are the filament lengths at radius r and R respectively. Hearle [45] then develops an expression for the mean

normalized axial stress on the yarn resulting from an imposed small

elastic axial yarn strain. This normalized mean axial stress is a useful

quantity, for when it is multiplied by the axial stress which would occur

in a parallel set of similar fibers and by the area (or count, depending on

the units in which the modulus is expressed) it leads to a prediction of yarn tension for a given axial yarn strain. This same quantity equals the

ratio of yarn axial tensile modulus to fiber tensile modulus. Hearle [7]

provides a most useful table (his table 4.2) showing calculated yarn

stress parameters for twisted systems with outside helix angles varying

from 0 to 50° and with yuta varying from —0.25 to + 0.50. The parameters

calculated include the normalized axial fiber stress at the yarn center

as well as the normalized transverse stress at the center, and also the mean normalized tension for iJiy=0 and 0.5 respectively. The simple

parameter, cos- Q, is seen to predict reasonably well the values which

are obtained by the more sophisticated analyses at outside helix angles

below 30°. Above 30° the predictions of cos- Q are increasingly in dis- agreement with the more rigorous analyses, with differences as high as 30 percent at 40° and over 50 percent at 50°. Finally, it is worth noting the simplified expressions which have been

derived for lateral pressures by others corresponding to eq (1.3.115,7).

Machida [55], for example, used the typical free body diagram such as pictured in figure 1.3.36 to obtain the equation for force equilibrium in

Figure 1.3.37. Variation of stress with radial position in yarn.
124 TIRE CORD STRUCTURE AND PROPERTIES

the radial direction. His basic relationship, obtained after neglecting

higher order terms, was

8(Tt _ — Efa€y cos^ q sin^ q

dr r (1.3.118)

which, when integrated, led to the following approximate expressions:

(Ttr — (Ttc (1.3.119)

CTtc Efn€y sin^ Q

or

(Tt, Efa€y sin- Q

[-(i)"]

(1.3.120)

(1.3.121)

where crtr is the lateral (radial) stress at radius r and crtc is the lateral pressure at the center of the yarn. Considering eq (1.3.104) and taking

(/), the packing, as unity, we have:

atc = 27rPT'

(Ttr = 27rPT'•Hi)']

(1.3.122)

(1.3.123)

where P is the tensile load acting along the yarn axis and T is the yarn

twist. Now the average pressure over the yarn is obtained from integration of:

ft- Jo leading to:

(1.3.124)

(1.3.124a)

Grosberg [56] derived eq (1.3.124a) along the way of predicting resistance

to fiber slippage in a low twist sliver. Sullivan derived an expression

similar to eq (1.3.122) but in his analysis the effective lateral pressure

did not fall to zero at points close to the yarn surface.

It is interesting to note that multiplication of eq (1.3.117) by Efa€y

provides an expression for crt,- which shares many of the characteristics of eq (1.3.121), as would be shown if one were to plot the parabolic drop in pressure from center to outside of the yarn. At the yarn center, eq

(1.3.117) differs from eq (1.3.120) by the constant ^/4 versus V2. Yet it is useful to have the rule of thumb which states that for a given yarn
CASE WITH LARGE EXTENSIONS 125

tensile strain Cy and a given fiber modulus £"/, the average tension along

the yarn axis will vary as the cos- Q and the average (or central) lateral pressure will vary as sin- Q.

The Case with Large Extensions

Considering the fact that many yarns in commercial usage, particularly of synthetic fibers, have rupture strains between 15 and 30 percent, it would seem desirable that large extension analysis be introduced in studies of yarn breaking strength. On the other hand, one might question

the importance of breaking strength as a quality parameter for textile materials. Certainly, in the case of conventional yarn usage, bending

and mild tensile action often predominate, at strain levels of a few

percent. Nonetheless, in matters relating to processing efficiency

textile yarns are still judged on tensile strength qualities, among others,

and the importance of the strength and rupture energy of tire cord is universally acknowledged.

Hearle [7] has derived more rigorous expressions for local fiber strains as a function of yarn tensile strain and yarn geometry. And

he reports = ey(cos^ — /Xy sin- — ie|(l + />ty) ^ sin^ q\ cos^ (1.3.125)

where q\ is the local helix angle in the strained yarn, and /Xy is the

contraction ratio of the yarn. Note that eqs (1.3.108) and (1.3.125) have

the same first term. And if /Xy = 0, this first term equals eq (1.3.100), for the case of no lateral contraction.

For constant volume deformation Hearle [7] derives: = (cos^ 9i — i sin^ ) — 3e| sin ~ qi (cos- — i sin^ q^ ) (1.3. 126)

and in his calculations for eqs (1.3.100), (1.3.108), (1.3.125), and (1.3.126)

he shows that suitable accuracy is obtained in using eqs (1.3.100), and

(1.3.108) up to 10 percent yarn extensions. But with yarn extensions of 30 percent it is reasonable to use eqs (1.3.125) and (1.3.126) above.

Finally, he notes that the difference between use of /Xy=0 and the use,

at high strains, of the more exact constant volume expression, is small

except at strains of 30 percent and helix angles above 25°. For the case of large extension, and lateral contraction, Hearle suggests substitution of eqs (1.3.125), and (1.3.126) in the equations for yarn

force prediction, i.e., (1.3.99) through (1.3.104). And he normalizes the

non-Hookean stress-strain curve: ^= a' + b'ef (1.3.127)

where 077, is the rupture stress of the fiber. Then eq (1.3.127) is combined

with eqs (1.3.125), (1.3.126), and (1.3.99) through (1.3.104) to obtain

detailed expressions for yarn specific stress, normalized to fiber tenacity
126 TIRE CORD STRUCTURE AND PROPERTIES

YARN EXTENSION/FIBER BREAKING EXTENSION Figure 1.3.38. Theoretical stress-strain curves of yarns assuming Hooke's Law, and

regarding broken regions as merely inoperative, after Hearle [7]. as a function of the stress-strain parameters a and b and the geometry

of the yarn. Note, the yarn stress expressions furnished by Hearle for large extensions with lateral yarn contraction and non-Hookean be- havior are equations for the stress-strain curves of the yarn up to the

point of first fiber rupture, which Hearle states will occur at the center- most fiber at yarn strain, ey = efb, the rupture strain of the fiber. Beyond

this point, Hearle reasons, the yarn will rupture catastrophically or, if the broken regions are merely considered inoperative, the stress strain curve will sink linearly as shown ixi figure 1.3.38. Note that the above

stress analysis has not been extended by Hearle to provide for the effect of transverse forces, although the latter combined problem is treated

by energy methods.

There is, however, an interesting sidelight to the treatment of trans- verse pressure under conditions of large strains. It has been considered

by Machida, and shows the introduction of eq (1.3.102) into the differential equation for lateral pressure, as well as for axial yarn tension. Solving

for the center pressure, he obtains

(Ttc = ^\n (l + tan2^)+|ey(sin2^) (1.3.128)

P = ^M (l + tan2(3)+^Jsin2^) (1.3.129)

hence

(rc = 27rT'P (1.3.122)

the same expression shown above to apply in the case of Hookean

materials.. But in treating the high end of the stress strain curve, one must take into account the effect of ey on T, the twist at a given yarn

strain. If To is the original yarn twist.

(1.3.130)
YARN ANALYSIS USING AN ENERGY METHOD 127 whereupon for a yarn load elongation curve of P = A+Bey (1.3.131)

substitution into eq (1.3.122), differentiating P with respect to ey and

equating to 0 gives the Cy at which dtc is maximum, which turns out to be

Thus, for yarns with varying load elongation curves as in figure 1.3.39a,

the central pressure versus yarn elongation curves will vary as in figure

1.3.396, with maximum pressure occurring at ey = 0 for case (4) and

way off the graph to the right for A=0. It is interesting to note that

(Ttc has the same values at 25 percent strain since from figure 1.3.39a we note that aU four load elongation curves converge at 25 percent strain. Yarn Analysis using an Energy Method

Treloar and Riding [59] turned to one of the energy methods for the

prediction of the load-extension relationship of a twisted yarn. There are three theorems related to elastic energy, as Den Hartog states [60].

theorems of "virtual work", of "Castigliano" and of "least work".

Den Hartog points out that the theorem of virtual work is applicable to non-Hookean materials so long as no energy is dissipated in heat. Castigliano's theorem does not hold for nonlinear materials, nor does

the theorem of least work. The theorem of virtual work states that

"if an elastic body in equilibrium is given a small displacement or deformation, then the work done by all external forces acting on the

body equals the increase in elastic energy U stored in the body." (For a recent review of energy methods, see Oravas and McLean [67].) Stated

simply.

(1.3.132)

dU = Pnd8,

or

n (1.3.133)

0.1 0.2 ELONGATION,

(a)

0 0.1 0.2 ELONGATION , 6y

(b) Figure 1.3.39. (a) Assumed load-elongation curves for different yarns, (b) Central pressure

vs. yarn elongation curves.
128 TIRE CORD STRUCTURE AND PROPERTIES

where the elastic strain energy U is expressed entirely in terms of

displacements 8 of the body.

Treloar ajid Riding [59], using the method of virtual work, treat a

filament material whose stress-strain properties are represented by the

general function

(Tf=i\f(\) (1.3.134)

where ay is the force per unit original area along the fiber axis and X

is the extension ratio Ijlo = 1 + e/. The work performed in stretching a

unit of filament to X/r is Wf=\'/'H^)dK (1.3.135)

which equals the elastic strain energy U stored in the unit volume.

Now the increment of strain energy in the increment of yarn volume

27Trodro is dU = 2TT(t)rodro \ \\f{\)d\ (1.3.136)

where A/r is the filament extension ratio at the radius r, <f) converts

the increment of yarn volume to an increment of fiber volume and the

radius, To, is measured in the unstrained condition. The total energy in a

unit length (when unstrained) of yarn is then

C/=277(/) /"[/ i\){K)d\ TodVo (1.3.137)

Differentiating eq (1.3.137) with respect to the yarn axial extension ratio \y as in eq (1.3.133) should then provide the axial yarn force at any

extension ratio, and as Treloar shows after some manipulation,

P = 2tt^ f''>(M ^rodro (1.3.138) Jo "'^y where ^{)^fr) is in units of stress based on the unstrained cross sectional

area, i.e., nominal or engineering stress rather than true stress based on

strained cross-sectional area. Comparing eq (1.3.138) and Piatt's eq (1.3.103), one is tempted to designate the quantity {a + bey cos^ q) as equivalent to (//(V) the

quantity cos^ q as equivalent to 8\/r/6Xy, which they are if tJiy = 0. But,

for the case' which takes lateral pressure into account and yarn volume is constant during straining (also 0, constant), Treloar shows

r' = rll\y

(1.3.139)

(1.3.140)
ROLE OF FIBER UNIFORMITY 129 where R vs. Ro and r vs. ro are the radial positions after and before yarn

straining. In a yarn of unstrained unit length the length of a filament at helix angle qro is sec qro and the length of the corresponding filament in the strained state of the yarn (yarn extension ratio being \y) is Ay sec q,-. From eq (1.3.2) and eq (1.3.140), Treloar shows the filament extension

ratio to be

sec qro (l + tan"^^o)

Now, by definition,

\=l + e (1.3.142) so Treloar substituted eq (1.3.142) in eq (1.3.141), expanded and re- tained only first order terms in e to give

V 1 -f tan^ qo J where the quantity in the parentheses is seen to be the dk/r/dky of eq

(1.3.138).

Now, for the case of small strains,

iJj{kf)=Ef{kj-l) =Ej€f (1.3.144)

whereupon substituting eqs (1.3.144) and (1.3.143) into eq (1.3.138)

Treloar reports the nominal average stress cry of the yarn (in g/denier

of the unstressed yarn corresponding to the case where i// and E are ex- pressed in similar units).

d-y = Ef€y [f cos2 Q-3 cot' Q log sec ^ + i] (1.3. 145)

which is shown to be equivalent to Hearle, El Behery, and Thakur's

[47] results as corrected [61] for the stress analysis at small strains, in which lateral pressure is taken into account. And Treloar shows that reductions of modulus at yarn twists "which are not too large" are about

50 percent greater on the above theory than in the case of Piatt's theory.

Analyzing the Role ofFiber Uniformity

All of the tensile behavior analysis discussed in detail above was

predicated on the constancy of fiber properties, both along each fiber and between fibers. However, some reference was made earlier to

Piatt's [50] treatment of the case in which fiber properties varied from

fiber to fiber, and this section is intended to summarize the results of

his analysis. But first we note that variable fiber behavior in tension may

follow from variable fiber geometry, or from actual variation in specific

local properties such as fiber modulus, or breaking stress, or breaking

strain. And the source of variation may lie in differences between fibers.
130 TIRE CORD STRUCTURE AND PROPERTIES

each of which is uniform in geometry and properties along its length.

Or, it may derive from variation along the length of each of the fibers. Chaikin [62] was concerned with the effect of nonuniform cross-sectional area (along a fiber length) on the stress-strain-time behavior of the fiber. This was the case of a series loading of a nonuniform specimen, whereas

Piatt's study was occupied with the case of parallel loading of fibers with properties varying from fiber to fiber. Chaikin [62] showed that the strain over the entire length Iq subjected

to load Pis

l-lo I f'o / P \

where z is the position along the fiber axis, A{z) is the local cross- sectional area, and / is a known function.

Now,

dz= lo hence from eq (1.3.146)

(1.3.147)

where /is the mean value of over the length of the unstrained

specimen. Chaikin goes on to indicate that for the strain-stress distribu- tion with the form

Strain = B (stress)" (L3.148)

where B is constant and ^ > 0, the strain for the uniform specimen,

€«, and the strain for the nonuniform cross section, e«, become:

BP^

A{z)_

(1.3.149)

(1.3.150)

and for the cases of a truncated cone whose end radii are in the ratio of

2 to 1 and a cylinder whose cross-sectional area is equivalent to that of

the cone, the ratio of strains at a given axial load, P, is:

enu = BP"

A(z)

^-2.5forn = 3; ^ - 1.17 for ;i= 1.
ROLE OF FIBER UNIFORMITY 131

For the case where the cross-sectional area of the specimen is normally

distributed with a mean A and standard deviation, cta, and where the

coefficient of variation cta/A is 20 percent —=1.33 for 71 = 3; —=1.04forn=l.

Chaikin goes on to treat the effect of specimen nonuniformity on stress- strain properties of certain fibers, on the creep behavior under constant

load, and on stress relaxation behavior at constant extension.

The case where variation in tensile behavior is noted from fiber to fiber was treated by Piatt [24, 50]. His first case was that of variable property

fibers subjected to tensile strain in parallel, without interaction between

filaments. The bundle load, Pb, in such a case is simply the sum of the

loads on individual fibers Pi , P2 , . . . , Ph , i- e.

,

P,=Pl+P2+P3, . . .,P» (1.3.151) or Pb = NfP (1.3.152)

where Nf is the number of fibers and P the average fiber load. Now, if aU the fibers in the bundle had been tested independently for their individual load-elongation curves, it would be possible to calculate an

average load P for any given strain, e/, and by plotting P against €/ an average stress-strain curve could be obtained. If P is calculated on the

basis of the number of unbroken fibers at a given €/, it is possible to extend the average load extension curve over the entire range of strains. Likewise, if {Nfnu) = Nfu, the number of unbroken fibers, is substituted

in eq (1.3.152) and Pis the average load on the unbroken fibers taken from

the average load elongation curve, then eq (1.3.152) is valid in the

region where fiber rupture is taking place.

The same data used to obtain the average load elongation curve for the fibers can be used to calculate average breaking load, average

breaking elongation, and standard deviations of these parameters. The

variation in breaking load can be attributed to variation in material

properties or in fiber geometry (such as cross section). Since the case

treated was for independent parallel filaments, Piatt was concerned

primarily with the variation encountered in fiber rupture strain. As in- dicated in section 1.3.3.1 above, he showed a normalized frequency

distribution for fiber rupture elongations in figure 1.3.31, then drew a general curve (fig. 1.3.32a) of the percent unbroken fibers versus €/ (from the cumulative distribution of broken fibers versus €/). The following power series was used as an approximation to the curve of figure 1.3.32a: nu ^ 100 (0.496 - 0.442^ - 0.0979^^) (1.3.153)
132 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.40. Average load strain curve, after Piatt [50]. where t is the number of standard deviations of strain from the mean

rupture strain (as plotted in figure 1.3.31). By definition, then,

l^. = ((Tj€m) 100, (1.3.154)

(1.3.155)

where is the elongation of the bundle, €m is the mean elongation to rupture in the fiber population and (Xt is the standard deviation of the

rupture strain of the fiber population. Since (cre/em)100 is the coefficient of variation, v^, of rupture strain of the fibers, it follows that eq (1.3.153)

100

90

>-* U

z 80

UJo

ll. u.UJ 70-

LU

_)Q

z

60 - IDCD

cr 50- 4oL

10 20 30 40 50 60

beMV m= = bcr 100 Figure 1.3.41. Bundle efficiency vs. parameter m = he^Yj 100, after Piatt [50\.
ROLE OF FIBER UNIFORMITY 133 can be expressed as a function of e^,, €,« and v^. Now rewriting eq (1.3.152) as Pb = NfnuP (1.3.156)

and expressing Uu as a function of the above variables via eq (1.3.153)

the load P^, is differentiated with respect to e&, the bundle strain, and the

derivative set equal to zero. This permitted determination of the bundle

strain €& at which the bundle load was maximum, and this value of

€6 was substituted into eq (1.3.156) to obtain the maximum value of

Pft. The average load-elongation curve of figure 1.3.40 was expressed as P=b{j^-^e,^ (1.3.157)

which can be rewritten in terms of v^, e,,,, e^, and alb, the ratio of inter- cept to slope. The results of the calculated maximum bundle loads ex- pressed as a percent of the product of number of fibers and the average

breaking load is plotted in figure 1.3.326 and reflects the efficiency of

translation of average fiber strength into bundle strength. Piatt [50] later shows that parallel bundle efficiency can be expressed in terms

of a single parameter m, where

as plotted in figure 1.3.41. He points out that while this simple rela- tionship applies only to the case of untwisted structures, nonetheless

except for extreme cases of high twist and large changes in em, a lower

value of m indicates a better translational efficiency in any structure.

Finally, a study is made of the effect of the shape of the elongation to rupture frequency curve compared to the results obtained with the

Gaussian distribution of figure 1.3.31. It was found that considerable

departure from normality of the distribution could be suffered with only

small deviations in the number of unbroken fibers with increase in

strain. Basically, the above analyses had not been intended for treatment

of twisted structures. Meanwhile, in the relationships expressed in eqs (1.3.103) and (1.3.104), Piatt had assumed that "all fibers in the

yarn are bearing load hence (they) are valid only for yarn elongations

less than those necessary to cause the lowest elongation fiber to rupture.

For these low elongations yarn characteristics such as modulus, ex- tension, and energy absorption are independent of variability." To

combine the statistical and geometric (twist) approaches, Piatt modified

eq (1.3.155) so that the bundle strain, read eycos- q, the strain of the

fiber lying in the helix at radius r. Then, introducing this modification

in eq (1.3.153) to give n'u and using the linear fiber stress-strain relation- ship, (a + 6e/), he rewrote eq (1.3.103) as P = 2 j \b (^^ ey cos^ q^ cos' q-^ rdr (1.3.159)
134 TIRE CORD STRUCTURE AND PROPERTIES

where n^^ is a function of Cy, cos^ q, e^, and v^. From eq (1.3.159) Piatt

determined specific maximum values of P for different values {ajb),

fe, and Q{=0), and expressed them as a ratio of the average assembly

strength when tested one component at a time. This ratio, as plotted in figure 1.3.42, is the strength efficiency of the yarn — combining the effects of yarn structure and component variability. It is worth noting that the

effect of the stress-strain relationship, (a/6), is a dominant one when the mean rupture elongation is small (e^ = 2 percent) as shown in figure 1.3.42a. This is true even when the fibers are perfectly uniform, i.e., Ve = 0. But the alb effect becomes less important at Cw^lO and 20

percent as shown in figures 1.3.426, c. The variability effect, however,

is shown as a major contributor to loss of yarn strength efficiency even

at helix angles below 20° For example, for c.v.'s of Ve = lO percent,

the fibers with e^'s of 10 percent and 20 percent lose about 20 percent

of their strength efficiency in low twist yarns, and for v^ = 20 percent

the loss exceeds 30 percent as seen in figures 1.3.426, c, and d. The

implication of these graphs is worth studying in any application of

textile fibers, natural or synthetic, into mechanical systems.

V = 0 7o • A/B'-0.8

=10% o =-0.4

= 20% © = 0.0

= 30% o = 0.4

0^ i5° 30° 45°

e, SURFACE HELIX ANGLE

Figure 1.3.42a. Yarn efficiency vs. helix angle (Em = 2).
ROLE OF FIBER UNIFORMITY 135

V = 0 % • A/B = - 0.8

= 10% « =-0.4

= 20% o = 0.0

30 I

,

, , 0=" 15° 30° 45° e, SURFACE HELIX ANGLE

Figure 1.3.42b. Yarn efficiency vs. helix angle (Em = 10). The variability of mechanical properties along the length of a single

fiber was the subject of Peirce's early studies [49] and the effect of the

variability of a fiber's geometry along its length on the average mechanical properties of that fiber was treated by Chaikin [62] as discussed

above. In contrast, Daniels [64], Rosen [65] and Scop and Argon [66] have considered the statistics of failure of fiber bundles. The latter two, working in the field of composite materials and laminates, have

assumed a failure model based on a chain-link scheme. Scop [66] pic- tures the parallel bundle failure model as shown in figure 1.3.43, with

N vertical sheets (or elements) and n horizontal links (or strips). If 6 is the vertical length of each link, then for a model L in length, n = Lib.

Scop assumes that if a single vertical element (sheet or filament)

breaks, its failure is limited to a very small region in the neighborhood

of the break. Beyond this region, the vertical element is still useful for carrying its fraction of the load, as the glue sets up shear tractions along

both sides of the sheet. The stress in the sheet thus increases from zero

at the break and at some distance d on either side of the break it reaches

its maximum value, equal to the average applied stress. Thus, the
136 TIRE CORD STRUCTURE AND PROPERTIES

^"^0° 15° 30° 45°

d, SURFACE HELIX ANGLE

Figure 1.3.42c. Yarn efficiency vs. helix angle (Em = 20). presence of the glue limits a single vertical element break to a region

8 = 2d. And 8 is taken as the length of the link in the chain-link model.

Clearly, rupture of the entire assembly does not occur when one vertical element or sheet breaks. It does not even occur when each of the ele- ments has broken somewhere along its length, but only when each of the vertical components has broken at the same horizontal strip. (Thus

the analysis does not allow for inclined propagation of a rupture, such

as is often observed in yarns or in fabric structures.)

Scop defines ^(cr) as the flaw stress distribution, equal to the number

of cracks (or weak spots) per unit area (in a sheet of given thickness)

with a strength ^ a. In practice ^(cr) is determined by experiment. (For

a yarn or a fiber, repeated tensile tests on a given length would do it, breaking the original, then the two broken segments, then four seg- ments, etc. The curve f(cr) vs. cr is obtained by integrating the strength

frequency distribution curve for the yarn. This method assumes that a given rupture level is not affected by previous loadings. If it is, the

strength distribution may be obtained by successive testing of many

very small specimens.)

Knowing f (cr) one may then determine the cumulative strength dis- tribution Qs{(t) for individual sheets. This function has the form

(1.3.160a)
ROLE OF FIBER UNIFORMITY 137 Em=30

ol \

^ 1 ^0° 15° 30° 45°

^.SURFACE HELIX ANGLE

Figure 1.3.42cl. Yarn efficiency vs. helix angle (En, = 30).

Figure 1.3.43. Failure model. where W is the width of a sheet and L is its length. If a shorter length

such as 8 be substituted for L in the above equation, then ft(cr) is the

probability that a single sheet (or fiber) one link, 8, in length will fail when the local stress ^ cr. Now, consider a horizontal strip or link, 6 in length, as shown in figure 1.3.43 which consists of parallel segments.
138 TIRE CORD STRUCTURE AND PROPERTIES

8 in length. It remains to calculate the cumulative strength distribution

F(cr) for all the A'^ elements in parallel in the link, 6. This is not a simple

calculation, for one "must take into account all the possible ways that the sheets can fail within the strip and compute the probability of

failure in each of the failure modes. In statistical terms, each of the

failure modes represents an event which is mutually exclusive so that F(cr) equals the sum of the probabilities of failure in each of the failure modes.." Scop shows that for A'^ sheets in a strip 8 long there are a total of 2^~^ possible failure modes. For the case of A^=3 a simple example

serves to illustrate the implications of the theory. At any given externally

applied stress the local stress in each parallel element is ctq. If one of the three elements breaks, the stress, cti, in the remaining two becomes

3/2(To= CTi and if two of the three elements are broken, the stress, (72, in the third becomes S(To=(T2- The modes are then: (1) all three elements

breaking at a stress ^ (To, which can happen in only one way; (2) two

sheets break at a stress ^ (To and the third sheet breaks between cro and

(72, which event can happen three ways, i.e., with one each of the sheets

possibly outlasting the cro level; (3) one of the sheets fails at a stress ^ (To and the remaining two fail at a stress between cro and (7i, which

event can happen three different ways; and (4) one of the sheets breaking

below (7o, one between cri and 0*2, which event can happen A^! or six ways.

The probability F{(7) that this single (bundle) link will fail at a stress ^ cro is a function of Q{crk), for the single (sheet) links, where

Equation (1.3.1606) is obtained from eq (1.3.1606) by substituting 8 for L and using the relationship between the local stress, the sheet thickness

and the adhesive stress between sheets, i.e.

Finally, the stress redistribution factor is used to redefine cro after the

A:th break among the N sheets

(1.3.160b)

8t= crt (1.3.160c)

A^

(Tk = (1.3.160d) N-k

and the value of F(cr) for the given example becomes:

F(o-) = (33(o-„) +3(?^(o-o) [(3(0-2) -(?(o-„)]

+ 3<?(o-„)[«3(o-,)-(3(o-o)]2 (1.3.160e)

+ 6(3(o-„) [(?(o-,) -(3(o-o)] [(?(o-2) -(?(o-,)]

reflecting the summation of probabilities of all possible modes of failure of the three sheets across the strip or (bundle) link 8 in length.

Looking along the entire laminate, or bundle, one determines the
SINGLES YARN TENSILE BEHAVIOR 139

cumulative strength distribution function G(cr)

G(cr) = l- [l-F(o-)]« (1.3.160f)

where n is the number of links in the chain. As Scop points out, g{cr) = dG{(T)!dcr is the strength distribution function of the laminate,

i.e.. g{(j)d(j is the probability that the laminate fails when the external

stress lies between cr and a+dcr.

It is emphasized that the above treatment assumes that when a single

element is broken, its prebreak stress is distributed uniformly over the

remaining parallel elements. The likelihood that this occurs in filament

bundles is questionable and it may be expected that fibers in the neighborhood of the first break will suffer greater overloading than those far removed from the scene of the first rupture.

Scop and Argon [66] have carried the study further to allow for deter- mination of a stress enhancement factor, Kr, (where the subscript r stands for the number of adjacent fractures to be bridged by the two

bonding elements). And this factor was used to modify the stress levels

in statistical determinations such as those described above. Modified

stress levels for elements in the vicinity of the broken elements were

calculated for two conditions; (1) the plastic matrix, and (2) the elastic matrix.

1.3.3.3. Experimental Observations of Singles Yarn Tensile

Behavior

The validity of the geometric factor cos^ ^ as a basis for predicting

translation of fiber properties into yarn properties was checked at an

early stage. Tensile strength as predicted by eq (1.3.104) with the rupture strain of the fiber substituted for ey was shown by Piatt to be in close agreement with experimental values as reported in figure 1.3.30. Likewise, the eq (1.3.107) prediction of the relative yarn tensile modulus,

EyIEfq, was checked experimentally by Hearle [7] and found to give

reasonable results over a range of twists as shown in figure 1.3.44,

although it was clear that a more extensive analysis based on eqs (1.3.116)

and (1.3.117) provided an improved correlation between theory and ex- periment. This latter analysis took into account the effect of lateral yarn

pressure and Poisson contraction of both fiber and yarn. And for the cases

plotted in figure 1.3.44, both fibers and yarns were assumed to deform

with constant volume (^tM = )Ltfa= 0) which permitted simphfication by

Hearle of the expression for relative yarn modulus to f^^FiQ. M„)=| + 5

^' +

7I37?y (1.3.161)

where c is cos Q. Equation (1.3.161) plotted as the lower curve in figure

1.3.44 appears to be closer to the center of the data spread while eq

(1.3.107) lies more or less along the upper bounds of the experimental

results.
140 TIRE CORD STRUCTURE AND PROPERTIES

1.2

1.0

CO3

§ 08 o

:e<

Gj 06

0.4

0.2

o Viscose 75 75

D 100 24

100 40

• 300 100 ^ Tenasco 400 -180 A M 1650 -750 o Acetate 100 -28

$ 100 -48 300 -78

Nylon 100 -34 840 -136

+ Terylene 100 -48

F(a,cr|,cry)

a «

\

N

0° 10° 20° 30° 40° TWIST ANGLE , degrees 50=

Figure 1.3.44. Comparison of experimental values of modulus with theoretical relations.

Hearle's [7] treatment of large extensions has been referred to briefly

and his more rigorous expressions for strain have been reported in eqs

(1.3.125) and (1.3.126). The approximation to the non-Hookean region of

the fiber stress strain curves was given in eq (1.3.127) and these were

developed to obtain expressions for normalized specific yarn stress as a function of fiber stress strain parameters a and b and of yarn geometry.

Hearle's laboratory studies, however, led him to the conclusion that there is not very good quantitative experimental confirmation of his theories developed for large extensions.

It turns out that tensile tests over the full range of yarn extension can

be best predicted by the energy method of Treloar as, for example, via

eq (1.3.138). In testing a number of yarns, Treloar and Riding [59] and,

later, Riding and Wilson [68] showed close agreement between eq

(1.3.138) and experiment for high tenacity rayon (see fig. 1.3.45a), nylon

(fig. 1.3.456), low tenacity polyester (fig. 1.3.45c), high tenacity polyester

(fig. 1.3.45c?), triacetate and Teflon. (Their method required the knowledge of the geometric parameters of twist and yarn radius measured in the unstrained state of the yarn. Since Ro was somewhat difficult to measure directly, it was calculated from yarn retraction during twisting, as previously done by Riding [23]). "It is found, however, that for Forti- san and high tenacity Terylene there is serious disagreement between

experimental and calculated curves, for which there is no obvious ex- planation," according to Riding et aL [68], but a quick look at one of these "serious disagreements" for high tenacity polyester (in fig. 1.3.45c?)

shows that the discrepancies occur at tan Q (or a) values of .66 and .78.
SINGLES YARN TENSILE BEHAVIOR 141

PERCENTAGE EXTENSION Figure 1.3.45a. Experimental and calculated stress-strain curves for yarns with various

twists (represented by tan a,n).
142 TIRE CORD STRUCTURE AND PROPERTIES

At tan Q = .39 and less, the agreement is quite good, i.e., at outside

helix angles of 21° and less. At this point it is worth referring to a few studies reported in the

literature on experimental determination of fiber properties such as those required for the prediction of yarn tensile (and bending) behavior.

While studies of fiber tensile properties, such as tensile modulus, breaking tenacity, breaking elongation and, indeed, the entire fiber stress

strain curve, are too numerous to mention, published data on compres

sive moduli, and Poisson's ratio of textile fibers are Hmited. Backer

[69] and Miles [70] measured the initial modulus of compression of nylon monofilaments. Some work has been reported on the tensile vs. compressive moduli of keratin, but most of the fiber studies as such

have compared the tensile modulus with the bending modulus (in effect, a composite value relating to tensile and compressive moduli).

A most useful report has been given by Elder [71] on the tensile, compressive, and also bending moduli of nylon, polyethylene, polypropylene, saran and polyester monofilaments. In general. Elder found that the

initial modulus of these many monofilaments in tension, compression,

and bending showed little variation, i.e., they were fairly constant in

all deformation modes. The monofilaments which he tested were shown

to be reasonably well oriented.
SINGLES YARN TENSILE BEHAVIOR 143

There are also a few papers reporting tests of the Poisson's ratio of

textile fibers. Higuchi and Takai [72] tested nylon 6 fishing fine and found

that the ratio was .385 to .386 for longitudinal strains up to 11 to 13

percent. Beyond that point, the Poisson's ratio goes as high as 0.446 at the rupture point. Frank and Ruoff [73] obtained the average of .392

as a Poisson's log ratio for nylon at a log strain of 6 percent. Their re- sults may be considered in agreement with the above.

The prediction of twisted yarn properties relating to rupture is most

difficult for a number of reasons. As has been pointed out above in the

discussion on migration, the conditions of twisting significantly affect the degree of migration, the level of local strains during twisting, the

extent of local filament recovery and/or buckling and the retraction of the yarn itself. Thus, one encounters a varying stress history from fiber

to fiber within a given yarn structure, as well as a significant difference

in local geometry for yarns which are nominally the same, i.e., with the same fiber composition, the same twist, and the same yarn count. The simplest prediction of breaking strength is expressed in eq

(1.3.104), based on the assumption of ideal yarn geometry (no migration) and all fibers at zero stress when ey = 0. It is further assumed that the yarn wiU rupture completely when Cy = e/r, where e/r is the fiber rupture strain. AU of these assumptions are frequently observed to be

violated, yet Piatt shows excellent agreement between eq (1.3.104) and

experiment in figure 1.3.30. And Hearle shows good agreement in figure
144 TIRE CORD STRUCTURE AND PROPERTIES

1.3.46, between measured relative tenacities and cos^^ as Q (or a)

varies. But this time, a significant improvement in prediction of breaking strength is not observed as he uses eqs (1.3.116), (1.3.117), and

(1.3.161), taking into account lateral yarn pressures and both fiber and

yarn lateral contraction at constant volume. So, it appears that the

simple model of yarn tensile behavior is very close to reality in a number

of cases insofar as tensile strength of yarns is concerned. Yet it is worthwhile looking at other behavioral developriients which follow from the

simple model to see if these modes of behavior are frequently observed

in practice.

First, the simple model and its fiber strain/yarn strain relationship

of eq (1.3.100) suggest that when the yarn is extended, first fiber failure

will take place at the yarn center, then rupture will take place in the

fibers of the first ring about the center, the second ring, and so on. It

is difficult to demonstrate this mode of rupture in a simple filament yarn,

since, as Piatt predicted, the twisted filament yarn tends to rupture

precipitously once the first break takes place. Hearle [7], on the other

hand, has shown that by varying yarn twist and the gage length of the

yarn specimen being tested, he can arrest the process of rupturing midway. But detection of the specific mode of rupture initiation and propagation thereafter is difficult. Backer, Monego and Machida [74] have shown the use of mechanical

tracer elements as a means of studying rupture mechanisms in con- tinuous filament twisted structures. They have worked with model

plied structure, twisted under "no migration" conditions of 91 singles yarns. The bulk of the singles were 70/34 polyester filament yarns, and

the tracer yarns were 79's cotton, dyed various colors for identification. Because of the low rupture elongation of the cotton (— 8 percent) compared to the 30-35 percent rupture elongation of the polyester, the

cotton was prone to rupture at an early stage without leading to total
SINGLES YARN TENSILE BEHAVIOR 145

failure of the surrounding Dacron. The technique also permitted study

of propagation of the rupture from cotton to cotton tracer, and even from

cotton to polyester. And the tracers were used to confirm experimentally

the distribution of strain and of lateral pressure through the twisted

model as it was extended axially.

A fairly simple system was used to record the location and frequency

of component breaks at each extension level for each gross model. A

series of parallel lines was drawn in groups corresponding to the number

of components in each ring of the model cross section. These lines are shown in figures 1.3.47a-c. Each group is arranged symmetrically

about the center line or axis of the model. The lines were then numbered

arbitrarily from right to left. Thus, each numbered line corresponds to a numbered position in the cross section of the model shown in figure

1.3.47c?. It now becomes a simple matter to identify the location of the

cotton component by numbering the appropriate cross sectional location

as in figure 1. 3Aid. The unmarked locations represent polyester yarn

positions for the given specimen. In recording breaks in the component

yarns, one places a horizontal mark on the appropriate line at a longitudinal position corresponding to the location of the actual break in the

actual 8-inch gauge length. Six and often eight strain levels were used in the test program and break counts were made after each test. A few

sample results follow. Figure 1.3.47 shows data taken in tests of a 2-cotton/89-polyester component model with the cotton located at positions #8 and #52, as shown

in figure 1.3.47c?. The twist multiple was 2.19. After extension of the

model yarn to the 11 percent level, component #52, located in the second

ring from the core, evidenced five breaks in its 8-inch gauge length, while

component #8 in the fifth or outer ring showed no breaks, as seen in figure 1.3.47a. (In another similar specimen examined after a 7 percent

extension, the cotton yarn one ring from the core showed but three

breaks.) When another specimen of the material containing cotton components #8 and #52 was extended to 15 percent, it evidenced 19 breaks

in #52 and two breaks in #8, as shown in figure 1.3.476. Figure 1.3.47c,

representing the effects of 25 percent extension, shows the presence of

44 breaks in #52 and 13 in #8. These results and data taken from tests

at other strains are plotted as (dotted and solid) vertical lines on the

actual stress strain curve shown in figure l.3Ald. Thus, figure 1.3.47c? provides a complete history of rupture propagation superposed on the

load elongation curve of the material, and permits the reader to judge

the interaction between local rupture and total yarn resistance to further extension. Better examples of this interaction are given in figures

1.3.48-50, for figure 1.3.47o? shows little effect of the rupture of only

two cotton components on the overall load elongation behavior of the

model. Rather, these two components have served, in effect, as mechanical tracers. As mechanical tracers, components #52 and #8 show in figure 1.3.47

that the model yarn break does initiate in the more centrally located low

elongation component. Secondly, they clearly illustrate the dependence

of multiple break frequency on location, hence on local strain level and
PERCENT EXTENSION

Figure 1.3.47. Diagram of break location in a 2-cottonl89-polyester component yarn

model.

(a) At 11% extension. (c) At 25% extension.

(b) At 15% extension. (d) Composite plot of yarn load-elongation and break frequency.

on local pressures. For component #52 breaks in multiple before #8

breaks even once; further, #52 breaks far more frequently than #8 at

all levels of model strain. The presence of multiple breaks along a single cotton tracer illustrates the invalidity of the assumption that a broken fiber in a yarn ceases to contribute to yarn strength. For even though the cotton tracer fails a
SINGLES YARN TENSILE BEHAVIOR 147

few times (as did yarn #8 in figure 1.3.476, it is subjected to the full level of the yarn strain at positions removed from the broken end. This

buildup in local stress (or strain) from zero at the position of the break

to the average yarn strain at a given radius, occurs through a frictional mechanism such that (Tf= fjiatls (1.3.162a)

where ay is the fiber tension, fi the fiber coefficient of friction, at the

lateral force per unit fiber length and Is the slip distance from the broken

end. Thus, if ay equals the stress in the unaffected parts of the yarn, it is likely that yarn #8, for example, can rupture elsewhere as the yarn

is further extended. And this is what we observe in figure 1.3.47c. Now, if ay, rup is the average rupture tension of the fiber (or tracer component), it follows that the minimum length into which the tracer element can be broken is Lin = ^O-f. ruJf^O-tr (1.3.162b)

For a fixed value of /jl between the tracer element and the yarn matrix we will then expect that the frequency of breaks will be proportional to the local pressure in the yarn. Machida [55] used eqs (1.3.123) and

(1.3.162) to calculate the minimum lengths of the tracer segments in such models as described above. The agreement which he found be- tween the predicted minimum lengths and the measured lengths served

to validate eqs (1.3.123) and (1.3.162) and underscore the importance of

lateral stress transfer of load between ruptured elements in the yarn

and the surrounding matrix.

The tracer yarn method is used with the expectation that rupture of the cotton will not affect the load elongation behavior of the remaining

yarn. But the technique can be modified so that the proportion of the

tracer element is increased to such an extent that the tracer participates

in true rupture of the yarn. At this stage it is no longer a tracer, but a

full fledged blend component. But even though the model system is now a blended one, it can provide mechanistic illustration for the single

fiber twisted filament yarn. For now the low elongation fiber element

can be considered to represent that portion of a uni-fibered yarn which was overstrained during twisting, either due to excessive overall spinning tension or due to restriction of the stress relieving mechanism of filament migration. With this application in mind, the range of examples

becomes unlimited.

Take, first, the case of a low twist 40-cotton element model whose

tensile behavior is shown in figure 1.3.48. One notices here that the

initial rupture of the centrally located cotton components is soon prop- agated to many of the other cotton yarns in the same vicinity. And this failure concentration appears to loosen the model structure to the extent that the cotton contribution to yarn strength is lost. This is seen

as a severe drop in load in the load-extension curve of figure l.SASd

for the model (with a 2.17 twist multiple). Figures 1.3.48a-c shows the

break history at 8 percent, 12 percent, and 30 percent extensions. The
148 TIRE CORD STRUCTURE AND PROPERTIES

small number of breaks per yarn in figures 1.3.48c and 1.3.48c? versus

those shown in figures 1.3.47c and l.3A7d for comparable model strains

is significant; for it confirms the fact that the matrix of the 40-cotton

component model of 2.17 twist multiple has loosened up upon initial rupture of its cotton elements and cannot grip them sufficiently to cause

their breakage into shorter segments.

If the twist of the 40-cotton component model is increased from 2.17

to 3.26 twist multiple, the initiation of rupture of a few inner cotton
SINGLES YARN TENSILE BEHAVIOR 149 components is followed by rapid, concentrated, propagation of the break

to most of the cotton yarns in that region. At this point, the load shed so

precipitously by the cotton exceeds the additional load-bearing capacity

of the polyester and the entire model fails in a manner characteristic of

a 100 percent cotton model. The remaining elongation of the polyester

is not reahzed, as is seen in figure l.3A9d. The concentration of the break

propagation is seen in figures 1.3.49a-c for extension steps of 11 per- cent, 12 percent and 13 percent.

In a case where extreme straining takes place in the outer fibers during

twisting, resulting in their reduction in e/r, the model pictured in figure 1.3.50 is appropos — or the case where the central fibers were buckled,

hence their effective C/r was increased. In figure 1.3.50 we note the

10-

AVERAGE NUMBER OF BROKEN YARN SEGMENTS

10 =

I 38- YARNS BROKEN

15 20 25 30 35 PERCENT EXTENSION

Figure 1.3.49. Diagram of break location for a medium twist {3.26 TM) yarn model with

40-cottonl49-polyester composition.

(a) At 11% extension.

(b) At 12% extension.

(c) At 13% extension.

(d) Composite plot of yarn load-elongation and break frequency.
150 TIRE CORD STRUCTURE AND PROPERTIES

low elongation fiber in the outer two rings and observe their early rupture at about 10 percent. But with their first rupture pictured in figure

1.3.50a, the lateral pressure in the cotton sheath was completely re- lieved and there was no further gripping to effect subsequent breakage.

The result was a significant drop in tensile load of the model, and from

the elongation of 10 to 33 percent, the model behaved like a 100 percent

high elongation yarn of reduced size. In still another case where the low elongation elements were located

70 COTTON YARN BREAKS RELATIVE TO EXTENSION AVERAGE NUMBER OF YARN PIECES I (01 INCH-I PIECE

;

(a)

15 20 PERCENT

25 30

EXTENSION

35 40 50

Figure 1.3.50. Diagram of break location for a cotton-dacron yarn model where extreme

straining of the outer fibers was induced during twisting.

Composition 56% cotton, 44% dacron with 2.17 TM.

(a) Composite plot of yarn load-elongation and break frequency.

(b) Diagram of break location at 10% extension.
SINGLES YARN TENSILE BEHAVIOR 151

asymetrically in the model structure of figure 1.3.51, the first rupture

quickly spread to the adjacent regions of the cotton as seen in figure

1.3.516. But the model's strain levels were not high enough to rupture

the remaining high elongation (polyester) elements. Yet, the low elongation elements remained gripped in the high elongation matrix. As a

result there was no major drop in the model tension at 10 percent ex- tension, and the tension increased slightly as the model was extended

further to 30 percent while the low elongation element was ruptured

many times as seen in figure 1.3.51c.

40 30 10 /5 5 eS 7S &S SS 46 40 JO K) IS 5

iiillii.lil (0

60 SO 4S

1 1 T 1

Figure 1.3.51. Break location for a 22% cotton-78% dacron yarn model with asymmetrically located low elongation elements. TM— 1 .09.

(a) Composite plot of yarn load-elongation and break frequency.

(b) Diagram of break location at 10% extension.

(c) Diagram of break location at 15% extension.
152 TIRE CORD STRUCTURE AND PROPERTIES

Finally, it is shown in figure 1.3.52 that twist strongly influences the

load transfer and rupture propagation and thus, by affecting component

independence, it significantly affects the level of tensile strength, the

reaching of (or failure to reach) full fiber extension potential, the overall shape of the stress strain curve and its general slope and, finally,

the work to tensile rupture. Figure 1.3.52a shows the load elongation curves of the 100 percent Dacron model, 1.3.526 the 56/44 cotton/Dacron

model, and L3.52c the 100 percent cotton model. It should be emphasized that the Hkelihood of multiple stage rupture (i.e., with sudden

partial load relief during a constant elongation test) is increased at lower twists. But it is also increased as the elongations to rupture of

the filament components differ either due to processing history or to original material selection and placement.

Hearle [7] reported test results on 100 den. nylon yarns at various

twist levels and illustrates the occurrence of multiple filament breaks

at zero twist levels, but at twists of 10 to 70 turns/inch, the ruptures were singular, sudden and "clean." He also showed the effect of sample

gauge length on the shape of the stress strain curve and on the mode of

rupture propagation. The transition from a complete to a partial sharp

break is not only dependent on gauge length, but also on speed of jaw

separation. And he observes that "the change from one mechanism to

0.54 TWIST MULTIPLIER

1 .07 " 2.15 3.24

4.29

100% COTTON

J \ L

5 10 15 20 25 30 35 40 PERCENT EXTENSION Figure 1.3.52. (a) Load-elongation curves for 100% dacron model yarn, (b) Load-elongation curves for 56% cottonj44% dacron model, (c) Load-elongation curves for 100% cotton model.
CONTINUOUS FILAMENT PLIED YARNS AND CORDS 153

the other as the gauge length is ahered is due to the amounts of elastic energy stored in the shorter specimen being less, and insufficient to complete the breakage. Similarly, in the slower tests, the stored elastic energy will have decreased due to stress relaxation."

Hearle [7] hypothesizes five modes of break propagation for the

partial break phenomenon: (1) break starting at center, moving out symmetrically to stop partway through the yarn; (2) break starting at outside, then moving in symmetrically half-way; (3) break starting at center, then moving outward in one-half of the yarn; (4) break starting

at one side only, in the outer ring, then moving in to the center; (5) break starting half-way out from the center at one point, then spreading

over half the yarn. He then provides convincing mechanistic arguments

as to twisting conditions and yarn structures which will cause these

various modes to occur, and proceeds to relate the departure of yarn

structure from the ideal model to the variation of yarn rupture behavior

with twist. Specifically, consideration is given to the change in breaking extension with twist for various fibers, including viscose rayon,

acetate, nylon and polyester. And the effects of filament buckling,

twist-process fiber deformation, and migration on breaking elongation

are analyzed. Finally, a report is given of the influence of twisting ten- sions and twisting machine types on yarn load-elongation properties.

The inescapable conclusion is that the properties of a yarn cannot

be predicted solely from a knowledge of its fiber content, and yarn

geometry, but that the complete processing history must be included.

1.3.3.4. Analysis of Tensile Behavior of Continuous Filament

Plied Yarns and Cords

Simple Fiber Tension and Idealized Geometry

Piatt, Klein, and Hamburger have made significant contributions

in the use of stress analysis for the prediction of tensile properties of

ideal yarn geometries [43] and also in use of rupture statistics to predict

the breaking strength of nonuniform fiber bundles and yarns [50]. Their

studies were later extended to the cases of cordage strands and ropes

[63] where the more complex geometry of plied yarns became an essential part of their analyses. Their later starting point was eq (1.3.159)

which applies to the singles yarn, taking into account the translation

of fiber strength into yarn strength on the basis of inherent fiber properties and yarn geometry. The factor n^ accounts for the effect of fiber

variability. It will be given consideration after the simple effects of geometry have been dealt with below.

Starting back with a simple structured plied yarn as pictured in figure 1.3.4c or 1.3.4/ and figure 1.3.3, it is seen that the contribution

of each singles yarn to the plied yarn load is Pp^Py cos Qp (1.3.163)

where Pp is the tensile load along the plied yarn axis, Py is the tensile
154 TIRE CORD STRUCTURE AND PROPERTIES

load along the singles yarn axis and Qp is the helix angle of the plied yarn. For the plied yarn eq (1.3.2) is simple

tanQp = 27rTpRp (1.3.164)

where Tp is the ply yarn twist and Rp is the plied yarn helix radius (corresponding to r in figure 1.3.3). For the case of noncircular singles (as

they lie in the plied yarn) Piatt [63] defines Rp as the distance from the center of the plied yarn to the center of mass of the given singles yarn.

Piatt then assumed the same expression to hold for the helix angle,

ayp, of any fiber in the singles yarn as it lies in the ply. tan ayp = 27TTypay (1.3.165)

where Typ is the number of turns per unit length of the singles as it lies in the ply, and ay is the radius of the singles. Clearly, from eq (1.3.44a) we note that ayp will vary from the side of the singles bend to the outside

(that is as (f) or A.^ varies), but Piatt points out that the use of eq (1.3.165) over (1.3.44a) introduces negligible error in the results of the mechanical analysis.

A factor of more importance to the mechanical analysis is the creation

of excess fiber lengths in the case where the plying twist works directly

to reduce the number of turns in the singles yarn as it lies in the ply.

(Of course, if the original twist direction of the singles yarn coincides

with that of the ply twist, the local singles twist will be increased during

plying. But the conventional twisting procedure, certainly for tire cords,

is to use opposite singles and ply twists so as to effect torque balance,

and this procedure will reduce the original singles twist.)

Since Tp is the ply turns per unit length of plied axis, then Tp cos Qp

is the ply turns per unit length of singles axis as it lies in the ply and

eq (1.3.165) can be rewritten as tan ayp = 27T{Tyo — Tp cos Qp)ay (1.3.166)

where Tyo is the original turns per unit length of the original singles yarn. Remember that the original singles yarn helix angle «yo is related

to the original yarn twist, Tyo, via its tangent as per eq (1.3.165). Piatt assumes that the change in singles twist during plying does not affect the singles axis length. Further, the fibers lying at the center of the

singles have zero helix angle both before and after plying, hence their

length remains constant. However, the helix length of a fiber lying at a

local radius, a, and at an original singles helix angle of per unit

length of the singles yarn axis, is

S tto L If- = l + 47r2a%o

cos - ' ^

1/2 (1.3.167)

and if the singles yarn has been formed with the simple ideal geometry

(no migration and no local fiber average strains), it can be said that this
CONTINUOUS FILAMENT PLIED YARNS AND CORDS 155

helix length // is equal to the unstrained length of that particular fiber.

If. due to pMng. the local helix angle changes as per eq (1.3.165) and the

singles yarn axial length does not change, then the helix path length

for that same fiber (of eq (1.3.167)), per unit length of singles yarn is 1/cos a or Vl -\- 4^77- a

-Tip. The difference between the original fiber length and the new "in situ" heUx length is the excess of fiber length,

or crimp caused by the local untwisting of the singles during plying,

which is seen to be the difference between the reciprocals of the singles

helix angle cosines before and after plying.

Now let A// be the total fiber path extension resulting from an extension A/y of the singles yarn axis. From eq (1.3.100) it foUows that Mf= My cos a (1.3.168)

but only a portion of this fiber helical extension comes from actual load

bearing extension (A//)p of the fiber. The rest comes from taking up the

fiber crimp, thus

which can be expressed in terms of load bearing strain of the fiber (e/jp (COS ex \

1 -] (1.3.170) COS a / If the reader will check eq (1.3.159), it will be noted that €y cos^ q rep- resents (in the reasoning leading to its derivation) the load bearing

strain of the given fiber which, in terms of eq (1.3.170), is (e/)p. Thus, one may substitute (e/)p of eq (1.3.170) for Ey cos- q of eq (1.3.159) to formulate an expression for the tensile resistance of the yarn, Py, to a given

yarn extension Cy. And if €y of the singles equals €p cos- Qp of the plied yarn, then summation of the PyS from eqs (1.3.170) and (1.3.159) and

use of eq (1.3.163) will allow calculation of plied yarn tensile resistance

Pp to a plied yarn strain €p. Because of the improbability of integrating eqs (1.3.159) and (1.3.170) Piatt employed a piecewise summation to obtain approximate solu- tions for several specific cordage structures. He did this by dividing

the singles yarn cross section into five parts by inscribing four circles with radii of (ay/5), 2(ay/5), 3(ay/5) and 4((3y/5). Then for a given value

of Cy, he calculated (e/)p for each ring from eq (1.3.170) and from eqs

(1.3.165) and (1.3.166) determined ao and a for each ring. From the cal- culated value of {€f)p and the fiber stress-strain curves, he calculated

the axial load of the fibers in each ring (or in simplified form from eq

(1.3.157)). And with knowledge of (e/)p, em and he used eq (1.3.153)

to determine Uu. the fraction of unbroken fibers in each ring. Finally,

he summed the axial tensile contributions of the fibers in each ring, to obtain the total tensile resistance of each singles yarn. This, in turn, was converted to plied yarn tension via eq (1.3.163). The entire pro-
156 TIRE CORD STRUCTURE AND PROPERTIES

cedure was then repeated at successively higher values of Cp, until a maximum value of Pp was obtained. This maximum tensile resistance

of the fiber assembly was divided by the product of the average strength

of each fiber and the number of fibers to give the strength efficiency, as was done earfier for preparation of figure 1.3.42 for the original yarns.

If the excess fiber lengths occurring in the singles yarns as a result of plying are neglected, the strength efficiencies can be more easily deter- mined for twisted singles and plied yarns of the "parallel bundle" type,

i.e., wherein each single traces the same geometric path (as in the case of two-ply or three-ply structures). One need only apply the results of figures

1.3.42a-c, using eq (1.3.166) to convert values of Tyo and Tp to the sur- face helix angle, ayp (or 0) of the singles yarn as it lies in the ply for entry

into the graphs of the figure, then using cos Qp to convert from singles

efficiencies to ply yarn efficiencies. The results of these conversions are shown in figures 1.3.53a-G?. The graphs express the theoretical yarn

efficiency 17 as a function of Tpay (or, as marked, NpRy). The curves

plotted are for constant values of {Typ-\-Tp)ay (indicated as {Nyp-\-Np)Ry).

Thus, one enters the graph at the product TpGy, then selects the desired

Typtty and adds it to Tpay. This sum determines the proper curve inter- section with the vertical at Tpay, and one reads off the resultant effi- ciency 7] at the intersection. For these calculations it has been assumed

that there is no variability between rupture elongations of the singles yarns. However, the calculations do take into account the variation, V,

in elongation at rupture of the individual fibers constituting the singles yarns. The a/b characteristic of the fiber stress-strain curve is also ac- counted for, along with the mean elongation to rupture of the fibers. Checking the many assumptions in the above analyses, Piatt et al. [63]

show that the in situ singles-in-plied-yarn helix angles calculated from

eq (1.3.165) are quite close to those calculated from the more rigorous

equations of Chow [16]. The major percentage difference between the
CONTINUOUS FILAMENT PLIED YARNS AND CORDS 157

VARIABILITY = 10%

VARIABILITY =20%

VARIABILITY =30%

o (Nyp + Np)Ry = 0.04

• (Nyp+Np)Ry = 0.08

<S> (Nyp+Np)Ry =0. 12 e {Nyp+Np)Ry =0. 16

40 I i ^ 1 ^ ^ ^ ^ 1 0 0.02 O04 0.06 008 010 0.12 0.14 OI6 NpRy Figure 1.3.53b. Theoretical three-ply yarn efficiency. Yarn rupture elongation

variability = 0. Fiber properties: a/b=5, eni=10%.

VARIABILITY = 10%

VARIABILITY =20%

VARIABILITY = 30%

30 J 1 \ 1 1 1 1 1 1 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 NpRy Figure 1.3.53c. Theoretical three-ply yarn efficiency. Yarn rupture elongation

variability = 0. Fiber properties: a/b = 0, em = lO%. two calculated values occurs at low values of ayp where differences are

of small importance. Further, these differences decrease as Qp decreases.

Piatt [63] also compared the required path length of a fiber in a plied

yarn per unit length of the plied yarn, l/p, expressed as

100

90

lfp = -. TTT^T T (1.3.171) (cos Qp) (cos ayp)

with the corresponding Chow expression using Chow's more rigorous

equation for ayp, and Piatt reports that the differences in calculated excess lengths are small over the range of practical singles and plied

yarn twists.
158 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.53d. Theoretical three-ply yarn efficiency. Yarn rupture elongation vari- ability =0.

Fiber properties: V = 0, en, = 5%. Finally, Piatt compared his simplified expression for fiber strain, €/,

€/ = €p cos'^ cos^ ayp (1.3.172)

with a more rigorous expression derived from Chow's analysis:

€f = ) ^ + tan^ Qp

tan2 Qp

V

1 +

tan a Uy (1 + ep) tan Qp

1 + epCos-Qp rp (l + ep)^ + tan^

VI , r «y tan Qp T

(1.3.173)

and found that values calculated from the two equations differ little in the ranges calculated. The values calculated via eq (1.3.172) represent

average strains. It is interesting to note, however, that Chow's equations can be used to calculate local strains, within a single heHx, as were cal- culated by Backer [80] in his study of local strains incurred during the

bending of twisted yarns. Piatt el al. shows that for a case of average

helix strain of 7.5 percent calculated via eq (1.3.172) the local strains

ranged from 4.7 at (j) = 7r to 11.5 percent (at ^ = 0). It is worth recording

their comments concerning the factors which tend to mitigate the severity

of these local conditions.

"(1) If local differences in strain are admitted, then local differences

in excess lengths must also be considered, and the high excess length

regions occur just when high strain would be predicted, i.e., at <^ = 0;
CONTINUOUS FILAMENT PLIED YARNS AND CORDS 159

(2) any freedom of motion of fibers whatsoever would tend to equalize

strains, and some freedom of motion must exist; (3) any distortion of the

yarn from circularity will tend to remove excess lengths, and by a direct

extension of this reasoning it is easy to see that unequal strains along

the length of a given fiber will cause an equalizing distortion. These

distortions can occur either before or during the loading of the structure,

or at both times. Distortions before loading can occur as the result of the presence of excess lengths. Such distortions tend to alter the local

fiber radii of curvature considerably from that for circular yarns. The

local strains are critically dependent upon local radii of curvature,

whereas the average strains are not so dependent. Examples of distor- tion under loading, for a circular cross section yarn with no excess

lengths, would be: (a) local flattening in the region of high fiber strain,

(b) higher freedom of fiber relative motion in the region of low fiber

strain, with the attendant opportunity of fiber length travel to the region

of high fiber strain. Both mechanisms will equalize fiber strain. Thus,

differences in local strain may safely be considered negligible, and the

average values used."

Piatt concludes from experimental strength and rupture elongation

results "that large differences in strains do not exist, since large differ- ences would produce mucb lower strand and rope efficiencies and elon- gations than those observed." Further comment on this point will be

saved for discussion of experimental results on lubricated versus nonlubricated yarns in tension.

Paul [75] likewise used the stress analysis approach for prediction

of tensile properties of yarns, twines and ropes possessing ideal geometry. In his survey of the literature in the mechanics of twisted structures he focusses on the differences between wire ropes, yarns and twines

and fiber ropes. He treats the idealized three strand rope model, start- ing with the geometry of the strands before and after laying onto the

rope structure and considers the influence of the laying process on

strand twist (during which strand twist can be varied independently

of rope twist during the laying process). He also considers the bending

strains in rope twines after rope manufacture.

Paul then stretches the rope model theoretically and calculates local strand elongation as weU as local rope twine elongation. He does consider the effect on local strains of lateral contraction during rope extension. The yarn force-strain diagram is then used to convert local strains

to local forces. Then the sum of the axial components of yarn and strand

forces provides the calculation of average rope tension at each level of

rope elongation. His computations were conducted on a CDC 1600

computer. Paul carried out sufficient computations so as to predict the

full load elongation curves of rope structures with various geometries,

various material stress strain properties, and differing "foreturn" levels

(in the rope laying process). He then calculated the theoretical breaking

strength as that which occurred when the most highly strained components of the system reached their rupture elongation, as did Piatt

in his earlier paper [43]. In contrast to the more complex treatments used by Piatt [50] [63]
160 TIRE CORD STRUCTURE AND PROPERTIES

and, later, by Paul [75] on plied yarn, cord, strand, and rope structures

of "cordage fibers" and even wire, Symes [76] proposes a greatly simplified approach for nylon (continuous filament) tire cords. His theory

does not assume any particular form of cord geometry. But, rather, on

the basis of measured values of cord tex (count) and of filament properties, he predicts cord breaking load, extension at break, and contraction on heating. He focuses on the concept of average cosines-of-inclinationangles for the individual filament as it lies in the cord, and this permits

significant simplification of the treatment — for it obviates the consideration of the singles yarn and its geometry. Symes' treatment follows (with

his symbols converted, where possible, to those used earlier in this chapter):

/3o = inclination of filament to the axis of the unstrained cord,

j8s = inclination of filament to the axis of the strained cord, = tensile extension of filament (fractional),

erc = tensile extension of cord (fractional),

Df = sum of filament counts (in tex),

Dc = cord count (in tex),

e/c = contraction of filament (from heating),

Ccc = contraction of cord (from heating),

// = filament length,

/c = cord length.

Symes assumed that all filaments in the cord would behave alike, that

all filaments have the same average inclination angle and are strained to the same degree in relationship to the cord strain. He further assumed, as did Piatt above, that the strain along a given filament would be uniform,

implying freedom of longitudinal slippage. Finally, he considered that no change in volume would take place during straining of the cord. Considering a segment 8// of the filament before straining, and its projection on the cord axis 6/c. Then

8/c = 8// cos j8, (1.3.174)

and the mean value of cos /3o is defined

(1.3.175)

whence

If Dc

(1.3.176)

For the strained configuration eq (1.3.174) becomes

( 1 + e^c ) 8/e = ( 1 + €</•) cos Psdlf (1.3.177)

or
CONTINUOUS FILAMENT PLIED YARNS AND CORDS 161

(l + e^c) cos IBo = (! + €(/•) cos /3s (1.3.178)

It remains to determine how (So changes to /3s with cord strain etc. If a short straight length of fiber, //, starts from the point A in figure 1.3.54

h £{l+£tc)

Unstrained State ' Strained State Figure 1.3.54. Assumed geometry of cord extension. and goes to D, while AB is drawn so as to lie parallel to the cord axis (in the unstrained state), then for the strained state AD becomes A'D' and

AB becomes A'B'. The tangents of the angle (So between AB and AD and

angle /8s between A'B' and A'D' are

tan (3o = y; tan (B. b' /c(l + e,c) (1.3.179)

where Ic is the length of cord axis corresponding to AB in the unstrained

state. Now b' will relate to b as r'c is to Tc, where r'c is the radius of the

strained cord and Tc that of the unstrained cord. From the assumption of constant volume

or

nlcr^ = tt/c ( 1 + €tc)rc^ ^=(l + e,e)-« =

fc ^0 and combining eqs (1.3.179) and (1.3.181) we obtain: tan /3s = tan (Boil ^ etc)

-^''^

which when combined with eq (1.3.178) gives:

(1 + ey) cos Ps cos jSo tan^^2/3

tan /3s

(1.3.180)

(1.3.181)

(1.3.182)

(1.3.183)

Equation (1.3.183) is rearranged to read:

sin /3s cos^/2 (1 + etf)-^''' sin po cos"^ (3o (1.3.184)

in which Symes has expanded the sines and cosines as power series
162 TIRE CORD STRUCTURE AND PROPERTIES

and multiplied them together. Then, neglecting cubes and higher

powers, he shows

(3s = l3oil^etf)-'l'' (1.3.185)

which for €tf of 0.15 and Po ^ 0.75 radians gives as error ^ 3.5 percent

when compared with eq (1.3.184). Substituting eq (1.3.185) in eq (1.3.178)

he obtains

(1 + e,c) = (1 + e,/) {sec Po cos [po(l + e,/)-3/2] }. (1.3.186)

Equation (1.3.186) can be used directly to predict cord breaking strain

if the value of filament breaking strain is substituted for €tf and if the

bracketed function { } in eq (1.3.186) is a Hnear, or nearly Hnear, function

of cos Po for the /Bo's expected in the cord, thus permitting the use of cos Po from eq (1.3.176). In terms of yarn count before and after twist- ing eq (1.3.186) becomes

( 1 + e,c) = ( 1 + e,/) (De/Df) cos [ ( 1 + etf) '^'^ cos"! Df/Dc) ](1.3.187)

For a case of thermal contraction in the cord, where €cf is the filament

contraction or shrinkage and Ccc the cord contraction,

(1 -€cc) = {l-€cf) sec Po cos [Po{l-€cf)-"''] (1.3.188)

Symes considers that the contribution that an inclined tension loaded

component makes to the cord is P cos p. If cos Ps is the average cosine

of the angle of inclination of the filament at the instant at break, then

the breaking strength of the cord is simply the breaking strength of the

component yarns times cos ps- Thus

Pcr= nyPyr cos [Po{l + €tfr)-^l^] (1.3.189)

where €tfr is the rupture strain of the fiber and Per and Pyr are the breaking strengths of the cord and yarn component respectively. There are

Jiy yarn components. And since the cos function in eq (1.3.189) is found

to be very nearly linear in cos Po, it is reasonable to use cos po obtained

from eq (1.3.176). The strength efficiency 7)c of the plied yarn or cord is the strength Per of the cord divided by the sum of the singles strength

rtyPyr, hence

Vc = -^=cos [(l + e/r)-^/2 cos-i {DflDe)] (1.3.190)

Tlyryr

plied Yarn Analysis using an Energy Method

Treloar extended his earUer use of energy methods [59] in deriving the

stress-strain properties of continuous filament yarns to the case of multi-
PLIED YARN ANALYSIS USING AN ENERGY METHOD 163

ply cords [77]. As he stated, the essential features of the theory were (1) "that it enables the whole of the stress strain curve of the twisted yarn to be predicted from a knowledge of the stress-strain curve of the filament

material," (2) that its calculations based on strain energy of filaments in the whole structure, are simpler than for the case of stress-analysis, and

(3) that its use of numerical methods permits direct use of the full stress- strain curve of the fiber without the need for simplification. This same

approach was then applied to the somewhat more complex geometry of

plied yarns and so-called tire cord structures with two, three, and seven components. The two- and three-ply structures correspond to Piatt's [63] parallel bundle case and will be considered below. The seven-ply

yarn, a special case with center yarn behaving differently than the six sheath yarns, has been treated by Treloar [77] and Piatt, as well [63],

but we will not consider it at this time.

Going back to the singles yarn analysis, using the energy method, we

note in eq (1.3.138) that Tq represents the distance of a given filament

from the singles yarn axis and dX/r/dky represents the change in filament

extension ratio at ro with change in yarn extension. Equation (1.3.138)

reflects the force P required to extend the singles yarn axis to an extension ratio ky and was obtained by differentiating the total strain energy

(per unit length of unstrained singles yarn) which developed in the yarn

extended to Xy, as expressed by U of eq (1.3.137).

If the tensile strain energy developed in the singles yarn as it lies in the

plied yarn is indicated by Usp and \sp is the extension ratio of that singles as the plied yarn is extended Xp, then the expression corresponding to eq (1.3.138) for the tension in the singles, Psp is:

'., = ^=27r<t> ^(X,.)

^

rjro (1.3.191)

Of^sp JO OAsp

where (j) is the usual packing factor. Since Ugp is based on the unit length

of the singles as it hes in the ply, the Up, the strain energy per unit length

of the plied yarn axis (i.e., tire cord axis) is simply

Up=Uso sec Qpo (1.3.192)

(that is, for one of the singles components). The tension contribution of

this one component along the plied yarn axis is then

dkp ^^""^'^dXp '^"^ dXsp dkp (1-3.193)

Combining eqs (1.3.191) and (1.3.193) gives:

P, = 2770 sec ^

p"" ^{\fr) 1^ rodro. (1.3.194)

To complete the analysis it is necessary to express the quantities dkspldXp
164 TIRE CORD STRUCTURE AND PROPERTIES

and 8kfrl8\siJ in terms of directly measurable parameters, as is done

in the paragraphs to follow. Recall that the total plied yarn force at extension kp is obtained by multiplying eq (1.3.194) by the number of

parallel structured components or plies.

Treloar assumed that extension of the singles in the plied yarn takes

place at constant volume, hence

Rsp = Rs,Jkl^' (1.3.195)

where Rsp is the radius of the extended singles as it lies in the plied yarn. For a two-plied yarn, the radius of the ply helix Rp would be Rsp and for a three-ply yarn, 2RsplV3. This assumes no flattening of the singles

during plying or during ply extension. Since, as is seen in figure 1.3.15,

flattening is to be expected, the ply helix radius Rp is taken as an independent measurable parameter. It is assumed, however, that RspIRp is independent of strain, i.e., the shape of the ply axis is unchanged during

cord extension. Thus

/?p = /?/>o/Xii>2 (1.3.196)

Now

tan Qpo = 27rRpoTpo, (1.3.197)

and in the strained state

tan Qp = 27TRpTp =^-^^' (1.3.198)

The ratio of the strained singles yarn axis as it lies in the ply to the un- strained singles yarn as it lies in the ply is ksp=kp^^^ (1.3.199) sec Qpo

and from the above it is shown that

kl = Xi„ sec Qlo - (1.3.200)

Asp

By difl'erentiation of eq (1.3.200) one can obtain dkspldkp, one of the key

quantities required for solution of eq (1.3.194). It remains now to deter- mine a suitable expression for dk/r/dksp in eq (1.3.194).

When a straight singles yarn with an original local twist, expressed

in radians as 27rTs, is ply twisted into a multiple structure, its local twist as it lies in the ply must be corrected for the change in singles axis

length (due to plying) and for the geometric torsion of its ply-helical path.

These corrections are expressed simply as
PLIED YARN ANALYSIS USING AN ENERGY METHOD 165

27rn,„ = 27rn„;^ + 5iil%^2i^ (1.3.201)

where the subscript o refers to the unstrained state. For the strained

state

Iso

^ sin Qp cos Qp

Isp Rfj

(1.3.202) _ 2ttTsoIso

^ sin Qp cos Qp

^splspo f^P Making use of the above expressions Treloar showed the last term of

eq (1.3.202) could be expressed as a function of Qpo, Rpo, and Xsp from

which he showed

Now from eq (1.3.56) Iso is

ho =

(2iTTsoYm ^ ^ (2^n/?.) 'yi'-l}' (1.3.204)

He then showed that Ispo can be expressed in directly measurable

parameters as:

Ispo = —A2(l + 27tTsoRsoIsoU, + Ul

)

- i2^27TTsoRsolsoUi)] (1.3.205)

where

ui = sin Qpo cos Qpo. (1.3.206)

\^spo/

Now we proceed further towards an expression for dkfr/dksp, noting that

in accordance with previous notations the local helix angle between

filament axis and singles yarn axis as it lies in the ply is originally ao and a for the strained state. Let So and S be the unstrained and strained

length of a filament at a radius distance r (measured from the singles

yarn axis). Then

So=sec ao (1.3.207)

S = ksp sec a (1.3.208)

therefore the filament extension ratio X/r is sec ct
166 TIRE CORD STRUCTURE AND PROPERTIES

and from the constant volume assumption

Rsp RspJ^p- (1.3.210)

From the identities: sec2a«= 1 + 477271^^1,spo (1.3.211)

sec2 a=l + 477-2p/?2 (1.3.212)

and using eqs (1.3.201) and (1.3.203), Treloar obtains ~ (1.3.213)

From eq (1.3.213) one obtains 8Xfrl8ksp required for solution of eq

(1.3.194). Treloar emphasizes that in the above treatment the curvature

of the singles lying in the ply is taken into account in deriving values of

the singles axial length in the ply and its helix torsion, but the effects of

this curvature are omitted in derivation of filament strain. Analysis of the Role ofSingles Yarn Variability in Plied Yarn Mechanics

Piatt et al. [63] point out that "variability of elongation to break among

singles yarns in a plied yarn cannot, in general, be treated in the same manner as variability among fibers in a yarn." They present a method of predicting the effect of variability on the strength of small groups of

fibers or yarns, and use the method to define the minimum size of a sample for which the large sample method developed earlier may be

used. The method can be applied directly to cords and plied structures where all the components considered follow parallel (though helical)

paths. Note that the ply helix angle serves as a geometric factor for strength efficiency calculations, but it does not enter in the statistical determinations.

Consider a population, say, of singles yarns each of which has a some- what different stress strain behavior than the others. By testing the

population one obtains a distribution of singles' elongation at rupture.

Now, form a plied yarn from two or more of these singles and attempt

to predict rupture behavior of the plied yarns. Clearly, the lowest elon- gation singles will govern the occurrence of the component break in the pHed yarn, and it can be said with certainty "that the mean load

for the first break in each component will also be very nearly the mean maximum load." It is therefore important that one knows the distribu- tion of first unit breaks for various bundle or plied yarn sizes. Piatt [63] considered the population of singles with a mean rupture

elongation Im and a normal distribution of elongations to break with a C.V. = V. He defined the probability of any unit breaking at an elongation

in excess of the normalized extension t as
ROLE OF SINGLES YARN VARIABILITY 167

(1.3.214)

where f(t) is the relative frequency of break at {t). Now h is defined as the relative frequency of 1st unit break at elongation t. If n parallel

component units are strained to t, then the probability of having n — \ components unbroken is p"~^ An the probability that the other component shall have just ruptured is f{t), therefore for one rupture and no

others at t, the probability, A, ish=f{t)p''-' (L3.215)

By assigning various values to t and using eqs (1.3.214) and (1.3.215)

Piatt was able to obtain the distribution of first unit breaks for various

plied yarn sizes and these are plotted in figure 1.3.55a. The mean value

for t in each of these curves is plotted versus n in figure 1.3.556. This mean value is equivalent to an efficiency value, Piatt points out, if one

uses a linear approximation to the stress strain curve. That is, using

the value of t from figure 1.3.556, one may read off the load on the Hnear

curve corresponding to the indicated then divide this by the mean

breaking load of the components to give the strength efficiency of the

plied yarn (still to be corrected for Qp).

15 -

-3-2-1 0 I 2 3

t= BUNDLE ELONGATION

Figure 1.3.55a. Distribution offirst unit rupture in a bundle vs. bundle elongation.
168 TIRE CORD STRUCTURE AND PROPERTIES

2 3 4 5 6 7 8 9 10 II 12 13 14 15 16

n = NUMBER OF UNITS PER BUNDLE

Figure 1.3.55b. Mean value of t vs. bundle size n for first unit break.

Piatt uses various values of m = bae = bemV described earlier eq

(1.3.158) together with the t values at first break in figure 1.3.556 to calculate efficiency versus bundle size for m = 10, 20, and 30 as shown in fig- ure 1.3.56. Note the solid lines joining the crosses are the efficiencies to be expected for different sized "parallel" groups whose tensile resist- ance does not exceed the bundle load at the occurrence of first break.

Note also the effect of b times CTe or of m on the shape of first break

efficiency curves. As long as the number of parallel components in the structure is small,

there is a "very low probability of ever attaining a higher load than that

100

80

m =0

y 60

40

X FIRST UNIT RUPTURE O MAXIMUM BUNDLE LOAD

8 10 12 n= NUMBER OF UNITS Figure 1.3.56. Yarn efficiency vs bundle size n.
ROLE OF SINGLES YARN VARIABILITY 169

reached when the first unit broke (except in cases of very high variability).

Conversely, if n is very large, say 100, the maximum bundle load will almost certainly not be reached until several units have been ruptured."

The factors defining attainment of maximum load in the latter case were

formulated analytically by Piatt et al. [63] as follows: Assume that at extension t, the first of n units breaks and also that the

load in the bundle never subsequently exceeds the level it reached at this instant, i.e., ng{ti), where ^(^i) is the tension per unit at the strain

ti. This means that the instant before the second unit breaks, g{t2), the

maximum tension achievable in the unbroken units at strain t2 is deter- mined from the relationship {n — l)g{t2) = ng{ti). And the limiting load,

^(^3), at the occasion of the third break is {n — 2)g{t3) = ng{ti), etc. for

^4, ^5, . . Thus, if ti is selected arbitrarily and the average stress- strain function g{t) is known, one can then calculate all tis from the

above relationships. It is also possible to treat the case where the maximum load occurs at the second break, or at the third break, etc., Piatt's treatment is summarized below.

The probability of any unit breaking in the elongation interval ti-i to

ti (where ?o = — °°) is:Pi= P f{t)dt = ^ P e-^'I'^dt (1.3.216)

where values of this definite integral are to be found in probability handbooks. For the case of maximum load occurring at the first break, it is desired to plot a curve of relative frequency versus t. This can then be

done for maximum load at second break, etc. Towards this end h{t)i is defined as the relative frequency of occurrence of a maximum bundle

load at t when the Ith unit breaks at t. From probability theory, Piatt

points out, the probability of various combinations of events occurring in a total of n events is designated by the term

Ap^^^P2^ p„9« (1.3.217)

where ^^qi=n (1.3.218)

1=1 Here pi is the probability of event #1 (rupture of any unit between to

ti) occurring in a single test or run, p2 is the independent probability that event #2 (rupture of any unit between ti and 12) will occur in a single

test. The exponents q\, 92, etc. indicate how many times event #1 and

event #2 occur in the particular combination of n events designated by

eq (1.3.217). As stated, the numerical value of eq (1.3.217) is the probabihty of occurrence of that particular combination of gi, ^2, • • Qn values. It is worth repeating that pi is the probabifity of a unit breaking

in the strain interval to ti and p2, the probability of a unit breaking in the interval ti to ^2, etc. The expression eq (1.3.217) is seen to be the

individual terms in the expansion of

(P1 + P2+. . .+Pn)" (1.3.219)
170 TIRE CORD STRUCTURE AND PROPERTIES

To determine h{t)u the relative frequency of occurrences of a maximum

bundle load when the first unit breaks, one must multiply the relative frequency /(fi) of any unit breaking at ti by the ''probability of the exist- ence of a breaking array among the remaining units such that the first break at U is a maximum.'' A modification of eq (1.3.219) is necessary

to deal with the breaking array among units remaining after the first unit has broken. The necessary expression is seen to be

(P2 +P3+ . . . -\-Pn)"-' (1.3.220)

and product is hit), = CfM{p2-\-p,-^ . . . +Pn)«-^ (1.3.221)

where C is a normalizing constant such that the total probability is unity.

It can be shown that certain terms of the expansion in eq (1.3.221)

must be discarded to meet the conditions of having the load at first break

be the maximum load as well. This restriction can be seen to be equivalent to the condition

Xqi>{k-l), /c = 2, 3, . . n (1.3.222) In short, eq (1.3.222) specifies the "necessary and sufficient conditions

such that each of the terms of the expansion satisfies the assumption

of maximum bundle load being attained at first-unit break." Thus, to determine the h{t)i curve involves: (a) "selection of a series of arbitrary

values of and the corresponding values of /(^l); (b) for each value of

ti so chosen, the determination of the corresponding limiting values of

t2, ts, etc. from the g{t) equalities previously described; (c) the deter- mination of p2, P3, etc. from the probability integral for each value of

ti and its corresponding ^2, ^s, etc."; (d) the evaluation of the expansion

terms of eq (1.3.221) retaining only those in accordance with condition

eq (1.3.222).

Piatt suggests that the function h{t)i can be plotted as a curve of

strength versus ti. The strength is the product of n (the total number of

units in the bundle) and the load per unit ^(^i). For the case where the

load elongation curve of the units is a straight line passing through the

origin, Vt

(1.3.223)

where g{t) is expressed as a fraction of mean load per unit. The average

t is taken by inspection from the curve of h{t)i vs. ti leading to the aver- age bundle load, g{ti). Thus, the maximum bundle loads resulting from

these average bundle loads of first unit ruptures is ng{ti). In a similar way one formulates aU values of h(t)i as shown for 1 = 2

and 3:
l! ROLE OF SINGLES YARN VARIABILITY 171

= 2 qi^{k-2), k = 3, 4, . . ., n

(=3 "

/1(03 = C/(?3)(P1+P2+P4+P5+ . . . +P«)"-'

A- ^1 = 0; ^1 + 92=2; ^ Qi^ k — S A; = 4, 5, . . ., w

Values of h{t)i and h{t)2 can be plotted as curves of frequency vs. bundle break as done in figure 1.3.57. The strength here is the product

of the number of intact units times the appropriate g(t). Thus, for first break maximum, then second break, then third break maxima, we have:

(1.3.224)

(1.3.225)

(1.3.226)

(1.3.227)

ngM; {n-l)git2); (^-2)^(^3), etc. (1.3.228)

Taking and h from curves as in figure 1.3.57 and using eq (1.3.223)

where V is given, one calculates a weighted mean breaking strength of

the bundle where each unit breaking load (first unit break maximum,

second unit break maximum, etc.) is weighted by the relative area of its h{t)i curve and by the number of units as yet unbroken. The areas under

each of the h{t) curves representing the relative number of all bundle

breaks which produce maximum load at first break, at second unit break,

etc. The sum of the areas is then the total number of breaks.

Dividing the weighted mean breaking load of the bundle by the mean

strength of the individual units times the number of units, n, is nor- mally the procedure for determining the breaking efficiency of the bundle.

"5p

t Figure 1.3.57, Relative frequency of maximum bundle load vs. t,for n = 4, m = 20.
172 TIRE CORD STRUCTURE AND PROPERTIES

But in the procedure described above the g{t) according to eq (1.3.223)

is the fraction of mean load per unit, hence the efficiency concept is introduced automatically.

The bundle breaking efficiencies have been calculated by Piatt et al. for different values of n and m according to the above procedure and

plotted in figure 1.3.56. The calculations were first based on first unit

break values, then on maximum bundle loads. It is seen that as n be- comes smaller, the efficiency can be best characterized by the first unit break efficiency. As n reaches about 15 an infinite sample can be

assumed insofar as these calculations are concerned.

Finally, we point to the similarity of the approach taken by Piatt [63]

and later by Scop [66] in considering the probabilities oi rupture in suc- ceeding units following failure of the first unit, i.e., the similarity of eq (1.3.160e) and eq (1.3.217). Yet there were differences, one treating

load transfer, the other strain variation. It is clear that Scop went beyond

considering the initial break and its subsequent lateral propagation to occur in one chain link comprised of many parallel units. For he also treated the probability of rupture of other links in the system. In short, in Scop's system, the failure of a single parallel unit at one link did not

rule out its «^trength contribution at another link location above or below the original location. In Plait's analysis, a single unit break

ruled out any subsequent contribution by that unit. As we know from

other evidence, such as that shown in figures 1.3.47-52 inclusive,

the broken unit in a large sample of parallel twisted elements can continue to participate in resistance to subsequent extension, but this con- tinued participation is dependent on lateral pressures, frictional conditions and strength reserves at the local region of break propagation.

Cord Behavior in Compression

Considering the early emphasis put on the compressive part of the

fatigue cycle in testing of tire cord, it is surprising that work was not

initiated earlier on the simple axial compressive behavior of tire yarns

and cords. So much has been done, as indicated in this chapter, on the

tensile properties of fibers, yarns, and cords. The difficulty, of course, was in the matter of procedure for compression testing of yarns or, most

specifically, in the arrangement of the mechanical boundaries in a given

test. A similar difficulty was encountered in measurement of the axial compressive properties of fibers and the planar compressive properties

of fabrics. In 1963 Clark [79] studied the plane elastic characteristics of cordrubber laminates and deduced that the modulus of elasticity of the cords was much greater in tension than in compression, although the compressive modulus of the twisted cords was not found to be insignificant, and

such cords were observed to carry appreciable compressive loads when

properly encased in rubber and when prevented from buckling.

Wood and Redmond [78] two years later ran similar tests while

emphasizing the role of cord geometry in determining compressive behavior. Their analytical approach is summarized briefly below with symbols changed to match terminology of this chapter.
CORD BEHAVIOR IN COMPRESSION 173 Viewing a single filament helix loop (in a yarn) with helix radius r, twist r, and helix angle q. Wood considers the filament axial strain e/c which occurs as a result of an applied compressive strain, e^c, with an

allowed lateral strain of eyt. His primary relationship between lateral strain and twist is seen to be equivalent to eq (1.3.108)

where

Cy (cos- q — iJiy sin^ q)

l^y— €ytl €yc.

(1.3.108)

If buckling is considered to take place before any appreciable fila- ment strain takes place, then €/ = 0 and

€yt= eycl tan'^ q (1.3.229)

A plot of eytleyc versus T in figure 1.3.58 shows the powerful effect of

twist on lateral yarn strain (remember, tan q=27TrT).

Wood then views the longitudinal compression of a tire cord embedded

in rubber stock taking into account the axial fiber compressive forces,

the lateral pressures of the laterally expanding yarn, the pressures of the

rubber, and most important, the geometry of the cord structure. The rela- tionships of the lateral expanding pressures to the axial fiber compressive force are the same as those used initially to relate lateral contractive

pressures to axial fiber tensile forces (as in eq (1.3.118)). Starting with the

average compressive force,/, on each of filaments in a yarn, he pro- poses a level of lateral pressure, p

Nf{2sm'q) 2Nf {TrTdyY

dyicos q){dy) [l+{7TTdyV] 1/2 (1.3.230)

where dy is the yarn diameter, T the yarn twist, q the helix angle of the

filaments and / the compressive (axial) force in each filament.

hi

120

100 8060

40

20

d =0.C)07 in

21/2 5 7Vz 10 121/2 15 TWIST, turns per inch Figure 1.3.58. Lateral yarn strain per unit imposed longitudinal cord strain vs. twist.
174 TIRE CORD STRUCTURE AND PROPERTIES

Through a succession of simpHfying assumptions (which warrant

close examination) Wood arrives at a relationship between the lateral pressure, p, developed through yarn expansion in contact with the sur- rounding rubber hole and the local strain levels, i.e., ^^EAe^

(1.3.231)

where €i is the longitudinal tensile or compressive strain of the cordrubber assembly, and et is the transverse spreading of the filaments,

with Er being the effective rubber modulus (or spring constant acting

against transverse spreading of the filaments). The filament compressive

stress-strain curve is simply

whence

and

f=-Efef (1.3.232)

/ ^ [i^{7TTd,yyi^[i-{7TTd,y]

mEf [\^-{TTTdyYyi^^^j^ {TrTdyY

[_\ + (iTTdyYyi^+^^{7TTdyy

(1.3.234)

Values of eq (1.3.233) and eq (1.3.224) are plotted against turns per inch,

r, of the yarn for arbitrary values of dy2TTldlEf and Er (corresponding to his experimental data). See figure 1.3.59. From the figure it is seen that as twist increases, the resistive force per unit of longitudinal strain falls off significantly, leading to the expectation of lower modulus in high

twist embedded cord systems. This reduction is much larger than that expected in a tensile modulus variation with twist. And if filament

modulus in compression is actually lower than that in tension, it may be

Figure 1.3.59. Theoretical curves showing filament stress and folding separation vs. twist. See eqs (1.3.233) and (1.3.234).
TENSILE BEHAVIOR OF PLIED YARN 175

expected that the compressive softening with twist of the embedded

cord will be even greater.

Wood does observe the occurrence of buckling in his compressive

tests on embedded cord, but he provides no insight as to its mechanism.

Actually, the buckling of fiber assemblies has received more attention of

late in connection with studies of fabric aesthetics, but it still remains

for someone to work out the specific mechanics of compressive buckling

of a twisted structure embedded in an elastomeric matrix. We can only

refer at present to several recent studies which deal with various phases

of the problem. The work on buckling of fabrics, taking into account

elastic behavior and inter-fiber friction is summarized by Grosberg [7] and, more recently, by Olofsson [81, 82], Sadowsky et al. [83] treat the

buckling of microfibers in composite materials. Haringx [84] dealt with

compressive buckling of helical springs, which in a way simulate the

structure of a two- or three-ply yarn, and he also considers the effect of

lateral elastic restraints. Amirbayat and Hearle [85] have studied the

buckling of reinforcing fibers in an elastomeric matrix as a result of

repeated flexing a companied by slippage. And, finally, Menzies [86]

has treated the buckling of twisted structures under the combined in- fluence of torsion and tension. He was able to show the usefulness of the classical torsional buckling relationship for an elastic rod as applied

specifically to the case of a twisted strand, i.e., Mj = iEIP + ^^YTTT- (1 -^-^SS)

where Mt is the torque at buckling, EI is the bending rigidity of the yarn, P is the tensile load on the yarn, L is the yarn length between torsional

jaws. Menzies attempted to derive an effective EI on the basis of yarn

geometry and fiber properties, but the full success of such an effort must

await further results in the elastic-viscoelastic-friction treatment of

bending in twisted fiber assemblies. Some review of progress in this

latter area wiU be given in the following section.

1.3.3.5. Observations of Plied Yarn Behavior in Tension and

Compression

Tensile Behavior

In presenting quantitative data on the stress strain properties of

yarns and cords, one generally includes breaking tenacity, elongation

at rupture, initial (tangent modulus), secant modulus and energy to rupture — aU based on a first time load elongation test in uniaxial tension.

The major part of the analysis incorporated in this chapter is directed

toward this limited view of the stress strain behavior of raw, that is non-dipped, tire cord. It is hoped that the survey of mechanics here

recorded, provides the reader with starting concepts and directions of

analysis useful in treating the geometry of twisted structures and the

interaction of fiber properties with said geometry as it determines the
176 TIRE CORD STRUCTURE AND PROPERTIES

stress-strain properties of the raw cord. We will say something about

other stress-strain properties at the end of this section, but for now we

focus on the more common properties of strength, elongation and

modulus.

Mechanical properties may be listed in table form with separate

columns for the structural variables of yarn counts, diameter, singles

twists and ply twists, and this was done at an early date by Gurney

et al. [87] for tire cords made of cotton. A most extensive series of ex- perimental studies were undertaken during the period 1931 to 1943 by

Coulson and Dakin [88] (although not published until 1957) on the

mechanical properties of plied yarns — again cotton. They provide ex- tensive tabular data for 20s/2, 30s/2, 40s/2, 45s/2, 60s/2, and 80s/2 ply yarns, but unfortunately, the material is in staple form and has application to continuous filament tire cord yarns, only in limited regions of

singles and ply twist. Yet it is instructive to provide a graphical picture

of the effect of structural variables on mechanical properties of staple

plied yarns as is done in figure 1.3.60. Here we see a topological plotting

of tenacity and elongation at rupture at different singles and ply twists

for 42s/2 cotton (1 Vs" carded). The maximum tenacity of 1.55 grams/grex

occurs at a singles TM of 4.0 and ply twist of 30 and the tenacity slopes

off as one moves southeast, south, southwest or west from the tenacity

summit, as reflected in the solid isotenacity lines. The isoelongation

I 2 42/2, U.S.A. COTTON, I 1/8". CARDED3 4

T.M. OF YARN

Figure 1.3.60. Tenacity and elongation at rupture- as a function of twist multiple of the yarn and turns per inch of the ply for 42sl2 cotton.
TENSILE BEHAVIOR OF PLIED YARN 177

lines likewise reflect a positive slope as one moves up vertically, al- though the data do not reflect a distinct summit of elongation values. The work of Piatt et al. [50, 63] on cordage fibers was referred to at length in earlier discussions of singles and plied yarn mechanics, for their

studies emphasized the interaction between fiber properties (including

variability) and structural geometry. The following data reveal their success in predicting efficiencies of cordage yarns and cords based on

the analyses of this chapter. Their experiments had to do with such yarns

as abaca, henequen and sansevieria and therefore cannot be directly

applied to the continuous filament rayon or nylon. But the good agreement

obtained in the following table between theoretically calculated and

experimentally determined efficiencies underwrites the validity of the

mechanics and statistical treatment reported by Piatt. The tabular results that follow were obtained by determining the

pertinent fiber data, picking points off the curves in figures 1.3.42a-c

and comparing theoretical strength efficiencies with the experimental

results. Table 1.3.2. Comparison of theoretical and experimental efficiencies

Yarn type em a/b V 6 Yarn Efficiencies Theoret. Experim.

(%) {%) deg (%) (%) Blended abaca 3.00 -0.73 22 16.5 54.0 56 ±3

3.00 .73 24 20.0 50.0 50 ±3

3.00 .73 31 24.5 47.0 43 ±3

3.00 .73 32 29.4 42.0 38 ±3

Abaca strand yarns 3.00 2.10 24 14.0 62.0 64 ±3

3.00 1.51 24 : 18.0 58.5 56 ±2

3.00 0.91 24 24.0 52.5 50 ±2

Henequen 5.00 2.10 20 22.0 59.0 63 ± 1 5.00 2.10 20 28.0 54.0 56 ±2

5.00 2.10 20 35.0 45.0 39 ±3

Sansevieria 3.00 0.00 30 17.0 55.5 56

It should be pointed out that certain corrections were necessary for calculation of either theoretical or experimental efficiencies shown

above. These included corrections for oil pickup, for twist take-up in the yarns, and for mean fiber strength (as against stress at mean fiber elongation). In another set of calculations Piatt determined the efficiencies of

translation of Manila Abaca fiber strength into bundles containing a small number of units. The theoretical efficiencies taken from figure

1.3.56 are compared below with experimental efficiencies and show

reasonable agreement.
178 TIRE CORD STRUCTURE AND PROPERTIES

Table 1.3.3. Efficiency of strength translation of manila abaca fibers Number of

fibers in bundle

Efficiencies Theoretical Experimental

2 90 88.3 ±7

3 81 81.0±5

7 73 76.0 ±8

15 68 69.8 ±7.5

Piatt summarizes all his information on the structure and efficiencies of abaca strands (in effect, a multi-ply yarn) in the following cited table:

Table 1.3.4. Comparison and theoretical and experimental strand efficiencies

Strand code

Cosine

strand

helix angle

Strand efficiency No excess

length Excesslength

Experimental

SSS 0.931 %

40.7 %37.4 %41.4±2

SSM .897 35.5 32.1 37.1±2

SSH .880 34.3 29.8 31.0±2

SMS .945 38.3 33.6 40.6 ±3

SMM .917 37.7 31.2 43.5 ±2

SMH .878 33.1 26.6 29.6 ±2.5

SHS .943 34.6 31.4 31.8±2

SHM .942 34.5 31.4 33.2 ±5.5

SHH .931 33.7 30.7 33.4 ±6

It was pointed out that Symes [76] took a greatly simplified approach

for nylon tire cords, in analyses covered by eqs (1.3.174-190). He emphasized the effect of changes in cord denier (or tex) to indicate changing

radii, helix angles, and structure. And in eq (1.3.187) he related cord

strain to fiber tensile strain, in eq (1.3.189) cord strength to rupture strain of the fiber and in eq (1.3.188) cord contraction to filament shrinkage in heating. Figures 1.3.61a-flf show the agreement between theory and

experiment— shown to be good except in the prediction of elongation

to rupture. Symes states of the latter that "Experimental extensions are higher than the predicted values by 3 or 4 percent (strain). Most sur- prising is the lack of agreement between the low twist, low tex, extension

of cord at break and the extension of the yarns extracted from the cords.

The difference is perhaps due to structural or coil looseness in the cord,

that is, the filaments in the cord when the cord is inserted in the clamp
TENSILE BEHAVIOR OF PLIED YARN 179

for tensile testing are not straightened to the same extent as when test- ing yarn. Higher tensions when making the cords might reduce extensions

of the cord or give a better fit than that shown."

The use of energy method to analyze the tensile stress strain properties of multiple yarns was reported by Treloar [77] following on his earlier work with continuous filament singles yarn [59]. Experimental

verification of the cord, or multiple yarn analyses was provided by

Riding [90]. Recall that Treloar expressed the load on the ply yarn Pp

in eq (1.3.194) as a function of the fiber stress strain curve [i//(A./r)], the

helix angle of the ply {Qpo), the position of the fiber in the singles (ro), and the change ratio of singles strain to ply strain, and fiber strain to singles strain. The latter two ratios had to be developed separately from

eqs (1.3.200) and (1.3.213) respectively. Thus, eq (1.3.194) becomes the

vehicle for predicting the ply yarn load at any given ply yarn extension.
180 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.62d. Stress-strain curves: tivo-fold.

It says nothing about breaking loads or breaking extensions — but can

be used to predict the full load elongation curve of the ply yarn in the

"unbroken" region. Riding did just this for a two-ply, a three-ply, and a seven-ply yarn, with good agreement between theory and experiment,

as shown in figure 1.3.62 for the two- and three-ply. Equal agreement was observed for the seven-ply, not shown. The mean breaking point

determined experimentally is shown on the broken curves with a cross.

It was observed, however, that relatively low values of the experimental

curves (versus the calculated ones) occurred at low strains, and particularly for high twists. As Riding points out, this difference was also observed in comparison with theory and experiment on singles yarns, hence it is not surprising to find the effect occurring again in cords made

from the singles. The unfortunate feature is the focus of the difference on the slope of the curves; in other words, on the initial tensile modulus,

the parameter so important to the engineer working with tire structures. Riding [90] replotted the curves of figure 1.3.62 as shown in figure

1.3.63 (only for case of two-ply yarns). This form should be more useful

in tire studies.

The reader should be aware from earlier discussions in this chapter

that data such as that plotted in figures 1.3.62 and 1.3.63 can be misleading unless full details are provided of the experimental techniques

for measuring singles and plied yarn structure. Riding provides such

details; unfortunately, many other reports in this field do not. He de-
EFFECT OF TEMPERATURE ON TENSILE BEHAVIOR 181

1.5 - Cord Twist three-fold (turns /cm)

EXTENSION, %

Figure 1.3.62b. Stress-strain curves: three-fold.

0\—j , 1 , < , , , 1 I -4 -2 0 2 4 6 8 10 12 CORD TWIST, turns/cm Figure 1.3.63. Nominal specific stress vs. cord twist: two-fold. termines (see fig. 1.3.%, also see eq (1.3.96)) the yarn radius indirectly

from measurements of retracted length per unit filament length. Singles

twists are determined on 30 specirnens of the singles yarn. The mean
182 TIRE CORD STRUCTURE AND PROPERTIES

stress strain curve of the original yarn was obtained from 20 tests. The

basic parameters to be determined in the two- and three-ply cords are the cord twist and the ply axis radius. Since it is virtually impossible to obtain satisfactory direct measurement of the ply axis radius, Riding

uses instead an "effective" radius, defined as the radius required to give a calculated value of cord retraction in agreement with an ob- served value for corresponding, measured values of cord or ply twist. A note of caution pertains to the determination of the stress strain properties of the fiber as it lies in the singles, or of the singles as it lies in the ply. Any prediction of strength translation into higher order

twisted structures must take into account changes in fiber properties

which occur as a result of the twisted (and winding) operation. Riding

[91] has shown "that the twisting of a viscose rayon yarn followed by

a subsequent detwisting, produces a reduction in tenacity and extensibility of the constituent filaments, and that the magnitude of this effect is influenced by the type of twisting machine, twisting tension,

amount of twist, and the number of times the operation is repeated,

but is unaffected by the time for which the yarn remains in the twisted

state." He has also shown that this effect was caused by a form of fila- ment surface damage, produced by inter-filament contact. We cannot

provide a corresponding reference concerning loss in filament tensile

properties due to twisting of other tire cord materials, such as nylon or the polyesters, yet this factor must be kept in mind in any attempt to predict yarn or cord properties on the basis of single filament stress- strain curves.

Effect of Temperature on Tensile Mechanical Properties

The change in filament properties due to damage incurred during

twisting is only the first of many variations in filament behavior which

must be taken into account in predicting cord strength and elongation.

The most obvious change which must be dealt with is the influence of temperature on mechanical properties of the basic filament. At first, one must consider the effect of tire running temperature on the properties of the fiber, hence of the cord. Then consideration must be given

to the effect of temperatures during cord stretching treatments and,

finally, to the effect of temperatures of vulcanization. Takeyama et al. [89] have provided an excellent survey of the literature of tire cord materials and treatments and there is no point of repeating their extensive

reporting of heat effects on tire cord behavior. But for convenience, we

reproduce one of their summary curves in figure 1.3.64, which shows

the effect on breaking strength and energy of temperatures ranging

from 100-300° F, with both low speed tests and high speed tests. At

low testing speeds it is noted that decreases in strength and breaking

energy of nylon are more rapid than those of rayon and polyester. At

high testing speeds decreases in strength and breaking energy of nylon

are greater than those of rayon and polyester. In fact, rayon shows an

increase in breaking energy with temperature. To be sure, in the ab- sence of full identification of the types of rayon, nylon, polyester and

glass involved in the experiments reported by Takeyama in figure
EFFECT Of STRAIN RATE ON TENSILE BEHAVIOR 183

80

7060

504030-

cr

I-coI

BO S 70

a:CD 60

5040-

30

POLYESTER

200 300 GLASS

POLYESTER NYLON

1 I

300 400

POLYESTER RAYON

NYLON

GLASS

J ! I 200 300 400 100 TEMPERATURE {°F) 200 300 400

Figure 1.3.64. Effect of temperature on breaking strength and energy. A and B: low speed

test. C and D: high speed test.

1.3.64, and in the absence of full data as to dipping or stretching conditions, caution in ranking fiber behavior is necessary.

Effect of Strain Rate on Tensile Mechanical Properties

The influence of strain rates on the mechanical behavior of individual

filaments is a subject to itself, and it is to be expected that an increase

in testing speed which will significantly affect the stress-strain curve

of the individual filament will correspondingly change the mechanical

behavior of the plied yarn or cord. And presumably the same analyses

leading to prediction of plied yarn properties, as covered in this chapter,

will prevail provided that the altered filament stress-strain properties

are inserted in the equations used in the prediction of cord behavior.

From experience we know that fiber modulus will generally increase

as the strain rate goes up and elongation to rupture will go down (for most filament materials) at higher testing speeds. But another factor

to be considered is the effect of higher testing speeds on the slope of the stress-strain curve in the region of filament rupture. Piatt et al. [50] analyzed the effect of this slope on conversion of fiber properties

into bundle, or yarn properties. And Takeyama [89] refers to the effect of this "latter curve" slope on the yarn to cord conversion efficiency for polyester yarns — the higher the slope of the upper part of the stress- strain curve, the lower becomes the conversion efficiency. What in- fluence strain rate has on the "latter curve" slope is evident in the

many stress-strain curves reported by Krizik et al. [92] at strain rates of

0.33, 2025 and 2775 percent per second. Of particular interest is their

data showing the effect of strain rate on the loop strength efficiency of several textile yarns — for here the role of "latter curve" slope and slope

change with strain rate is of utmost importance.
184 TIRE CORD STRUCTURE AND PROPERTIES

Of course, yarn history, particularly that of drawing, will influence

the shape of the stress-strain curve, hence the conversion efficiency in

tire cords. Takeyama discusses the effect of polyester drawing on con- version efficiency as shown in figure 1.3.65. The interaction with ply

twist factor is quite evident.

An important reference text on the impact, or high speed testing of

fibers, yarns and fabrics was written by Lyons [94]. Lyons provides an

historical review of instrumentation developed in this special field of

testing, including the falling weight devices, the rotary disk methods,

ballistic methods and the more recently developed pneumatic and

hydraulic apparatus. He summarizes the theories of deformation and

rupture under high-speed impact and reports considerable experimental data on fibers, yarns, cords, ropes, woven cloth, and special

body armor fabrics. A good collection of impact data is provided by Hall [95] who studied

the impact behavior of various fibers at strain rates of 330 sec~^ and

at 8.3 X 10~^ sec~^ for low twist yarns. A sampling of his data taken

from Takeyama is provided below.

Table 1.3.5. Effect of rate of extension on mechanical properties of various yarns

Yarn

Extension breakingTenacity Initial modulusEnergy to rupture Critical velocity

Low* Hi** Low Hi Low Hi Low Hi Nylon 900 20.1 12.7 84.9 110.0 510 1900 87.1 67.5 360 Nylon 300 20.5 13.8 60.6 93.1 360 700 71.7 59.7 350 Nylon 100 30.4 13.7 48.2 65.4 350 1400 103.0 51.7 260

H. T. Enkalon 26.8 15.3 73.9 90.8 320 1500 88.0 56.5 290 Tenasco 105 15.2 10.5 35.6 50.5 900 1800 29.8 32.8 190

Textile Tenasco 19.2 15.5 26.1 39.5 730 1700 28.1 39.1 220

Standard Rayon 23.2 19.6 18.9 29.8 400 1200 27.8 38.9 210

Ulstron 19.7 14.6 78.9 82.8 1080 1100 98.5 75.9 270

Polypropylene

multifil 54.0 16.5 43.0 65.1 610 1100 193.0 69.8 300 Courlene X3 19.7 7.5 37.4 63.0 640 1300 57.6 25.7 210

H. T. Terylene 12.8 7.3 60.8 77.7 1290 2000 54.5 34.4 220 M. T. Terylene 31.0 17.6 41.4 43.9 900 1900 98.0 67.5 150 *Low = 8.3 X 10-^ sec-i strain rate. **Hi = 330 sec-' strain rate.

Lothrop worked with dipped rayon and nylon cords, measuring tensile

properties as a function of temperature and rate of extension. His data

are shown below in table form. According to Takeyama, for both oven

dried rayon and nylon, breaking strengths (measured by Lothrop) de- crease as temperature increases at a given rate of extension [96].
EFFECT OF STRAIN RATE ON TENSILE BEHAVIOR 185

^.90

>-

PLY TWIST FACTOR Figure 1.3.65. Effect of yarn drawing on tenacity conversion efficiency for polyester.

Table 1.3.6. Breaking characteristics* of dipped rayon and nylon cords as a function of temperature and rate of extension

Rayon Nylon

Strain Cond., Oven-dried, °F Cond., Oven-dried, °F

rate Iteir °F °F %/sec

75 75 150 225 300 75 75 150 225 300

1 A 32.3 35.0 ' 31.9 27.8 24.0 28.3 29.2 24.0 21.1 16.7 B 13.5 10.2 10.4 10.8 11.3 19.9 20.8 18.9 20.5 22.2 C 2.45 2.08 1.94 1.69 1.51 2.50 2.69 1.92 1.93 1.82 1000 A 38.5 41.7 40.0 36.6 33.0 32.9 33.4 29.1 25.2 21.3 B 13.8 10.4 11.4 12.5 13.5 17.6 21.1 20.9 21.1 21.3 C 3.24 2.66 2.70 2.78 2.75 2.80 3.12 2.59 2.51 2.05 3000 A 39.2 43.6 40.6 38.3 36.3 34.6 33.4 29.2 25.1 22.2 B 14.2 10.4 12.7 13.3 14.2 18.5 20.6 20.1 19.5 20.5 C 3.17 2.76 3.07 3.08 3.12 2.83 2.94 2.58 1.98 1.91 6000 A 41.0 44.0 42.2 40.3 38.1 35.3 34.7 29.3 26.3 22.4 B 13.7 10.3 12.4 13.3 15.0 17.5 21.0 19.8 19.5 19.0 C 3.21 2.55 3.07 3.18 3.32 2.75 3.01 2.35 2.18 1.79

^Breaking characteristics A= strength, lb B = elongation, %

C = energy, in. lb/in. of cord.
186 TIRE CORD STRUCTURE AND PROPERTIES

Compression and Buckling

We now briefly turn to observed axial compressive behavior of tire cords so as to complement earlier discussion on the mechanics of compression. Wood [78] has attempted an analysis of the structural inter- action of tire cords subjected to axial compression and his experiments

bear out the shape of his predictions in a general way. Figure 1.3.66a

shows the effect of twist on the lateral/longitudinal strain ratio in compression. The theory is indeed different than the experimental values

at twists below five T.P.I. The effect of twist on the compressive stress- strain curve is shown in figure 1.3.666, while the effect of twist on compression modulus of cords in rubber is shown in figure 1.3.66c. An added

comparison between rubber embedded cord behavior in tension versus compression is provided by Clark [79] in figure 1.3.67 for rayon cords.

He reports similar data for embedded nylon cords, reflecting higher

moduli in tension than in compression. Finally, it is worth noting the

appearance of twisted, embedded structures subjected to axial compression. Wood used a transparent rubber slab for his compression

tests and was able to show two distinct mechanisms of cord compression. First, at low twists he observed distinct cord zig-zag buckling.

He reported that when the cord reinforced rubber slab was bent for the

first time, it exerted a large resistance, which suddenly decreased as the

low twist cords collapsed and buckled. At the point at which the cords

buckled, adhesion breakdown occurred, this being so marked that it was actually heard as the slab was first bent. In regions of high twist,

cord buckling became less pronounced and eventually disappeared.

At this twist level and above, separation of the cord plies was observed,

with each of the singles forming the plied yarn compressing in a manner similar to that of a spiral spring. This expansion of cord diameter

during compression was also noted by Busby et al. [93]. The buckling pictured by Wood was a sudden, catastrophic one,

localized in nature and occurring primarily at low twists. In general,

however, buckling takes place in twisted structures in a weU-defined

relationship, as per eq (1.3.235), between torque and axial load. And

Menzies found the form of the experimental relationship was linear for Manila cordage as shown in figure 1.3.68 and as expected from eq

(1.3.235). But the difference in behavior of his samples as against Wood's

was that torsional buckling was a gradual process whereby the straightaxis of the twisted structure assumed a helical configuration, then the

helix localized into a loose loop, which then tightened on itself. The

end point was one of complete local distortion and strain concentration, such that when- the specimen was straightened and tested again,

early buckling would reoccur at the same position.

Spectrum, of Stress Strain Properties

As pointed out above, the stress strain properties most often reported

in the textile literature include breaking tenacity, elongation at rupture,

initial modulus and energy to rupture. In the analyses cited in previous

section 1.3.3.4, numerous methods are presented for calculating the

conversion of these properties from fiber to cord structures. And in the
COMPRESSION AND BUCKLING

9% Longitudinol compression 12%

14% "

2'/2 5 7'/2 10 12 '/z 15 TWIST, turns per inch (a)

0 2 72 5 71/2 10 12 '/2 15 TWIST, turns per inch (C) Figure 1.3.66. Cord compression characteristics.

(a) Lateral cord strain per unit longitudinal strain vs. cord twist.

(b) Compression-load curves for rubber blocks containing 30 rayon cords.

(c) Compression modulus of cord in rubber.
188 TIRE CORD STRUCTURE AND PROPERTIES

-700-

600- -I

500 4-

— /inn 4-

r

TRESS.

PS j—

200-

(

-.010 -.0 )8 -.0( STRAIN

)6 -.0( 14 -.0

100-

)2 oc 12 .004

- 100

-200 Figure 1.3.67. Continuous stress-strain curve through compression and into tension for rayon cords embedded in rubber. Stress based on total cross-sectional area of specimen.

section L3.3.5 on Observations of Plied Yarn Behavior in Tension and

Compression, considerable numerical data on cord properties are pro- vided under each of the above-named principal properties. The data

serve two purposes, first to validate the analytical methods, as well as the geometric assumptions on which they were based and, second, to furnish the technologist or engineer with experimentally determined

quantities as a basis for product design. But the tire manufacturer and

the fiber producer responsible for improving cord quality cannot function

without considerably more information concerning cord behavior.

Knowledge of a range of stress strain properties is necessary, including

creep, flatspot index, shrinkage impact toughness, and fatique. These

additional properties constitute but a small part of the more than fifty

stress-strain properties listed in the Thesaurus of Textile Terms [8], yet

they add immeasurably to the challenge of developing analytical methods

for predicting or designing total tire cord mechanical behavior. We will make no effort to deal with these topics at present, since they are covered

so recently in Takeyama's treatment [89] and we simply extract pertinent

summary data in the following table based on work by Schroeder and

Prettyman [97] (see Takeyama also).
SPECTRUM OF STRESS STRAIN PROPERTIES 189

200

180

160 S 140 zi

120 o 100 7 80

E" 6040

20

0

0 12345678 P =L0AD , lbs. Figure 1.3.68. Buckling torsion of twisted structure. Table 1.3.7.

Cord construction

Rayon

(current) 1650D/2

12X 12

Nylon

840/D2

12X 12 Nylon

(modified)

Polyester

840D/2

12X 12

2200D/3

8.5 = 8.5 1260D/2

lOx 10 840D/2

12X 12 1260D/2

lOx 10

Tenacity, gpd 4.5 7 6.5 6

Elongation at break, % 9 21 15 13

Initial modulus gpd 60 27 40 70 Creep, % 3 4 3 2 Shrinkage, % 0.6 6 7 7 Flatspot Index, mills 50 170 115 50 Impact toughness,

erg/cm D 225 525 400 375 C-F fatique min 400 1400 600-1200 500

Adhesion, lbs min 100 90 70 65

Aging, % 70 95 70-95 i 70
190 TIRE CORD STRUCTURE AND PROPERTIES

Effect of Lubricants

Takeyama [89] et al. point out that the yarn to cord tenacity conversion

efficiency of rayon cords is quite low and therefore the choice of tire cord lubricant is important. They point out that a lubricant which

decreases the friction coefficient of the yarn and improves the conversion

factor (as shown in the accompanying table) has to be developed.

Table 1.3.8. Correlation ofcord strength with yarn friction coefficient Specimen

1650/2 rayon

Friction coefficient Bone dry

strength per

Static Dynamic 100 denier A 0.207 0.197 263 B .200 .184 266 C .160 .150 272 D .132 .158 280 E .107 .144 296 F .118 .133 306

Roder [98], who originally reported the above data, points out that the manner of twisting of the cord will likewise have significant effect on

cord strength, but with regard to finish, he feels that a low static coefficient will help each filament in the yarn find its equilibrium position

during cord twisting, which leads to maximum strength in the cord. If one reviews the possible effects of friction between fibers on the ease of

migration during twisting, on the ease of singles flattening and cord

radius reduction, on the ease of longitudinal slippage of fibers during

the bending of the singles inherent in the plying operation, and on the ease of longitudinal fiber slippage to accommodate varying strain levels along a single filament during axial tensioning of the cord, then it is easy to understand the results of Roder's experiments. It remains, however, to determine which, of the above-named mechanisms, fiber to fiber friction affects to the greatest degree, at least insofar as breaking

strength is concerned.

Finally, it may be expected that lubricants should show a far greater

effect on the bending rigidity, as against tensile strength, of fiber as- semblies whose bending is accompanied by so much fiber slippage.

And this bending rigidity, in turn, should play an important role in matters of flex fatigue and fatigue in general. This leads directly to the

topic of bending mechanics of twisted structures, a subject which has

received considerable attention in the textile literature of the last five years.
GEOMETRY OF BENT YARNS 191 1.3.4. Bending Mechanics of Twisted Structures

1.3.4.1. Geometry of Bent Yarns

Anyone who has seen a tire in action is aware of the fact that extensive bending takes place in the "footprint" region. Depending on where

the cord is located, it may be in tension, in compression, and/or in a bend. It follows that during the full rotation of the tire the cord ex- periences a tension-compression flex cycle, the magnitude of which is controlled by the deflection of the tire. The frequency is, of course,

dictated by the speed of the vehicle, with the average tire rotating about

800 times per mile, or for a speed of 60 mph, at 800 cycles per minute.

As Kovac [10] points out, the average passenger tire will be subjected

to about 20 million flex cycles during its lifetime. The bending of twisted structures has been given very little attention

in the textile literature before the 1950's. although civil engineers have

been concerned with bending mechanics of wire rope used in heavy

construction. And in the late 1940's such authors as Czitary [99] and

Wyss [100] pumished reports on the subject. More recently, the topic

has been considered by Lutchansky [101] in connection with problems

on bent submarine cables. Tovey's [102] excellent review article is a good starting point for review

of the textile literature on the subject of bending and bending recovery — a subject of great interest in the commercial market as it is related to the

fabric qualities of wrinkle resistance. Hamburger et al. [103] were con- cerned with the analysis of bending strains in yarns at the fabric crease, and they proposed that average bending strains be computed with the

assumption of freedom of motion between fibers. They showed that experimental recoveries from tensile strain calculated for bent struc- tures correlated well with experimental bending recoveries of creased

fabrics. This led to the conclusion that tensile strains in bent fabrics were of the order of only a few percent, even though the yarns concerned were being bent around themselves — an exercise which should produce

outer-fiber strains of the order of 50 percent if the fibers were restricted

from sHpping past one another. At a later date, Piatt, Klein and Hamburger [14] examined the structural behavior of singles yarns in bending,

using a more rigorous analysis to predict bending rigidity of the twisted

structure on the basis of bending, as well as twisting moments of the

individual fibers. This approach warrants further consideration. And

still later, they [104] used a similar approach to the mechanics of torque

development in a singles yarn system.

Backer [80. 105] emphasized the geometric aspects of bent yarn

mechanics, and as was seen in eqs (1.3.90) through (1.3.95), showed that the differential geometry of the double helix could be greatly simplified

if the helix angle of the primary helix angle went to 77/2. Thus the primary

helix became instead a planar circle, and the singles yarn assumed the

form of a torus as shown in figure 1.3.69. The vector treatment then

follow ed the course of eqs (1.3.32) to (1.3.44) above, except that Q became
192 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.69. Torus form of bent yarn.

7r/2 and all equations were somewhat shortened. In particular eq (1.3.44a)

became

tan a = ——— (1.3.236)

r — a cos At^ remembering that \9=^ is the angle shown in figure 1.3.69 locating the

fiber position at P in the yarn cross section. When (/) = 0, we have the

fiber at the inside of the bend, and when 4>=7t, the fiber is at the outside

of the bend. At (/) = 7r/2, the fiber is at the pseudo-neutral plane of the

yarn. For these particular fibers the local helix angles are:

for (t> = 0, tan ab = -^= , ^

, (1.3.237)

r — ar/a—l

forc/) = 77/2, tSinao = ^= ^^y^=27raT (1.3.238)

for 0 = 77, tan at = = . ^, -, (1.3.239)

r-\-a rla-\-l

since X can be shown equal to 2TrrT. Note that 27TaT is equal to the

tangent of the original helix angle of the unbent yarn, hence no change

in helix angle takes place at the pseudo-neutral plane during bending

of a singles yarn. However, considerable change is seen to take place

at the top and bottom of the bend. This is quickly seen if we take the case of a yarn bending around itself, i.e., where r=2a, which simplifies
GEOMETRY OF BENT YARNS 193

the expressions for tan a to: A. at the bottom, \/2 at the middle (equal to the tangent of the unbent yarn helix angle) and A./3 for the top as re- flected in figure 1.3.69. Notice the heavy horizontal line drawn along the center of the yarn in figure 1.3.70a. The yarn is then bent concavely

upward (plane of curvature perpendicular to the paper) and the inside

of the bend is seen to experience a significant increase in helix angle as seen in figure 1.3.706. In figure 1.3.70c, one views the top of the yarn

bend, which evidences a reduction in helix angle. As a result of these

helix angle changes, the originally connected horizontal heavy line seg- ments of figure 1.3.70a are rotated, each around its center point, as shown

in figures 1.3.706 and 1.3.70c, and clearly the separation which is now

shown between the ends of these segments represents relative slippage

between fibers, 8s, calculated to be

Ss = 2p/(cot a - cot ao) (1.3.240)

where p/ is the fiber radius, ao is the original helix angle and a is the

local helix angle in the bent state. The drawings of figure 1.3.70 were

presented by Backer [80] and actual photographs v/ere furnished later

[112]. These changes in local helix angle take place every time a yarn is bent. And they are present in the singles yarns bent and twisted in the

ply configuration of a plied yarn, as shown in figure 1.3.71. Here for the case of a three-ply nylon yarn, the high helix angles at the inside of the

bend and the low helix angles at the outside of the bend are evidenced

by the low and high degree of fiber sectional ellipticity in these two

locations.

Figure 1.3.70. Change in local helix angle due to yarn bending.
194 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.71. Cross section of three-ply, 102-filament nylon yarn. Note that fibers at the center of the ply lie at a steep angle with the ply axis, as evidenced by their elliptical appearance. {Courtesy of E. R. Schwarz [13]). One of the more interesting parts of this study on bent yarn geometry

[80] was the analysis of local strains in the bent yarn. Starting with

eq (1.3.41) it was possible to show for Q = 7r/2 that dS=[{r- a cos \d) 2 + a2\2] 'l^dO (1.3.241)

If no slippage of the fiber is permitted to occur because of frictional constraint, then the local strain of the fiber lying in the angular increment

when the yarn is bent is dS — dSp

dS

(1.3.242)

where dS is the length of that fiber segment after bending and dSo is the length before bending. Noting that from the geometry of the unbent

yarn

ad(l) sin ao =

dSo

(1.3.243)

and for the bent yarn
GEOMETRY OF BENT YARNS 195

X = 27TrT = - tan (1.3.244) a where oto is the heUx angle of the given fiber in the unbent yarn. Thus

eq (1.3.242) becomes:

(r/a - cos (/))2 + X2

reflecting zero strain at (/) = 7r/2, the pseudo-neutral axis. Values of e/ are plotted against (/) in figure 1.3.72 as (/) varies from 0 to 180°. Symmetry

conditions apply in the range 180° < (/> < 360°. The parameter rja varies

from 1.5 to 2.5, the normal range of diameter variation encountered in ordinary woven textile structures. (But in ordinary ply structures the

efi'ective rja for high twist two-ply yarns with helix angles of 30° would

be of the order of 4.0 and even higher for lower ply twists. Thus fiber bending strains in straight plied yarns would be still lower than those

pictured in the right hand graph of fig. 1.3.72). In each of the three

diagrams of figure 1.3.72 the range of curve slopes from maximum to minimum correspond to A. = .5, .75, 1.00, and 1.25 respectively. Re- member that X for a given r/a is proportional to tan ao, the helix angle

of the unbent yarn. In short, the analysis suggests that for yarn bending in the absence

of fiber slippage, the fiber strains will be zero at the neutral plane and

will increase nonlinearly away from the neutral axis. If X = 0, i.e., the

original yarn twist is 0, then eq (1.3.245) becomes the expression for the

fiber strain, e/p, in a parallel bundle with no freedom of slippage.

.6

.4

LU<

/I r/a = 1 50 „y/y/ r/a=2.00 J

r/a = 2.50

el 1 1 1 1 1 1

1 1 1 1 1 1 1 1 \ 1 1 0 60 120 180 0 60 120 180 0 60 120 180 ANGLE ^ OF FIBER SEGMENT Figure 1.3.72. Local fiber strain caused by yarn bending, assuming no freedom of motion

between fibers.

In each of the three diagrams of figure 1.3.2, the range of curve slopes from maximum to minimum correspond to A.= .5, .75, 1.00, and 1.25, respectively.
196 TIRE CORD STRUCTURE AND PROPERTIES

r/g — cos (f) I — ^ 4> _c

r/a r r (1.3.246)

where c is the distance of a given fiber from the netural axis and r is the radius of curvature. This is, of course, the simple formula for bending

strain in an elastic prismatic beam bent to small curvatures. If A. = {rja) tan a is substituted in eq (1.3.245), then e/becomes:

€/= V(l-e/p)2 cos2 a+sin^ a:-l (1.3.247)

At the same time [80] it was shown that length of the fiber helix before

and after bending of the yarn is constant, which means that the average

strain of the fiber above the neutral axis just equals that of the fiber below the neutral axis, except for the sign, i.e., tensile above and compressive below. The yarn system will then attempt to cancel this fiber strain (and hence fiber stress) differential along a given fiber by slipping

material from the compressive region into the tensile region. This material interchange does not take place in the simple prismatic elastic beam — it occurs, however, in all twisted (nonbonded) structures, because

of the direct connection along the same fiber element in the two zones

strained in compression and tension respectively. If complete freedom

of motion exists, the slippage will be complete and the average strains

will subside, with only the bending strains within each fiber remaining.

If AS is the difference in path length in the outer and inner parts of the

bent yarn fiber loop, the following expression is obtained:

^ J(t> = TTl2 J<t> = 0 or AS^ 2ar 2aV\

2 "\(aU2 + r2)i/2 3(V^IX^T7^

then it can be easily demonstrated that the slippage of fiber from bottom

to top of the bend, past the neutral axis is half of eq (1.3.249) or A5/4.

The first term of eq (1.3.249) is plotted in figure 1.3.73 and it is seen that

this approximate value of A5/2 drops rapidly when A. increases. To

convert the value of AS/2 in figure 1.3.73 it is necessary to multiply by r, expressed in the same units as "a" and 'T". For fixed radius of bend, r, and a fixed twist, T, the slippage decreases as "a" becomes smaller and

vice versa. Slippage of this nature must be calculated in order to deter- mine the amount of work which will be expended due to friction during

bending of a twisted yarn. More will be given about this later. 1.3.4.2.' Bending Rigidity of Twisted Yarns (with no Friction)

Generally speaking, the work to bend a twisted structure will depend

on its material modulus £"/, its geometry, and, in the case of partial

slippage, the friction between its elements. Putting slippage friction

(1.3.248)

(1.3.249)
BENDING RIGIDITY OF TWISTED YARNS 197

2.6 24 w 2.2 i 2 .0 o

5 i.8 i

1.6 o_iy 1.2 z

°= 1.0

\

\\ y-r/o

//—r/o

-- 1.50

= 1.75 /A//— r/o =2.00 \ /O^^r/o = 2.25

. r /o = 2. 50

1 1 0.6 0.7 0.8 0.9 1.0 TWIST RATIO, X Figure 1.3.73. Difference in path length of upper and lower parts of the fiber loop indicated

as a function of r/a and k. Here r is taken as the unit of length, and AS/2 is expressed in terms of this unit. aside, the analysis of the contribution of geometry and material properties is somewhat more extensive than the prediction of tensile properties alone. Consider first the case of complete constraint, i.e., no

longitudinal slippage along the fiber axes. Assuming that plane sections

of the yarn remain plane during bending, it is noted in eq (1.3.246) that

€/p in a parallel bundle with no freedom of slippage is c/r= {a cos (/))/r. For the case of a twisted yarn of radius a, bent to the radius of curvature

r, the strain €/ in a fiber lying at an angle q, is

e/p cos^ q a cos (/) cos- q

(1.3.250)

and for (Tf = Ef€f, it foUows that the force dPy resulting from the stress CTy normal to the yarn axis acting on an element of yarn sectional area dAy{= ad(j)da) is: dP, J

. Era cos (b cos^ q , , , (TyClA y = aacpda

The corresponding element of bending moment is dM,

£/a cos^ 6 cos^ Q 1 i 1 cdPy = — dchda

r

(1.3.251)

(1.3.252)

which is first integrated for (f) or the limits 0 to 27r and then for a or the

limit 0 to fly, where ay is the radius of the yarn. Thus, Backer showed

that when the packing factor was unity, [105]

n/r _ E/n-a'^y i In SGC - Q—sinQ

(1.3.253)
198 TIRE CORD STRUCTURE AND PROPERTIES

r

1 In sec ^ ^—sin ^ Q

where Mc ~

2 tan4 Q

which for small Q becomes

Efly M,-^ (2cos2 Q) (1.3.254)

r Equation (1.3.253) above was independently derived by Piatt [14] to permit prediction of yarn bending rigidity from fiber properties and actual

numbers of fibers in the yarn Nf. Thus, rather than writing the Enal of

eq (1.3.253) above, he utilized the fiber cross-sectional area, Af, and the

number of fibers, Nf, so that Enat ^NfEfAfall^) (1.3.255)

where ayo was the cross-sectional area of the untwisted yarn (with no change in packing factor, 0, as a result of twisting). The expression for the ratio of the twisted yarn radius to the untwisted yarn radius (i.e., GylAyo) was shown to be [107] ^ ^ (1.3.256) 2(1 -cos Q)

An equivalent expression was derived independently [105]. Remembering

the definition of 0(= ^Af/Ay) , one can show

Enat ETr(f)al (1.3.257)

leading to the suggestion that </) be included in both eqs (1.3.253) and

(1.3.254).

The response of yarns with completes freedom of relative fiber motion was treated by Piatt [14] in considerable detail. If the fiber segment

located at position (/) and helix radius r is considered bent to a curvature Kf as a result of yarn bending, then the required bending moment vector

for such a fiber configuration would lie along the binormal, b, to the fiber

at the given segment. The component of this bending moment vector perpendicular to the plane of yarn bending represents the contribution

of the given fiber to resistance to yarn bending and is seen to be SMy^

where

bMyb = EflfKf cos q cos (1.3.258) or for the fibers in the incremental area dAy where

dAy = a- da - d(t) (1.3.259)
BENDING RIGIDITY OF TWISTED YARNS 199

and the number of fibers in the dAy is

dNf

Nf{adad(f>) cos q

ira (1.3.260)

yo where ayo is the radius of the yarn before twisting, with = constant during twisting. Whereupon dM^ per incremental area dAy is obtained

through the product of eqs (1.3.259) and (1.3.260). Kf may be obtained

from eq (1.3.92). But Piatt has elected to treat Kf vectorially, thus elimi- nating the need for the terms cos q cos (/> in eq (1.3.92). For in the cartesian

system i, J, k where k is the unit vector corresponding to the direction

in figure 1.3.69, the "yarn bending" component of the binormal fiber bending vector Kjb is simply its k component. Piatt thus obtains the

expression for Kjk:

Kf cos q cos (f) = Kfj = [(a cos (/) — r)^ + 0^X^(1 H-sin^ 0)

-arX2 cos (1.3.261)

where u = {a cos cf) — r)^ + a^X.'^; k = (r/a) tan Q. Then, combining eqs

(1.3.258-261), he integrates through (f) and a to obtain for the bending

rigidity of the yarn (with approximations)

(EI) ys,rn — ^byr (with freedom to slip

sec2 (?-3 sin2 (?+ (l-cos^ Q)

NfEflj j" 2 1n tan Q

which can be shown to equal:

(EI) free y; irn — ^fEflf (4M„ tan^^ + cos'*^) f-^V

\ CLyo /

(1.3.262)

(1.3.263)

Now, comparing eqs (1.3.253) and (1.3.263), we note that the bending

rigidity of the "no fiber slip" yarn is that of the no twist no slip yarn

corrected by the factor 4<Ma. The free fiber slip yarn has a bending

rigidity equal to that of a zero twist "free slip" yarn corrected by the

bracketed quantity in eq (1.3.263). It can be shown that with freedom of

fiber slippage

m EtI

free yarn Nf L (4M„ tan2 (J + cos^ Q) (1.3.264)

where ^/ is the collective (nontwisted) fiber cross-sectional area, and

is the twisted yarn cross-sectional area. Thus, it appears that the

rigidity of the free to slip twisted yarn is related to that of the "not free

to slip" untvvisted yarn:
200 TIRE CORD STRUCTURE AND PROPERTIES

(1.3.265)

ee For (? = 30°, AflAy = .lS, the ratio is about .6/A^/. If A^/=l, which means

there is no slippage, the reduction in stiffness is about 40 percent. If ^ = 30° and A(/=100, clearly the reduction in rigidity is influenced' by

the factor of 100 and the relative loss due to twist is small by comparison.

Piatt points out that if the yarn being bent has been "set" after twist- ing, then the curvatures incurred during yarn bending do not determine

local bending moments but, rather, they are determined by the changes

in fiber curvature from the straight yarn state to the bent yarn state. Now, if the yarn has been relaxed after twisting, there is a likely effect on the fiber modulus, therefore a new quantity Ef must be introduced to the above equations. Piatt has calculated the effect of yarn relaxation on bending rigidity of a singles yarn, leaving aside the modulus effect from setting, and concludes that "aside from its possible effect on

changing fiber modulus, relaxation of fiber bending stresses during yarn

twisting diminishes yarn bending rigidity. The effect is slight, ranging

from no effect at zero twist to 4.3 percent at a surface helix angle of 35°.

The effect of torsional moments (acting on individual fibers) on the

bending moment of the yarn has been treated by Piatt. For while the

k components of the fiber torques, Mt, are seen to cancel out in the

twisted, straight, yarn and additional fiber torque is cancelled out by

local fiber rotation in a "free to slip" bent yarn, yet unrelaxed torque

due to yarn twisting can have an effect on bending rigidity of that yarn.

Piatt has carried through an analysis similar to the above, this time

taking fiber torque components; and he concludes that for fiber tor- sional rigidity equal to or less than fiber bending rigidity, the effect of

fiber torque on bending rigidity is relatively small and since for most

drawn synthetics it is .2 to .3, it can be safely neglected in such cases.

Finally, Piatt has looked at the important question of "clustering" — defined as the joint action of groups of fibers acting thereby as single fibers. The two cases of complete freedom of motion, and complete restraint of slippage are, of course, extremes which do not occur often in the

application of fibers in textile systems. In most cases, there is some

slippage with frictional, but not total, restraint and such instances fall between the extremes discussed above. The clustering phenomenon

represents another case between the extremes wherein some fibers are

free to slip (without friction) and others are totally restrained from

slipping in groups or clusters. The effect of clustering is similar to using

fewer heavy (high denier) filaments as contrasted to more low denier

filaments in yarns of the same total denier. The analysis of bending

rigidities of yarns containing clustered fibers does require determination of bending rigidities of the individual clusters. And since the cross

section of the clusters is noncircular, and the sectional moment of in- ertia depends on cluster orientation, Piatt assumes a random distribution of orientations and calculates an average moment for insertion in
BENDING RIGIDITY OF TWISTED YARNS 201

the above equations. Clustering significantly effects the bending rigidity of a yarn and if x is the number of fibers in a cluster, then the bending

rigidity of the clustered yarn is approximately x times the rigidity for the "completely free to slip" yarn. If the entire yarn is clustered, i.e., X = Nf, then its bending rigidity is Nf times the bending rigidity of the

"completely free to slip" yarn. Geometric effects aside, this is shown

in eq (1.3.265).

Zorowski [106] took an alternate approach to the "cluster" condition

in the bending of continuous filament yarns. He assumed either complete fiber freedom to slip or complete restraint in various parts of a yarn cantilever. The "no slip" region is shown as cross-hatched in figure 1.3.74 and is seen to cover the entire cross section of the canti- lever for a length lo from its free end. In the region at a distance x from

the free end, where lo < x < /, there was an elliptical central core of no slip fibers, surrounded by free slip fibers. The minimum radius of the ellipse was R, and its maximum radius was Ro = Ry, the nominal

radius of the round yarn.

w luniTorm WT. OT cora; ^J/////A

0 U —

y Section AA

Figure 1.3.74. Proposed equivalent bending model of Zorowski [106]. For the cantilevered homogeneous isotropic beam bent under the

action of its own weight, one would expect an end deflection of 8

SEI

(1.3.266)

where w is the weight per unit length. This relationship is based on

small deflection theory but Zorowski points out that it is actually within

16 percent of agreement with large deflection theory for 3/1 ratios up

to 0.5. It would indicate that 8// should vary linearly with Actual tests by Zorowski on cantilevered 840 denier, two-ply yarns of different fiber deniers are reported in figure 1.3.75 which does indicate a linear rela- tionship for 8// up to values of 0.5 to 0.6. But, as seen in the insert of

figure 1.3.75, the curves do not go through the origin. Determining the

slope, m, from figure 1.3.75, where m = 1^18/1, one obtains the bending

rigidity (Eh) from:

_mw

(1.3.267)
202 TIRE CORD STRUCTURE AND PROPERTIES

0 0.25 0.50 0.75 1.0

Deflection Ratio - Figure 1.3.75. Characteristic results from bending tests. Now from the definition of the conditions of complete freedom and

complete restraint

Eh = NEf {EI) . = Ei^ (1.3.268)

and as was shown above

(EI)o = ^{EI)^ (1.3.265)

Using eqs (1.3.267) and (1.3.268), Zorowski reports that for selected

plied yarns (cords) the measured {EI)b were 5 to 20 times the zero

friction rigidity. However, the infinite friction rigidity is an order of

magnitude 30 to 200 times the measured value. And the infinite friction

rigidity was times the rigidity of zero friction case since both were

calculated from eq (1.3.268).
BENDING RIGIDITY OF TWISTED YARNS 203

Zorowski assumed that difference between zero friction fiber strain and infinite friction fiber strain which exists at the position x = lo will be

constant over the region /« < xl. The maximum strain for the friction case at /« is

, , yM Ry ( ^ \wll 2w ( II \

while the zero friction maximum strain is:

where Nj is the number of fibers. Rewriting eqs (1.3.269) and (1.3.270) for lo < X < I and letting the strain difference be constant at each

location, i.e., (6. - €/).. = (e. - ef)i^ (1.3.271)

Zorowski showed /? as a function of x: (!')=©{l-VNf+VNf (1.3.272)

which provides the equivalent dimensions of a nonhomogeneous propertied beam for which gravity deflections of a cantilever are to calculated. He then applied strain energy principles and the unit load method

of deflection analysis to determine the end deflection of the cantilever

beam of figure 1.3.74. The deflection 8, is then:

5 = lfw''^ (1.3.273)

where / = Nflo for 0 < % < /© = NfIo (^)'for lo<x<l

and the factor is the geometric factor developed by Piatt as described

above. Substituting eq (1.3.272) into eq (1.3.273) provided an expression

for 8 as a function of fiber properties and yarn geometry. Then calculating from eq (1.3.267) and from m^/^S,

(EI), =—^—. (1.3.274)

Using the calculated value of 8, Zorowski showed
204 TIRE CORD STRUCTURE AND PROPERTIES

where K was a function of /, lo and A^. Using eq (1.3,275), Zorowski compared theoretical deflection curves and those measured in the canti- lever test in figure 1.3.76. The agreement between experiment and

theory was not as good as might be desired, he concluded, but the

general characteristics of the observed behavior are reproduced and the

numerical magnitudes are not completely out of line.

0 .10 .20 .30 .40 .50

Deflection Ratio ^/a Figure 1.3.76. Comparison between predicted and experimental bending behavior.
BENDING RIGIDITY OF TWISTED YARNS 205

1.3.4.3. Bending Rigidity of Twisted Yarn (with Friction)

An early attempt to include the role of friction in the analysis of bending rigidity of twisted yarns was that of Mellen [107] et al. who con- sidered the case of a ring of cords helically wound around an inflated

cylinder. The cords were, in effect, the reinforcing elements required

to prevent excessive expansion during inflation of the cylinder. Since

the inflated cylinder was intended for incorporation into a fully pres- surized suit for space walking, it was necessary to minimize the change

in cylinder dimensions with inflation, and at the same time to reduce the

bending rigidity of the inflated cylinder.

If the reinforcing cords are considered to have a tensile modulus

infinitely greater than the modulus of the material of the cylinder, and

if they are wrapped equaUy in sets of right hand and left hand helices,

it can be shown that they will assume an equilibrium helix angle of 35°. The assumption is then made that the cylinder does not flatten during

bending, a reasonable condition for small curvatures, but one which

has been observed to fail at large curvatures. Now viewing the wrap reinforced bent cylinder in figure 1.3.77 we

consider two cases: (1) where cord stress develops with no cord slippage

along the cylinder surface, and (2) where the cord wrapping does slip, as indicated in figure 1.3.77, along its length. For the case of no slippage, we can show that the bending moment Mb for an applied curvature K=\lp, is: Mb = 2 \ JcriwRdk = 2 ] J{Ec€c cos'' Q) (R cos X)RdX

Jo Jo (1.3.276)

where 1/7 is the helical separation of the cords, Jcri is the longitudinal

stress along the cylinder axis provided by the tensile force ac in the cord,

ec is the tensile strain of the cord and p, w, Q, R and k are geometric

parameters as shown in figure 1.3.77. Since the inclined cord strain is related to the longitudinal strain by the factor cos- Q and the longitudinal strain is w/p or R cos X/p, eq (1.3.276) becomes

2JEcR^cos'Q r 2 2JEcR' cos' Q{7t/2) Mb = cos^ kdk =

P P

(1.3.277)

It was reasoned that smce the local strain at the neutral plane is zero from eq (1.3.245) or from eq. (1.3.247), the stress there would also be

zero, while it would be maximum at k (or 0) =0 or .t (fig. 1.3.72). Thus,

if no slippage occurred, the local stress of an Hookean cord would be

pictured in figure 1.3.78, with ctc as a function of /, (as per fig. 1.3.77, or s in fig. 1.3.2). To prevent slippage, dcr/dl must not exceed p.A^ where

fji is the coefficient of friction, and is the normal force between cord

and cylinder per unit length of cord. Since fxN can be assumed constant, one can expect that

(Tc = ixNl = Ec€c (1.3.278)
206 TIRE CORD STRUCTURE AND PROPERTIES

Figure 1.3.77. Geometry of wrap-reinforced bent cylinder. Since / in figure 1.3.77, or s in figure 1.3.2 is a function of A., as shown

in figure 1.3.78, or of figure 1.3.2, one can use eq (1.3.278) to plot a "slippage allowable" €c versus (f) in figure 1.3.78, and for small curvatures this will be a straight line passing through the origin. Thus, as the

cylinder is first bent, the Ciocai versus (f) curve will be dictated by the no

slip condition as in figure 1.3.78. But as the curvature, 1/p, increases,

the no slip curve rotates counterclockwise until it contacts the "slip

allowable" curve. Since the slope of the "no slip" curve cannot exceed

that of the "slip allowable" curve, then the change in the actual Ciocai versus 0 curve can be thought of as a "freezing" or flattening of the

"no slip" curve against the straight "slip allowable" curve. Wherever

the flattening or freezing takes place, local slippage will be present.

And from the shapes of the curves of figure 1.3.78, it will be expected

that slippage will first take place at the neutral axis at X = 7r/2, then

propagate both upward and downward to values of kd, or 0 = 0, tt re- spectivelv. Note this is the X of figure 1.3.78.

If now, / is zero at X = 7r/2, it follows for low curvatures that / can be

taken as (A. — 7r/2)/?/sin Q and it can be shown from eqs (1.3.276) and

(1.3.277) that the bending moment incurred for an applied curvature

of 1/p is: Mt= I 2J[fjLN{k - 7r/2)/?/sin Q] cos^ Q{R cos k)Rdk (1.3.279)

•'0 = JfJLNR^cos^QlsmQ
BENDING RIGIDITY OF TWISTED YARNS 207

which, surprisingly, shows to be independent of Ec, the cord modulus, and even independent of p, the radius of curvature of the cyUnder.

In short, for the conditions assumed, the versus K curve will be flat and at a level independent of Ec\ rather, its level will be dependent on

the cord frequency, J, the cube of the cylinder radius, and the product

of coefficient of friction fx and normal pressure A^. Note that the K or

1/p region considered here was the one in which slippage was propa- gated throughout the structure. At smaller X's we would expect partial

slippage and partial restraint as shown in figure 1.3.78. At still smaller K's we would expect complete restraint. There was no effort made in

this early study to consider the very lower curvature cases, nor to consider the contribution of cylinder tensile-compressive modulus to the

bending moments at "all slip" curvatures, nor the contributions of the

bending rigidities of the slipping cord to overall assembly rigidity in the "all slip" curvature region. (Clearly, the contribution of the cylinder

modulus and of the EI of the cords will be dependent on K, hence the

total Ml) versus K curve would no longer be flat in the all slip region.
208 TIRE CORD STRUCTURE AND PROPERTIES

but this is hindsight based on results of subsequent multilayer beam

bending studies.)

Treatment of a multifilament yarn in bending calls for a more realistic model than that of figure 1.3.77 and the obvious requirement is for the

introduction of multiple helical layers of fibers, with slippage between

layers or cylindrical rings during bending. But since the direction of

slippage must be in opposite directions, at the top and bottom of each

layer of fibers, it follows that eq (1.3.278) must be expressed in terms of

8N, the normal pressure differential across each fiber. As is seen from

eq (1.3.123) or atr is larger at the inside surface of each ring than the

outside surface, when the yarn is under some tension during bending.

Hence from eq (1.3.123) one can express 8N as a function of radial position r and of fiber diameter 8r and integrate dMb once again as in eq

(1.3.279) to provide the frictional bending moment in a twisted yarn when

bent under tension as, for example, in the case of a cord running over a

pulley, or the bending of a tire cord in an inflated tire. The form of the

expression for the predicted frictional moment obtained by this stress analysis method agrees with the moment predicted by the work methods

to be discussed below, specifically with eq (1.3.288). It is interesting to note that the "work" solution for bending behavior of the multilayered

twisted structure with friction actually preceded the stress analysis

solution for bending of a multifilament yarn while in the case of tensile

properties the stress analysis treatments preceded the energy treatments. Using as a model yarn a series of straight circular filaments stacked

in layers so as to form a rectangular cross-sectional bundle, it has been

shown [110] that the local slippage, S, between layers pressed together

by a pressure p (per unit fiber length) is a source of work loss, JF/, during

bending, i.e..

where ff is the friction per unit fiber length constant throughout and

equal to jJipdf (p the lateral pressure being constant throughout the

model), w is the layer width, df the fiber diameter, and x the length from

the fiber center. The local slippage, S, between fibers for the model

shown in figure 1.3.79 is shown to be

where A:, the layer curvature, is constant throughout the model. Substituting eq (1.3.281) into eq (1.3.280) and integrating, we have the

frictional work loss at a single interface

(1.3.280)

S = dfkx (1.3.281)

ffWkl^

4

(1.3.282)

From beam theory it is shown that the work to bend one layer to a curvature k (not considering friction, but only elastic forces) is
Figure 1.3.79. (a) Simplified model of yarn bending, (b) Slippage between layers offibers.

We

Eirdjwlk^

128

(1.3.283)

since /layer equals Ifw/df, i.e., rrdjwl^^. The sum of Wf and We is the

total work Wt to bend each layer to curvature k. From eq (1.3.282) it is ^een that Wf is linear in /c, i.e., dWf/dk is constant. Now, if Mf is the

moment required to do the frictional work,

dWf=MfdO = Mf{l)dk

fwl

(1.3.284)

hence Mj is constant and equal to —j— . From eq (1.3.283) We is seen

linear in k'^ hence Me is directly proportional to k, as one would expect

for elastic behavior. Thus, the total bending moment Mt can be shown as: dffwl EjrdfWk

MT = Mf+Me = ^^-^

4 J64

(1.3.285)

where / is the frictional shear stress between fiber layers {=dfff). If V

is the total vertical force spread uniformly between fiber layers, eq
210 TIRE CORD STRUCTURE AND PROPERTIES

(1.3.285) can be expressed as: Mf=^ Vfidf (1.3.286)

as was shown independently by Grosberg [108] and by Popper [109]. From eq (1.3.285), one anticipates a bending moment versus curvature curve as shown in figure 1.3.80. The recovery portion of the curve results from the change in sign of the second term of eq (1.3.285) during unloading. Clearly, the final recovery (at zero moment) from the applied curvature will be reduced as the frictional moment increases relative to the

elastic rigidity of the layered system.

Popper [109] takes special pains to analyze the initiation of slippage

in a layered beam, showing that it must start at the center of the beam

at a local region where vertical shear force and bending moment prevail,

and then it wiU propagate longitudinally into the region of pure bending

moment. Grosberg [108] and Popper [109] independently analyze the

forces required to maintain yarns in the crimped configuration of a woven

fabric, considering that the total normal force between warp and filling

yarns is equal to the sum of the normal forces necessary to bend each of

the fibers comprising the yarn. An important element of the analysis

has to do with the fact that the pressure between fibers in each yarn

set is maximum at the fabric center, but it decreases linearly with each

fiber layer (say in a rectangular cross section yarn) until the last interface

where the interfacial normal pressure is simply enough to keep the

outside fiber layer bent. Grosberg [108] makes the important point that the force between yarn sets in a fabric should not be calculated on the

CURVATURE

Figure 1.3.80. Effect of sliding friction on moment-curvature relation.
BENDING RIGIDITY OF TWISTED YARNS 211

basis of the in situ curvatures of the fibers in the fabric, but rather on

the difference in curvatures between the fibers as they He in the yarn in the fabric and the same fibers as they He in the yarn removed from the

fabric. This use of curvature change serves to account for yarn bending

relaxation after weaving or that due to setting treatments. Popper [109, 111] then shows that when a twisted yarn, with a constant normal force between fibers, is bent, the friction moment, M/, is

where rii is the normal force per length at each fiber interface, Nj is the

number of fiber interfaces, and Lt/2 is the length of fiber in half a turn

of twist. Note that Lt/2 corresponds to the characteristic length / in the

Mf term of eq (1.3.282). The similarity of the expressions is obvious.

Popper [111] treated the case of bending a yarn subjected to axial tension.

The pressure distribution between fibers was taken according to Machida

[55] (eq (1.3.123)). This pressure distribution was then translated into a normal force per unit length on each fiber interface and this was combined with the relative fiber motions to give the work loss. The essential

difference between this analysis and the cases for the layered beams or square fiber area with constant pressure between layers is that in this case the work loss varies with radial position. For this reason, the total frictional work loss involves summing the work losses throughout the

structure after correcting for the differences in number of interfaces

at each radial position. The resulting frictional moment is approximated

by Popper [109] as

where (j)y is the yarn packing factor and P is the axial tension on the yarn. And it is interesting to note that eq (1.3.288) can be obtained from

eq (1.3.277), taking into account the indicated pressure distribution result- ing from the axial tension P and calculating the tension build up along

each yarn due to the incremental friction. This incremental friction fol- lows from the incremental resultant of normal pressure forces at one

sliding surface of each fiber. In terms of predicting the bending moment resulting from bending

a yarn to a known curvature, the Popper method of determining eq

(1.3.288) is more direct, but as is the case with energy, and work methods, it loses some of the detail of the stress analysis method. In particular, if we wish to calculate the local force which builds up along a

fiber in a yarn, due to slippage friction (with a known pressure dis- tribution from layer to layer) we can do so simply by using eq (1.3.278).

And this knowledge will permit a determination of the local fiber stress and strain level, which when compared with tensile (or if available,

compressive) recovery data for the fiber, will permit calculation of yarn bending recovery taking elastic behavior, friction, and also viscoelastic behavior into account. Of equal importance, the knowledge of

Mf=^ IxniNidf^ (1.3.287)

Mf=micf)yfjiPT (1.3.288)
212 TIRE CORD STRUCTURE AND PROPERTIES

local fiber stress and strain levels will permit the use of fatigue data

based on straight fibers and low twist yarns to predict fiber fatigue

incurred during the bending of highly twisted structures. Clearly, much

more work remains to be done in this problem area. As a final comment on the mechanics of bending fiber assemblies, we present in figure 1.3.81 a diagrammatic summary of the different

strain distributions which can occur in the cases of: (1) complete freedom

to slip; (2) no freedom to slip; (3) frictional resistance to slippage. While numerous experimental data have been reported on the substance of these analyses, we include but one set by Popper [111], (fig. 1.3.82)

showing a comparison between theoretical and measured deflection curves for a layered beam with constant pressure between all layers.

Here the stiffness, i.e., the vertical load V per unit deflection, is plotted

against the position of loading y along the beam. The value of y is 0 at the free end of the multilayered cantilever, whose layers slip with constant friction.

(3b) Figure 1.3.81. Diagrammatic summary of different possible strain distributions.

1. Complete freedom to slip.

2. No freedom to slip.

3. (a) Constant pressure at all interfaces.

(b) Constant pressure change across each layer.

I
REFERENCES 213

X

Figure 1.3.82. Multi-layer beam stiffness vs. position of load.

References

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214 TIRE CORD STRUCTURE AND PROPERTIES

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[39] Bunting, W. W., Compact Interlaced Yarn, U.S. Patent 2,985,995, May 30, 1961.

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[42] Hearle, J. W. S., and Merchant, V. B., Interchange of position among components

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aspects of stress analysis of textile structures — continuous filament yarns, Textile Research Journal 20, 1-15 (1950).

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(1961).

[48] Galileo, Galilei, Dialogues Concerning Two New Sciences, Leyden (1638) trans- lated by A. De Salvio and A. Fabaro, Evanston, Illinois, (1914).

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of textile materials. Part IX: Factors affecting translation of certain mechanical

properties of cordage fibers into cordage yarns, Textile Research Journal 22,

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[51] Backer, S., and Petterson, D. R., Some principles of nonwoven fabrics, Textile Research Journal 30, 704-711 (I960).'

[52] Petterson, D. R.. and Backer, S., Mechanics of nonwovens: orthotropic behavior.

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[53] Gurney, H. P., Distribution of stresses in cotton products. Journal of the Textile

Institute 16, 269-289 (1925).

[54] Sullivan, R. R., A theoretical approach to the problem of yarn strength, Journal of Apphed Physics 13, 157-167 (1942).

[55] Machida, K., Mechanics of Rupture of Blended Yarns, M.S. Thesis, M.I.T. Cambridge. Massachusetts, 1963.

[56] Grosberg, P., and Smith, P. A., The strength of slivers of relatively low twist. Journal

of the Textile Institute 57, 15-23 (1966).

[57] Petterson, D. R., On the Mechanics of Non Woven Fabrics, Sc.D. Thesis, M.I.T.

Cambridge, Massachusetts, 1958.

[58] Hearle, J. W. S., and El-Behery, H. M. A., The transmission of transverse stresses

in fiber assemblies. Journal of the Textile Institute 51, 164-171 (1960).

[59] Treloar, L. R. G., and Riding, G., A theory of the stress strain properties of con- tinuous filament yarns. Journal of the Textile Institute 54, 156-170 (1963).

[60] Den Hartog. J. P., Advanced Strength of Materials, pp. 212-234 (McGraw-Hill

Book Company, New York, 1952).

[61] Treloar, L. R. G., and Hearle, J. W. S., Mechanics of twisted yarns, a correction. Journal of the Textile Institute 53, T446—T448 (1962).

[62] Kenny, P., and Chaikin, M., Stress strain time relationships of non-uniform textile materials. Journal of the Textile Institute 50, 18-40 (1959).

[63] Piatt, M. M., Klein, W. G., and Hamburger, W. J., Mechanics of elastic performance

of textile materials. Part XI: Factors affecting the translation of certain mechanical

properties of cordage fibers and yarns into cordage strands and ropes. Textile Research Journal 24, 907-939 (1954).

[64] Daniels, H. E., The statistical theory of the strength of bundles of thread. Proceedings of the Royal Society (London) 183A, 405 (1945).

[65] Rosen, B. W., Tensile Failure of Fibrous Composites, J.A.I.A.A. 2, 1985 (1964).

[66] Scop, P. M., and Argon, A. S., Statistical theory of strength of laminated composites.

Journal of Composite Materials, Part 1 1, 92-99 (1967). Part II 3, 30-47 (1969).

[67] Oravas, G. A., and McLean, L., Historical development of energetical principles in elastomechanics. Part II: From CotteriU to Prange, Applied Mechanics Reviews

19, 919-933 (1966).

[68] Riding, G., and Wilson, N., The stress strain properties of continuous filament

yarns. Journal of the Textile Institute 56, 205-214 (1965).

[69] Backer. S., Compressional behavior of textile fibers, Textile Research Journal 30,

405-407 (1960).

[70] Miles, J. B., Compressional behavior of textile fibers. Textile Research Journal 30,

408-409 (1960).

[71] Elder, H. M., The tensile, compressive, and bending moduli of some monofilament

materials. Journal of the Textile Institute 56, 8-14 (1965).
216 TIRE CORD STRUCTURE AND PROPERTIES

[72] Higuchi, K., and Takai, H., Stress strain diagram, Young's modulus and Poisson's

ratio of textile fibers, Journal of the Textile Machinery Society of Japan, 7, 4-12

(1961).

[73] Frank, F. I., and Ruoff, A, L., A method of measuring Poisson's ratio of fibers. Textile Research Journal 28, 213-217 (1958).

[74] Backer, S., Monego, C. J., and Machida, K., Mechanisms of tensile rupture in blended

yarns. Journal of the Textile Institute (in press).

[75] Paul, W., Untersuchungen iiber das Kraft-Dehnungsverhalten dreilitzger Seile aus

synthetischen Fasern, Dissertation (Doktor—^Ingenieur), Technischen Hochschule,

Hannover, 1967.

[76] Symes, W. S., The prediction of cord properties, Journal of the Textile Institute 50, 241-248 (1959).

[77] Treloar, L. R. G., The stress-strain properties of multi-ply cords. Part I: Theory,

Journal of the Textile Institute 56, 477-488 (1965).

[78] Wood, J, O., and Redmond, G. B., Tyre cord behaviour under compressive stress. Journal of the Textile Institute 56, 191-204 (1965).

[79] Clark, S. K., The plane elastic characteristics of cord rubber laminates. Textile Research Journal 33, 295-313 (1963).

[80] Backer, S., The mechanics of bent yarns. Textile Research Journal 22, 668-681

(1952).

[81] Olofsson, B., The rheology of textile fabrics. Chapter 9 of Rheology Vol. 5, Edited by

Eirich, F. R. (Academic Press, New York and London, 1969).

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[83] Sadowsky, M. A., Pu, S. L., and Hussain, M. A., Buckling of microfibers, ASME

Paper 67-WA/APM-4 (American Society of Mechanical Engineers, New York

City, 1967).

[84] Haringx, J. A., On Highly Compressible Helical Springs and Rubber Rods and Their

Application for Vibration Free Mountings, Doctoral Thesis, Technical University

of Delft, Netherlands, 1947.

[85] Amirbayat, J., and Hearle, J. W. S., Properties of unit omposites, Fiber and Science Technology 2, 123-154(1969).

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of Mechanical Engineering, M.I.T., Cambridge, Massachusetts, 1951.

[87] Gurney, H. P., and Davis, E. H., The elastic properties of yarns with special refer- ence to tire cords. Journal of the Textile Institute 21,463-480 (1930).

[88] Coulson, A. F. W., and Dakin, G., Doubled yarns. Parts I to V, Journal of the Textile

Institute 48, 203-232, 233-257, 258-292, 293-320, 321-332 (1957).

[89] Takeyama, T., and Matsui, J., Recent developments with tire cords and cord to rub- ber bonding. Rubber Reviews, Division of Rubber Chemistry of the American

Chemical Society 42(1), 159-256 (1969).

[90] Riding, G., The stress strain properties of multiple cords. Part II: Experimental,

Journal of the Textile Institute 56, 489-497 (1965).

[91] Riding, G., The effect of yarn twist on filament properties, Journal of the Textile

Institute 56, 109-120 (1965).

[92] Krizik, J. G., Mellen, D. M., and Backer, S., Dynamic Testing of Small Textile Structures and Assemblies, Final Report, QMC R and D Center, Natick, Massachusetts,

TD-93-1961.

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[95] Hall, I.'h., Journal of Applied Polymer Science 8, 237 (1964).

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[99] Czitary, E., On the bending strains of wire cables, Asterreichisches Ingenieur Archiv 1,342-350 (1947). [100] Wyss, Th., Einfluss der sekundaren Biegung und der inneren Pressungen auf die
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Lebenselaner von Stahldraht-Litzenseilen mit Hanfseele, Schweiz-Bauzeitung

Nr. 14, 15, and 16 (1949). [101] Lutchansky, M., Axial stresses in armor wires of bent submarine cables, ASME Paper

No. 68-WA/DE-l (ASME Headquarters, New York City, 1968).

[102] Tovey, H., Wrinkle resistance and recovery from deformation. Textile Research

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at low strains. Textile Research Journal 22, 695 (1952). [104] Piatt, M, M., Klein, W. G., and Hamburger, W. J., Mechanics of elastic performance

of textile materials. Part XHI: Torque development in yarn systems: Singles Yarn,

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Varying Conditions of Stress, Sc. D. Thesis, Department of Mechanical Engineering, M.I.T., Cambridge, Massachusetts, 1953.

[106] Zorowski, C. F., and Chen, C. S., Cantilever bending behavior of continuous filament

cords. Textile Research Journal 35, 529-538 (1965). [107] MeUen, O. M., Backer, S., and Park, B. J., Research on Materials for use in a Full Pressure Suit, WADD Technical Report 60-311, M.I.T., Cambridge, Massachusetts, 1960.

[108] Grosberg, P., Mechanical properties of woven fabrics. Part II: Bending of woven

fabrics. Textile Research Journal 36, 205-211 (1966). [109] Popper, P. G., Bending of Fiber Assemblies, Report TD-153-66, Fibers and Polymers

Division, M.I.T., Cambridge, Massachusetts, 1966. [U.S.D.A. Grant 12-14-100-

7650 (72)]. [110] Popper, P. G., The Effect of Friction on the Bending Behavior of Textile Materials, Report TD-106-62, Fibers and Polymers Division, M.I.T., Cambridge, Massachusetts, 1962 (unpublished). [Monsanto Company Project.]

[Ill] Popper, P. G., Bending of Fiber Assemblies, Sc.D. Thesis, Mechanical Engineering

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twill direction as related to fabric structure, Textile Research Journal 26, 87-107

(1956).

1.4. Recent Developments with Tire Cords and

Cord-to-Rubber Bonding

T. Takeyama ^ and J. Matsui ^ Page 1.4.1. Physical properties of tire cords 220

1.4.1.1. Introduction 220

1.4.1.2. High tenacity rayons 221

1.4.1.3. High tenacity nylons 225

1.4.1.4. Polyester cord 238

1.4.1.5. Fiber glass cord 241

1.4.1.6. Steel wire cord 243

1.4.1.7. Miscellaneous cords 244

1.4vl.8. Comparative analysis of various tire cords 244

1.4.1.9. Impact resistance of tire cords 246

1.4.1.10. Fatigue resistance of tire cords 253

1.4.1.11. Effects of twist on cord properties 261

1.4.2. Rubber-to-cord bonding 265

1.4.2.1. Introduction 265

1.4.2.2. Outline of bonding methods 266

1.4.2.3. Adhesive treatment of nylon and rayon — RFL

treatment 274

1.4.2.4. Polyester to rubber bonding 279

1.4.2.5. Adhesive treatment of miscellaneous tire cords. . . 290

1.4.2.6. Evaluation of adhesion 291

References 299

Industrial Products Laboratory, Fibers and Textile Research Laboratories, Toray Industries, Inc., Otsu, Japan. 219
220 TIRE CORD ADHESION

1.4.1. Physical Properties of Tire Cords

1.4,1.1. Introduction

The first pneumatic tire was made in 1888 by J. B. Dunlop with Irish

flax as the reinforcing material. This fiber was one of the strongest at that time. It was gradually replaced by cotton, since Irish flax was expensive.

Cotton tire fabrics in the early stages, about 1910, were plain fabrics. As requirements imposed by the severity of tire operating conditions

increased, these fabrics were gradually replaced by the present tire cord

fabrics, which were first devised by J. F. Palmer in 1892. Prior to World War II, cotton was the sole textile used to any large extent in pneumatic tires. However, this cotton tire cord fabric also

failed to meet requirements imposed by increasing severity of tire operations, so that tire engineers began to consider man-made fibers. The first rayon tire cord tenacity was about two grams per denier, and was produced in 1923. Du Pont started to manufacture high tenacity rayon, Cordura, in 1933 [1] ^ and Courtaulds also started to manufacture

high tenacity rayon, Tenasco, in 1936 [1], but by 1940 high tenacity

rayons had only a small portion of the total tire cord market. In the late 1940's, use of high tenacity rayon tire cords increased rapidly in the

United States and Europe. In Japan, rayon truck tires were first manufactured in 1951 [2-3]. Initially rayon cords were used only for truck

tires. The rayon cord tire had improved carcass performance and its life was increased 30 to 60 percent [2-3]. This improvement of tire quality was utilized more for truck tires than for passenger car tires. Increased

power of automobile vehicles, however, gave birth to troubles in rayon

cord (truck) tires. New, tougher materials were required for tire cords,

especially for heavy duty tires. In 1947, nylon cords (nylon 66) were examined as reinforcing materials

for truck tires in the United States, and it was confirmed that nylon truck

tires have excellent properties, especially under severe operating conditions. In the late 1950's, rayon cords were gradually replaced by nylon

cords, especially in the United States. Japanese tire manufacturers made extensive efforts to use nylon 6 for

tire cords. Nylon 6 was available from domestic suppliers (nylon 66 was more costly) and nylon 6 tires, when post cure inflation was used, showed

no difference from nylon 66 tires in practice [2], [3]. Mass production of nylon tires in Japan started in 1958. In the first five years, 60 percent

^ Figures in brackets indicate the literature references at the end of this chapter.
HIGH TENACITY RAYONS 221

of all tire cords had been replaced by nylon cords. In 1967 nylon cords were used for 90 percent of the tires made in Japan.

In Europe, rayon cord still has a large portion of the tire cord market.

Goodyear started to produce polyester tires for passenger car service

in 1962. In 1969, production of polyester tires has rapidly increased in the United States, and polyester cord has become very important in the field of the original equipment tires for passenger cars. Mass production of polyester tires in Japan started in 1967 and their production is gradually increasing. In the United States, most polyester cords are used in fiber glass

belted-bias construction tires, whereas in Japan they are used as rein- forcing elements for bias construction tires. Steel cords, which made their appearance in France in 1936, have

been used extensively in Europe in radial tires. Glass fiber cords, one of the most promising types for the belt ply of

radial tires or belted-bias tires, recently appeared in tires in the United

States. 1.4.1.2. High Tenacity Rayons

For more than thirty years, high tenacity rayons have been used as tire cords. The tenacity of the first rayon tire cord in 1923 was only two grams

per denier as previously mentioned. In the 1930's, commercial production

of high tenacity rayons, Cordura and Tenasco started in the United States and in England, respectively. Tenacity of these cords was about 2.3 grams per denier.

Thereafter, tenacity and durability of rayon cords was improved and

super series rayons, e.g. Super I, Super II and Super III were developed.

More recently, newer types of rayon cords, compressed rayon Dynacor

[4-6] and extra high modulus rayon were developed [7].

The improvement of cord tenacity of rayons is illustrated in figure

1.4.1 [8]. Cord tenacity of rayon now has reached five grams per denier

or higher.

6

5 Q

O

3

1950 1955 1960 1965 1970 Figure 1.4.1. Evolution of rayon tire cord strength.
222 TIRE CORD ADHESION

In spite of nylon's and polyester's assault, rayon cords still keep their

position in the original equipment market for passenger car tires in the

United States and Europe. In Japan, however, the future of rayon tire cords is not promising.

The production process for high tenacity rayons differs in several re- spects from that for regular rayons. High tenacity rayons have to be

made denser and more uniform in fine structure than regular rayons. High quality wood pulp, and high quality, homogeneous viscose solu- tion are usually used in the production of high tenacity rayons. The degree of polymerization of cellulose in the production of high

tenacity rayons ranges from 400 to 600, which is considerably greater

than that of regular rayon, 300. A higher concentration of carbon disulfide and sodium hydroxide, and lower concentration of cellulose are employed to improve the solubility of viscose.

The high yarn strength can be obtained upon application of stretch at high temperature.

Cox [9] first found that some additives (retardants) in the spinning

bath were effective for this purpose. These modifications by retardants

became very important in the production of super series rayons. These

retardants have the effect of retarding the regeneration of the cellulose from the xanthate and of increasing the tenacity of the fiber [1]. Fila- ment cross section obtained by this method is circular and more uni- form, as demonstrated by absence of skin and core regions, which can be

seen in cross sections for earlier high tenacity rayons. Many reports were published on the effects of retardants on regeneration and coagulation mechanisms. It was explained that these retardants react with

zinc ions to form stable colloidal chelate compounds, suppress diffusion of the coagulant and then slow down the rate of regeneration.

Yarn-to-cord tenacity conversion efficiency of rayon cords is quite low. Therefore, choice of cord lubricant is important. A lubricant, which

decreases the friction coefficient of the yarn and improves the yarn-tocord tenacity conversion, has to be developed [10-11], see table 1.4.1 [10]. Detailed investigations of fine structure of super high tenacity rayon

filaments show that while overall change in total crystallinity is very

Table 1.4.1. Correlation of cord strength with yarn friction coefficient [10]

Specimen

(1650/2 rayon cords)

Yarn friction coefficient Bone-dry

strength per 100 denier

Static Dynamic

A 0.207 0.197 263 B .200 .184 266 C .160 .150 272 D .132 .158 280 E .107 .144 296 F .118 .133 306
HIGH TENACITY RAYONS 223

small, the average crystallite size has been reduced. Thus, a fine, even

textured filament permits the load to be distributed more evenly across

its structural elements.

Internal structure of a rayon filament is simply expressed in terms of

lateral order distribution [12-15] which can be measured by accessibility of the cellulose internal structure to various chemicals, e.g., esterifica- tion by formic acid, dissolution by alkali solutions and so on. Skin or less-ordered regions can be dissolved by alkaline solution more easily

than core or ordered regions.

The change of lateral order distribution in super series rayon filaments

is illustrated in figure 1.4.2 [16]. The improvement of super series rayons

explained by change of the lateral order distribution to a low ordered, narrow distribution.

Figure 1.4.2. Change of lateral order distribution of super series rayons. Dynacor rayon {compressed rayon [3-4]): Dynacor rayon, manufactured by all Tyrex members, has the same chemical composition as conventional rayon tire cord, but differs in physical characteristics and

in the way it is processed. It is made of low modulus, high tenacity and

twisted greige yarn dipped at minimum tension in a standard RFL (Resorcinol Formaldehyde Latex) adhesive and immediately stretched 15

percent prior to drying.

This processing technique increases dip penetration into the cord. Stretching aligns the filaments and compacts the bundle. Physical

locking-in of the dip and uniform stress distribution among the several

thousand individual filaments in the cord are obtained. In addition, it is said that longitudinal air wicking, considered by many tire experts to be

a major cause of ply separation, is reduced to a minimum and cord-to- rubber bonding made more durable. Equally important, this cord is claimed to cost no more than regular rayon and requires little or no modification of processing equipment.

Performance results together with indoor wheel tests [3] are claimed

to indicate that this rayon cord has greatly improved resistance to ply

separation, and that in the accelerated test for tread separation, Dynacor

tires run three to four times longer than regular rayons, with separations

generally occuring at a rubber to rubber interface.

SUPER 4

ORDER
224 TIRE CORD ADHESION

Extra high modulus rayon [5]; Extra high modulus rayon is a polynosic type of filament yarn. Extra high modulus rayon yarn has a tenacity

of more than 9 g. per denier bone dry and approximately 8.5 g. per denier

in the conditioned state. Elongation at break, 4 to 5 percent bone dry and

5 to 6 percent conditioned, is extremely low, while wet strength reaches

7 g. per denier and wet elongation at break only 6 percent.

Young's modulus is very high, see figure 1.4.3 [7], and its temperature

dependence is low. This indicates dimensional stability for extra high

modulus rayon in tire construction, particularly for radial ply tires. Effects of twist on cord strength of Super III rayon and extra high modulus rayon are indicated in figure 1.4.4a [7]. The curve for Super III indi- cates increase in strength to 22 kg. in the twist range of 20 to 40 turns per

10 cm., followed by a downward trend because of increasing conversion

losses. With low-extension, extra high modulus rayon cord, the curve

starts out at 26 kg. and then shows a steady downward trend over the

entire range represented to reach 15 kg. In one range of twist extra high

modulus rayon cord has a greater breaking load than Super III, while in the higher range the breaking load of Super III is superior.

Results of the Firestone flex fatigue test are illustrated in figure 1.4.46 [7]. The shape of the curve shows that higher cord twist can in- crease fatigue resistance. A twist level of approximately 47 turns per

10 cm. gives an optimum balance between pre- and post-flexing strength.

STEEL

ELONGATfON ( %

)

Figure 1.4.3. Comparison of stress-strain curves of extra high modulus rayon and other

cords.
HIGH TENACITY NYLONS 225

30

20

o

UJ

a: 20

10

Jehm

"si ^•-^

BEFORE -LEX (a)

S

AFTER 360p00aa^•A

/ /

/

/ ^ / EHM (b)

20 30 40 50 60

TWfST (TURNS/IOCM)

Figure 1.4.4. Effect of twist on cord tensile strength and fatigue resistance of Super III rayon and extra high modulus rayon [7].

(a) Tensile strength vs. twist.

(b) Residual tensile strength after flexing with Firestone Flex Tester vs. twist. The curve for extra high modulus rayon cord shows great similarity to that for Super III, except that retained strength at all twist levels is 5 kg.

lower. In the commercial twist range, breaking loads both before and

after the flex test are lower with extra high modulus rayon than with

Super III. Use of this low-extension rayon cord for conventional tire construction seems out of the question. On the other hand, its high

strength and low elongation may make extra high modulus rayon cord a choice for reinforcing belts in radial ply tires, where it is not subjected

to such severe flex compression.

1.4.1.3. High Tenacity Nylons

Nylon cord is the strongest tire cord. In the standard twist construction 840D/2, 47X47 (turns per 10 cm.), the cord strength of current nylon

6 and nylon 66 are 8 g. per denier and 7.5 g. per denier, respectively.

As compared with rayon cords, nylon cords give much longer tire

life, particularly when used in heavy duty truck tires. For this reason, nylon cord was initially applied in heavy duty truck tires and then was

gradually adopted for light truck tires and replacement tires for passenger cars. Up to now, nylon tires are the largest fraction of automobile
226 TIRE CORD ADHESION

tires, especially in Japan and the United States. Almost all nylon cords

used in Japan are nylon 6. On the other hand, nylon 66 cords have a greater portion of the nylon cord market in the United States. Nylon 6

cord differs slightly from nylon 66 cord in regard to tire cord characteristics. Nylon 6 cord has greater tenacity, is more economical and more

shock resistant than nylon 66 cord. Nylon 6 cord balances so well with carcass compounds that dynamic adhesion fatigue is highly improved

[2]. Furthermore, Allied Chemical stated their nylon 6 tire cord, Caprolan, has better heat aging resistance, adhesion to rubber, and flex fatigue

resistance than nylon 66 [1]. On the other hand, nylon 66 has better thermal stability than nylon 6

due to polymer melting point difference. Thermal shrinkage at fairly

high temperatures such as tire curing temperatures is greater than for nylon 66, figure 1.4.5 [17]. This requires minor changes in processing

conditions in manufacturing tires with nylon 6 cord. Higher stretch rate must be applied in the heat treating process for nylon 6 cords due to low modulus.

Nylon 6 loses tensile strength more than nylon 66 during high temperature cures. Accordingly, nylon 6 cord requires lower tire curing tem- peratures or enough cooling when the tire is removed from the mold.

130 150 170 190 210 TEMPERATURE (° C ) Figure 1,4.5. Thermal shrinkage of nylon 6 and 66 tire cords as afunction oftemperature.

840 D/2 (47 X 47 turns/lOcm.) [17]. Nylon 6 and nylon 66 in common show flat spotting phenomena and

lower modulus than rayon and polyester. Many efforts have been made to improve these properties, but aU of them failed to achieve commercialized materials.

The improvement of strength is one of the currently important problems of nylon cord. The tenacity of recently developed nylon tire yarn

has reached ten grams per denier or higher [276].

Nylon monofilaments are currently under investigation for tire service

[276].
HIGH TENACITY NYLONS 227 To obtain high tensile strength, higher molecular weight polymer is used and a higher draw ratio. Tensile strength increases progressively

with increasing molecular weight but dimensional stability of greige and

treated cord decrease at high temperature. Therefore, molecular weight

has to be selected to obtain the best balance of tensile strength and

dimensional stabiUty. Hot stretching at a higher temperature and other means to improve dimensional cord stability [18-21] are adopted in pro- duction processing of the yarn, e.g. multistep stretching, stretch and

relax, etc. Heat aging resistance is also an important property for tire service.

Various additives are usually used in yarn production to improve aging

resistance of the tire cord. Thermal degradation of nylon cord in hot air decreases tensile strength.

This change becomes important at temperatures reached in heat setting

cords and in operating tires [22-24].

Decrease in tensile strength runs parallel to reduction in length of

polyamide chains, reduction in the number of primary amine end groups

and increase in the number of carboxyl end groups. In order to improve heat and oxidation degradation properties, various

additives are used before or after polymerization. These additives are

often organic, e.g. some amines, some phenols, some haloaromatic acids,

diarylamine-ketone condensation products, 2-mercepto benzoimidazole

and so on. Sometimes they are inorganic, e.g. phosphorous compounds

and sometimes metals and their compounds, e.g. copper and its salts, organotin compounds, cobalt chelating compounds and so on [22-23].

There are many reports concerned with heat treatment of nylon cords

and dipping and heat-stretching machines for tire cords or fabrics

[17, 27-32].

Nylon cords, or more generally, thermoplastic fiber cords, must be

dipped and heat stretched to improve dipped cord properties, e.g. to improve tensile properties, to reduce growth, (to increase Young's modulus and to decrease creep), to improve dimensional stability, and to impose cord-to-rubber bondabihty. A wide variety of single- and multiplestep treatments have been adopted by various tire and fiber producers.

Typical heat-treatment machines are diagrammed in figures 1.4.51 to

1.4.54. Cord or fabric treating conditions used by the tire industry at large vary somewhat. Yet, within this variance, cords or fabrics produced

under different conditions all serve the precise purpose of the individual

company.

Table 1.4.2 shows examples of variations that exist in conditions for treating nylon 66 cords [30].

We discuss the effect of treating conditions on the physical properties

of nylon 66 according to Du Font's data [27].

Figure 1.4.6a [27] shows the change in breaking strength that occurs

in a single step process as net stretch is increased.

Cold growth, heat shrinkage, and heat shrinkage tension are also de- pendent upon net stretch as shown in figure 1.4.66 [27]. Heat shrinkage

increases with increasing net stretch up to a point, after which little
228 TIRE CORD ADHESION

further change is seen. On the other hand, shrinkage tension increases in a fairly linear manner with increasing net stretch at least to 20 percent

net stretch.

Table 1.4.2. Nylon 66 processing conditions for drying zone and single, dual or triple zone treatment [30]

Tension, Residual Company Time Temp. lb/cord Stretch H2O, % Tenacity, lb Drying conditions A 1.5 min 270° F 1 1% 1% 24-26 B 2.5 min 240° F 0.75 1% 1% 24-26 C 3.5 min 180° F 0.5 1.5% 7-10% 23-25

Heat set conditions A 20 sec 425° F 10 12-14% 0% 29-30 B 30 sec 410° F 10 10-12% 0% 28-29 C 45 sec 390° F 10 10-12% 0% 27-28

Normalizing conditions A 20 sec 400° F ? 8% 0% 30 B 30 sec 370° F 9 8% 0% 30 C 30 sec 350° F ? 7% 0% 29 Reheat stretch conditions A 20 sec 425° F 9 12-16% 0% 31.5 B 30 sec 410° F 9 12-15% 0% 31.0 C Does not use triple zone treatment

Basis -840/2 nylon 66 -Type 700. Temperature levels for nylon 6 materials are approximately 30° to 40° below the nylon 66 values shown.

Figure 1.4.6c [27] shows the results for a two-step process chosen to give the same net stretch as the optimum one-step, with considerably

better heat shrinkage. Heat shrinkage tension does not improve. There

are patents on similar processes [33].

Our similar results with nylon 6 are shown in figure 1.4.7 and table

1.4.3 [17, 32]. Good dimensional stability can be obtained by heat treating at high

temperature.

Drying conditions are also important. Loss of tensile strength in the

heat stretching process is marked when water in the cord has not been

removed sufficiently in a dryer. This loss, however, can be avoided by

adopting a predip process (water dip prior to RFL dip [34]).
HIGH TENACITY NYLONS 229

9

lio.3

X 1x1 to I— ]

.3

I 6

2 4 6 8 10 12 U 16 18 20 22 aio°f/x lb/60 sec

2 4 6 8 10 12 U 16 18 20

AioV/eoLB/eosEC - AltfF/XLB/120SEC

4 6 8 10 12 14 16 18 20 NET STRETCH ( % ) Figure 1.4.6. (a) Correlation between net stretch and breaking strength for nylon 66 cords

(single step stretching) [27]. (6) Correlation between net ,stretch and cord stabilityfor nylon

66 (single step stretching) [27]. (c) Correlation between net stretch and cord stability for nylon 66 (two step stretching) [27]. Table 1.4.3. Correlation between stretch and thermal shrinkage for nylon 6 cord; two

step stretching [17, 32]

1st Stretch, 2nd Stretch, Total stretch, Elongation at Thermal % % % 4.5 kg, % shrinkage, %

4 4 8 9.5 4.0 6 2 8 9.4 3.8 8 0 8 9.4 3.7 10 -2 8 9.3 3.2 12 -4 8 9.5 2.8 14 -6 8 9.6 2.7
230 TIRE CORD ADHESION

Figure 1.4.7. (a) Correlation between stretch and cord strength for nylon 6 [17-32]. (6) Correlation between stretch and elongation at 4.5 kg. for nylon 6 [17-32]. (c) Correlation between stretch and thermal shrinkage for nylon 6 [1 7-32]. (d) Correlation between treat- ing temperature and thermal shrinkage for nylon 6. 840 D/2, 47 X 47 turns/10 cm. [17-32). Among heat treating machines other than the usual types mentioned

above, a fluid bed process is worthy of mention [35]. This process, in which small glass beads are used as the heating medium, was developed

in England.

In high temperature curing of nylon tires, particularly nylon 6 tires by

the Bag-O-Matic process, it is considered important to improve the ther- mal stability of the tire cord to withstand the curing conditions. Loss of strength is more serious in the portion around the bead than at the crown.

Figure 1.4.8 indicates Pieper's result [36], in which strength loss of nylon 6 is compared with that of nylon 66. Strength loss of nylon 6 is greater than that of nylon 66, particularly at high curing temperatures.

However, adoption of cooling before removing from the mold is effective

in avoiding the strength loss. We also have examined loss of strength in

tire cures with nylon 6, using a steel mold like Reegan and Sabos [37], to simulate the typical actual shrinking behavior of cord which is expected

in a tire curing process [38]. The test diagrams of shrinkage are shown in figure 1.4.9. Strength loss and thermal shrinkage in the curing process are illus trated in figures 1.4.10 and 1.4.11, respectively.

From these results, it was concluded that strength loss of the cord is largely to be attributed to rapid shrinkage when the work is removed

from the mold at high temperature and that a slow shrinkage of about 10

percent, which the cord suffers during the curing process, has little
HIGH TENACITY NYLONS 231 ol5

» X13 »-o

UJ 1 1

i 7^

NYLON 6 NYLON 66

"TlJ 11 Jl nil

140 150 160 170 180 11 11 11 11 li

lAO 150 160 170 180 CURING TEMPERATURE C'C ) MOLD OUT Wl THOUT COOLING MOLD OUT AFTER COOLED DOWN TO \UO(i

Figure 1.4.8. Strength loss of nylons in vulcanization.

840 D/2, 50 X 50 turns/ 10 cm. [36].

effect on strength. Therefore, it is advisable to cool the cord and then remove it from the mold. These phenomena have also been confirmed by

actual tire curing experiments. The strength loss of the cord is also affected by heat treating conditions of cord. Molecular mechanisms of strength loss with (rapid) thermal shrinkage

have been studied by various investigators. Dismore and Statton [39]

concluded from their study on nylon 66 that strength loss with thermal

shrinkage at high temperature is related to introduction of folded chains.

Fujimoto stated that there were good correlations between strength

loss with shrinkage at high temperature and the 002 lattice spacing of

crystalline regions or their change with thermal shrinkage as shown in

METHOD AB

PRESS OUT POST STRETCH

Figure 1.4.9. Laboratory test diagrams of shrinkage used for evaluating strength loss in curing.
232 TIRE CORD ADHESION

N

N

I 1 1 1L ucTU/in

E

^ B

1V—

A

160 170 180 190 200 CURING TEMPERATURE ( " C ) Figure 1.4.10. Strength retention of nylon 6 {T-781S) after curing as a function of curing

temperature.

figures 1.4.12a and 1.4.126 [40]. Furthermore he stated that when

strength loss was great, a large number of filaments were ruptured with

sharp edges inclined approximately 35 degrees to the fiber axis. There was a good correlation between strength loss and number of bias breaks,

figure 1.4.13. From these experimental results, he suggested that mechanisms of strength loss of thermal shrinkage and fatigue failure have

something in common as wiU be mentioned later.

160 170 180 CURING TEMP rC)

Figure 1.4.] 1. Total shrinkage of nylon 6 (7-7818) through curing cycle.
HIGH TENACITY NYLONS 233

0 \

— ^Sr^ 60

A

3.74 3.76 3.78 0.2 0.3 0.4 d(002) A Ad (002) A Figure 1.4.12. Strength retention of nylon 6 after curing as a function of lattice spacing

and change of lattice spacing [40].

(a) Strength retention vs. lattice spacing.

(b) Strength retention vs. change of lattice spacing.

Co 100

20 3 0 4 0 5 0 60 7 0 PERCENT OF FILAMENTS

WITH BIAS BREAKS Figure 1.4.13. Strength retention as a function of bias filament breaks [40]. Mechanisms offlatspotting: It is generally assumed that the tire cord is primarily responsible for flatspotting, in spite of other variables [41-42]

in tire construction and manufacture which affect flatspotting. Therefore, almost all articles which deal with test methods or mechanisms of

flatspotting are based on the viscoelastic behavior of the component

fiber. Some papers have dealt with laboratory tests on cords (or yarns) to simulate the phenomenon of flatspotting in tires as affected by choice of

tire cord [43-51]. Other papers have been concerned with quantitative

measurement of flatspotting in tires on indoor wheels [52-53].

Here, we will consider how tire cords behave in tire service. Figure

1.4.14 shows the deformation of a tire under load. Cord strain in the

footprint is smaller than in other parts. All volume elements of the tire, however, spend an equal time in the footprint during a revolution; their

average strain per revolution is the same. This situation no longer exists

after the tire stops. When the tire comes to rest under load after long

running which raised the tire temperature, the volume elements in the

footprint cool to ambient temperature under less strain than other ele-
234 TIRE CORD ADHESION

FOOTPRI NT

^lllllllllllllllflllllflllllllll Figure 1.4.14. Deformation of tire under load.

merits. When the tire starts to rotate again, persistence of this strain dif- ference causes the flatspot. As the tire rolls and consequently is heated,

this difference decreases with time and eventually becomes negligible.

The qualitative phenomena can be investigated in the laboratory; the

conditions which we examined are illustrated in figure 1.4.15 [54].

Two cords are used in the experiments. First, they are given thermal

and mechanical conditioning steps to approximate vulcanization. Thereafter, these cords are given a thermal and mechanical history to simulate the behavior of cords in actual tire operation; one cord represents a cord located in the footprint when the tire was brought to rest and the

second cord represents a cord in other parts of the tire. The difference between A and B, see figure 1.4.15, is an index of the

magnitude of flatspotting.

Finally, to simulate the run-out of flatspotting, both cords are reheated.

Tippetts explained the magnitude of flatspotting in terms of the dif- ference in Young's modulus of the cord at tire operating temperature and

at room temperature [44]. He concluded that magnitude of flatspotting is proportional to (1/M/y) X [1— {MhIMr)'], where Mh is Young's modulus

at tire operating temperature and Mr is Young's modulus at room

temperature.

Flatspotting becomes large when the operating temperature exceeds

the glass transition temperature Tg of the cord, because then {IIMh)

[1— (MhIMr)] becomes large rapidly.

Relationships between flatspot index and tire operating temperature,

which were examined by Papero and co-workers, are illustrated in figure 1.4.16 [46]. In both cases, nylon and polyester flatspot indices increase

rapidly when tire operating temperature exceeds the Tg of the cord. Flatspot index decreases as cord increases, as shown in figure 1.4.17

[54]. Moisture regain of the cord also has important effects on the flat- spot index of water sensitive cord because absorbed moisture lowers the

Tg of the cord, figure 1.4.18 [54]. The flatspot index rapidly increases,

then passes through a maximum and finally decreases with moisture

content. Reduced flatspot nylon, modified by blending of aromatic polyamide, is claimed to exhibit good flatspotting resistance at low humidity

but loses its merit at high humidity.
HIGH TENACITY NYLONS 235

I 600G

OG

160C

lOMINl

80 C 60 Ml N — P.C.I.

1200G

25C

30MIN

1600G

80 C 30MIN

RUNNING

1 600G

500 G

80 C 15MIN

1600G

500 G

25 C 5MIN

PARKING

1200G

500 G

25 C 30MIN

a1200G

25 C IMIN

RUNNING

900 G 900 G 900G A 900G

65 5 C TO EQUILIBRIUM

-— RUNNING —

^^G 0 G b 900G

25 C

2 sec

-> RUNNING

65 5 C 15MIN 25C

TO EQUILIBRIUM PARKI NG

G 840 1260 1260 1260 { \

c 105 25 105 25 MIN 60 30 60 30

840

105

1 6HR

840 840 640 840 r r

420 420

25 105 105 25 30 20 5 20 RUNNI NG —

•

•PAR KING

Figure 1.4.15. Laboratory testing methods for flatspot index of cords [54].

A. Du Pont method.

B. Modification of Goodyear method.

C. Modification of Allied method. AO-

_l I 1 1 L_ AO 60 80 100 120 1A0 TEMPERATURE (°C ) Figure 1.4.16. Flatspot index as a function of running temperature [46].
236 TIRE CORD ADHESION

-A

» —

100 110 120 130 KO 150

2.0

16

1.2

0.8

METHOD B

» 0 —o

100 110 120 130 UO 150 METHOD C

_o NYLON 60R66-AROMAT1C NYLON COMBINATION

NYLON 6-PET COMBINATION

100 no 120 130 KO 150 TtanSmax CO

Figure 1.4.17. Flatspot index as a function ofT (tan 8majc){as measure o/Tg) [54]. Reduced flatspot nylons:

(1) Diisocyanate crosslinked nylon [55]

Cipriani and co-workers found that short, vapor phase treatments

with diisocyanates, particularly tolylene diisocyanate, produced nylon 6

with greatly improved flatspotting resistance, without undesirable side

effects.

(2) Cyanuric chloride crosslinked nylon [56]

It was reported that treatment with cyanuric chloride improved the

high temperature porperties and flatspotting resistance of nylon 6 cord.

(3) N-44 [48] , [57-59]

N-44 is Du Font's reduced flatspot nylon, a melt blend yarn of nylon 66

and aromatic polyamide. The process is disclosed in Du Font's patents
HIGH TENACITY NYLONS 237

N A4 \ \nylcIN 66

/

c NYLC)N 6

1.0 1.3 2.7 MOISTURE REGAIN {%)

Figure 1.4.18. Flatspot index as a function of moisture regain [54].

for blending aromatic polyamide, Tg of which is higher than 140" C, with

aliphatic polyamide, e.g. nylon 6 and nylon 66 to improve flatspotting

resistance. A typical example of an aromatic polyamide is polyhexamethylene isophthalamide (61). This blend yarn loses its good properties at high humidity. In another Du Pont patent, it is mentioned that use

of an aromatic polyamide made from ^-butyl isophthalamide or from

polymethylene diamine having a longer polymethylene chain than hexamethylene diamine, is effective in overcoming this demerit at high

humidity.

(4) X-88 [60]

Monsanto's X-88 nylon is nylon 66 modified by aromatic polyamide

based on terephthalic acid.

(5) EF-121 [46, 49, 57]

Allied Chemical's EF-121 nylon is a melt blend yarn of 30 percent

polyethylene terephthalate and 70 percent nylon 6.

(6) NF-20 [57, 61]

Firestone's NF-20 nylon is said to be a block copolymer of polyamide

and polyester.

(7) Merged fiber [62]

Kovac and co-workers investigated merged fibers of various materials.

They reported that the merged fiber combination of nylon-polyester is very effective in flatspotting resistance without undesirable side effects, as shown in table 1.4.4.
238 TIRE CORD ADHESION

(8) Aromatic polyamides

Aromatic polyamides which have high melting and glass transition

temperatures, are excellent in their resistance to high temperature and

flatspotting. Typical examples of these aromatic polyamides are poly

(/n-phenylene isophthalamide) (Nomex [63]) and poly (hexamethylene

terephthalamide) (6T fiber [64]).

Table 1.4.4. Cord properties of merged fibers [62]

Cord Tenacity, Durability,

3%

Modulus, DS^

gidenier kc lb Polyester 1000/2 7.3 500 9.5 2.2 0.35 Rayon/PE 1650/1000 5.0 410 11.4 2.1 0.35 Rayon 1650/2 4.1 290 17.7 2.0 0.40 Nylon/PE 840/1000 7.0 520 6.8 3.5 0.50 Nylon 840/2 7.3 500 3.3 7.3 0.90 Glass fiber/PE 3040/2000 6.0 150 14.0 0.9 0.10 Glass fiber 1520/2 9.1 50 40.0 0.0 0 Nomex/PE 1200/1000 6.5 450 10.0 2.0 0.35 Nomex 1200/2 5.8 400 4.2 1.9 0.38

^ Dimensional stability " Flatspot

1.4.1.4. Polyester Cord [65-69]

Almost all polyester yarns used in the tire industry are filament yarns

of polyethylene terephthalate.

As rayon and polyamide dominate the tire cord market an improved

reinforcing material is expected to meet the requirements imposed by

increasing severity of tire operating conditions. A fiber, to be a satisfac- tory tire cord material, must possess a rather special, balanced combination of properties.

Polyester's high modulus and low elongation reduce tire deformation

and growth under service conditions and lead to better high speed per- formance and tread wear, reduced tread cracking and better steering

characteristics. In addition, polyester's dimensional stability allows

manufacturing a more uniform tire. High strength or tenacity is required to provide adequate carcass

strength and in this respect, nylon is superior to current commercial

polyester, which in turn is superior to rayon. Polyester is considerably superior to rayon in fatigue resistance. In summary^ polyester yarns have a better balance of fundamental properties for efficient use in modern tires than does rayon, and wiU replace a

large portion of rayon cord in the future. Recently, Dunlop U. K. announced that they were introducing an all polyester radial tire utilizing polvester macrofilament yarn in the belt [274].
POLYESTER CORD 239 Differences in production processes for high tenacity polyester yarn

from those for regular polyester yarns are explained below. To improve

the tenacity of cords, molecular weight of yarns and draw ratio in the

production of the yarn are important factors. Intrinsic viscosity, IV, (a measure of molecular weight) of current tire yarn is 0.8 to 0.95, while that of regular polyester yarn is under 0.6. Draw ratio in tire yarn production

is also much higher than that in regular yarn production.

Draw ratio in the production of the yarn has a large effect on both

yarn tenacity and the yarn-to-cord tenacity conversion efficiency. As can

be seen in figure 1.4.19 [65], the results effectively demonstrate the

lower conversion efficiency of the higher tenacity materials. This de- crease in conversion efficiency eventually reaches such proportions (as

yarn tenacity is increased by increasing yarn draw ratio) that cold tenacity

II actually begins to decrease in spite of higher starting yarn tenacities,

} figure 1.4.20 [65].

The slope of the upper part (near the breaking point) of the stress- strain curve significantly affects yarn-to-cord conversion efficiency [34,

65, 70]. The higher the slope of the upper part of the stress-strain curve

the lower becomes the conversion efficiency. This tendency is explained

in terms of homogeneity of stress distribution in the cord by twist theory

[71]. The slope of the upper part of the stress-strain curve is greatly

affected by the draw ratio in the production of yarn, but other factors are

also significant.

Tenacity of the cords can be increased without increasing the slope

of the final part of the stress-strain curve or decreasing the yarn-to-cord

conversion efficiency by increasing the molecular weight.

Using a suitable lubricant, which decreases the static friction coeffi- cient of the yarn, is also effective for improving the yarn-to-cord conversion efficiency, as in the case of rayon. Furthermore, heat aging resistance and hydrolytic resistance of the

yarn are important factors. To improve heat aging resistance the diethyl- ene glycol content of yarn must be reduced to a low level. On the other

hand, as the carboxyl end group content decreases hydrolytic resistance

increases [34, 73-75].

8 9 10 n

PLY TWIST FACTOR Figure 1.4.19. Effect of yarn drawing on tenacity conversion efficiency for polyester [65].
240 TIRE CORD ADHESION

-X

y^

y' y y

•

y

y

y yf

y^

-y 12 3 4 5 6 7 8 9 YARN TENACITY ( G/D ) Figure 1.4.20. Cord tenacity vs. base yarn tenacity for 1100 D/2, 51 X 51 turns/ 10cm. polyester cords [65]. Heat treating conditions of polyester cords have a pronounced effect on their properties. The heat treatment temperature of polyester cord

is considerably higher than that of nylon or rayon. Stretch rate for polyester is considerably lower than that for nylon. Typical heat treating

conditions of polyester cords are compared with those of nylon and

rayon cords in table 1.4.5 [38]. In the case of polyester cord, fatigue

resistance is particularly sensitive to the heat treatment.

Table 1.4.5 Comparison of Typical Heat Treating Conditions for Rayon, Nylons, and

Polyester [38]

Set temperature Net stretch

°C °F %

Passenger tire Rayon 155-165 310-329 1.5-3.0 Nylon 6 205-210 401-410 5.0-9.0 Nylon 66 220-230 428-446 4.0-7.0

Polyester 235-245 455-473 0-4.0 Truck tire Nylon 6 207-212 404.6-413.6 6.0-10.0

Aitken and co-workers reported that although stretching ratio affects the fatigue rating, treating temperature is the single most important

factor in achieving high fatigue rating with polyester tire cord, table

1.4.6 [65]. They concluded from their results shown in table 1.4.7 [65] as follows; strength retention of the cords after heat treatment is best for the

highest IV samples, which increases the advantage of high IV material over low IV material. This effect is most marked at high heat-treating

temperatures. The higher the cord IV, the higher the fatigue rating
FIBER GLASS CORD 241

which can be achieved. A reduction in Mallory rating occurs at all IV

levels when treating temperatures are taken to 490° F. High IV cords

suffer less than low IV cords in this respect, however.

At 450° F, the Mallory ratings of the low IV samples exceed those of the high IV samples. At the two higher temperatures investigated.

475° F and 490° F. this effect is reversed. In our experiments dealing with higher molecular weight (0.85 and

0.95 IV) yarn, rather high temperature heat treating may give a good

fatigue rating and good dimensional stability but results in low strength

cord and in a stiff" fabric, which may cause difficulties during tire building

[34], Table 1.4.6. Optimization of heat-treating conditions for polyester cord[65] Oven

temperature. ^ °F

Heat treatment Mallory

fatigue

First oven rating, kc

stretch, %

Second oven

stretch, %

A 400 3 0 11 B 400 5 -2 12 C 450 3 + 2 24 D 450 5 -2 48 E 475 4 0 134 F 475 4 -3 196

^ 2/1100, 13 X 13 cord made from Type 24 "Terylene" yarn. Dwell time 90 sec in all cases.

Table 1.4.7. Effect ofIV and treating conditions on treated cordproperties for polyester [65] Greige Treating Breaking Mallory Yarn IV breaking temperature, °F

load fatigue

load, lb retained, % rating, kc 490 76 8

0.60 28 475 86 132 450 92 122 490 81 21

0.69 30 475 87 159 450 95 61 490 85 97

0.73 31 475 91 175 450 95 23

kt

1.4.1.5. Fiber Glass Cord

Simply stated, the most important problem of fiber glass cord is effective use in multifilament structures of superior single filament properties

shown in table 1.4.8 [76]. Characteristically, single-filament properties

are reduced by more than one-half when formed into strands or yarns.
242 TIRE CORD ADHESION

Early developments showed that excellent flex life of single-filaments

disappeared when they were combined as a cord. For years, many attempts have been made to solve these problems.

Phenol formaldehyde, elastomeric materials and more recently, some of

the organo-silicon compound coatings were found to be effective to improve these properties.

Some inorganic coatings such as metals were also effective in improving abrasion resistance and flex life of multifilament glass fiber structures. However, lack of rubber-to-cord adhesion and insufficient filament-tofilament stress transfer resulted in unsatisfactory performance.

Table 1.4.8. Physical properties of single filaments of glass and of organic fibers [76]

Glass Rayon Nylon Polyester

EGG" T-130 T-A05 T-52

Tensile strength

ultimate, psi... 500,000 79,000 126,000 139,000 Tenacity at break, gpd 15.32 4.05 8.65 7.89 Ultimate elongation, % 4.76 11.74 19.93 14.92 Modulus, gpd 322 35 43 53 1000 psi 10,500 680 630 940 Toughness, gpd... 0.365 0.312 1.05 0.720

psi 11,900 6,100 15,300 12,800

"Sp. gr., 2.55; elastic recovery, 100%; softening point, 1555° F; coefficient of thermal expansion (T), 2.8 X 10.6; and water absorbency, 0.3%.

Various surface finishes were investigated to improve glass-to-rubber

adhesion. A combination system of a compatible surface finish and an RFL was also developed as an impregnation material to improve filament

to filament abrasion. With these developments it becomes possible to maintain single filament properties in impregnated multifilament glass

cord, table 1.4.9 [76]. To achieve the aging characteristics associated with glass filaments,

a melt was formulated in which oxides of boron, aluminum, and magnesium were combined with silicon dioxide, table 1.4.10 [76]. At "present

this aluminum-boron-silicate composition, known as E glass, is used

for tire cord. E glass has a three dimensional, completely crosslinked structure. The structure consists of a continuous silicon dioxide network with some

silicon atoms from other networks forming oxides such as aluminum.
STEEL WIRE CORD -MISCELLANEOUS CORDS 243

Table L4.9. Physical properties of cords ofglass and organic fibers [76] Glass ECG-150

10-lA/O

Glass ECG-150

10-1A/3

Rayon

T-130

1650/2

Nylon

T-130

840/2

Polyester

T-52

1100/2 Tensile strength

(ultimate), psi Af\'7 AAA 4U / ,UUU OfiC AAA A/I AAA V4,UUU 1 OO AAA 1ZZ,UUU 1 f\A AAA iU4,UUU

Tenacity at break. gpa 1 O /I T

11. i / A m O Q "7 8.3/ 0.86 Ultimate elongation,

Vc A OO A OA 4.84 A O 9.0 1 A Q 19.0 IOC 18.0 A QAO U.oUZ A OOA A CAA

psi 9,900 8,300 5,800 10,200 9,900 Modulus, gpd 259 231 49 43 32

1000 psi 8,450 7,540 960 630 570 Breaking strength, lb.. 79 219 39.1 33.2 32.1 Impact resistance.

(It/ID Xiu ^)/aenier.. 0 AC

Z. lb 1 c? 1.8/ A AO 4.U8 OA! 3.41 Diameter, mils 17 35 26 21 24 Number of filaments. .

.

2,040 6,120 8,000 280 500

Specific gravity (fiber). 2.55 2.55 1.53 1.14 1.38

Table 1.4.10. E glass formulation [76] Silicon dioxide 52--56%

Calcium oxide 16--25%

Aluminum oxide 12 -16%

Boron oxide 8 -13%

Sodium and potassium oxide... 0 -1%

Magnesium oxide 0 -6%

Glass cord seems to be a promising material for belt ply cord for radial tires and belted-bias tires [80-84]

.

1.4.1.6. Steel Wire Cord [85-89]

The principal use of wire cord tires is in truck and bus tires and in some cases for off-the-road tires. Mileage in most cases is high.

The wire cord tire differs not only in cord properties but also in construction of the carcass plies and in method of tire manufacture.

Conventional tire designs are used in some cases, but the wire characteristics allow radial ply construction.
244 TIRE CORD ADHESION

Typical cord constructions currently used are illustrated in figure

1.4.21. Wire cord has high strength, extremely high modulus and high

fatigue resistance.

Most wire cords are brass or zinc plated to improve cord-to-rubber

bonding. This is done in the production process of the cord wires by

continuous electrolytic methods after the wire has been drawn.

The wire is usually wound on convenient spools and is shipped to tire companies in a package containing a desiccant.

(7X3) (5X7)+(1X3) FOR PASSENGER CAR FOR TRUCK Figure 1.4.21. Typical cord constructions of steel wires.

1.4.1.7. Miscellaneous Cords

In addition to the cords mentioned above, Vinylon [16a], polyolefins,

and polycarbonate [90] are being investigated as tire reinforcing materials.

Vinylon is used to some extent in bicycle tires but not in automobile

tires because of inferior heat aging resistance.

Kurashiki Rayon has recently developed a new type of high modulus,

high tenacity PVA filament yarn, which could be good material for the

belt ply of radial tire [275]. Among polyolefins, polypropylene fiber is also being examined for bicycle tires [91] but its future as tire cord material does not seem to be promising because of poor high temperature properties and rubberto-cord bondability.

1.4.1.8. Comparative Analysis of Various Tire Cords

A number of articles have appeared which are concerned with comparative analysis of various tire cords. The results which are summarized

in table 1.4.11 were reported by Schroeder and Prettyman [92]. In increasing order of tenacity and breaking elongation are rayon, polyester,

modified nylon and nylon. Rayon and polyester have higher initial moduli

than do either of the nylons. Rayon possesses greater thermal stability

than polyester and nylon. Rayon and polyester have higher flatspotting

resistance than nylons. Nylon shows a definite superiority for both impact

toughness and compression-flex fatigue resistance. Adhesion tests place
COMPARATIVE ANALYSIS OF VARIOUS TIRE CORDS 245

.CONDITIONED

RT 600 rOVEN DRIED

PERLON HR

NYLON 130HR DtOLEN

170 170 190 170 170190 170 170190 170 170190 ^ VULCANIZATION TEMPERATURE!' C)

*^ 30 45 18 30 4 5 18 30 45 18 30 45 18 VULCANIZATION TIME (MIN)

17 30 45

17 30 45 17 30 45

17 30 45

RT600F NYLON 130 PERLON DIOLEN

HR HR Figure 1.4.22. Comparison of strength loss in vulcanization [93]. RT 600F is rayon, 1650 D/2, 47.2 x 47.2 turns/lOcm. Nylon 130HR is nylon 66, 840 D/2 , 48 X 48 turns/lOcm. Perlon HR is nylon 6. 840 D/2, 48 x4S turns/lOcm. Diolen is polyester, 1000 D/2, 50 X50 turns/lOcm.

(A) Effect of moisture on strength loss of cords in vulcanization.

(B) Effect of vulcanization time, 17, 30 and 45 min. on cord strength; Vulcanization temperature, 170° C; Conditioned

at normal humidity.

rayon on top, followed in order by nylon, modified nylon, and polyester,

although all are at a high level with modern dip systems.

We can conclude from these results that any current commercial cord

cannot satisfy all cord properties required.

Schroeder and Prettyman also indicated the desired directions for cord properties as shown in table 1.4.12. Daimler investigated the strength loss of various tire cords during tire curing under various conditions as shown in figure 1.4.22 [93]. For both rayon and nylon 66, no serious strength loss was observed

under any condition investigated. Nylon 6 loses strength when it is allowed to shrink rapidly under high temperature, high humidity conditions, because of physical causes. On the other hand, polyester loses

its strength because of hydrolysis at high temperature and high humidity.

Strength loss of polyester depends greatly upon curing time, while

that of nylon 6 does not.
246 TIRE CORD ADHESION

Table 1.4.11. Physical properties of various tire cords [92]

Rayon Nylon Nylon Polyester

(current) (modified) Cord construction 1650 D/2, 840 D/2, 840 D/2,

1 0 V 1 0 1 9 V 1 9 iZ X iz IZ X iZ 9900 Tl/^ 1 9^0 D/9 1IZ.9 y^ 1iZ9 IZDU LflZ, 8.5 X8.5 10 X 10 10 X 10

Tenacity, gpd 4.5 7 6.5 6

Elongation at break % 9 21 15 13

Initial modulus gpd 60 27 40 70 Creep % 3 4 3 2 Shrinkage % 0.6 6 7 7 Flatspot Index mils 50 170 115 50 Impact toughness,

erg/ cm D 225 525 400 375

C-1^ latigue, mm 400 1400 600-1200 500

Adhesion, Ibs/m 100 90 70 65 Agmg, % 70 95 70-95 70

Esthetic rating of tires r ar Even

Roughness Par ++ ++ Even

Ride Par Even

Noise, smooth road Par

Noise, brick road Par

Handling Par Even Even Even

Furthermore, strength loss of polyester was affected by the rubber

recipe; some amines, some moisture sensitive fillers, and thiuram

accelerators in a rubber compound degrade polyester cords [34, 94]. Ebert [7] also compared various textile cords with steel cord as the

reinforcing material for the belt ply of radial tires, see table 1.4.13. Extra high modulus rayon and steel are promising materials for this purpose from the viewpoint of initial modulus.

Some polyesters also seem to be acceptable.

1.4.1.9. Impact Resistance of Tire Cords [95-99]

Impact failure of tires and tire cords, which will be considered here,

is one of several failures that may occur in high speed operations. In

general, impact resistance of a tire is determined either by road or simulated roajd tests, or has been inferred from laboratory tests at low speeds

and room temperature.

Road tests are difficult and expensive and are limited in their ability to provide specific technical information for the design and modification

of the design of tires. It is expected from the well-known behavior of
IMPACT RESISTANCE OF TIRE CORDS 247

Table 1.4.12. Characteristics ofsuper-goal tire cord [92]

JT 1 upci L y suggested

1 lie lCo|JU115>CS ICiiUCCU,

Tenacity: 75° F to 400° F High Impact breaks, heat breaks, heat buildup.

high speed failure. High Impact breaks. Tensile modulus High Flatspotting, growth, high speed failure. to medmm Impact breaks, tlex lailure, roughness.

Bending modulus Low Flex failure, roughness.

Creep Low Growth, tread wear, sidewall checking,

tread cracking, high speed failure, flat- spotting, tire dimension and uniformity

variations. Thermal shrinkage Low Tread concavity, bead distortion uniformity

to variations. high High speed failure.

Flatspotting index Low Flatspotting. Impact toughness High Impact breaks. Flex fatigue resistance High Flex breaks. Adhesion High Tread and ply separation. Heat aging resistance High Impact breaks, heat breaks, flex breaks.

polymers that deflections, breaking forces, and breaking energies under

high speed, high temperature conditions will differ greatly from those

determined at low speeds and room temperature.

Laboratory impact tests of tires and tire cords are important as a design base for modern high speed tires. Many superior laboratory methods of testing impact resistance of

tires and tire cords have been proposed but we will not discuss them

in detail here. Hall [95] investigated the impact behavior of various fibers at a very

high rate of straining, 330 sec.~^ and at a normal rate, 8.3 X lO""^ sec."\

for a range of lightly twisted yarns covering all commercially important

fiber forming polymers. Details of his experimental results are given

in table 1.4.14. From these data, we see that at the high rate, breaking

stress is always greater and with one exception, breaking extension less than at normal rates. Energy to rupture increases with rate for wet spun

fibers and with one exception and decreases for thermoplastic fibers. Similar experiments performed with three of the yarns highly twisted,
248 TIRE CORD ADHESION

Table 1.4.13. Physical properties of cords for radial tire construction [7]

Elongation

at 20% of breaking

load

Adhesive powerImpact

energy Bending

life stiffness Heat

growth

% tons cm • kg cycles cm • g %

Steel 1 0.4 7 230 40 1,400 0.2 EHM 2 0.7 16 450 i,9UU OA80 0.3 3 0.5 15 700 1,000 60 0.3 Rayon 4 0.5 18 730 1,500 100 0.4 5 0.9 14 1,700 10,000 25 0.9 Polyester 6 1.0 1,800 4,200 700 0.5 7 1.1 20 1,600 5,200 140 0.5 8 1.9 14 1,400 5,600 30 0.5 Nylon 9 5.0 19 1,800 26,500 17 0.2

the data of which are given in table 1.4.15 and 1.4.16 [95J, showed that ranking of the yarns according to a particular property could be altered

by insertion of twist. Lothrop [96] determined the tensile properties of rayon and nylon

cords as a function of temperature and rate of extension. His testing

equipment was designed and built to be capable of measuring the properties of tire cords over a range of temperature from 75° to 300° F at rates of extension of 1000 percent to 6000 percent/sec. (10 to 60 sec."^).

His complete data for treated rayon and nylon cords are summarized

in table 1.4.17. Characteristically, for both oven-dried rayon and nylon,

breaking strength decreases as temperature increases at a given rate of

extension. Likewise, for a given temperature, breaking strength increases

as rate of extension increases. For oven-dried nylon cord, breaking

elongation is reasonably constant throughout the complete range of

testing conditions. Oven-dried rayon, on the other hand, shows an in- crease in breaking elongation with increase in temperature as well as with increase in testing speed so that the maximum breaking elongation was observed at a temperature of 300° F and a rate of extension of 6000

percent/sec. (60 sec.~^).

Breaking energy of the oven-dried nylon cord decreases with in- creasing rate of extension at the lower test temperature but not at the

higher test temperatures.

Breaking energy of oven-dried rayon decreases with increasing temperature at low rate of extension but increases with increasing temperature at high rates of extension.

Recently, Lothrop published another report [97], which confirmed the
IMPACT RESISTANCE OF TIRE CORDS 249

Sensitivity

to

strain

rate

" fOr^OOOOONO'O^cO-^'^LOt^r—l?^^r^cOvO"00^000^ csi--u^co^i-Hco^OiC^oou^fOfO_i_i-HTf<LOi-Hr~--i_ovocs

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_ 0 — '5

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250 TIRE CORD ADHESION

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IMPACT RESISTANCE OF TIRE CORDS 251

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252 TIRE CORD ADHESION

above results by tire plunger tests at high rates and high temperatures.

He used a test machine which consisted of a pneumatic gun for pro- pelling the plunger at an inflated tire, mounted in a temperature-controlled cabinet, and photoelectric devices for measuring plunger velocity.

The minimum kinetic energy required to cause failure of the tire fabric

is taken as a measure of carcass breaking energy. Data on rayon and

nylon cord tires indicate that the breaking energy of a tire is dependent

on both speed and temperature and that the relationship between

energy, speed, and temperature depends on the cord-reinforcing material. As a result, it is impossible to predict the relative impact re- sistance of tires under service conditions on the basis of laboratory

test results obtained with the standard static plunger test, in which

the plunger penetrates the tire at 2 in/min. at room temperature.

A comparison of data obtained on 7.75-14, two-ply rayon and nylon

cord tires at three different speeds is shown in figure 1.4.23. O'Neil, Dague, and Kimmel [98] also reported test results which

include results of an individual cord impact test and three dynamic

tire tests using a pendulum, a ballistic plunger, and resiliometer bruise.

Figure 1.4.24 shows the stress-strain curves of the four cord materials used.

Effects of temperature on strength and breaking energy of these

cords at low speed are shown in figure 1.4.25a and 1.4.256, respecTESTING SPEED — •RAYON 2 IN/MIN — 63MPH — 100MPH

o NYLON 2 IN/MIN

7 63 MPH " —a 100MPH

125 175 TEMPERATURE (** F ) Figure 1.4.23. Breaking energy of 7.75-14, two-ply, rayon and nylon cord tires as a function of temperature.

Speed: 2 in/min., 63 mph, and lOO'mph.
FATIGUE RESISTANCE OF TIRE CORDS 253

lively. Decreases in strength and breaking energy of nylon are more

rapid than those of rayon and polyester.

Effects of temperature on strength and breaking energy of these cords

at high speed are shown in figures 1.4.25c and 1.4.25fl?, respectively.

Also in this case, decreases in strength and breaking energy of nylon are greater than those of rayon and polyester. On the other hand, rayon

shows an increase in breaking energy with temperature.

_ 60 m 50

^ 20 ^ 10

3LYESTER

5 10 15 STRAIN Figure 1.4.24. Stress-strain curve, low speed, 12 inlmin. [98].

80

70 m

-J 50 X40 §30

^20 O

z

280

LlI 70 ^ 60

A

1— ^ B

6 GLAS^ 5 ^POLYESTER

'^'N^^^^NYLON

POLYESTE R NY .OhT m "^'^^-^..^RAYON

1 ^

1 GLASS

J ICX) 2 DO 3 )0 IC 0 2C 0 300 4C)0 GLASS ^ 7 O -"^^Ol|y ESTER

RAYON

CC 6 ^5 RAYON

LU 5 y

ESTER 3 NY .ON

:5k;P0LY NYLON

I

50

300 400 100 200 TEMPERATURE ( *F )300 400

Figure 1.4.25. Effects of temperature on breaking strength and energy [98]. A and B. Low speed test. C and D. High speed test. They also examined the impact resistance of tires. Table 1.4.18 shows the comparison between the low speed test and the high speed

tests. The nylon cord tire has the highest failure energy at low speed,

and the polyester cord tire has the highest value at high speed.

1.4.1.10, Fatigue Resistance of Tire Cords

Fatigue resistance of tire cord is an important property but difficult to assess. A variety of cord fatigue tests and laboratory tire wheel tests
254 TIRE CORD ADHESION

Table 1.4.18. Plunger, Pendulum, and projectile tests of tires [98]

Rayon Nylon Polyester

Plunger (slow) (I) energy, in-lb 2260 3450 3120 Pendulum (impact) energy, in-lb 9160 3390 3900 change over (I), % + 40 -2 + 25

Projectile (impact) energy, in-lb 3470 3600 4430 change over (I), % + 53 + 4 + 42

are currently available for evaluating relative tire durability. However,

none of these tests can be considered adequate for characterization

of road performance and therefore, in the last analysis, a road durability

test is required.

Numerous papers are concerned with fatigue resistance of tire cord; some give phenomenological treatments of fatigue under cycUc tension

[99-103], biaxial rotation [104] or flexing of cords in air [101, 105-109], some present statistical treatments of fatigue phenomena [110-112],

[lOSfl?, e.g.], and others are treatments of fatigue under compressive or

flexing conditions in rubber blocks or in actual tires. Here, we will be mainly concerned with fatigue in rubber blocks or

in the tire itself, which is most important for durability of tires. Various views on the mechanism of cord fatigue in tires have been

presented in the literature and diverse opinions have clearly indicated

need for further research. For example, Williams and co-workers [113]

concluded that cords in tires lose their strength linearly with mileage

due to broken filaments. On the other hand, Entwistle and co-workers,

and Klein and co-workers [114], see figure 1.4.26, said [115] that cords

did not become progressively weaker until failure but fail suddenly by

an undefined catastropic process. Fatigue failure of cords in tires generally occurs at special localized

points. Williams and co-workers [113], and Patterson and Anderson

z 28

ui

cc lU Qa 20 8

RAYON

FLEX CENTER

SIDE WALL

20 40 60 80 100

Xo 28

-z. oe

t;^ 24 D

g20

K «

-•— r

a

-NYL ON e D )40/

>> •

r

20 40 M I LE5

Figure 1.4.26. Cord strength as a function of tire mUes, taxi fleet test [114].
FATIGUE RESISTANCE OF TIRE CORDS 255

reported that cords lose their strength more rapidly in inner plies than

in outer plies. Klein and co-workers [114] also reported from tests of Tyrex rayon and nylon cord tires, that cord strength loss is greater in the flexing or side waU region than in the crown or center region, figure

1.4.26. Patterson and Anderson [115] found that strength loss of nylon cords

in tires was affected by the direction of tire rotation and the direction

of cord bias and that cords in opposite sidewalls of tires lost strength

at markedly different rates. Strength loss was higher for the half of the

cord that led into the load bearing region of tire as it rotated (leading

half) than for the other half of the cord (trailing half), as shown in figures

1.4.27 and 1.4.28 [115]. Two regions of high strength loss were found in each tire sidewaU by

breaking short segments of cords; at the shoulder, and at a point about

two inches above each bead. Figure 1.4.29 [115] shows breaking strength

at these points in first-ply cords from unfailed tires as a function of miles run on the test wheel. The point of lowest cord strength, in the range

from 12,000 to 18,000 miles where tire failures began to occur, is shown to be in the leading shoulder. However, tire failures were always in the

sidewall about two inches above the bead. This seemed to indicate that

tires do not fail where the cords are weakest.

TRAILING HALF OF CORD

LEADI NG HALF OF CORD

Figure 1.4.27. Identification of cord sections to side wall and rotation direction [115\

O

012 2

10

TRAILING HALF

LEADING HALF

'LY 1

1

^ 12

10

1000 2000 3000 0 MILEAGE

PLY 3

1 1 1000 2000 3000

Figure 1.4.28. Cord Strength as a function of tire miles at test [115].
256 TIRE CORD ADHESION

o12

2

aj g a

t—

V) 8

7

6

TRAILI NG

BEAD.

TRAILING SHOULDER

NG SHOULDER

k 8 12 16 MILEAGE

Figure 1.4.29. Average breaking strength of sections offirst ply cords {115\.

Patterson [116] concluded from the above discussions and microscope examination of broken ends of cords, that fatigue failures in nylon

filaments were mainly related to cyclic flexing associated with compressive loading of inner ply cords. In addition to the above fatigue failure of cords, it is generally ac- cepted that cord-to-rubber adhesion failure [114] and rubber failure [117] play an important part in fatigue failure of tires. The general

levels of adhesion (as measured by the stripping test) before and after a long run are shown in figure 1.4.66. Highest adhesion loss is observed

in the region of maximum compressive flexing.

Microscope examination of broken ends of cords from failed nylon

66 tires showed a large number of filaments broken at a certain angle

to the fiber axis [115, 116]. Polyester and nylon 6 filaments showed

similar bias ruptures [115, 118-120].

Bias breaks are most prevalent at the point of cord rupture but are

also found occasionally along the entire cord length. Patterson and

Anderson stated that the number of bias ruptured filaments in flex tested

tires increased linearly with severity of fatigue damage to the cord, see figure 1.4.30 [115]. These bias breaks were not produced by tensile loading of filaments. Patterson dealt with the mechanism of bias rupture in another paper

[116], a number of tests being run on filaments and cords to produce

bias rupture.

A variety of tensile loading conditions applied to single filaments

failed to produce bias ruptures. These results indicated that possibly

unique or complex loading conditions were imposed on cords during

flexing in a tire. To establish what these conditions might be, an in- vestigation was made of the behavior of cords in transparent rubber

under a variety of loading circumstances.

Examination of the cords during alternate tension and compression

showed that the filaments undergo bending and sometimes buckling

when in compression. The severity of bending increased with reduc-
FATIGUE RESISTANCE OF TIRE CORDS 257 015

10 85

O

V

0 20 40 60 80 100 PERCENT OF Fl LAMENTS

WITH BIAS BREAKS

Figure 1.4.30. Cord strength as a function of percent of bias filament breaks {115].

tion of twist. Similar results were also obtained by Wood and Redmond

[121]. The observed increase in filament bending at low twist and the

accompanying poorer fatigue resistance indicated that bias filament

ruptures might not depend on the complex stress associated with a twisted structure but more directly on simple bending associated with

compressive loading of the specimens.

This hypothesis was tested with Mallory and Goodrich Disk Fatigue

Testers. These tests established conclusively that bias rupture was

produced by simple bending.

Further investigation showed that bias rupture began at the compression side of the bend in filaments subjected to repeated cyclic

bending. Similar findings were indicated for ductile metal.

Fujimoto [40] investigated the correlation between fatigue life and

molecular structure of nylon 6. He found that fatigue life in compressive loading decreased linearly with the second moment of the NMR

peak of the noncrystalline part, and then concluded that long fatigue

life is to be attributed to weak intermolecular interaction and mobility

of molecules in amorphous regions, see figure 1.4.31a. Furthermore,

he mentioned that fatigue life decreased linearly with lattice spacing

d (002), see figure 1.4.316, and suggested that mechanisms of fatigue

failure and strength loss in the curing process, as mentioned before,

have something in common.

Each tire company has its own methods of assessing tire cord fatigue

resistance. Laboratory machines designed to examine fatigue resistance

of cords fall into two groups; those flexing cords with air as medium

and those flexing cords in rubber. Among typical examples of the former,

there are cycHc tension types (e.g., U.S. Rubber Fatigue Tester, Du Pont

Dynamically Balanced Tester [103] and Goodrich Tension Vibrator [102]

and flexing types, e.g. Goodrich Flexing Tester [106]. Dillon classified these test methods according to the types of applied

stress or strain as figure 1.4.32 [122]. Of the latter group, generally regarded as the more important, there
258 TIRE CORD ADHESION

X

^ o

A B

1 1,A 1.6 1.8 2.0 IX 2.U 3.74 3.76 3.78 3.80 <AH^>A GAUSS^ d(002)A

Figure 1.4.31. (A) Fatigue life of nylon 6 as a function of second moment of NMR peak

for amorphous region. (B) Fatigue life of nylon 6 as a function of lattice spacing d{002)

[10].

(C)

MAA/

STRAIN

TIME TIME

Figure 1.4.32. Classification of cyclic tension fatigue tests [122]. are several methods, as we will mention below; Mallory Tube [123-125], U.S. Rubber Tube [124], Goodrich Disk [124-127], Firestone Compression Flex [124-125], Dunlop and De Mattia Flex [128-131].

(1) Mallory Tube Test

The testpiece is a rubber hollow cylinder in which the cords to be

tested run parallel to each other and to the axis of the test cylinder and

are arranged to have the required number per inch.

The flexing principle is illustrated in figure 1.4.33. The test tube is bent and tightly clamped on two spindles of the flexer. Thereafter, air pressure inside the tube is increased and then the horizontal spindle is rotated. Cords in the tube undergo alternatively compression and tension.

The number of revolutions until failure is the measure of fatigue

resistance.
FATIGUE RESISTANCE OF TIRE CORDS 259

DRIVE

PULLY COUNTER

ROTARY UMION

REGULATED AIR PRESSURE SUPPLY Figure 1.4.33. Schematic drawing of Mallory Tube Fatigue Tester. ROTATING ARM

/

'ROTARY UNION

REGULATE AIR PRESSURE SUPPLY

Figure 1.4.34. Schematic drawing of U.S. Rubber Tube Fatigue Tester.

SPECIMEN

ROTATING SHAFT MAVABLE mmROTATING SHAFT

FIXED V

SPECIMEN Figure 1.4.35. Schematic drawing of Goodrich Disk Fatigue Tester.

(2) U.S. Rubber Tube Test

This method is basically similar to the Mallory Tube Test but the

flexing principle differs somewhat, as shown in figure 1.4.34.

(3) Goodrich Disk Test

The flexing principle is illustrated in figure 1.4.35. The testpieces are rectangular rubber blocks in which test cords run parallel to the long

axis. The testpieces are firmly secured into the periphery of two canted

disks, the cords passing cross the gap between them. When these disks
260 TIRE CORD ADHESION

are motor driven, each cord in a rubber block suffers simple longitudinal

extension and compression. Strength loss after a certain number of revolutions is examined to evaluate the cords.

(4) Firestone Compression Flex Test

The testpiece is a rubberized belt which contains two plies of cords

in parallel planes, one ply of test cords and the other of steel cords.

Compressive fatigue is produced by flexing the testpiece over a spindle, figure 1.4.36. The inextensibility of the steel ply causes the

layer of rubber itself and the spindle to be compressed and the tire cords

are situated in this layer.

(5) Dunlop Fatigue Test

Essentially, the testpiece is an endless belt made up of five plies of rubberized cord. Counting from the inside, the second and fifth plies are comprised of the cords to be tested, these cords running along the

length of the endless belt and being arranged in the required number

per inch. The belt is tensioned between two pulleys as shown in figure

1.4.37. The test usually consists of running belts for known times, then

extracting the cords from the two test plies and measuring their breaking strength.

(6) De Mattia Flex Test

The testpiece is a rubber block, in which the cords to be tested run

parallel to each other and to the long axis of the block. The testpiece

is firmly clamped on the two heads of a De Mattia Flex Tester which

is usually used in rubber flex fatigue tests, figure 1.4.38. The test consists of flexing the rubber blocks and measuring their residual strength.

Figure 1.4.36. Schematic drawing of Firestone Compression Fatigue Tester.
EFFECTS OF TWIST ON CORD PROPERTIES 261

DRIVEN PULLEY

TEST BELT

IDLER

APPLIED LOAD

Figure 1.4.37. Schematic drawing of Dunlop Fatigue Tester.

]h-^ORD

Figure 1.4.38. Schematic drawing of De Mattia Flex Fatigue Tester.

1.4.1.11. Effects of Twist on Cord Properties

Many papers have been published deaHng with effects of twist on

cord properties, some are theoretical, others experimental.

Tire and tire cord producers throughout the world do not agree on which twist gives the best results. Some of them choose a symmetrical

twist and others prefer an asymmetrical twist. They also adopt different twists for different materials; usually

polyester cords are twisted more highly than nylon cord. Generally

speaking, as twist increases, cord strength, initial modulus, and fatigue

resistance in cyclic tension decrease, while elongation at break, rupture energy, and fatigue resistance in compression increase.

Kemmnitz and co-workers have reported extensive studies on rayon

cord dealing with effects of twist on cord properties. Figures 1.4.39

to 1.4.46 [132] show their experimental results. Cord construction was llOOD/2.

The effects of twist on strength were observed when the ratio of
262 TIRE CORD ADHESION

397 (TURNS/ lOCM)

r

0 20 AO 60 80 100 120 SINGLE TWIST (TURNS/lOCM)

Figure 1.4.39. Effect of twist construction on breaking strength of raw cords [732].

o

0 5 10 15 20 25 30 TWIST SHRINKAGE (V.

)

Figure 1.4.40. Breaking strength of cord as a function of twist shrinkage [123].

0 10 20 TWIST SHRINKAGE (V.) Figure 1.4.41. Impact rupture energy as a function of twist shrinkage [132].
EFFECTS OF TWIST ON CORD PROPERTIES 263

12 o 10 z

K.i'

/

k 8 12 16 20 24 TWIST SHRINKAGE( V.) Figure 1.4.42. Energy damping as a function of twist shrinkage [132].

A 8 12 16 20 22 TWIST SHRINKAGEC/.)

Figure 1.4.43. Energy loss as a function of twist shrinkage [132].

— 70

PLY TWIST

50 TURNS/ 10CM

59

80 120 SINGLE TWIST (TURNS/ 10CM)

Figure 1.4.44. Residual strength after 10'^ and 10^ cycles of flexing (De Mattia Tester)

[132].
264 TIRE CORD ADHESION

'7100

• a ^80

UJ ^ ov

1-u. O

< ^60

OS

UJ

LlI _iI o o tn UJ :i2o

or

t—

I/)

i1 +

1

/

/zo 10 5 ZV

01 02

Figure 1.4.45.

1 ZV TWIST LOSSC/O

Correlation between strength loss after 10^ cycle of flexing and twist loss

[132]. ply and cord twist was not equal to unity. Cord strength is indicated

as a function of twist shrinkage in figure 1.4.40. In figure 1.4.41, rupture energy is shown as a function of twist shrinkage. Figures 1.4.42 and 1.4.43 show the relationship between dynamic

properties and twist shrinkage.

These cord properties were stated to be uniquely determined by a simple function of twist shrinkage in the range examined. In figure

1.4.44, effects of twist construction on fatigue resistance are shown as residual cord strength measured after a known period of flexing on

the De Mattia Flex Tester. The relationship between residual strength after a known period

of flexing, AF, and twist loss of cord ZV (Zwirn Verlust) are indicated

in figure 1.4.45. AF decrieases linearly with ZV and the optimum residual strength

is obtained by suitably balancing AF with ZV.

Figure 1.4.46 also shows the relationship between amount of twist

and cord strength after flexing with number of load cycles as a parameter

NUMBER OF

CYCLES

48 51 60 TWIST (TURNS/ lOCM)

Figure 1.4.46. Residual strength of cord after flexing on De Mattia Tester as a function of ply twist and cycles offlexing [132].
RUBBER-TO-CORD BONDING 265

[72, 133]. Twist construction was symmetrical twist of 1650D/2. Obviously

the symmetrical twist 49 turns per 10 cm. reached its peak strength

already after application of few load cycles. Similar results have also been obtained with Disk Fatigue Tests and tire fleet tests [113], figure

1.4.47. Furthermore, it is well known that fatigue resistance measured by

the Mallory Tube Test also increases with amount of twist in the ordinary

twist range [65, 66]. Fujimoto stated that fatigue life in cyclic tension decreases with the

amount of twist in the ordinary twist range, while flexing and compressive

fatigue life increase, figure 1.4.48 [134]. 1.4.2. Rubber-to-Cord Bonding

1.4.2.1. Introduction

Composite rubber and textile products are a considerable portion of

all rubber products. For example, in Japan new rubber consumed for

tires, belts, hose, and coated fabric was about two-thirds of the total

TIRE FLEET TEST DISK TEST

9 10 11 12 13 14 15 16 TWIST (TURNS /INCH)

Figure 1.4.47. Effect of twist on cord strength before and after flexing in actual tire and

Goodrich Disk test [773].

1000

100 EXFEMSION

BENDING

MALLORY TUBE

) (POLYESTER)

2 A 6 8 10 12 14 16 18 TWIST (TURNS/ INCH)

Figure 1.4.48. Fatigue life of cord as a function of twist [134].
266 TIRE CORD ADHESION new rubber consumption in 1967. Particularly, new rubber consumption

for tires reached about half the total. Adhesion is an important factor in using textile materials together

with rubber, as are the individual properties of each material. The roles of adhesion may be: (1) to give desirable properties, (2) improve durability,

and (3) maintain the shape of the composite material.

Tires, conveyer belts, and transmission belts are used under severe

conditions, and a very high level of adhesion is required. In general,

strong adhesion is obtained through adhesive treatment of the textile or through addition of bonding agents to the rubber compound. Hose

and coated fabric do not require such high adhesion. Therefore the

mechanical keying effect of textile to rubber gives sufficiently good

results for these uses. Though several methods are available to bond

textiles to rubber, it is necessary to select an adequate method according

to the raw materials and the application.

For instance, looking back upon the progress of tire cord adhesion,

in the early period (1890-1920's) cotton was used with no adhesive

treatment or with rubber cement treatment. As loads, horsepowers,

and operating speeds of automobiles increased, needs grew for stronger

and cooler running tires. Therefore rayon filament was introduced.

Rayon has good mechanical properties, but lacks adhesion. Initially,

reclaimed rubber-casein-latex adhesive was tried to improve this defect, but it failed to satisfy the requirements of the tire industry. Thereafter, studies of adhesives based on latex and thermosetting resin

progressed, and led to finding resorcinol-formaldehyde-latex adhesive,

commonly called RFL. RFL has met the requirements for modern

tires, and has gotten wide acceptance in the tire industry since 1935. In 1947 nylon 66 was introduced as a tire reinforcing material in the

United States. RFL adhesive is also effective with nylon. At present

this adhesive is used extensively for bonding rubber to rayon and

nylon cord. In 1936 (for the first time) steel wire cord tires were pro- duced in France. Adhesion of wire can be gotten by zinc or brass plating.

Recently polyester fibers and glass fibers have been used in the tire industry. From the rubber side, natural rubber has been gradually re- placed by synthetic rubber and new materials, such as cis-l, 4-polybutadiene and ethylene propylene rubber, have been introduced in the

rubber industry. New adhesives and new methods have been developed

to bond these new materials satisfactorily.

1.4.2.2. Outline of Bonding Methods

Methods of rubber to textile bonding should be varied according to the application. It is convenient to divide the methods into the follow- ing three groups:

(1) adhesion based on mechanical keying effects,

(2) adhesive treatment of textile material,

(3) addition of bonding agents to rubber compound.

Each of the methods is explained briefly in this section, and details
! OUTLINE OF BONDING METHODS 267

of adhesive treatments of rayon, nylon, polyester, and other tire cords

will be described later. Mechanical interlocking, which is brought about by protruding

fiber ends or fabric openings, has an important role in rubber to textile bonding. Borroff and Wake investigated the function of staple fiber in adhesion [135-136]. The fabric tested was a five-end satin weave which

consisted of rayon filament as warp and a staple yarn-filament yarn

combination as weft. Fabrics were coated with a rubber cement, and

then vulcanized in an oven. Adhesion was measured by the direct

tension method. As shown in table 1.4.19, bond strength is remarkably

affected by amount of staple yarn as well as type of rubber cement

and kind of textile material. The effect of staple yarn may be due to mechanical keying around the protruding fiber ends.

Also, rubber to textile adhesion varies with the nature of the weave. Adhesion to a fabric woven from continuous filament yarn is related to

its interfacial surface area. Differences in surface area can be secured

by differences in weave. The dependence of adhesion on interfacial

surface area is obvious from the results shown in figure 1.4.49 [137]. Use of very bulky yarn enables rubber to bond to fabric without any

adhesive treatment [138]. In this case adhesion may be brought about

through an anchoring effect of Httle loops formed by the air bulking

method. Also insertion of coarse fabrics into rubber is effective for reinforcing the composite material. This method is applicable to materials which are difficult to bond since rubber bridge formation through

openings of the coarse fabric contributes to adhesion. For example re- inforcement of silicone diaphragms has been achieved by insertion of

multiple layers of nylon nets into the rubber plate, since silicone rubber

is a favorable raw material because of its chemical and heat resistant

properties, but it is very difficult to bond to textiles. Rubber cement adhesives: Rubber cement adhesives have been employed, preferably in the belt and coated fabric industries. These ad- hesives are based on rubber solutions to which very reactive chemicals

are added just prior to use. Preferred reactive chemicals are polyisocya- nates such as diphenylmethane diisocyanate, tolylene diisocyanate, or triphenylmethane triisocyanate. Commonly used solvents are inert

organic solvents, such as gasoline, toluene, halogenated hydrocarbons,

or ethyl acetate. A high level of adhesion is obtained by immersing a

textile material into rubber cements of low concentration or spreading

with doughs of high concentration. The adhesive treated textile is then

dried in an oven, and vulcanized with rubber compound. Immersion is suitable for cotton and rayon. Spreading is preferred for nylon and poly- ester [139]. This method is simple and available for every textile material

including cotton, rayon, nylon, and even polyester [139-140]. By choosing

suitable rubber compounds this method is available not only for natural

rubber, but also for polychloroprene, nitrile rubber, etc.

It has been recognized from general experience that triisocyanate is more effective. However, Meyrick and Watts [141] reported that the effect on adhesion was much the same for di- and triisocyanate. Adhesion is affected by drying conditions after immersion. As shown in figure 1.4.50,
268 TIRE CORD ADHESION

(U !-> 3 p ja

Z Oh Z

422-398
OUTLINE OF BONDING METHODS 269

long dwell time or too high oven temperature is detrimental [34]. In

general, severe heat treatment causes much crosslinking of the rubber

component in the adhesive. Hence, miscibility of adhesive with the

unvulcanized rubber to be bonded decreases, and adhesion decreases.

Optimiirr: conditions should be determined empirically.

However, use of this adhesive is limited by the following disadvantages: (1) necessity for solvent recovery, (2) danger of fire, (3) toxicity,

(4) long-drying time, and (5) sensitivity to moisture and short life. The

isocyanate group — NCO is easily decomposed by moisture and loses

its activity as an adhesive. Blocked isocyanates have been used instead

of free isocyanates to improve this defect [142-143].

RFL adhesives: If latex alone is employed as an adhesive, good

rubber to textile adhesion cannot be obtained because of lack of active

groups in the latex and weak tensile properties of the adhesive coating

film. Addition of a resin was studied to improve the mechanical prop-
270 TIRE CORD ADHESION

erties of latex film. The reclaimed rubber-casein-latex system initially

used provided improved adhesion, but failed to satisfy requirements of

today's automobile tire [144]. Next, several thermosetting resins were

used for this purpose. Phenol formaldehyde [145], urea formaldehyde

[146], aniline formaldehyde [147], and ketone formaldehyde resins

[146], were patented. Among them, resorcinol formaldehyde resin was selected to be used as an aqueous adhesive incorporated into latex because of its superior adhesion and ease of processing [148]. RFL is useful to bond rubber to textiles, such as rayon, nylon 6,

nylon 66, Vinylon and easy-to-adhere types of polyester i.e., Trevira

GPA, Diolen, DSP and Diolen V75 [93, 149, 150]. Also, RFL is used as a base for polyester adhesives. By suitable choice of the latex component, RFL provides good adhesion for many kinds of rubber as described later. A typical RFL treatment will be outlined below. The tire cord or

tire fabric is immersed in an RFL dipbath, and squeezed by passing

through a squeeze roU unit, vacuum unit, or beater to adjust pickup

of adhesive. Then the fabric is introduced into an air oven where drying and baking of the RFL is accomplished. Heat treatment is indis- pensable for nylon and polyester to give the preferred tire cord

properties. Baking of RFL can be done at the same time. Widely accepted

dipping machines for cord and fabric are illustrated in figures 1.4.51 through 1.4.54 [151]. A predip unit is included in this equipment. Water

predip before RFL immersion is widely accepted to improve physical

properties of dipped cords [151-152].

Double dipping systems are often employed for polyester. Polyester tire fabric is initially immersed in the first dip solution which

consists, for example, of aqueous polyepoxide solution, and heat treated. This treated fabric is then passed through the second dip solution

which is RFL, and heat treated again. A new dip equipment which in- cludes two dip units and two heat treatment zones has been introduced.

An example of this type equipment was described by Kersker and

Kovac of Goodyear [66]. A lucid review on the dipping machine was

presented by C. A. Litzler [277]. The treated cord or fabric may be stored a long time if it is sealed

in black polyethylene film to shut out light. It is said that addition of wax [201] or ketene dimer [202] to RFL results in good resistance to lowering of adhesion during storage.

Mechanism of rubber to cord bonding through RFL adhesive will be

outlined below. Although there have been numerous opinions concerning

the nature of the fiber to rubber bond, only a few papers have been

published about this problem. It is generally recognized that there are two primary considerations, nature of the RFL to fiber bond and nature

of the RFL to rubber bond.

The nature of the fiber plays the most important role in RFL treat- ment of tire cords. In practice it has been recognized that rayon and

nylon tire cords are easy to bond to rubber by RFL treatment, but with

polyester cord this is extremely difficult. This should be attributed

to the nature of the polymer structures. Rayon and nylon contain reactive
OUTLINE OF BONDING METHODS 271

HOT

DRYER STRETCH NORMALIZING

ZONE ZONE ZONE

LET SPLICE PRE DIP TENSION WIND

OFF PRESS DIP BATH ROLLS UP

BATH Figure 1.4.51. Schematic drawing of Ross fabric type heat set equipment [151]. groups such as hydroxy groups -OH and amide groups -CONH-. But poly- ester has a less reactive structure, and it does not have so-called active

hydrogen in the polymer chain. It was also suggested that ability for hydrogen bonding of polyester is less than that of polyamide [153]. Iyengar and Erickson deduced that lower adhesion of polyester is attributed to inaccordance of the solubility parameter of polyester

with resorcinol [278]. There are many opinions on how RFL functions with rayon and

nylon: hydrogen bonding between phenolic hydroxy groups in RFL

resin and electronegative groups in fibers, condensation reaction beFiGURE 1.4.52. Schematic drawing of Litzler fabric type heat set equipment [151].
272 TIRE CORD ADHESION

OVEN

q:

ID ROLLS

TENSION COMPENSATOR

WINDER DIP BATH PREDIP Figure 1.4.53. Schematic drawing of Kidde cord type heat set equipment. TIME Z0NE1 TIME ZONE 2

ADHESIVE A LET OFF WIND UP Figure 1.4.54. Schematic drawing of Goodyear continuous double dip equipment.

tween methyl group of RF resin and active hydrogen in the fiber, dipoledipole interaction, molecular entanglement [152-157]. However, the

evidence has been insufficient for any of these bonding mechanisms to gain wide acceptance. Also many opinions have been presented for the

rubber to RFL bond: interdiffusion between RFL and rubber, covulcanization of carbon-carbon double bonds in RFL with rubber, ionic interaction, chemical reaction between RF resin and rubber [158-162]

.

Rubber to textile adhesion can also be achieved by addition of bonding

agents to the rubber compound, thus eliminating adhesive treatment of

textile materials. Isocyanate derivatives have been used for this purpose

[163], but more widely accepted chemicals are combinations of resorcinol

and formaldehyde donors [164], such as hexamethylenetetramine or
OUTLINE OF BONDING METHODS 273

paraformaldehyde. Many modifications of this method have been

patented. For example, trimethylolnitromethane [165], trimethylolacetoaldehyde [166], or hexamethylolmelamine [164] are used. Encapsulation

of formaldehyde donors is also patented [167]. By addition of resorcinol formaldehyde donors to the rubber com- pound the best adhesion can be obtained even with untreated nylon

and rayon. But good results are not obtained for polyester. However

polyester to rubber bonding by this system is markedly improved by

polyepoxide pretreatment. Easy-to-adhere polyesters, such as Diolen

DSP or V75, bond very well, according to the Bayer's test results [168]. On the other hand, addition of p-chlorobenzoquinone together with

resorcinol and formaldehyde donor can be used to bond conventional

polyester (without pretreatment) to rubber [169]. The complex forming

property of chlorobenzoquine with aromatic compounds may function

in the specific action with polyester [170]. On the rubber side, silica enhances adhesion. A typical tire carcass

formulation may contain 40 phr^ carbon black. To use these systems

with this rubber, 15 phr of silica are added and the carbon black is reduced to 25 phr to keep the total fiUer loading the same. Optimum

amounts of other components are 2.5 phr resorcinol and 1.5 phr hexamethylenetetramine. According to Bayer's test results, the effect of

silica is evident.

Table 1.4.20. Bond strength of a typical fire hose compound with and without suitable bonding additives

Additives, phr

Carbon

black

Silica " Resorcinol Formaldehyde

donor * Cotton Rayon Nylon Polyester

60

60

30

30

2.7 4.6

3.8

8.6

1.5 2.5

3.2 15.8

1.1 3.3 2.4 17.5

1.0 2.3 2.4

5.6 30

30

2.5 2.5

2.5 2.5

Bond strength, kg/25 mm

Vulcasil S. Bonding Agent 7110. At present these methods are used in manufacturing conveyer belts,

transmission belts, hose, and footwear. Pittsburgh Plate Glass, Degussa,

and Bayer are all making efforts to develop these systems and they are working with rubber companies to use these adhesives for tires.

It is not well understood why these systems work. Resorcinol and the

formaldehyde donor form resorcinol-formaldehyde resin in rubber. The

resin may combine with both rubber and textile as described before.

phr = parts per hundred rubber.
274 TIRE CORD ADHESION

Besides the above methods, addition of resorcinol and azomethynes

[171], resorcinol-aldehyde polymer and azodioxybicyclooctane [172],

methylene donor and acceptor to the rubber compound are patented

[173]. Recently many practical details of the method have been disclosed

[279-281].

1.4.2.3. Adhesive Treatment of Nylon and Rayon — RFL Treatment

Many factors are known to affect the bond strength secured with RFL

adhesives. For example, composition of RFL, method of RFL preparation, amounts of adhesive applied to cords, its distribution on the cord,

heat treatment after dipping, method of storing dipped cord, composition

of rubber compound, and vulcanization conditions may be factors affect- ing the cord to rubber bonding. Participation of each factor is explained

in the following sections.

RFL adhesives are usually prepared by reacting resorcinol and

formaldehyde under alkaline conditions prior to addition to latex. The

mixture of RFL resin solution and rubber latex is further aged before

use. An example of RFL preparation is as follows [174].

Sodium hydroxide, resorcinol, and formaldehyde are dissolved suc- cessively in water. After complete solution is effected, a reaction takes

place which should be allowed to continue for six hours at 25°C.

Resin solution

Water 238.4 g

Resorcinol 11.0 Formalin, 37% 16.2 Sodium hydroxide 0.3 The RFL recipe should be varied with the textile material. The follow- ing formulations have been found to give optimum adhesion with rayon

and nylon.

Fabric

Gentac latex, 41%

SBR latex, 40%

RF solution, 6.5%

Water

Rayon Nylon

52.5 g 428.0 g 215.0

284.0 465.0

467.8 107.0

The recommended maturing condition for the above RF-latex mixtures is six hours at 25° C. Precondensation of RF resin is, however, not indispensable when

synthetic latex is used [154]. Also resorcinol-formaldehyde precondensate, such as Penacolite

resin of Koppers Co., is available to prepare RFL [159]. Adhesive is prepared by successively mixing the following components:
ADHESIVE TREATMENT OF NYLON AND RAYON 275 Water

Sodium hydroxide, 10% aq. sol. Penacolite resin R2170, 75%

Formalin, 37%

Vinylpyridine latex, 40%

407.7 g

8.0 26.7

20.3

250.0

The mixture should be aged 18 to 20 hours before use. Adhesion is affected by the composition of RFL and maturing conditions. Resorcinol to formaldehyde ratio, pH of solution, concentration, maturing time and temperature have an effect on the structure and molecular weight of resorcinol formaldehyde resin. Choice of

latex component is very important on the viewpoint of affinity of RFL

for rubber. Latex to resin ratio is also influential in adhesion. Strength

of RFL film and balance of affinity to both fiber and rubber depend on L/RF resin ratio. Nowadays, the composition of RFL for nylon and rayon has been

decided empirically. R/F is in a range of V2 to V4 in molar ratio, and

L/RF resin is in Vs to Vis in weight ratio of solid components. The

most widely accepted latex component for natural rubber and/or SBR

compound is vinyl pyridine-styrene-butadiene terpolymer latex. In some cases, mixture of vinyl pyridine latex with SBR latex on natural

rubber latex is used [34, 154, 159, 176-194].

Adhesion is affected by conditions of dipping and of heat treatment

after dipping.

Effect and control of RFL pickup: Pickup of adhesive on a cord and

penetration conditions of the adhesive into the cord interior affect adhesion. From general experience, bonding strength increases with

pickup of adhesive. It is, however, important to consider not total pickup, but effective pickup since penetrated RFL does not contribute

to adhesion. It is observed by microscopic inspection of a cross section

that RFL penetrates into the cord interior to a certain extent. Penetration may be controlled by squeezing conditions [154], con- centration of RFL, viscosity of RFL, cord tension in the dipping bath

[195], and presence of water predip. However, the contribution of penetration of RFL to adhesion has never been quantitatively analyzed.

The dependence of adhesion on RFL pick-up was reported in many

papers. Adhesion increases gradually with pickup and reaches a saturation

point. An example is shown from Dietrick [159], and illustrated in figure

1.4.55. Too much pickup should be avoided because the amount of adhesive affects stiffness of the cord [197]. Also a pickup level should be

decided upon by balancing cost against adhesion level. It is generally

recognized that an adequate level of RFL pickup is from 6 to 8 percent

of the cord weight.

Pickup level is affected by the kind of textile material, textile lubricant,

concentration and viscosity of adhesive solution, and conditions of squeezing. Looking at the kind of textile material, nylon takes up the

adhesive with more difficulty than rayon because of its hydrophobic

nature. Polyester has still less pickup than nylon. A comparison for
276 TIRE CORD ADHESION

Figure 1.4.55. Effect of adhesive pick-up on pull through load [159]. rayon and nylon is shown in table 1.4.21 [1541. If dipping is done under

the same conditions where concentration of RFL and squeeze pressure

is held constant, rayon cord has more RFL pickup, and adhesion is superior to that of nylon cord. Concentration of adhesive should be

changed with the textile material to adjust the pickup level. For example,

it is recommended that suitable concentrations for nylon and rayon are 20 and 12 percent respectively [174]. Table 1.4.21. Relationship between squeeze conditions and pickup [154]

Latex RFL

RFL

cone, %

Squeeze

roll, lb

Rayon Nylon

Pickup,

%

Pull-through

load, lb Pickup,

%

Pull-through

load, lb

Butyl latex 12.5 0 8.5 17.2 3.3 11.4

12.5 15 2.2 14.5 2.8 10.7

12.5 35 3.5 12.5 2.2 10.6 SBR latex 0 7.3 26.4 3.2 17.7 15 4.9 23.9 2.9 15.9 35 3.7 19.2 2.5 14.2

Even if the textile material is the same, different lubricants cause

different pickups and different bond strengths. Gillman and Thoman

[198] reported on this problem using the casein-latex adhesive system

with rayon tire cord. When effects of waxy ester and sulfonated oil are compared, hydrophilic cord lubricated with sulfonated oil has higher

pickup than hydrophobic waxy ester cord. Materials that have been

used in tire cord lubricants are white mineral oil, petroleum sulfonate,

triglyceride, ethoxylated and sulfonated derivatives, and so on. Generally, two or more of these materials are blended to produce a proprietary

formulation. Petroleum sulfonate is often used to enhance cord to rubber

adhesion [199]. The dependence of adhesion on lubricants is, however,

a very complicated phenomenon, and cannot always be explained solely

by the hydrophile-hydrophobe nature of the lubricants.
ADHESIVE TREATMENT OF NYLON AND RAYON 277

Also, pickup level depends on the adhesive concentration and squeeze

conditions as shown in table 1.4.22. Change of concentration is more

effective than change of squeezing conditions.

Table 1.4.22. Effect of hydrophilic nature of lubricants on adhesive pickup and adhesion

Adhesive cone, %Viscosity at 25 °C, cp Waxy ester on cord Sulfonated oil on cord Pickup, % Adhesion, lb Pickup, % Adhesion, lb 5 1.67 0.4 7.8 1.7 9.0

7.5 1.94 0.5 8.0 2.6 10.6 10.0 2.46 0.8 8.1 3.9 10.9

12.5 3.53 1.2 9.4 6.1 12.2 15.0 8.36 1.9 10.9 8.4 13.8

17.5 26.5 2.5 11.0 9.5 14.1 20.0 57.2 5.9 12.0 22.0 17.4 22.5 83.5 21.1 14.8 46.3 19.8

Effect of heat treatment after dipping: Heat treatment conditions

should be decided from the kind of textile material, adhesive composition, and procedure for the condensation reaction of the RF resin. Since each heat treatment equipment has its own heat efficiency, it is impossible to estabUsh identical heat treatment conditions. Roughly

saying, commonly accepted conditions are: Rayon 155° to 165° C 2 to 3 min.

Nylon 6 205° to 210° 0.5 to 1 min.

Nylon 66 220° to 230° 0.5 to 1 min.

During heat treatment, a highly crosslinked structure is formed in RFL and strong interaction between adhesive and textile is achieved.

Weak heat treatment causes the RFL coating to have inferior tensile

properties and there is lack of interaction with the textile. Strong treat- ment impairs compatibility with the rubber. Both over- and under8

HEAT TREATMENT AT

220°C 200^0 ISO'C

0 2 A 6 8 10 HEAT TREATMENT TIME (MIN.) Figure 1.4.56. Variation of adhesion with heat treatment conditions.
278 TIRE CORD ADHESION

heat treatments are undesirable for adhesion. These features are shown

in figure 1.4.56. Iyengar reported the similar results on nylon 66 and

6, [282]. Examples of heat treatments used by United States tire manufacturers for nylon 66 tire fabric were given in a previous section, table 1.4.2. During the subsequent processes in building tires, rubber-to-cord

adhesion is affected by calendering temperature and vulcanization

conditions, but mostly by composition of the rubber compound.

To obtain a high level of adhesion, it is important for rubber and

the adhesive on the cord to have an opportunity for contact. The op- portunity relates to fluidity of the rubber compound during processing.

For example, adhesion increases with rising rubber temperature in the calendering process. However, at too high temperatures, cross- linking of the rubber proceeds with passage of time, and its fluidity

gradually decreases. Too high calendering temperature causes rather

inferior adhesion for this reason [203]. The opportunity for interaction

also depends on time of contact. As scorch time of the compound in- dicates the time which it is fluid, opportunity for contact can be rep- resented as a function of scorch time. Dependence of adhesion on

scorch time with various rubber compounds is shown in figure 1.4.57,

[204]. The rubber compound with a long scorch time, that is, with high

probability of contact, showed high adhesion in both static and dynamic

evaluations.

Adhesion is associated not only with opportunity of contact, but

also with polarity of the rubber compound. Polarity of a compound

may be varied by choice of rubber and filler. It has been stated that higher adhesion attained by substituting channel black for furnace

black was primarily due to surface polarity of the channel black [204]. Rubber compounds with highly polar silica filler showed a very high

level of adhesion if they were compounded to have a very long scorch

time. Iyengar suggested that long scorch time and high polarity were

two requirements for development of good adhesion.

o

40

UJIQ

<

J. STATIC/^

DYNAMIC

0 5 10 15 20 SCORCH TIME (MIN.) Figure 1.4.57. Variation of adhesion with scorch time of rubber compounds [204].
POLYESTER TO RUBBER BONDING 279 When adhesion is at a very high level, it is observed that a test specimen in the pull-through test is broken by failure of the rubber. In these

cases, pull-through value is determined by the properties of the vul- canizate. Meyrick and Watts reported on this problem using rubber

compounds with various fillers. They found that there was some rela- tionship between hardness and modulus (stiffness) of the vulcanizate

and adhesion level [141]. The properties of vulcanized rubber are, of course, affected by vul- canization conditions. Adhesion is also affected by these conditions. If adequate properties of the unvulcanized rubber are secured by adjustment of the cure cycle high adhesive strength is obtained even at high

vulcanization temperatures [34].

1.4.2.4. Polyester to Rubber Bonding

The potential of polyester fiber as a tire reinforcing material was

recognized as described before, but its use has been delayed by lack of

a suitable bonding method. This situation has, however, been improved

by introduction of new adhesive systems such as the double dip system.

Nowadays, use of polyester tire fabrics is increasing steadily in the United

States and Japan.

Looking back on the progress of polyester to rubber adhesion, attempts were first made to modify existing RFL adhesives. But this approach

proved inadequate because the adhesion between fiber and RFL was

insufficient [66]. Poor adhesion may be due to reduced ability to form

hydrogen bonds [153], lack of reactive hydrogen in the polyester molecule,

and its hydrophobic nature [205]. Therefore many efforts have been made

to secure a high level of adhesion both by improvement of the fiber and

improvement of the adhesive.

Modification of polyester fiber can be attained by solvolysis besides

copolymerization or blended spinning. Immersing the polyester fiber in alkaline solution causes it to undergo hydrolysis and many hydroxyl

groups and carboxyl groups are produced on the surface [206]. These

groups give sites for the condensation reaction and hydrogen bonding

with RF resin. Contribution of sodium hydroxide pretreatment of the

polyester to rubber adhesion is obvious from results shown in figure

1.4.58. Only ten seconds immersion in 20 percent aqueous sodium hydroxide solution at 65° C produced Terylene tire cord to rubber bonding after RFL treatment [207]. Amines are avilable too. In this case, amide and hydroxyl groups are formed on the polyester surface [206, 208]. Too long immersion causes somewhat decreased adhesion [34]. This phenomenon may be caused

by deterioration of the surface layer of the cord. The polyethyleneimine-RFL double dip system of N. V. Onderzoecking

Inst. Res. may be based on aminolysis [209]. Polyethyleneimine contains

primary amino groups to the extent of about 30 percent of the total amino

groups [210]. These primary amino groups attack ester links, and polyethyleneimine is combined tightly at the polyester surface. Imino-HN-,
280 TIRE CORD ADHESION

20

Q 15 <o

10

''^CONTf

Figure 1.4.58.

0 12 3 IMMERSING TIME INTO SODIUM HYDROXIDE SOLUTION (MIN.) Effect offiber immersion in sodium hydroxide solution on rubber to Terylene adhesion after RFL treatment. and amino-NH2 groups in polyethyleneimine may have the same affinity for RFL as do the hydroxyl groups of rayon. Pretreatment by aqueous solution of a carboxyl-group-containing

polymer such as polyacryiic acid was also proposed [211]. Now these solvolysis methods have been abandoned because adhesion

is not sufficient in practice, and there is strength loss of the fiber. Several new adhesives have been found as the result of major efforts. These adhesives may be classified as follows by the nature of the functional material.

(1) Isocyanates, blocked isocyanates

(2) Ethyleneureas

(3) Modified polyvinylchloride

(4) Polyepoxides

(5) N3,Pexul

In this section, details of each method will be explained.

Isocyanate rubber cement is useful in polyester to rubber bonding

as described before. The isocyanate group, -NCO, has a specific action

for both polyester and rubber. Once polyester is treated by isocyanate,

which is used as an organic solution, adhesion to rubber is facilitated remarkably after RFL treatment. The process and experimental results with this method are shown in what follows [141].

Polyester cord

Polyisocyanate

in organic solvent

Drying RFL Treatment

Resulting strength of adhesion

Isocyanate/RFL double dip system 36.3 lb/in. RFL single dip system 8.8 lb/in.

Ail
POLYESTER TO RUBBER BONDING 281 Bond strength also depends on pH of the RFL. Satisfactory adhesion can be achieved over a pH range of 9.8 to 10.7. A further interesting feature is that cord treated with polyisocyanate can be stored for prolonged periods under laboratory conditions, e.g. six months, without fear of loss of adhesion. Patents were issued for these methods [185, 186, 212]. They are simple and superior for adhesion,

but requirements of solvent recovery and ventilation, as well as toxicity, limit their wide acceptance in the tire industry. Another drawback is its sensitivity to water. Since isocyanates react easily with water and

lose their activity [213], they cannot be used in an aqueous adhesive

system as is. Stabilization against water is accomplished by encapsulation [214], dimerization [215-218], or blocking of isocyanate [215]. Utilization of blocked isocyanates is commoner. Phenol is widely ac- cepted as a blocking agent. Phenol blocked isocyanate is not decomposed

further in water, and regenerates free isocyanate when it is heated at 140 to 170° C. Some blocked isocyanates are sold on the market, for example, phenol blocked diphenylmethane diisocyanate is marketed

by DuPont as Hylene MP.

The adhesive solution is prepared by mixing ball-milled Hylene MP,

latex, and thickener. A mixture of Hylene MP and RFL is also a good

adhesive for polyester [219-224].

Ethyleneurea is a derivative of an isocyanate, but does not act as an

isocyanate generator [225]. Ethyleneurea is prepared through the addition reaction of ethyleneimine to isocyanate.

Originally ethyleneureas were used as an adhesive for wood [226], or were incorporated into rubber cement as a bonding agent of cellu- lose fiber or polyamide fiber to rubber [142, 143]. Application of ethyl- eneureas for polyester-to-rubber bonding was developed by Japanese

tire manufacturers and fiber producers [222, 227-231]. They are avail- able for both single and double dip systems. A single dip system combined with RFL was, however, developed purposefully.

As low-priced isocyanates are limited to tolylenediisocyanate and

diphenylmethane diisocyanate, only aromatic ethyleneureas have

practical significance. They are insoluble in water and settle out easily.

Several ways have been developed to improve mechanical stability of the resulting dip solution such as addition of thickener, introduction

of methylol groups into ethyleneureas [232], and so on [231]. With the

thickened dip system, maturing conditions of the RF resin should be

very carefully controlled to avoid gelation of the dip solution [233-234].

Bond strength also depends on the heat treatment [34]. The ethyleneurea-RFL single dip system had been developed in Japan, but it did

1 not find practical use in the tire industry because of poor dynamic

I performance at tire operating temperatures and inferior adhesion

with high vulcanization temperatures. As to the action of ethyleneurea

with polyester, Timmons speculated that it functioned the same as isocyanates, i.e., isocyanate should be regenerated from ethyleneurea

[240]. But this does not seem to be correct since ethyleneurea changes

thermally to an isomerized product or a polymer, but never dissociates

I to an isocyanate and ethyleneimine [225]. Kigane's concept that the
282 TIRE CORD ADHESION

adhesion is dependent on reaction of polyester polymer chain ends

and aziridine rings is fairly acceptable [235]. In 1958 a new aqueous double dip system was invented at CIL (Canadian Industries Limited) [236, 237]. The first dipping solution is coded TR-5, and consists essentially of a poly(vinylchloride) latex and

a reactive polyamide. The TR-5 emulsion is nontoxic and its composition is as follows: Geon 151 (PVC 50% emulsion) 40.0

Versamide 125 5.0 Dioctylphthalate 12.0 Catanac SP 4.0 Triton X-100 0.6 Acetic acid 0.3 Isopropanol ; 2.8 Water 60.0

Versamide 125 is a low molecular weight polyamine resin containing

many free amino groups. High speed emulsifying equipment and specialized techniques are required to prepare TR-5. Since recommended solid pickup of TR-5

is 0,5 percent concentration, the above concentrated TR-5 solution is diluted 7 : 1 with water just prior to use. The second dipping solution is RFL as commonly applied to rayon and nylon.

Adhesion depends on the heat treatment, especially after the first dipping, see figure 1.4.59. Relatively low temperature heat treatment,

150° C, after the second dipping gives satisfactory results [238]. This

process was used in some commercial polyester tire production in Canada, but it did not find wide use. It is said that lack of wider accept- ance was caused by the necessity of using special techniques in preparing

a stable emulsion and by stiffness of the dipped cord [240]. The original use of polyepoxide for rubber to polyester bonding is found in patents secured by N. V. de Bataafsche Petroleum and Shell

Development. Many other systems have been developed from these

systems. They may be classified into three groups shown in the following

schemes, see figures 1.4.60-1.4.62. Group 1: The adhesive is composed of polyepoxide, hardener, and

latex. This class is exemplified by using the TRL-12 system of Du Pont

[69]. First dip composition is the following mixture:

Glycidylether of glycerin 12.5 g

Lauryl sulfate 0.5 g Water 125 cc Gentac, 41% 50 g Second dip solution is prepared by mixing the following components:

Metaphenylenediamine

Water

Gentac, 41%.

6.3 g

138 cc 195 g
POLYESTER TO RUBBER BONDING 283

r'olyester cord is immersed in the first dip solution and then heat treated

at 232° C for 1.5 minutes. This treated cord is immersed in the second

bath and heat treated at 232° C for 1.5 minutes again. While the cord

treated only with the first dip shows relatively poor adhesion, 2 lb., the doubly dipped cord has strong adhesion, 10 lb. This double dip system

is superior to the D-15 blocked isocyanate system for adhesion as shown in table 1.4.23. Drawbacks are lower adhesion at high temperature

and difficulty of handling in subsequent processing because of stickiness

of the dipped cord. Group 2: All methods are double dip systems in which the first dipping

solution is a mixture of polyepoxide and hardener, and the second is GROUP 1

N.V. DE BATAAFSCHE PETROLEUM MAATSCHAPPIJ BRIT P. 799381 09*)

FIBER LATEX PaYEPOXlOE

HARDENER

HEAT TREATMENT

TOYO RAYON T. TAKEYAMA ar^l M EKCK)

Poor ESTER GRAFTED LATEX AMirc HARDENER

HEAT TREATMENT

DU PONT (M.SCHEPARO) TRL 12 SYSTEM LLSP 3l30ep07 (l9€7) JAB^N P 38-1 J^lS (1963) LATEX

POLYEPQXI OE OPTIONAL LATEX HEAT DRYTNG HAROE-NER TREATMENT

PIRELLI SOCIETA PER AZIONI BRIT. P. IP 36951 (1966) POLYESTER GMA-BD CgPQLYMER ZINC OXIDE SULFUR DRYING

LATEX

ZINC OXIDEf HARDENER

HEAF

TREATls^ENT

TOYO RAYON (T. TAKEYAMA and M. ENDO) JAPAN P-i2-ieAI0 (1967)

POLYESTER GRAFTED LATEX HEAT TREATMENT

LATEX

AMINE

Hardener HEAT 1 TREATMENT^

Figure 1.4.60. Flow sheet of dipping methods. Group 1 systems.
284 TIRE CORD ADHESION

GROUP 2 SHELL DEVELOPMENT (CW SCHROEDER)

POLYESrER

DU PONT

POLYESTER

l.CI, PaYESTER

FIRESTONE

POLYESTER

DU PONT

POLYESTER

(H R KRYSl-XK) POLY EPOXIDE

LACTAM

NYLON 66 SALT

POLYEPOXIDE VERSAMIDE

0RGA<-J!C SOLVENT

POLYEPOXIDE LATEX

ZINC FLUQROBORATE

U.S.P 2.902.398 (19 59) POLYEPOXIDE HEAT HARDENER TREATMENT RFL

U.S.P 3.222,238 (1965) HEAT

TREATVIEI^T RFL

BELG.P 662.282 (1967) HEAT

TREATMENT RFL

BELG.P. 585,065 (1967) HEAT TREATMENT RFL HARDENEP (CJSHOAF) DA17 SYSTEM

HEAT

TPEATMENTl

HEAT

TREATMENT

HEAT TREATMENT

HEAT TREATMENT

POLYEPOXI DE

ISOCYANATE DERIV HEAT

TREATMENT

U.S.P 3J07^66 (1967) JAPAN P A2-11,A82 (1957) RFL HEAT

TREATVENT BRIDGESTONE PaVESTER GOODYEAR

POLYESTER

( R FUJISAKI.R YASUHARAardT HONDA) JAPAN P 42-9,004 (1967) POLYEPOXIDE ITHYLENEUREA LATEX

lEAT TREATMENT RFL

HEAT TREATMENT

(J A CAROINA) U.S.P. 3,190.764 0965)

IpOLY EPOXIDE ^FL

JAPAN R 40 - 9.622 (1%5) DRYING RFL HEAT TREATMENT Figure 1.4.61. Flow sheet of dipping methods. Group 2 systems. RFL. The D417 system of Du Pont containing isocyanate as hardener

is a typical example of this class. The composition of each dipping solu- tion is tabulated as follows [261]:

First bath

Triton X-100 0.4 cc Phenol blocked

diphenyl methane

diisocyanate 16.0 g Water 400 cc Diethylaminoethyl

methacrylate,

0.5% aq. sol 25 cc

Glycidylether of

glycerine 4.8 cc Heat treatment after immersion was 218° C for 45 seconds.

Second bath

Resorcinol 73.7

Formalin, 37% 40.0

Vinyl pyridine latex,

41% 148

Water 480

in the first and second dip solutions

A series of test was conducted varying the proportion of the polyepoxide and isocyanate from 0:100 to 100:0. Results are shown in figure 1.4.63. For comparison ethylenediamine was used as a hardener. The results are shown on the same graph.
POLYESTER TO RUBBER BONDING

Gf?OuP 3

285

GOODYEAR (J. A. CAADINA) U.S.P. 3 247,043 (1956} J ARAN P 39-I05R (I 9 6 4)

POLYESTER DEERI NG M ILLI KEN

PaYESTER

POLYEPOXrOE

HEAT TREATMENT RFL

HEAT

TREATMENT DEERI NG M ILLI KEN RESEARCH COPP ( W C PRUI T T and W.J.SCHRQEDER) USP. 523l/il2 (1966) POLYEPOXIDE

VP LATEX HEAT

TREATMENT

? RFL

HEAT TREATMENT

TDYO RAYON (E KATO r TAKEYAMA and M. ENDO ) JAPAN P. 42-3548(1967)

POLYESTER

POLYESTER

POLYEPOXIDE

LATEX HEAT TREATMENT

(T TAKEYAMA and MENDO) JAPAN P GRAFTED LATEX HEAT TREATMENT

RESORCINQL

HEAANtTHVLENE -TETRAMINE

LATEX

HEAT TREATMENT

RFL HEAT TREATKtNT

Figure 1.4.62. Flow sheet of dipping methods. Group 3 systems.

Table 1.4.23. Comparison ofdip systems _ \

Dip TRL-12 Isocyanate RFL

Test method \

\ Rubber NR SBR SBR SBR

PuU-through load, lb/0.25 in Cold Hot

Strip force, lb

28.8

10.8

4.2

22.0

9.5

7.9

19.3

7.4 6.3

16.0 6.0

1.5

It is well known that proper combination of the isocyanate and polyepoxide increases the adhesion, and amine hardener is less effective

under these experimental conditions.

Q 6

0

100

ISOCY/ NATE

AMIN E

20

100 EPOXIDE

0 HARDNER Figure 1.4.63. Effect of polyepoxide to hardener proportion on adhesion.
286 TIRE CORD ADHESION

Although this system has many disadvantages such as requiring a double dip, necessity of a time consuming ball-milling process, settling out tendency of the dispersion, and lack of good dynamic performance

at high temperatures required for use in tires [65], many tires have been

built using this system, and it still provides a commercially available

method.

Group 3: This class is also a double dip system. Polyepoxide is used

in the first dipping solution without hardener, and the second is RFL.

An example of this class is Deering Milliken's process. Polyester cord

is passed through the first dip solution. The recipe is as follows:

Vinyl pyridine latex ,41% 125 g Eponite 100 « 31.2 Petrowet 1.25 Water 312

" Eponite 100 is diglycidyl ether of glycerine (Shell). Wet pickup is adjusted to give 3.3 percent solid pickup by passing

through a squeeze roll unit. The dipped cord is then passed into a heat

treatment oven at 232° C for 96 seconds. The resulting tacky cord is immersed in an RFL bath and heat treated again at 230° C for 120 sec- onds with slight stretching. The second dipping solution is RFL which is prepared by maturing the following composition for 96 hours at room

temperature:

Resorcinol 45 g

The resulting cord is substantially free of stickiness and processes well. Adhesion of this doubly treated cord to rubber is excellent. Table L4.24

gives the adhesion obtained at each step of this process. In this method

the adhesion depends upon the composition of the first dip solution and

the heat treatment. The proper range of concentration of polyepoxide

gives a high level of adhesion as shown in figure L4.64. Higher temperature treatment is preferable to give a high level of adhesion. This cor- relation is common to most methods utilizing the polyepoxide. The

following results shown in figure L4.65 were obtained with Deering

Milliken's method.

A similar correlation is found with the polyepoxide-Versamide/RFL

double dip system of ICI ^ (Belg. 662,282) and with the D417 system,

i.e., polyepoxide-isocyanate/RFL double dip system of Du Pont [261].

^ Imperial Chemical Industries.

Formalin, 37%

Water

Isopropanol

Vinylpyridine latex, 41%,

Sodium hydroxide, 10% . Optional amount to adjust pH to 9 to 9.3

, 22 219

5.8 144
POLYESTER TO RUBBER BONDING 287

Figure 1.4.65. Effect of heat treatment temperature on adhesion of Deering Millikens

system.

Table 1.4.24. Comparison of adhesion at each step of Deering Millikens system

Step Pull-through

load, lb

Untreated 0 to 4 Only first dipped 14 to 16 Only second dipped 10 to 12 Doubly dipped 27
288 TIRE CORD ADHESION

Table 1.4.25. Effect of heat treatment on polyester cord to rubber adhesion

Polyepoxide-Versamide/RFL system

First temp, °C

20 45 90

205° 7.12 7.53 234° 5.44 8.71 10.57 246° 7.85 9.53 260° 9.85 10.70 11.66

Time, sec Pull-through load, kg/6.35 mm

The most popular epoxide for this purpose is glycidyl ether of glycerine

since it is easy to handle because of its water solubility. But it is necessary to have regard for the life of the dipping solution because the epoxide

groups gradually undergo hydrolysis in an aqueous solution, and activity

drops by half in a week [34]. Compositions of RFL used for the second dip solution are either the same for nylon and rayon or a special one for polyester. RF resin of lower

formaldehyde content is found in several patents for this purpose al- though a lower formaldehyde RFL is inadequate for nylon and rayon. Examples of special recipes are shown in table 1.4.26. Recently easy-to-adhere type polyester yarns have appeared in West

Germany such as Trevira GPA [150], Diolen DSP [93] and V75 [149]. Cords or fabrics of this type polyester give a good rubber to textile ad- Table 1.4.26. Examples of REL recipes for the second dip solution

A B C

Resorcinol/formaldehyde

molar ratio 1/0.66 1/0.68 1/0.73 Latex solid/RF solid weight ratio 3.3/1 1.7/1 2.2/1 Resorcinol 48.4 10.3 10.0 Formalin, 37% 23.3 50.5 54 VP latex, 40% 505 646 SBR latex, 40%

Sodium hydroxide, 10%

445

6 Optional amount Water 463 330 200 Source Goodyear Deering Milliken Du Pont

U.S. 3,247,043 U.S. 3,231,412 U.S. 3,307,966
POLYESTER TO RUBBER BONDING 289

hesion with the same RFL treatment as nylon or rayon. RFL treated

Diolen DSP cable cord (1000D3/3) gives a pull-through load of over 20kg/cm.. but the normal type only lOkg/cm. A comparison of two Diolen

types using two vinylpyridine latexes for the RFL component, and two

adherent rubber compounds is shown in table 1.4.27. Table 1.4.27. Comparison of two Diolen types [93]

Pull-through load, kg/cm

Type of Diolen BSP Normal

Kind of VP latex A B A

Kind of rubber R D R D R D

Heat treated at 175 °C Heat treated at 195 °C15

16 2222

232625

29

8

7 10

9

The manufacturing process for these modified polyesters is not dis- closed. Deducing the method from patents of Glanzstoff-Fabriken [241],

it is supposed that undrawn polyester fiber is treated with a spinning

preparation which contains lubricant agents, polyepoxide, and amine

hardener, then stretched and heat treated at the same time. Since exist- ing dipping equipment can be employed for these polyesters, demand for them will grow if adhesion is satisfactory for tire cords.

This trend has been followed by fiber makers of the United Kingdom,

Canada, and Japan. Little of MiUhaven presented information at the

Akron meeting in 1970 that T 790, an easy-to-adhere type of polyester,

shows satisfactory adhesion even in tests at elevated temperatures. [274]. CIL has developed a new polyester to rubber adhesive system [65,

240-246]. It was called N3 and consisted of a reaction product of triallylcyanurate, resorcinol, and formaldehyde of the following composition.

Resorcinol 100 parts Triallylcyanurate 24 Red lead catalyst 0.25 Formalin, 37% 28 Water 400 Ammonium hydroxide. 28% 28 N3 is available both for single dip and double dip processes. It is also applicable to nylon.

Resorcinol to formaldehyde ratio of the RFL used with N3 affects the adhesion of polyester to rubber, see table 1.4.28. For this adhesive

system high heat treatment temperature is necessary. Typical conditions
290 TIRE CORD ADHESION

Table 1.4.28. Effect of RFL composition on adhesion with the N3-RFL single dip system A B C

Gentac latex, 41% 53.0 29.2 58.5 SBR latex, 41%

Resorcinol

7.3

3.1 2.1 5.0 Formalin, 37% 4.9 3.4 2.8 Sodium hydroxide, 5% 5.7 3.7 Water 33.3 54.3 33.7 Pull-through load, lb/0.25 in Rubber compound

A 15 17 23 B 22 22 "27 C 12 19 19

" Rubber failure. are drying at 104° C for 120 seconds and heat treatment at 254° C for 45

seconds. Addition of sodium carbonate or diethylenetriamine to the N3-

RFL mixture results in better adhesion after less severe heat treatment. Bonding force also depends on adhesive pick-up and preferable pick-up

is in a range of 5 to 7 percent on the weight of cord. ICI has developed a new single dip adhesive system. The additive

is called Pexul, formerly H7, and the dip solution is prepared by mixing

Pexul with RFL of a special recipe. The composition of Pexul has not been disclosed, but it can be surmised to be 2,6-6i5-(2,4'-dihydroxyphenylmethyl)-4-chlorophenol from the patent Hterature [248]. The working

mechanism may be sorption of Pexul into polyester.

There are several single dip adhesives which could be attributed to sorption to polyester other than N3 and Pexul. They are furfural modified

resorcinol formaldehyde latex of AKU ^ [247], alkylated resorcinol

formaldehyde latex [278], and I-RFL of Toray [25]. 1.4.2.5. Adhesive Treatment of Miscellaneous Tire Cords

Steel Wire Cord: A special method has been used to bond rubber to steel wire cord. A high level of adhesion can be obtained by vulcanizing

the rubber compound in contact with brass or zinc plated steel cord. Most tire wire is plated at the wire mill, after drawing, by continuous

electrolytic methods. Amount of brass and zinc on the wire is controlled

in the ranges of 4 to 8 g/kg. and 2 to 3 g/kg., respectively. It is mentioned

that the preferable brass composition is 70 percent copper and 30 percent

zinc [249]. Choice of the type of plating is decided by the nature of the

rubber compound. And it is said that adhesion is affected by ingredients

of the rubber compound [250, 283-286].

Besides the plating method, addition of isocyanate or halogenated

rubber adhesives to the rubber compound is also employed [113].

' Algemene Kunstzijide Uni.
TREATMENT OF MISCELLANEOUS TIRE CORDS 291

Glass Fiber: Glass fiber cord is pretreated with an adhesive prior to treating with RFL or rubber cement [87, 77]. The pretreatment is appHed

during the fiber forming process in combination with a variety of lubri- cants which give improved fiber properties, especially abrasion resist- ance, to glass fiber. Since unsaturated silanes such as vinylsilane are

effective for bonding glass to rubber, the pretreatment agent may contain such a compound [251-252].

Polypropylene: Polypropylene to rubber bonding is one of the most

difficult problems in this field. Several methods are patented. For ex- ample, polyepoxide/RFL double dip was claimed to be effective [253]. Adhesion of polypropylene for isocyanate rubber cement is improved

by graft polymerization of vinylacetate, followed by hydrolysis [254].

It is mentioned that polypropylene to ethylene propylene rubber bonding

is attained simply by vulcanization together if a peroxide curing system

is used [255]. Vinylon: Vinylon is made bondable to rubber by the same treatments

as for nylon or rayon. Since Vinylon has a very hydrophilic nature, easy

penetration of RFL adhesive into the fiber interior causes stiffening of the dipped cord. Addition of water-repellent agents to RFL is used to reduce this tendency [256]. Also pretreatment of Vinylon by softeners,

a type of surfactant, is effective for softening the dipped cord [257]. 1.4.2.6. Evaluation of Adhesion

Adhesion has been evaluated by both tire tests and laboratory methods.

In both cases, measurements of the bonding strength and observation

of the failed state are important to judge the level of adhesion. And since

adhesion falls gradually during tire operation, it is necessary to simulate

a change of adhesion with running of the tire. To meet the requirements,

static measurements, dynamic measurements, and combinations of them

have been developed. However, there is no method of satisfactory evaluation on a laboratory scale. Klein and co-workers [114] stated that stripping adhesion decreased

with mileage, and the highest adhesion loss occurred in the region of maximum flexing (between 3.5 and 4 inches along the cord from the center

of the tire), figure 1.4.66. Kenyon [258] mentioned that deterioration of adhesion of rayon tires after running is greater than for nylon tires. Also, reduction of adhesion

of Ply 1 to 2 is greater than that of Ply 3 to 4 on four-ply tires. Eccher [117] investigated the failed state by microscopic observation

of tire cross sections. He reported that damage was concentrated in the zone beneath the tire shoulder, and consisted of rubber tears and

separation between cord and rubber.

Numerous methods have been proposed to estimate tire cord to rubber

adhesion. They are briefly explained here. Static Tests Pull-through type tests; H, T, and U-tests: Adhesion is represented

by the force required to pull an embedded cord through and out of a
292 TIRE CORD ADHESION

F 1

NEWJIRE AFTER/'INDOOR

AFTER FLEET

TEST

TEST

0 2 A 6 8 10 DISTANCE FROM CENTER OF CROWN (in.) Figure 1.4.66. Deterioration of adhesion after fleet test and indoor wheel test.

CORD

JIG

MOLD

JAW

H TEST T TEST U TEST

Figure 1.4.67. Test specimens and testing features of pull-through type test.

rubber block in the H-test, T-test, and U-test. The various test names come from the shapes of the test specimens, and are used only for con- venience, see figure 1.4.67. In these methods, the force is affected by embedded length of cord,

rate of loading, and temperature [258-259]. Measurement is frequently

at 110° to 130° C since temperature in a running tire may be this high.
EVALUATION OF ADHESION 293

NO LOAD ONE END

C71

INITIATION PROPAGATION PROPAGATION PROPAGATION COMPLETE RECOVERED

OF FAILURE OF FAILURE BOTH ENDS TWIST FAILURE SAMPLE ONE END Figure 1.4.69. Distortion of "/" test specimen during load application [154]. Dotted line

is the cord.

I-test: The I-test is made by pulling both ends of an I-shaped test speci- men in which a sample cord is embedded as shown in figure 1.4.68. The

cord is represented by a dotted line. The force-deflection curve is re- corded. Adhesion is represented by the second peak force of figure 1.4.69. Distortion of the I-test specimen during loading is shown in figure 1.4.68

[154].
294 TIRE CORD ADHESION

Pop-test: The pop-test is made by compressing a cylindrical rubber

test specimen in which a sample cord is embedded in the middle as shown

in figure 1.4.70. The test specimen is deformed on application of load,

from shape A to B. When failure of adhesions occurs, appearance of the

specimen changes to C or D. Adhesion is represented by the compression

force required to deform the test specimen to the C or D stage [260]. Cord stripping test: A form of stripping cord adhesion test is practiced

in Du Pont, the Single End Strip Adhesion (SESA) Test [261]. The force

necessary to strip the cord from a rubber sheet is determined and is reported as pounds force per single end of cord. The test arrangement is shown in figure 1.4.71. ABC D

Figure 1.4.70. Deformation of Pop test specimen during load application [260].

Figure .1.4.71. Testing arrangement for Single End Strip Adhesion test.
EVALUATION OF ADHESION 295

Fabric stripping test: Adhesion of tire fabric and belting duck is evaluated by stripping a two-ply specinen which is a sandwich structure

of rubber cover-test fabric-inner rubber-test fabric-rubber cover [262-

263]. In evaluating tire fabric, high cord end counts are used to minimize

rubber bridge formation. The force is also affected by thickness of the

inner rubber and rate of loading [264]. This method has been adopted

to estimate the level of adhesion in tire carcasses. Timmons [240] sug- gested that visual inspection of strips frequently correlated with the

appearance of cords in tires after high speed tire testing. The testing

arrangement is shown in figure 1.4.72. Dynamic: Adhesion gradually deteriorates with repeated deformation.

Dynamic evaluation of adhesion is made by counting cycles of defor- mation to reach a limiting value. Testing equipment is commonly de- signed to cut off when bonding force reaches a specific value. Type of

deformation may be classified as: (1) compression, (2) shearing, (3) dynamic strip, and (4) flexing.

Compression type [265-266]: A test specimen is repeatedly compressed

as illustrated in figure 1.4.73. The test cord is under tension by means

of a weight. When the test specimen is compressed the rubber to cord

interface is subjected to shear force. Adhesion deteriorates with compression cycles, and failure occurs when the bonding force becomes less than the weight suspended on the cord. The Goodrich Flexometer has

been conveniently used for this purpose. Besides, the Goodrich Disk

Figure 1.4.72. Testing arrangement for two-ply strip test.
296 TIRE CORD ADHESION

Figure 1.4.73. Principle of Compression Adhesion Tester.

Figure 1.4.74. Schematic drawing of Compression-Extension Type Tester and test specimen.

Fatigue Tester is available, in which the specimen is subjected to both

compression and extension [267]. Compression-extension deformation

may be used with the equipment diagrammed in figure 1.4.74 [268]. A

test cord is embedded in the middle of a dumbbell shaped test specimen.

In these tests, failure occurs just inside the rubber where there is severe heat degradation. Therefore, it is said that these methods are inadequate

to examine adhesion [195]. This method is available not only for cord

but also for fabric [269-270].

Shearing type: Continuous vibration is applied to a rubber specimen

as illustrated in figure 1.4.75. This is an example of this type testing

equipment, the Dynamic Shear Adhesion (DSA) Tester. Adhesion is represented by cycles to failure of the cord-to-rubber bond and this is automatically recorded [204]. Dynamic strip type: In the case of tire fabrics or belting ducks, a twoply strip testpiece is subjected to continuous vertical vibration under a

certain weight. Adhesion is expressed by cycles required to strip a unit
EVALUATION OF ADHESION 297

length. This type of equipment has been on the market and called a Scott

Tester, see diagram in figure 1.4.76.

TEST SPECIMEN

R' TIME

CUT-OFF Figure 1.4.75. Schematic drawing of Dynamic Shear Adhesion Tester.

HEATER

Figure 1.4.76. Schematic drawing of Dynamic Strip Adhesion Tester (Scott Tester). Flexing type: Dipped cord to be evaluated is embedded in a cylin- drical rubber specimen. Flexing is by deformation of the specimen

between a stationary anvil and a plunger, see figure 1.4.77 [271]. Compression deformation can be given to the specimen at the same time by

adjustment of the position of the stationary anvil. An alternative flex adhesion fatigue testing equipment was proposed. The test strap has a
298 TIRE CORD ADHESION

ECCENTRIC DRIVE

UPPER ANVIL

LOWER ANVIL N|^^_^^^E5T SPECIMEN u^COR D WEIGHT /

Figure 1.4.77. Principle of Dynamic Flex Adhesion Tester.

band of cords passing longitudinally through the middle. Affixed at one end to a stationary mounting, the strap passes horizontally from there to a pair of little rollers which are mounted in a reciprocating carriage.

The strap follows a sigmoidal path around the rollers, and continues

horizontally to its free end on which hangs a weight. Figure 1.4.78 gives

a schematic view of the equipment. Adhesion is evaluated by a compari- son of pull-through load before and after flexing [272]. In evaluating fabrics, the test specimen is built as a two-])ly sandwich

structure. Compressive fatigue is produced by flexing it over a spindle.

The principle is illustrated in figure 1.4.79 [125, 209, 273]. Adhesion is estimated by either comparison of strip adhesion before and after flex- ing or cycles for ply separation [200].

TEST ROLLERS STRAP

Fl(;URE 1.4.78. Schematic side view of Roller Flex Adhesion Tester.
REFERENCES 299

Fi(;l;re 1.4.79. Schematic drawing of Belt Flex Type Dynamic Adhesion Tester.

References

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300 TIRE CORD ADHESION

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302 TIRE CORD ADHESION

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REFERENCES 303

General Tire brochure, Gentac. Wake. W. C. in Adhesion and Adhesives, Vol. 2, 2nd ed., Eds, Houwink, R., and

Salomon. G.. p. 418 (Elsevier Publishing Co., New York, 1967). Dovle. G. M.. Trans. I.R.I. 36, 177 (1960). Dubrisay, R., and Papauh, R.. Compt. Rend. 215, 348 (1942). Le Bras, J., and Piccini. I., Ind. Eng. Chem. 43, 381 (1951). Uzina. R. V.. et al, Sov. Rubber Tech. 20(7), 18 (1961). Levitin, I. A., et al. Sov. Rubber Tech. 21(1). 41 (1962). Stedrv. P. J.. Ind. Eng. Chem. 43, 2372 (1951). van Giles. G. E., Polymer Preprints 8(1), 508 (1967). Du Pont. Brit. 1.106.920 (1968). Esso. Brit. 803.182 (1958): Brit. 823.282 (1959): Brit. 861,814 (1961). Burlington. U.S. 3.240.659 (1966). Burlington. U.S. 3,240.649 (1966). Stamicarbon. Belg. 680.492 (1966). Dunlop Rubber. U.S. 3.268.386 (1966). Du Pont. U.S. 3,276.948 (1966). Farbwerke Hoechst. Japan 42-17642 (1967). Montecatini. Japan 42-23632 (1967); Brit. 1,071.528 (1967). Reeves. L. W.. Rubber Worid 132. 764 (1955). Reeves, E. V., Adhesive Age 6(3), 28 (1963). Hannel. J. W., Du Pont Tire Yarn Tech. Rev., p. 27 (1958). James. D. I.. Wake. W. C. Trans. I.R.I. 39, 103 (1963). FMC Corporation. Japan 42-6668 (1967). Murphy. R. T.. Baker. U. M.. and Reinhardt. Jr.. R., Ind. Eng. Chem. 40, 2292

(1948). GiUman. H., and Thoman. R., Ind. Eng. Chem. 40, 1237 (1948). Turner. J.. Cohen. N.. and Janow. S., Rubber World 153(3), 67 (1965). Kigane. K.. Teijin Times 38(2). 36 (1968). Nippon Rayon, Japan 42-15765 (1965). Nippon Rayon. Japan 41-12228 (1966). Maizels. M. G.. and Raevskii, V. G., Sov. Rubber Tech. 19(6), 16 (1960). Iyengar. Y.. Rubber World 148(6), 39 (1963). Garetovskaya. N. L.. and Belyaeva, N. V., Sov. Rubber Tech. 24(7), 20 (1965). Ludewig, H.. Polyesterfasern. Chemie und Technologie, Chap. 11 (AkademieVeriag. Berlin. 1965). RABRM. Brit. 804.005 (1958). Zahn. H.. and Pfeifer, H., Polymer 4, 429 (1963). N. V. Onderzoecking Inst. Res., Japan 40-1713 (1965): Belg. 612,198 (1962). Leibnitz. E.. Konnecke. H., Gawalek, G., J. Prakt. Chem. 6,289 (1958). ICI. Belg. 663.578 (1966). ICI, Japan 35-13125 (I960): Brit. 816,640 (1959). For example. Shkapenko, G., Gmitter, G., Gruber, E.. Ind. Eng. Chem. 52, 605

(1960). Farbwerke Hoechst, Japan 41-12109 (1966). Bayer, O., Angew. Chem. A59(9), 257 (1947). Tanabe, T., et al., Kogyo Kagaku Zasshi (J. Chem. Soc, Japan, Ind. Chem. Sec.) 66,

821 (1963). Farbwerke Hoechst, Belg. 656,656 (1965); Belg. 672,260 (1966). Yokohama Rubber. Japan 42-784 (1967). Thompson. W. L., and Parke, L. W., Rubber World 138, 588 (1958). Du Pont. U.S. 2,994,671 (1961). Du Pont. Instruction Manual for D-15, 1957.

Teijin. Japan 38-13236 (1963). Goodyear Tire and Rubber, Japan 40-21429 (1965); U.S. 3,268,467 (1968); Japan

40-10723 (1965): U.S. 3.226.276 (1966). Toyo Rayon, Japan 38-20961 (1963). Iwakura. Y.. Nabeya, A., and Nishiguchi, T., Yuki Gosei Kagaku Kyokai Shi (J. Soc. Organic Synthetic Chem. Japan) 26, 101 (1968).

[226] Iwakura, Y., in Setchaku, High Polymer Series 1, p. 185 (High Polymer Soc. Japan

ed., Maruzen, 1965).
304 TIRE CORD ADHESION

[227] Toyo Rubber, Japan 42-2896 (1967). [228] Toyo Rayon, Japan 42-2275 (1967); Japan 40-26548 (1965). [229] Teijin, Japan 37-3123 (1962); Japan 37-3124 (1962). [230] Toyo Spining, Japan 42-543 (1967). [231] Takeda Pliarmacy, Japan 42-8221 (1967); Japan 42-541 (1967); Japan 42-860 (1967). [232] Teijin, Japan 39-10513 (1964); Japan 40-28937 (1965); Japan 41-5260 (1966); Brit. 953,415 (1964), U.S. 3,235,333 (1967); I.C.I. Belg. 674,529 (1966). [233] Teijin, Japan 40-922 (1967). [234] Mitsuboshi Belt, Japan 40-16135 (1965). [235] Kigane, K., Japan 38-20956 (1963). [236] Little, J. S., Can. Textile J. 78(19), 57 (1961). [237] Canadian Industries Ltd., Japan 36-5740 (1960); U.S. 3,051,594 (1962); Brit. 867,187

(1961); German 1,259,093 (1968). [238] Chapman, R., Rubber Plastic Age 47, 769 (1966). [239] ICI, Manual for Pexul, The use of pexul for bonding terylene to rubber, 1967.

[240] Timmons, W. D., Adhesive Age 10(10), 27 (1967). [240a] Law, C, Rubber World, 158(4), 43 (1968). [241] Vereingte Glanzstoff-Fabriken, Brit. 1,025,310 (1966); U.S. 3,297,467 (1967); Brit. 1,035,299 (1916); Japan 42-19921 (1967); Brit. 1,025,309 (1966); U.S. 3,297,468

(1967); Brit. 1,035,220 (1966); Japan 42-19922 (1967); Brit. 1,012,935 (1966); Brit. 1,026,221 (1966). [242] Canadian Industries Ltd., U.S. 3,318,750 (1967). [243] ICI, Belg. 674,528 (1966). [244] Smolin, E., and Rapoport, L., The Chemistry of Heterocyclic Compounds, S- Triazines and Derivatives, p. 400 (Interscience Publishers, New York, 1955).

[245] Clampitt, B. H., German, D. E., and Galli, J. R., J. Polymer Sci. 27, 515 (1958). [246] Saunders, J. H., and Frish, K. C, Polyurethanes Chemistry and Technology, Part

1, p. 116 (Interscience Publishers, New York, 1962).

[247] Algemene Kunstzijide Uni, Fr. 1,481,392 (1967); Belg. 682,153 (1966). [248] ICI, Belg. 688,424 (1967); Fr. 1,496,951 (1967); Neth. 6,614,669 (1967); Belg. 688,425

(1967). [249] Wake, W. C, Reference [175], p. 413.

[250] For example. Hicks, E., and Lyon, F., Rubber Chem. Tech. 40, 1607 (1967). [251] Vanderbilt, B. M., and Clayton, R. E., Rubber Chem. Tech. 38, 379 (1965). [252] Esso, U.S. 3,307,967 (1967). [253] Toyo Rayon, Japan 42-3947 (1967). [254] Toyo Rayon, Japan 42-542 (1967). [255] Montecatini, Japan 42-26830 (1967). [256] Kurashiki Rayon, Japan 36-7742 (1961); Japan 37-12119 (1962). [257] Mitsuboshi Belt, Japan 36-7742 (1961). [258] Kenyon, D., Trans. I.R.I. 38, 165 (1962). [259] Wood, J. O., Rubber Chem. Tech. 40, 1014 (1967). [260] Lessig, E. T., and Compton, J., Rubber Chem. Tech. 19, 223 (1946). [261] Du Pont, U.S. 3,307,966 (1967). [262] JIS K-6301-1962, Physical Testing Methods for Vulcanized Rubber.

[263] ISO Recommendation R36, Rubber Chem. Tech. 31(1), xxi (1958). [264] Barroff, E. M., Elliot, R., and Wake, W. C, Rubber Chem. Tech. 25, 391 (1952). [265] Uzina, R. V., Gromova, L. S., Rubber Chem. Tech. 32, 898 (1959). [266] Gardner, E., and WilHams, P., Trans. I.R.I. 24, 284 (1949). [267] Kenyon, D., Trans. I.R.I. 40, 67 (1964). [268] Khromov, M. K., Reznikovskii, M. M., and Lazareva, N. K., Sov. Rubber Tech. 2 1(6), 25 (1962). [269] Levitin, I. A., Korablev, Yu. G., Kornev, A. E., and Babitskii, B. L., Rubber Chem.

Tech. 32, 889 (1959). [270] Tsydik, M. A., Lukomskaya, A. I., and Slonimskii, G. L., Rubber Chem. Tech. 33,

42 (1960). [271] Pinman, G. A., and Thornly, E. R., Trans. I.R.I. 25, 116 (19^9). [272] Lyons, W. J., Anal. Chem. 23, 1255 (1951). [273] ASTM Standards, Part 28, Rubber; Carbon Black; Gaskets, Method D430-59,

Dynamic testing for ply separation and cracking of rubber products, p. 233 (Ameri- can Society for Testing and Materials, Philadelphia, 1968).
REFERENCES 305

[274] Little, J. S., Akron Rubber Group Symp., Akron, Ohio (Jan. 23, 1970).

[275] Kawai, K., Akron Rubber Group Symp., Akron, Ohio (Jan. 23, 1970).

[276] Hannell, J. W., Akron Rubber Group Symp., Akron, Ohio (Jan. 23, 1970).

[277] Litzler, C. A., Akron Rubber Group Symp., Akron, Ohio (Jan. 23, 1970).

[278] Iyengar, Y., Erickson, D. E., J. Appl. Polymer Sci. 11, 2311 (1967). [279] Creasey, J. R., and Wagner, M. P., Rubber Age 100(10), 72 (1968). [280] Creasey, J. R., Russel, D. B., and Wagner, M. P., Rubber Chem. Tech. 41(5), 1300

1968).

[281] Rawe, A., and Fitko, C, J. Polymer Sci. Al 4, 2533 (1966). [282] Iyengar, Y., J. Appl. Polymer Sci. 13, 353 (1969). [283] Adler, O. E., Adhesive Age 12(1), 30 (1969). [284] Hicks, A. E., and Lyon, E., Adhesive Age 12(5), 21 (1969). [285] Sexsmith, F. H., Adhesive Age 13(5), 21 (1970). [286] Sexsmith, F. H., Adhesive Age 13(6), 31 (1970).
I
CHAPTER 2

Properties of Cord-Rubber Laminates

Samuel K. Clark ^

Page 2.1. Elastic Properties of Cord Reinforced Rubber 308

2.2. Strength and Failure Mechanisms for Cord Reinforced

Rubber 341

References 351

Department of Engineering Mechanics, University of Michigan, Ann Arbor, Michigan 48104. 307
308 PROPERTIES OF CORD-RUBBER LAMINATES

2.1. Elastic Properties of Cord Reinforced Rubber

The increased use of more sophisticated structural configurations in pneumatic tires during the last few years has resulted in a greater interest

in the elastic properties of cord-rubber laminates. Much of the reason for

this may be found by considering the basic nature of redundant struc- tures, in which the loads carried by individual cords are in part deter- mined by the elastic characteristics of the entire system. In such

situations, a knowledge of elastic characteristics becomes important to good structural design practice. As a second reason for increased atten- tion to the elastic properties of such materials, one might cite the body

of work which is now developing in the general area of filamentary rein- forcement of materials. A knowledge of elastic characteristics is important in obtaining optimum reinforcement properties, and such studies

inevitably lead to a clearer understanding of the internal stress states of all reinforced materials. One result of this research is that much of the work done in the areas of fiberglass and whisker reinforcement has

increased the general understanding, in a broad way, of the action of

cord reinforced rubber. The main aspects of the problems are similar,

although major differences exist in the structure of the reinforcement

itself and in the fact that in many reinforced elastomers the cords are anchored to some relatively rigid structural member.

The elastic properties of cord-rubber materials are understood today

much better than they were even 10 or 15 years ago. General reviews

which describe various aspects of current knowledge have been published by Hashin [1],^ Hofferberth and Frank [2], and by Clark [3]. Historically, the role of material properties in pneumatic tire design

and analysis has been approached in three distinct ways:

a. First, the anisotropic nature of such materials may be completely

ignored, and all tire structures treated as membranes whose stresses are determined entirely from membrane equilibrium considerations. This

is a statically determinate shell approach, and in some regions tire stresses, and cord loads, may be found rather accurately by this tech- nique when loadings are simple, such as in the case of inflation of a tire. However, proper elastic properties are necessary for adequate stress determinations near edges or boundaries in all shell problems, and in the case of a pneumatic tire it is necessary to know the material characteristics in order to find stresses near the bead and in the sidewalk

In more complex loadings where bending is an important factor, such

as in deflecting a tire against a flat surface, the elastic properties of the

material are needed for estimation of cord loads, and an isotropic approximation is not adequate.

Quite apart from stress analysis problems, which are always governed

by force and moment equilibrium and hence can sometimes be less

- Figures in brackets indicate the literature references at the end of this chapter.
ELASTIC PROPERTIES 309 dependent on material properties, all deformation characteristics of cord-rubber structures are entirely determined by their elastic constants

so that an isotropic approximation might be characterized as a very

crude one in most cases.

b. An interesting and unique attempt to calculate cord loads in a cordrubber structure has been provided by Rivlin and his co-workers [4-6]

and by Adkins [7-11]. They both utilize the concept of an inextensible net made up of the reinforcing textile cords in a normal cord-rubber

structure, with the assumption that in such a net the cross-over points

between cords in adjacent lamina act as knots which do not slip. This

analysis has much to recommend it. Basically, in most situations, the

reinforcing textile cords do indeed carry a major share of the tire structural loads. Thus neglect of the surrounding rubber matrix is not generally serious, although situations exist when the surrounding rubber

can contribute substantially to the load carrying ability of the composite,

as has recently been discussed by Clark and Dodge [12]. The mathematics of such cord networks is interesting, and represents a fertile field for the applied mathematician. Its most important contribution is un- doubtedly as a stress analysis tool, where many solutions are now known.

However, it has not been possible to use it to obtain the elastic constants

of cord-rubber composites since the absence of the rubber characteristic

results in a completely rigid structure.

c. A third and somewhat later general approach to the elastic properties of cord-rubber laminates utilizes the overall elastic stiffnesses of

both the cords and rubber treated as a two-dimensional orthotropic

material. This requires that the concentrated cords and distributed

rubber be viewed over dimensions much greater than the cord spacing,

so that the average elastic modulus of a single lamina may be obtained,

as opposed to the local or microscopic modulus which varies widely with

position. The primary reason for this is that it is difficult to mathematically

describe the large variations which exist in elastic stiffness of a cordrubber laminate. It is much more convenient to work with an average

property.

Basically a single ply or lamina of cord-rubber material may first be

considered in terms of its constituent properties. These are discussed

in detail elsewhere in this volume, but it is worthwhile to review the fact that for large deformations both the cord and rubber properties are highly

nonlinear as well as visco-elastic. For example, quasi-static force extension curves for a typical rubber are given in figure 2.1 for large extension

ratios, while for small strains in rubber the force extension characteristic

is essentially linear as shown in figure 2.2. Stress-strain curves for typical

tire cord materials are given in figure 2.3. From these, it might be con- cluded that a nonlinear theory would be necessary to completely describe

the action of combined cord in rubber, and this is undoubtedly true for large strain. For small strains, which fortunately are the primary condition in tire usage, material characteristics are much more linear. This

is illustrated in figure 2.4, where a stress-strain curve for rayon cords
310 PROPERTIES OF CORD-RUBBER LAMINATES

Figure 2.2. Region near origin offorce-extension ratio curve for rubber.
ELASTIC PROPERTIES 311

10

0 5 10 15 20 25

STRAIN (%)

(1) RAYON (5) NYLON -66

(2) NOMEX (6) STEEL

(3) POLYESTER (7) GLASS

(4) NYLON

TESTED AT 70*F.,657o R.H. INSTRON - 100% STRAIN RATE

Courtesy of FABRIC RESEARCH LABORATORIES, INC. Figure 2.3. Stress-strain characteristics of typical tire cords.

imbedded in rubber is given, where here the direction of tensile stress

is parallel to that of the cords. Aside from the soft stretch portion at very

low loads, the cord action may be approximated by considering the stress- strain curve to be linear. One might thus hope to treat cord-rubber

laminates as linearly elastic materials.

Before attempting to examine the elastic properties of laminates made

up of a number of plies of cord-rubber sheets, it is first necessary to review the basic characteristics of a single layer of such material. It is common to align a parallel series of evenly spaced textile cords, and to calender these with unvulcanized rubber, in such a way as to form a

single layer of cord reinforced material as illustrated in figure 2.5a. This

figure shows each of the reinforcing cords as a circle, although in actuality

the cord is somewhat irregular in shape. It is made up of a multiplicity

of filaments twisted into singles yarns (strands) and two or three of these

are then twisted in the opposite direction, for balance, to form the cord.
312 PROPERTIES OF CORD-RUBBER LAMINATES

2500

0 .002 .004 .006 .008 010 .012 .014 STRAIN Figure 2.4. Stress-strain curve in tension of rayon cords embedded in rubber, where stress

is based on total cross-section area of specimen.

y

Figure 2.5a. End view of a series of parallel textile cords embedded in rubber.

Prior to calendering with rubber, the cords are usually passed through a dip which promotes adhesion to the rubber. One may visualize each of the reinforcing cords as a complex twisted structure made of many monofilaments having this irregular cross section shape, and being imbedded

in rubber which encases the cords completely in order to prevent mechanical chafing of one cord against another. This is shown in the photograph of figure 2.5b. Due to the complexity of such a structure, the

prediction of the elastic characteristics of such a single lamina is in

itself a difficult job. Fortunately, the cord diameter and spacing are almost

always small compared with typical dimensions of a pneumatic tire, so
ELASTIC PROPERTIES 313

Figure 2.5b. End view of a single textile cord. that the average elastic properties will suffice in many structural

applications.

Such a sheet as shown in figure 2.5a may be further idealized considering it to be essentially a two-dimensional orthotropic body, where

the properties through the thickness are taken to be isotropic. This

simplifies discussion of the elastic properties considerably, and is not

particularly restrictive since most structures associated with pneumatic

tires have very small bending rigidity, compared to their membrane

rigidity. This means that the coupling of plane tension to twist is very

weak in most cord-rubber laminates, and can safely be neglected. This

is not the case for general filamentary or fiber reinforced materials, since

when the matrix is of sufficient stiffness and monofilament reinforcement

is used, then tensile and compressive moduli are about equal and the

coupling between plane effects, such as tension and shear, and bending

effects, such as twist and curvature change, may be significant.

Insofar as the two-dimensional orthotropic properties of the sheet are concerned, primary interest usually centers on determining the elastic characteristics in the so-called principal directions, which in figure 2.5a

are given as the x and y directions. For such a definition one may utilize either a direct or inverted form of Hooke's law, as given alternately by

eqs (2.1) and (2.2).

(2.1) ss^xy-
314 PROPERTIES OF CORD-RUBBER LAMINATES

O'y E, Fxy ^y Ey Fxy

O'xy

. Gxy

(2.2)

The fact that the same constant Cxy appears in both the first and second

of eqs (2.1), and similarly the constant Fxy in the first and second of eqs

(2.2) may be demonstrated by drawing upon the single-valued character

of the strain energy stored in such a sheet. This has been derived in the Hterature [14].

700-

600-

500-

TRESS,

PS

200

-.010 -0( -r 1)8 -.OC

i H

STRAIN

)6 -OC H 1)4 -OC

100-

OC>2 004 £ RUBBER

CURVE

STRESS'S- RAirj^

- 100

-200 Figure 2.6. Continuous stress -straincurve through compression and into tension for rayon

cords embedded in rubber.

Stress based on total cross-section area of specimen.
ELASTIC PROPERTIES 315 The use of such a law as given in eqs (2.1) or (2.2) is only valid if the

materials involved are linearly elastic. In some applications this is almost self evident, such as in whisker reinforced metallics and ceramics,

and fiber reinforced plastics. In the case of cord reinforced rubber this

is not nearly so assured, and experimental data must be used as confirmation. Typical experimental dat^ from Clark [3] is given in figures

2.6 and 2.7. From these it is seen that such a mathematical simplification

is only justified under the following conditions.

a. The cord is loaded in tension at all times.

b. The value of the cord tension is large enough so that the region near

the origin, commonly called the soft stretch region, is not encountered.

c. Viscoelastic effects are ignored.
316 PROPERTIES OF CORD-RUBBER LAMINATES

cr

en

a:

e STRAIN

Bi-linear Stress Strain Curve Figure 2.8. Idealized bilinear stress-strain curve.

Unless these conditions are fulfilled, typical stress strain curves such as shown in figures 2.6 and 2.7 indicate that some other more complex

structural model for a tire cord must be chosen. It has been suggested

that a bilinear model, such as shown in figure 2.8, would be appropriate

but little evidence appears in the literature that such a model has been

useful for analysis for practical problems.

In reviewing the implications of experimental data such as shown in figures 2.6 and 2.7, and associated idealizations such as shown in figure

2.8, it should be noted that the textile cords used in both figures 2.6 and

2.7 will carry appreciable compressive loads, since- the stiffness of the

associated rubber predicts only negligible compressive resistance. The

specimens in question were thick-waUed hollow cylinders made from

ordinary pneumatic tire carcass material. The contribution to stress or load of the surrounding rubber is shown, and it is seen that the textile cords carry surprisingly large loads in compression. When a twisted

textile cord is compressed, it tends to deform as a spiral spring, each

ply yarn behaving as a monofilament. The lower the helix angle of the

plied structure, the greater the initial radial growth during compression.

This structural growth leads to a very high structural Poisson's ratio of the cord structure (not the cord filaments) when the cord is compressed.

The compressive resistance is attributable to the rubber, the proximity

of neighboring cords and the pressure normal to the laminate resisting

cord growth. In any event this type of response leads one to conclude

that even for small strains it would be necessary to consider twisted

cords in rubber as being represented by a bilinear stress strain curve

as a first approximation, as shown in figure 2.8, for all cases where cord

tension could not be maintained. The practical effect of this is to force

the description of the elastic constants of the single sheet of cord rein-
ELASTIC PROPERTIES 317

forced rubber into two distinct parts, one in which the cords are in tension

and another in which the cords are in compression. This would appear

to be the ideahzation of greatest benefit, and in the event that one wished

to work in both tension and compression regions there will be just twice

as many elastic constants applicable to a cord reinforced rubber sheet

as to a sheet of wood or fiberglass reinforced plastic. Thus, eqs (2.1) and

(2.2) actually have counterparts in the compressive region so that, as a

first idealization, eight elastic constants actually exist for each laminated

cord-rubber ply, four constants pertaining to conditions in cord tension

while the other four pertain to conditions of cord compression.

In view of the similarity of the mathematics for each case, i.e., tension

and compression, further discussion will be restricted to a single set of equations, since the form for the other set is identical. The four constants C,j in eqs (2.1) represent the stiffness of the composite material in the x and y directions, while the constants Ex-, Ey and

Gxy in eqs (2.2) are the conventional Young's moduli in x and / directions

and the conventional shear modulus of the material along jc-y boundaries,

and the quantity Fxy represents a so-called cross-modulus of the material.

From this point of view some authors have preferred eqs (2.2) as a standard form, since the conventional elastic constants appear directly in it, while other authors have used eqs (2.1). A complete discussion of the

fundamentals of anisotropic elasticity is given in the book of Hearmon [15].

It would be desirable to be able to calculate either set of elastic constants given in eqs (2.1) or (2.2) in terms of the elastic properties of the

textile cord and its surrounding rubber matrix. Theoretical methods

for calculating the elastic constants of multi-phase materials have been

published by Hashin and Rosen [16], and other similar methods have

been reviewed by Hashin [1], based on energy principles to establish

upper and lower bounds for elastic constants. These principles have

not proved to be of much practical value since they do not bound the constants closely enough for the type of constituents being considered

here. However, from this has resulted a generalization of the so-called law

of mixtures which has validity when fiber and matrix have the same

Poisson's ratios. One of the difficulties of applying this type of theory

to a cord-rubber structure is the noncontinuous nature of the cord itself, and due to the large values of possible cord contraction there may be

substantial differences between the contraction characteristics of the

cord and its surrounding rubber. The law of mixtures has been utilized by Whitney [17] in computing the elastic modulus Ex parallel to the

cord direction in a sheet of material reinforced by parallel cords. This

results in the expression

Ex = Ec'X + EH(l-k) (2.3) where Ec represents cord modulus, Er represents rubber modulus, and

A is the fraction of the cross sectional area occupied by the cord, this cross section being taken at right angles to the cord direction. This type
318 PROPERTIES OF CORD-RUBBER LAMINATES

of expression may hold well for tension loads in the cords, but probably

breaks down when the cords are in compression since then the effective

cord Poisson's ratio becomes very large, much larger than that for rubber.

The other elastic characteristics of a single sheet of parallel cords

imbedded in rubber are not so easily obtained. Methods of determining

them on a semi-empirical basis have been investigated by both Ekvall

[18] and by Clark, Field, and Dodge [13]. Both of these later attempts

utilize simple volumetric fraction proportioning of the individual elastic constants of the constituent materials, and were partly successful by

this approach. The elastic modulus parallel to the cord reinforcement

direction can be predicted quite well by such principles, so that eq (2.3)

is usually accurate enough. It is considerably more difficult to calculate

the modulus Ey, referring to deformation at right angles to the direction

of cord reinforcement. Shear modulus Gxy of the lamina shown in figure

2.5a is also difficult to predict. Test data reported in reference [19] seems

to imply that a basic difficulty in calculating the elastic constant Gxy of a single ply involves the unknown role of the shearing deformation of the

textile cord reinforcement itself. This test data was obtained using

thick-walled tubes, with cords parallel to the tube centerline, tested in torsion accompanied by pre-tension in the longitudinal direction. The

specimen is shown in figure 2.9. During such a test the cord length will not change, at least for relatively small deformations, so that the cords

are exposed to pure shear. Apparently such a textile cord can contribute

substantially to shear deformation, and this fact must be taken into

account in constructing any simple theory for calculation of the overall shear property. It is difficult to visualize the contribution to shearing

deformation of the twisted textile cord and it is difficult to include this factor into a stiffness proportioning scheme. Finally, the elastic constant Fxy is obtained by considering contraction of the sheet in the y direction

due to the extension in the x direction. Experiments designed to deter- mine values of Fxu for various combinations of cord-rubber sheets indi- cate that the cords themselves apparently participate strongly in the

process of contracting at right angles to their axial direction under ten- sion load. Attempts to measure such effects have shown that the effective

Poisson's ratio of normal textile cord materials used for pneumatic tires

is about 0.7, as reported by Budd [20] for tension measurements, but

Figure 2.9. Test specimen used to determine cord shear properties.
ELASTIC PROPERTIES 319

can reach high values for cords in compression. Piatt and Butterworth

[37] report a Poisson's ratio of 3.3 for a two-ply nylon tire cord in rubber

compressed to 15 percent strain. While such a semiempirical approach allows one to predict single

sheet elastic constants over a limited range, it gives little insight into the possibilities of optimum reinforcement or into the basic mechanisms

of load carrying.

As an illustration of elastic constants of a single ply of tire material,

table 2.1 gives experimental values taken from a set of experiments

from reference [19]. Tables 2.2 and 2.3, taken from the same source,

give measured values of Ey and Gxy for similar tire materials. It may be

seen from table 2.1 that the simple approximation of considering the

entire stiffness to be furnished by the reinforcing cords is quite good,

Table 2.1. Comparison of measured and calculated values ofY,y^ Type of End count Ply {AE)c E, Error, cord n thickness

h (measured) (measured) %

Rayon 16 0.054 338 100,000 92,000 8 Rayon 25 .035 371 265.000 271,000 2 Rayon 14 .074 493 93.000 87,000 7 Nylon 34 .029 92 10,800 10.800 0 Nylon 16 .058 93.5 26.000 26,000 0 Nylon 33 .042 107 81,000 75.000 7 Rayon 34 .030 209 236,000 250,000 6 Rayon 34 .032 209 236,000 176,000 34 Nylon 15 .034 263 116.000 98,000 18 Nylon 15 .030 263 131,000 100,000 31

n ~ cords per inch measured perpendicular to cord direction.

h ~~ ply thickness in inches. {AE)c ~ product of the cord area and cord modulus, outside the "soft-stretch" region. This is measured by determining the total spring rate of a cord, in lbs. per inch, and then dividing by the length of this cord in inches. calculated using eq (2.3) with £"« =416 psi. Ex measured in tension on a strip specimen 1 inch wide and 12 inches long.

Table 2.2. Measured values ofE y Type of End count Cord Ply E Ey

cord n diameter d thickness h (ERubber) (Measured)

Rayon 25 0.032 0.059 416 2021 Nylon 28 .026 .047 416 1300 Nylon 29 .021 .037 416 1134 Nylon 29 .021 .036 416 1134 Rayon 25 .026 .042 416 1145 Rayon 32 .021 .040 416 1514
320 PROPERTIES OF CORD-RUBBER LAMINATES

Table 2.3. Experimental values ofG. Cord Cord Ply Rubber Gxy

Cord diameter. spacing, thickness. shear measured

material d s h modulus, experi- G, mentally

Rayon 0.028 0.034 0.039 275 470 Rayon .028 .034 .039 275 442 Rayon .028 .034 .039 275 447 Rayon .028 .034 .039 275 431 Rayon .029 .041 .039 406 553 Rayon .029 .041 .039 406 573

i> yion .UZo .yJoo t D Nylon .028 .033 .039 275 439 Nylon .026 .033 .039 275 463 Nylon .021 .029 .027 406 569 Nylon .021 .029 .027 406 578 Wire .032 .123 .071 406 446 Wire .031 .123 .071 406 438

since it gives results not far removed from measurement. On the other

hand, such a simple approach for calculating modulus Ey and the shear

modulus Gxy is obviously not possible, and considerable work needs to be done in this area in order to provide rational expressions for these

quantities.

A number of important references bear on the problem of predicting

the elastic characteristics of a multiplicity of cord reinforced lamina

bonded together. A direct approach to such a complex structure was

published by Hofiferberth [21] in which an attempt was made to de- compose forces into vector components lying parallel to the reinforced

cords, and to allow these cords to carry the component loads. From

such an analysis, Hofferberth was able to predict certain characteristics

of the elastic constants which appear to agree in a general way with

the experimental evidence obtained from other sources, but Hofferberth

himself gives no experimental evidence to validate his theoretical

development.

A very general discussion of the composite properties of filamentresin systems has been given by Herrmann and Pister [22]. However,

the models which Herrmann and Pister use are primarily concerned with

multiple layers of parallel cords, which , does not generally correspond

to the case with cord-rubber laminates.

The elastic properties of plywood have many similarities in theory

to the elastic properties demonstrated by cord reinforced rubber laminates, with the exception of the fact that normally the compressive modulus of wood does not differ much from its extension modulus, so that the problem of the bilinear representation does not exist in wood as it does in textile cord reinforced rubber. The plywood problem has been

extensively studied and is thoroughly reviewed in the work of Hoff

[23] and in a recent report from the Forest Products Laboratory of the
ELASTIC PROPERTIES 321

Figure 2.10. Schematic view of a small section of two-ply laminate. U.S. Department of Agriculture [14]. From such a background, it is not difficult to construct a theoretical prediction of the elastic characteristics of symmetric cord-rubber laminates provided that the elastic properties of a single sheet or lamina of the material are known. In what

follows, a brief review will be given of such a theory.

Attention here will be centered on layered, or laminated, structures

built up by bonding together plane sheets of rubber-coated fabric. In the case of pneumatic tires this fabric is simply a series of parallel

tire cords. In addition, the class of laminated structure considered here

is that of a two-ply combination in which the cords are separated angularly in the two plies by an included angle 2a, as shown in figure 2.10. In that illustration, the heavy diagonal lines representing typical textile cords are shown being bisected by the coordinate axes f and 17, two arms of the orthogonal ^, 17, ^ system.

For the present, attention will be confined to those structures in which

stresses in the ^ direction are negligibly small compared with those in the f and 7) directions. This assumption will generally be true for bodies

that are large in the f and 17 directions compared to their thickness

in the ^ direction. While discussion here is limited to the characteristics of a two-ply

system, the ideas and techniques developed allow fairly easy extension to any number of plies provided that the structure remains orthotropic as defined below.

Neglecting effects through the thickness of the two-ply structure

direction), one may define an orthotropic material as one possessing

two planes of symmetry at every point. For example, in figure 2.10,

the planes of symmetry are ^ — 17 and ^ — ^. From this it is seen that not only will the two-ply construction be orthotropic, but so will a multi-ply

construction made up of a series of similar two-ply structures, and so will a number of special cases such as illustrated in figure 2.11. In figure 2.10,

^1

Figure 2.11. Typical orthotropic laminates.
322 PROPERTIES OF CORD-RUBBER LAMINATES

it may be seen that f and r) are principal directions, in that normal

stresses applied in these directions do not produce shearing distortion, nor do applied shearing stresses in these directions result in normal

distortion. A convenient general form for Hooke's law about principal axes for such orthotropic material is

(T-n

(2.4)

where E^, E-q, F^v, and G^-n represent the four orthotropic elastic constants of such a laminate. These equations may be used in the structural

analysis of devices made from such materials. In order to determine the four elastic constants just mentioned, it is necessary to consider the construction of each individual ply making

up the laminate. The laminate is formed by embedding a series of parallel,

straight cords lying in a plane into a sheetlike matrix of more elastic material. This is illustrated in figure 2.12. It is seen that maximum and

minimum moduli of elasticity are to be found in the x and y directions

and that this single sheet is also orthotropic about x and y, with result- ing elastic constants Ex, Ey, Fxy, and Gxy. These are assumed to be

known. Generally they would be associated with a state of tension in the

cords. However, they could be associated with cord compression as previously discussed.
ELASTIC PROPERTIES 323 The generalized form of Hooke's law may be written for the sheet

of figure 2.12 using the stresses as independent variables with respect

to any desired coordinate axes, as shown in eqs (2.5). These are basic

to this development and will be treated in some detail. — fliicr^ + auCTrj + aiscr^T?,

€7? = a2icr^+ ^220-17 + a23cr^T7, (2.5)

Reference [19] derives expressions for the elastic constants ay as functions of the angle of inclination a of the ^-r) axes with respect to the

x-y axes, and of the orthotropic elastic constants of a single sheet

Ex, Ey, Fxy, and Gxy- These expressions are

cos^ a sin^ a ( \ 2 \ . ,

_ aii = —7^ \ 7^ ~F<— sin- OL cos- a,

, ^^3^ Oi

, / 1 2 \ . , «22 = — \ F r It:; 7;— I sm" a COS- a,

<]) \^xy ^ xy/

s \n* a , cos^ £7

^ / 1 , 1 , 2 \ . , (cos- q: — sin^ a)2

^33 = 4 ( —+ —+ — sm^ a cos^ a^- (2.6)

\t!jx t^y xyl ^xy

0-21 = ai2 = [-p,—h —— I sm^ a

\£jx t^y ^xy/

sin^ aH- cos"* a

cos^ a =; .

tt31 = «13 =

2 cos^ a sin a 2 sin^ a cos a Ex En

(cos^ a — sin^ a) sin a cos a,

^32 — ^23 2 sin^ a cos a 2 cos^ a sin a Ex Ey +

\^xy ^ j-y/ (cos^ a — sin^ a) sin a cos a.

Equations (2.5) and (2.6) now allow the properties of an orthotropic sheet

(for example, one ply of cords embedded in a matrix) to be predicted

in any direction f-ry at an angle a with the cord direction x. This knowledge can be used to construct a two-ply structure. Consider now a sheet of the type shown in figure 2.12 but inclined at some angle a to the axes ^—r). This is illustrated in figure 2.13. Imagine
324 PROPERTIES OF CORD-RUBBER LAMINATES

Figure 2.13. Single ply of cord embedded in rubber at angle a to the vertical ^ axis.

that this sheet is extended in the ^ qnd iq directions by means of normal

stresses acting on the edges. Since the orthotropic axes x and y do not coincide with the ^-rj axes, this is not possible; distortions e^v will inevitably accompany the application of any set of normal stresses cr^ and ctt). For that reason one must admit the existence of shearing stresses

(T^Tj as necessary for distortionless extension of an element such as that of

figure 2.13, where the reference and orthotropic axes are not the same. With this provision in mind, one may go directly to eqs (2.5) and presume

no distortion under loading; that is, the loads will be adjusted to prevent

distortion. Then, eqs (2.5) become = ai icr^ + ai2cri7 + aiacr^i?

,

€17= a2iO-|4- a220-i?-|- a23cr^i?, (2.7)

From these it may be seen that the only stress which is not independent can be chosen to be cr^ri since it can be expressed in terms of cr^ and ctt? from the last of eqs (2.7), and this expression can then be substituted

into the first two of eqs (2.7). On doing this, one obtains

o-^v = — («3i/«33)o-^— («32/ 033)0-17, (2.8a)

(«i3«3i/«33)]cr^+ [«i2— (ai3«32/a33)]crT,, (2.8b)

ev = [a21— («23«3l/«33)]0"^+ [^22— («23«32/«33) JcTt,, (2.8c)
ELASTIC PROPERTIES 325

where a^v is the stress that must be furnished from some external source to obtain a distortionless extension. In view of the previously

determined symmetry ai2 = «2i, «i3 = «3i, a32 = «23, it is seen by comparison of this set of equations with eqs (2.4) that Ei = [an — (af3/a33)]~S

Ev= [a22— (a23/«33)]"\

F^v = — [ai2 —{aizCL2zl a33)]~^.

(2.9)

In connection with eqs (2.9), if one were to return to eqs (2.6) and uti- lize the compression characteristics E'^, E'y, etc., a completely different

set of constants ajj, agg, etc. would exist, and this would result in a set of moduli E'^, E^, and F'^-q completely different from those using the elastic constants associated with tension.

It is now necessary to consider the last elastic constant G^v- Two

different possible shear moduli G^-q exist, one associated with all cords

in compression and the other with all cords in tension. These two

cases may be worked out in a very straightforward way and will be

presented here. To obtain the G^-q associated with either cord compression or cord

tension, it is necessary to postulate an extension-free distortion, or pure

distortion e^rj, which occurs as the result of the proper application of the stresses cr^, ctt?, and cr^r,. Equations (2.4) become, when applied to

this case, — ano-f + ai20-Tj + aucr^rf = 0,

€17 — 0-210'^ + a220'Tj + a23(T^i7 — 0,

€^17 = asiCT^ + a32cri7 + a^sCTiv-

(2.10)

The first two of these may be used to express cr^ and err) in terms of a^v- This results in 023— («12«13/«ll)

(TV ^22— [aiJan)

a\2 «23— («i2ai3/aii) CL\z\ ail 0^22 — {dlJ ^ii) aii\

(2.11)

cr^T,

and finally these may be used to obtain r \ r/ai2 ^

/

ai2ai3 — ana23 \ , ai3l , r«i2«i3 — OuOmI G^T,= j-a3i — H +a23

^ I L\aii/\ «ii«22 — a^2 / «iiJ L aii«22 — ai2 J + a33

(2.12)

Equations (2.9) and (2.12) are extremely important as they allow the

elastic constants of an orthotropic structure to be calculated in any
326 PROPERTIES OF CORD-RUBBER LAMINATES

direction as a function of the terms an, a22, . • 033. These, in turn, are determined completely by the elastic properties of the single sheet in the directions of the orthotropic axes, that is, parallel and perpendicular

to the cord directions, as well as the angular orientation of the sheet. As

mentioned previously, eq (2.12) represents both the possible shear

moduli of a laminated orthotropic structure. It can be used to represent

the shear modulus for those structures where all cords are either in tension or compression, depending on whether Oy or is used.

It is seen that the elastic properties given by eqs (2.9) and (2.12) are identical to the elastic properties of the compound orthotropic structure of figures 2.10 and 2.11 provided that the two plies which make

up figure 2.10 are identical in all respects, and provided that provisions

are made for obtaining the necessary shear and normal stresses such

as given by eqs (2.8a) and (2.11). In general, it must be considered that these necessary shear and normal stresses are provided by the bond

between adjacent plies in a laminated multi-ply structure, and that these stresses may be used to define the load intensities acting on the

adhesives linking the plies together.

320|— \ \ 1 -T 1 1 ,

0 .010 .020 .030 .040 .050 .060 AXIAL STRAIN, IN. /IN Figure 2.14. Typical load-axial strain curve for a cylindrical tube subjected to axial

tension.
ELASTIC PROPERTIES 327 Reference [19] gives comparisons between calculated and measured

values of such elastic constants for a series of four-ply cylinders made up

of rayon cord reinforced rubber. The construction of these cylinders was

orthotropic, in that alternate plies were oppositely directed at the same

angle with respect to the centerline of the cylinder. Such specimens were

tested under various combinations of internal pressure and end load, and

the resulting stress-strain curves were surprisingly linear. An example

of one of these is given in figure 2.14, from which it may be seen that the

resulting composite body exhibits linear stress-strain relationships to within a close approximation, and hence elastic constants for such struc- tures are meaningful.

10* r i ^ \ 1 \ 1

O Experimental Data for — Theoretical Curve A First Approximation Correction for Angle

Chonc 9 of Experimental Dota

30" 45« 60« CORD HALF ANGLE -a

75« 90* Figure 2.15a. Modulus E^ as predicted from eg {2.9) vs. cord half-angle a, along with experimental values for E^.
328 PROPERTIES OF CORD-RUBBER LAMINATES

A comparison between the modulus in the axial direction of the

cylinder, as predicted from eqs (2.9) and as measured on the rayon

cord-rubber cylinders, is given in figure 2.15a while a similar comparison

for the so-called cross modulus F^t? is given in figure 2.15b. In connection

with figure 2.15b it is interesting to note that as the half-angle between

textile cords in adjacent plies is increased, the imposition of pure tensile

loads on the cylinder eventually causes compressive forces to be set up in the reinforcing cords. In the absence of any cord preload, this means that at this particular point, which occurs at approximately 56°, one must begin to think in terms of the compression elastic properties.

-io«

-10'

-10*

Calculated Curve Using Compression

Properties fnr r.nmhirxitiftnc \A/ hero a > 55»

Calculated Curve Using all Tension

Properties for all Angles

1 o Experimental Data for F^^

1

11

11

1

1

//

/

\ tS

11 A 1 1 1 J — 'A a/-

\ ( ) //

//

//

y <

I5» 30° 45" 60" 75° 90° CORD HALF ANGLE - a Figure 2.15b. Modulus Ffii, as predicted from eq (2.9), vs. cord half-angle a, along with experimental values forFiv-
ELASTIC PROPERTIES 329 This has been done in figure 2.15b and the resulting plot actually illus- trates the tension and compression portions blended together. This demonstrates very clearly the necessity for utilizing all eight elastic constants

of such cord-rubber laminates when conditions require, since it is clear

from figure 2.15b that calculations made on the basis of tension characteristics alone would not correspond to experiment.

One of the basic elastic constants used in describing the elasticity of homogeneous isotropic materials is the ratio between the lateral and the

axial strain caused by an axial stress. This quantity, called Poisson's

ratio, has a value ranging from approximately 0.1 to 0.5 for most ordinary

materials, with rubber having a value very close to 0.5. One may physically

associate the property of infinite bulk modulus, or incompressibility,

with the value of Poisson's ratio of 0.5 for an isotropic body. As has been

pointed out previously, a cord-rubber combination is not homogeneous

nor is it isotropic. It is not, therefore, surprising to find cord-rubber

combinations exhibiting values of Poisson's ratio outside those ordinarily

obtained from homogeneous materials possessing isotropy. In fact, linkFiGURE 2.16. Poisson's ratio as predicted from eq (2.9), vs. cord half-angle a, along with experimental values for Poisson's ratio.
330 PROPERTIES OF CORD-RUBBER LAMINATES

ages or structures can be arranged in such a way that their overall effec- tive Poisson's ratio is large, and could easily be 5 or 10. It is probably more correct in this case to consider the combination of monofilaments

into singles strands, the strands into cord, and the cord in rubber as a structure rather than as a continuum. As an example, previously quoted,

Piatt and Butterworth [37] report a Poisson's ratio of 3.3 for a two-ply

nylon tire cord in rubber compressed to a 15 percent axial strain. Poisson's ratio may be obtained in terms of the notation used here

by taking the ratio E^/F^v, which is the ratio of lateral to axial strain. This has been calculated from the experimental data in reference [19]

and is given as a function of the cord half-angle in figure 2.16. Again, a comparison is made between experimental data and the theoretically

predicted Poisson's ratio using eqs (2.8) and (2.11). The results indicate

that the corresponding shapes of the curves are at least of the same type.

It is interesting to note that Poisson's ratios in excess of 1 are possible

and have been observed. This means that at the proper cord angles, a tubular cord-rubber specimen will contract more in its diameter than it will elongate when subjected to a tensile load in the axial direction.

This is undoubtedly the physical basis of the well-known Chinese finger

puzzle, as weU as various gripping devices derived from it.

50x10'

30x I0~

1 1

1 Calculated from-£q.( IZ) O Measured from Experiment with Cords Preloaded in Tension

/ < ) \

( Y^AII Co

\ Tens

rds in ion

/ c Cords in Cornpression

60° 75' .90° Figure 2.17. Shear modulus G^v, as predicted from eq {2.12), vs. cord half-angle a, along

with experimental values for Gfi?.
ELASTIC PROPERTIES 331

Torsion tests of cylindrical tube specimens have also been reported

over a range of half-angles from 0° to 60° where all cords were pre- loaded into tension. The results of these experiments are compared

in figure 2.17 with calculations made on the basis of eq (2.12). It may be

seen that the agreement between the experimental data and the calculations is acceptable. It should also be noted that in figure 2.17 the

calculated value of shear modulus, assuming all cords to be in a state

of compression, is also presented for comparative purposes. This

condition results in shear stiffnesses considerably lower than obtained

when all cords are in a state of tension.

Reference [24] shows that the elastic constants of a symmetric

laminated series of sheets may be expressed in dimensionless form in terms of a single dimensionless parameter ExIGxy, and also in terms of

the cord angle of the individual lamina with respect to the principal

axis of elasticity. This is accomplished by making simple approximations for the four elastic constants of a single sheet of cord reinforced

rubber, on an elementary strength of materials basis, and by using

these approximations to obtain numerical values of some of the ratios of elastic constants. This particular simplification has only been worked

out for textile and wire cords embedded in rubber and may not be particularly appropriate for fiberglass-epoxy or other materials. From such

an approximation the elastic constants of a multi-plied orthotropic

structure may be determined and plotted as a function of the cord half angle a and the single numerical quantity ExIGxy, representative of the

degree of anisotropy of a single sheet of material used in the laminate.

This analysis also assumes that all sheets used in the laminate are identical, although in a more general case this need not be so. It should

be noted that the ratio ExIGxy is convenient to represent the characteristics of a single sheet of material since this ratio takes on a value of

3 for an isotropic incompressible material. Values for this ratio as high

as nearly 10^ have been measured for twisted wire cords reinforcing

rubber. The notation used here is illustrated in figure 2.18. Reference [24] shows that for those cases where all plies of the

multi-ply laminate are loaded in such a way that their cords are in tension, or alternately, when all plies are loaded in such a way that their

cords are in compression, then exact expressions for the moduli of the

Figure 2.18. Left, single sheet offabric; right, even number of laminated fabric sheets.
332 PROPERTIES OF CORD-RUBBER LAMINATES

resulting orthotropic structures can be found. These moduli are ex- ^ pressed in terms of the cord half angle a and the four elastic constants

of a single sheet of the material used in forming the laminate. Figure i, 2.19 gives values of the dimensionless extension ratio E^IG^y for such ' laminates over a wide range of anisotropics, while figures 2.20 and

2.21 give similar dimensionless information concerning the cross modulus and the shearing modulus of such laminated structures.

j It is seen that cord angles play a most dominant role in the variation of

j these elastic constants. In particular, the variation of extensional i modulus with cord angle is extremely steep. This indicates that

j there might be many situations in which a laminated structure can be ! used to control the elastic characteristics over a wide range. i In most applications for pneumatic tires, the loadings are such that the material constants are those associated with cord tension. Occa10000

100

_

1

1 1

1

1 Ex /Fxy » 1/2 or 1/3 6.«/E« -1/3

1- T

-Gxy/Ex •( 0-*

Gxy/Ex =1 o-»

LlJ_L.

\^Gx,/E X «IO-« Y

^Gxy/Ex = 10-' —

1 1 1

1

1 20 40 60 80 100 CORD HALF-ANGLE - a Figure 2.19. E^/Gxy vs. half-angle a for various values o/Gxy/Ex-
ELASTIC PROPERTIES 333

Ex /f'xy = 1/2

1

Gxy/Ey >: 1/3

/Ex = I0-*

1 1 ^Gxv

1/Ex ' 10"' /

=\"Mi— -f—

/E, = 10'* //

\\

,Gxy/Ex ' 1 )-'

•

1 1 I I I I I I

I 20 40 60 80 100 CORD HALF-ANGLE - a Figure 2.20. F^n/Gxy vs. half-angle a for various values of G^y/E^.

sionally, under large tire deflections, cord compression appears in the tire sidewall and the elastic constants defining the stiffness then

take on values appropriate to this, as previously discussed. Under

both large deflection and significant tractive effort or braking force,

tire sidewalls can wrinkle. Insofar as is known there is no systematic

technique in the Hterature for defining the conditions under which this occurs, but recently Stein and Hedgepeth [25] have proposed methods

for analyzing partly-wrinkled structures, and have shown that membrane structures retain much of their stiffness at loads substantially

above the load at which wrinkHng first occurs. The stiffness of cord rubber laminates in which cord layers take on three or more different directions is of importance in the design

of breakers or belts for the tread band reinforcement of tires. With this application in mind, such elastic properties have been studied by
Figure 2.21. G^v/G^y vs. half-angle a for various values of Gxy/Ex.

Gough [38] and co-workers. They propose a method of analysis in which

one takes a rectangular sample of finite dimensions and applies a tensile

strain ei parallel to one axis and in the plane containing the cords. It

is then supposed that there must be a complementary strain 62 in the

perpendicular direction, also in the cord plane. Tensions in the cords

are then deduced from the magnitude of ei and €2, and the cord angle. It

is assumed that the component in the €2 direction of the cord tension is balanced by the stress in the rubber in the same direction, since no

external forces are presumed to be applied in the €2 direction. This

condition allows €2 to be eliminated from the equations. The contribution to the total force in the ei direction made by the effects of the

rubber and by the cord tension is then computed, and from this value

and the, strain ei the equivalent Young's modulus is obtained.

Such a computation is based on the following assumptions:

(a) That the long edges of the sample remain straight and parallel as if the sample is part of an infinite sheet.
ELASTIC PROPERTIES 335

(b) That strain is homogeneous throughout rubber and cord, so that the strain in either part is identical with the strain of the whole.

(c) That the stress along the cord length is proportional to the cord

modulus, but that the cords behave as part of the rubber in shear

and transverse stresses.

(d) That the finite thickness of the cords does not in itself influence

complementary stresses in the calculations, although it is taken

into account when computing the sectional area of the rubber and

cord in both principal directions.

In carrying out such an analysis, one must account for the additional

physical facts imposed by the type of structure to be encountered. For

example, in the construction of pneumatic tires it is common that the

structure be symmetric, or nearly symmetric. Within that restriction,

and presuming similar but oppositely directed cord angles ±ct, an additional cord at an angle of zero is often considered to form the third

direction of a three directional cord system. The two bias cord plies may

have a different modulus from the single zero degree ply, and the ratio of these moduli is an important new parameter which must appear in any computation of elastic stiffness of the plane sheet. The product of cord end count and cord modulus may be used as a measure of the

local stiffness density in each lamina or ply, measured parallel to the

cord direction. The ratio of these stiffness densities may be thought of

as an overall cord stiffness density ratio for a typical three-directional

composite. It is clear that composites of four or more directions could

become even more complex, and most interest is centered on the symmetric cases of these possible combinations.

Gough [38] has computed elastic stiffness characteristics for a number

of four-ply combinations, a few of which are shown in figures 2.22, 2.23

and 2.24. Similar computations may be done to estimate the effect of breaker angle and modulus on stiffness of the breaker region of tires,

and these are illustrated in figure 2.25. Reference [38] also gives plots of stiffness of breakers using cords in three directions. Once again the

need is demonstrated for the presence of a pair of low angle layers, in the region of 20°, for maximum stiffness. Due to the difficulties of presenting information on multi-ply structures

in concise form, considerable additional work could very well be done in the area of bringing to the tire designers simple and concise information on the elastic stiffness of various cord angle and cord density ratio combinations. Along this line, Gough and co-workers studied tire to ground forces and movements of the tire elements across the ground,

using as a model for the tire an elastic beam on an elastic foundation.

From such studies, they deduced that treadwear is minimized if the edgewise stiffness of the tread band region is maximized. From such studies, one may conclude that low crown angles and tread reinforcement insure

high beam stiffness, as might be expected, but they also show the rather

unexpected conclusion that maximum beam stiffness may be contained
336 PROPERTIES OF CORD-RUBBER LAMINATES

Figure 2.22. Crown stiffness of tires with a 4-ply rayon casing and 2-ply breaker of ply

stiffness (EcAn)/T = 15,000 Ib/inK

by other than zero angle breaker plies, as illustrated most clearly in figure 2.23. The role of the individual reinforcing cord in load carrying is not well

understood in dealing with the elastic properties of cord reinforced

rubber. Hedgepeth [26] has shown that sudden failure of one cord in an array of parallel cords can result in dynamic load overshoot of neighboring cords by a factor of about 1.25. Clark and Reid [27] have examined

the nature of the cord loads in the neighborhood of a broken cord by means of a small cord force transducer inserted into one cord in the center of a strip of parallel cords. This strip was placed in a state of

tension. The load in the instrumented cord was measured, and then the
ELASTIC PROPERTIES 337

instrumented cord was cut, first at a great distance from the location

of the force transducer and then closer and closer to it. After each cut, cord load was again measured, and from this one could obtain information on the way in which cord load is built-up along the cord from its free end.

This information is summarized in figure 2.26, where it is shown that for the typical tire carcass materials used here about 1 inch of cord length is needed before the cord carries most of its intended load. Weiner and Gogos [28] have studied the damping characteristics

of cord rubber laminates by means of sinusoidal excitation in bending of

a specimen which is pretensioned. Their results were presented in the

form of curves showing the ^-factor, which may be interpreted for simple
338 PROPERTIES OF CORD-RUBBER LAMINATES

0° 10° 20° 30° 40° 50° 60° 70° 80° 90°

BIAS ANGLE OF BREAKER PLIES ( ± ^9°)

Figure 2.24. Crown stiffness of tires with a 4-ply rayon casing and 2-ply steel breaker of ply stiffness (EcAn)/T = 123,000 Ib/in^.

harmonic motion in terms of the dimensionless damping f by means of the relation

Their conclusions generally were that the damping of such bending waves

in fabric reinforced rubber appears to be caused primarily by energy

loss in shear, and that such loss characteristics tended to increase

rather markedly with the frequency above a level of about 100 Hz.
ELASTIC PROPERTIES 339

1800

1600

1400

1200

1000 800 600

400

200

0

BREAKER MODULUS EcAn leoo

T 23,000 Ib/in^

000 lb/in'

1600 1400

1200

1000 800

600 400

200

BREAKER MODULUS EcAn T

1 23,000 Ib/in^

0 10 20 0 50 60 70 0 10 20 30 40 50 60 70

/3°-BREAKER ANGLE /3-BREAKER ANGLE

4 PLY CASING ANGLE (£ = 90° 4 PLY CASING ANGLE OC = 60° 1800

1600

1400 1200 1000 800

600

400

200

0

^ BREAKER MODULUS EcAn

\ T L- 123,000 I b/in^

80,000 Ib/in^

40,000 Ib/in^ W - 15,000 lb/in'

0 10 20 30 40 50 60 70 -BREAKER ANGLE

4 PLY CASING ANGLE 01= 45°

Figure 2.25. Effect of breaker angle and modulus on stiffness of breaker region of cross bias cover.

Some of their results are shown in figure 2.27, while the influence of membrane tension on energy dissipation in damping is shown more

clearly in figure 2.28.
340 PROPERTIES OF CORD-RUBBER LAMINATES

2 3 4 5 APPROXIMATE LOAD/CORD (LBS.) Figure 2.26. Transducer load vs. distance of load transducer from free end of cord.

Figure 2.27. Q offabric-reinforced rubber specimens with two plies of cord as a function of rubber thickness. 200] 1—I I I I I 1

1

\ \—[—n 1

r ^ 34568 234568 23 20 100 1000 4000 FREQUENCY IN CYCLES PER SECOND Figure 2.28. Q of fabric-reinforced rubber specimen with one ply of cord as a function of tension.
STRENGTH AND FAILURE MECHANISMS 341

2.2. Strength and Failure Mechanisms for Cord

Reinforced Rubber

The failure mechanisms of cord reinforced rubber are exrtemely

complex and have recently been the subject of considerable active

research. For purposes of application to pneumatic tires, most of the

emphasis lies on the fatigue failure characteristics of these materials,

since service requirements demand that a pneumatic tire undergo a very large number of loading cycles. This inevitably forces the tire de- signer to proportion the carcass textile reinforcement in such a way

that ultimate static strength is many times the inflation pressure or the

applied load. Large static factors of safety are the rule in tire design.

When ultimate tensile strength is to be calculated, most approximations

consider that cord failure results when cord loads reach a tensile value

equal to that which they exhibit in simple direct tension tests. In other

words, ultimate strength of cord reinforced rubbers is equivalent to cord strength uninfluenced by the surrounding rubber, so that the rubber

neither increases nor decreases the ultimate tensile strength of the

cord. To that extent the cord-rubber system may be thought of as un- coupled in regard to ultimate strength. For example, bursting of a tire due to inflation would be predicted on a cord tension basis without

regard to presence of the rubber.

The more common loading environment involves the fatigue characteristics of cord reinforced rubber, and here the situation is much more

complex. First of all, it is necessary to recognize that the complete

cord reinforced rubber system consists of three phases: first, the textile cord itself, secondly, the adhesive which is applied to the cord surface

in order to promote adhesion to the rubber, and finally the surrounding

rubber matrix. Details of the rubber and cord characteristics are given

in previous chapters of this volume, but it should be noted that the cord

is made of a multiplicity of filaments which are first twisted to form

a strand, followed by the twisting in opposite directions of two or more

strands to form a cord. For purposes of pneumatic tire construction

two and three strand cords are the most commonly used.

Since many service failures seem to occur as a result of fatigue, a

great deal of attention has been directed to mechanisms of tire fatigue

failure. In order to describe such mechanisms, it is probably best to do

so in some systematic order roughly parallelling that which would

occur in an actual tire. For such a description, one may draw upon the

work of Eccher [29], Patterson and Anderson [30], Patterson [31],

Uzina and Basin [32], Butterworth, Davis, and Piatt [33]. A number of other authors and references will be cited in addition, but the basic

mechanism of fatigue failure as it is now understood is well delineated

by these authors.

First of all, two primary concepts regarding fatigue failure should

be stated clearly:

a. For cord reinforced rubber, the so-called fatigue failure is, in fact,
342 PROPERTIES OF CORD-RUBBER LAMINATES

a system failure in which the failure is initiated at or very close to the

cord-adhesive-rubber interface.

b. A normal fatigue process, such as occurs in metals, does not exist for the cord-adhesive-rubber combination as a whole. However, there

exist fatigue mechanisms both for the rubber and for the rubber to adhesive bond, which then directly or indirectly act on the strength

of the fabric. The actual initiation of the processes which precede the fatigue

failure of a cord reinforced elastomer has been studied by Eccher

[29], by use of the Mallory tube test and by microscopic examination,

and by Uzina and Basin [32], using fluorescence analysis as well as photomicrography. Uzina and Basin studied dipped viscose. Nylon 6, cotton and polyamide cords under fatigue conditions and observed the

following forms of initial failure on tires which had been run to the point

of obvious ply separation:

a. Cord separation accompanied by a considerable part of the adhesive adhering to the rubber, coupled with some adhesive remaining on the cord. Such a mechanism clearly indicates separation occurring

within the adhesive layer, but is cohesive in nature.

b. Separation in which a thin film of the adhesive remains attached to the rubber, while in some areas a portion of the rubber remains with the

adhesive. This type of separation is much closer to a separation along

the adhesive-rubber boundary than type a., and can be termed a composite separation. It is also cohesive in nature.

c. Separation occurring without traces of adhesive film being left on the rubber or of rubber on the cord. Consequently, this separation occurs cleanly at the adhesive-rubber interface and is purely adhesive

in nature. Some of the details of such a cord-rubber-adhesive system are pre- sented here. Table 2.4 gives typical penetration depths for RFL^ dipped

cords.

Eccher [29] found by photomicrography that both Mallory test tubes

as well as examination of tires implied very similar types of failure

initiation. Butterworth, Davis and Piatt [33] have accumulated extensive evidence

to show that the types of failures just described are promoted and ac- celerated when the cord is forced to carry compression in the pneumatic

tire structure. Their studies agree with those of Patterson [31], and both

authors report that compressive loading of typical tire cord structures

in transparent rubber allows one to clearly observe the strain concentra-

^ RFL is the common abbreviation for resorcinal-formaldehyde-latex, which is the basic tire cord adhesive. Most

commercial adhesives use this as a basis for more sophisticated, proprietary adhesives.
STRENGTH AND FAILURE MECHANISMS 343

Table 2.4. Depth to which latex-resorcinal-formaldehyde

compositions penetrate various cords when dipped

Viscose 1M 1 JNylon Cotton

Cord diam. (mils*) 35 20 35

Radial dip penetration

in fiber diameters 4 3 4

Radial dip penetration

(mils) 1.9 3.54 2.8 Fiber diam. (mils) 0.485 1.18 0.7

*1 mil =0.001 inch.

tion effects of compressive forces on the cord-rubber interface. In particular, the reentrant angle caused by the twisting of the several strands

together to form the cord seems to be a point of particularly high strain

concentration, and a point which is prone to initiate adhesive system

failure under compressive loads. Cord degradation resistance is deter- mined by the ability of the surrounding rubber matrix to control the

compressive axial deformation of the cord. Where the adhesive interface

fails, this control is lost and filaments traversing the reentrant angle in the cord geometry tend to buckle in an uncontrolled manner. This

leads to severe relative deformation between the surrounding rubber

matrix and the cord structure. This relative deformation increases in severity because, with continued cycling, the cord geometry opens up

(reduction in cord packing factor) leading to increased filament buckling

and an increased interaction with the matrix. A reduction in cord packing

factor leads to the induction of higher tensile strains in filament lengths

having a low angle of inclination to the cord axis — mainly filaments on

the external surface of the cord. Test data in general indicates that

cord fatigue is greatly dependent upon the presence of compressive

stresses in the cord, as reported by Illingworth [35]. Following the initial failure somewhere in the cord-adhesive-rubber

system, present evidence points toward the occurrence of increased relative motion between the cord and rubber, as well as among the cord

filaments themselves. This may result in chafing or rubbing of the cord

filaments against one another, with the consequent possibility of initi- ating sites of microscopic mechanical damage on the surface of the

individual filaments. Patterson [31] has studied this in some detail and

concludes that the mechanism of filament rupture in the fatigue of in- dividual nylon monofilaments is directly traceable to the presence of some small surface defect. Such a defect serves as an initiator of fatigue

failure, which then propagates due to alternating bending of the monofilament. Patterson found that if the imperfection occurs at the point of maximum compression in bending, the fatigue life of a monofilament was

greatly decreased. Simple bending seems to account for the mechanism
344 PROPERTIES OF CORD-RUBBER LAMINATES

Figure 2.29. Bias ruptured filament.

of failure since the appearance of broken filaments taken from pneumatic

tires and broken filaments fatigued in bending are essentially identical. Figure 2.29 shows an enlargement of a filament taken from a tire after some running.

This same problem has recently been studied by Piatt and Butterworth.

They conclude that the creation of a filament buckle and the location

Figure 2.30a. Filament failure modes. {Courtesy of Fabric Research Laboratories, Inc.)

Failure initiated on inside of bend flex zone. Nylon monofilament whose diameter is equivalent to a standard 2 ply nylon cord ~ 20 mils.
STRENGTH AND FAILURE MECHANISMS 345

Figure 2.30b. Filament failure modes. {Courtesy of Fabric Research Laboratories, Inc.) Two bend buckle of 20 mil nylon monofilament.

Figure 2.30c. Filament failure modes. {Courtesy of Fabric Research Laboratories, Inc.)

Partially failed nylon tire cord. Degraded in rubber on Goodrich test machine.

3 percent tension cyclic axial 15 percent compression cord strain Note inception of filament buckle failure at cord ply line.
of the buckle angles is determined by the position of the element of filament, its degree of association with neighboring filaments and tri- axial stress state to which it is subjected. They conclude that it is unlikely

that all filament failures are initiated by a defect of a filament leading

to a buckle failure. They observed that when monofilaments of diameter

equivalent to a conventional tire cord are subjected to simulated tire use, the monofilament tends to buckle in fashion similar in shape and

character to buckles noted in individual filaments taken from fatigued

tire cords. Such buckles are illustrated in figure 2.30. In particular, the

two angle buckle shown there failed initially on the compression side of the buckle but ultimately failed in tension.

The process of tire degradation therefore apparently involves failure

first at the surface of the cord, by one of the three forms proposed by

Uzina and Basin, allowing detachment of the cord from the surrounding
STRENGTH AND FAILURE MECHANISMS 347

rubber. This is followed, according to Butterworth, Davis, and Piatt [33], by mechanical working of the cord leading to a reversion of its tensile properties to those of the undipped, unstretched cord. Failure of in- dividual filaments next occurs, in a sequential fashion, finally leading

to total cord failure. This hypothesis seems to be in accord with the

experimental evidence obtained by studying the characteristics of tires which have been run on both road and test wheels, since most observations have found that tire cords which have lost a fraction of their strength,

due to monofilament fracture in fatigue also exhibit reduced adhesive

bond strength to the surrounding rubber, as measured by strip-out tests. The process of cord degradation goes forward by means of individual

monofilament fatigue fracture. The nature of this process is clearly

statistical and is not well understood. Patterson and Anderson [30] carried

out an extensive study of the residual strength of nylon tire cords in tires which had been run on test wheels, while similar studies on rayon cords

in highway service have been carried out by Klein, Piatt, and Hamburger
348 PROPERTIES OF CORD-RUBBER LAMINATES

Figure 2.30f. Filament failure modes. {Courtesy of Fabric Research Laboratories, Inc.)

Partially failed nylon tire cord.

[36]. Both authors conclude that cord strength does not Hnearly degrade

with mileage. As a matter of fact, it is this particular area which repre- sents one of the substantially unknown regions in the process of tire cord

degradation, since both these studies indicate that after an initial, rela- tively rapid loss of cord strength there follows a long period in which the

rate of loss is rather small. Typical data are shown in figures 2.31 and

2.32. The reason for this initial loss is as yet uncertain, but it has been

postulated that it is due to physical and geometric rearrangement. Even

in fatigue-failed tires, cords adjacent to the failure are in general of normal strength. It must be concluded that the cord failure process, once

initiated, is somehow catastrophic and runs a rapid course, similar to the

effects found in propagation of a fatigue crack in metals.
I STRENGTH AND FAILURE MECHANISMS 349

Figure 2.30g. Filament failure modes. (Courtesy of Fabric Research Laboratories, Inc.) Cord cross section photomicrographs. Failure sequence A-F.

AVERAGE "

CORD

BREAKING '0 STRENGTH, KGMS 9

LEADING SHOULDER M 1 11 I '0 2 4 6 8 10 12 14 16 18 MILES ON TESTWHEEL,THOUSANDS

Figure 2.31. Average breaking strength of sections offirst-ply cords.
350 PROPERTIES OF CORD-RUBBER LAMINATES

o

40' ^ ^ ^ 1 L 20 40 60 80 100

THOUSANDS OF MILES

Figure 2.32. Percent residual cord strength vs. tire mileage.

There are substantial differences between cord failure rates in different

parts of the tire. Patterson and Anderson report the most critical regions

to be at the shoulder and shortly below the turn-up region, as shown in figure 2.33, and further depends on the direction of the cord with respect

to the tire direction of motion, as previously shown in figure 2.31. Eccher [29] has discussed in some detail the use of various tests, and

particularly the Mallory tube test, as a mechanism for simulating the

behavior of the tire materials in service insofar as such fatigue characteristics are concerned. He concludes that the Mallory test, which uses

a tube bent through a 90° arc, held under internal pressure and rotated

at high speed, is valuable in reproducing cord-adhesive-rubber system

failure similar to those observed in tires. A sketch of the Mallory appaREGION OF HIGH STRENGTH

REGION OF HIGH STRENGTH

LOSS AT END OF FLIPPER

Figure 2.33. Locations of maximum cord damage in tire after flexing test.
REFERENCES 351

COUNTER

REGULATED AIR PRESSURE SUPPLY

Figure 2.34. Sketch of Mallory tube test.

ratus is shown in figure 2.34. A number of other test devices have been

designed to induce cord fatigue, and since this is an area related but not

particularly pertinent to tire mechanics, the reader is again referred to Eccher [29] for a more thorough discussion of various test methods.

Once a cord has failed, the loss of strength throws additional load to neighboring cords in such a way that under favorable conditions an ad- hesive separation of one of the types previously mentioned may begin

in the neighboring cord, followed by individual filament rupture and

eventual cord tensile fracture. Under sufficiently severe loads this process can result in the propagation of a failure. Finally, it might be of some brief interest to indicate the magnitude of the problem of fatigue failure of tire materials. This can only be esti- mated roughly, but the data of Starks [34], taken on the British Motorway

M.l in 1962-63, showed that the percentage of total tire failures was

caused almost exactly half by puncture and half by burst, the latter representing a fatigue failure of some sort or another. Therefore, the

question of the fatigue of cord reinforced rubber is not only complex but

very practical indeed.

References

[1] Hashin, Z., Theory of mechanical behavior of heterogeneous media, Appl. Mech. Rev. 17 (1), 1-8 (1964).

[2] Frank, F., and Hofferberth, W., Mechanics of the pneumatic tire. Rubber Chem. Tech. 40 (1), 271-322 (Feb. 1967).

[3] Clark, S. K., A review of cord-rubber elastic characteristics, Rubber Chem. Tech. 37

(5), 1365-1390 (Dec. 1964).
352 PROPERTIES OF CORD-RUBBER LAMINATES

[4] Rivlin. R. S., Plane strain of a net formed by inextensible cords, J. Rati. Mech. Anal. 4, 951 (1955).

[5] Rivlin, R. S., Networks of inextensible cords. Nonlinear Problems of Engineering, W. F. Ames, ed. (Academic Press, New York, 1964).

[6] Genensky, S. M., and Rivlin, R. S., Infinitesimal plane strain in a network of elastic cords. Arch. Rati. Mech. Anal. 4, 30-34 (1959).

[71 Adkins, J. E., Large elastic deformations. Progress in Solid Mechanics, Vol. II, I. N. Sneddon and R. Hill, eds. (North Holland Publishing Co., Amsterdam, 1961).

[8] Adkins, J. E., Cylindrically symmetrical deformations of incompressible elastic materials reinforced with inextensible cords, J. Rati. Mech. Anal. 5, 189 (1956).

[9] Adkins, J. E., Finite deformation of materials exhibiting curvilinear aeolotropy, Proc. Roy. Soc. A229, 119 (1955).

[10] Adkins, J. E., Finite plane deformation in thin elastic sheets reinforced with inextensible cords, Phil. Trans. Roy. Soc. (London) A249, 125 (1956).

[11] Adkins, J. E., and Rivlin, R. S., Large elastic deformations of isotropic materials, X:

reinforcement by inextensible cords, Phil. Trans. Roy. Soc. (London) A248, 201

(1955).

[12] Clark, S. K., and Dodge, R. N., A comparison of net and continuum theory for cord

reinforced laminates. Textile Res. J. 38 (9), 931 (Sept. 1968).

[13] Clark, S. K., Dodge, R. N., and Field, N. L., Calculation of the elastic constants of a

single sheet of rubber-coated fabric. Report 02957-13-T, (Office of Research Administration, The University of Michigan, Ann Arbor, Mich., Oct. 1961).

[14] Stress-strain relations in wood and plywood considered as orthotropic materials, Rept. No. 1503 (U.S. Dept. of Agriculture, Forest Products Laboratory, Forest Service, Madison, Wise, March 1956).

[15] Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity (Oxford University Press, London, 1961).

[16] Hashin, Z., and Rosen, B. W., J. Appl. Mech. 31, 223 (1964).

[17] Whitney, J. M., Basic mechanics of fiber reinforced composite materials, Textile Res.

J. 37, 1056-1062 (Dec. 1967).

[18] Ekvall, J. C, Elastic properties of orthotropic monofilament laminates, ASME Paper

No. 61-AV-56, (1961).

[19] Clark, S. K., The plane elastic characteristics of cord-rubber laminates, Textile Res.

J. 33, 295 (1963).

[20] Budd, C. B. (Private communication).

[21] Hofferberth, W., The elasticity law for composite rubber-textile bodies, Gummi Asbest

9, 240 (1956).

[22] Herrmann, L. R., and Pister, K. S., Composite properties of filament-resin systems, ASME Paper No. 63-WA-239, (1963).

[23] Hoff, N. J., The strength of laminates and sandwich structural elements, Chapt. 1, Engineering Laminates, A. G. H. Dietz, ed. (John Wiley and Sons, New York, 1949).

[24] Clark, S. K., Internal characteristics of orthotropic laminates, Textile Res. J. 33, 935

(1963).

[25] Stein, Manuel, and Hedgepeth, J. M., Analysis of partly wrinkled membranes, NASA

TN D-813 (National Aeronautics and Space Administration, Washington, D.C.,

July 1961).

[26] Hedgepeth, J. M., Stress concentrations in filamentary structures, NASA TN D-882

(National Aeronautics and Space Administration, Washington. D.C., May 1961).

[27] Clark, S. K., and Reid, F. R. (UnpubHshed data).

[28] Weiner, F. M., and Gogos, C. M., Rubber Chem. Tech. 34 (1), 158 (1961).

[29] Eccher, S., Typical damage in Mallory tubes and tire carcasses. Rubber Chem. Tech. 19,299 (1966).

[30] Patterson, R. G., and Anderson, R. K., Fatigue failure in nylon reinforced tires, Rubber

Chem. Tech. 38 (4) (Nov. 1965).

[31] Patterson, R. G., Mechanics of bias filament rupture in fatigue of nylon. Rubber Chem.

Tech. 39 (5) (Dec. 1966).

[32] Uzina, R. V., and Basin, V. E., Study of the nature of failure in the system cord-adhesive-rubber, Sov. Rubber Tech. 19, 27-33 (July 1960).

[33] Butterworth, G. A. M., Davis, S. J., and Piatt, M. M., The response of nylon tire cord

in rubber systems to cyclical strain (Fabric Research Laboratories, Inc., unpublished

report).
REFERENCES 353

[34] Starks. H. J. H.. T>Te failures in accidents on the motorway M. 1 in 1962-63. Inst. Auto. Assess. 17'(1). 17-25 |1966).

[35] lUingAvorth. J. W.. J. Textile Inst. 44, 328 (1953).

[36] Klein. W. G.. Piatt. M. M.. and Hamburger. W. J.. Cord fatigue in fleet tested tires (Fabric Research Laboratories. Dedham. Mass.).

[37] Piatt. M. M.. and Butterworth. G. A. M. (Private communication).

[38] Cough. V. E.. Stiffness of cord/rubber^ constructions. Kautschuk und Gummi. 20,

469-482 (Aug. 1967).
i
CHAPTER 3

Structure of the Tire

V. E. Gough 1 Page 3.1. General considerations 356

3.2. Pneumatic tire structure — general features 360

3.3. Flexible filament and soft matrix constructions 369

3.4. Tire construction methods 379

3.5. Cord path— practical factors determining selection 385

3.6. Calculation of cord length 386

3.7. Analysis of manufacturing methods by R vs. cos (J) charts... 390

3.8. Mechanism of load carrying— infinitely flexible membrane... 392

3.9. Mechanism of load carrying— tire structure 398

References 404

Formerly with the Dunlop Company, Birmingham, England (Retired. Present address: Croyde, Devon, England.) 355
356 STRUCTURE OF THE TIRE

3.1. General Considerations

That the function of a tire is to transmit the forces which drive, brake,

and guide the vehicle as well as to carry the load is now accepted by

all— it is, of course, the only component of the vehicle which makes

contact with the road surface. Also, it is now being recognized that the

tire not only has to absorb local road surface irregularities over a wide

range of types of road materials but also has to provide a vibration-free

motion on smooth roads comparable with that given by a circular solid wheel on a perfectly straight guideway. There is a conflict in these

differing requirements. Flexibility is required for the absorption of road

rugosity. Constancy of effective dimensions is needed for the exacting

requirements of constant axle height, straight line motion and uniformity

of effective rolling radius on a flat, smooth road as illustrated in figure 3.1. This conflict is further heightened by the fact that basically a pneumatic

tire structure is a surface of revolution, usually of curved cross section

in a radial plane, and consequently is a surface of double curvature. A

surface of double curvature is, from a geometrical point of view, a non- developable surface and yet it must deform to give an area of contact on a plane road surface.

These conflicting geometric and mechanical requirements demand

that a vehicle tire is either:

a. a solid tire made to a high geometrical precision from relatively

low modulus, highly elastic material capable of substantial deformation, or

b. a gas inflated envelope with certain exacting requirements to be

met.

Q. b. Figure 3.1. Conflicting requirements which a tire has to meet.

(a) ability to absorb surface irregularities — this requires flexibility. (b) constant axle height and effective rolling radius on smooth roads -this requires uniformity, and is most easily obtained with a rigid wheel.
GENERAL CONSIDERATIONS 357

It will be seen later that this envelope of double curvature usually is an

anisotropic hollow structure made of flexible filaments of high modulus

material such as textiles, metal or glass, embedded in and bonded to a matrix of low modulus material such as rubber or a rubber-like polymer.

The orientation of the filaments in this envelope has to be such as to meet certain contrasting structural requirements. These are:

a. no appreciable change of size upon inflation,

b. an ability to envelop obstacles without sustaining damage,

c. the ability to deform from a surface of double curvature to a plane

surface, and

d. enough rigidity to develop substantial forces in any direction.

It is this type of structure in which we are interested.

Inflated shell structures having surfaces of double curvature made

solely of a single isotropic material have in the past failed to meet all of the several requirements outlined above. The capabiUty of having a long

fatigue life when continuously flexed from a surface of double curvature

to a flat surface and back again — and at the same time having adequate

structural rigidity to carry the vehicle load and resist the drive, brake

and side force for a practical automobile, and also at the same time pro- viding a constant dimensional size nearly independent of the inflation pressure, have not been met by an isotropic thin or thick shell of materials

yet known.

Uninflated thin shell structures of double curvature made from high

modulus materials such as metals or the semi-rigid or rigid plastics,

including aU known forms of high modulus polymers, have not been

practical since the necessary flat contact cannot be developed without

stressing the material beyond its yield point or at least to the point where

fatigue life is very short. The same limitations apply to open frame structures, unsupported by

gas inflation, made from a single material or a combination of materials,

as anyone who has had experience in designing spring wheel alternatives

to the pneumatic tire will agree. Such structures can, however, be de- signed to meet the requirements associated with low gravitational fields,

e.g.. moon vehicles. Consider the design of a tire-like space frame consisting of a series of steel rings of appropriate dimensions for use on an

automobile on earth, such as shown in figure 3.2. It is found that either

the rings are thin enough to be capable of deflecting the required amount

but can only carry a trivial load, or, they are stiff enough to carry a useful

load but too stiff to change their shape to a flat one without exceeding

the yield point or fatigue stress limits of the metal or other material being

used. Practically the only type of spring wheel design to get further than

the drawing board stage is a series of helical springs supporting a chain

belt or flexible ring as shown in figure 3.3. This is a sort of solid tire in which the low modulus material is replaced by suitably fashioned high
358 STRUCTURE OF THE TIRE

Figure 3.2. Space frame and spring wheels built of steel hoops in any configuration can

be designed for either load requirement or deflection requirement, but not both for a practical earth vehicle.

Figure 3.3. One form of spring wheel surviving initial tests, but which does not compete

with pneumatic tires.
GENERAL CONSIDERATIONS 359 modulus material to get a long length of metal torsion bar in the space

available. Such devices have, however, never seriously challenged even

solid rubber tires, let alone the pneumatic tire. Consideration of these various concepts and designs may seem to be

somewhat removed from the structure of a pneumatic tire, but they have some relevance to the fact that a pneumatic tire casing is a practical

working device precisely because it is comprised of high modulus flexible filaments embedded in and bonded to a low modulus matrix. Such an

anisotropic shell structure, although unable to carry much load by itself, can carry substantial loads when inflated by a gas under pressure, and

can still meet the geometric and mechanical requirements stated earlier. One reason for this is that the ground contact pressure within the contact patch is primarily determined by the pressure of the gas inflating

the tire structure, so that the load carried by the tire is not limited by the

conflict of material properties in the same way as it is in the case of the

solid rubber tire. In the case of the solid tire the entire load is carried by

only a small fraction of the total tire volume, in fact, to a first approximation it is carried by a volume made up of the tire cross-sectional area

times the length of the contact patch. To carry a greater load at no greater

tire stress requires a wider, heavier tire. The total weight of the tire is thus approximately the ratio of the tire periphery to the contact patch

length multiplied by the weight of the material being locally stressed.

In a pneumatic tire, on the other hand, such a calculation is not pertinent

since load carrying ability is controlled by contact patch area and infla- tion pressure. Increasing inflation pressure contributes negligibly to

tire weight.

All these points and several others lead to the fundamental importance

of the character of the structure of the pneumatic tire. Before describing in outline the character of the pneumatic tire struc- ture it must be stated that full or complete mathematical analyses of these structures in the form of closed functions are unattainable precisely

because of the highly redundant nature of the structure. Useful analyses, usuaUy approximate in character, of a number of

aspects of major importance are possible. This is so in spite of the fact

that the relevant mechanics problems are usually considered difficult subjects, such as finite deformation of a nondevelopable surface of revolution, nonlinear elastic and time dependent or nonlinear hysteretic

properties of the materials, and added complexity due to anisotropy of

the basic structure. The problems are also fundamentally difficult be- cause of the fact that the final structure is highly redundant, even if none

of these effects were present, and all materials were linear and the law

of superposition was applicable.

The fact that the pneumatic tire is essential to the basis of twentieth

century living and that innumerable forms, types, and ranges of sizes are manufactured on a mass production basis without any noticeable

competition from alternative devices shows that tire designers have

succeeded in producing viable and efficient product designs using the

simple but theoretical analyses or concepts which are available to them.

The objective in this chapter is to try to examine the fundamentals of
360 STRUCTURE OF THE TIRE why the pneumatic tire is made from filaments embedded in a lower

modulus matrix and to point out the principles of the consequences of

this fact, along with the fact that the structure requires gas inflation to ; be an effective load carrying device. It is not intended to discuss current design rules and practice.

3.2. Pneumatic Tire Structure — General Features

A pneumatic tire has certain essential structural elements. The most

important is the casing or carcass made up of many flexible filaments of high modulus cord, of natural textile, synthetic polymer, glass fibre, or

fine hard drawn steel embedded in and bonded to a matrix of low modulus

polymeric material, usually natural or synthetic rubber.

The flexible high modulus cords are usually disposed as multi-filament

layers. One, two, or more such layers of parallel cords are used according

to the design requirements. The number of layers is decided in the first instance by the tire type, the tire size and the inflation pressure to be

used. The first commercially successful tires were made from woven

fabric as shown in figure 3.4a, and even sheet leather was proposed in the first recorded design. It was only when layers of parallel cords, not woven, were introduced that tire endurance life reached a reliable level and adequate tire performance could be taken for granted. This is shown

in figure 3.4b. Although the principle is well established that cords in a given layer must be in one direction only, and cords in another direction

must be in another layer or ply, the parallel cords in a ply are frequently

connected by light weight wefts, see figure 3.4c. These fine wefts are not

Figure 3.4. Filament arrangements which have been used successfully in pneumatic

tires, at one time or another.

(a) Woven cord (obsolete).

(b) Weftless cord.

(c) Cord material with light wefts.
PNEUMATIC TIRE STRUCTURE -GENERAL FEATURES 361

Stress carrying members and serve only to hold the cords in their appropriate relationship during manufacturing processing. The need for these

wefts is principally during any dipping or adhesion treatment prior to rubbering, and during the feed stage of the rubbering process on the

calender. It is essential that the direction of the wefts are not diverted

substantially from their normal direction of lying at right angles to the

main cords of the ply during these process stages because, during the

subsequent shearing within the ply which occurs during the shaping

stage of tire manufacture, the wefts could damage or take a position so

as to cause damage to develop during tire life. Sometimes to avoid this problem the wefts are so designed that they fracture during the tire shaping process during manufacture.

It would appear that a continuous ring like a hose would make a suit- able tire construction, provided that the tubular tire was so attached to the wheel on which it is mounted that it could not roll off the wheel under

the action of sideforce. While constructions of this type have been made

for some special applications, these constructions do not form the basis

of ordinary pneumatic tires. There are several reasons for this. During

the long period of time when all tires had a separate inner tube to contain the inflating gas — and also for the quite large numbers of tires still of this type — it was and is essential that the tire casing can be demounted

from the rim of the wheel to enable the inner tube to be fitted and to allow

Figure 3.5 Essentials of bead construction.

(a) Low turn-up construction.

(b) High turn-up construction.

(c) Overlap construction.

(d) Detail of a typical bead.
362 STRUCTURE OF THE TIRE

for repair after being punctured during road use. Manufacture of the

normal demountable casing is far easier, better, and cheaper than any

form of tubular tire. Specifically, a tubular tire has to be made substantially in its final shape so that molding and consolidation of the inner

surface and plies of the tire are difficult. These layers are readily consolidated by the diaphragm or curing bag in the normal casing of the

demountable tire — a device which cannot be employed in the manufacture of a tubular tire as it would be virtually impossible to remove from

the finished tire. It is for these reasons that tubeless tires have the same

type of demountable casing, although it might have been thought that puncture repair by plugs would cause a trend away from the conventional

demountable casing towards the tubular type of tire. Tubeless tires, that is tires which are designed for use without a separate (removable) inner tube, have a thicker inner lining inside and

integral with the tire casing— usually of a material of low permeability.

In order that the tire casing can be demountably fitted to the wheel

rim, the layers of the high modulus cords or filaments are turned around

bead coils made of a number of turns of high tensile, hard drawn steel wire, located at the inner edge of the tire sidewalls as shown in figure 3.5.

d e f Figure 3.6. Bead wires.

(a) Single bead wire for cycle tires.

(b) Single wire wound to produce a multi-turn coil.

(c) Tape of several wires wound in layers.

(d) Multi-wire woven tape.

(e) Cable bead.

(f) Tape bead.
PNEUMATIC TIRE STRUCTURE - GENERAL FEATURES 363

Although in cycle tires a single wire bead coil can be used, as in figure

3.6a, beads for pneumatic tires are usually comprised of many turns of hard drawn steel wire. There are several reasons for this fact. The wiredrawing process ensures that the tensile strength (100-150 tons/sq. in.)

is several times that of the undrawn material. The multiple wire feature ensures a degree of flexibility essential to tires fitted on a weU base or drop-center rim, and is a great help for ease of fitting and seating home

on the rim base even for tires fitted on detachable flange or divided wheels,

as shown in figure 3.7. The multiple wire bead coil can be made from an

appropriate number of turns of one length of wire as in figure 3.6b, or an

appropriate number of layers of a tape of several parallel wires embedded

in hard rubber as in figure 3.6c, or an appropriate number of layers or turns of woven tape, in which the several main bead wires are held to- gether by an interlacing wave-wound binding wire shown in figure 3.6d.

Figure 3.7. Means of demounting pneumatic tires and extracting or inserting inner tube.

(a) Split or divided wheel.

(b) Well in the rim base or drop center rim.

(c) Demountable flange on the rim.

(d) Collapsible rim.
364 STRUCTURE OF THE TIRE

All of these bead wire systems have the wires embedded in hard rubber,

the wire having been treated to achieve a bond to the hard rubber, usually

by a copper dipping process. The ends of the various wires in these

multiple wire coils are not joined in any direct way; the overall strength

of the bead is achieved by having an adequate overlap to ensure that the

tension is built up through the adherence to the surrounding hard rubber

and consequently to the adjacent layer of wires as indicated in figure 3.6c. The single wire coil of a cycle tire is usually butt-welded to form a continuous loop and often the welded joint is reinforced by a thin sleeve

soldered to the wire as in figure 3.6a. Another form of multiple wire bead coil is the cable bead made from

a single length of wire in which each layer of the winding is helically

wound at the opposite hand to the immediately previous layer shown in figure 3.6c. This forms a more flexible type of bead than the other types

of bead coils and it also makes the ply turn-ups easier. Since the tension developed in the coils of the upper layers of the

layered bead coils shown in figure 3.6e depends on the compression

modulus of the intermediate layers of hard rubber, these coils may be

under lower stress than the coils at the base of the bead. To minimize

such effects when particularly strong beads of low bulk are required,

beads have been made of hard drawn flat tape as in figure 3.6f. Other forms of bead have been used at times in the past but they are

of historic interest only, as they were either superseded by the wired-on

type or they failed to replace the wire bead.

The tensions in the casing cords set up by the inflation pressure are

resisted by the tension developed in the steel wire bead coil. The material

in the bead which encases the steel wire coils is pressed against the rim

flange of the wheel upon which it is mounted by the inflation pressure

within the casing, some of this force being the axial component of the

casing cord tensions due to the direction of the sidewall at the bead as shown in figure 3.8. This reaction between rim flange and tire bead en- ables traction and braking forces to be transmitted by friction between

tire bead and rim flange. The magnitude of available traction or braking

force is augmented by the radial pressure between tire bead base and

Figure 3.8. Forces pressing bead against rim flange on the wheel to obtain driving and

braking reactions.
PNEUMATIC TIRE STRUCTURE - GENERAL FEATURES 365

wheel rim seat due to the load on the tire and also by the use of an inter- ference fit between bead and rim dimensions.

The height to which the plies are taken after they have passed round

the bead coil can be low as in figure 3.5a, or high as in figure 3.5b, or the

plies can extend across the crown as in figure 3.5c. In this case of crown

overlap the plies can be arranged to give extra casing strength in the crown region compared with the sidewalls.

The choice between these alternatives rests on experience of fatigue

life of tires in the field and to some extent on manufacturing considerations. In all cases where a number of plies finish in relatively close

proximity the endings are staggered as in figure 3.5d, or a variant of

this, so as to cause a gradation of the stress concentration and stress transfer to the main part of the plies. The integrity of the tire structure depends on the security of attachment of these ply ends to the main part

of the plies, which form the tire casing, and this depends on the bond

between the high modulus cords or filaments and the low modulus material in which the filaments are embedded, and the bond between the

various layers of low modulus materials in that region of the tire. The steel bead coils may have a canvas wrapping. The bead almost

always has a form of packing above it to give a suitable path of return

for the edges of the plies to the main part of the carcass, as in figure 3.9a.

If this packing, or filler, or flipper as it is variously called, were absent

the tension would not build up smoothly from zero at the end of the cords

to the fuU value at the bottom edge of the plies of the inner part of the

sidewaU. If the plies were wrapped round the bead coil only without the

fiUer being present, as in figure 3.9b, only the cranked portion, that is the portion wrapped round the bead coil, would be of use in resisting the

a b c Figure 3.9. Conventional layout to give filler with good return transition for ply end.

(a) Satisfactory design with filler. (b) Unsatisfactory design with no filler. (c) The effectively stressed part of design (b).
366 STRUCTURE OF THE TIRE

cord tensions, and the construction might as well be as in figure 3.9c. In

this case the only way that the cord tensions would be resisted and the

movement of the plies prevented would be by the stresses set up in the

bead coil acting as a ring with a torsional moment applied along its periphery. The resisting moment and the torsional rigidity of even a solid

ring, when loaded in this manner, is not great and consequently the usual

multiple wire bead coil would be quite inadequate to resist the ply tension. Hence plies are turned up round the bead and passed across the

bead filler to adhere to the main plies as in figure 3.9a. In tires which have a large number of plies, two or even three bead

coils are employed, with each bead usually having about the same number

of plies around it as in figure 3.10. Multiple bead coils are not used in

tires with less than eight to twelve plies and so they appear only in tires for trucks, earthmovers and aircraft. The bead region commonly has a layer of textile material, such as woven fabric, wrapped around it before the outer rubber covering is applied. This is shown in figure 3.5d. The function of this chafer, as it is called, is to minimize the movements of the tire bead surface in contact with the steel rim flange in the region in which the tire cords are caused

to move within the tire wall due to the application of the load on the road. \\ In the area where the tire bulges due to load there are radial and longitudinal movements of the tire wall which are constrained by the chafer,

and so abrasive wear is eliminated at this point. In some tire designs

rubber chafers of appropriate hardness are suitable for this function. In a large proportion of tire designs, security of the attachment of tire to rim is augmented by an interference between bead and rim diameters.

This interference is always employed on tires fitted to one piece rims. It is essential for tubeless tires, to ensure an airtight seal between bead

Figure 3.10. Multiple bead coils for tires with a large number of plies.
PNEUMATIC TIRE STRUCTURE -GENERAL FEATURES 367

Figure 3.11. Axial tension in rim base.

(a) Tire and wheel.

(b) Tubular tire.

and rim, and even with tube tires it is more necessary with modern wide

base rims than it was in some of the earlier narrow rim widths, because

of the relative magnitudes of the axial component of casing tension. The

interference causes compression in the material under the steel bead

coils, and this in turn sets up a tension in the coils which adds to the

effects of inflation. Sometimes dimensions are so chosen that the tension

in the bead coils due to interference fit is greater than that due to casing

inflation tensions. In a large proportion of designs for low pressure tires,

such as passenger car tires, the interference fit is so chosen that the tire has to be inflated to a pressure substantially higher than its operating

pressure in order to elastically stretch the coils sufficiently to permit the

bead to move up the seat. This ensures that the beads seat home against

the rim flange during the fitting of the tire on the rim.

The axially outward component of the casing tensions press the bead

against the rim flanges and set up a tension in the rim base, shown in figure 3.11a. In the little-used tubular tire design this tension would be

taken either whoUy or in part by the casing cords on the inner periphery

of the toroid, as illustrated in figure 3.11b. However, it should be noted

that tubular tire designs require anchorage points to the wheel, spaced

apart as in the manner of the beads of the conventional form of tire. Without such anchorage points the tubular tire would roll sideways and

so provide only low resisting forces to sideway drift on the road. It would

in consequence have only poor guiding properties. The apparent digression to discuss the almost unused tubular tire design serves to draw

attention to important aspects of the function of the bead and of the

high modulus wire coils in the bead of the conventional tire. Demounting of a conventional tire is made possible by either the provision of a weU in the rim base, also known as drop center, or by the use

of a detachable flange, or by having the wheel in a divided or split form.

These are shown in figure 3.7. The shape of the tire bead and rim on to which it fits can be divided

into two main categories, the 5° and 15° taper rims shown in figure 3.12.
368 STRUCTURE OF THE TIRE

Figure 3.12. 5° and 15° taper rim flanges drawn in similar relation to the wheel axis of

revolution.

The 15° taper rim design is the more recent design, having been intro- duced in medium size truck tires in the United States about 15 years ago. Variants of the more generally used 5° taper rim have existed in the past,

since it developed from a cylindrical base, or 0° taper. There have been

one or two examples in the truck tire range with a taper angle of 8° to

13°. Although such rims may still exist in service, the tires are designed

for mounting on the 5° taper.

The flange height of the rim is higher for the 5° rim than for the 15°,

and the tire beads have some necessary differing disposition or internal

components. Tire sectional heights are shallower for the 15° rim than the

5°. For example, the 22.5 in. rim shown in figure 3.12 uses a tire of the same outer diameter as the 20 in. rim also shown there. AU tires in regular use have two equal size bead coils but there is no

theoretical reason why the two bead coil diameters should not be

different. One could produce technical arguments in favor of such

designs, and in fact various tire designs of this type have been the

subject of patents but none have gotten beyond the experimental stage.

One practical objection against them, and an objection raised against > any asymmetrical or directional tire feature, is that they have to be

fitted one specific way. If the unequal rim diameter is coupled with a directional tread pattern feature then two different moulds and tire types are required for any given vehicle. { The technical arguments which have been put forward for the use of a i, smaller diameter bead seat on the side of the wheel away from the

vehicle, and a larger diameter bead seat on the inner side are the lower

risk of curb damage with a long sidewall on the outside, and greater ^ diameter for the brake on the inside of the wheel. Further arguments 6 stem from the fact that during cornering of a vehicle the wheels on the \ outside of the curved path, which are the ones which carry the greater 1 load, are usually caused to camber by the suspension linkwork and this v
FLEXIBLE FILAMENT AND SOFT MATRIX CONSTRUCTION 369 camber is in the direction to increase the deflection of the outer sidewall

of the tire, and hence to lessen the deflection of the inner sidewall, the

terms inner and outer being in relation to the vehicle here. The fact that the tires on the inside of the curve are usually cambered so as to shorten

the short inner waU more than the long outer wall of the same tire is not important, it is argued, since the load on this tire is lower and so the

percentage deflection of the short inner wall of the tire on the inside

of the curve need not be any greater than the percentage deflection of the longer outer wall of the tire on the outside of the curve. Even so, no such tires have gotten beyond experimental investigation.

So far, tire casing and beads have been discussed in a general descriptive way. The third important component of a tire is the tread. The tread performs several functions. It is the only part of the tire to come into contact with the road surface. It provides a wear-resistant

layer and also protects the casing. It provides frictional contact with

the road sufficient to transmit driving, braking, and cornering forces. These frictional forces may reach a value equal to the load carried by

the tire. The tire tread carries a pattern of such character and detail design as to ensure adequate removal of water and other contaminants

from the road surface, so as to maintain an adequate level of frictional adhesion between tire and road over a wide range of operational

conditions.

For tires intended for operation on soft ground, the character of the

deformation of the ground and the laws of soil mechanics determine the

form of the tread pattern.

The only type of material which has been successfully used as tread

material is rubber or a rubber-like material, that is, a long chain molecular

material or polymer of a molulus comparable with that of the matrix in which the filaments are embedded in the rest of the structure. The

tread polymer has to be reinforced with suitable ingredients such as carbon black to obtain the required abrasion resistance but this is a subject in itself and wiU not be discussed here because detail changes in

this material do not significantly alter the tire structure required. Experience in the development of spring wheel devices shows that rubber,

natural or synthetic, has no competitor as a tread material.

These materials automatically give good friction on dry roads and some aid the achievement of good friction on wet roads. This will not be

discussed here since it is covered elsewhere in this book.

3.3. Flexible Filament and Soft Matrix Constructions

Although the patent literature contains a wide variety of alternatives

to the use of filaments or cords in tires, all commercially successful tires are now built as a series of layers of flexible high modulus cords encased

in a low modulus rubber or rubber-like material, the cords in each layer

being in a given path or direction and substantially equi-spaced and

parallel.
370 STRUCTURE OF THE TIRE A conventional cross bias casing or carcass is built of two or more,

usually an even number, and most commonly four, layers of parallel cord

filaments, the direction of the cords in a layer being at an angle to the

principal axes of the tire. Half of the layers have the cords at a positive

angle and half at a negative angle as shown in figure 3.13. Usually alter- nate layers have cords at the opposite hand, thus giving the cross bias

effect between adjacent layers at all points in the tire as in figure 3.14a. However, tires have been made with adjacent layers of cords parallel. In the case of four-ply tires the middle two plies are parallel, and in the case

of more than four plies, the ply arrangement is, sa.y, AABBAABB etc, or, ABBAABBA. Some of these are shown in figures 3.14b and 3.14c. Tires

have been made with an odd number of plies, and even tires with three

plies have been commercially tried but they are the exception rather

than the rule. Usually the crown angle of the cords in the two directions are equal in magnitude. The crown angle is the angle between the path of the cord

and the line along the tire periphery defined by the intersection of a plane at right angles to the axis of rotation of the tire and the highest

point of the tire cross section, as shown in figure 3.15. Tires of conventional cross bias construction are rarely made with a crown angle exceeding 38°. Lower angles are used, particularly when

high speed or other special performance characteristics are to be

achieved. Tires intended for vehicles on public roads are rarely less than 10° lower than the above figure, although tires with crown angles of

Figure 3.13. Conventional cross-bias tire.
FLEXIBLE FILAMENT AND SOFT MATRIX CONSTRUCTION

Figure 3.14. Cord ply arrangements.

(a) ABAB-the usual,

lb I AABBAA.

(c) ABBA.

(b and c are called parallel ply constructions.)

Figure 3.15. Cord crown angle.
372 STRUCTURE OF THE TIRE

the order of 20° lower may be used for very high speed vehicles restricted

to use on tracks. Conventional cross bias tires with a crown angle substantially higher

than 38°, although capable of giving a good comfort, are not used because

of the low cornering power and consequent poor guiding characteristics

of such tires. In order to give increased protection to the casing under the tread

and/or to increase the casing strength in the crown region of a conventional cross bias tire, one or two layers of cords are sometimes incorporated substantially parallel to the cords in the other plies but extending

only approximately the width of the tire tread. This is shown in figure 3.16. Such layers are called breakers. Sometimes the cord spacing in the

breaker pHes is greater than in the main plies, sometimes the spacing

between layers is greater than between the main plies, and sometimes

insulations of differing modulus are incorporated above, between, or below the breaker cords. These variants are decided from service experi- ence and relate to the fatigue life rather than the mechanical behavior of the tire as a structure. A conventional tire of cross bias casing construction with a breaker of the same angle as the casing is not detectably

different in mechanical behavior from a tire without a breaker but other- wise of similar construction, i.e., cord angle, number of plies, etc. The above comments relate to the conventional cross bias tire, which

has been used from the beginning of the automobile era. However, if the

Figure 3.16. Breaker layers in conventional cross-bias tire. The cords in the breaker layers are at the same crown angle as the casing cords.
I FLEXIBLE FILAMENT AND SOFT MATRIX CONSTRUCTION 373

cord angles in the breaker layers are substantially different from those

i in the main pHes, and the breaker is made of either higher modulus cords

j than the casing or of more layers than the casing, then the breaker con-

'I struction has an important bearing on the mechanical properties of the

tire.For about 20 years in Europe, and more recently in North America,

tires of a construction very different from the cross bias construction

' have been produced and successfully used in large quantities. In these

j

' newer tires the cords or filaments in the casing are disposed in a radial,

I or substantially radial, direction giving a 90° bias or crown angle in relation to the axis of rotation of the tire. They also use a breaker or

I belt of several plies of cords fitted on top of the casing under the tread,

1 and laid at various crown angles, two of the layers at least having a low

j crown angle of the order of 20° as shown in figure 3.17. These tires are

I commonly called radial tires although a more correct description is rigid breaker, radial ply tires. An alternative name is belted radial ply

tires. The rigid breaker or belt is essential to the functioning of the tire. Without it a radial ply casing can become unstable. When inflated to a high pressure, determined by the magnitude of the slight irregularities

in cord spacing, differing extension of the rubber between the cords

i at different points around the tire permits the cords to move out of a

radial plane and the tire periphery develops into a series of severe

i buckles. Such an instability is to be expected if the comparison is made

Figure 3.17. Radial ply rigid breaker tire.
374 STRUCTURE OF THE TIRE

with a tall pile of interspersed rubber and metal washers under compressive load. The concept of a radial casing with a reinforcing band or belt on it in the crown region, although developed successfully only recently, is much

older since a British patent dated 1913 describes a tire in which the rein- forcing belt is placed under the casing cords. These early designs were

never brought to a successful development state. Tires are also made with low crown angle belts on conventional cross

bias casings (of crown angles greater than that of the belt). Usually the

cords in the belts of these tires are of higher modulus materials than the

casing.

These belted-bias tires have at times been wrongly called semiradial because of the common use of the term radial— ignoring the es- sential belt on the radial casing tire. The objective of the belt or breaker of low cord angle is to provide

rigidity to the tread against the distortions in the lateral direction which

are set up during cornering, and so to reduce tread wear under this condition of use. The conflict between a nondevelopable surface, such as a surface of

revolution of curved cross section, which in consequence is a surface

of double curvature, and a flat plane surface is an important factor in the decisions of designers to employ cords embedded in rubber for the

tire structure. The simplest tire-like form which can be expressed in simple geometry

is a toroid. If the curved cap is flattened from its original shape to a

flat plane without altering any other part of the tire periphery, as shown

in figure 3.18, then the length along the surface is the direction of the

Figure 3.18. Toroid in contact with flat plane.
FLEXIBLE FILAMENT AND SOFT MATRIX CONSTRUCTION 375

longer axis of the patch, originally aD, is compressed to a length 2a. Now

d' = b(D-b)

and

Noting that

and by putting

a = sin (I)

sin _ x^ Sx"^ X ~ 6 40

2a we have

Original — length length along arc aD _ _ . , ^

;—= :—\ —= = i + Longitudinal btrain New length length along flat Za

^

^ 2 8{D-d) ^ 6 8HD-8y

3 Z)2 ' 5 Z)^

28

Consideration of the dimensions across the patch leads to

^TT ~ IH- ^= 1 + Transverse Strain

Zo 6M

It follows that the surface may undergo compression in both directions

at all points in the area under discussion while simultaneously the size of the elliptical contact area of this patch is unaltered. It also follows

that if the cap were flattened without compression in either of its principal axes, the outer edges of the elliptical patch must extend or tear. In the case just discussed it was assumed that the part of the tire outside the contact patch is undisturbed. This might be the basis of a challenge that the example, as discussed, does not establish the point

being made. It is not as can be seen by considering a sphere. If the

structure under consideration were a sphere made of isotropic material,

instead of a toroid, the contact patch would be circular, not elliptical.

The actual circular contact patch (ignoring transition curves due to shell bending stiffness, which in an appropriate case can be considered to be a second order effect) will lie between that given by the intersection of
376 STRUCTURE OF THE TIRE

the sphere and the "contact plane" and a circle of diameter

(-1)

times as large, as shown in figure 3.19. The former corresponds to a hypothetical case in which all the distortion is in the cap or part of the

shell within the contact patch, and there is no tension along the periphery

of the contact patch. The latter case corresponds to the hypothetical case

in which there is no radial compression in that part of the shell forming

the cap or contact patch, but in this case the periphery of the patch is extended in the ratio (1 + 28/3S). This implies that the part of the sphere

outside the contact patch is strained. In the actual case the compressions and extensions will lie between

zero and 28/S, and at least one of the strains will be of that order if another is near zero. If they are all of similar magnitude they will be of the

order of 8/3S. If the sphere diameter S is similar in magnitude to the

diameter of the cross section of the toroid discussed above, and d/S or d/H is taken to be of a magnitude of acceptable tire operating deflection,

i.e., 0.15 to 0.35, then the resulting strains are greater than any crystalline

metallic material can withstand within its elastic range or fatigue limit.

Figure 3.19, Sphere in contact with flat plane.

Actual contact patch radius will be between the values ab and cd, where d is the intersection point of sphere and

plane and ab equals arc of ad.
FLEXIBLE FILAMENT AND SOFT MATRIX CONSTRUCTION 377

Clearly if the sheet or surface were isotropic it would have to be made of a material having a high extensibility before fracture. This demands a low

modulus, and such a structure would change dimension on inflation rather like a toy balloon. The alternative possibility of a material with a medium to high modulus would require a very high breaking strength

and a very high yield point, and no such material has been found. It is for this reason that proposals for thin shell toroidal spring wheels of metal or plastic have all so far been abortive. In order to get a rough idea of the deformation of a shell structure of double curvature made of high modulus filaments embedded in a matrix

of low modulus material, consider the following case chosen for ease but

reasonably accurate discussion. Figure 3.20 represents a two-ply sheet,

made of one ply of filaments parallel to two parallel sides of the rhombus

ABCD, and the other ply with filaments parallel to the other pair of sides. Imagine that this sheet of composite material undergoes a biaxial extension of the order of magnitude which would arise at the principal axes of

a toroidal indentor stretching the originially flat sheet to the shape of the

cap of double curvature discussed above. Consider two filaments AB and

DC in one layer or ply, and two filaments BC and DA in the other ply,

before stretch. If one assumes that the filaments are inextensible, then

after extension the filaments move to a'b' and d'c' in one ply and b"c"

and d" a" in the other ply. These movements are permitted by the low

modulus matrix, which develops shears as indicated by the tilt of the

lines a' a" , b' b" , c' c" , and d' d" from their original ^4 '^4 " , B' B'\C'C"

,

D'D" as shown in figure 3.21. Within a ply there is shear and extension

indicated by comparisons of a'b' and A'B' etc. Change of cord length, assumed not to occur in this example, would of course exist in many practical cases. This would cause some modification of the strains in the matrix but in detail only.

d" d' Figure 3.20. Tivo-ply high modulus filament embedded in low modulus matrix under

biaxial extension.
378 STRUCTURE OF THE TIRE

As the local flattening of a surface of double curvature involves a process the reverse of the above, the problem is analogous but results

in the initial tensions being reduced during the process instead of being

increased. In some cases the cords will buckle in the process but because

of their close juxtaposition within a layer, and because of the layered

construction, the buckles are of a controlled character. The cords are

elastically supported by the matrix in relation to the neighboring cords,

and the wavelength of the buckles is determined by the consideration of

a cord as an elastic beam on an elastic foundation.

This arbitrary example is artificial because it has been considered in the flat plane only, for ease of discussion and presentation. However, it does show how plies of high modulus cords or filaments embedded in a low modulus matrix can permit distortion in a structure which would not be possible if the structure were formed by an isotropic sheet of that material.

It is clear that in changing from a surface of double curvature to a plane surface the high modulus filaments must bend. They must also be

flexible enough to be turned round the bead coils during the manufacturing process. Furthermore, during tire usage, conditions arise where

flexibility or the ability to bend is vital. The most severe bending would

probably be during accidental running of a tire in a deflated state. Reasons such as these require that the high modulus filaments must

be flexible high modulus filaments.

Flexibility of the high modulus filaments is obtained by restricting the

diameter of the filament or/and using a multiple fiber or multiple strand

cord in which each separate strand of the high modulus material is sufficiently thin to keep the stresses below both the permanent set and

the fatigue limit under any condition which might be met — even if the
TIRE CONSTRUCTION METHODS 379

; occasion were infrequent. Typical cord cross sections are shown in

' chapter 1 of this book.

j It is suggested here that a general requirement might be expressed in

I the following form. The high modulus filaments shall have, by the choice

I of strand or fiber diameter and construction or cabHng, sufficient flexij bility that they are not overstressed as judged by any appropriate

\ criterion, including fatigue and permanent set, if bent as much as it is possible to bend the low modulus matrix of the size and character into which they are embedded. It follows that small tires with thin walls need

the finest strand filaments in cabled cords, and that coarser strands may

be permissible in the thicker walls of large tires.

3.4. Tire Construction Methods

The conflicting requirements of Hmited expansion of the structure during inflation, coupled with the need for internal shear movements

within the sheet forming the surface or envelope when a tire structure

of double curvature is flattened at the ground contact patch, necessitate

the use of a multiplicity of relatively inextensible, but flexible fllaments

embedded in an easily distortable matrix. Accepting this, it can be seen

that, in theory, there are a number of alternative ways of building a tire. All methods of making a tire of these composite materials involve a building process followed by molding and vulcanization. The building process is the stage where the materials or components are placed in the required appropriate relative position one to the other, and will be

discussed further below.

The molding process is needed for the compaction and consolidation

of the various components, for example embedding the higher modulus

filaments in the low modulus matrix to eliminate voids and to obtain the

initimate contact necessary to secure adhesion between filaments and

matrix, and also between layers of the matrix itself. The vulcanization process is required for changing the state of the rubber or rubberlike polymer used for the low modulus matrix, from the uncured plastic

state necessary for the building and molding stages of the manufacturing

processes to the elastic state essential for effective tire performance in the final product, and for the formation of the adhesive bonds between

filaments and matrix and between all layers of matrix within the tire. The uncured plastic state of the rubber-like polymer has a feature of value during the building stage and also the molding stage, that of "self-tack" whereby two contacting surfaces adhere and, even if separated, readhere on recontact.

It should perhaps be mentioned at this point that although a fairly

wide range of materials have been successfully used as the high modulus

filaments e.g. cotton, rayon, nylon, polyester, glass, and steel, which

range from organic, long chain polymeric to crystalline materials,

the low modulus matrix has always been a long chain polymer of rubber or

rubber-like characteristics. It is this material which requires vulcanization. Although many of the materials used in the filaments are affected
380 STRUCTURE OF THE TIRE

by the heat used in vulcanization, not all are, and in practically all cases the vulcanization stage is not required to bring the cord material

into its required final physical state. The molding process also fulfills at least one other function, namely

the formation by molding of the tire tread pattern. In many of the more modern methods of building, the shaping of the tire from the assembled

shape to the required final shape is carried out during the molding

process. The alternative ways of building a tire casing can be enumerated as

follows: A. The tire could be manufactured on a former or pattern substantially of the shape of the desired finished shape of the tire, as shown in figure 3.22a. Let this be denoted process A.

B. The tire could be made on a former of a shape other than that of the desired finished tire shape. The former could be a simple cylinder,

as shown in figures 3.22b and 3.22c. In this case the change of shape can be effected as a separate process prior to the molding stage, called

process 51, or during the molding stage, called process B2, using the same molding equipment in which the vulcanization is carried out. C. Part of the tire could be manufactured on a former different from

that of the final shape, and that part of the tire shaped to substantially

that of the final tire shape, and then the remainder of the tire components could be added. Let this be called process C.

In each of the above cases the tire casing can be built in one of the

following alternative ways, or some components built in one way and some in another.

a. The tire casing could be assembled from sheets of high modulus

filaments or cords embedded in a low modulus matrix, the sheets

being produced by a separate previous process which ensures that the filaments are straight, parallel and equispaced, within the

(a) (b) (b,)

Figure 3.22. Manufacturing processes I.

(a) Casing built on "core" former substantially the size and shape of the finished tire. (b) Casing built on a "low crown" former— a different shape from the finished tire, and distorted to the required shape

in a subsequent process.

(bi) Casing built on a flat or cylindrical former and distorted to the required shape in a subsequent process.
TIRE CONSTRUCTION METHODS 381

Figure 3.23. Manufacturing processes II— tire casing building methods.

(a) Plies of material with multiple cords cut to dimensions and wrapped around a tormer.

ai, single ply.

a:;, pocket or paired plies at opposite bias angles.

(b) Plies with cranked or curved cord path.

(c) No illustration (see text). (d or e) Single cord wound on former.

d-, as a simple winding.

d:), as a wave winding,

(e) No illustration but as (d) using narrow strip of several cords.

(0 Strip of material wound on former,

(i) Peg winding— ii cords at one bias.

\z, cords at cross bias. rubber. These sheets can be laid on the building former shown in figure 3.23a either

(1) individually, as shown in figure 3.23ai,

(2) in pairs having been previously put together so that their

angles are complementary, as shown in figure 3.23a2,

(3) in pairs having been previously put together so that their

angles are asymmetric,

(4) in four or more layers having previously been assembled so

that alternate layers are at complementary angles, or

(5) in four or more layers having been previously assembled

with their angles in some relationship other than alternately

complementary.
382 STRUCTURE OF THE TIRE

Figure 3.23. Manufacturing processes II— tire casing building methods.

(g) Band ot pre-rubbered cords wound around and through the bead coils held apart.

(h) Same, but for single cord.

b. The rubbered multifilament sheets used for the assemblies (listed

above in a.) could be distorted so that the cord paths were curved

from the straight, as shown in figure 3.23b.

(1) Before assembling into pairs.

(2) After assembling into pairs.

c. The rubbered multifilament sheets could be used as in a, but the

curved effect or an asymmetric angle effect could be gotten by a

relative rotation of one tire bead in relation to the other during

building, shaping or molding.

d. The cord or filament could be wound as a single rubbered cord on a former appropriate to building method A or B, p. 381;

(1) as a simple uniform winding.

(2) Great circle winding, with a slight advance of location at each

turn, resulting in spaced winding, as in figure 3.23d2, or

(3) controlled wave winding, as in figure 3.23d3. In all cases of d the cord can be rubbered at the building machine

by passing it through an extruder die.

e. A narrow band of pre-rubbered continuous cords can be wound

on the former as a strip, with all of the possible variations listed in d above.
TIRE CONSTRUCTION METHODS 383

f. A band of two or more layers of cords can be wound on the former

as shown in figure 3.23f:

(1) with the constant angles differing at different places across

the former.

(2) with differing numbers of layers at various places.

(3) with all the possible variations in d listed above.

g. A band of pre-rubbered cords may be wound around and through

the bead coils suitably held apart, the bobbin carrying the band

of material passing through the bead coils during the winding

process. This is shown in figure 3.23g.

h. As in g but using a single cord rubbered at the building machine.

i. A single cord wound from bead to bead, the cord being located by

a series of pegs or slots:

(1) the cords all being parallel to one another in each layer as shown in figure 3.23ii

,

(2) the cord progressing forward at the opposite bias at each peg,

as shown in figure 3.23i2.

j. As in i but using a narrow band of several cords. The above classification seems to cover all possible ways of building

a tire. All of these cases and combinations of cases seem to have been

the subjects of patent applications. Many have been tried and used for some tire application or another at various times.

The vast majority of passenger car and truck tires were originally made

using processes A and a2. but quite early in the history of the industry

the process became Bl and a2, and quite soon afterwards B2 and a2. Today the vast majority of car tires, truck tires, earthmover and aero

tires, and some belted cross bias casing tires are made this way. All production radial ply belted tires are made by processes C and al, a2

or a3. Most of the other methods have not gotten beyond the experimental

stage, and practically all lack the flexibility of the current standard

methods. One method which has become standard, that of g and B2 has

been used extensively for the manufacture of cycle tires. It is restricted,

by its character, to tires whose section width is much smaller than the

bead diameter. The bobbin carrying the total material for the tire must

be capable of passing through the bead coils while they are held apart and

tensioned by either four drums or a system of arcuate guides, so that the

portions of the bead coils between are straight and sufficiently tensioned

so that the winding of the rubbered cord does not cause the beads to sag

towards each other. There is another restriction of this method however,

since the choice of bias angle is linked with the strip width being wound

and the bead coil periphery. Furthermore, the bead coils must be suffici- ently flexible for the passage around the roller system. The problems

specific to this method have been mentioned as an example of the fact
384 STRUCTURE OF THE TIRE

b,

Figure 3.24. Manufacturing processes III — Molding and vulcanization.

(a) Shaping and fitting curing bag on the bladder as separate operation.

(b) Shaping as part of the diaphragm molding and vulcanization process.

that most of the alternative,methods set out above have restrictive characteristics peculiar to each one. The shaping and molding stages are illustrated in figure 3.24. There are two main alternatives. The curing bag or bladder is inserted into the al- ready shaped tire, as in figure 3.24ai, or into the unshaped tire as in figure 3.24a2, during shaping by a machine process. Afterwards the

tire is vulcanized in a mold in a press.
SELECTION OF CORD PATH 385 The more modern process is one in which the unshaped tire is intro- duced into a special press in which a cyHndrical diaphragm shapes the

tire as the press, carrying the two half molds, closes as illustrated in figures 3.24bi and 3.24b2. The inflated diaphragm consolidates the tire during the molding and vulcanization process in the heated mold.

3.S. Cord Path — Practical Factors Determining

Selection

Although in theory any cord path is possible in any molded shape there

are several factors which restrict the range of practicable cord paths.

At the outset a choice must be made as to whether the cords are to be

coated with rubber (I) collectively, that is in large numbers as a sheet of

material, or (II) individually, or (III) in a group relatively few in number

to produce a band or tape of rubbered cord. In the first case (I), sheets of several feet in width and hundreds of yards in length, comprising a large number, say one or two thousand,

cords across the width are produced in a continuous calendering process

separate from the tire building. Sheets, called plies, of appropriate

dimensions are bias cut from the calendered sheet so as to give the

required angle of cord direction to the cut edge. This is the most common

method and is a very flexible process, since a wide variety of ply dimensions can be produced as required. The only real limitation is that the

initial rubbering process must be made with the cord spacing at its re- quired amount, and the correct thickness of rubber applied.

In the other two cases (II) and (III), the cord would be rubbered as it is wound on a former of appropriate shape at the tire building machine.

The factors which control the speed of covering the high modulus

cords with low modulus rubber-like compound are the temperature and

temperature-time cycle and the vulcanization and scorch characteristics

of the rubber-like material, along with the characteristics of the bonding

process of rubber to cord. These place an upper limit to the speed at which the cord can pass through the rubbering stage. Clearly, without

going into secondary detail consequent on the differences in the various

processes involved, rubbering a thousand cords simultaneously is more

expeditious than rubbering one or a score of cords at once. It is for this reason that the processes using calendered sheet material dominate in commercial tire production.

Processes involving curved cords, or other paths than straight,

in the initial flat sheet even if produced initially straight and subsequently

cranked in a special process, raise the problem that limitations on the

closeness of spacing in the low angle cord region demands relatively

wide spacing in the high angle cord region. The geometric necessity for

this is clearly seen in figure 3.25.
386 STRUCTURE OF THE TIRE

Figure 3.25. Cranked cord limitation on cord spacing.

The number of cord ends per inch in the high angle region is Hmited by the limitation of closeness of cord spacing

in the low angle region.

3.6. Calculation of Cord Length

The integrals for finding the length of a line drawn on a solid of revolution are known explicitly only in very special cases, such as the constant angle helix on a cylinder, or a great circle on the surface of a sphere. In the case of a tire, even if the tire cross section were circular, which

it never is, the integration does not lead to a solution in a closed form.

In the general case of a noncircular cross section the integral cannot be

written explicitly, let alone evaluated and its solution stated. On the other hand, if the path of a cord is traversed step by step,

and the length and direction of each of the steps is recorded, it is possible, provided the cross-sectional shape of the tire is known, in numerical, graphical or mathematical terms to resolve the progress along the

path into any definable coordinate system. This can be done whatever

the cord path and whatever the tire cross-sectional shape.

This resolution into a known coordinate system is analogous to navigation at sea, with the difference that the map is not a plane, or a sphere,

or generalized three-dimensional space. It is the surface of the solid of revolution of a cross section defined by the tire shape specific to the case in hand. Whatever the path or whatever the cross section, the

progress in the desired coordinate system is recorded by resolution of the known step in the known direction, followed by simple summation

in each coordinate. Integrals expressed as mathematical functions do not

arise. In effect, the simple summations of the coordinates correspond

to, and replace the need for, the integration along the path itself. There are two simple relations between /?, the radius, and cos 0, both

as defined in figure 3.26, which hold over all possible cord paths in a

tire. For this reason a graph of R vs. cos (/> is an important diagram.

Note here that (/) is defined as the angle between the cord tangent and a

parallel on the tire surface, while R is the radius to the axis of rota- tional symmetry. The chart obtained by plotting R versus cos (/>, or a formula relating the two quantities, or a numerical tabulation of pairs

of values, is the full description of a cord path or as appropriate, of the movement of a point on the cord during processing. Pantographing

movements and geodetic movements are linear and hyperbolic relations
CALCULATION OF CORD LENGTH 387

Figure 3.26. Definitions of radius R and cord angle </).

respectively on this chart. On the other hand, the geometry of the tire surface is completely described in terms of R versus X, where X is the

distance along the meridian, that is the distance along the surface measured from the tire crown in the plane of a cross section of the tire as shown also in figure 3.26. Given the cord path by means of R versus cos (/) on a former or mold

of known R vs. X, then if the tire is shaped by a process involving movements expressed in terms of an appropriate transformation of vs. cos 0,

it is immediately possible to determine the R vs. cos (/) of the cord path

in the finished tire.

It is also known that the R vs. cos </> chart can be used to diagnose

the character of the cord path in a tire, or to elucidate the character of

the processes occurring during the shaping and molding stages of tire manufacture.

Consider any surface of revolution on which any cord path is drawn.

Consider a tire made of two layers, one layer comprising a large number

of cords of the same cord path and these cords being equispaced and

encompassing the whole surface of this layer, and the second layer of the same system of cords but of opposite hand. Furthermore, consider

that the cords in these two layers are pin-jointed one to the other at their crossing points. It has been shown in reference [1]- that the shape

of this network can be changed, and provided that the surface is maintained to be a surface of revolution around the original axis, the trellis- sing or pantographing of the cord structure will ensure that all times

cos (/)

as shown in figure 3.27, where C\ is a constant.

If the solid of revolution is a cylinder and if the helix angle is constant, R and (/) are the same at all points on the path, so that one point on the R vs. cos </> graph represents the whole helical path. This corre-
388 STRUCTURE OF THE TIRE

COS(^

Figure 3.27. R-cos (f) chart.

Plots for conventional and geodetic paths. spends to the path on a cyHndrical building drum. If the tire casing

is shaped up by pantographing without sUppage or extension of the

cord then the cord path will be described by a portion of the straight

line through this initial point and the origin, as illustrated also in figure

3.27.

It follows from a study of geodesies on any surface of revolution that the relation between R and cos cj) for points on such a path, which

always follows thw shortest route between any two points on the surface,

is R cos 0 = C2

where Cz is a constant. The fact that both of these relations may be displayed simultaneously on the R vs. cos (/) chart makes such a chart important. The plot on the

R vs. cos (f) chart is independent of the cross-sectional shape, since a curve obtained by trellissing or pantographing always lies along a line through the origin, while a geodesic path always appears as a hyperbola

asymptotic to the coordinates of the chart. This is shown in figure

3.27. This perhaps is made clearer if Log R is plotted against Log cos (f), as in figure 3.28. Here, the two lines are straight at 45" to the coordinates,

and are orthogonal to one another.

If in the pantographing system the cords extend, then the R vs. cos (f) values which are obtained lie on a new straight line through the origin

with a slope of (1+cord strain) times the slope of the line describing

the original unstretched cord path, so that R = R{l-\- e) at a given value

of cos cj). This is shown in figure 3.29.
CALCULATION OF CORD LENGTH 389

LOG COS <^ Figure 3.28. Log R-log cos 4> chart.

COS 4>

Figure 3.29. R-cos 0. Effect of cord stretch.

If the cord length changes in the ratio (1 + e) the R-cos 4> line changes from a to b where the slope of b is (1 + e) times a. The above discussion relates to the characteristics of an actual

path of a cord on a surface of revolution. The presentation of information on a set of coordinates of R and cos (/) has another usage, that of

depicting the changes or movement during shaping of the tire casing,

or other manufacturing processes. Movement of the point on the R vs. cos </) graph, representing a specific point on the actual tire cord, can be

examined if the following facts are noted. As the cord lattice pantographs,

the point on the R vs. cos (/) chart moves along a straight line through the

origin, as illustrated in figure 3.30. As a cord stretches, the point on the

chart moves along the hyperbola passing through the point, as in fig- ure 3.30b. If slippage occurs the angle remains constant but the radius R

changes. This results in a vertical displacement of the point, as in figure

3.30c. For further discussion of this concept see reference [1].^

-Figures in brackets indicate literature references at the end of this chapter.
390 STRUCTURE OF THE TIRE

COS 4>

Figure 3.30. Laws of change of R-cos 4> at a point of the lattice.

(a) During pantographing.

(b) During cord stretch.

(c) During ply slippage.

3.7. Analysis of Manufacturing Methods by

R vs. cos <^ Charts i The fact that the R vs. cos (/) presentation of changes of cord path

enables the character of the changes to be understood, and the fact that the R vs. cos (/) representation of the actual cord path together with ! the R vs. X diagram for the surface of revolution on which the cord path

lies completely defines the cord path in space, enables a system of cord

path dimensions, and changes under manufacturing conditions to be < set out. I Consider three sets of data expressed in the form of five graphs or t tabulations, as shown in figure 3.31. The graphical form is easiest to understand, although the numerical form enables the required precision

to be met. The curves of figure 3.31 labeled I show the cord path characteristics

or R vs. cos (/) information, curve a, on the tire building former, which

is a surface of revolution defined by the R vs. X statement determined

by the generator of the surface, curve b. The curves labeled II show the changes due to trellissing or pantographing, changes in cord extension, and any slippage of the plies at the

trellissing pivot points, during manufacture, curve c. The curves labeled III show the final cord path characteristic, or R vs. C03 curve d, in the finished tire casing of shape defined by its R vs. X information, shown in curve e. Given any two sets of information out of the three, I, II, III, the other can be found explicitly, or, given any four of the five items of information (a), (b), (c), (d), (e), the fifth can be found exactly.
ANALYSIS OF MANUFACTURING METHODS 391

Figure 3.31. Representation of R-cos 4> method.

I Original plied material on former.

II Shaping process.

III Final tire cord path.

It is possible that a bad choice of the given data will result in the

remaining calculations taking on an impractical form. This is not a limi- tation or fault of the method, but rather the reverse — it shows the limita- tions of what can be done in the process of manufacture.

The calculations do not involve difficult mathematical functions, in- complete integrals or mathematical singularities, as would a method in which the length of a path is determined analytically from the coordinates. This method is, as mentioned earlier, one of step by step resolution of the cord path into the coordinates of the surface, analogous to navigation.

The details of the application of the method outlined here, including

the choice of the incremental steps along the path of the cord so as to avoid systematic cumulative errors, along with examples of applications,

are given in reference [1]. This reference also gives information on

the R vs. cos (/) characteristics of paths other than the pantographing

and geodetic cases. In addition to the above uses, the R vs. cos 0 method of presenting

information can be used as a diagnostic tool to investigate why a given

tire does not conform to the expected cord path. The departures from

the expected path can be interpreted in the light of the observed departures of the R vs. cos (/) from the normal characteristic. Cord crowding

due to insufficient spacing in the initial material, cord extension, tight

cords, and cord or ply slippage each have a specific effect on the chart.

It is a general truth that the most likely change during the shaping

process is pantographing, since departures from pantographing and

uniform stretch are caused by local effects, such as the tire contacting

the mold at one region before another and the subsequent movement of

parts of the casing being restricted by this contact with the mold.
392 STRUCTURE OF THE TIRE The fact that the high modulus cords are embedded in a low modulus

matrix which adheres to them is the reason why pantographing, cor- responding to cos (/)//? =Ci, occurs most readily. The low modulus

matrix provides a low resistance to the shearing movements involved in pantographing, while the adhesion between the high modulus filaments

or cords and the low modulus matrix or rubber resists slippage of one

ply across another, and ensures pin-jointing or pivoting at filament

intersection points between two adjacent layers of opposite bias. Since a cylindrical former belongs to the group of axisymmetric

surfaces which are developable, so that they can be made from a flat surface without internal distortions in the plane, then plies laid to- gether on the flat wrap round the cylinder without pantographing. If the

form is of any shape other than a cylinder, including those regions of a so-called flat (cylindrical) former which depart from a true geometrical

cylinder, pantographing will occur appropriately.

3.8. Mechanism of Load Carrying — Infinitely

Flexible Membrane

To understand the fundamentals of the mechanism by which a pres- sure inflated structure carries a load, consider cases of infinitely thin

and flexible membranes. The flexibility is to be such that the forces

set up by bending of the film are so low that they can be ignored. Thin

soap films are often quoted as examples of such membranes.

Consider a tube of these characteristics, held in friction-free contact between two parallel flat plates whose distance apart is such as to compress the tube as in figure 3.32. At any point in the wall of the tube

not in contact with the plates, an element of the surface is indistin- guishable from the surface of a complete tube of the same material and

characteristics free of contact with any plates, but of diameter 2r, the

separation distance of the two plates in the first case. It follows that the tension in the free wall of the tube held compressed between the

two plates is t = pr, from elementary equilibrium. Since this holds right

up to the actual point of contact with a plate, it follows that the free wall

is truly semi-circular of constant radius of curvature r, and it meets

the plates tangentially at the four contacting points. The total reaction on plane AB is p-{AB), and the tension in the free walls at A and B is

Figure 3.32. Flexible cylindrical tube compressed between two parallel flat plates.
MECHANISM OF LOAD CARRYING 393

each t =pr, and also AB = 2r-\-2a where 2a is the length CD of the

film on the plate. Now, from equilibrium arguments, the total reaction on CZ) = total reaction on AB minus the resultant, in the direction

normal to CD, of the pressures on the curved walls AC and DB. This

leads to p-iCD)=2p{a + r) -2pr=2pa

This is exactly what is to be expected when it is noted that the gas pressure acts in a direction normal to the plate CD, and to the flexible membrane in contact with the plate. Furthermore, as this surface of the

tube is parallel to the plate CD, the tension in that part of the tube

does not matter since it has no component normal to the plate. In the case

just discussed, the tension in the wall just as it meets the plate tangentially is t = pr, and this must be resisted by the tension in the film across the plate which is in friction-free contact with the plate, so that the tension in the film or membrane in contact with the loading plate

is t = pr. If the friction between the loading plate and tube were not

zero, then the tension in the membrane across the plate would be

modified according to the friction laws pertinent to the contacting

surfaces, and possibly the manner of approach of the surfaces, as well

as the elastic moduli of the tube wall and platen. But in any event

this would not alter the load carried normal to the plate.

The main point to be noted is that as compared with the free tube,

the use of the tube to support load on the parallel plates reduces the

tension in the free walls of the tube, and in the case discussed the

load carried is precisely equal to the gas pressure times the contact

area between tube and plate.

If the wall of the tube could not be assumed infinitely flexible, then

the bending stiffness and/or wall shear rigidity would cause the load

to be supported by the platen to be greater than the pressure times the

actual contact area. The stresses in the wall would be modified near

the contact region, locally the wall curvature might vary from the semicircular arc, and the tension in the membrane across the contact might

be different. At this moment these effects are not relevant to the matter

directly under discussion but are pertinent later.

It will be noted that in the above case, if the material is inextensible

as well as infinitely flexible, the overall periphery of the tube remains

constant so that 277r-f 4a = 277-/?. As the load carried is 2ap = L, it can

be deduced that the plates approach towards each other by an amount

^2L_

irp or put another way, at a deflection x the load carried is

where L is the load per unit length of the tube.
394 STRUCTURE OF THE TIRE

Figure 3.33. Sketch of spherical element and nomenclature.

Now consider the case of a complete spherical membrane or balloon,

flexible yet inextensible, compressed between two parallel flat plates.

The principal radii of curvature of the surface are n radius of curvature

of the cross section in a plane radial to the axis of symmetry of the

surface, and r-z, the distance from the surface to the point where the

radius n extended meets the axis of symmetry of the surface, as in figure 3.33. As the surface of that part of the baUoon or membrane in contact with a plate is flat, r-z at a point immediately adjacent to the edge of the contact patch is infinite and we have from the equation of normal equilibrium (see fig. 3.34).

^ ri rz ri oc n

Since p is constant throughout the interior not in contact with the wall,

including the region of the equator between the two plates where the

Figure 3.34. Membrane in contact with plate.
MECHANISM OF LOAD CARRYING 395

Transition^'^ Vrad=R/2

/

/ rad = R

Figure 3.35. Transition of membrane surface of edge of contact with plate.

membrane is nearly spherical and has a tension t, approximately the same in both t\ and t-z directions, then

^l 2t

where R is the radius of curvature of the spherical wall. Now ti = t because, at any point on the curve representing the cross section in the

radial plane (fig. 3.34) nothing contacts the membrane except gas infla- tion or the plate, and so just at the point where the membrane leaves

the plate, h=t and so n =RI2, as in figure 3.35. Therefore the radius of curvature of the cross section at the point

in the membrane where it touches the plate is R/2 and the plate is tangential to the membrane surface arc at this point. The surface

follows a transition curve from this point to the main region of the

sphere, as in figure 3.35. Because of these facts the radius b of the actual contact is less than a, the radius of the intersection of the sphere and

plane in figure 3.34. As stated earlier, the load on the plate is equal to the product of the

internal gas pressure and the actual contact area of the membrane

on the plate. The only other loading which could come on to the plate is a shear loading within the thickness of the membrane around the perimeter of the contact patch, and this could only arise as a consequence of bending stresses in the membrane. Because of the initial assumption

of infinite flexibility, these stresses cannot be present in the problem

as discussed.

It also follows that the two contact patches are equal in area and that

their midpoints are on the same normal to the flat plates.

Now consider a toroidal or tire-like structure of the same material as the previous balloon membrane but with a rigid tubular rim for the

central zone or bore of the toroid, as in figure 3.36a. Assume that the

junction between the thin flexible membrane and the rigid tubular

rim or base has zero bending rigidity. Inflation of the structure puts tensions in the membrane and it takes

a shape as determined equilibrium and compatability conditions.

The membrane tensions are resisted by reactions at the edge of the

tubular rim, and for our present purpose these can be discussed in terms

of two components viz, radially outward tension and tension in an axial direction (that is paralled to the axis of rotational symmetry or rotation

of surface generators) at each point around the edge of the rim.
396 STRUCTURE OF THE TIRE

(a) (b)

Figure 3.36. Toroidal membrane on cylindrical rim.

If a flat plate is pressed against the membrane while the structure is supported by the rim a reaction will develop between membrane and

plate in the manner already discussed for the sphere and the cylinder, as in figure 3.36b. By the same arguments the load will be equal to the

product of the actual contact area and the inflation pressure. The contact area will be approximately but not exactly elliptical, but its geo- metrical shape is of importance only for the discussion of the point

about to be made.

The question now arises, how does the reaction develop at the rim? | The air pressure is uniform and the rim base width is constant, and

it follows that the resultant of the air pressure reactions on the rim is zero even when the plate is pressed against the membrane.

Consideration of the structure shows that the only possible way

in which the reaction can develop at the rim is by the changes in magnitude and direction of the membrane stresses at their points of attachment

to the rim, in the region of the membrane near the point where the ! plate is pressed against it. As can be seen from figure 3.37 the curvature of the wall of the membrane increases in the region between the loading plate and the adja- cent rim. Hence, because of the increased curvature the membrane

stresses in this region are lower than elsewhere in the membrane walls,

since t = p • r. The deflection also causes the membrane to distort locally,

increasing the angle between the direction of the wall and a line normal

to the plate from the rim; this is true whatever the cross-sectional shape,

shown in figure 3.38. This increase of angle reduces algebraically the

cosine of the angle between the wall and the line of action of the applied - load on the plate. The net effect of the reduced tension and reduced

component at the deflected region is to develop the required reaction.

The rim, in efl^ect, hangs in the tensions of the undeflected walls as shown in figure 3.39. The radially outward components of the wall

tensions are greater in the undeflected regions than in the deflected

region. The system of load transmission is analogous to that of a cycle

wheel where the hub hangs by the steel wire spokes from the top of the } rim, which is loaded at the bottom as illustrated in figure 3.40.
Figure 3.39. Polar plot of radially outward component of wall tension of membrane

toroid on inner cylinder.
398 STRUCTURE OF THE TIRE

The above analysis assumes that the membrane structure is so flex- ible that the load which would be carried by it when uninflated is trivial.

If, as is usually the case, the structure had some bending stiffness, this would contribute some load support, and to do this it would need

to develop bending stresses in the membrane, particularly around the

edge of the contact patch where it contacts the loading platen. Bending stresses may also, but not necessarily, develop at the rim edge, where

the presence of a flange on the rim would, of course, set up bending

moments depending on its shape and the bending stiffness of the membrane at that region.

The objective of this discussion of some apparently academic cases

is to get principles clear, because the actual tire case involves them and some further complications, particularly in the bead region because of i the steel bead coils. ; 3.9. Mechanism of Load Carrying — Tire Structure

In an undeflected tire the cords are tensioned by the excess pressure

of the inflating gas over the external or atmospheric pressure. The

tire casing takes up its equilibrium shape, which is determined primarily by the cord paths, perhaps modified somewhat in local regions
MECHANISM OF LOAD CARRYING 399

by the presence of extra rubber, ply turn-ups, flippers, filler, etc. For

clarity and ease of discussion we will ignore these features and their

effects and start from the simple theoretical equilibrium shape.

As the tire is pressed against a flat roadway the tread rubber is compressed and at the same time the tire casing locally loses its axial symmetry and takes on a substantiaUy flattened contact patch. If there were

no tread rubber on the tire the casing would be flat over the area actually

contacting the ground.

Let us continue our examination of the problem assuming that, for the present, there is no tread rubber. The cords in the area of the casing

in actual contact with the ground will also lie in a flat plane parallel to the ground plane and hence, viewed along any line parallel to the ground

plane, the line of the cord is straight. It foUows that the tension in the

cord bears no relation to the internal inflating gas pressure, in this particular region of the tire. To make this point emphatically clear it will be noted that the basic law determining the relation between tension

in the casing cords and the internal gas pressure which it resists is based

on the simple laws of statical force resolution, and in such cases a path

of infinite radius of curvature results in zero resultant force opposing

the gas pressure. The tension in the cords across the flat part on the contact patch is therefore determined primarily by the cord tension

transmitted from the adjacent free wall of the tire, modified by the

effects of the transition curvature around the perimeter of the flat contact patch. It also follows that the contact pressure between the

tire casing and the ground will be equal to the inflation pressure, modified

around the edges of the contact patch by the extra pressures set up

by the bending stresses within the transition zone, as shown in figure 3.41. In the case of practical tire designs the presence of tread rubber

of differing thickness, of tread pattern design, and such factors as the

bending stiffness caused by the multiple layers of cord cause the actual

tire contact pressure to be locally greater than the inflation pressure, and in fact to differ in different parts of the contact patch. The contact pressure at the sides of the contact patch, under the shoulders of the

tire tread, is often higher than the general contact pressure because

of the reaction necessary to develop the bending stresses in the transition zone around the contact area. This is most marked if the shoulders

of the tire tread are thicker than the tread at the crown of the tire.

Figure 3.41. Effect of casing bending stiffness on contact pressure.
400 STRUCTURE OF THE TIRE

For these reasons the actual load carried by the tire is higher than

the product of the overall contact area and inflation pressure. The extra reaction is carried by the tire acting as a structure. The load carried by these two processes seems to be linearly additive.

The load-deflection data of figure 3.42 can be analyzed by redrawing

the various load-deflection curves for the range of inflation pressures

as a lattice plot, as in figure 3.43. In this plot the load-deflection curve

for a given pressure is started from a zero determined by the value of the

inflation pressure, the zeros being located along the abscissa on a

linear pressure scale. Points on each curve for a given combination of load and deflection are taken from figure 3.42 and are joined to produce a line of constant

deflection. This process produces a family of constant deflection curves

1 2

INS DEFLECTION

Figure 3.42. U.OOX 20 tire deflection data over a range of pressures.

Figure 3.43. Lattice plot of ll.OOx 20 tire data offigure 3.42 showing both load-deflection curves at various pressures and also load-pressure curves at various deflections, thus sepa- rating pneumatic load and direct structural load.
MECHANISM OF LOAD CARRYING 401

laid across the original family of constant pressure curves. For any

combination of pressure and deflection, the load developed can be

read off. Interpolation at intermediate values is easy. The constant deflection lines can be extrapolated to produce a curve

for zero inflation pressure, shown as a dashed line in figure 3.43. The

load at any specified pressure and deflection is seen to be the linear sum of the part dependent on the inflation pressure and the part due to the structural stiffness, or the stiffness at zero inflation pressure. It should be noted that the slope of a constant deflection line, shown

chain dotted, is an effective area and is of the order of the contact area,

as would be expected from the discussion above.

The stifl'ness of the structure can be expressed as an equivalent

effective pressure, the value of which is determined by extrapolation of a constant deflection line through the line for zero inflation pressure. The horizontal intercept from the foot of the ordinate of the point where

the given deflection line cuts the zero inflation line, to where the given

deflection line cuts the zero load base line, determines an effective

inflation pressure equivalent to the stiffness of the structure as shown

in figure 3.44. Construction of curves of this type is described in ref- erences [2-4].

So far, the development of the load at and in the immediate vicinity of the tire-ground contact has been discussed. The load must be trans- mitted to the wheel and this involves not only the transmission of forces

between bead and rim but also the transmission through the structure. The area of the wheel rim base acted upon by the air pressure is axially symmetrical and is a cylinder of constant length corresponding

to the rim width, so the uniform inflation pressure cannot produce a

resultant force on the rim.

To postulate how the force set up by the inflation pressure acting

through the flattened contact region can produce stresses in the struc- ture having a resultant equal to the load carried, consider two mechanisms of force transmission acting in parallel.

i
402 STRUCTURE OF THE TIRE

Figure 3.45. Load transfer fror% ground to wheel by effects of inflation.

First of all, consider a band approximately the width of that part

of the tire tread and casing in contact with the ground, and with the same properties and construction as that region, extending around the

whole periphery of the tire. This is shown in figure 3.45. At all points

the inflation pressure acts on this band, and, at all places except that part in contact with ground, it is curved and so the tensions set up by

inflation pressure resist and equilibriate the inflation pressure. In

that portion in contact with the ground the inflation pressure forces are transmitted by compression through the band without producing a result- ant reaction on the band or ring. The net result is that the absence of forces on this part of the band leaves the forces due to inflation pres- sure on the top sector of the band unresisted by an opposite force on the

bottom sector, while at other parts of the tire outward forces have zero

resultant, as shown in figure 3.45a. The resultant upward force on the upper half of the ring, being greater

than the resultant downward one on the lower half, causes the sidewall

tensions in the upper half to be greater than the lower half, as shown

in figure 3.45b, and this force pulls the bead coil against the base of the wheel rim above the contact area, thus transmitting the upward

force to the wheel, as shown in figure 3.45c. The other mechanism of force transmission from the ground to the

wheel is analogous to that discussed earlier for the cases of a cylinder

and a sphere, where the deflection of the walls at the contact region

lowers tension forces in the walls in that region, and here, because the
MECHANISM OF LOAD CARRYING 403 bead coil has a high modulus, it bears on the base of the rim via the

ply material around it with a force just sufficient to make up the total load, as shown in figure 3.45. Both mechanisms lead to the same kind of transmission of force

from tire to rim — the wheel rim hangs in the bead coils, which in turn hang in the tensioned casing cords which have lower tension in the

contact region than elsewhere.

In addition to the load carried by a combination of these two mechanisms, some load is carried by the structure of the tire in the region of the contact, as would be expected if it were a solid tire. This load is carried by the structure of cords and rubber from the contact patch to the rim as a compression in the wall held stable by the inflation pressure

and curvature, but its contribution is small. The structural load may

also be transmitted via the bead ring system already discussed as a mechanism for the inflation load. The true stress pattern in the tire structure is the sum of these several

effects. An analysis of the problem will require the application of numerical methods, possibly based on strain energy methods, although finite element methods seem to be the most likely method of obtaining an

estimate of the total structural stiffness. Two points will be made,

however. The true estimate of structural stiffness will require the de- velopment of true finite displacement analysis — the analysis of lattices iwith substantial deformations so that the compatibility of the frameFlGURE 3.46. Points of high stress in breaker and other regions.
404 STRUCTURE OF THE TIRE work is not violated, even where the framework is distorted substantially from its original state of axial symmetry. The second point is that even when satisfactory structural stiffness and load carrying capacity

analyses have been established, the use of finite element models for stress estimation will be difficult and open to serious doubts because

of the assumption of uniform stress in each element, basic to the method.

One cannot estimate the peak stresses at the ends of high modulus

elements, and these are the ones which determine failure starting

points and fatigue life. The complexity of the problems to be faced are

illustrated in figure 3.46. The stress is difficult to obtain at points such

as ^, the free end of a cord in a region where the stress in the low modulus

matrix is changing rapidly, and B, the closest points on the surfaces of nearby crossing cords. Element sizes will need to be small to cope with

this, and this may make the problem too difficult to handle.

Small changes in relative proportions and dimensions will have

a large effect on the stresses at the cord ends. When it is noted that analyses by finite element methods of stresses at the ends of even

single embedded wires in tension only involve a considerable number

of elements, the truth of the statement earlier is clear. Practical size computations may be inadequate to estimate the important stresses but adequate to obtain overall structural stiffness. References

[1] Gough, V. E., Cord path in tyres. Trans. I. R. 1. 40 (1), T20 (1964). [2] Gough, V. E., Tyre to ground contact stresses, Wear 2, 107-126 (Nov. 1958). [3] Cooper, D. H., Radial stiffness of the pneumatic tyre. Trans. I. R. I. 40 (1), T58

(1964). [4] Gough, V. E., Tyres and air suspensions, Advances in Automobile Engineering, Tid- bury, ed., p. 59 (Pergamon Press, 1963).
CHAPTER 4

Tire Stress and Deformation

Joseph D. Walter 1 Page 4.1. Introduction 406

4.2. Analytical techniques 407

4.2.1. Solutions for inflation loads 407

Inflation cord tension 418

Inflation bead tension 419

Cord-rubber interface shear stress 419

4.2.2. Solutions for centrifugal loads 421

4.2.3. Methods for analyzing the tire in service 426

4.3. Experimental techniques 430

4.4. Principal notation 439

References 440

' Central Research Laboratories, The Firestone Tire & Rubber Company, Akron, Ohio 44317. 405
406 TIRE STRESS AND DEFORMATION

4.1. Introduction

The conventional automobile tire possesses three distinct structural

components: the rubber matrix which contains the air and provides

abrasion resistance and road grip; the cords (textile, steel, or glass)

which provide tensile reinforcement for the rubber and carry most of

the load applied to the tire in service; and the steel beads which cir- cumferentially connect the tire to the wheel of a car. These three components, with air under pressure, form a thin-walled composite toroidal

shell which is both highly flexible and relatively inextensible. The

purpose of this chapter is to discuss those methods of stress analysis

which are applicable for the calculation or measurement of the stresses and strains developed in the three structural components of the presentday automobile tire. Stress analysis is that branch of mechanics which is concerned with

the quantitative determination of internal stresses and strains pro- duced in a body as the result of external loads and deformation. Its methods are both theoretical and experimental. The complete stress analysis of the tire should establish the magnitude and direction as well

as the type of stress at all points in the tire under each loading condition

of interest. This information defines the so-called state of stress. Such

knowledge, when combined with the material properties of the rubber,

cord, and steel can be used to predict or explain the behavior of the

tire in service.

The material properties of the three structural components of the

tire are widely different. For example, the Young's moduli obtained at room temperature from statically conducted tests are approximately:

300-3000 psi for the rubber, 100,000-800,000 psi for the textile cords,

and 30,000,000 psi for the steel bead wire. In addition, the final geometries into which these components can be combined are diverse. Presently, there are three distinct cord arrangements in automobile tire construction: bias, belted-bias, and radial. These material and geo- metric factors, coupled with the fact that the tire as a whole is an anisotropic body subjected to finite deformations which are rate and temperature dependent, have made the theoretical and experimental stress analysis of the tire more difficult than that associated with the majority

of engineering structures.

It is not the purpose of this chapter to compile a bibliography containing every publication related to the subject of the stress analysis

of pneumatic tires. Rather, the intent is to discuss the important topics

and to present them in a logical order. The references to the literature are mainly directed to elucidating the textual material. However, every

attempt has been made to settle questions of priority.
SOLUTIONS FOR INFLATION LOADS 407 4.2. Analytical Techniques

4.2.1. Solutions for Inflation Loads

The simplest stress analysis problem to consider from a theoretical

standpoint is the calculation of the shape taken by and the stresses developed in an inflated but otherwise unloaded bias tire. The first rigorous solution to this problem was obtained by Purdy [1]- in the United

States in 1928.'^ However, it was not until 1956, when Hofferberth [3] pubhshed his results in Germany, that such work became readily

accessible to all tire engineers. Biderman's investigations [4] treating

the same subject appeared in 1957 in the Soviet Union. In the remainder

of this chapter it is convenient to refer to the solution of the inflated tire shape equation independently obtained by these three investigators as the PHB solution. The importance of this shape equation is due to the

fact that once the geometry of the tire is accurately known, the stresses developed in it can be calculated. Before developing the analytical details for this particular problem,

it should be noted that some investigators have found it convenient

to a priori assume that the inflated tire cross section can be adequately

represented from bead to bead in the meridian plane by a portion of an

elHpse since the PHB solution deviates only slightly from such a shape.

While this assumption is valid for estimating such quantities as the

surface area of a tire and the volume enclosed by a tire, it can easily lead

to large errors in stress analysis since these stress calculations are very

sensitive to small changes in geometry. However, since the basic equations employed in the PHB analysis are the same as those used when

the shape is a priori assumed, it is useful to present the details of this

latter, though less exact, approach before discussing the PHB solution. A portion of a pressurized but otherwise unloaded toroidal shell is shown in figure 4.1. The axial coordinate z is an axis of revolution. Be- cause of the very small bending stiffness of the tire walls, and the

rotational symmetry of both the applied load and shell geometry, in-plane

internal reactions develop which are constant through the shell thickness.

These internal reactions, denoted by and Ne, are usually called

membrane stress resultants, and they act in the meridional and circumferential directions, respectively. Physically, the stress resultants represent forces per unit length.

The applied load (inflation pressure p) is just balanced by the internal

reactions and TV^— that is, force equilibrium exists — so that summining forces normal to the shell surface leads to the well known equilibrium equation —+ = p (4.1)

r<p re

^ Figures in brackets indicate literature references at the end of this chapter.

' Purdy's original report is contained, in many places word for word, in Chapters I and II of his 1963 monograph [2].
408 TIRE STRESS AND DEFORMATION

r

Figure 4.1. Portion of a pressu rized toroidal shell. where and re are principal radii of curvature for the doubly curved

toroidal surface. Both the stress resultants and the radii of curvature

are functions of the radial distance r. A second equation of equilibrium can be obtained from figure 4.1 by summing forces along the axis of revolution to obtain:

N, = PIi(!^ZJ^.

,4.2)

Equations (4.1) and (4.2) are two equations for the two unknown stress resultants and they can be immediately solved when the meridional

geometry of the toroidal shell is given. For example, in the case of an

elliptic meridian, as in figure 4.2, it is known that _[6^+(a^-6^)(r-r»;)^]3/2

_ rW+{a'-b')(r-r^Y]

re — 7 ^ 1/2

Thus, the stress resultants are found from eqs (4.1) and (4.2) by eliminating rv and re to obtain

M'-+r„)W+{a^-b^)(r-r^)^^l-'

p[2r(a^-6^)(r-r«,) + 6'']
SOLUTIONS FOR INFLATION LOADS 409

r

Figure 4.2. Cross section geometry ofa toroid with an elliptic meridian. Lorenz [5] obtained these equations for the stress resultants devel- oped in a pressurized toroidal shell of elliptic cross section in 1910. Since then, several investigators [6-9] have employed these relations

in their analyses of pneumatic tires. While the value of N>p given above

deviates only slightly from the value obtained from the PHB solution,

the value of A^^ deviates considerably. In fact, inspection of the above

results indicates that Ne can be negative in some regions of the shell,

depending on the eccentricity of the ellipse. This condition is difficult,

if not impossible, to realize in a symmetrically loaded fabric structure such as an inflated tire. Also note that eq (4.2) defining the meridional stress resultant A^.^ for a toroid with an arbitrary meridian, is indeterminate at the radial location r= Fw where the circumferential radius of curvature re is undefined.

I This indeterminancy has the form 0/0. However, by recognizing that any

i equation for re applicable to a toroidal shell always contains the quantity

I {r—rw)~^ and by employing appropriate limiting processes, the indeI terminancy is removed regardless of meridional geometry and it presents no problems in membrane stress analysis — i.e., A^v, as well

as Ne, is everywhere finite and continuous. Other arguments can be

used to show that N>p is always tensile and always has the same value

at the radial position r=rc for any pressurized toroidal shell. It has

been known for some time that the displacement analysis for this particular problem presents analytical difficulties at r= r^, whenever the usual

J relations are employed to connect strains and displacements for shells

I of revolution [10]. In particular, the displacement in the z direction is singular at r=rw in the linear membrane theory of toroidal sheUs.

This anomalous behavior can only be explained by resorting to a non-
410 TIRE STRESS AND DEFORMATION

linear membrane theory [11-14]^ or a linear bending theory [15-17]^

either of which involves considerable mathematical complexity. It should be emphasized, though, that these analytical difficulties in the

displacement analysis do not keep us from predicting reasonably

accurate membrane stresses and cord forces in inflated tires. For actual tire structures, relations must be employed in addition to eqs (4.1) and (4.2) to account for the following:

a. The load carrying capability of the cord and rubber components

of the tire.

b. The path of the cords from the bead to the crown in the tire.

c. The unknown geometry of the tire cross section.

The first matter (a) needs to be discussed in some detail since at the

present time there is no easy way to rigorously account for the combined

load carrying capability of the cord and rubber. In the PHB analysis of

tires the rubber is treated as a nonstructural liner whose only purpose

is to contain the air and transmit load to the cords, i.e., its presence is entirely neglected. This engineering approximation, which greatly simplifies analysis, is more nearly realized at high than low inflation pressures. At very low pressures — below those used in service — the rubber is just as effective as the cord in supporting load and the approximation is not

valid. For mathematical simplicity, then, it is assumed that the pressure

is great enough to stabilize the tire shape and this shape is unchanged

as the pressure is further increased, and the cords carry all of the applied

load so that the surface to which all tire dimensions are referred is the

surface midway between the plies. We consider a typical cord network

element which is bounded by two adjacent meridians and two adjacent

parallel circles as in figure 4.3. Note that the cord angle j8 is the angle

between a meridian and a cord.^ Using figure 4.4, the resolution of

forces in the meridional and circumferential directions, respectively,

requires that N^=nT cos^p,

N6 = nT sin' where n is the product of the (even) number of plies in the bias tire multiplied by the number of cords crossing a line of unit length perpendicular

to the cords, and T is the tension in an individual cord. These equations can be combined to yield either

^ These cited references are only a few of many studies treating this subject.

^ Another common convention is to define the cord angle as the angle between a circumferential line and a cord, i.e., V2 7r-)8.
SOLUTIONS FOR INFLATION LOADS 411

z

Figure 4.3. Typical geometry ofa pressurized bias tire. which is the basis of a failure law for filamentary composite materials

under plane stress conditions [18], or f^=tan^0, (4.3) which is the basis of the so-called netting analysis of composite materials.

This netting concept is often referred to as a condition of balanced design

[19] or equilibrium design [7]. Equation (4.3) is a special case of a more

general relation first derived by Haas [20] in 1912 and, like eqs (4.1) and

(4.2), it is based only on equilibrium considerations. Thus, we have three

equations for the two unknown stress resultants, and these three equations can be simultaneously satisfied only if the tire (or any other fabric

structure) is in its "equilibrium configuration." The concept of the

equilibrium configuration is sufficiently important in the stress analysis

of pneumatic tires to justify further consideration. A simple demonstration of this concept is afforded in the case of a pressurized cord-rubber

cylinder as in figure 4.5. Equations (4.1) and (4.2) give with re= rc= r= R,

N0 = pR

N^=y2pR

so that eq (4.3) produces

tan2 p = 2 or /3 = ±54.7°.
Figure 4.5. Pressurized cord-rubber cylinder with cords at equilibrium angle {— 54.7°]
SOLUTIONS FOR INFLATION LOADS 413 The implication of this resuh is: only if the cylinder is constructed with

a helical cord path at ±54.7° with the longitudinal (z) axis will it retain

its cylindrical shape when pressurized. If the cord angle jS is less than or greater than this ideal angle, pressurization causes the cords to tend to seek their preferred or equilibrium orientation, but the rubber, since it has some strength, restricts this action. Thus, when j8 7^±54.7°, the

shape of the structure changes with pressurization and it is no longer

cylindrical. As an extreme case, consider an initially uninflated cylinder

containing cords parallel to its axis (j8= 0°) as shown in figure 4.6. After

inflating, the resolution of forces gives

Ne=nT sin' p = 0 (a) while the membrane theory of shells gives

Ne = pR (b) Since these membrane forces are determinate, eq (b) must be satisfied. From eq (a) no A^^ is generated by cord loads T, hence No must come

from stresses in the rubber, which is ill-suited for such stressing since

it results in excessive deformation. For this reason such designs are generally avoided. When the cyfinder of figure 4.6 is gripped at each

end and inflated, it will exhibit large radial strains, associated with A^^ acting on the rubber matrix alone, at all regions of the cylinder except

at the ends where it is restrained by grips. Under similar loading, a

cylinder with a 54.7° cord path would exhibit little dimensional change.

UNINFLATED INFLATED

Figure 4.6. Cord-rubber cylinder with longitudinal cords (= 0°) before and after inflation.
414 TIRE STRESS AND DEFORMATION

The concept of the equilibrium configuration has been extensively

used in the design and manufacture of fabric hose [21] and filament

wound rocket motor cases [22]. These same ideas, of course, are valid

for the more complicated geometry of tires, and Midgley [23] was aware

of the implications of this concept as long ago as 1920. He recommended,

without specifying the mathematics, that tires with cords having large

values of ^ should be cured in a low oval mold while tires with cords

having small values of P should be cured in a high oval mold. Tires pro- duced in this fashion, i.e., close to their equilibrium configuration, would

not tend to change their shape in service and would be free of unnecessarily imposed inflation stresses in the rubber matrix.

The second matter (b) is most easily disposed of by noting that a reasonable approximation to the cord path in tires of conventional construction ^ is given by the so-called trellising or pantographing law [24]

which relates the local cord angle P with its radial location r in the

finished tire

r r _ constant (4.4) sin j8 sin /3, where the subscript c denotes a quantity evaluated at the crown {r= rc) of the tire. Modifications of this equation, as well as some nonconventional cord paths, will be referred to later. As a consequence of prescribing the cord path through eq (4.4) and of employing the netting concept

of eq (4.3), the meridional or cross-sectional shape of the tire is not

arbitrary. Similarly, if the meridian curve were to be specified, there is no freedom as to the choice of the cord path [19]. The third matter (c) is taken care of by expressing the principal radii of curvature for a toroid in their differential form in rectangular cartesian

coordinates [25], i.e., ..^ii^^, (4.5)

re=- ; , (4.6) z where z' and z" denote the first and second derivatives of z with respect

to r, respectively.

Equations (4.1-4.6) are the basic relations needed for the determination of the equilibrium contour of the bias tire. Eliminating all variables

that depend on the radial distance r {N>p, Ne, r<p, re, and /3), and combining the results leads to a second order nonlinear differential equation

involving z" and z '

.

' .-+.'[i+(.-)-]{ /t!^- -^1=0.

Corventional construction means that the cords in each ply alternately form right and left handed helices on a cylin- drical building drum before the tire is expanded into its final toroidal shape.
SOLUTIONS FOR INFLATION LOADS 4L5 By the usual methods of ordinary differential equations [26], this relation can be reduced to a first order equation which, in turn, can be expressed

as the indefinite hyperelliptic integral

r

,) (r,^-r^sin^j3c)'^'rfr ^^^^

[

( r? - r^,) V| cos^ /3o - ( -4) Mr| - sin^ /3c ) ] Equation (4.7) defines the equilibrium contour, z = z(r), of the conventional bias tire and is equivalent to the results obtained independently

by Purdy, Hofferberth, and Biderman, i.e., the PHB solution, except

that Hofferberth included an average but unknown cord elongation in

his equation. It cannot be integrated in closed form except for the

special case of radially disposed cords [27], or j8c = 0°. Its solution, then,

generally requires the methods of either numerical or graphical integration. This integration is easily effected for given values of the parameters

Tc, rw, and f^c even though eq (4.7) is singular at the crown r=rc. Purdy

[2] has plotted typical equilibrium contours for different values of rc, r^;, and l^c using graphical techniques. These graphical methods of in- tegrating the tire contour integral are discussed in detail by Biderman

[4, 28] and Day and Gehman [27]^. Ellis and Frank [29] have shown how

these same calculations can be carried out on an analog computer.

Biderman has developed an extensive set of nomograms based on

graphical [4, 28] and numerical [30] procedures which permit one to calculate the equilibrium dimensions of an inflated tire knowing three

quantities: the coordinates (2&, r^) which locate the so-called rim point on the inside tire surface, shown in figure 4.3 and which are used in eq (4.7) making it a definite integral; the length of the cord from bead

to crown, L, which is given by

ds cos /8

' where the element of arc length, ds, is expressible in terms of eq (4.7); and the crown angle fie In effect, the first two items provide two equations for the determination of the two unknowns rc and rw which are then

used with fic in the calculation of the Biderman nomograms. Similar

nomograms have been machine plotted by high speed digital computers

[31].The importance of having a relationship like eq (4.7) is that it allows

tire engineers to carry out parameter studies without actually building

tires. For example, it is easy to show that if the cord length and rim

dimensions are held constant and only the crown angle is allowed to vary, low values of fic lead to tires with tall, narrow equilibrium cross

sections while high values of fic lead to tires with low, wide equilibrium

cross sections. The study of these equilibrium contours is of great

importance as a guide to good mold design.

^ The tire contour integral given by Day and Gehman [27] contains a typographical error; specifically, an unneeded r appears in the denominator of their eq (4. 13).
416 TIRE STRESS AND DEFORMATION

The lowest inflation pressure at which the assumptions inherent in this analysis are valid can be determined for a given tire by plotting a

typical dimension, such as section width, as a function of pressure and

then noting the pressure at which this dimension remains (relatively)

unchanged as the pressure is increased. Biderman [4, 28] has done this for a 9.00-20 truck tire and has shown that the section width for this tire no longer changes, for all practical purposes, at an inflation pressure

well below that recommended for use in service. In fact, if there is a

significant change in a tire dimension during the early stages of inflation, this is an indication that the molded contour of the tire is difl"erent from

its equilibrium contour. Several refinements can be added to the basic analysis presented here

for calculating the equilibrium configuration of a bias tire. The conventional cord path relation, eq (4.4), can be modified to include the effect of cord extensibility [3, 4, 32, 33] which means that the tire contour inte- gral given by eq (4.7) will then include directly the material properties

of the cord and indirectly the magnitude of the internal pressure. These

factors, cord properties and internal pressure, always influence tire shape

to some extent and in some cases, such as a nylon tire operating near its bursting pressure, cannot be neglected. A further modification of the

preceding analysis is to completely replace eq (4.4) by another cord path

of interest. For example, a geodesic path is defined by the theorem of Clairaut [34] as rsin (3 = rc sin (Be = constant,

and has the remarkable properties that the distance between any two

points on a cord is minimum and the tension at all points along a cord is constant. Many other types of nonconventional cord paths possessing some unusual or desirable properties when compared to the conventional cord path, eq (4.4), have been discussed by Purdy [2], Day and

Gehman [27], and others [35-37]. In all cases, modifying the theory to include either the cord elongation or a nonconventional cord path leads

to a singular hyperelliptic integral similar to eq (4.7) which, in general,

does not possess a closed form solution. Since an accurate prediction of meridional geometry is a prerequisite

for calculating tire stresses, the validity of eq (4.7) can be demonstrated

on an existing tire by determining the parameters r^, and by

measurements and then using these value to plot the equilibrium

contour, z = z{r). The comparison between a theoretically predicted

and an experimentally measured contour is shown in figure 4.7 for an 8.25-14 two-ply polyester tire [38]. These calculations were carried out numerically using a digital computer, which is a quicker and more ac- curate method than using graphical integration. Lauterbach [39], Mikell

[40] and Lauterbach and Ames [41] have plotted comparable results

for an 8.00-15 four- ply nylon tire using numerical integration techniques.

In general, the agreement between theory and experiment is good. Deviations between the predicted and measured coordinates are largest in the bead region where the theory consistently predicts smaller axial
SOLUTIONS FOR INFLATION LOADS 417

Figure 4.7. The equilibrium contour for an 8.25-14 two-ply polyester tire theoretically

predicted by eg {4.7) and experimentally measured {rc^ 13.12 in., ru = 10.14 in., 6c^55°)

dimensions (z) than those observed. These deviations are expected near

the bead since membrane theory is known to be inadequate [42] near

shell boundaries, i.e., at the point of attachment of the tire to the rim

and in areas where large changes in rigidity occur, i.e., at the location of the steel bead and the ply turn-ups.

Equation (4.7) can be used directly, without integration, to calculate

the stress resultants using eqs (4.1) and (4.2):

Ne

Prdrl rl) cos 13c 2r{4-f' sin' ficY 12 prrc(fl — r%) cos j8c sin^ fS, 2{rl-r' sin' f^c^l' and the principal radii of curvature using eqs (4.5) and (4.6): rc{rl-ri){fl-f' sin' Pc)'" cos pc . 2r(r2 - r' sin^ fic) - r(f' - rl) sin' (3c

'

re rrc(fl-f'J cos /3c {r'-fi){rl-f' sin' PcY''

From the preceding analytical developments, it is possible to calculate

the cord tension, the bead tension, and the cord-rubber interface shear

stress developed in an inflated but otherwise unloaded bias tire. By

themselves, these inflation forces and/or stresses are not responsible

for the failure of any of the tire components (cord, rubber, steel) in service; rather, they are important because all loads that a tire receives in operation are superimposed on these initially developed inflation forces
418 TIRE STRESS AND DEFORMATION

and/or stresses. The derivations of the fomulas for cord tension, bead

tension, and interface shear stress are all based on the tire contour integral, eq (4.7). However, the lengthy manipulative details involved in each

derivation are not presented here since they are well documented in readily accessible literature; rather, we list the pertinent equations, give an example of their use, and discuss their limitations. Inflation Cord Tension: The tension T developed in the cords of an

inflated bias tire is given by the formula

P^rlirj-rl) cos (3, N{ri-r'' sin2 pc)

(4.8)

For a given tire, cord tension is minimum at the bead where the radial

distance ris minimum and is maximum at the crown where r is maximum.

Equation (4.8) does not involve the modulus of the tire cord because of the inextensibility assumption used in the derivation of the tire contour

integral. For the special case of radially disposed cords, /8c=0°, which

is also a geodesic path, cord tension becomes

p7T{rl-rl)

N

Equations equivalent to the above were first derived by Purdy [2]; many

investigators have subsequently discussed such cord load formulas [4, 27, 28, 41]. A typical result obtained for the cord tension distribution,

T= T{r), for an 8.25-14 two-ply polyester tire at 24 psi inflation pressure

is shown in figure 4.8 [38]. The equilibrium contour z=z{r) for this tire

is shown in figure 4.7. For comparison with the theoretical predictions,

experimentally measured cord forces are also shown. These measureFlGURE 4.8. The cord tension distribution in an 8.25-14 two-ply polyester tire theoretically

predicted by eq {4.8) and experimentally measured (rc = 13.12 in., ru: = 10.14 in., (3c = 55°, p=-24 psi, 'N=2350) [38\.
SOLUTIONS FOR INFLATION LOADS 419 ments were made using miniature force transducers described later in

this chapter and shown in figure 4.16. Lauterbach and Ames [41] plotted the cord tension distribution in an

8.00-15 four-ply nylon tire at 24 psi inflation pressure and compared

their predictions with the experimental measurements of Weickert [43]. These studies show that the maximum inflation cord load developed in typical automobile tires at rated pressure is approximately 10 percent

of the cord breaking strength as determined from a room temperature,

slow speed tension test of the greige cord. This large strength reserve is needed in order to provide adequate fatigue resistance and impact

strength to the tire under service conditions [44]. Inflation Bead Tension: The tension B developed in each bead of an

inflated bias tire is given by the formula

p(r,^-4)cos/3e ^ 2 cos /3, ^^'^^ where j8& is the cord angle at the "rim point" as illustrated in figure 4.3 and is nominally equal to the angle of the cord on the building drum

(bias angle). For the special case of radially disposed cords, Pc = I3b = 0°

which indicates that larger inflation bead forces occur in radial tires with cords normal to bead than in bias tires with cords incHned to bead.

If the cords are disposed tangentially to the bead, no tension develops

there as the tire is inflated. Equation (4.9) is the form recommended by

Biderman [4]. The formula derived by Purdy [2] and discussed by Day and

Gehman [27] is difficult to use since it involves differences of nearly equal

large numbers. As has been previously discussed, the equilibrium contour of the inflated tire is not predicted accurately in the bead region.

Equation (4.9), then, should be used with some caution since it is based

on the tire geometry. Bead tension is plotted as a function of inflation pressure in figure 4.9 for an 8.25-14 two-ply polyester tire. The equilibrium contour and the cord tension distribution for this tire are shown in figures 4.7 and 4.8, respectively. Pugin [45] has shown that eq (4.9) is in error on the high side, that is, the actual bead forces in this tire due to

inflation pressure are less than those predicted and shown in figure 4.9. Irodov [58] has attributed these differences to friction forces existing

between the bead and rim which are neglected in eq (4.9). Cord-Rubber Interface Shear Stress: When a filamentary composite

structure such as a tire is loaded in some arbitrary fashion, the rela- tively weak matrix transfers most of the load to the stronger reinforce- ments. The mechanism available for this load transfer is interface shear

stress which is generated over the contact surface existing between the

cord and rubber. The bonding agent or adhesive that connects the cord

to the rubber is not usually thought of as a structural component of the

tire, but if this bond ruptures the tire will quickly fail in service due to
420 TIRE STRESS AND DEFORMATION

2000

Figure 4.9. Bead tension as a function of inflation pressure using eq {4.9) for an 8.25-14

two-ply polyester tire irc = 13.12 in., ru= 10.14 in., (Bc = 55°, ^^ = 2872°).

the relative movement of cord and rubber. In an inflated bias tire, the

shear stress r developed at the cord-rubber interface in the direction

of the cords is given by the formula _ pre sin' Pclirl r^'r^ cos^ /3c - (r^ - r" sin' if^-f' sin' Per

^^-^""^

Equation (4.10) involves the rate of change of cord tension with cord

length; thus, for radially disposed cords (j8c = 0°), the cord-rubber

interlace shear stress vanishes since the cord tension is everywhere

constant between the bead and the crown of the tire. The preceding

expression differs from the equivalent result first obtained by Purdy

[2] and discussed by Day and Gehman [27] by a factor sin (3 since in these two studies r was interpreted as a shear stress existing circumferentially between the plies rather than being developed in the cord

direction between cord and rubber. In any case, the values of the

interface shear stress calculated from eq (4.10) at inflation pressures

used in service are well below the rupture shear stress values generated

in a cord-rubber pull-out test [46], just as the cord tension developed at normal inflation pressures is well below the breaking strength of the

cords. A typical distribution of the shear stress T=T{r) is given in figure 4.10 for an 8.25-14 two-ply polyester tire at 24 psi inflation pressure. The equilibrium contour, the cord tension, and the bead forces

associated with this tire are shown in figures 4.7, 4.8, and 4.9 respectively. Note that r vanishes at the crown and is maximum at the shoulder,

a result that holds true for all bias tires.
SOLUTIONS FOR CENTRIFUGAL LOADS 421

C..I6 2

RADIAL DISTANCE r-IN. Figure 4.10. The shear stress distribution in an 8.25-14 two-ply polyester tire according

to eg {4.10) (r^ = 13.12 in., r^, = 10.14 in., (Be = 55°, p = 24 psi) . At the present time there is no generally accepted theory for accommodating a "belt" into the equations which govern the inflated equilibrium contour of tires. Consequently there is no way to predict the

stresses developed at all locations in such tires. The circumferential

constraint imposed by a belt causes a change in tire dimensions to occur compared to comparably constructed nonbelted tires. All of the published work in this area has dealt exclusively with the radial belted or, simply, radial tire. No definitive studies of a theoretical nature have been

published for the belted-bias tire. Slyudikov [47] has made measure- ments of various radial tire dimensions as a function of belt properties

and has shown that no simple linear relationships exist between the

parameters. A formula used by Biderman, et al. [48], to predict the forces

in the belt of an inflated radial tire gives values that are two to three

times the values measured experimentally. Frank [49, 50] has made what

is perhaps the most comprehensive study to date of the problem of pre- dicting the equilibrium contour of the radial tire. This work includes as a

special case the nonbelted bias tire. Bohm's method [51] for analyzing the

radial tire possesses an acceptable degree of rigor and is much simpler to apply than the method of Frank. It includes an empirically determined

"girdling" or "belting" function. When this function is zero, the contour

integral for the conventional bias tire is obtained. Without the belt, the

equilibrium geometry of the tire with radially disposed cords is easily

determined [27, 52]. Belted tires are also the subject of a separate chapter

in a Russian monograph [53] of 1963. 4.2.2. Solutions for Centrifugal Loads

Any tire subjected to loads that are distributed in a rotationally symmetric fashion can be analyzed using the techniques given in the pre- ceding paragraphs for the inflation problem. Such loads primarily produce

a membrane response in the tire with negligible bending effects. Because
422 TIRE STRESS AND DEFORMATION

of symmetry, the meridional or cross-sectional geometry of the tire is identical at all circumferential locations and only ordinary, rather than

partial, differential equations need to be employed to effect a solution

suitable for stress analysis. Cord load formulas for axisymmetrically

loaded tires are characterized by the absence of ply-to-ply variations in cord tension at a given location in the tire. Of course, for tires under

actual operating conditions where the applied forces and moments lack

rotational symmetry, deflection at the contact zone and ply-to-ply effects

in the tire are important features which cannot be taken into account

using this restricted theory. The most important case amenable to solu- tion in the category of axisymmetrically loaded tires, other than the

inflation problem, is the combination of internal pressure and centrifugal

force. However, a tire inflated inside a constraining cylinder, i.e., a uniformly deflected tire, can be studied within the framework of this theory [54, 55]. Even the tire in a mold during vulcanization is loaded

thermally and mechanically in an axisymmetric fashion. Neither of these

last mentioned problems have received as much attention as the inflated,

rotating tire. The first analytical study of the bias tire subjected to both inflation and centrifugal loading was made by Bukhin [56] in 1960 who used a

Ritz energy method to find the dynamic equilibrium contour. Bukhin

discovered that at high angular velocities, the location at which max- imum cord tension occurs in the tire can be displaced from the crown

toward the bead. Wutzler [57] studied the same problem by working

directly with the differential equations of the membrane theory of shells. For the rotating tire Wutzler concluded that bending as well as membrane effects can be important at low inflation pressures. Walter [38]

developed equations for the dynamic equilibrium contour and the cord

forces which reduce to the corresponding formulas of the PHB solu- tion in the static case. These studies show that while the static equilibrium contour of the inflated tire is independent of pressure, the

dynamic equilibrium contour of the inflated, rotating tire depends both on pressure and angular velocity.

The effect of a centrifugal load on an inflated, rotating tire can be

investigated using the same techniques as employed in the analysis

of the tire loaded solely by inflation pressure, except that eq (4.1) must

be replaced by the more general equilibrium equation

where m is the mass per unit area of cord and rubber in the tire which

is a function of position, and w is the angular velocity of the tire. Equa- * tions (4.2)-(4.6) given earher remain unchanged. Again, numerical or ) graphical techniques must be employed to integrate the resulting con- ; tour integral. Typical curves obtained by Walter [38] for the dynamic

equilibrium contour z = z{r) and the cord tension distribution T = T{r)

(4.1')
I SOLUTIONS FOR CENTRIFUGAL LOADS 423

are shown in figures 4.11 and 4.12, respectively, for an 8.25-14 two-ply

polyester tire at 24 psi inflation pressure and angular velocities up to 1600 rpm (about 120 mph). Figure 4.11 shows that as the angular velocity

is increased, the th-e diameter increases, while the section width and

sidewall curvature decrease. Figure 4.12 shows that centrifugal effects can produce a twofold increase in cord tension compared to the static

results. Up to 800 rpm, the change in tire shape and the increase in cord tension are negligible. Other features of this particular tire have

been previously given in figures 4.7-4.10.
424 TIRE STRESS AND DEFORMATION

Formulas for the bead tension and cord-rubber interface shear stress can be developed for the case of centrifugal loading just as they have

been for the inflation problem. Bukhin [56] has plotted the tensile force

in the bead ring of a 7.50-16 tire as a function of speed up to 250 km/hr.

Wutzler [57] has plotted the cord strain developed in a 5.20-14 tire up

to speeds of 175 km/hr. by assuming that the tire cord obeys Hooke's

law. An equation for the cord-rubber interface shear stress for a centrifugally loaded tire has not yet been published in the literature. The study of the inflated and rotating but otherwise unloaded tire

is related to the investigation of traveling waves in tires, even though

this latter problem involves roUing at high speed on a road or drum,

which is a condition of nonsymmetric loading. However, since eq (4.1') given previously can be used to qualitatively explain this phenomenon

(even though in the strictest sense it is not applicable to nonsymmetric

loads), it is appropriate to discuss this topic now.

Traveling waves ^ first appear at a certain critical roUing velocity,

VcT, and are produced in that portion of the tire which has just left contact with the road or drum. Below this critical velocity no waves are

present. The length and amplitude of the waves, as well as the rolling

resistance and temperature of the tire, increase drastically as the

velocity is increased above the critical value. This destructive process

quickly produces tire failure. This failure is due to both stress and

temperature extremes resulting from the severe mechanical and thermal

loads acting on the tire. The first meaningful theoretical study of the traveling wave problem was made by Turner [59] in 1954. He modeled the tire as a flat flexible membrane, infinitely long in the circumferential direction and bounded

by rigid steel beads on each side. When this model is subjected to external excitation, a displacement perpendicular to the membrane

is propagated as a transverse wave with a velocity c given by

c = {Nelmyl''

which is recognized as the classical result for the speed of a trans- verse pulse moving along a stretched string. Turner emphasized that traveling waves are not a resonant vibration phenomenon (as are standing waves with their attendant nodes and antinodes), meaning that one

cannot drive through the critical condition merely by increasing speed

so that the tire again runs smoothly. As a consequence, every portion

of rubber on the same circumferential line undergoes identical trans- verse motion when the tire rotates at a velocity greater than or equal

to the critical value (c ^ Fcr). To find the critical velocity for a bias tire, solve eqs (4.1') and (4.3) for the circumferential stress resultant to obtain

* Traveling waves, traction waves, and rippling are used synonymously in the tire meciianics literature. Many times,

traveling waves have been incorrectly referred to as standing or stationary waves. \
SOLUTIONS FOR CENTRIFUGAL LOADS 425

Using Turner's value for the speed of transverse wave propagation

c = {Ndlmy - with the above equation and letting Vcr = c= ra>, one finds

Fcr=tan;8(pr^/m)i/2 (4.11)

which is a result that is in qualitative agreement with experimental

findings. For example, in 1951, Gardner and Worswick [60] in an ex- tensive testing program showed that the critical velocity of a tire could

be increased — and therefore its endurance at high speed could be in- creased—by increasing the inflation pressure p, decreasing the tire mass m, increasing the meridional radius of curvature r^, or increasing

the angle p between a cord and a meridian. Thus, eq (4.11) expresses

the real nature of the traveling wave phenomenon even though it does

not take into account the effect that the forces in the contact zone have

on the stress resultants. However, the usefulness of this relation strongly depends on the values of the parameters employed since ^, r<^, and

m are a function of position, and, additionally, /3 and change with

velocity while p increases above its initial value due to heat build-up.

As an illustration of the use of eq (4,11) for calculating critical tire velocity and the onset of traveling waves, consider the 8.25-14 two-ply

polyester tire treated in figures 4.7-4.12. For simplicity we use the

static tire dimensions at 24 psi inflation pressure and evaluate all functions of position at the crown {r=rc). From the relationship for the

meridional radius of curvature given earlier in this chapter with the

dimensions and parameters given by Walter [38], we have

r^c — 4.46 in.

TUc = 71 XlO-6 lb.-sec.2/in.3

/3c - 55°. Thus, eq (4.11) gives

Fcr= 1750 ips (=99.5 mph).

This predicted critical velocity is below that visually observed in drum

' testing, but considering the simplicity of the calculation, it does pro- vide a reasonable first approximation to the true experimental value. Refinements in Turner's original work have been made primarily

by investigators in Japan and the Soviet Union. These more advanced

studies are general enough so that one can calculate, in addition to the

critical velocity, the length and amplitude of the traveling waves as the

speed of the tire is increased above the critical value. To date, the

best agreement between theory and experiment can be found in a paper

published in 1968 by Akasaka and Yamagishi [9]. They obtained critical

I velocities and wave lengths at various inflation pressures for 5.60-13

and 7.00-13 bias tires. Ames [61] has made an appraisal of world lit- erature on the traveling wave phenomenon for both bias and radial
426 TIRE STRESS AND DEFORMATION

constructions. Frank and Hofferberth [50] have briefly discussed this problem and the related problem of tire vibration in their review article on tire mechanics.

4.2.3. Methods for Analyzing the Tire in Service

At the present time, no investigations have been published that adequately treat the theoretical determination of either stress or strain

in the cord, rubber, and steel components of a tire under service conditions. Thus, most of our knowledge of problems in this category has

been obtained through the use of experimental methods which are discussed in detail later in this chapter. In this section, then, we point out why there is a scarcity of literature on this subject and discuss those

few attempts that have been made to cope with the problem.

Recall that the tire shape, the stress resultants and the cord forces can be reasonably predicted in axisymmetrically loaded tires using

only the equilibrium equations of the membrane theory of shells. The

strain-displacement equations for a toroid and the stress-strain rela- tions for an anisotropic body were not employed in the study. However, these are the factors that must be included in any stress analysis

of the tire in operation but they are also the ones that are most difficult to analytically incorporate into the theory. These strain-displacement

and stress-strain laws cannot be neglected without causing appreciable

error in those regions where bending effects might be important, such

as in the bead region, near the belt edges, or in the contact zone of a

rolling tire. Incorporating these factors into shell theory, as it is known

today, proves to be the major difficulty hindering the rational development of the theoretical stress analysis of the tire under service load- ing conditions.

An extensive body of literature exists treating the structural mechanics of thin shells [10, 42]. However, most of this literature was

written for analyzing the stresses occurring in conventional rigid struc- tures made from steel or concrete. As a consequence the theory is formed

around the concept that the shell geometry remains essentially fixed

during the loading process. While this is a valid assumption for the kinds

of structures just mentioned, it is not valid for pneumatic tires since their geometry undergoes substantial changes during loading. Some measure

geometry undergoes substantial changes during loading, some measure

of the magnitude of the displacements involved in statically deflecting

a 10.00-20 radial ply truck tire is given by Frank and Hoff"erberth [50]. In the inflated but otherwise unloaded condition the meridional radius

of curvature r^p varies between 100 and 200 mm. depending on location. The total thickness of the tire at the crown is 32 mm. of which 13 mm.

is tread thickness; the sidewall thickness is 11 mm. At rated load the

deflection at the crown is 36 mm. and the section width increases by

15 mm. It is apparent that these displacements are appreciable compared

to the tire thickness and the initial values of the meridional radius of curvature. Thus, if conventional sheU theory is to be applicable to the

deflected tire problem, it can only be used as an approximation for those
METHODS FOR ANALYZING THE TIRE IN SERVICE 427

cases where the loading is such that relatively small displacements are produced.

It should also be noted that in shell structures made from isotropic

materials, a sound experimental basis exists for the Kirchhoflf assumption which states that a perpendicular edge of the undeformed shell element remains perpendicular to the shell surface in the deformed

state, i.e., plane sections remain plane. This assumption is usually

associated with the requirement that the thickness of the shell be small

compared to the principal radii of curvature. For the cord-rubber materials used in tire construction, there is at the present time no experimental evidence that the Kirchhoff assumption is justified. In fact, some

evidence exists for glass reinforced plastic structures that plane sections

do not remain plane under load [62]. This means that until further

information is available, it is not possible to definitely relate bending

strains to displacements, which is a crucial step in the analysis of any

structure, without danger of error. Thus, the adaptation of conventional

shell theory may represent yet a further approximation to the real situa- tion existing in loaded tires. A further difficulty peculiar to pneumatic tires is associated with their

toroidal geometry. It has been mentioned earlier in this chapter that a

singularity occurs in the displacement at the sidewall, using linear

membrane theory, which can only be removed at the expense of considerable mathematical complexity [11-17]. Thus, even a small deformation analysis of the tire treated as a toroidal shell will involve this compHcation.

Up to this point we have only discussed geometric factors which cause

difficulties in the stress analysis of the tire in operation. These factors appear as nonlinearities in the strain-displacement equations, with the

requirement that the equilibrium equations be written for an element of

the tire in the deformed state. There will also be complications in the

stress-strain relations because of the anisotropy of the cord-rubber

structure. Even if one assumes that stress and strain are linearly related

through an orthotropic law, the behavior of typical materials used in

tire construction is different in tension than in compression [63]. Also, it would be desirable to reflect rate and temperature effects in the stressstrain law by using a viscoelastic constitutive relation. Thus, in addition

I to geometric nonlinearities in the shell equations, an accurate formula-

( tion of the problem will also contain complex stress-strain relations if the

( theory is to be applicable to tires. The usual procedure in structural mechanics is to combine the equilibrium equations, the strain-displacement equations, and the stress-strain relations to obtain a differential equation or system of such equations whose solution is obtained subject

; to certain boundary conditions. Frank and Hofferberth [50] note that this

] approach, when applied to anisotropic toroidal shells, has never met with

success in the field of tire mechanics.

Ames and Lauterbach [64] have circumvented all of the difficulties

I just mentioned by assuming that a deflected tire can be adequately

described by a membrane state of stress. They avoided the formidable

problem of predicting the deflected tire contour at a given load by meas-
428 TIRE STRESS AND DEFORMATION

uring the loaded tire shape directly. Reasonable analytical approximations were made for the pressure distribution in the contact zone, based

on experimental measurements. Then they solved the differential equations applicable to shells of revolution on an analog computer to predict

the stress resultants and the cord forces in a four-ply nylon tire. Since

a deflected tire is not a surface of revolution, a more sound approach

would have been to use the differential equations applicable to shells of arbitrary shape subjected to the same loads (inflation pressure and

static deflection). Also, their calculated cord tensions did not show ply-toply cord force variations which, from experimental measurements

[65], are known to be appreciable for this problem.

Gregory, et al. [66], have found an approximate solution for a tire loaded by internal pressure and subjected to a normal load over a small

portion of its outer surface on the basis of differential equations set up

by Biderman for a cord-rubber cylinder. Equations in closed form were

obtained for the tire deflection, the pressure in the contact zone, and the

tension in the cords in terms of inflation pressure and normal load. It was shown analytically that the maximum cord load occurs at the crown

of the tire in the center of the footprint; this conclusion is not in agree- ment with experimental cord force measurements [65]. In those cases where the primary interest is in the tire displacement rather than the stresses, using energy methods to obtain solutions can be more successful than attempting to directly solve the differential equations governing the displacements. To use the energy principles,

approximate functions are assumed for the displacement patterns which

satisfy either the boundary conditions or the differential equations.

These functions are then quantitatively determined by using techniques

such as those of Ritz, Galerkin, or Trefftz [67]. One of the difficulties with the energy methods is that stresses are usually not calculated as accurately as displacements since the stresses depend on the derivatives

of the displacements. This means that if the energy function for a pneumatic tire can be written, this principle can be used to determine the

displacement characteristics of the tire under the action of external loads,

and from these displacement characteristics, the stresses may be cal- culated. Frank and Hofferberth [50] give an outline of this procedure

for the case of small displacements for shells of revolution. These authors

are also of the opinion that if answers to problems associated with the

tire in service are ever to be obtained in a closed mathematical form

useful for further applications, it will only be through the use of the

energy methods. Knowles and Reissner [70] have obtained an energy

expression for thin elastic shells using rigorous geometrical relations. Their energy function contains certain terms not normally used in shell theory and, as a consequence, it should be more accurate for those cases where displacements are highly sensitive to local factors, as in a tire rolling under load. Clark, et al. [68], have attempted to apply the

energy expression of Knowles and Reissner to the statically deflected

tire problem but their solution is valid, at most, only for small displace- ments. The contributions of investigators in the Soviet Union who have

used these methods have been reviewed by Biderman and Bukhin [69].
METHODS FOR ANALYZING THE TIRE IN SERVICE 429

At the present time, these energy principles have only been successfully

appUed to axisymmetrically loaded tire problems, and these are the

problems that are more easily solved using less involved techniques

such as the direct solution of differential equations. However, energy

methods do have the advantage of allowing computation of tire shape as

a function of inflation pressure, which cannot be accounted for in eq (4.7). The shell equations governing the problem of the tire rolling under

load will be exceedingly complex, and based on the preceding comments

there is some doubt as to whether such equations can be accurately

formulated and solved. For this reason, it appears that direct numerical

procedures based on more elementary principles might meet with greater success in attacking this problem. Among the more promising of these

techniques is the finite element method [71] in which a continuous struc- ture such as a tire is replaced by a finite number of elements inter- connected at a finite number of nodal points.^ This type of idealization

is inherent in the conventional analysis of frames and trusses. Loads

acting on the tire would be replaced by statically equivalent concentrated

forces acting at the nodal points of the finite element system. Displacements are related to loads using matrix equations in such a way that large

numbers of elements are joined together. Digital computers capable of

solving thousands of algebraic equations simultaneously are required to implement the finite element method effectively. While no definitive results have yet been published in the field of tire mechanics, the related

problem of the elastic contact of a hollow sphere with a flat plate has been

solved using this method [72]. A further aid in the study of axisymmetrically loaded tires is the ease

with which the applied loads can be mathematically represented for the

problems of practical importance, such as inflation pressure and centrifugal force. These loads are smoothly distributed over the tire and

cause no abrupt changes in the toroidal geometry. For the tire in service, one has to consider radial, lateral, and circumferential forces and

moments distributed over the tire which, besides being difficult to represent mathematically, cause significant changes in the geometry

of a rolling tire, especially in the contact zone. In fact, at the present

time there are no known methods available for predicting the displacements which occur at all points in a loaded tire. Analytical investigations

have been mainly concerned with predicting tire deflections at one

point — such as the crown — as a function of load [73]. From the previous

discussion of the inflated but otherwise unloaded tire, we know that an

accurate description of the entire geometry of the tire is needed in order to calculate the stresses. Additionally, most of our knowledge

of the performance characteristics of tires has been experimentally ob- tained in a form suitable for chassis engineers concerned with suspension

design who treat the tire as a "spring" which connects the vehicle to the

road [74]. Thus, while it is experimentally straightforward and useful

to measure such quantities as the lateral force on a tire in a cornering

The finite element method is also referred to as the extended Ritz method.
430 TIRE STRESS AND DEFORMATION

maneuver, tire engineers need to know for stress calculations the actual

distribution of this lateral force and not the integrated result of this distribution at the wheel axle. This force distribution, either experimentally measured or theoretically predicted, would then be used as one

of the inputs into the equations which govern the stress state of the

loaded tire. In order to study the tire in operation, stress analysts at the present

time will have to be satisfied with investigating either highly ideal- ized mathematical models of the whole tire or isolated components

of part of the tire. For example, analytical investigations of anisotropic

cylinders have established the fact that the plies should be layered in an

order different from that usually employed in tire building [75]. That is, to have a symmetrical four-ply construction, the cords should be oriented

in the order ±j8, T^^^ rather than the usual ±j8, ±/3. If these concepts

hold true for toroidal shells, such ply arrangements would inhibit coupling

between bending and membrane effects in the cord reinforced tire structure. Interestingly, these studies show that all two-ply structures are unsymmetrical. The stabilizer plies of belted tires and the beads of all tires are examples of components that can be reasonably studied in- dependently of the rest of the tire structure. Thus, for radial tires, the

behavior of a steel belt in a cornering maneuver [77] and the shear stress distribution between the carcass and the belt [78] have been studied

using the classical equations of infinitesimal elasticity.

4.3. Experimental Techniques

Since theoretical methods are not sufficiently developed to allow tire engineers to predict the stresses generated in the cord, rubber, and steel components of a tire under service conditions, a great deal of effort has been expended developing experimental procedures for this purpose.

These stress and strain measurements are important since they may sug- gest changes in tire design which will make more efficient use of construction materials and/or improve performance, especially with regard to

tire cord fatigue phenomena. Additionally, experimental data can be

used to validate analytical predictions in those cases where theory exists. Finally, experiments are often valuable in that they provide guidance in the development of an accurate mathematical model governing the tire loaded in some particular manner, i.e., experiments are useful in for- mulating theories. In experimental mechanics, there are two broad classifications of methods for making strain measurements in stressed bodies, whether

such bodies be tires or other load bearing structures. These two methods

are commonly referred to as "point" and "whole-field" techniques [79]. In point methods, a strain gage (which could be a mechanical, optical, or electrical device) is used to measure the average strain in a particular

direction over some given (usually small) gage length. It is apparent that

Woods [76] refers to this cord lay-up pattern as "paired" ply construction.
EXPERIMENTAL TECHNIQUES 431

in many studies, large numbers of strain gages must be used to map a region of interest since such devices function only at a point.

Whole field methods yield data at many points simultaneously. They

are ideally suited for locating regions where large strain gradients or

stress concentrations exist. In the experimental stress analyses of tires,

whole field methods have not been as commonly used as point methods.

The two earHest published articles covering the subject of tire stress analysis, Schippel's work [6] of 1923 and Hencky's work [80] of 1935,

make no mention of experimental techniques tor measuring strains m

tires. In the period 1959-1962, three survey articles appeared which

discussed many of the experimental methods known at that time [81-83].

Especially interesting is the work of Kern [81] and of Barson and Gough

[83] who point out the Hmitations of several of the techniques especially

developed for use with tires. Methods for measuring strain which are applicable to pneumatic

tires are not particularly different, in principle, from those methods

developed in other areas of experimental stress analysis [84]. However,

it has been necessary to design special transducing systems for tire engineering, since service strains occur in cord and rubber which are

at least an order of magnitude greater than those which occur in the

usual structural materials. In addition, the low elastic modulus of cord

and rubber, compared to metals, causes many conventional strain measuring instruments to locally stiffen the area being measured due to their

inherent mechanical rigidity. This stiffening causes extremely large meas- urement errors. For this reason special transducers with low mechanical

impedance have been devised to be elasticaUy compatible with cord

and/or rubber.

Before discussing the more recent of these specialized techniques, we note that some relatively simple point methods of strain measure- ment have been used in the past. One of the easiest, but not very accurate or rapid methods for making strain measurements on tirecord is to remove rubber locally from the tire in the region of interest so that bare cord is exposed. By putting "bench marks" on the cord, strains can be calculated from the measured increase or decrease of the dis- tance between bench marks due to an applied load. Furnas [85] is known

to have used this method in 1933 to measure outer ply cord strains in the sidewall of an inflated and statically deflated tire using bench marks

one inch apart. Loughborough, et al. [86], employed this same tech- nique in 1950 to measure outer ply cord strains in the sidewall region

of tires of various sizes. Both studies established the fact that in in- flated and deflected bias tires, cords in the contact zone have their initially imposed inflation tension reduced all along their length.

An ingenious scheme was employed by Buckwalter [87] in 1932 to determine the strains in the innermost ply of an inflated and deflected

tire. Buckwalter fastened small knots along the cords at equal intervals

(0.4 in. bench marks) to form a repeated diamond pattern completely

covering a section of the tire about 18 inches long. He then inserted an

inner tube provided with an uncured section which had been swelled

with carbon bisulfide and fiUed the uncured section with portland
432 TIRE STRESS AND DEFORMATION

cement. The tire was then inflated, the uncured section took the impression of the knots and was cured by the heat of setting of the cement.

This operation was then repeated with the tire inflated and deflected. The rubber covered cement casts obtained from this procedure pro- vided accurate maps of the diamond pattern of knots and made possible

fairly precise measurements of the cord length changes which took

place due to the deflection of the inflated tire. Buckwalter was mainly

interested in relating his strain measurements to tirecord fatigue phe- nomena and showed that cords are weakest along their length at the

location where maximum compression occurs. Lauterbach and Ames [41] measured the cord strain arising from the

vulcanization and expansion process in an uninflated tire by laying one inch strips of self-adhesive tape on the inside of the cured tire from

bead to bead. The tape, when pulled away from the tire, had imprinted on it a replica of the first ply cord network. This replica, in conjunction

with a mold drawing, permitted cord length changes arising from cure and expansion to be calculated. For a four-ply nylon tire, it was established that cord elongation occurs in the crown region and cord compression occurs in the bead region. Of course, the sign and magnitude of such

cord strains will be greatly influenced by the parameters of the tire on the

building drum relative to the mold geometry, as well as by the thermal

shrinkage characteristics of the cord at the tire curing temperature.

Several special transducers have been successfully developed and

extensively used by many European investigators for measuring surface

rubber strains in tires. Biderman, et al. [88] , reported data in 1957

obtained from a transducer which is a modification of the well-known

clip gage used for measuring the Poisson contraction during the tensile

testing of metallic specimens [89]. It is constructed by bonding electric resistance strain gages to the upper and lower sides of a piece of

channel shaped spring steel. Legs with pins extend from the backbone

of the gage and small pieces of vulcanized rubber are pushed on these

pins, so that a definite spacing from the tire surface is achieved (see

fig. 4.13). This device has good temperature compensation characteristics and because of the mechanical attenuation of the signal it can

STEEL CLIP

JJ_^ RUBBER WASHER^^^^^^ I I . T^PiN

Figure 4.13. Clip gage for measuring large deformations [88].

STRAIN GAGES
EXPERIMENTAL TECHNIQUES 433

Figure 4.14. Rubber wire transducer for measuring large deformations [94].

measure strain magnitudes of 20-30 percent, or larger, depending on

the length of the legs. Kern [81, 90-92] has used this transducer, as well as other types, to obtain strain measurements radially, circumferentially, and in the cord direction on the inside and outside rubber

surface of automobile tires. He was the first investigator to study the

effect of camber and cornering on tire strains and to map the principal

strain trajectories in the contact zone of a deflected tire. Pugin [93]

has used this clip gage to study the effect of crown angle on the side- wall deformation of bias tires rolling under load. Another special transducer developed for measuring rubber strains was first described by Biderman and Pugin [119] in 1958 and later by Biderman [94] in 1959 and is shown in figure 4.14. It consists of a prestressed rubber thread with a diameter of 1.5 mm. onto which a coil of fine constantan wire has been wound with a small helical pitch.

Compression or elongation of the rubber thread brings about an increase

or decrease in the stress in the wire due to an increase or decrease in the diameter of the thread. This stressing changes the electrical resist- ance of the wire. Prestress of the rubber thread is necessary in order

that the wire remain under mechanical load when the thread is extended

to its maximum elongation. In order to make compressive measurements, the thread must be supported over its entire length on some

sort of base to avoid buckling. For best results the rubber thread should

have a low stiffness and small hysteresis. Calibration and tempreature

compensation of this gage are difficult. Pugin [82] , however, recommends

this transducer for measuring deformations under dynamic conditions

in preference to the clip gage previously described. Such rubber-wire

gages have been used to measure the circumferential and radial inflation strains in the proximity of artificial cuts introduced into the sidewall

of agricultural tires [95] and to measure the sidewall strains developed

in radial tires rolling on a drum [96]

Two published studies have dealt with the determination of internal

cord and rubber strains in statically loaded tires using X-ray photography.

Loughborough, et al. [86], used fine steel wires wrapped tightly around

the cords as markers. Two markers were put on the same cord about

one-half inch apart and the change in length between markers due to
434 TIRE STRESS AND DEFORMATION

inflation and deflection loads gave the strain. Weickert [43] treated one

strand of a twisted cord with a metal salt and used the metal-shadowed

cord as the marker. He also used one-sixteenth inch diameter steel balls as markers to detect carcass rubber strains between the second and third

ply of a four-ply nylon tire. X-ray photography can also be used to measure the change in crown angle in the contact zone due to tire deflection

using metal-shadowed cord [97].

Liquid metal gages have been developed to measure the large strains

which occur in rubber. These devices consist of a small rubber tube or a

cylindrical cavity in a rubber block filled with mercury. Their principle

of operation is similar to the bonded electric resistance strain gage, i.e., a change in resistance due to change in geometry is a measure of strain. Hurry and Woolley [98] first described this type of strain gage in 1953

but presented no data from actual tire testing. Gregory, et al. [66],

measured inflation strains and some dynamic strains at various loca- tions on the inside and outside surface of an aircraft tire using mercury filled capillaries as the sensing elements. Similar devices have

been developed to measure strains in human tissue [99].

Strains in the metallic bead rings of tires have been measured by Alek- seev [100] and Pugin [101] using resistance wire strain gages. For these

measurements a solid steel bead of appropriate diameter replaced the

many strands of small diameter bead wire normally used in tire construction. Bead tension measurements were reported for various tire loadings (inflation, rotation, and rolling under load). Resistance foil strain gages have been applied to one wire in the bead bundle of an air- craft tire following the removal of a small portion of rubber and cord

[66]. Strains due to wheel mounting, bead seating, inflation, and static vertical load are discussed. In statically loaded truck tires with steel cords, Forster [102] used resistance wire strain gages to measure

the average strain across four adjacent outer ply cords at various loca- tions between the bead and the shoulder.

Kern [103] , using a method not described, compared the meridional

and circumferential strain magnitudes developed on the inner carcass

surface of a tire as a result of loading on a plane and a drum, and showed

that larger strains are produced when a tire is deflected against the

curved surface. Biderman [28, 104] has shown that for a 7.50-16 tire the meridional surface rubber strain pattern 50 mm. from the crown

is considerably different in shape at 150 km/hr. than at 50 or 100 km/hr.

The unusual high speed deformation pattern occurred because the tire was operating above its critical velocity with traveling waves present.

Miniature force transducers using resistance foil strain gages have

been developed by Clark and Dodge[105], Walter [106], and Patterson

[107] which permit the direct measurement of the cord loads in a

tire under operating conditions. These devices are placed in series with

the cord and are embedded in the tire during building. In service, they

provide a reproducible and easily monitored electrical signal which is an accurate analog of tirecord load. These transducers are much smaller

than either the clip gage, the rubber-wire gage, or the liquid metal gages

previously discussed.
EXPERIMENTAL TECHNIQUES

METALLIC TUBE

Figure 4.15. Cordforce transducer used by Clark and Dodge [105, 108].

Figure 4.16. Cord force transducer used by Walter and Hall [65] {lead wires and strain gages not shown; dimensions in inches). The force transducer used by Clark and Dodge is shown in figure

4.15; it is a 0.375 inch long thin walled beryllium-copper tube with

tirecord bonded at each end. Extensive cord force measurements

made at the crown, shoulder, and sidewall in the innermost ply of two- and four-ply nylon tires at different loads and pressures have been

reported using this device [108].

The basic geometry of one of two types of load transducers used by

Walter and Hall [65] is shown in figure 4.16; it is a 0.375 inch long aluminum alloy billet of rectangular cross section which averages force between two adjacent cords in a tire. This kind of transducer was used to measure cord forces in all plies of bias and belted bias tires in straight

ahead rolling and cornering; the effect on the cord force pattern of wheel load, inflation pressure, obstacle impact, tire speed, rim width,

and tire-road interface was also studied. Significant ply-to-ply cord

force variations were detected, and the cord force patterns observed in the first ply as the transducer passed through the footprint were nearly

a mirror image of those observed in the second ply at the same location. The cord tension developed at various locations in an inflated and

rotating, but otherwise unloaded, two-ply polyester tire was measured

with this device up to angular velocities of 1600 rpm — about 120 mph

[38]. Patterson [107] used a force transducer 0.460 inch long of rela- tively complex geometry to measure the cord loads which occur during
436 TIRE STRESS AND DEFORMATION

shaping, cure, and postinflation of a 7.50-14 two-ply tire as shown

in figure 4.17. These are the only published data treating this subject.

Some measure of the usefulness of these miniature force transducers

is afforded in the investigation of cord loads in the ply turn-up region,

i.e., near the bead. In this area, changes in tire stiffness and the closeness

of the rim render theoretical stress analysis difficult if not impossible.

Even surface rubber strains on the inside of the tire would be difficult to obtain in this otherwise inaccessible region. In the turn-up area, design

variations exist among manufacturers because, for tires with two body

plies, three different methods exist for wrapping fabric around the beads:

a 2-0 tie-in with a step-in construction, a 2-0 tie-in with a step-out

construction, and a 1-1 tie-in with a bead reinforce construction. These

different constructions are shown in figure 4.18. Cord force data [109]

.460

.195 .200

.065 1-,

T.011

037

.010

.168- .254

.460

Figure 4.17. Cord force transducer used by Patterson [107] {lead wires and strain gages

not shown; dimensions in inches).

.TRANSDUCERS TRANSDUCERS 'TRANSDUCERS

STEP-IN

"2-0"Tie-ln STEP-OUT

"2-0"Tie-ln BEAD REINFORCE

"I- 1" Tie-In Figure 4.18. The three constructions for wrappingfabric around the bead of two-ply tires.
EXPERIMENTAL TECHNIQUES 437

obtained in the middle of the step and at the edge of the reinforce are shown in figure 4.19 for otherwise identically constructed G78-15 belted

bias tires with two polyester body plies and two fiberglass belts at rated

load (1380 lb.) and pressure (24 psi) in straight ahead rolling at low

speed (2 mph). In each case, the magnitude of the initially imposed

inflation tension and the peak-to-peak cord force values are consistent

with the type of turn-up design.

All of the experimental techniques discussed so far have been point

methods. We conclude this chapter with a discussion of whole field methods, i.e., techniques which simultaneously give the stress, strain, or displacement field over some region of interest. There are four

methods of whole field stress analysis that have been used or are potentially useful for measuring strains on the surface of a tire: surface grids,

photoelasticity, moire, and holography. The biggest problem concerning

the efficient use of these whole field methods is the difficulty in quantitatively interpretating data which occur simultaneously over a doublycurved surface. This difficulty no doubt accounts for the scarcity of

literature on this subject which is applicable to tires. Strains are measured with surface grids by placing reference lines on. say. the sidewall of a tire, measuring the distance between fines before and after loading, and computing the strain as the change in length divided by the original length [110]. If the reference lines are

in the form of a continuous rectangular grid pattern, sufficient infor- mation becomes available to determine the principalt strain magnitudes

and directions at every point. That is, the strain can be measured in

Figure 4.19. Cord forces measued at rated load and pressure in each ply in the turn-up

region for G78-15 belted-bias tires [109] {see fig. 4.18 for transducer locations).
438 TIRE STRESS AND DEFORMATION

three directions by noting the change in length of the sides of the rec- tangular grid and the change in length of a diagonal; then the principal

strains can be calculated from the usual rosette equation for strain at a

point. In principle, this method is just a two dimensional extension

of the technique used by Furnas [85] and Loughborough, et al. [86] , to measure one dimensional cord strain. Barson and Gough [83] state that a grid method was their most successful technique for determining

surface strains on tires. The grids were drawn on the rubber using white

poster paint and were transferred to a flat glass surface for measurement

using transparent self-adhesive tape. They report data obtained from

statically deflected 5.20-13 four-ply tires. Strains are measured photoelastically by detecting changes in the

index of refraction of light passing through a birefringent material.

The principle is based on the fact that polarized light, passing through

a transparent medium under stress, will split into two polarized beams

which travel in the planes of the principal strains. These beams have

different velocities, and the resulting retardation is a measure of strain

in the body. A photoelastic model is analyzed by use of an optical ap- paratus called a polariscope. With this device it is possible to determine

the differences in the two principal strains from the isochromatics and

the directions of the principal strains from the isoclinics. The specialized techniques of photoelasticity and the solutions to many plane two

dimensional problems of the theory of elasticity are discussed in the

treatise of Frocht [111]. Angioletti, et al. [112], have studied rubber

birefringence and have analyzed the stresses in various rubber products

photoelastically. Oppel [114] investigated the strains in the tread rubber

of tires using this method. Recently, low modulus photoelastic coatings

and adhesives which eliminate the need for models have been developed.

These are suitable for use in the sidewall region [113] though no data

obtained from tires have been reported in the literature. The word "moire" is a French word that describes a silk screen

effect and which is now used to denote an optical phenomenon which

can be employed for surface strain measurement. The moire fringe

method of experimental stress analysis is used to measure the relative

displacements on a specimen by the mechanical interference of closely

spaced lines. This technique has been reviewed by Theocaris [115]

and is also referred to as mechanical interferometry or the photoscreen

method. To use this technique a grid must be applied to the area of

interest and a fringe pattern produced by straining must be measured.

The fringes represent contours of equal displacement, and strains are calculated from the measured displacements through the use of the

strain-displacement equations. The resulting measurement does not depend on a change in resistance (as with an electrical resistance

strain gage) or index of refraction (as with photoelasticity). Thus, many

of the problems inherent in the usual transducing systems (cement

creep, stability, zero drift, etc.) can be avoided. Potter [116] has used

the moire fringe method to measure strains in the range 1-10 percent

in a pressurized neoprene rubber cylinder using two orthogonal sets of parallel Hnes with a spacing of 200 lines per inch. He discussed the
PRINCIPAL NOTATION 439

application of this technique to tires using long, thin grids with the

long axis of the grid placed radially on the tire sidewall.

A technique that has a great deal in common with moire is the method

of holographic interferometry based on wavefront reconstruction [117]. Wavefront reconstruction is a method for recording specific data about

a three dimensional object in a medium such as photographic film, and

then reconstructing a three-dimensional image of the object from this recording. Holographic fringes, like the moire fringes, result when

load is applied to a body, and they also represent a locus of points

of constant displacement. Its advantages compared to other whole-field

methods are increased sensitivity (displacements as small as 10 microinches can be measured) and elimination of the need for mechanical

attachments of any sort to the tire. At the present time holography is much more expensive than the other methods of experimental stress analysis partly because it requires a coherent fight source such as a laser. This technique may have more potential as a nondestructive testing

tool than as a quantitative analysis device since non-uniformities in the

surface fringe patterns can be qualitatively interpreted as regions of

internal stress concentration in the tire [118]

Recent developments in the entire field of experimental stress analysis are discussed in current issues and volumes of such publications as Experimental Mechanics and Proceedings of the Society for Experimental Stress Analysis.

4.4. Principal Notation

B = bead tension.

c = wave velocity.

m = tire mass per unit surface area. n — number of plies in tire multiplied by the number of cords per

unit length in a direction perpendicular to the cords. N = total number of cords in tire. N>p = meridional stress resultant or force per unit of circumferential

length.

Ne = circumferential stress resultant or force per unit of meridional

length.

p = inflation pressure.

r = radial coordinate (distance from axis of revolution to an arbitrary

point on shell meridian).

Tft = radial distance from axis of revolution to "rim point" (see fig. 3).

Tc = radial distance from axis of revolution to crown.

fu; = radial distance from axis of revolution to widest part of shell meridian.

r<p = meridional radius of curvature.

re = circumferential radius of curvature. T — cord tension.

F(;r= critical rolling velocity of tire. z = axial coordinate (half-width at an arbitrary point on shell meridian).

2ft = axial distance to "rim point" (see fig. 3).
440 TIRE STRESS AND DEFORMATION

z = first derivative of z with respect to r. z" = second derivative of z with respect to r. P = angle between the cord and a meridian plane at an arbitrary point on the tire.

jSft = angle between the cord and a meridian plane at r=ri) (equal to drum bias angle).

jSc = angle between the cord and a meridian plane at the crown.

T = shear stress at cord-rubber interface.

0) = angular velocity.

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Desidlei, L. V., Khrapunov, L. G. and Rekitar, M. I., Carcass strength and its effect on tyre life, Soviet Rubber Technology 21 (6). 22-24 (1962). Pugin, V. A., Stresses in the bead rings of tyres, Soviet Rubber Technology 21

(10), 8-12 (1962). Wood, J. O., Pull-through adhesion test. Transactions Institution of the Rubber

Industry 32, 1-18 (1956). Slyudikov, L. D.. Cross-sectional profile of tyres having transverse carcass cords, Soviet Rubber Technology 21 (8), 24-28 (1962); Gummi, Asbest, Kunststoffe 16,620-623 (1963). Biderman, V. L., Bukhin, B. L., Pugin, V. A. and Levkovskaya, E. Y., Calculation of

initial forces in cords of the breaker strip of radial tyres, Soviet Rubber Technology 24 (3), 29-30 (1965). Frank, F., Theorie und Berechnung der Statischen Krafte und der Querschnittsform von Kreuzlagen-und Gurtelreifen, Paper presented at the Rubber Conference of the DKG, Berlin, 1968.
442 TIRE STRESS AND DEFORMATION

[50J Frank, F. and Hofferberth, W., Mechanics of the pneumatic tire. Rubber Chemistry and Technology 40, 271-322 (1967).

[51] Bohm, F., Zur Statik und Dynamik des Gurtelreifens, Automobiltechnische

Zeitschrift 69, 255-261 (1967).

[52] Biderman, V. L., Tyres with transverse carcass cords, Soviet Rubber Technology

19 (7), 10-12 (1960).

[53] Biderman, V. L., et al., Radial Cord Tires (Russian), Automobile Tires (Construction,

Design, Testing and Usage), pp. 238-253 (State Scientific and Technical Publishing House, Moscow, 1963). [54] Rivlin, R. S., The deformation of a membrane formed by inextensible cords. Archive

for Rational Mechanics and Analysis 2, 447-476 (1959). [55] Gehman, S. D., Calculated deflected contours and load-deflection curves for tires. Rubber Chemistry and Technology 41 , 977-987 ( 1968)

.

[56] Bukhin, B. L., Calculation of Stress and Deformation Appearing in Pneumatic

Tires on Rotation (Russian), Raschet na Prochnost No. 6, pp. 56-65 (Moscow,

1960). [57] Wutzler, G., Die Deformation des Luftreifens unter Rotationssymmetrischer

Belastung, Plaste und Kautschuk 14,679-683 (1967).

[58] Irodov, V. A., Strength calculations on the bead rings of pneumatic tyres, Soviet Rubber Technology 20 (2), 17-18 (1961). [59] Turner, D. M., Wave Phenomena in Tyres at High Speed, Proceedings Third Rubber Technology Conference (London), pp. 735-748 (Heflfer, Cambridge,

1954). [60] Gardner, E. R. and Worswick, T., Behavior of Tyres at High Speed, Transactions

Institution of the Rubber Industry 27, 127-146 (1951).

[61] Ames, W. F., Wave Phenomena in Tires, Contract CST-435 with National Bureau

of Standards, University of Iowa; Rep. No. 1 — Dec. 31, 1967; Rep. No. 2 — July 15, 1968; Rep. No. 3 -Oct. 15, 1968. [62] Tarnopol'skii, Y. M. and Roze, A. V., Distortion of cross sections during deformation

of oriented glass-reinforced plastics. Polymer Mechanics 1,69-73 (1965).

[63] Clark, S. K., A review of cord-rubber elastic characteristics. Rubber Chemistry and Technology 37, 1365-1390 (1964).

[64] Ames, W. F. and Lauterbach, H. G., Stresses in Deflected Tires, Proceeding

International Rubber Conference (Washington), pp. 50-56 (American Chemical

Society, 1959). [65] Walter, J. D. and Hall, G. L., Cord load characteristics in bias and belted-bias

tires, SAE Paper No. 690522 (1969). [66] Gregory, R. K., Rastrelli, L. U. and Minor, J. E., Tire Structural Design Improvement,

Contract AF33(657)- 12861 with Air Force Systems Command, Southwest Re- search Institute, 1968 (ASD-TR-68-12).

[67] Collatz, L., The Numerical Treatment of Differential Equations, (Springer, Berlin,

1959). [68] Clark, S. K., Tielking, J. T., Bobbins, H. R. and Schreyer, H., Analysis of a pneumatic tire under load. University of Michigan ORA Report 02957-17-T (1963). [69] Biderman, V. L. and Bukhin, B. L., Methods of calculating stresses and deformations in the components of pneumatic tyres, Soviet Rubber Technology 20 (3), 15-19 (1961).

[70] Knowles, J. K. and Reissner, E., On stress-strain relations and strain energy expressions in the theory of thin elastic shells. Journal of AppHed Mechanics 27,

104-106, (1960).

[71] Zienkiewicz, O. C. and Cheung, Y. K., The Finite Element Method in Structural and Continuum Mechanics, (McGraw-Hill, London, 1967). [72] Rumbarger, J. H. and Herrick, R. C, Analysis of the Elastic Contact of a Hollow

Ball with a Flat Plate, Contract F33(615)-68-C-1203 with U.S. Air Force, FrankHn

Institute Research Laboratories, 1968 (AFFDL-TR-68-123) ; also published as ASME Paper No. 69-LubS-5, 1969. [73] Rotta; J., Zur Statik des Luftreifens, Ingenieur-Archiv 17, 129-141 (1949).

[74] Nordeen, D. L. and Cortese, A. D., Force and moment characteristics of rolling

tires, SAE Paper No. 713A, (1963). [75] Whitney, J. M. and Halpin, J. C, Analysis of laminated anisotropic tubes under combined loading. Journal of Composite Materials 2 , 360-367 ( 1968)

.

[76] Woods, E. C, Pneumatic Tyre Design, p. 45 (Heff'er, Cambridge, 1952).
REFERENCES 443

Longmore. D. K. and Goodall. J. R.. Elastic Behavior of Radial Tyre Breakers,

Proceedings International Rubber Conference (Brighton), pp. 389-404 (Universities Press. Belfast. 1967). Bohm, F.. Mechanics of the belted tire. University of Michigan ORA Translation No. 5. (1967): Ingeniur-Archiv 35, (1966). Dove. R. C. and Adams. P. H.. Experimental Stress Analysis and Motion Measurement, p. 18 (Merrill, Columbus. Ohio, 1964). Hencky. H.. Stresses in rubber tires. Mechanical Engineering 57, 149-153 (1935). Kern. W. F.. Strain measurements on tires by means of strain gauges. Revue

Generale du Caoutchouc 36, 1347-1365 (1959): Kautschuk und Gummi 13,

WT59-WT68(1960).

Pugin, V. A.. Electrical strain gages for measuring large deformations, Soviet Rubber Technology 19 (1), 23-26 (1960). Barson. C. W. and Cough. V. E.. Measurement of strain of materials of large extensibility. British Journal of Applied Physics 13,168-170 (1962). Hetenyi. M. (ed.). Handbook of Experimental Stress Analysis, (Wiley, New York,

1950). Furnas. A. R.. Firestone Tire & Rubber Co. Research Report, (1933) (unpublished).

Loughborough. D. L., Davies, J. M. and Monfore, G. E., The measurements of

strains in tires. Canadian Journal of Research 28 (F), 490-501 (1950); Revue

Generale du Caoutchouc 29, 712-716 ( 1952)

.

Buckwaher. H. M.. United States Rubber Co. Research Report, (1932) (unpubHshed): ASTM Committee D-13 Meeting, Providence, Rhode Island, (1933). Biderman. V. L.. Drozhzhin. P. K.. Pugin, V. A. and Shchaveleva, V. F., Experi- mental Investigation of Deformations in Elements of Pneumatic Tires (Russian), Transactions Tire Research Institute (NllShP) 3, pp. 5-15 (State Scientific and Technical Publishing House, Moscow, 1957). Perry. C. C. and Lissner, H. R., The Strain Gage Primer, p. 275 (McGraw-Hill,

New York. 1962). Kern. W. F., Uber die Cordbeanspruchung in Reifen, Faserforschung und Textil- technik 11,401-408 (1960). Kern. W. F., Uber Verformungsmessungen an Kraftfahrzeugreifen mittels spezieller Dehnungsgeber, Automobiltechnische Zeitschrift 63 , 33-41 ( 1961 )

.

Kern, W. F., Hafner, K. and Nippold, H., Uber die Auswertung von VerformungsMessungen an Reifen, Kautschuk und Gummi Kunststoffe 16, 619-627 (1963). Pugin, V. A.. Effect of the angle of the cord on its deformation in the area of the

tyre waU. Soviet Rubber Technology 22 (3), 21-24 (1963). Biderman. V. L.. Deformation of tyre elements during rolling. Revue Generale du Caoutchouc 36, 1366-1371 (1959). Dzhafarov. I. D.. Deformation in the damage zone of a tyre, Soviet Rubber Technology

26 (3). 38-39 (1967). Biderman. V. L., Pugin. V. A. and Fil'ko, G. S., Deformation and stresses in the cover rubber of the sidewall of a radial tyre, Soviet Rubber Technology 24

(7), 18-20 (1965). Hinden. H. B.. The unique characteristics of the radial wire tire, SAE Paper No.

278A.(1961). Hurry. J. A. and Woolley. R. P., A new high range strain gage. Rubber Age 73,

799-800 (1953). Gibbons. G. E.. Strandness, D. E. and Bell, J. W., Improvements in design of the mercury strain gauge plethysmograph, Surgery, Gynecology, & Obstetrics 115,679-682 (1963). Alekseev, P. I.. Stressing of the bead rings of a tyre under various conditions of deformation, Soviet Rubber Technology 19 (2), 18-22 (1960). Pugin, V. A., Stresses in the bead rings of tyres, Soviet Rubber Technology 21

(10), 8-12 (1962).

Forster. M. J., Firestone Tire & Rubber Co. Research Report, (1952) (unpublished). Kern. W. F.. Observations on the relation between laboratory and test stand measure- ments of tire treads and their behavior on the road. Rubber Chemistry and Technology 29, 806-828 (1956).

[104] Biderman. V. L.. Critical Speed of a Rotating Pneumatic Tire — Basic Experimental Data (Russian), Raschet na Prochnost No. 7, Part 3, pp. 324-349 (Moscow, 1961).
444 TIRE STRESS AND DEFORMATION

105] Clark, S. K. and Dodge, R. N., Development of a textile cord load transducer, ORA

Report 01193-1-T, (University of Michigan, 1968). 106] Walter, J. D., A Tirecord Tension Transducer, Textile Research Journal 39,

191-196 (1969).

107] Patterson, R. G., The Measurement of Cord Tensions in Tires, Rubber Chemistry and Technology 42 , 812-822 ( 1969)

.

108] Clark, S. K. and Dodge, R. N., A Load Transducer for Tire Cord, SAE Paper No.

690521,(1969).

109] Walter, J. D., Firestone Tire & Rubber Co. Research Report, (1969) (unpublished). 110] Parks, V. J., The Grid Method, Experimental Mechanics 9 (7), 27N-33N (1969). Ill] Frocht, M. M., Photoelasticity, (Wiley, New York, 1941 (Vol. I), 1948 (Vol. II)). 112] Angioletti, A., Eccher, S., Polvara, O. and Zerbini, V., Rubber birefringence and

photoelasticity. Rubber Chemistry and Technology 38, 1115-1163 (1965). 113] Anonymous, Modern photoelastic stress analysis. Bulletin B-1200-1, (Photolastic

Inc., Malvern. Pennsylvania).

114] Oppel, G., Konstruktionsprufung von Gummifederungen fur Motorfahrzeuge mit dem Einfreverfahren und mit Spannungsoptik, Automobiltechnische Zeitschrift 51,77-84 (1949). 115] Theocaris, P. S., Moire Fringes: A Powerful Measuring Device, Applied Mechanics

Reviews 1 5 , 333-339 ( 1962 )

.

116] Potter, J. M., Measurement of Strain on Curved Surfaces Using the Moire Method, M. S. Thesis, University of IHinois, (1968). 117] Haines, K. A. and Hildebrand, B. P., Surface-Deformation Measurement Using the Wavefront Reconstruction Technique, Applied Optics 5, 595-602 (1966). 118] Grant, R. M. and Brown, G. M., Holographic Nondestructive Testing (HNDT)

in the Automotive Industry, SAE Paper No. 690051, (1969). 119] Biderman, V. L. and Pugin, V. A., Wire Strain Gauge for Measurements of Large

Deformations (Russian), Zavodskaya Laboratoriya 24 (7) , 874-875 ( 1958)

.
CHAPTER 5

The Contact Between Tire and Roadway

Samuel K. Clark ^ Page 5.1. Introduction 446

5.2. Contact area 447

5.2.1. Experimental measurements 447

5.2.2. Theory 454

5.3. Slip between tire and roadway 463

5.4. Contact pressures 471

References 498

Department ofEngineering Mechanics, University of Micliigan, Ann Arbor, Michigan 48104. 445
446 THE CONTACT BETWEEN TIRE AND ROADWAY

5.1. Introduction

The purpose of this chapter is to examine both the experimental data

and the appropriate mathematical models for three characteristics of a

rolling elastic tire. These characteristics are:

a. The area of contact between the tire and road.

b. The slip or relative motion between the tire and road.

c. Normal and tangential contact stresses between tire and road.

Since the forces needed for vehicle support, guidance and maneuvers

all arise in the tire contact area, study of these characteristics and

application of the resulting theory should improve both the mechanical

properties of tires and the control of vehicles. These characteristics are

all influenced by the operating parameters of a tire, such as its inflation pressure, its rolling velocity, its path of rolling relative to its midplane,

its load as well as its size and shape. For this reason it is quite difficult to give a complete definitive description of all possible interacting factors,

and for the most part it will be necessary to rely on rather insufficient experimental evidence which merely indicates trends and magnitudes,

as well as some relatively simple theoretical ideas which from time to time may prove helpful in explaining the general characteristics of

observed phenomena.

In addition to the influences which are listed above, it is obviously

true that the characteristics of contact between a tire and roadway

depend as well on the particular type of roadway surface to be studied,

as well as on any possible contaminants between the tire and roadway.

For the most part one is interested in the influence of water in relatively

small depths, and its effect can be marked on contact processes,

particularly at high velocities. Roadway surfaces usually are constructed

to be relatively smooth compared to tire tread patterns, and there is

little if any information available in the literature on the influence of road roughness on contact characteristics.

The quantities of interest here are shown in the sketch of figure 5.1.

Figure 5.1. Total reaction from ground to tire expressed as the combined effect of two

forces, one normal and one tangential to the ground plane.
EXPERIMENTAL MEASUREMENTS OF CONTACT AREA 447 5.2. Contact Area

5.2.1. Experimental Measurements

The shape of the contact area between the tire and roadway depends

on the tire cross section shape and structure. For example, the contact area between an aircraft tire and a flat surface usually appears to be

nearly elliptical in shape. This is illustrated in figure 5.2. This is generally

the kind of shape associated with a tire which has little if any external

tread or shoulder region, but is primarily a toroidal carcass with small

additional tread rubber.

For an automotive tire a somewhat different set of relationships exists due to the fact that the usual construction involves the use of a relatively

heavy tread, particularly in the shoulder region. In this case any significant contact spreads over the entire width of the tire between shoulders

so that the contact area tends to have essentially straight parallel sides,

and the width of this contact area is nominally independent of tire deflection. A typical contact area is shown in figure 5.3.

AIRCRAFT TiRE

Figure 5.2. Contact patch of a typical aircraft tire.
448 THE CONTACT BETWEEN TIRE AND ROADWAY

8:00x14 Automotive Tire Four ply Rayon Bios Ply Construction

24 psi inflation 1.25" Deflection

1350 lbs load Figure 5.3. Typical contact area for an automotive tire.

C )

(1

<C •

N n

1 c • »

1 • - 13 1

1 ^

» I

t ' 1

1 • 8:00x14 BIAS F o 7:50x14 BIAS F A 5:90x15 BIAS F

LY

>LY A

n

'LY

4» 7:50x14 STEEL RADIAL

• 1 55x15 TEXTILE RADIAL

1 1 1

50

45

40

35

30

25

20

15

105

.25 .50 .75 TIRE

1.00 1.25 1.50 1.75 DEFLECTION - INCHES2.00

Figure 5.4. Gross contact area vs. tire deflection.

Data relating gross contact area to tire deflection is shown in figure 5.4

for a typical set of automotive tires of different size and manufacture.

It is seen from this that the relationship between tire deflection and gross contact area is nearly linear, accounting for some experimental scatter

in the data. Similar results have been reported by Hadekel [If previously.

' Figuies in brackets indicate literature references at the end of this chapter.
EXPERIMENTAL MEASUREMENTS OF CONTACT AREA 449

10

ALL DATA TAKEN AT 1" DEFLECTION

0 7:50x14 STEEL RADIAL

1 CONSTRUCTION

7:50x14 BIAS PLY CONSTRUCTION

0 10 20 30 40 50 PRESSURE (PSD

Figure 5.5. Plot of contact patch length vs. pressure for two automotive tires at fixed

deflection.

Experimental evidence indicates that tire deflection is the most

important variable governing the area of contact between the tire and

roadway. If inflation pressure and load are simultaneously varied so

as to maintain constant tire deflection, the contact area of the tire will remain eff"ectively constant. This conclusion was previously reached by

Michael [2] on the basis of aircraft tire data, but is also shown here in figure 5.5 from unpubUshed data [3]. This tends to lend credibihty to the

inextensible membrane or sheU bending models for contact phenomena.

There is not too much evidence available in the literature concerning

the role of curved surfaces in forming contact areas with pneumatic

tires. For example, the contact of a pneumatic tire with a cylindrical

surface is of considerable interest due to the large amount of indoor

tire testing carried out on cylindrical steel roadwheels. Some prehminary

data is available on this eff'ect from the work of Dodge [3] and this is shown in figure 5.6, where the variations of contact patch length with

surface radius are shown for two different automotive tires. The contact area of a slowly rolling tire may be slightly different from

that of a standing tire, but it is not clear that such differences would be

large. In some cases carcass deformation would be slightly different in standing as opposed to pure roUing due to the presence of friction be- tween the roadway and the wheel. Experimental data on this point has
450 THE CONTACT BETWEEN TIRE AND ROADWAY

i

2' ' ^ 1 ^ ^ ^ ^ ^ ^

0 .025 .050 .075 .100 .125 .150 .175

1 ^ ^ ^

^ CURVATURE (inches)-' J 72 48 24 12 6 RADIUS OF CURVATURE (inches)

1* IGURE 5.6. Flot of contact patch length vs. roadwheel curvature for two different tires and two different deflections.

been obtained by the U.S. Army Engineer Waterways Experiment

Station [4] for a single truck tire. This data is summarized in table 5.1. There is only limited data available on the influence of velocity on the area and shape of the contact region between a tire and roadway. Seitz [5] gives footprint data taken on a roadwheel. In figure 5.7 is shown the contact area for a textile belted tire at various speeds at constant load, while

similar data for a steel belted tire and a bias-ply tire is shown in figures 5.8 and 5.9 respectively. Figure 5.10 shows contact areas as a function of tread radius, while figure 5.11 shows the bias-ply tire contact area again

as a function of velocity. Figure 5.12 gives a similar presentation for a belted tire whose contact area is a function of velocity. These measure- ments were made on a cylindrical roadwheel, but still give a valuable

indication of the general influence of velocity on contact area. While
! EXPERIMENTAL MEASUREMENTS OF CONTACT AREA 451

the trends are not completely clear, it appears that increasing velocity

of travel generally tends to increase the contact area slightly, all data

being taken at constant load. It is known that increasing velocity mav

Table 5.1. Static and rolling contact patch areas and dimensions {slow rolling conditions) Test

CUIIU-ILIUII Inflation Load Deflection Contact area

psi pounds inches % length width area

Stationary 60 4500 1.56 Li .6 11/1 11.4 6.4 6/

Rolling 60 4500 1.56 17.3 11.4 6.6 67

Stationary 60 3000 1.12 IOC 9.9 6.0 5z

Rolling 60 3000 1.12 12.5 10.2 6.0 58

Stationary 60 1500 O.oO

r r D.D 4.6 on

Rolling 60 1500 .50 5.5 7.8 4.8 31

Stationary 30 4500 0 99 9/1 A 10.

u

Q 9 O.Z 1 9c: iZD

Rolling 30 4500 2.22 24.6 15.8 8.5 115

Stationary 30 3000 1.75 19.4 12.4 7.4 83

Rolling 30 3000 1.75 19.4 12.4 7.1 81

Stationary 30 1500 0.88 9.7 9.9 5.6 47

Rolling 30 1500 .88 9.7 9.2 5.3 43

Stationary 15 4500 5.00 55.4 20.0 9.3 188

Rolling 15 4500 5.00 55.4 20.2 10.5 161 Stationary 15 3000 2.75 30.4 16.1 8.2 130

Rolling 15 3000 2.75 30.4 15.5 8.2 115

Stationary 15 1500 1.31 13.1 10.5 6.6 63

Rolling 15 1500 1.31 13.1 11.0 6.2 63

cause the center of normal pressure to move slightly forward in the contact area, since the power loss characteristics of the rolling tire may

result from such a forward shift. This apparently is not accompanied by a geometric shift of the actual contact area, according to the data of Seitz. Finally, some photographic information is available on the nature of an

actual contact patch under rather slow cornering conditions. This was

obtained by photographing through a glass plate as the tire is run over

the plate. A still picture from such a movie is given in figure 5.13, showing

typical cornering distortion of the contact patch.
452 THE CONTACT BETWEEN TIRE AND ROADWAY

mm

80

7060

5040

30

20

100

1

1 r

-

1

1 1

/ 3

ll

\'

1 J...

1 —

V = ll9.8(km/h) V = 60.5(km/h) V = 30.0(km/h)

p. = 1.7 (kp/cm2)

(inflation) P= 360(kp)

(load)

LEADING EDGE TRAILING EDGE Figure 5.7. Contact area for a textile belted tire at various speeds, taken at constant load.

) — —

) —

) — 1 —

\\ r

!/ y ) —

\

) — \-t— >

) —

) —

7 V ) —

)— / /

1 H

\

/

V= 30(km/h)

V = 60{km/h)

V =l20(km/h)

p. = l.7(kp/cm2)

(inflation) P = 360(kp)

(load) 30 50

LEADING EDGE TRAILING EDGE Figure 5.8 Contact area for a steel belted tire at various speeds, at constant load.

—j— i-f

—1

/, If

i -

i ~\~

-i- —/-/- 4/

r _\ —\- - \\_ W 1 ir^i , 1

1 1 1 1 1 1

mm

80

70

60

50

40

30

20

10° -70 -50

LEADING EDGE •30 -10 10 30

V = 120.2 (km/h) V = 6I.3 (km/h) V = 30.2 (km/h)

Pi=l.7(kp/cm)'

(inflation) P=360(kp)

(load) 50 70 mm

TRAILING EDGE Figure 5.9. Contact area for a bias ply tire, with 37° crown cord angle, at various speeds,

at constant load.
EXPERIMENTAL MEASUREMENTS OF CONTACT AREA 453

P = 370(Kp)(load) mm. Pj =l.7(Kp/cm2)(jnflation)

Figure 5.10. Contact areas for a bias ply tire with 37° crown cord angle as a function of

tread radius, taken statically at equal loads.

V = 120.5 (km /hr) V = 60.7(km/hr) V= 30.5{km/hr)

p.= l.7( kp/cm*^)

-70 -50

LEADING EDGE 50 70 mm

TRAILING EDGE

(inflation) P=360(kp)

load)

Figure 5.11. Contact area of a normal tire as a function of velocity, at constant load.

TRAILING EDGE

LEADING EDGE

V = l20(km/h) V = 60(km/h)

p. = 2.5(kp/cm2)

( inflation) P = 60(kp)

20mm('o°d)

Figure 5.12. Contact area of a steel belted tire at high inflation pressure and extremely

light load, at constant vertical load.
454 THE CONTACT BETWEEN TIRE AND ROADWAY

5.2.2. Theory

No single unified theory exists for describing the area of contact of a

tire with a rigid roadway or test wheel. Most of the reasons for this lie in the complexity of the tire construction, since from the point of view of mathematical modeling of the tire its shell-like structure and rather soft serrated tread are difficult to represent analytically. Nevertheless,

several specialized theories are available for approximately describing

the dimensions of the contact area of the tire with a roadway or test wheel, and these are useful in explaining some of the phenomena which

are observed.

One type of problem which has been studied in some detail is that involving the contact of isotropic solid elastic bodies with geometrically

simple surfaces. This solution obviously has application in various

kinds of bearings and friction drive devices. For the case of small

deformations of solid bodies in contact with one another, the contact

Figure 5.13. Photograph of contact patch of a haded tire. {Courtesy B. F. Goodrich Company.)
THEORY 455

areas have been expressed in particularly useful form by Whittemore and

Petrenko [7], who conclude that

a. Contact area between an infinitely rigid plane and an elastic body of double curvature, with principal radii of curvature Rc and Rt, as shown in figure 5.14, is an ellipse.

b. Upon experiencing a deflection 6 against a rigid plane, as shown

in figure 5.15, the semi-major and semi-minor axis of the contact

ellipse are given by

§V2 d = (5.1) where A is given by

Rt Rc

(5.2)

Table 5.2.

6

0° 10° 20° 30° 35° 40° 45° 50° 55° a 00 6.612 3.778 2.731 2.397 2.136 1.926 1.754 1.611 0 0.319 0.408 0.493 0.530 0.567 0.604 0.641 0.678

K .851 1.220 1.453 1.550 1.637 1.709 1.772 1.828

6 60° 65° 70° 75° 80° 85° 90° a 1.486 1.378 1.284 1.202 1.128 1.061 1.00

/8 0.717 0.759 0.802 0.846 0.893 0.944 1.00 X 1.875 1.912 1.944 1.967 1.985 1.996 2.00

and a, (3 and k must be taken from table 5.2 in terms of the angle 6, this angle being defined by

From these results it is seen that the isotropic contact theory, often

known as the Hertz contact theory, predicts linear dimensions which are dependent upon the deflection to the exponent 0.5. Since the contact area

is an eUipse, it becomes linearly proportional to deflection in the form
456 THE CONTACT BETWEEN TIRE AND ROADWAY

A = TTcLc = 8 (5.4) where the factor in parentheses is a function of the relative values of

tire radius Rc and tread radius Rt. While this type of solution certainly does not represent an automobile

tire completely, it is interesting to note that another situation does

exist in practice where the contact area can also be shown to be linearly

dependent on deflection by quite a different form of analysis. This occurs

in an aircraft tire where the tread is relatively thin compared to automotive tires. Here the cross section of the tire is close to being circular,

with a radius of curvature approximately equal to half of the section

width w. In this case, it has been demonstrated by many experiments

that the contact area between tire and roadway is also approximately an ellipse. Here it is common to treat the tire as a carcass of zero bending

rigidity, and of membrane characteristics such that it can map itself onto the flat roadway by appropriate membrane compression of the

carcass. A side view of this assumed tire geometry is shown in figure 5.16,

Figure 5.14. Radii of curvature Rc and Rt of tire tread surface.

///////^m/W//////f///'Jl/'^'f//^/'/%0////

Figure 5.15. Idealized contact geometry between a torus and a plane.
THEORY 457 where the tire with original running band AOB is presumed to map

itself onto the plane CC by assuming the shape AO'B. The geometry of

this deformation is such that the right triangle DO'E is governed by the

relation

/?|=(R.-6)2+Q'. (5.5) where it is assumed that the original and final tire shape is the same

outside of the contact area. From this one may obtain the contact length

^=S'i'(Dc-Syi^ (5.6)

If one considers the section of the tire at right angles to that shown in figure 5.16, as illustrated in figure 5.17, then it is seen that an analogous

Figure 5.16. Tire contact symbols.

Figure 5.17. Cross section view of a tire in contact.
458 THE CONTACT BETWEEN TIRE AND ROADWAY

geometry governs the mapping of the tire in this cross-sectional dimension. Using the same type of analysis as previously, the width of the contact ellipse h is given by the expression

|=8V2 {w-byi\ (5.7)

Since the contact area is elliptical in shape, its value is given by

A = Tv^^=7rb\/(w-b){Dc-b)

(5.8) ~ 7Tb ^DcW

Thus, for an aircraft tire, one might also anticipate a linear, or nearly

linear, relationship of area with deflection if the membrane carcass theory is valid. Experiments with aircraft tires on this point have been

conducted by Michael [2] as quoted by Hadekel [1]. This data appears

to be extremely linear up to quite large deflections, and to be essentially

independent of inflation pressure, so that there appears to be some

physical basis for acceptance of the membrane theory for aircraft tires. Very similar relations have been obtained from the data of Smiley and

Horne [6], with slight modification in the numerical constants associated

with the linear relationship.

S/D^- 0.85 Ns/Dc-( S/Dc)^ 26x6.6 - I2PF^-2II -R23 -A 26x6.6-l2PR-3Zn -R23-B 40xl2-l4PR-3m -R24-A

40x 12 - 1 4PR-3Zn -R24-B 56xl6-24PR-3Zn-R22-A

56x16-24 PR-3ni-R22-B 57x20-16 PLY-I(56-INCH)-R53

I I I i I I I I I 0 .02 .04 .06 .08 .10 .12 .14 .16 VERTICAL-DEFLECTION PARAMETER, S/D^

Figure 5.18. Variation of footprint-length parameter with vertical deflection parameter

for several types I and VII tires (aircraft).
THEORY 459

i/Dc= 0.85^S/Dc-( S/Dr)^ ~

o 10x3^

I I I o 25x9 I E^pe''"T»ental Data A 27x 10 I ^""^^ Reference 45

V 28x7 J I I I .02 .04 .06 .08 .10 .12 .14 VERTICAL DIRECTION PARAMETER,8/Dc

Figure 5.19. Variation of footprint-length parameter with vertical deflection parameter

for several German aircraft tires.

For an automotive tire the presence of the shoulder region causes

the contact area to be close to a rectangle whose length varies with

deflection of the tire. Two different methods have been proposed for calculation of the contact patch length. The simpler one, which is useful mainly for static or slow rolling considerations, utilizes an experimentally determined fraction of the geometric contact length given by

eq (5.6). Experimental data is given by Smiley and Horne [6] for this using aircraft tires in figures 5.18 and 5.19, along with a curve showing

the best fit obtained by using a fraction of the geometric intersection

length, in this case 85 percent of it. A more complicated theory has been proposed by Bohm [8], Fiala [9], and Clark [10] which attempts to take into account the dynamic characteristics of the contact patch. This model visualizes the running band

of the tire as consisting of a circular shell on an elastic foundation, the

elastic foundation representing the effect of sidewall support. In refer- ence [10] mathematical techniques are given for transforming these

shell equations into moving coordinates so that the shell may be visual- ized as moving down a flat plane at some rolling velocity Vq. In both

cases, it is possible to calculate the length of contact patch by solution

of the governing differential equation.
460 THE CONTACT BETWEEN TIRE AND ROADWAY

On the assumption that the running band undergoes pure bending

and is inextensible, one may solve the equations governing displacement

of the cylindrical shell on an elastic foundation for the specific case of contact with a rigid frictionless plane, provided that one assumes that the initial geometry of the shell continues to be valid. Under this as- sumption, it is possible to assign values to the bending stiffness of the

running band of the tire, and to the elastic support given to the running

band by the pressurized sidewalls. This allows direct numerical calculation of the contact patch length as a function of vertical tire deflection.

Such a typical calculation is given in figure 5.20, compared with experiments taken from a tire used as a model to obtain the stiffness and elastic

Contact Pofch Lengths

'O 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 VERTICAL TIRE DEFLECTION - INCHES

Figure 5.20. Plot of contact patch length vs. vertical tire deflection.

spring support data. Dodge [11] has compared the results of calculations

and experimental tire contact patch length for seven different automotive

tires, five bias ply and two radial. These comparisons are given in figure 5.21. The foregoing discussion has concerned itself with gross contact area. The presence of grooves and kerfs, designed to channel or wipe water,

will substantially reduce this gross contact area so that the net contact area is considerably less. The exact value of this must be determined for each individual tread pattern. A representative set of different automotive

tires yields data as shown in table 5.3.
I THEORY 461

14 Z 12 iio

(3 8

u 6

X 4

TIRE -#1 8:00x14 BIAS PLY

"TIRE #2

7:50x14 BIAS PLY

o

o' ^8'

o EXPERIMENTAL

X CALCULATED 0.25.50 1.00 1.50 2.00 0.25.50 1.00 1.50 2.00

DEFLECTION- INCHES DEFLECTION-INCHES

' 10 X

zy 6

I 4

0

TIRE [ 5:90x15 BIAS PLY

0 .25 50 .00

"TIRE _7:50x

#4

14 STEEL

RADIAL

. 1

0 .25 .50 1.00 1.50 2.( DEFLECTION -INCHES DEFLECTION -INCHES

xio

TIRE '«*5" 155 mm xl5 TEXTILE I _ RADIAL I /-?-^o" 78^

0.25.50 1.00 1.50 2.00 DEFLECTION -INCHES

Figure 5.21. Contact patch length- deflection characteristics for five automotive tires.

1(
462 THE CONTACT BETWEEN TIRE AND ROADWAY

CO o CM CO CO On 00 CO o

CM o

cK On" cK

(TO CO

vO lO

LO co' lO 00

00 cr> CO CO

c« CO O 2 O

=^

Z 3

CM CM UO d d CO CO

CO CO

CO LO CO

CO CO CM CO CM

<\< zb

P (h CO CO O On CO CO csi CO CO CM CM CM

Csl lO uO LO

00 eg CO

CM CM CM

1j

03 2 2

<v

cn LO ^

X X X X O o o o 03 LO CD On lO

CO LO
SLIP BETWEEN TIRE AND ROADWAY 463 5.3. Slip Between Tire and Roadway

In all of the discussion which has been given here, nothing has been

said of the role of slip in the contact patch. Since this becomes important

for purposes of describing the yawed rolling tire, as well as in braking and

accelerating a vehicle, it is necessary to discuss this question briefly.

The term "developable surface" is generally used to denote a curved

surface which can be formed by simple folding or bending of a flat sheet, without extension or compression of this sheet. A cylinder is an example of a developable surface, as is a cone. The concept of a developable surface is important since it represents a kind of surface which may

be placed in finite area contact with a flat plane by simple bending

processes without stretching, tearing or compressing the sheet. Both the

cylinder and cone exhibit this kind of behavior.

A surface which is not developable can only be formed from a flat sheet by means of stretching or contracting the sheet in its plane. A

toroid, which is generally the shape of most tires, is not developable.

Thus, when a tire is brought into contact with a flat roadway such a geometric process can only take place by both simultaneous bending and

compressing of the tire surface. This means that tread elements in contact with the roadway will generally undergo a small deformation in the

plane of the carcass as such tread elements pass through the contact patch of the rolling tire and exit out the trailing edge. This process takes

place in every tire. It is possible to minimize the membrane stretching

and contraction by making the carcass structure of the tire as rigid as

possible. This is generally the principal behind the radial or belted tire constructions. It is also possible to minimize such membrane distortions

by use of extremely stiff materials for the tire carcass, such as steel wire

or glass. Since the membrane deformations actually take place in the carcass of the tire, it should be possible to prevent motion of the tread on the contact surface by choice of an appropriately soft tread compound.

Unfortunately this normally counteracts attempts to secure a long wearing tread material, so that it is usually not possible to make the shear

stiffness of the tread itself low enough to prevent all motion. This description just given applies primarily to straight line motion of the tire, and describes what might be thought of as secondary slip between tread

and roadway, that is to say, slip whose magnitude is of the order of a few

thousandths of an inch while a particle on the surface of the tire moves

through the contact patch. The direction of such secondary slip may be

partly longitudinal and partly transverse, depending on tire construction

and local friction conditions.

Over and above secondary slip exists a phenomena which might be

classified as primary slip. This normally takes place independent of the

nondevelopable nature of the toroidal tire, and can be thought of as a function of such parameters as steer angle, braking torque, and driving

torque acting on the tire. The most common of these phenomena is that

of steer angle. Here, the contact area of the tire with the roadway is first of all distorted due to the presence of the steering effort, and finally is separated into two regions due to this steering effort. The first region is
464 THE CONTACT BETWEEN TIRE AND ROADWAY

one of essentially static contact between the tire and the road surface,

but nevertheless a region in which secondary sHp may exist. The second

region, adjoining the first, is the region of primary slip where the tan- gential surface stresses necessary to maintain geometric distortion of the

tread surface exceed the local frictional stresses available, so that the

tire surface exists in part in a state of slipping in a gross sense. The order

of magnitude of such primary slip might be in the neighborhood of 0.1 inches or more.

It is difficult to measure the contact areas under such conditions of

yawed rolling, but these have been observed on model tires through a

glass plate by Saito [12]. Drawings are given in figure 5.22 showing the

growth of the primary slip area with increasing steer angle.

A - adhesive area (secondary slip area) B - partial skidding area (primary slip area)

Figure 5.22. Slip areas as a function of steer angle <f> as determined by visual observation.

It should be noted in figure 5.22 that the primary slip is confined to the rear of the contact patch in all cases. Effective stationary contact between tire and roadway, marred only by secondary slip, occurs always

at the leading edge of the contact patch as the tire rolls. In regard to straight line motion of the tire under braking or tractive

effort, there is probably always a certain amount of primary slip when the

wheel is in this condition, but the magnitude of this slip is variable. The

size of the zones of slip may also change, depending on the magnitude

and direction of the applied wheel torque. At the point where a driven

wheel comes into contact with the road and also at the point where it leaves the road, the tire surface attempts to slip in the direction in which

the wheel is turning, while at the center of the contact area there is a region where the tendency is to slip in the opposite direction. Novopol'skii

and Nepomnyashchii [13], present diagrams showing the change in the

positions of tread elements along the contact length of a tire relative to
I SLIP BETWEEN TIRE AND ROADWAY 465

Figure 5.23. Displacement of tread elements along contact length of tire: (a) free rolling,

(b) driving, (c) braked.

Zone I Longitudinal tangential stress acting from tire to roadway in direction of motion. Zone II Longitudinal tangential stress acting from tire to roadway opposite to direction of motion.

the positions which they occupied at the instant of entering contact. This is shown in figure 5.23. The sUpping of a driving or braking wheel

is also determined by the magnitude of the tangential forces acting in the plane of contact. Figure 5.23 also shows displacement curves for the

tread of a driving wheel and for the tread of a braked wheel. Note that

in both of these latter cases the tendency for primary slip is confined to the rear of the contact patch, just as in yawed rolling shown in figure 5.22. This type of behavior is exemplified by measurements taken on an

automotive tire and presented in reference [13]. This data is given in figure 5.24 where it is shown that the slip measured at the surface of a

tread element is essentially in accord with what one would expect from

the idealized representation of figure 5.23. In these cases the differentiation between primary and secondary slip

regions is not particularly clear. For example, the first plot of figure 5.24

shows a freely rolling wheel without applied torque. Here, essentially

all of the tire is at a slip level low enough so as to be classed as secondary

slip. On the other hand, the last plot of figure 5.24 shows the slip of a braked wheel in which slip values are large enough so that they might be
466 THE CONTACT BETWEEN TIRE AND ROADWAY

mm . Sf

a

75 50 25 0 25 50 75 mm

b 4 B

100 75 50 25 0 25 50 75 100 mm

c '

' ' ^2,5mm

S, Figure 5.24. Longitudinal slipping in the contact zone of a 6.00-16 tire.

(a) driven wheel-zero torque, (b) driving wheel-torque = 25Kgf m., (c) braked wheel-torque = — 25Kgf • m.

Contact length measured along the abscissa, with A the point of first contact and B the point of last contact. Ordinate

is longitudinal displacement of a tread element from its initial position. Downward is in the direction of turning of the wheel. Curve 1, middle of tread; Curve 2, 17mm. from tread centerline; Curve 3, 34mm. from tread centerline; Curve 4, 51mm.

from the tread centerline, i.e., edge of tread.

thought of as primary slip toward the end of the contact region. It is clear, of course, that in the case of steering under fairly large steer angles the slip at the rear of the contact patch might be primary slip, of the order of centimeters, while under complete locked wheel skidding

the slip values would be very large, and the entire contact patch would

be in a state of primary slip.
SLIP BETWEEN TIRE AND ROADWAY 467 One comment should be made concerning the measurements of slip

commonly available in the literature. They are obtained in almost all cases by rolling the tire over a plate which has protruding from its smooth

surface a small pin of low mechanical impedance. The pin contacts the

tread as the tread element moves into contact with the roadway, and

from that point onward registers a signal proportional to slip as the

tread element moves through the contact patch. Thus all measurements

are made with respect to the leading edge of the contact patch, which

serves as a zero reference. The data previously quoted have pertained

to slip in the longitudinal direction, i.e., slip in the direction of travel of the rolling wheel. Biderman, Volodina, and Pugin [14] have used similar

techniques to measure deformation of the tire carcass. While this is not the same as the slip between tread elements and the roadway, it nevertheless takes on the same character. Transverse slip, at right angles to the direction of travel of the wheel, also takes place in the contact patch.

Its nature may be inferred from figure 5.25, which gives data on meridianal displacement of the tire carcass itself as the tire rolls through the contact area. mm

LENGTH ALONG CONTACT AREA, mm

Figure 5.25. Meridional displacement of carcass.

Distance from crown: 1-lOmm; 2-20mm; 3-40mm; 4^50mm; 5-55mm. Straight line slow rolling. 6.00-16 tire, longitu- dinal ribs 2mni high.

It is clear that the question of slip during the rolling process is inti- mately connected with the normal pressure distribution which exists in the contact area. The normal pressure distribution does not suddenly

rise at the edge of the contact area from zero to a finite value, but rather

increases, however gradually or suddenly, at a finite rate. This means

that inevitably there will be a narrow ring or boundary layer around the

edge of the contact patch on which the normal pressure distribution is very small, so that consequently the available adhesion limits are very

small. At these regions it would be natural to expect that at least secondary slip would occur simply due to the fact that horizontal friction forces

are unable to further restrain elastic displacement of the rolling tire. The experiments of MindUn [39] tend to substantiate this.
468 THE CONTACT BETWEEN TIRE AND ROADWAY

An analogy to this has been given by Reynolds, and is quoted by

Hadekel [1]. Imagine a belt on a pulley, transmitting a definite power,

and therefore having a difference in tension between entry and exit, and

hence also a difference of strain between these points. A given element

of the belt undergoes some change in length from entry to exit, which is possible only if there is relative sliding. Reynolds concluded that there

must be some sliding near the edges of the contact region, even in the

absence of horizontal forces other than rolling resistance.

The question of slip in a rolling tire has been studied by Novopol'skii

and Tret'yakov [15] who conclude:

a. When a tire rolls, the slip areas are where contact begins and

ends.

b. Torque greatly increases both the area of the slip and the amount

of slip where contact ends. There is only an insignificant change

in the amount of slip where contact begins.

c. Both the carcass design and the type of tread pattern have a real

effect on the slip but the part played by the carcass design is of

greater importance.

One of the commonest cases in which an external influence changes

the slip picture dramatically occurs when the tire runs on a wet roadway.

This problem has been studied in some detail for aircraft tires by Horne

et al. [16], due to its importance for the landing of aircraft on wet runways.

It is also of some importance in automotive tires, since the presence of water can materially decrease the region in which secondary slip, or

effective tire contact, is present. As long as at least some of the tire is in effective contact with the roadway, then lateral and guiding forces

are available for maneuvering the vehicle. As soon as such effective contact is lost then these forces are also lost and complete overall skid may

result.

It is generally realized that as speed increases, the removal of water

in the contact area becomes progressively more difficult. The volume of water to be removed per unit time, and the inertia and viscosity forces

necessary to do this, all increase. Below the so-called hydroplaning

speed, defined as that speed at which all effective contact is lost, the

ground contact area may be idealized into three zones. These are shown

in figure 5.26, taken from Kelley and Allbert [17]. The effective length

of zones A and B in figure 5.26 progressively increase with vehicle

velocity, thus reducing the size of the contact zone C until, at hydroplaning velocity, the entire contact area is water borne. Two phenomena

are mainly responsible for this progressive penetration of the water layer

into the contact zone.

(1) Hydrodynamic pressure due to the impact of the tire on the water

layer. This impact on the stationary water film results in a hydrodynamic wedge which forms between the inclined tire tread

surface and the ground, immediately ahead of the contact zone. Hydrodynamic pressures which result from the change in mo-
SLIP BETWEEN TIRE AND ROADWAY 469 mentum of the surface water produce an upward thrust on the tire tread just forward of the contact area. The magnitude of this thrust

depends mainly on the depth of the water layer, the tread width

and the forward velocity of the tire. The hydroplaning speed has

been approximated by Horne as Fp= 10.35 [P] 1/2 where is the minimum hydroplaning speed in miles per hour

and P the tire inflation pressure in psi.

CONTACT AREA

FRONT REAR

FRONT

HYDROPLANING CASE — CONTACT AREA —

REAR

A- BULK ZONE-UNBROKEN WATER FILM B-THIN FILM ZONE -REGION OF PARTIAL BREAKDOWN

OF WATER FILM Figure 5.26. The three-zone concept of contact area under wet conditions.

A. unbroken water film; B, thin film zone, partial breakdown of water film; C, contact zone. (2) A squeeze film in the vicinity of zones A and B. As shown in figure 5.26, the thickness of the water film decreases progressively

towards the rear. Considered independently of the "water wedge",

the size of zones A and B depends on the relation between the

times taken by each element of the tread entering the contact area to squeeze through the film and to travel the full length of

contact. For a particular tread element the time to squeeze through

the film depends on the dimensions of the tread element, the

thickness and viscosity of the water film and the surface texture. This time is independent of speed. The second time, that necessary to travel the length of contact, varies inversely with speed.

It is thus possible that at some speed the time necessary to penetrate the water film will exceed the time in which the tread

element is in the water film, so that an unbroken water film will exist through the entire contact area.
470 THE CONTACT BETWEEN TIRE AND ROADWAY

The combined effects of the water wedge and viscous lubrication may

result in minimum hydroplaning speeds considerably lower than that given in the previous equation.

WATER FILM: 0.04 TO 0.06 in. HARSH TEXTURED ASPHALT SURFACE

185-15 AUTOMOBILE TIRES-RADIAL PLY

0.6

u. 0.4

0.2

r- HYDRO i

PLANING

0 SMOOTH0.25 RATIO:

0.50 GROOVE DEPTH

0.75 WORN

1.0 NEW

GROOVE DEPTH NEW

Figure 5.27. Effect of tread pattern groove depth on braking grip, under flooded road conditions. Wear maximizes hydroplaning tendencies.

Figure 5.27 shows the effect of tread groove depth on braking characteristics under flooded road conditions. On smooth surfaces, under wet

conditions, tread pattern designs showing the least loss in braking force

coefficient, and the highest hydroplaning speeds, have three main

features:

(1) Wide, preferably straight and unobstructed grooves or channels

to facilitate removal of water trapped in the wedge immediately

ahead of the contact areas.

(2) Narrow ribs or block elements which, by effectively reducing

flow distances, shorten the time required to squeeze thin water

films from between the tire and the smooth surface.

(3) Slots or knife-cuts in the surfaces of the main rib which act as low pressure reservoirs and also effectively shorten the flow

distances.

These three design characteristics refer mainly to high speed conditions.

They conflict somewhat with both low speed wet holding performance

and with other tire requirements such as pattern stability and durability.
CONTACT PRESSURES 471

Table 5.4. Factors influencing effective braking friction between tire and wet road {100

mph maximum)

Factors Level of variability due to factor considered

Tire factors

1. Tread pattern design. Scale of effect recorded during assess- ment of practical tire pattern designs.

Up to 4 :

2. Tread materials. Maximum effect due to tread material changes

within range of materials currently available Up to 1.5:

3. Patterned tire. Smooth tire Up to 8:

Road factors

1. Road surface characteristics. Scale of effect recorded during assessments of practical tire pattern designsUp to 5:1

2. Water depth. Effect of variation in depth of water film covering

a lUcLil SUlidCC Hi lllc Idll^c \J*\JO \J,^\J 111. Up to 3:

Vehicle factors

1. Speed. Reduction in braking force coefficient due to an increase

in speed from 30 to 80 mph (patterned tires only) Up to 10:

2. Braking system. Perfect nonlocking system versus locked wheel

braking

Up to 3:1

Note: Cases of braking force coefficients below a value of 0.1 have been excluded. Kelley and Allbert have attempted to assess the variabiUty inherent

in the various factors affecting braking friction between a tire and the

wet road. These are given in table 5.4. From the results of this table, it is seen that the most important characteristics in regard to total braking

force lie in the patterned versus smooth tire, and in the speed at which

the braking force is measured. The pattern remains the main design

variable which can be controlled by the tire designer, so that it would

be anticipated that significant wet road improvement could be obtained

by use of proper tread patterns which are capable of channeling water

away from the contact region at maximum flow rates.

5.4. Contact Pressures

At the interface between the tire and the roadway an element of tire surface area is acted upon by a force vector which can be expressed as two components, one perpendicular to the contact surface, called the

normal component, and one tangential to the contact surface. This is
472 THE CONTACT BETWEEN TIRE AND ROADWAY

shown in figure 5.1. This latter component may be further decomposed

into two components, each lying in the contact plane, but one parallel to the central plane of the tire and the other perpendicular to it. These components in the contact plane are commonly called the shear components.

With equal validity they could be expressed as components parallel and

perpendicular to the direction of travel of the wheel. Either decomposition

would be useful for describing the shear effects. In this section attention is directed first to the normal pressure

distribution components caused by contact of the tire with some other

surface. As a basic primary concept, one might state that

p = po+/(Tire structural characteristics, tire driving or braking

torque, tire side forces, tire velocity, etc.) (5.9) where p is the vertical pressure component at any point, po is the infla- tion pressure of the tire and / is some general functional relationship

which insofar as is now known is extremely complicated, and can best

be described in a qualitative sense. In eq (5.9), we postulate that the net pressure distribution at any point

depends primarily upon the inflation pressure, and there is considerable

experimental evidence to indicate that this is indeed true. Such a conclusion was used by Hadekel [1] as the basis for one of his estimates of the net contact pressure of a tire. On the other hand, subsequent studies

have shown that the characteristics determining the function / also play an important part in defining the detailed tire pressure distribution. In general, these characteristics may be thought of as being divisible into two major parts:

a. Tire structural characteristics.

b. Tire operating variables.

Of these two types of influences, the use of theory is most beneficial

in discussing the role of structural parameters in modifying the basic

inflation pressure distribution. Some of the most important aspects of the various simple theories will be reviewed here. In the first instance, consider the contact against a flat surface of an

inflated membrane with vanishing bending stiffness. No matter what the

tension in the membrane, the fact that it is in geometric contact with the

flat surface means that the contact pressure distribution is exactly equal

to the inflation pressure inside the membrane, and it is this line of reasoning which allows one to postulate that the primary component of

tire vertical contact pressures is the inflation pressure. On the other

hand, if the tire is in contact with a cylindrical road wheel then this statement is no longer true, since the cover tension now plays some role

in defining the net contact pressure. This is expressed analytically as

Po + I

(5.10)

where represents the shell membrane force per unit length while R

represents the radius of the road wheel contacting the tire. From this it
CONTACT PRESSURES 473 may be seen that the road wheel test may generally be expected to exhibit slightly higher average contact pressures than acutally exist on

the highway.

It is of interest to attempt to classify the various structural parameters

of the tire which affect the vertical contact pressure, in order to obtain some idea of the influence of these parameters.

First of all, the tire is basically a structural shell. To an even greater

approximation, one may roughly consider a tire to be represented by an

elastic running band elastically supported by the tire side walls. This

type of model is particularly applicable to a radial tire, although it also can be used to express the characteristics of a bias ply tire. Using such

a model, attention may be primarily directed to the fore-and-aft variation

of the pressure distribution, since effects of right angles to this will be very similar. This will allow description of the pressure distribution

as a function of only one dimension, namely that measured down the

length of the contact patch, which will considerably simplify the discussion which follows. The tire structural components of most importance in describing

the effect of the tire carcass upon normal contact pressures are:

a. Elastic support of the tread by the sidewall.

b. Bending of the tread.

c. Shear deformation of the tread.

d. "Snap through" buckling of the tread, defined as the tendency of the tread to seek a deformed equilibrium position due to membrane

compression.

e. Normal compliance or stiffness of the tread. The role of these various components has been discussed from the

theoretical point of view by Clark [37] who also performed model experiments designed to illustrate some of these effects. If the tire is thought

of as a running band supported by an elastic foundation, then deformation

of the elastic foundation requires a pressure distribution which should

be very close to being proportional to the amount of radial deformation

undergone by the tire. The interference of the tire running against a flat plane is known from the geometry, and the form of pressure distribution

associated with deformation of the side wall may be expressed as

where pn represents the contact pressure needed to deform the elastic sidewall, k is the sidewall foundation stiffness in units of pressure, while ^0 represents the angle of geometric interference as shown in figure 5.16. This type of pressure distribution is illustrated in figure 5.28,

(5.11)
474 THE CONTACT BETWEEN TIRE AND ROADWAY

while in figure 5.29 an actual pressure recording down the length of the contact patch is given by a model wheel made with an isotropic annular

band. This is a nearly pure elastic foundation and exhibits a pressure

distribution similar to that predicted by eq (5.11).

Figure 5.28. Idealized elliptic pressure distribution.

POSITION IN CONTACT PATCH

Figure 5.29. Measured pressure distribution of a silicone rubber ring with rigid hub.

The bending of the running band itself represents an interesting form

of pressure distribution, since in considering the deformation of a

circular band in contact with a flat plane, a finite bending moment is required to change the original curvature 1/r of the running band to a zero curvature, which it must have when it conforms with the plane. In

addition, this change in curvature, and hence the bending moment, must

be constant within the region of contact. The only pressure distribution

satisfying these equilibrium and bending moment requirements is made

up of a pair of concentrated forces, located at each end of the contact

patch. This is illustrated in figure 5.30 schematically, while figure 5.31 shows an actual pressure distribution recording taken from a model test, in which a running band free of elastic support is rolled over a pressure

transducer buried in a steel plate.
CONTACT PRESSURES

Q Q

475

Figure 5.30. Idealized pressure distribution for a circular ring.

1

-> I 1 POSITION IN CONTACT PATCH

Figure 5.31. Measured pressure distribution for a circular metal band on a flat plane.

The case of pure bending of the running band is an idealization. In any real situation some transverse shear deformation of the running

band must be present. The problem of the curved ring in contact with the

flat plane, utilizing both bending and shear deformation of the material,

has been studied by both Robbins [18] and Akasaka [19]. Both of these

authors conclude that the role of transverse shear deformation is to change the idealized concentrated force distribution into a continuous

one. The form of this pressure distribution is illustrated in figure 5.32,

in a schematic sense, while a pressure recording taken from a worn auto- motive tire is given in figure 5.33.

Figure 5.32. Theoretical pressure distribution for a circular ring, with shear deformation,

in contact with a rigid plane.
476 THE CONTACT BETWEEN TIRE AND ROADWAY

3

NEDGE

\C0N

OF

TACT

1

l_

A

1 CENl

* CON!ER OF ACT

\\

\

\ ^\

\

i J

—\ -1

\ 1

POSITION IN CONTACT PATCH

Figure 5.33. Measured vertical contact pressure on a 7.50-14 automotive tire with worn

tread.

Buckling of the tread, or "running band" usually would occur at somewhat larger deflections. This phenomena has been studied extensively from the point of view of structural shell theory, although often

using concentrated forces or uniform pressure loadings in order to obtain

stability criteria. Experimental evidence gives a good idea of what happens here, although it is clear from the nature of the phenomena itself. Imagine an internally inflated membrane subjected only to inflation and

buckling efl^ects. The pressure distributions for small and large deflec- tions respectively are shown in figure 5.34. Due to buckling, the pressure

is decreased markedly in the center of the contact region, as would be

expected. Unfortunately the form of decrease of the contact pressure near the center of the contact area is similar to that predicted by shear

deformation, and it is difficult to distinguish between the two causes of smaller contact pressure near the center. Such a phenomena may also be observed on model tests by increasing the deflection substantially, as is shown in figure 5.35.
I CONTACT PRESSURES 477

POSITION IN CONTACT PATCH Figure 5.34. Idealized pressure distributions due to buckling.

1 MODEL TEST LARGE

S DEFLE

\

\

:CTION /

/

/

/ \

\\ ^SMALL

\ DEFL ECTION

\

\

\

\

\

\1-

1

1

1

1

1

1 POSITION IN CONTACT PATCH

Figure 5.35. Measured pressure distribution on tire model at small and large deflections. The role of normal compliance, or stiffness of the tread, is not well

understood in forming the total pressure distribution pattern, since few

pertinent theoretical studies involving shell theory interacting with nor- mal stiffness exist. The general role of normal compliance can be seen by referring to the well-known cylinder contact solutions of Hertz [20]

for an isotropic body. This approximates the type of contact pressure ob- tained when a low normal stiffness exists, since most of the deformation

due to contact is confined to a relatively thin surface lamina. The Hertz

solution for a cylinder in contact gives a semi-elliptical pressure distribution as shown in figure 5.28. Unfortunately, it is nearly identical in shape

to the pressure distribution required to deform the elastic foundation for the tread, and it is difficult to distinguish between them. However, figure 5.35 shows a recording of the pressure distribution of a model having a

relatively soft isotropic outer band simulating the tread of the tire. This

illustrates the general role of normal compliance, and illustrates further
478 THE CONTACT BETWEEN TIRE AND ROADWAY

that pressure distributions such as shown in figures 5.33 or 5.34 do not

rise abruptly, but rather exhibit some sort of gradual build up of pressure

from the edge of contact. Many of the phenomena just discussed can be observed clearly on

pressure distribution recordings taken from various kinds of pneumatic

tires. Some of this data has been reviewed briefly by Hadekel, with particular emphasis on the early work by Kraft [21], and by Markwick and

Starks [22] who dealt with automotive tires. More recently, much more

detailed measurements on truck tires have been reported by Bode [23],

who reports data such as shown in figures 5.36-5.40.

z> 6

CO 8

lij Of 10

Q.

y7

/ a.

0 50

LEADING EDGE

100 150 200

LENGTH mm. 250 300 mm

TRAIUNG EDGE Figure 5.37. Pressure distribution down the length of a tire with no tread pattern, at con- stant 15 Km/hr velocity, for two loads: (a) static load 1680 Kp., inflation 6.5 atm; (b) static load 3 740 Kp. , inflation 6.5 atm.
CONTACT PRESSURES 479 0

2 u1 4

^ in

IT^ 12

1 \)

c M

100 80 60 40 20 0 20 40 60 80 100 mm

POSITION ACROSS WIDTH

Figure 5.38. Influence of load and internal pressure on contact pressure distribution under

a tire with no tread pattern.

Constant velocity 15 Km/hr measured across the tire width, (a) Load 1680 Kp, inflation 3.5 atm; (b) load 1680 Kp,

inflation 5.0 atm; (c) load 1680 Kp, inflation 6.5 atm; (d) load 3740 Kp, inflation 6.5 atm.

LEADING EDGE TRAILING EDGE

. 4

bJ§ 6

111 O

100 50

POSITION

100 150 200 250 300 mm

IN LENGTH OF CONTACT PATCH

FlGtiRE 5.39. Influence of the tractive force on pressure distribution under a smooth tire, taken down the length of the tire. Load 1680 Kp; inflation 6.5 atm. (a) 0.6 m/s- acceleration V=10.2 Km/hr; (b) 1.6 m/s^ acceleration V = 8.4 Km/hr;

(c) 4.1 m/s^ acceleration V = 8.9 Km/hr.

Figure 5.40. Influence of the braking force on pressure distribution under a smooth tire, taken down the length of the tire. Load 1680 Kp; inflation 6.5 atm. (a) -4.1 m/s^ deceleration V = 10.0 Km/hr; (b)-2.7 m/s^ Load 1680 Kp; inflation 6.5 atm. (a)-4.1 m/s^ deceleration V = 10.0 Km/hr; (b)-2.7 m/s^ deceleration V = 35.7 Km/hr;

(c) -2.0 m/s^ deceleration V= 14.5 Km/hr.

1
480 THE CONTACT BETWEEN TIRE AND ROADWAY

A very complete set of data on normal stresses taken in various soils has been reported by Green [24]. A typical pressure contour plot of a

tire in sand is shown in figure 5.41, while the total soil pressure distribu- tion is given more accurately by figures 5.42-5.44. A further comparaDISTRIBUTION OF NORMAL STRESSES

PROJECTED ON HORIZONTAL PLANE

TOWED WHEEL IN SAND

11.00-20, 12-PR SMOOTH TIRE 3000-LB LOAD 15-PSl INFL PRESS.

0- TO 6-IN. CONE INDEX = 30 Figure 5.41. Distribution of normal stresses under a towed tire in sand.
CONTACT PRESSURES

EM AT AXLE; (o)C'^r)-CbKT) = 0

b

ASSUMPTIONS: AXLE FRICTION NEGLIGIBLE NORMAL AND TANGENTIAL

FORCES ON A TOWED

PNEUMATIC TIRE

IN SOFT SOILS

Figure 5.42. Stresses in a towed wheel in soft soil.
482 THE CONTACT BETWEEN TIRE AND ROADWAY

DISTANCE FROM AXLE IN INCHES STRESS IN PSl

_^ POINT BELOW AXLE ^ ^^'^'^^^^^^^^^

/ ~~;^.,c^:^^—^^~--~STIIESS AT i — "vL^iT-ZJfii AT 3.TS IN. OFF I

STRESS AT I BOW WAVE

STJfESS AT 3.75 ORIGINAL SURFAC E

HORIZONTAL COMPONENTS

VERTICAL COMPONENTS NORMAL STRESSES AT TIRE SURFACE IN SAND

TOWED WHEELS

11.00-20, 12-PR SMOOTH TIRE Figure 5.43. Normal stresses at tire surface in sand.

OrSTANCE FROM AXLE IN INCHES

DIRECTION OF TRAVEL

MAX <t OEFLECTIONj HARD SURFACE =112 IN TOWED IN SAND = 0 SI IN POWERED IN SAND = 0 42 IN. SOW WAVE AND BOTTOM OF RUT MEASURED AT Ct OF TIRES CROSS

SECTION

3.18 IN ORIGINAL T SURFACE

80T TOM OF RUT HORIZONTAL COMPONENTS NORMAL STRESSES AT TIRE SURFACE IN SAND

TOWED AND POWERED WHEELS

11,00-20, 12-PR SMOOTH TIRE 46-PSI INFLATION PRESSJPE Figure 5.44. Normal stresses at tire surface in sand.
CONTACT PRESSURES 483

Figure 5.45. Vertical stresses along principal axes of contact ellipse of three tires of

dissimilar construction under constant 890 lb. load but with different inflation pressures.
484 THE CONTACT BETWEEN TIRE AND ROADWAY

tive study between various types of tires has been reported by Freitag

and Green [25] which is in part summarized in figure 5.45 where pres- sure distributions taken both along the length and width of three common

types of automotive tires are presented. These tires were tested in contact with an essentially rigid surface. They illustrate the rather inter- esting point that the presence of a four-ply structure with its greater

bending rigidity tends to emphasize the pressure spikes located at for- ward and leading edges of the contact patch, as might be expected from

the simple theoretical discussion of the bending of a curved ring, as pre- viously outlined and illustrated in figures 5.31 and 5.33. A similar but

less detailed study has been presented by Hofelt [26] as shown in figure 5.46.

Id ps: ;r.flatior. 108= B. 24 psi inflation 1085 lbs. C. 32 psi inflancr. ICo^

(a) A. 875 lbs. load 24 psi

inflation. B. 1085 ibs. load 24 psi

inflation.

Jvli

C. 1513 lbs. load

inflation.

Figure 5.46. Effect of inflation and load on normal contact pressure.

(a) Effect of inflation at constant load, (b) effect of load at constant inflation pressure. A number of theoretical investigations and mathematical models

have been proposed for explaining or calculating contact pressure distri- butions between various types of bodies and a rigid plane. HofFerberth

and Frank [27] mention the work of Martin [28] as well as that of Vlas- sov [29], In the latter case a variational method is suggested for the

treatment of two-dimensional contact problems involving plates and

shells, while in the former case symmetric two-sided flattening of cylindrical and spherical shells has been considered. Basically, the problem

usually reduces to the matching of two equations, one equation describing the free surface of that portion of the plate or shell not in contact with

the rigid plane, under known external loads but with unknown deformations. The second region describes the area of the plate or shell in contact with the rigid plane, and this defining equation usually is written in terms of known deformations but unknown load distribution. Continuity

of displacement must take place and hence the two sets of equations
CONTACT PRESSURES 485 must be matched, usually at the unknown boundary of contact. This

introduces an additional degree of freedom into the system, and often

makes such a matching process amenable only to trial and error solution. Bohm [30] has extended Martin's theory and developed it for the one- sided flattening of the toroidal shell with circular cross section, but

without numerical results. Martin found theoretically that the contact pressure distribution is rectangular under a flattened cylinder, with a value smaller than that of the inflation pressure providing the cylinder

has bending stiffness. Equilibrium is maintained by a pair of concentrated bending moments which act at the forward and leading edges of the contact patch, these bending moments arising due to the fact that

it is not possible to match bending moment and shear at the junction

of the contact region with the free surface. On the other hand, the inclu- sion of shear deformation by Akasaka and Robbins leads to theoretical

pressures as shown in figure 5.47.

5* 10** 15° 20* 25*' 30*» POSITION IN CONTACT PATCH, 0-^ "

Figure 5.47. Calculated pressure distributions under a flattened cylindrical shell, under

internal pressure p, with different degrees offlattening. The angles given are characteristic for half of the flattened, originally cylindrical portion of the arc. A great deal of interest has been shown in the role of velocity in modifying the contact pressure distributions just discussed. A number

of investigations have examined this, including the previously discussed

work of Bode as well as the work of Zakaharov and Novopol'skii [31]. In general, the results seem to show that increasing speed causes an

increasing vertical contact pressure at the forward end of the contact patch and a decreasing value at the rear portion of the contact patch.

This is schematically illustrated in figure 5.48, and has been experimentally confirmed by Zakaharov and Novopol'skii [31]. Attempts have

been made by Clark [10] to utiHze a relatively simple shell and elastic foundation theory to predict dynamic contact pressures, but due to the

inability of such a simplified model to accurately provide for continuity of shear and bending moment through the edge of the contact patch, the
486 THE CONTACT BETWEEN TIRE AND ROADWAY

Direction

Pressure

Figure 5.48. /dealized pressure distribution as influenced by speed oftravel.

Direction

of Travel

I - Inflation Pressure n - Elastic Sidewall Support

^ HI a - Forward Edge

Shear Disconuity

nib - Trailing Edge

' ' Shear Disconuity

Figure 5.49. Calculated dynamic pressure distribution from simple elastic shell tire model, without shear deformation. corresponding dynamic pressure distributions are predicted in the

form shown in figure 5.49. It is clear that shear deformation must be

added to such a theory before reaHstic contact pressure distributions

will be available. However, the general trends of such dynamic pressure

distributions are apparent.

The necessity for including shear deformation effects, and for dis- carding the restrictions of the Kirchoff bending hypothesis, were also discussed by Essenburg and Gulati [32] who studied the axisymmetric

contact of plates. They concluded that only by such a relaxation of con- ventional bending theory could the contact problem be successfully

approached.

All measurements on automotive tires show a relatively high normal

contact pressure region in the shoulder area of the tire tread, due to the

heavy tread shoulder conventionally used on such tires. This means that generally one finds higher pressures at the shoulder than at the center of

contact. This has been illustrated in the experimental data given, but

bears repeating.

Vermeulen and Johnson [33] have discussed the contact of nonspherical elastic bodies transmitting tangential forces, and have by such an

effort given at least a beginning to a formal theory for tractive forces in combination with the contact problem. Vermeulen and Johnson pre- sume that the presence of tangential forces does not, to a first approximation, change or alter the normal pressure distribution. Their discussion

further presumes that the contact surface of two rolling bodies trans- mitting a tangential force is divided unsymmetrically into a region of
CONTACT PRESSURES 487

slip and a region of no slip. The no-slip region is adjacent to the leading

edge of the contact boundary, independent of the direction of the torque.

Elliptical contact regions are assumed, and it is also assumed that regions

of no slip are located adjacent to the contact ellipse at the leading edge.

The tangential component of the contact stress between a tire and

roadway has been studied rather extensively due to its close association

with the processes of tire wear and skid. Hadekel [1] quotes from the

early work of Martin [28], Markwick and Starks [22] and Kraft [21].

Most of these early measurements were made with mechanical dynamometer systems imbedded in a flat plate, which allowed a rolling tire to actuate a recording system and to measure the resulting tangential sur- face stress in terms of its components in the direction of travel and at right angles to the direction of travel. In particular, Martin has idealized

these measurements into two vector components, each of which must be

doubly symmetric about the centerline of the contact area for a stationary

tire without side or longitudinal force acting on it. This is illustrated in figure 5.50. These components combine in such a way as to form shear

stress resultants which are roughly directed toward the center of the contact zone.

Figure 5.50. Shear stress distribution in the contact patch of an aircraft tire. The distribution of tangential forces as given in figure 5.50 is ideal- ized, since there the tire is presumed to have been loaded statically and

not rolled into position. In this case it is clear that the deformation of the

nondevelopable tire surface, so as to cause contact with the flat roadway,

will almost surely result in both bending and membrane deformations of the tire carcass with consequent symmetric strain distribution. The

friction coefficient between the tread and roadway prevents the free con-
488 THE CONTACT BETWEEN TIRE AND ROADWAY

traction or expansion of the tread surface, so that local shear stresses are

set up in the tread elements between the essentially rigid roadway and

the deforming or straining tire carcass. It is these shear stresses which on

the road surface become the tangential force components during straight

line rolling, and at the same time serve to bring the elastic tire carcass

into the state of compatibility of deformation with the entire roadway and

tire structure system. Were it not for the coefficient of friction between

the tread surface and roadway, the tire carcass would freely deform and

there would be no tangential stress distribution. Were it not for the elastic characteristics of the tire tread elements and the tire carcass, there

would be complete slip throughout the contact region as deformation

took place. In truth, the actual phenomena lies somewhere between

these two conditions. Under conditions of static deformation of the tire against a roadway, with appreciable friction coefficient, most of the contact surface is relatively free of slip, while in cases of heavy braking or yawed rolling, substantial amounts of primary slip are observed.

Detailed experiments on the friction of rubber indicate that, potentially, friction coefficients are available which exceed those normally ob- served in a typical pneumatic tire in contact with a roadway. It is felt that the cause of this probably lies in the fact that small amounts of slip

(secondary slip) exist almost entirely throughout the contact patch for most practical cases of tire rolling, so that in effect the measurement is of the coefficient of friction of rubber in a state of secondary slip, and not

in a state of maximum friction coefficient. The apparent symmetry of the static shear stress distribution is shown

in detail by some of the early experimental results of Martin, given in figure 5.51, for the shear stress component perpendicular to the midplane of the wheel, i.e., in the lateral direction.

Figure 5.51. Shear stress distribution across the width of an aircraft tire. The exact distribution of the lateral component of the tangential contact stress is not well understood even for a stationary tire. The work of
CONTACT PRESSURES 489 Biderman, Volodina and Pugin [14] seems to indicate that for a standing

or slow rolling tire the lateral (sideward) component of the tangential contact stress starts from zero at the forward edge of the contact area, reaches a maximum some distance away from this point and then de- creases to zero at the trailing edge of the contact area. This is inferred

from meridional displacements of the carcass, but is in complete accord

with figure 5.51. Novopol'skii and Nepomnyashchii [13] have studied the longitudinal

component of the tangential stress vector in some detail, and have

proposed two causes for this stress component in a rolling tire without

steer angle. These are:

a. Deformation of the carcass with respect to the contact plane,

giving rise to a shear stress approximated by the expression

I- sin 2^0 -sin 2d\ (5.12)

"o J where t= longitudinal shear stress

A- = carcass stiffness constant and the other symbols are defined in figure 5.16.

b. A uniformly distributed shear stress caused by tire rolling

losses, which for commercial automotive tires is a negligible factor

compared to carcass deformation.

The form of the shear stress variation down the length of the contact patch predicted by eq (5.12) is given in figure 5.52.

Figure 5.52. Distribution of longitudinal tangential stresses along the contact length between a tire and the road while the wheel is in motion.
490 THE CONTACT BETWEEN TIRE AND ROADWAY

The action of a driving torque changes the magnitude and distribution

of the tangential forces over the plane of contact. No equation has yet

been found to connect the tangential force and the torque, but a qualitative picture can be drawn by considering the rotation of an elastic wheel.

Figure 5.53 shows the rotation of a wheel transmitting torque Mt. As

a result of the torque transmitted through the wheel, two sets of forces

act upon it — the reaction of the wheel axis Pk, and equal to it in the

opposite direction the reaction of the road acting in the plane of contact. As a first approximation it may be assumed that the reaction of the road

Pk is evenly distributed over the area of contact. The component of

tangential stress from the reaction Pk is denoted as Tp (see fig. 5.53).

Figure 5.53. Rotation of a driving wheel {the distribution of longitudinal tangential

stresses in the contact zone of driven, driving, and braked wheels).

(1) total component of the longitudinal tangential stress t,„ due to the driving torque; (2) component Tm due to the braking

torque; (3) distribution of longitudinal tangential stress along the contact length of a driving wheel; (4) ditto, for to of a free wheel; (5) ditto for a braked wheel. There is a certain amount of adhesion over the area of contact, and

because the tire possesses tangential elasticity the torque will compress

the tread elements in the zone immediately before contact (denoted

by the sign —) and at the same time will stretch those elements in the area just after contact has been released (denoted by the sign +). Thus,

an initially compressed element, Ajc, of the tread, is released from this compression as it passes through the contact area, reverting to its normal

state Ajci, then it undergoes stretching and emerges from contact in a stretched state Ajc2.
CONTACT PRESSURES 491

Since the elements, as they pass into the contact area, are in direct contact with the road, any change in their dimensions is prevented by

the force with which each element grips the road surface, and tangential

stresses Tk arise in the plane of contact. As a first approximation it is assumed that the stress Tk changes linearly along the length of the contact zone, at the center of contact t/c=0 (see fig. 5.53). Thus tangential

stress Tm, which is the sum of Tk and Tc is also linear and is represented

by straight line 1 in figure 5.53. When the wheel is braked the situation is reversed, i.e., immediately

before contact the tread elements are stretched, and they contract at the point at which contact is released. In this case it is line 2 (fig. 5.53)

I which represents the tangential stress Tm.

' Thus, the magnitude and direction of the longitudinal tangential

stresses, r, acting in the plane of contact of a driving or braked wheel are determined by the sum of the stresses created in the rolUng of a free wheel, To, and the additional stresses Tm created by the application of a

torque. The lines representing the resultant tangential stresses along the

length of the contact zone of a driving or braked wheel are shown at the

bottom of figure 5.53. The precise way in which these stresses are dis- tributed over the contact area depends on the design of the tire, the

radial load, the internal air pressure, and the adhesion to the road. These

tangential stresses can cause local slipping of the tread elements on the

i| road. It is clear that radial, bias and other tire constructions would exhibit

different detailed shear stress distributions, even for the same tractive

force.

1
492 THE CONTACT BETWEEN TIRE AND ROADWAY

Extensive experimental work on the longitudinal component of the contact shear stress has recently been published by Bode [23]. Here, an

effort is made to assess the role of traction and braking in forming the

longitudinal component of tangential contact stress. Figure 5.54 shows a three-dimensional plot of such a stress component over the entire contact area, while figures 5.55 and 5.56 from Bode show that the influence

of either braking or tractive forces is to throw the major part of the longitudinal component of tangential stress to the rear of the contact patch, as one might anticipate. The direction of this component of the tangential

stress is dependent on the presence of tractive or braking forces. Even

Acceleration 0.96 m/s2

1.28 m/s2

1.53 m/s2

2.02 m/s2

3.22 m/s2

5.66 m/s2

Velocity

V = 4l.7km/h

V= I7.0km/h

V = 25.8km/h

V= I7.7km/h

V= 7.4km/h

V= 8.9km/h

0 50 100 150 200 mm

LENGTH ALONG CONTACT

Figure 5.55. Distribution of shear in the running direction of a smooth tire under tractive force.

Static load 1590 Kp., inflation 6.5 atm.

1.0 0.0

1.0 2.0

3.0 4.0

5.0,

SHEAR OPPOSING

TRAVEL DIRECTION

a..-.

s b 1 s \

'4 N

\

1

deceleration -0.4 m/s2

- 1.23 m/s2

- 1 .5 m/s2

-2.45 m/s2

-2.78 m/s2

-4.3 m/s2

Velocity

V= 17.6 km/h

V= 17.7 km/h

V= 16.9 km/h

V = l5.4km/h

V = I6.9 km/h

V= 17.2 km/h

Slip ,= 1.2 %

= 3.0%

= 4.8 7o

= 6.9 %

= 10.1%

= 19.3%

50 100 150 200 250 mm

LENGTH ALONG CONTACT

Figure 5.56. Distribution of shear in the running direction of a smooth tire under braking

force.

Static load 2610 Kp., inflation 6.5 atm.
CONTACT PRESSURES 493 moderate values of such forces are sufficient to completely change the

distribution of tangential stress, again as might be expected.

It is possible to separate out the effects of traction on shear from the

normal tire deformation effects on shear. This is done in figure 5.57,

where the heavy lines show the net shear distribution taken in the longitudinal direction down the length of the contact patch, and caused

directly by the tractive effort. This is obtained by subtraction from the

data of figure 5.55, while a similar construction can be performed for the

effects of braking on shear. This is shown in figure 5.58, again taken

from Bode. From this it is seen that the effects of traction and braking are

essentially mirror images of one another insofar as they influence the

5.0

[3 3.0

01

I- 2.0

1.0

0.0,

SHEAR OPPOSING

DIRECTION OF MOTION VA

H-\\

0 50 100 150 200 250 mm

POSITION ALONG CONTACT LENGTH

Figure 5.57. Distribution of shear along the length in running direction.

Super-position of shear due to deformation and acceleration. Heavy lines are shear stresses due to acceleration alone.

1.0

<\Je 0.0

: 1.0 ^ 2.0 <¥ 3.0

4.0,

SHEAR OPPOSING

DIRECTION OF TRAVEL

0 50 100 150 200 250 mm

POSITION ALONG CONTACT LENGTH

Figure 5.58. Distribution of shear along the length in running direction.

Super-position of shear due to deformation and braking, from figure 5.56. Heavy lines are due to braking alone.
494 THE CONTACT BETWEEN TIRE AND ROADWAY

> 60

40

20

Vt

tI-2

Tire Size: 6.70-16

Pressure: 16 psi Load: 61 cwt

Slip Angle: 4*>

0.1 0.2 0.3 0.4 SIDE SLIP (inches)

Figure 5.59. Cornering force intensity vs. tread lateral distortion.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 SIDE SLIP (inches)

Figure 5.60. Cornering force intensity vs. tread lateral distortion at slip angles up to 12°.

magnitude and distribution of the longitudinal component of tangential

force in the contact area. The experimental data given by Bode agrees with the measurements of Novopol'skii and Nepomnyashchii.

The lateral component of the tangential stress, at right angles to the

direction of tire travel, has been extensively investigated by Gough [34]

,

[35] and by Cooper [36] for the case of yawed rolling, where the lateral tangential forces are much larger than exist for straight-line rolling. In
CONTACT PRESSURES 495

the case of yawed rolling, one may think of the tangential stress distribu- tion as first associated with forcing the elastic nondevelopable pneumatic tire against a flat roadway, followed by the addition of effects due

either to yawed rolling, braking or acceleration. Viewed in this way, one

might postulate that the yawed rolling, braking or acceleration effects are

clearly dominant. Figure 5.59, taken from Cooper, shows the distribution

of cornering force intensity as a function of tread lateral distortion at the center of a yawed tire. Cornering force intensity is obtained by inte- grating the lateral component of tangential stress across the width of the

Figure 5.61. Cornering force intensity along the contact spot.

tire contact patch. A more complete set of such curves is shown in figure 5.60 from the same source. Figure 5.61 shows the cornering force in- tensity plotted against position in the contact patch, giving a much

clearer picture of the asymmetric nature of this force component due to yawing. In figure 5.61, the centroid of the area under the cornering force

intensity curve lies behind the geometric center of the contact area, here

denoted by 0, which gives the self-aligning torque to the tire. The dis- tance t in figure 5.61 is called the pneumatic trail. A more complete set of curves of cornering force intensity as a function of position in the contact patch are given for various yaw angles in figure 5.62. Finally,

the results of such measurements may be combined into a single plot

such as shown in figure 5.63. Due to the method of making these measurements, it is not possible to determine the exact distribution of the lateral (sideward) component of
496 THE CONTACT BETWEEN TIRE AND ROADWAY

CONTACT POSITION (inches)

Figure 5.62. Cornering force intensity at slip angles up to 12°.

500 - 400

300

200

100

Tire Size 6.7C)-l6 Pressure: 16 psi Load: 6.1 cwt

1 / 9 y

> /

A

— > \-<- ><.

1.2

1.0 0.8 «

o

0.6 3

0.4 <

2 4 6 8 10 12 14 SLIP ANGLE (degrees)

Figure 5.63. Cornering force and trail at slip angles up to 12°.
CONTACT PRESSURES 497

tangential contact stress across the width of the contact area. Only the

integrated value is given. In figure 5.63 it is seen that the pneumatic trail decreases substantially as the steer angle increases. This is related to the shape of the

cornering force intensity curve with increasing steer angle, since the

\

I BRAKING FORCE

864202468 10 LEADING EDGE TRAILING EDGE CONTACT LENGTH cm

TIRE^ 5.60- 14 4ply L0AD= 300kg INFLATION = I.55kg/cm2 Figure 5.64. Vertical force and braking force distributions.

greater slip in the contact patch results in a more symmetric cornering

force distribution. The interaction of vertical pressure distribution with braking and yaw

has been studied experimentally by Iritani and Baba [37]. They show

that there is little change in vertical pressure due to yaw, but a measureable amount due to braking. This latter effect is shown in figure 5.64.
498 THE CONTACT BETWEEN TIRE AND ROADWAY

References

[1] Hadekel. R., The Mechanical Characteristics of Pneumatic Tyres, S. and T. Memo

No. 10/52 (British Ministry of Supply, TPA 3/TlB, 1952). [2] Michael, F., Zur Frage der Abmessungen von Luftreifen fiir Flugzeuglaufrader (The

Problem of Tire Sizes for Aircraft Wheels), Jahrbuch 1932 der D.V.L., 3, p. 17. Available in English translation as NACA TM 689 (1932). [3] Dodge, R. N. (Unpublished data). [4] Stresses Under Moving Vehicles, Report 4 (U.S. Army Engineer Waterways Experi- mental Station Technical Report No. 3-545, July 1964). [5] Seitz, Norbert, Die Krafte in der Bodenberubhrungsflache schnell rollender Reifen,

Fortschritt-Berichte, VDI-Zeitschrift 12 (19), 46 (1968). [6] Smiley, R. S., and Home, W. B., Mechanical Properties of Pneumatic Tires with Special Reference to Modern Aircraft Tires, NACA Technical Note 4110 (Jan. 1958). [7] Whittemore, H. L., and Petrenko, S. N., Friction and Carrying Capacity of Ball and

Roller Bearings, Nat. Bur. Stand. (U.S.), Tech. Note 201 (1921). [8] Bohm, F., Mechanik des Giirtelreifens, Ingeniur-Archiv 25,82 (1966). [9] Fiala, E., and Willumeit, H.-P., Radiale Schwingungen von Giirtel-Radialreifen, A. T. Z. 68 (2), 33 (Feb. 1966). [10] Clark, S. K., The roiling tire under load, SAE Paper No. 650493 (SAE Midyear Meeting, Chicago, 111., May 17-21, 1965).

[11] Dodge, R. N., Prediction of pneumatic tire characteristics from a cylindrical shell model. Report 02957-25-T, (Office of Research Administration, The University

of Michigan, Ann Arbor, Mich., March 1966). [12] Saito, Y., A study of dynamic steering properties of pneumatic tires, Proc. 10th FISITA Congress, Tokyo, 1964. [13] Novopol'skii, V. I., and Nepomnyashchii, E. F., The interaction of a motor vehicle

tyre tread with the road surface. Abrasion of Rubber, D. I. James, ed., (Palmerton

Publishing Co., New York). [14] Biderman, V. L., Volodina, T. N., and Pugin, V. A., Determining the relationship between the deformation of a tyre carcass and the displacement of the carcass

relative to the road, Sov. Rubber Tech. 25 (7), 37 (July 1966). [15] Novopol'skii, V. I., and Tret'yakov, O. B., Slip of the elements of the tread pattern

in the contact area of tires, Sov. Rubber Tech. 22 (11), 25 (Nov. 1963). [16] Horne, W. B., Yager, T. J., and Taylor, G. R., Recent research on ways to improve

tire traction on water, slush, or ice. Paper presented at AIAA Aircraft Design and

Technology Meeting, Los Angeles, Calif., Nov. 15-18, 1965.

[17] Kelley, J. D., and Allbert, B. J., Tread design of tire affects wet traction most, SAE

Journal 76 (9), 58 (Sept. 1968). [18] Bobbins, D. H., The contact of certain elastic shells with rigid flat surfaces. Report

05608-7-T, (Office of Research Administration, The University of Michigan, Ann

Arbor, Mich., Sept. 1965). [19] Akasaka, T., On the bending of a tire, Series B-9, Report No. 19, Chuo University,

Tokyo, Japan.

[20] Hertz, H. J., reine angew. Math. 92, 156 (1886). See also Timoshenko, S. P., and

Goodier, J. N., Theory of Elasticity (McGraw-Hill Book Co., New York, 1956). [21] Kraft, P., Force distribution in the contact surface between tire and runway, NACA

TM 1365 (Aug. 1954). [22] Markwick, A. H. D., and Starks, H. J. H., Stresses between tire and road, J. Inst. Civil Engineers 16, 309 (1941). [23] Bode, G., Krafte und Bewegungen unter rollenden Lastwagenreifen, A. T. Z. 64

(10), 300-306 (1962).

[24] Green, A. J., Normal stresses at the tire-soil interface in yielding soils (U.S. Army

Waterways Experiment Station Report, Vicksburg, Miss.).

[25] Freitag,'D. R., and Green, A. J., Distribution of stresses on an unyielding surface beneath a pneumatic tire (U.S. Army Waterways Experiment Station Report, Vicksburg, Miss.). [26] Hofelt, C, Factors that influence skid resistance. Part V: Effect of speed, load dis- tribution, and inflation, Proc. First International Skid Prevention Conference,

Charlottesville, Va., Aug. 1959.
REFERENCES 499

[27] Frank, F., and Hofferberth, W., Mechanics of the pneumatic tire, Rubber Chem.

Tech. 40 (1), 271 (Feb. 1967). [28] Martin, F., Theoretische Untersuchungen zur Frage des Spannungszustandes in Luftreifen bei Abplattung, Jahrbuch der deutschen Luftfahrtforschung, Teil I, 470 (1939). [29] Vlassov, V. Z., Izvest. Akad. Tekh. Nauk. No. 6, 819 (1949). [30] Br»hm, F., Zur Mechanik des Luftreifens, Habilitationschrift th., Stuttgart, 1965. [31] Zakaharov, S. P., and Novopol'skii, Distribution of specific pressure of a tire on the road at high velocities (in Russian), Trudy Tauchno-Issledovatel'skoga Instituta Shinnoi Pronyschlennosti, Sbornik 3, Methody Rascheta i Ispaytaniya Automobil'nykh Shin: U.S.S.R. Nauchno, Moscow: 1957, p. 139-53; Rubber Abs. 36,

439 (Sept. 1958). [32] Essenburg, F., and Gulati, S. T., On the contact of two axisymmetric plates, J. Appl. Mech. 33 (2), 341 (June 1966). [33] Vermeulen, P. J., and Johnson, K. L., Contact of nonsphericai elastic bodies trans- mitting tangential forces, J. Appl. Mech. 31 (2), 338 (June 1964). [34] Cough, V. E., Tyre to ground contact stresses. Wear 2, 107 (1958/59). [35] Cough, V. E., Practical tyre research, SAE Trans. 64, 310 (1956). [36] Cooper, D. H., Distribution of side force and side slip in the contact area of the pneumatic tire, Kautschuk und Cummi 1 1 (10), WT 273-277 (Oct. 1958). [37] Clark, S. K., Untersuchung der Verteilung des Auflagerdruckes in Luftreifen (An

analysis of contact pressure distributions for pneumatic tires), Kautschuk und Cummi Kunstoffe 8, 433 (1969). [38] Iritani, S., and Baba, T., Forces on the contact patch of the tyre. Paper No. 3.1, Tenth

International FISITA Congress, Tokyo, Japan, 1964. [39] Mindlin, R. D., Mechanics of Cranular Media, Proc. Second U.S. Nat. Cong. Appl. Mech. (1954). Pubhshed by A.S.M.E., New York.

CHAPTER 6

Skid Resistance and Directional

Control

A. Schallamach ^ Page 6.1. Introduction 502

6.2. Theory 502

6.3. Cornering and side force 508

6.4. Braking and circumferential slip 515

6.5. The speed dependence of rubber friction 518

6.6. Hydrodynamic effects on wet roads 520

6.7. The velocity dependence of the braking coefficient on wet

roads 523

6.8. The effect of tread pattern and carcass construction on tire friction 526

6.9. Tread compound effects on wet friction 531

6.10. Aquaplaning (hydroplaning) 534

6.11. Winter tires 539

References 544

' The Natural Rubber Producers' Research Association, Welwyn Garden City, Hertfordshire, England.

501
502 SKID RESISTANCE AND DIRECTIONAL CONTROL

6.1. Introduction

Speed and direction of a self-propelled vehicle are primarily controlled

by the forces between tires and road. These forces have an upper limit set by the available coefficient of friction; once the ratio between hori- zontal traction and normal pressure exceeds this limit anywhere in the contact area, local sliding occurs. It is important to remember that slid- ing friction has no preferred direction; effective control is therefore

lost when sliding extends over the whole contact area. This happens, for example, when the wheels are locked; then the mass center of the vehicle

slides to a stop in the direction in which it was last traveling when the

wheels were locked. It would appear at first sight that such a contingency arises whenever the circumferential velocity of the wheels rela- tive to the vehicle differs in direction or magnitude from the traveling

velocity, because the whole contact area should be expected to slide under these circumstances. This reasoning is, however, valid only for the

extreme case of an infinitely stiff wheel rolling on an infinitely stiff track. Possibly the most valuable property of an elastic wheel like the pneumatic tire is that it can travel at an angle to its plane and/or with a

velocity differing from its circumferential velocity without involving the

whole contact area in sliding motion.

Another Chapter [If describes how horizontal forces acting on the

wheel axle produce tractions and consequent strains in and near the contact area. These strains lead to differences between traveling and

circumferential velocities, but can at the same time ensure adhesion

between tire and road over at least part of the contact area (normally the

front part) and thus allow control to be maintained.

6.2. Theory

In order to facilitate further discussion, the relevant facts on the

connection between the forces applied to a wheel and changes in its velocity shall be briefly reviewed, and equations derived which allow a semiquantitative assessment of the functional relations. Aerodynamic

forces and rolling resistance will be neglected.

The treatment is made easier by introducing the concept of slip, s, which is a vectorial kinematic quantity defined by eq (6.1) s-(v-V)/|v| (6.1) where v is the velocity of the road relative to the wheel axle, and V is the circumferential velocity of the wheel in the plane of the contact area.

- Figures in brackets indicate the literature references at the end of this chapter.
THEORY 503

also measured relative to the axle of the wheel. When the wheel rolls in a direction making the angle 6 with its plane (side slip),

s=sinO. (6.2) Pure circumferential slip is given by eq (6.3) s=l-V/v. (6.3) When braking {v>V), s is positive with the maximum value of unity

for a locked wheel. During acceleration {v < V) , s is negative and be- comes negative infinity when the stationary wheel spins. Circumferential

slip has a simple physical meaning. During braking, kinetic energy is consumed in brakes, tires and road; even when the road is not appreciably deformed, it takes up a great part of the braking energy as heat. The sUp s is the proportion of energy lost in tire and road and clearly

cannot exceed 100 percent.^ Similarly, part of the engine power is lost in the tires during acceleration; s is the ratio between losses in the tires and road, and gain in kinetic energy of the vehicle. Hence its negative

sign and its infinite upper limit. The strains set up in the contact area of a slipping wheel are illustrated by the model experiment of figure 6.1 which shows the contact area of a

small, solid wheel on a transparent track. All cases have in common

that a circumferential element of the wheel on entering the contact area adheres to the track at first. As the element moves further into the contact area, the imposed slip produces a deflection which increases linearly

with increasing distance from the front edge. This is most clearly seen

for side slip, figure 6.1b, where the deflection is normal to the plane of the

wheel and increases at a rate equal to tan 0. The accompanying surface

stress increases in the same sense until the local value of limiting fric- tional stress is reached and the element begins to slide back toward its undeformed position.

A braking force, figure 6.1c, lengthens an element in the circumferential direction before entering the contact area, and the element adheres

at first to the track in this state of strain; the deflection of the wheel in- creases linearly with increasing distance from the front edge at the rate {v—V)IV= si (1— s). As with side slip, sliding starts (towards the front of the contact) when the ensuing stresses reach the local frictional stress

limit. A driving torque, figure 6. Id, produces contraction of an element

before entering the contact region; traction and deflections have the

opposite sign to those for a braked wheel, and an element finally slides out of the contact area. An exact calculation of the forces on a slipping wheel presents considerable difficulties for tires because of their complex structure, but

useful expressions can be derived for a simple model which replaces the

tire by a toothed wheel, it being assumed that the teeth can deform indePositive slips greater than 100 percent can, in principle, be achieved by making the rotation of the wheel oppose the

direction of motion.
504 SKID RESISTANCE AND DIRECTIONAL CONTROL

(a)

(b)

(0

(d)

DIRECTION OF TRAVEL

Figure 6.L Model experiment to illustrate the conditions in the contact area of slipping

wheels.

(a) Free rolling; (b) side slip (crab walk); (c) braking; (d) accelerating. Traveling direction from right to left. From Schallamach, ref. [2J.

pendently of each other and obey Hooke's law in their stress-strain rela- tionships [3]. The coefficient of friction must enter the calculations.

The discussion in chapter 1.2 [4] has emphasized the dependence of rubber friction on load, temperature and sliding speed, all of which vary

in the different parts of the contact. A constant coefficient of friction can

nevertheless be assumed in a first theoretical attempt without grossly

violating physical reality; some justification for this approximation will be given later in this chapter.
THEORY 505

Figure 6.2. (a) Horizontal traction in the contact area of a wheel with side slip; (b) trac- tion resultingfrom simultaneous side and circumferential slips. Hp is the limiting friction; the shaded area gives the side force in both cases. The tangential force ts per unit length of contact developed when the

model rolls with side slip is shown in figure 6.2a. Because of the postulated validity of Hooke's law, the traction increases initially at a rate proportional to the stiffness kg of the wheel,

ts = ksX tan 6 (6.4)

where x is the coordinate of the surface element relative to the undistorted wheel. The limiting tangential traction /mp is indicated by the semioval curve; /x is the coefficient of friction and p, the assumed normal

force distribution per unit of length. When ts = fJip at the point x —X,

sliding begins, and ts in the rest of the contact is given by

ts = fJip (6.5)

The shaded area represents the sideforce due to the slip sin 6. A diagram similar to figure 6.2a would depict the force distribution

tc due to circumferential slip, with eq (6.4) being replaced by eq (6.6)

tc = kcxsl{l—s). (6.6) The stiffness kc is considerably greater than kg.
506 SKID RESISTANCE AND DIRECTIONAL CONTROL

Sideforce S, braking force B and driving force A are therefore formally

given by the same type of expression.

If the normal pressure distribution is taken to be elliptical along the contact area, and constant across it, the calculation yields eq (6.7) in which F stands for S, B or A:

f=(,.L/;7) (sin--^ + y^) (6.7) L is the normal load, and c is defined by eqs (6.8) and (6.9):

77 c = — {ksa^lfJiL) tan 6 for side sHp, (6.8)

c = — {kca^l^L)sl{\—s) for circumferential slip, (6.9) o a = length of the contact.

It is easily seen that eq (6.7) reduces at low slip to

^small slip= y^ksO^O, (6.10)

^small slip ~^small slip" ^/2A:ca^5. (6.11)

The coefficient of friction has disappeared from (6.10) and (6.11)

because the contribution from the sliding region in the contact area becomes negligible and only the wheel stiffness matters. The linear

relation between force and slip reflects Hooke's law. At large slip and/or

a low coefficient of friction F tends asymptotically to the value fJiL. Figure 6.3 shows the theoretical dependence of FjfJiL on c. It is in- structive to compare the two components making up the force F, i.e., the contributions from the adhesion and sliding regions which have

also been plotted in figure 6.3, based on the elliptical normal pressure

distribution used to generate eq (6.7). The adhesion component, which

dominates F at low values of c, reaches a maximum at c = l/ VS, and is outweighed by the sliding component when c exceeds about 0.82. What

must be borne in mind is that the great contribution made to F by the

sliding region of the contact at large values of c is directionally controlled

only by the remaining adhesion in the front part of the contact patch, be- cause of the already mentioned direction-insensitive nature of sliding

friction. Tires have at times to sustain lateral and longitudinal slip simultaneously, and the forces resulting from such a condition differ from

those for one kind of slip only. Figure 6.2b illustrates the reason for

this effect by showing the influence of circumferential slip on the side

force. Circumferential tractions tc act then jointly with side tractions

ts in the contact area, giving the total force per unit length t in the form
THEORY 507

Figure 6.3. Force F on a slipping wheel as function of the variable c according to eq {6.7), and its two components.

(6.12)

and sliding sets in when this combined force equals ^p. The point X

is therefore shifted forward as compared with the case of only one of these tractions acting. Furthermore, the frictional force in the sliding

region is now taken up by both the lateral and longitudinal stiffness of the tire so that only part of the sliding friction is available for the side

force. The shaded area giving the side force is much smaller than the

corresponding area in figure 6.2a for side slip only. These considerations

apply to braking and driving forces which reduce the side force equally.

Theory gives the following equation for the side force S as function of

simultaneous side slip sin 6 and circumferential slip 5: S = ks sin 6

where

and

/jlL 77 (kl sin' O^k'is'Y'

2c 2c sm

1 + c^ l + c^ (6.13)

s = cos 0 — V/v

77 (A-i Sin2 0+^2^2)1/2 /^2

8 COS 0 — 5 KfJiL/ The ratio between side force and either braking or accelerating force is SIB = SIA = (ks sin d)l{kcs) (6.14)

The relevance of these considerations to the control of a vehicle is to be discussed now.
508 SKID RESISTANCE AND DIRECTIONAL CONTROL

6.3. Cornering and Side Force

The sketch in figure 6.4 shows a highly ideahzed vehicle with equal

front and rear weight distribution traversing a circular curve. It has

been assumed when drawing this picture that the car is conventionally

rear-driven, and that its mass center coincides with the geometrical

center. The vehicle has to counter the centrifugal force, marked as the vector C, with its components €« and Cp. To balance the centrifugal force, the

wheels have to make the slip angles 6/ and Or with the instantaneous velocity V in order to develop the side forces Sr and S/at right angles to their

Figure 6.4. Equilibrium offorces on a rear-driven vehicle describing a circular curve in the stationary state.'*

Rolling resistance, wind resistance, weight transfer and self-aligning torque have been neglected.

^ Continental spellings of such words as centre, tyre, behaviour, etc. have been changed in the text and figure captions

to the Americanized forms. It was not feasible, however, to change them within the figures themselves, where they have been left as originally submitted.
CORNERING AND SIDE FORCE 509

planes. The sum of these cannot by itself equilibrate C; an additional

force has to be supplied in the direction of the chassis in order to offset the drag forces D/ and D,- of the slipping tires. This force is the source

of the energy loss incurred when negotiating the bend and is, incidentally,

a measure of the wear accompanying cornering. It is, of course, delivered

by the engine. The front wheels, being free to rotate, cannot take up a circumferential force, and their contact area has only to cope with the

side force. The driving force D for both front and rear wheels has there- fore to be produced by the rear wheels, bringing about circumferential

slip in addition to side slip. If front and rear wheels carry the same load

and are inflated to the same pressure, the rear wheels need then a larger

slip angle than the front wheels to produce a given side force, for reasons

discussed at the end of the preceding section. When specifying different

inflation pressures for front and rear tires, the car manufacturer may

take this effect into account and also consider the weight distribution. In any case, the cornering maneuver is governed by the relation between

side force and slip angle, and by the influence of circumferential slip on

this relation. The full lines in figure 6.5a give experimental data on the slip angle

dependence of the side force under various road conditions and clearly

demonstrate the effect of the coefficient of friction on magnitude and deFiGURE 6.5a. Full lines, slip dependence of the side force of a smooth 5.00-17 tire on

asphalt under different weather conditions; load, 307 kg. The dotted lines are theoretical curves. From Fiirster, ref. [5].
510 SKID RESISTANCE AND DIRECTIONAL CONTROL

tailed shape of the side force curves. Figure 6.5a refers to a smooth tire, and the influence of moisture on the road is greater here than for patterned tires but this does not detract from the general validity of the

observed changes.

The upper three curves have the same initial slope, and the side force

at small slip is independent of the condition of the road. This is the region

where the lateral stiffness of the tire dominates the side force. The lowest curve in figure 6.5a appears to be an exception to this rule, but its initial slope is difficult to determine from the published figure. This curve is of

interest, though, because it shows the extremely low friction coefficient

(about 0.075) possible on a wet road after dryness and is often experienced in practice as a driving hazard. The most probable reason for this effect is the formation of a thin layer of viscous mud on the road which

leads to hydrodynamic effects dealt with in a later section. All curves are similar to the theoretical F vs. c curve in figure 6.3; the measure of quantitative agreement is shown by the dotted curves in figure 6.5a which have been calculated by means of eq (6.7). This required

a knowledge of the quantity ksd~ appearing in the expression for c, eq

(6.8). As this chapter is not concerned with the derivation of tire parameters from tire construction, the various factors in eqs (6.7) and (6.8) were

obtained by superimposing a doubly-logarithmic plot of the experimental

curve on a corresponding plot of the theoretical curve. The horizontal dis- placement to effect coincidence gives {7T/S)ksd^lfJiL, and the vertical

displacement gives fxL so that all factors, including kgd^ are found. The

resulting theoretical side force curve for the dry road is practically indis- tinguishable from the experimental curve at slip angles up to 10°. The

second theoretical curve in figure 6.5a (rain^ dirty) was calculated with

the same value of ksa~, and a coefficient of friction giving best overall fit. The semiquantitative agreement between theory and experiment shown

in figure 6.5a indicates that the theory can form the basis for further

reasoning.

Beyond the region of small slip in which the side force is determined by

the stiffness of the tire, the curves break away from the initially linear

side force vs. slip relation and approach their limiting values at slip

angles which become progressively smaller as friction decreases. Apart

from the obvious restriction on speed and turning radius imposed by low

friction, operating at side forces near the limit fxL can become difficult,

because of the following reason. The diagram in figure 6.4 of the forces balanced by the tires of a cor- nering vehicle shows that the normal component of the centrifugal force

acting on a front wheel, V4C,,, must equal the component 5/ cos 6>/of the

side force. As the two factors of this product vary with 6/ in opposite

senses, its value has a maximum at a certain critical slip angle dc. The

conditions are best visualized by a polar plot of side force vs slip angle,

figure 6.5b; the ordinate gives immediately S cos 0. The two curves in figure 6.5b have been drawn for a tire with a stiffness kga^lL of 11.32

and coefficients of friction of 0.91 and 0.25. The angle dc for maximum

(S/L) cos 6 can be calculated from the theoretical expression for the
I CORNERING AND SIDE FORCE 511

Figure 6.5b. Polar diagram of side force vs. slip angle for a passenger car tire. At large slip angles, the curves are circles centered in the origin.

side force given by eq (6.7), and is found to be about 21° for />t = 0.91,

but only about 10° for /Lt = 0.25 and 7.5° for /x = 0.15. The practical aspect of these findings is not only that the effective

cornering force S cos 0 becomes independent of slip near the critical

slip angle 6c but that at angles exceeding dc the turning radius at constant

velocity increases with increasing slip angle. This reversal in the usual

response of turning radius to changes in slip angle is likely to occur on

slippery or icy roads and, of course, severely impairs handling of the

vehicle. The complications which arise from braking in a curve come from

the reduction in side force by longitudinal slip, as explained in figure

6.2b. Figure 6.6 gives experimental examples of this effect; more comprehensive data for cross ply and radial tires on various wet surfaces

have recently been given by Holmes and Stone [47]. Apart from lowering the limiting side force, braking forces bring forward the departure

from the initial linear dependence of side force on slip to smaller slip

angles. Equations (6.13) and (6.14), though agreeing qualitatively with

these findings, predict smaller effects than found experimentally. The

probably reason is that theory assumed mutual independent of stresses due to simultaneous lateral and longitudinal slip.

The consequences of braking in a curve are shown schematically in figure 6.7. The left hand side shows a tire moving along the circular cruve

of radius R under equilibrium conditions, the centrifugal force C and

driving force D being balanced by the side force S. A braking force B

(right hand side of fig. 6.7) reduces the side force at the existing slip

angle to S' . The resultant of B and 5' has a smaller radial component
512 SKID RESISTANCE AND DIRECTIONAL CONTROL

0 2 4 6 8 10 12 Figure 6.6. Effect of braking torques {given in brackets) on the side force of a 4.75-17 tire at 30 km/hr. on dry asphalt; load 305 kg. From Fiirster, ref. [5].

than the original force S. Only the centrifugal force C can be balanced,

and the radius of the turn is increased to R'

.

A subject closely connected with side force characteristics of tires is oversteer and understeer. Figure 6.8 shows diagrammatically a car travel- ling straight in the direction v and suddenly hit by a disturbing force F

normal to v. The response of the vehicle to this force is shown for the case that the rear wheels have a lower side force coefficient and hence

drift off at a greater slip angle than the front wheels; after a short in- stant, the vehicle moves from position 1 into position 2. It is immediately

seen that it moves on an arc with the center of curvature on the right

hand side of the drawing. A centrifugal force is thus engendered in the same direction as F and reenforces its effect. A self-deteriorating state

is brought about which can make control difficult, particularly at high

speeds.

The response of a vehicle to a disturbing force sketched in figure 6.8

is a simple case of oversteer. The SAE (J 670 a) defines oversteer more

generally in terms of the change in turning radius on changing the steer- ing angle. The example given here is covered by the SAE definition as

limiting case when the turning radius changes without change in steer- ing angle. Understeer is the response of a vehicle to a disturbing force

opposite to that in figure 6.8. Extreme oversteer occurs when the rear wheels are locked and have

lost all directional function. If the car happens to make a small angle with

the travelling direction, side force on front wheels and frictional force on

rear wheels leave a couple around the mass center which increases the
CORNERING AND SIDE FORCE 513

Figure 6.7. Effect of an instantaneously applied braking force on the turning radius of a

cornering tire [schematic).

angle. This unstable state can, with inertia effects, make the car turn

front to rear, which is a stable condition. Tires for skid tests are therefore often mounted on the front wheels, and only these are braked.

This discussion of cornering and side force characteristics will be

concluded with a short description of a related phenomenon which can

affect handling of vehicles at large slip. The resultant of the side force

distribution along the tire-road contact area does not pass through its center (compare fig. 6.2a) but is displaced to the rear by a distance called

the pneumatic trail. Side force multiplied by pneumatic trail gives a couple tending to reduce the slip angle. This so-called self-aligning torque

is felt at the steering wheel as a measure of the cornering effort. The

self-aligning torque T for our model wheel is given by eq (6.15) 2 ixaL c

where c is defined by eq (6.8). Following Cough's suggestion [6], the

side force has been plotted in figure 6.9 as function of T, both quantities

being expressed in dimensionless form. The graph reproduces the essential features of experimental curves, in particular the maximum of T.
514 SKID RESISTANCE AND DIRECTIONAL CONTROL

Figure 6.8. Response of a vehicle with oversteer characteristics to a disturbing force F.

1.0

0.8

0.6

0.4

CO

r/a^L

0.01 0.02 0.03 0.04 0.05 0.06 0.07 Figure 6.9. Theoretical relation between side force and selfaligning torque according to eq {6.15).
BRAKING AND CIRCUMFERENTIAL SLIP 515 The self-aligning torque helps in the handhng of the vehicle because the

associated restoring couple on the steering wheel feeds back to the ex- perienced driver information on the side force; in particular, the change

in self-ahgning torque with changing steering angle allows him to judge

the approach to limiting conditions [22]. According to experience, the

self-ahgning torque should be at its maximum when the side force has

reached a value of 0.65 to 0.75 /itL [25].

6.4. Braking and Circumferential Slip

When a braking force acts on the wheels and slip develops, the vehicle covers a longer distance per unit time than corresponds to its rolling

radius and angular velocity; the opposite effect occurs with a driven

wheel. The shp is derived from the ratio between the angular velocity of the slipping wheel, o), and that of an otherwise identical, but freely rolling

wheel, 0)0, thus

5=1— oj/coo (6.16)

which expression comes directly from eq (6.3). In contrast to the slip

dependence of the side force, the slip dependence of the braking force

is, of course, of practical importance up to the maximum value of 100

percent.

Experimental work has been concentrated on braking performance on

wet roads because a great proportion of accidents can be attributed to insufficient braking force under such conditions. Figure 6.10 shows

typical curves obtained with patterned and smooth tires on two different

wet road surfaces. All curves have a maximum at a slip between 10 per- cent and 20 percent, followed by a gradual decrease of the braking force; a more sudden drop precedes locking of the wheels. Figure 6.10 exhibits

two other significant features. A smooth tire has a lower braking force

than a patterned tire but the difference vanishes at small slip; the effect of the road surface on the braking force can be great and outweigh

differences between patterned and smooth tires but this difference, too, disappears at small slip.

The common initial slope of aU four curves in figure 6.10 agrees with

elementary theory if it is assumed that the stiffnesses of patterned and

smooth tires do not greatly differ; the subsequent shape of the curves

does not conform to theory which predicts an asymptotic approach to a maximum. Attempts at fitting theoretical curves to the data for patterned

tires are shown in the graph. To do this, the tire stiffness was estimated

from the initial slope, which gave kca'^IL = this value is between four

and five times greater than the corresponding parameter for side slip, ksd^lL. The coefficients of friction were chosen to give best agreement

with the experimental curves at moderate slip.

The great difference between the longitudinal and lateral stiffnesses

of tires leads to an important quantitative difference between the slip

dependence of the braking and side forces. It is easily seen from eqs
516 SKID RESISTANCE AND DIRECTIONAL CONTROL

B is braking force, L is normal load.

(6.7) and (6.9), and from a comparison between figures 6.5 and 6.10, that the limit of the braking force B is approached at much lower absolute

values of the slip than the side force. When braking to a stop, the braking

force is therefore almost entirely determined by the available friction between tire and road, and the assumption of a constant coefficient of

friction will be shown to become untenable under such conditions. This

subject is treated in more detail in the next section but certain con- clusions drawn from tire mechanics have a bearing on the problem.

As the braking force increases rapidly with increasing slip, so does

the contribution to the total kinetic energy loss made by the tires, which

is given by the product Bs and increases initially as the square of B. The consequence of this energy dissipation is, of course, a frictional temperature rise in the contact area; there is no principal difference in

this respect between braking and cornering but the energy losses when

cornering at a comparable slip sin d are given by 5 sin 0 and are considerably smaller because the side force is so much lower than the braking

force. Another difference between braking and cornering is the slip depend- ence of the time a tire surface element spends in the contact area. When the wheels are locked (5=100 percent), the element produces

frictional energy continuously and suffers a temperature rise, the sta- tionary value of which is determined by the cooling effect of the road sur- face and the heat transfer from the free parts of the tire surface via

conduction through the tire. If, however, the wheel is still rotating, its dwelling time in the contact area is finite, and the frictional temperature rise is limited by the amount of sliding that has taken place during
BRAKING AND CIRCUMFERENTIAL SLIP 517

this period, the length of which is a/V= alv{l — s) , where a is the contact patch length.

Frictional temperature rise is the most likely reason for maxima in the braking force vs. slip curves observed on dry roads because the

friction coefficient of current tread compounds decreases with increasing temperature at sliding velocities in the range of travelling speeds.

On wet roads, hydrodynamic effects appear to be the predominant cause

of the maxima; the removal of water from the contact area becomes

increasingly difficult as slip increases and a slip velocity develops (see

sec. 6.6). When the brakes of a vehicle are suddenly applied, as in an emer- gency stop or during typical braking trials, the slip increases from zero

to 100 percent within a short interval, and the time dependence of the braking force follows curves similar to those in figure 6.10. The ex- ample given in figure 6.11 reproduces the main features of the curves in figure 6.10, the most prominent of which is again the maximum of the

braking force; it is known from braking experiments with simultaneous

recording of the slip that the wheels lock after the maximum has been

passed (compare fig. 23 in ref. [21]). The results of braking tests are

usually obtained in the form of records like figure 6.11, and the braking

performance of tires is specified by the two numerical values indicated

in that graph: the peak friction coefficient /Xp, and the sliding coefficient

jJLs which gives the constant braking force with locked wheels.

An advice often given for braking on sHppery roads is to "pump" the

brakes, i.e., to apply and release them in quick succession. The main

advantage of this technique is maintenance of directional control by

preventing the wheels from locking; it is also seen to afford the possibility, if used with skill, to operate in a sHp region of higher braking force

than with locked wheels. Aircraft are frequently equipped with braking

systems which automatically prevent locking of the wheels.

Time Trace

Vehicle speed —

Deceleration —

Base line

Deceleration caused by Un

Decelerotion *^ caused by/Xg

Deceleration - Time Curve Figure 6. .n. Time record of speed and deceleration during application of the brakes on a

6.40-15 tire. From Grosch and Maycock [8].
518 SKID RESISTANCE AND DIRECTIONAL CONTROL

The dependence of the coefficients /u,;> and /u-s on relevant parameters

such as driving speed, road condition and properties of the tread are to be discussed now in the framework of existing knowledge of rubber

friction and hydrodynamic effects in the contact area on wet roads. Confining the discussion to the two braking coefficients does not necessarily

restrict the generality of the findings. The limiting values of the side

force appear to be determined by the same factors as the braking force. Figure 6.12 shows that, at least at moderate velocities, the relation be- tween fjis and the side force at large slip is very nearly linear.

0.80

0.70 0.60

0.50 m 0.40

0.30

0.20

0.10O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 S/L SIDEWAY FORCE COEFFICIENT AT 30 MILES/H

(Tests made at 30niile/h with smooth treed

tyres on various wet surfaces) Figure 6.12. Relation between braking coefficient B/L and side force coeffix:ient S/L of smooth tires on various wet roads at low speed.

From "Road Safety", p. 518, ref. [9].

6.5. The Speed Dependence of Rubber Friction

An earlier section of this book has shown that rubber friction, whatever its detailed mechanism, is essentially a viscoelastic phenomenon,

and that its velocity dependence is interrelated with its temperature de- pendence through the rate-temperature equivalence principle characteristic of all viscoelastic processes. This principle states that a change

in temperature from the so-called Standard Reference Temperature 7^

(a material constant) to the temperature T is equivalent to a change in velocity— or rate in general— by a factor ar which is given by the

Williams-Landel-Ferry (WLF) equation. The validity of the WLF equation has been amply verified for dry friction. Recent laboratory work has

revealed a similar principal for lubricated rubber friction, but a change

in temperature is then equivalent to a change in velocity by a factor

ivrlvo) ' «T, where r)T and 170 are the viscosities of the lubricant at the
SPEED DEPENDENCE OF RUBBER FRICTION 519

1.5

0.5

r X57

Distilled Water

1 Jo'" Distilled Water

-8 -6 -2 Figure 6.13. Mastercurves for lubricated friction of unfilled acrylonitrile-butadiene rubber on a grindstone.

Reference temperature, 20° C. From Grosch [10]. experimental temperature and at the reference temperature [10]. Fig- ure 6.13 gives two "master-curves" for unfilled acrylonitril-butadiene

rubber sliding on a grindstone lubricated with distilled water, and with water containing detergent. With distilled water, the curve is very similar to results on dry, rough tracks [11], though friction is generally lower.

The detergent removes the hump on the low velocity side of the incipient

maximum and further reduces the friction level. The hump has, in the case of dry friction, been attributed to molecular adhesion, and the maximum to mechanical energy losses. If these explanations hold here, then

it appears from figure 6.13 that pure water in the interface allows a certain amount of molecular adhesion to be retained, and that adhesion is eliminated only by the detergent.

The velocity dependence of rubber friction is complicated in practice

by frictional temperature rise at any but the lowest sliding velocities; the experimental velocities in figure 6.13 did not exceed 10 cm/sec.

Once a relation between temperature and velocity dependence is estab- Hshed, this problem can be dealt with at least quaHtatively. It is easily

derived that the total velocity coefficient of friction, dfji/dv, is given by

eq (6.17)

dfji dv

d log {aTr)Tlr}o) d

dT

dn

dv\ mi)

where v is the sliding velocity and dT/dv the frictional temperature rise. As both the factors ar and T7r/T7o decrease with increasing temperature,

the frictional temperature rise is seen to reduce the actual velocity coefficient of friction from the value it would have at constant temperature.

This argument supports the assumption made earlier that a velocityindependent coefficient of friction is a reasonable first approach to the
520 SKID RESISTANCE AND DIRECTIONAL CONTROL

conditions in the contact area of tires at moderate slips on dry and wet

roads.

When braking, the slip and therefore the sHding velocity increase rapidly to a value not much below the original traveling velocity. (At 30mph,

the velocity has dropped by perhaps 10 percent when the wheels lock.)

The frictional temperature rise is exacerbated by the simultaneous

reduction in the angular velocity of the wheels, as described at the

end of the last section. The expression between square brackets in eq

(6.17) eventually becomes negative, and the observed velocity coefficient

of friction has then the opposite sign to that at constant temperature.

As {dfJi/dv)T of common tread compounds under operating conditions is positive, a decrease in friction during braking is to be expected in spite of the increase in sliding velocity. These considerations are necessarily

qualitative because the frictional temperature rise in the contact area

of a tire is difficult to estimate. Although the shape of experimental

braking deceleration vs. time curves is consistent with this explanation,

it is clear from the data in figure 6.10 that the temperature dependence

of rubber friction plays only a contributory role in wet skid resistance.

6.6. Hydrodynamic Effects on Wet Roads

Effective friction under wet conditions is obviously conditional

upon some form of contact between tire and road but it is difficult to define what constitutes a frictionally effective contact. Some of the current controversy about the elementary mechanism of wet skid

resistance revolves about this question but there can be no doubt

about the necessity of removing water from the contact area as far as possible.

When two bodies are brought together in a liquid medium, the liquid

resists their approach by hydrostatic pressures which originate from the

viscosity of the liquid flowing out of the gap between them and, at high

velocity, from the inertial forces necessary to accelerate the liquid.

Figure 6.14a represents schematically part of a tire surface about

to contact a flat-ended road asperity at the speed dh/dt through inter- vening water. Neglecting inertia effects, the force P required to bring

these two bodies together when their surfaces are approximately parallel

is -F = constant X r)iDVh^)dh/dt (6.18)

where /) is a length characterizing the lateral dimension of the asperity.

(For a review of the subject and derivation of eq (6.18), see ref. [12].)

P increases rapidly with increasing size of the asperity and with decreasing distance of the bodies; it is impossible in theory to remove all the

liquid between them. The pressure distribution across the gap is of particular importance. It has a peak and is, in the simple case of a cylindrical asperity, parabolic with a maximum twice the average value. In consequence, the tire surface is indented, and liquid can be trapped

between tire and asperity.
HYDRODYNAMIC EFFECTS ON WET ROADS 521

- D ^ ^ D

(a) (b) Figure 6.14. (a) Pressure distribution in the gap between two bodies approaching each

other vertically in a liquid medium at the velocity dhjdt; (b) Pressure distribution in the gap between an infinite plane and an inclined plane of limited extension moving relative

to each other with the velocity v. An instructive laboratory result is quoted in figure 6.15a [13]. A soft rubber hemisphere of 2.3 cm. radius was put on glass plate covered with

a viscous liquid (19 = 10 stokes) under a load of lOg. The profile of the

rubber surface in the contact region after 5 sec. and 5 min., as deter- mined interferometricaUy, demonstrates the trapped liquid bell and its persistence.

Squeeze films are drastically reduced in thickness by making the

lateral dimensions of the contacting bodies small, since these enter eq (6.18) to the fourth power. This remedy as applied to tires is, of course, to subdivide the tread into a tread pattern. The topology of a skid resistance road surface must fulfiU two requirements: an open

macrotexture to facilitate gross drainage, and microharshness of the

asperities to produce subdivision of the surface into sharp points which

can penetrate the remaining water film because of the large pressures

at the contact points [12]. With this in mind, it is useful broadly to

FILM THICKNESS FILM THICKNESS

5 SECS^

-v-/-\

11

1 5 WINS. \^—r 1 6 8 DIAMETER

(a)

6 8 DIAMETER

(b) Figure 6.15. (a) Profile of the surface of a rubber sphere in contact with a flat surface covered with a viscous liquid, ib) Profile of the same rubber sphere sliding toward the

left. Scales: abscissa. 3.5 divs = 1 mm: ordinate. 1 div = 2000 Angstrom units. From Roberts and Tabor [13].
522 SKID RESISTANCE AND DIRECTIONAL CONTROL

SURFACE

SCALE OF TEXTURE

MACRO

(LARGE) MICRO

(FINE)

A ^^^^

ROUGH HARSH

B ROUGH POLISHED

C SMOOTH HARSH

D SMOOTH POLISHED

Figure 6. 16. Classification of road surfaces. From Sabey [14].

classify road surfaces into the four prototypes shown in figure 6.16, due

to Sabey [14]. This classification, first suggested by AUbert and Walker

[22], is paralleled by the four basic surfaces proposed by Percarpio

and Bevilacqua [15] for the laboratory assessment of the skid resist- ance of tread compounds, using the Portable Skid Tester developed by

the British Road Research Laboratory [16]

.

A second manifestation of visco dynamic effects occurs when two

bodies slide over each other and their surfaces make an angle as sketched

in figure 6.14b. The same quantity of water intercepted per unit time at the wide side of the gap must also leave it at the narrow end and hence

acquire velocity during its passage through the gap. Even if inertial effects are neglected again, the viscosity of the liquid requires, to maintain motion, a hydrostatic pressure the distribution of which is shown in figure 6.14b for the conditions sketched there. For the basic

theory see reference [17]

.

The important aspect of the effect is that it is not contingent on a deliberately provided wedge-shaped gap. Experience shows that a

liquid wedge can automatically form even in nominal point contacts because the hydrostatic pressure developed in the wedge is great enough

to produce sufficient elastic deformation of the contacting bodies to accommodate it [18]. This is, in fact, the basis of hydrodynamic lubrication of machine parts.

A good example of how little is needed to create these conditions is given in the paper by Roberts and Tabor [13]. Figure 6.15b shows the center profile of their rubber sphere when sliding over plate glass at 0.1 cm/sec, and exhibits the liquid wedge in the interface; the coefficient of friction was only 0.04. The theoretical treatment of hydrodynamic lubrication needs numerical methods for its evaluation. Elementary theory predicts that the min-
VELOCITY DEPENDENCE ON WET ROADS 523 imum thickness of the liquid wedge, hmin (see fig. 6.14b), under a load

P should be given by an equation of the type

/imin= const. Xi)3/2 (^^/p)l/2 Qg)

From this, it is easily deduced that the "frictional force" is proportional

to (171;)^/^, as has been confirmed by Cohen and Tabor [19] when sliding a steel ball over lubricated rubber at sufficiently high speed.

Moore [20] has theoretically investigated the case of a 2-dimensional

rounded asperity sliding over rubber lubricated with water, and has

found very low values for the expected coefficient of friction (about 0.01). He comes to the conclusion that the much higher values of commonly

observed skid resistances must be due to microroughness of the road

asperities penetrating the water film. It will be seen, therefore, that both

the reduction of squeeze films and the prevention of hydrodynamic lub- rication require harshness of the road cover of the kind indicated by

surfaces A and C in figure 6.16. The eff'ect of squeeze films and hydrodynamic lift on the whole contact area of a tire rolling on a wet road can be assessed now. In the

absence of slip, a squeeze film is maintained under the rolling tire with an overall thickness which decreases towards the rear of the contact because sinkage increases with time of contact. The film will, in general,

be pierced by sharp points on the road. When the brakes are applied, or a curve is entered, and a slip velocity Av develops, a hydrodynamic lift comes into operation and the film increases in thickness according to eq (6.19), as (iqAv)^''^. It is as if water were pumped into the contact area. The water wedge intervening between tire and road is extended toward

the rear of the footprint. The number of individual contacts between tire and road asperities, which are naturally distributed in height, is reduced

but remains more concentrated at the rear than near the front part of the

footprint. The consequent loss in braking or cornering force increases

therefore with the slip and the traveling velocity.

Squeeze film thickness and hydrodynamic lift increase also with the

viscosity, as seen from eqs (6.18) and (6.19). This explains the severe

loss in friction on muddy roads even at moderate velocities, as exemplified by the lowest curve in figure 6.5a. The effect of inertial forces arising in the water film at high speeds

will be discussed separately in connection with hydroplaning.

6.7. The Velocity Dependence of the Braking

Coefficient on Wet Roads

The braking performance of tires is generally judged by the deceleration of the vehicle on which they are mounted when the wheels are braked

for a short time at various initial speeds. The test tires are usually fitted to the front wheels, and only they are braked in order to maintain directional control, as discussed in section 6.3 in connection with oversteer. The braking system is often modified by an adjustable needle valve in the

brake fluid line to give an approximately constant interval between appH-
524 SKID RESISTANCE AND DIRECTIONAL CONTROL

cation of the brakes and locking of the wheels. The deceleration is re- corded to give curves similar to that shown in figure 6.11 from which

the braking coefficients jJip and fis are determined by dividing the braking force by the load carried on the front wheels. Allowance is made for the load transfer during braking because the deceleration force acts on

the vehicle at the height of the center of gravity.

Maycock [21] has carried out skid measurements on various wet

road surfaces, using for his experiments smooth tires, tires with seven

straight ribs and similar seven-ribbed commercial tires with sipes. From

his data, we reproduce in figures 6.17 and 6.18 the results obtained with

oil-extended SBR treads on the following four road surfaces:

a. Quartzite, a rough, harsh surface resembling prototype A in fig- ure 6.16.

b. Round gravel, like surface B in figure 6.16.

c. Fine, cold asphalt, approximating surface C.

d. Polished concrete, with a texture intermediate between C and D.

All surfaces were kept wet by spray-bars; the open structure of quartzite and gravel afforded much better drainage than the other two

surfaces. The speed dependence of the braking coefficients was sensibly linear in the experimental range, with the exception of smooth

tires on the badly drained surfaces for which the curves were convex

0.8

0.6

0.4 02

Gravel s

1 Hbd

Sm.

mph. 20

Spd^

Concre e

1 Sm. 30 40 50 60 20 30 40 50 60 Figure 6.17. Velocity dependence of the peak braking coefficient of oil-extended SBR

tires on various wet roads.

From Maycock [21].
VELOCITY DEPENDENCE ON WET ROADS 525

Figure 6.18. Velocity dependence of the sliding braking coefficient of oil-extended SBR

tires on various wet roads. From Maycock [21].

to the speed axis. (The linear braking coefficient-velocity relations in these and later figures must not be extrapolated to higher velocities.)

The sliding coefficient of smooth tires on concrete was less than 0.05

and has been omitted from figure 6.18. The graphs in figures 6.17 and 6.18 forcibly demonstrate the inter- action between tire surface and road surface in determining the available

friction. Considering the ribbed and commerical siped tires together as patterned tires, and comparing their performance with that of smooth

tires, it is immediately seen that the difference in braking coefficient

is most marked on the asphalt and concrete surfaces. The tread pattern

raises the level of friction and, what is more important, reduces its speed dependence which, however, is still pronounced. In contrast, there

is little pattern effect on quartzite, a high level of friction being maintained over the whole speed range. Round gravel, though obviously an

unsuitable road surface, again produces only a small speed effect. The advantages of a tread pattern over a smooth tire are pronounced

on badly drained surfaces where the configuration of the tire surface

must be relied upon for the removal of water from the contact area. Sabey [14] has generalized this conclusion by stating that the effec- tiveness of a tread pattern is best assessed on slippery surfaces with

bad drainage.

The existence of a peak braking coefficient is understandable from

section 6.5 and, more particularly, section 6.6; a cause contributing to the effect is most probably tha;; water is periodically cleared from the

tread of a rotating tire. The substantial difference between peak and
526 SKID RESISTANCE AND DIRECTIONAL CONTROL

sliding coefficients evident from figures 6.17 and 6.18 shows that at least a vestige of vehicle control can be maintained even on a slippery

road as long as the wheels keep turning. The peak coefficient appears

also to be relevant for another aspect of road safety, the cornering ability of tires on wet roads.

Cornering, or "road-holding" tests, are carried out by determining

the maximum velocity at which the test car can describe a circular curve

of given radius. Considering that the wheels always rotate during a road-holding test, the influence of hydrodynamic effects on the results should be more nearly related to that under peak friction than under

sliding conditions in braking tests. Meades [28] has compared the braking coefficients of a commerical

radial tire on a wet slippery surface (brushed concrete) at speeds be- tween 30 and 85 mph with what he calls the maximum cornering force

but what seems to be the maximum centrifugal force attainable on cir- cular tracks. The results strongly suggest that the maximum supportable

centrifugal force is indeed similar in magnitude and velocity dependence

to the peak braking force when hydrodynamic effects play an important

role in determining friction at high speeds; the linear relation between

side force and braking force during sliding in figure 6.12 was found at a relatively low speed.

6.8. The Effect of Tread Pattern and Carcass Construction on Tire Friction

Most production tread patterns are based on circumferential ribs separated by grooves which must be wide enough to accommodate the

displaced water without developing a hydrostatic pressure. The rib width

should be as small as practicable, according to eqs 6.18 and 6.19, and pre- ferably straight so as not to impede the flow of water [22]. Figure 6.19

gives results obtained by Allbert and Walker [22] with radial tires having respectively four, five, and six grooves of equal width. These tests carried out on a smooth road show the improvement in friction at high

speed effected by increasing the number of ribs. Similar experiments by

Maycock [23] indicate that a further increase in the number of ribs up to 13 produces no significant advantage in braking performance, but All- bert and Walker [22] found the maximum cornering force on an indoor

rig to increase continuously with the number of ribs; the cornering coefficients were very low, though, not exceeding 0.2 at 30 mph. Maycock

also investigated the effect of groove width on the skid resistance of

tires with five ribs of equal width (0.5 in.); the brake coefficient reached

an asymptotic maximum for a ratio of groove width to rib width of about

0.4.The ribs of commercially produced tires are generally zig-zag shaped.

Kelley [24] points out that the obstruction of free flow of water along the

grooves of such tires is compensated by the wiping action of the rib edges

making an angle with the direction of motion. Experimenting with vari-
TREAD PATTERN AND CARCASS CONSTRUCTION 527

Figure 6.19. Effect of number of ribs on the braking coefficient of tires on wet, smooth

mastic asphalt; 185 X 15 radial tires. From AUbert and Walker [22].

ous patterns, he found that stopping traction could be optimized for cer- tain rib shapes to give friction values slightly higher than those of straightribbed tires. Kelley's findings on the effect of rib number on braking are

in broad agreement with those of the previously cited authors, but the

cornering power decreased slightly with increased number of ribs; comparison between his braking and cornering results are difficult because

the cornering experiments were carried out on a considerably rougher

road.

5.0

o

0= 0 0.2 0.4 0.6 0.8 1.0 APPROXIMATE LEVEL OF EFFECTIVE FRICTION (BRAKING FORCE COEFFICIENT FOR SMOOTH RIB PATTERN)

Figure 6.20. Effect of sipes on the braking coefficient of tires on seven wet road surfaces

at speeds up to 70 mph.

From Allbert and Walker [22].
528 SKID RESISTANCE AND DIRECTIONAL CONTROL

The influence of sipes on the braking coefficient is not too clear from

the data in figures 6.17 and 6.18 but a more detailed analysis of all re- sults shows them to be most effective on badly drained and hence slippery surfaces, and that the advantage increases with increasing speed.

Allbert and Walker [22] diagrammatically represent the achievable improvement in braking force derived from sipes by the graph in figure

6.20. Kelley [24] has given data for the stopping traction (at an unspecified speed) of two differently siped, and an unsiped tread with five zigzag ribs. Predominantly diagonal sipes in a five-ribbed tire improved

braking on a low-friction surface by some 80 percent as compared with

the plain-ribbed tire. A rather different tread pattern with only three ribs,

the center rib being very broad and having transverse siping, raised the

braking coefficient by some 220 percent. On a rough road, the differ- ences were very small, and the cornering power on a surface of inter- mediate roughness was slightly reduced by siping.

Sipes obviously influence tire friction by further subdividing the tread

surface and consequently reducing squeeze film thickness and hydrodynamic lift. In addition, the great number of edges introduced by siping

provides multiple wiping [25] similar to the action of zig-zag shaped ribs. Any tread pattern can function satisfactorily on a wet road only as long as grooves and sipes constitute a water reservoir of sufficient capacity, and its effectiveness decreases therefore with increasing wear of the

tread. The curves in figure 6.21, due to Sabey [26] , show the decline in sliding friction with decreasing tread depth which is, expectedly, more

pronounced on smooth, i.e., badly drained, than on rough roads.

The equations developed in section 6.2 for the ground forces in the contact area of slipping tires have shown the importance of the stiffness factors ks and kc in determining these forces at low and moderate slip.

Tire stiffness is dominated by the carcass construction of which two

types are in current use, the conventional cross-ply and the radial, or

belted, carcass. Radial carcasses have a number of variants but their

distinguishing feature is a relatively stiff breaker belt under the tread,

and a carcass proper with practically radial cords or wires from which

the construction derives its name. The breaker belt gives the tire great

lateral and longitudinal stiffness so that the necessary ground forces

are obtained at lower slips than with cross-ply tires. It has been stated

by Gough et al. [27], however, that very stiff belts have the disadvantage

of producing a peak in the cornering force vs. slip angle curve, and the

transition from the region where the cornering force originates mostly

from adhesion in the front part of the contact to the region where fric- tional sliding contributes most of the side force can become too abrupt

for satisfactory control; less stiff belts are therefore preferable.

The braking coefficients of treads with identical tread patterns on

cross-ply and radial carcasses differ only little from each other. According to data given by Allbert and Walker [22], the peak coefficient at higher speeds is somewhat greater, and the sliding coefficient lower on

the radial carcass. Kelley [24], comparing various tread patterns on

both radial and conventional carcasses, found only a marginal advantage

of radial tires in stopping power and cornering traction.
TREAD PATTERN AND CARCASS CONSTRUCTION 529

8

8 6 4 2 0

Tread depth (mm)

(a) 30mile/hr.

Tread depth (mm)

(b)50mile/hr.

Tread depth (mm)

(c) 80mile/hr Figure 6.21. Effect of tread depth on wet road skidding resistance at different speeds. From Sabey [26]. Meades [28] has measured the braking coefficients of five commercial

tires each with radial and conventional construction. The mean curves drawn through the experimental points for these two types of tire on

two different surfaces are shown in figure 6.22 and indicate that radial

tires have, on the whole, higher and less speed-dependent braking

coefficients than cross-ply tires on badly drained surfaces where hydrodynamic effects matter. The same author has extended the experimental

velocity range on another surface, brushed concrete, to 120 mph; the

sliding values of the braking coefficients are reproduced in figure 6.23

and show the coefficients for radial and cross-ply tires to become equal

at speeds over 90 mph. This graph is noteworthy for showing the level

of stopping power at very high driving speeds, and the breakdown of the linear braking coefficient-velocity dependence valid at lower speeds.

It is of some interest to estimate the stopping distance from 100 mph,
530 SKID RESISTANCE AND DIRECTIONAL CONTROL

30 40 50 60 70 80 30 40 50 60 70 80

mph mph

Figure 6.22. Average velocity dependence of the braking coefficients of commercial cross- ply and radial tires on two wet road surfaces. From Meades [28].

R

Cro

dial T

ss-ply

"yres - ryres .

1 1 1

' 20 30 40 50 60 70 80 90 100 110 120 130 Speed, mph

Figure 6.23. Average velocity dependence of the sliding coefficient offour radial andfive cross-ply commercial tires, all with synthetic treads, on wet brushed concrete. From Meades [28].

assuming that the velocity dependence of friction followed the curves

in figure 6.23 during deceleration. The result is 577 meters for radial,

and 734 meters for cross-ply tires. Allbert and Walker [22] make the general observation that radial car- casses allow more complex tread patterns and a wider range of tread

materials to be used and thus make it possible to achieve higher braking

coefficients than is practicable with cross-ply tires. The data in figure

6.24 presumably represent the optimum braking coefficients attainable on a smooth road with either carcass construction. From the sizeable

improvement in the peak coefficient at high speeds, one would expect

a similar improvement in cornering power to be effected by radial carcass

construction, according to the reasoning outlined at the end of the

last section.
TREAD COMPOUND EFFECTS ON WET FRICTION 531

Figure 6.24. Velocity dependence of the braking coefficients of production radial and

cross-ply tires on wet, smooth mastic asphalt.

From Allbert and Walker [22].

6.9. Tread Compound Effects on Wet Friction

The hardness of practical treads varies over a small range, and closer

consideration suggests that soft treads may have an advantage over harder ones on some rough roads because of their greater ability to wrap themselves around asperities and thus to produce a larger contact

area. Two natural rubber treads investigated by Grosch and Maycock

[8] differed in hardness by 6 IRHD units because of different levels in

filler loading. Both sliding and peak coefficients of friction of the softer tread were some 13 percent higher on gravel but 5 percent lower on

asphalt and concrete; the two compounds differed little on quartzite.

The two most common tread rubbers are natural rubber (NR) and sty- rene-butadiene copolymer (SBR). NR, formerly the only commercially

available rubber, is now generally reserved for the treads of heavy tires,

and passenger treads are, as a rule, made from SBR. Both rubbers are

often blended with butadiene rubber (BR); SBR, or its blends with BR,

are nearly always extended with a heavy oil. Figure 6.25 compares the braking coefficient of passenger tires with

treads of NR and oil-extended high-styrene SBR (OE H-SBR) [21]. The peak values of OE H-SBR are consistently higher than those of NR,

the improvement amounting to about 27 percent on all surfaces and over

the whole velocity range. The relation between the sliding coefficients of the two rubbers is more complicated because they depend less on

speed for NR than for OE H-SBR; the two compounds reverse ranking

on the badly drained surfaces at moderately high velocities.
532 SKID RESISTANCE AND DIRECTIONAL CONTROL

Figure 6.25. Velocity dependence of the braking coefficients of tires ivith treads of oil- extended high-styrene SBR and of NR on four different wet road surfaces. From Maycock [21].

The superiority in skid resistance of SBR-based treads over NR

observed under many conditions has been attributed to the higher

internal viscosity, or hysteresis, of SBR but authors differ in their

interpretation of these findings, as will be detailed later. The difference

in hysteresis between the compounds in figure 6.25 is accentuated by

the SBR being oil-extended and having a higher filler-level than the NR

(60 pph of super-abrasion furnace black as against 48 pph of highabrasion furnace black.)

A simple measure of hysteresis is derived from the resilience, i.e., the fraction of kinetic energy recoverable after impact on some pendulum

device. If p is the resilience, (1 — p) gives the hysteresis loss; at 20° C,

the losses in the OE H-SBR tread are about 50 percent higher than in the

NR tread as determined from their resiliences. A difficulty arises here

from the fact that hysteresis, a viscoelastic property, depends on rate and temperature. Apart from uncertainty about the local temperature

during sliding, the fundamental deformation period of the process

depends on sliding speed and average distance between the road asperities; this time may be short compared with the impact time of a technological resiliometer. Sabey and Lupton [29], comparing skid ratings on

the Portable Skid Tester [16] with resiliences, found correlation between

the two quantities to be most significant when the resiliences were deter- mined at temperatures some 40° C lower than the skid values. Lowering

the temperature has, of course, the same effect as shortening the impact

time, according to the rate-temperature equivalence principle.
TREAD COMPOUND EFFECTS ON WET FRICTION 533 The influence of hysteresis on wet skid resistance has been more

strictly confirmed by results with one rubber compounded to different

levels of hysteresis. The graphs in figure 6.26 refer to oil-extended NR

(OENR) and ordinary NR treads, both filled with 50 pph of high-abrasion

furnace black. The hysteresis loss at 20° C of the OENR was about 50

percent higher than that of the NR. Figure 6.26 bears a strong resemblance to figure 6.25 although the tread patterns were different in the

two sets of experiments; an intrinsically resilient rubber like NR is seen

to give wet skid resistances of the same order as a synthetic rubber when

compounded for low resilience. A feature shared by figures 6.25 and 6.26

is the more uniform increase in the peak than in the sHde coefficient of

braking when the hysteresis of the tread is increased. From what has

been said before, hysteresis of the tread may therefore be expected to improve cornering power over a larger velocity range than it will improve

stopping ability.

Gravel

Mn

1 mph

Conaefe

mph 30 40 50 60 70 80 20 30 40 50 60 70 80

Oil-Extended NR; NR Figure 6.26. Velocity dependence of the braking coefficients of tires with treads of NR and

oil-extended NR on four different wet road surfaces. From Grosch and Maycock [8]. A consistent theory of the influence of hysteresis on lubricated rubber

friction has been put forward by Tabor [30, 31]. He attributes the fric- tional force solely to mechanical energy losses which occur in the tread

surface when road asperities pass over it. The coefficients of friction found in the laboratory on well lubricated rubber are low; they increase

with the load but it needs about 2 kg. on a steel ball of Vs in. diameter to reach a value of only 0.25 [30]. While the theory is certainly true for near

ideally lubricated friction, rubber sliding on a wet road meets this condition only at speeds sufficiently high to produce a water film which pre-
534 SKID RESISTANCE AND DIRECTIONAL CONTROL

vents direct contact between tread and road asperities. The Tabor

mechanism will then contribute to the frictional force, and quite probably

helps to maintain the low but finite friction at high speeds in figure 6.23. Doubt has also been thrown on the validity of the mechanical energy

theory at moderate speeds by experiments with tires having a thin veneer

of resilient rubber on a tread of low resilience, and vice versa [32]. Service tests gave skid results representative of the surface veneer even when only 0.2 mm. thick. The mechanical energy losses thought to deter- mine wet friction involve, however, a deeper layer so that the hysteresis

of the underlying material should have shown up in the results. The account given by Bulgin, Hubbard and Walters [33] for hysteresis

effects on wet friction starts with an observation on dry friction. The

speed and temperature dependence of the friction of different polymers

are similar, with the curves shifted along the velocity or temperature

axis according to their glass transition temperature (butyl rubber is an

exception) [4, 11]. Maximum friction is reached for a combination of speed and temperature which depends on the glass transition temperature, and can be estimated from an empirical equation relating temperature of maximum friction to glass transition temperature and logarithm

of the sliding velocity. Experimenting with treads having various glass

transition temperatures, friction on wet and dry roads has been found

to increase with increasing glass transition temperature up to values of — 20° C, the operating conditions in the contact area apparently giving

then maximum friction. As treads with high glass transition temperatures become unduly stiff in cold weather (see also sec. 6.11), the best

practical compromise is, according to Bulgin and co-workers, treads

with glass transition temperatures between —30° and —40° C. It has not been established yet whether the improvement in wet skid resistance

brought about by oil-extension falls under this empirical rule. The similarity between the temperature dependence of dry and wet

friction leads then to the conclusion that tire friction on wet roads origi- nates from dry patches in the rear of the contact area where water has

been effectively squeezed out or wiped away (sec. 6.6). The influence of hysteresis on wet friction follows from work on the viscoelastic nature

of dry rubber friction [4, 11].

6.10. Aquaplaning (hydroplaning)

When water is displaced from the contact area at high speeds, inertia effects can no longer be neglected. The pressure in a liquid of density

y accelerated from rest to the velocity v (or stopped from the velocity

v),Pd, is

and should be contrasted with the linear speed dependence of pressures

arising from viscous effects. Moore [12] quotes eq (6.21) for the force

exercised by squeeze films when inertia is taken into consideration:
HYDROPLANING 535 P=-const.x(.,-j-{l-0.l(-j[-/.--/.^—/-j)

(6.21)

which shows that the lateral size D of the approaching bodies affect the

force in the same way as in the absence of inertia effects. Previous

remarks about the bearing of tread patterns and road surface topology on wet tire friction need therefore not be modified but hydrodynamic

lifts become greater.

The photographs in figure 6.27 show the contact area of an aircraft

tire passing over a flooded glass truck at different speeds [34]. As the

speed increases, the water wedge under the front part of the contact

is extended backward until only a small patch adheres to, or is at least

in the proximity of the track. Eventually, the tire is only supported by

water, with almost complete loss of control. This condition is called hydroplaning or aquaplaning. Horne and Joyner [34] give the empirical

equation (6.22) for the velocity Vp at which hydroplaning begins:

?;y>(mph) = 10.35 Vp/(psi) (6.22)

20x4.4 AIRCRAFT TIRE FOOTPRINTS

VERT. L0AD = 500LB;p = 30LB/SQIN.;WATER DEPTH = 0.5IN.

Vg=7I knots Vg=88 knots

Figure 6.27. Contact area on flooded glass of a 20 X 4.4 aircraft tire under 500 lb. load

at various speeds {1 knot^ 1.15 mph; water depth, 0.5 in.). From Horne and Joyner [34].

1
536 SKID RESISTANCE AND DIRECTIONAL CONTROL = inflation pressure) but the authors stress that this equation holds

only for smooth or close-patterned tires, and for ribbed tires when the

water depth exceeds the groove depth. The velocity at which the smooth

tire in figure 6.18 loses discernible friction on wet asphalt very nearly

agrees with eq (6.22). AUbert [35] has shown, however, that the minimum hydroplaning speed of a radial tire on an indoor drum increases

with increasing inflation pressure only as long as the tire deflection is kept constant; at constant load, hydroplaning speed decreases with in- creasing inflation pressure. Although comparison between eqs (6.20) and (6.22) indicates that hydroplaning can originate from the inertia of the water film, it must

not be forgotten that similar effects may occur at much lower speeds.

The danger of a viscous medium on the road has already been mentioned;

Allbert [35] cites instances of hydroplaning of worn truck tires on cer- tain road surfaces at half the velocity predicted by eq (6.22). The more complex conditions leading to hydroplaning of patterned

tires are illustrated in figure 6.28 which gives the speed dependence

of the water pressure under the center groove and adjacent rib of an

aircraft tire rolling over a flooded track. The groove pressure follows

closely the hydrodynamic pressure pa of eq (6.20). The pressure under

the rib rises much faster than pa at low speeds and then begins to level

out; rib and groove pressure become equal when hydroplaning sets in. Below this critical velocity, water can flow from under the rib into the

groove because of the higher rib pressure; at greater speeds, water can no longer safely escape. Horne and Joyner remark that the velocity

dependence of the rib pressure cannot be explained by inertia effects and ascribe it to viscous effects. Equation (6.21) suggests why this could

be so. The inertia term in eq (6.21) vanishes for thin squeeze , films (small h), and the water film under the rib will certainly be thinner than

the water depth in the grooves. A spectacular feature of hydroplaning is that tires stop rotating even though the vehicle moves at considerable speed; rolling resistance and

bearing friction outweigh the drag forces in the water layer. In conse- quence, a catastrophic loss of side force occurs which is quite as serious

as the loss in braking power. This "spin down" and accompanying

decrease in cornering force as observed on an indoor rig is shown in figure 6.29. The speed dependence of the cornering force of patterned

and smooth tires on two different concrete tracks given in figure 6.30

follows the same trend as the corresponding curve in figure 6.29, and

shows directly the importance of tread pattern and road surface topology on cornering power on wet roads. It also demonstrates how road-holding

ability of a patterned tire can disappear in deep water at a relatively

low speed.

Boness [36] has theoretically investigated how the thickness of the

water film under a smooth tire depends on the water depth when only

inertial forces are considered. He finds that for a tire of radius R, contact length a, and water depth H

h^,, = 0.25(a//?)«-66(prf/i^)ii(///i?)o-625 (6.23)
HYDROPLANING 537

240

a200

inv>ool20

c

>»O

-o 80 X

Ib/in2

40 —ci

1/2 yv,2 n

O CENTER GROOVE

ADJACENT RIB

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I Travelling Velocity /Hydroplaning Velocity Figure 6.28. Speed dependence of water pressure under rib and groove of a 32 'X craft tire; water depth, 1 in. From Home and Joyner [34].

150

100

S3 50

10 20 30 40

Drum speed, mph

70

Figure 6.29. Spin down at zero and 4° slip angle, and cornering force at 4° slip on a wet drum rig.

In the shaded region, tire will not spin up again if braked to a stand still. From AUbert [35]. where E measures the stiffness of the tire, and pd is given by eq (6.20). Equation (6.23) agrees well with experimental results for hmm obtained

by Gray [37] with an aircraft tire at 100 mph; it is difficult to see, though,

why smooth tires begin to hydroplane, according to figure 6.30, at about

the same speed in different depths of water. Whether high-speed braking coefficients of the order of 0.1 (fig. 6.23)

and similar values given by Horne and Joyner [34] for patterned tires indicate full hydroplaning, or still contain elements of friction proper
538 SKID RESISTANCE AND DIRECTIONAL CONTROL

Speed, mph Speed, mph — O— Textured Concrete

""^3— Smooth Concrete Figure 6.30. Speed dependence of corneringforce of smooth and patterned passenger tires on two different wet tracks. From Home and Joyner [34].

is a moot point. The length of the stopping distances given earHer are

serious by any reckoning. Nevertheless, given a well-patterned tire and

a good road surface, hydroplaning is hardly a problem at speeds to which

road traffic is limited but it constitutes a hazard for aircraft landing at high speed on wet runways. According to one U.S. Airline, quoted by

Horne et al. [38], loss of control due to this cause occurs every 500th

wet landing.

An effect sometimes accompanying hydroplaning is "reversion", i.e., damage to the tread with all the appearances of local overheating. In one

form, the tire surface has a scar roughly of the size of the contact patch

which is obviously due to frictional heating [38]. Surprisingly this damage
WINTER TIRES 539

has not been noticed after dry landings; the brakes are probably not fully

applied in that case so that the frictional heat is distributed over the

whole tread surface and, furthermore, damaged rubber is removed by

abrasion. The wheels will easily lock on a wet runway. Another type of

reversion is degradation of rubber just under the tread surface, sometimes leaving a top skin of apparently undamaged rubber which can

easily be peeled off [39]. This damage must be due to energy losses below

the surface when the tread slides over asperities (compare the wet fric- tion mechanism envisaged by Tabor, sec. 6.9). The energy density has

a maximum at a depth of about a quarter of the diameter of the contact

area [40]. Reversion has been observed on car and truck tires. AUbert [35] states that for reversion to occur, the inflation pressure must be greater than

50 psi, the wheels must be locked and start to slide at no less than 60

mph, and the water must be shallow with a depth of about 1 mm.

6.11. Winter Tires

The heavily profiled treads of winter tires are designed to provide grip on snow, and to facilitate clearance of the pattern when free of the road.

The Swiss Automobile Club have tested, in cooperation with the Technicum at Biel, traction and braking on hard snow and ice of 14 European

brands of winter tires, two radial tires, and a new and a worn ordinary

cross-ply tire [41]. Two of the winter tires had ice studs, and one tire was

furnished with snow chains. The shaded areas in figure 6.31 give the

range of driving and braking force coefficients at 10 km/hr. of the various

winter tires at different slips; values for the radial tires fall within the same region. The spread of the driving force coefficients at large slips is surprising; the appearance of the different tread patterns gives no immediate clue to the reason for the spread.

Braking force coefficients on ice remained practically constant at slips greater than 15 percent, and ranged between 0.1 and 0.2 for all nonstudded tires. The studded tires gave higher values but only one of them was markedly superior on ice. The effect of studs has been studied in more detail by Bird and

co-workers [42] with special reference to its temperature dependence.

The curves in figure 6.32 give friction coefficients derived from the

authors' calculated stopping distances from an initial velocity of 20 mph.

Studs are seen to improve friction most at temperatures from 0° C to — 10° C, their effectiveness tending to disappear at —20° C. An increase

in the number of studs was most effective if they were inserted in new

rows. An arrangement of 260 studs in eight rows, with 52 studs in the

outer rows and 26 each in the inner ones, raised the coefficient of friction on ice at about — 5° C by 150 percent at a sliding speed of 37.3 mph, and

by 90 percent at 12.4 mph. Sapp [43], who gives no details of the stud

arrangement apart from stating that the outer rows are most effective,

found an almost linear increase in skid rating with the total number of studs on front and rear tires. As one would anticipate, the advantage

derived from studs was less on snow than on ice.
540 SKID RESISTANCE AND DIRECTIONAL CONTROL

Figure 6.3 L Slip dependence of driving and braking force coefficients on hard snow at 10 kmlhr.

From Swiss Automobile Club [41]. The drawback of studs is that they loose grip when their points wear

round, and they damage the road suriface. Sapp gives data for the reduction in the friction on ice of studded tires with continued service. In

a test in which only the rear tires were studded, the improvement in traction over non-studded tires fell from an initial 83 percent to 9 percent

after 10,000 miles; the improvement in breaking power was reduced

from 34 percent to 14 percent.

Another observation of the same author is that studs increased

the cornering ability of cars only when both front and rear tires were

studded. Although this result could almost be deduced from first principles, it was worth practical demonstration.

An interesting point brought out by figure 6.32 in the increase in friction with decreasing temperature. This is an important effect characteristics of rubber friction on ice over a considerable temperature

range. Laboratory results obtained with the Portable Skid Tester [16] for an NR and an SBR tread compound on a wet and, below 0° C, ice- covered track are reproduced in figure 6.33 (the Skid Tester reading is roughly equal to 100 times the friction coefficient). On lowering the
WINTER TIRES 541

Ice Temperature ,X J \ \ \ -20 -15 -10 -5 0

Figure 6.32. Temperature dependence of sliding friction on ice of various combinations of studded and non-studded tires. From data given by Rosenthal et al. [42].

uj 40

8

o

\

ONR

• SBR

-40 -20 0 20 TRACK TEMPERATURE ( °C

)

Figure 6.33. Temperature dependence of the skid resistance of a NR and a SBR tread rubber on a wet or icy track; Portable Skid Tester. From Grosch [44].
542 SKID RESISTANCE AND DIRECTIONAL CONTROL

temperature, friction of both compounds drops to a low value at 0° C

and then rises again, but NR produces now greater friction than SBR,

in contrast to their ranking on the wet track. Service tests with tires have

given similar results [32]. The low friction at 0"" C is, no doubt, due to melting ice forming a well lubricated, smooth track; the temperature de- pendence of friction below 0° C is more difficult to understand.

If ice is taken to be a smooth, dry track, and the frictional behavior

of the two compounds is estimated from the existing laboratory evidence on dry friction [4, 11], NR should have a higher coefficient of friction on

ice than SBR at the speeds involved here (about 300 cm/sec), but the

temperature coefficient of friction should be positive for both rubbers.

This results from the fact that the temperatures in question lie on the

left-hand side of the peak in the friction vs. temperature curve for 300 cm/sec, and from the difference between the glass transition temperatures of the two rubbers. While the higher friction of NR on ice can be explained in this way, the negative temperature coefficients of fric- tion in figures 6.32 and 6.33 suggest extraneous effects. Bowden and Tabor [45] report a similar increase with decreasing temperature in the friction of skis and attribute it to ice or snow being melted

in the contact area by frictional heat. The estimated thickness of ice

melted under the test piece of the Skid Tester is between 2 and 4 X 10~-^ cm. if uniformly distributed over the nominal contact. The true contact area may easily be smaller by an order of magnitude, and thus be

covered by sufficient instantaneously melted ice to provide partial

hydrodynamic lubrication which will become more difficult, though,

with decreasing temperature. This mechanism qualitatively predicts,

therefore, a negative coefficient of friction on ice. French and Patton [32] have drawn attention to a negative correlation

between hardness and tire friction on ice (in contrast to friction on wet

roads) and say that treads with a hardness above 85 become practically useless. They could also demonstrate the effect by running tires for a short distance before the test to increase their internal temperature

and thus reduce their hardness. This procedure led to increased hill climbing ability. The effect, which is most probably connected with an increase in true contact area on lowering the hardness, would be

expected to work for the advantage of NR tires; NR hardens les.s than

SBR at low temperatures because of its lower glass-transition

temperature.

Whatever its explanation, treads based mainly on NR have markedly

higher friction on ice and snow than treads of similar SBR compounds.

Table 6.1 gives results of trials carried out with studded and unstudded

tires in Sweden [44]. All treads were oil-extended and blended with BR

in various proportions. It is interesting to note that the braking coefficients

of the non-studded NR tires exceeded those of the studded SBR tires; this finding was qualitatively confirmed by hill climbing tests. Sapp [43] points out the difficulties encountered when evaluating

winter tires, and the results in table 6.1 may not be reproduced exactly

under all winter conditions. However, more recent trials on ice, also in Sweden, showed NR and OENR tires to be about 17 percent better in
WINTER TIRES 543

skid resistance than commercial synthetic winter tires [46]. It was not

possible to distinguish with statistical significance between the ratings

of NR and OENR but cornering tests discriminated more sharply

between the treads. The results, obtained on ice, are given in table 6.2

in which the cornering power ratings are expressed in terms of the

maximum velocity achievable on a circular path of 31 m. diameter, and

in terms of the maximum radial acceleration. The NR tread is seen to have a substantial advantage in road-holding over the commercial tire; OENR. though worse than NR, still offers a useful safety margin over

the synthetic tread. Only a tentative explanation can be advanced for the greater difference

between the tread compounds in cornering power than in skid resistance;

it is based on the discussion of cornering on wet roads at the end of section 6.7. If melt-water is produced in the contact area, its removal

is easier when the tire rotates than when it slides. If, furthermore, the

true friction of NR on ice is greater than that of SBR (the main constituent

of synthetic winter tires), and this is suggested by work on dry friction,

then it can come into effect to a greater extent under a slipping than under

a sliding tire. Table 6.1. Mean braking coefficients on road ice at 10 mph and temperatures between -6° and -2° C

(From Grosch. [44])

Not-studded Studded

Ratio base polymer/BR

OENR OESBR OENR OESBR

100/0 0.279 0.213 0.288 0.257 80/20 .271 .196 .293 .255 60/40 .221 .194 .229 .217

Skid rating OENR vs. OESBR (%)

100/0 131 112 80/20 138 115 60/40 114 106

Table 6.2. Relative cornering power on ice (From Grosch et al. [46])

Tread Velocity rating Friction rating

NR 125 157 OENR 115 131 Commercial 100 100
544 SKID RESISTANCE AND DIRECTIONAL CONTROL

References

[1] Chapter 7 of this book.

[2] Schallamach, A., Rubber Chem. Tech. 33, 857 (1960).

[3] Schallamach, A., and Turner, D. M., Wear 3, 1 (1960).

[4] Ludema, K. C, Friction of rubber. Chapter 1.2 of this book.

[5] Forster, B., Deutsch. Kraftf. Forsch., Zwischenbericht 22 (1938).

[6] Gough, V. E., Auto. Engr. 44, 137 (1954).

[7] Giles, C. G., High. Res. Rec. 46, 43 (1964).

[8] Grosch, K. A., and Maycock, G., Trans. I. R. I. 42, T 280 (1966).

[9] Research on Road Safety (H. M. Stationary Office, London, 1963).

[10] Grosch, K. A. (Private communication).

[11] Grosch, K. A., Proc. Roy. Soc. A2 74, 21 (1963).

[12] Moore, D. F., Wear 8, 245 (1965).

[13] Roberts, A. D., and Tabor, D., Wear 11, 163 (1968).

[14] Sabey, B. E., Road Tar (in course of publication).

[15] Percarpio, E. P., and Bevilacqua, E. M., Rubber Chem. Tech. 41, 854 (1968).

[16] Giles, C. G., Sabey, B. E., and Cardew, K. H. F., ASTM Spec. Tech. Publ. No. 326, Philadelphia, Pa. (1962).

[17] Mitchell, A. G. M., Viscosity and Lubrication, The Mechanical Properties of Fluids, second edition (Blackie and Son Ltd., 1944).

[18] Archard, J. F., and Kirk, M. T., Paper 15, Proc. IME, (Lubrication and Wear Group

Convention, Eastbourne) (1963).

[19] Cohen, S. C, and Tabor, D., Proc. Roy. Soc. A2 91, 186 (1966).

[20] Moore, D. F., Int. J. Mech. Sci. 9, 797 (1967).

[21] Maycock, G., Proc. IME 180, 122 (1965-66).

[22] Allbert, B., J., and Walker, J. C, Proc. IME 180, 105 (1965-66).

[23] Maycock, G., Report LR 122, Road Research Laboratory, Crowthorne, Berks, England

(1967).

[24] Kelly, J. D., SAE Paper No. 680138, (Auto. Engr. Cong., Detroit, Mich., 1968).

[25] Gough, V. E., and French, T., Pro. First Inter. Skid Prevention Conf., Charlottesville, Va., Part I, p. 189(1959).

[26] Sabey, B. E., Road Tar (in course of publication).

[27] Gough, V. E., Jones, F. B., and Udall, W. S., SAE Paper No. 990A (Inter. Auto. Engr. Cong., Detroit, Mich., 1965).

[28] Meades, J. K., Report LR 73, Road Research Laboratory, Crowthorne, Berks, England

(1967).

[29] Sabey, B. E., and Lupton, G. N., Rubber Chem. Tech. 37, 878 (1964).

[30] Tabor, D., Engineering 186, 838 (1958).

[31] Greenwood,]. A., and Tabor, D., Proc. Phys. Soc. 71,989(1958).

[32] French, T., and Patton, R. G., Fourth Rubber Tech. Conf., London, 1962, p. 196

(L R. L, London, 1963).

[33] Bulgin, D., Hubbard, G. D., and Waher, M. H., Fourth Rubber Tech. Conf., London,

1962, p. 173 (I. R. I., London, 1963). Bulgin, D., Kaut. Gummi Kunstst. 20, 601 (1967).

[34] Home, W. B., and Joyner, U. T., SAE Paper No. 650145, (Inter. Auto. Engr. Cong.,

Detroit, Mich., 1965).

[35] Allbert, B. J., SAE Paper No. 680140 (Auto. Engr. Cong., Detroit, Mich., 1968).

[36] Boness, R. J., Auto. Engr. 58, 260 (1968).

[37] Gray, W. E., J. Roy. Aero. Soc. 67,302 (1963).

[38] Home, W. B., Yager, T. J., and Taylor, G. R., De Ingenieur 78, L39 (1966).

[39] Gough, V. E., Hardman, J. H., and MacLaren, J. R., Trans. 1. R. I. 32, 27 (1956).

[40] Timoshenko, S. PI, Theory of Elasticity, first edition, p. 344 (McGraw-Hill, 1934).

[41] Automobil Club der Schweiz, Berne, Winterreifen Priifung, (1962).

[42] Rosenthal, P., Haselton, F. R., Bird, K. D., and Joseph, P. J., CAL Report No. YM2360-K-l (Cornell Aeronautical Laboratory, Inc., 1967).

[43] Sapp, T., SAE Paper No. 680139 (Auto. Engr. Cong., Detroit, Mich., 1968).

[44] Grosch, K. A., Rubber Age 99 (10), 63 (1967).

[45] Bowden, F. P., and Tabor, D., The Friction and Lubrication of SoUds, p. 66 (Oxford: Clarendon Press, 1950).

[46] Grosch, K. A., Shallamach, A., Southern, E., and Swift, McL. P., Intern Rubber Conf,, Moscow (1969).

[47] Holmes, K. E., and Stone, R. D., Proc. IME 183, 3H-35 (1968-69).
CHAPTER 7

The Tire as a Vehicle Component

H. C. A. van Eldik Thieme ^ and H. B. Pacejka ^ Page 7.1. Introduction 548

7.1.1. Nomenclature 549

7.1.2. List of symbols 550

7.2. Straight line rolling experiments (H. C. A. van Eldik Thieme). . 553

7.2.1. Introduction to total force and moment measurements... 553

Historical 553

Laboratory tests 553

Towed trailer road tests 556

Six-component tire testers 559

Road platform tire tester 565

7.2.2. Load-deflection relationships 567

Vertical load-deflection relationships 568

Horizontal load-deflection relationships 574

Actual vertical dynamic tire forces 579

7.2.3. Effective rolling radius 584

Introduction 584

Methods of measuring effective radius 587

Parameters affecting the effective radius 588

7.2.4. Rolling resistance 591

Introduction 591

Methods of measuring rolling resistance 592

Tire conditioning 595

Parameters affecting the power loss ^96

Cornering... 600

Braking and traction 601

Temperature measurements 602

7.2.5. Braking and traction 603

Braking 603

Traction 613

7.2.6. Tire nonuniformities 616

Introduction 616

The tire manufacturing process 617

' Delft University of Technology, Delft, The Netherlands.
546 THE TIRE AS A VEHICLE COMPONENT

Page Treads 513

Cord angle and ply fabric , 519

Building and curing 520

Stability of materials 520

Wheel uniformity 521

Effect of tire nonuniformity on vehicles 521

Tire uniformity grading machines 524

References 528

7.3. Cornering and camber experiments (H. C. A. van Eldik

Thieme) 631

7.3.1. Introduction 531

Nonsteered vehicles 531

Steered vehicles 636

7.3.2. Cornering experiments 639

Photography of contact area deformations 639

Movable platform measurements with the Gough apparatus.. 640

Road and drum test measurements 649

7.3.3. Camber and cornering 663

Camber 663

Combination of camber and cornering 664

Actual slip and camber angle measurements 665

7.3.4. The influence of braking and traction on cornering 669

Dry surface measurements 669

Wet surface measurements 670

7.3.5. Difficulties in measuring forces and moments 675

General observations 675

The nature of the track surface , 676

Conditioning of the tire 685

Trailer road tests versus drum tests 686

Trailer road tests versus vehicle tests on the road 689

Tire characteristic variations due to manufacturing

fluctuations 690

References 691

7.4. Tire in-plane dynamics (H. B. Pacejka) 695

7.4.1. Low frequency properties 695

Normal force-deflection characteristics 695

Enveloping properties (obstacles). 702

Longitudinal slip , 711

Theory of the longitudinal force response to road wavyness,

tire nonuniformity, and vertical and longitudinal axle

motions 716

7.4.2. High frequency properties 726

Vertical and longitudinal vibration transmission 726

Tread band free vibrations 733

Standing waves 743

References 754

7.5. Yaw and camber analysis (H. B. Pacejka) 757

7.5.1. Steady state motion 763

Sideslip 763
INTRODUCTION 547

Page Camber and turning 782

Influence of braking and traction forces 788

Approximate mathematical representation of combined

effects 799

7.5.2. Nonsteady state motions 803

Horizontal motions (transient and shimmy) 804

Effect of a time- varying load 820

References 837
548 THE TIRE AS A VEHICLE COMPONENT

7.1. Introduction

The combination of road, tire, vehicle, and driver forms one entity.

The mechanical characteristics of the tire in contact with the road must

combine with the mechanics of the vehicle to help in producing operational characteristics of the tire-vehicle system which are satisfactory

to the driver. "The complexity of the structure and behavior of the tire are such that no complete and satisfactory theory has yet been propounded. The characteristics of the tire still present a challenge to the natural philosopher

to devise a theory which shall coordinate the vast mass of empirical data

and give some guidance to the manufacturer and user. This is an inviting

field for the application of mathematics to the physical world."

In this way Temple formulated the situation of more than one decade

ago (Endeavour, October 1956). Since then, in numerous institutes and

laboratories, the work of the earlier investigators has been continued.

Considerable progress in the development of tire mechanics during the

last decade has led to a better understanding of tire behavior. Owing

to the infinite complexity of the pneumatic tire and its interaction with

the road it does not appear at present, despite the progress made, that Temple's view will be altered in the foreseeable future. Thanks to new

and more refined experimental techniques becoming increasingly avail- able, and to the introduction of the electronic computer, the goal of

formulating more realistic mathematical models based on better insight

and leading to more reliable prediction of tire performance may be

achieved.

The authors of this chapter do not claim to have supplied a picture of

tire behavior which covers all knowledge achieved hitherto. A selection

of studies has been made in order to provide the engineer and the student

with background material necessary for the investigation and the understanding of tire and vehicle functional performance.

From the point of view of the engineer and the applied mathematician

the mechanical behavior of the tire must be systematically investigated

in terms of its reaction to various kinds of input related to vehicle motions

and road parameters.

With reference to the role of a tire it is convenient to distinguish be- tween symmetric and anti-symmetric modes of performance. First, the

tire supports the vertical axle load and transmits longitudinal braking or driving forces. Second, the tire is called upon to supply the lateral corner- ing and camber forces which are necessary for the directional control of

the vehicle.

The major portion of this chapter, consisting of an experimental and

an analytical part, has been subdivided according to these categories.

The experimental part (sees. 7.2 and 7.3) has been generally restricted

to steady-state or slowly varying motions, whereas the analytical part

(sees. 7.4 and 7.5) treats also high-frequency and nonsteady state behavior

of the tire. Experimental results have also been added.
NOMENCLATURE 549 Many of the investigations discussed in this chapter have been carried out at the Vehicle Research Laboratory of the University of Technology,

Delft, HoUand. The authors wish to express their appreciation to the

members of the staff of this laboratory: G. J. van der Burgt-; A. P. C. van Heesewijk^ and his design office; E. G. J. M. de Vries and his elec- tronic measuring department; P. Buis'^, D. A. Timan and J. H. M.

Rooney for their numerous tire experiments, as well as to H. M. Snijders

and P. J. Jillesma for their assistance in manufacturing various in- struments and apparatus.

7.1.1. Nomenclature

For both the experimental and theoretical investigations of tire behavior described in this chapter, we have attempted to use a uniform

system of notation. As a rule, the meaning of symbols has been explained

in the text. For this reason, only a list of the most important symbols

will be given below. The choice of symbols has been inspired by the

list which has been proposed by a SAE committee in 1965 (cf. ref. [3]"' indicated at the end of sec. 7.3). A number of changes and additions

appeared to be necessary in order to obtain a more or less systematic

and usable system of symbols adjusted to the specific subjects of this chapter.

Constant quantities describing construction, configuration and properties of the real tire or of the theoretical model are defined in such

a way that they become positive. In most cases, the positive sense of variable quantities are chosen in accordance with the (C, x, y, z) sys- tem of axes shown in figure 7.1.1. The origin C, defined as contact center,

is the point of intersection of the road-plane, the wheel center-plane

and the plane which is situated normal to the road-plane and which

passes the wheel axis. The jc-axis points forward and forms the inter- section of the wheel center-plane and road-plane. The z-axis points

downward and is directed perpendicular to the road-plane. Consequently,

the y-axis is the perpendicular projection of the wheel axis onto the road. In the same figure 7.1.1, the positive directions of forces and moments

acting from road to tire have been indicated, as well as the positive senses of the variables which describe the deviations of the position and

the motion of the wheel center-plane with respect to the rectilinear steady

state motion of the wheel center-plane, which_in that case coincides with

the (3c, z) plane of the coordinate system (0, x, y, z) fixed to the road

with the 2-axis directed vertically. As in figure 7.1.1, the road-plane has

in most cases been considered as a smooth horizontal surface. In some cases an alternative definition of positive sense has been felt to be preferable. In order to work with positive quantities, the tire normal load has been defined as Fn{= W) = — Fz' Similarly, the quantity

Fr = —fx has been introduced, denoting the rolling resistance force

- For his help and valuable suggestions.

^ Has left the Laboratory of Vehicle Research.

^ Figures in brackets indicate literature references at the end of this section.
550 THE TIRE AS A VEHICLE COMPONENT

normal to road-pLane

y y

steady state (stationary) rectilinear free rolling. general deviated

situation.

Figure 7.1.1. Nomenclature and coordinate system for a wheel on a plane surface. during free rolling, i.e., at constant forward velocity and without traction or braking torques. Also, the sense of the speed of rotation Cl of the for- ward rolling wheel has been defined as positive. Sometimes, the absolute

values of the longitudinal force Fx have been considered. They are designated as the braking force Fb{= — Fx) and the traction force

Fr{= Fx).

The lateral force acting from road to tire, Fy, has been provided with an additional subscript a: or y in cases when it has been felt necessary to express whether side slip or camber causes the lateral force. 7.1.2, List of symbols

L, F and T denote length, force and time units respectively.

RAD denotes radians. a half length of contact area (L)

b half width of contact area (L)

ct,c,r foundation stiffness per unit length in tangential (t) , lateral

(c) and radial direction (r) respectively (F/L^)

Cpx, y Stiffness of tread rubber per unit area in longitudinal (x)

and lateral direction (y) respectively (F/L^)
LIST OF SYMBOLS 551

Cs tensile tread band (carcass) stiffness per unit length (F)

Ccx,y carcass stiffness in contact region in x and y directions

respectively (FIL)

Cpa {=dFyjda at a = 0) cornering stiffness (cornering rate)

(FIRAD)

Cpy ( = dfyldy at 7 = a = 0) camber rate (cf. eq (7.5.61)) (F)

Cmu {= — dMJda at a = 0) cornering stiffness (aligning rate)

{FLIRAD)

Cr {=dMrldf at 8 = 8o) rolling resistance coefficient (F)

Cx {= — dFxIdxa or dFxIdnx at Fx= 0) longitudinal or tangential

stiffness in contact region of non-rolling tire (F/L)

Cy {= — dFyldya at Fy=0) lateral tire stiffness in contact region

of non-rolling tire {FIL)

Cz {= dWldf= dWjdza at 8 = do or Za = 0) normal tire stiffness {FID

Ck (=dFxldK at K^O) longitudinal slip stiffness (F)

Cj, {=— dMzldifj at M2 = 0) torsional stiffness about vertical axis of non-rolling tire (FLIRAD)

EI flexural rigidity of tread band (FL^)

Fb,t braking and traction force respectively (F)

Fn {=W=~Fz) tire normal load (F)

Fx,y,z longitudinal, cornering (= lateral) and normal force acting

from road to tire (cf. fig. 7.1.1) (F)

G{x, f) Green's function

Ix polar moment of inertia of wheel about wheel axle (FLT^)

I wavelength of standing wave (L); half of projected contact

length (cf. fig. 7.5.1 of sec. 7.5)

Mr (=My) rolHng resistance moment (cf. fig. 7.1.1.) (FL)

Mx, y, z moment acting from road to tire (cf. fig. 7.1.1), Mz = aHgning torque (FL)

n frequency of motion (Hz)

Px,y,z contact force per unit area acting upon tire in negative

y and z direction respectively (F/L'^) Pi inflation pressure (FjL^)

Qx,y,z contact force per unit length acting upon tire in negative

X, y and z direction respectively (F/L)

r {or R) tire radius (L)

re (or Re) effective radius of rolling (L)

ri (or Ri) loaded tire radius = wheel center height (L)

R radius of curvature {L)

s distance travelled (L); mode number

S tension force in tread band (F)

t time; pneumatic trail {= — MzlFy) {L)

u tangential (longitudinal) deflection (L)

V lateral detlection (L)

Vc, p lateral deflection of carcass and tread rubber respectively

(L) V ( = dsldt) speed of travel {LjT)

Vc,r,s creep (slip), rolling and sliding velocity respectively {LjT)
552 THE TIRE AS A VEHICLE COMPONENT

w

I'll /I .• / •• . radial denection (positive outwards) (L) w ( = — Fz) tire normal load (F)

A longitudinal horizontal force acting upon tire (F)

%, y, z coordinates with respect to moving system (hg. 7.5.1)

coordinates of contact center C with respect to system

hxed in space

ya, Zfi variation oi wheel center position with respect to steady

state motion

'^c, yc, 2c coordinates of contact point with respect to system fixed

in space

q; slip angle (ct. hg. 7.1.1); crown angle

P ( =arctan aylas) path angle (ct. hg. 7.1.1)

y camber angle oi wheel center plane (ct. hg. 7.1.1)

0 normal tire denection (positive toward the center) (L)

\A wavelength of motion (L)

K longitudinal slip value (ct. eqs (7.4.z7, 7.5.o4)

Kb, t percentage oi brake and traction slip respectively

coemcient ot triction

( — ailJlas ) spin = yaw rate {KAD}L)

yaw angle (ct. hg. 7.1.1)

X deviation trom steady state angle of rotation ilt (ct. hg.

/.4.1o)

P mass density oi tire tread band (r 1 ^iL^)

relaxation length for tire model without tread rubber (/>) relaxation length for tire model with tread rubber (L) frequency of motion {RADjT)

CDs reduced, spatial or path frequency (RAD/L)

a speed of rotation of wheel {RADjT)
STRAIGHT LINE ROLLING EXPERIMENTS 553

7.2. Straight Line Rolling Experiments

H. C. A. van Eldik Thieme

In this section part 7,2.1 is devoted to a description of the equipment

used to measure forces and moments required for the investigations as treated in sections 7.2 and 7.3. 7.2.1. Introduction to Total Force and Moment Measurements

Historical

A variety of measuring devices have been designed to determine the

forces and moments that arise in the tire contact area. Ahhough initially

the interest was in load-deflection relationships in a radial direction, later research emphasized the rolling resistance and the nonskid qualities of

tires (braking force coefficient). The need to evaluate the cornering force

coefficient and the slip angle was discussed as early as 1930 by Bradley

and Allan. The research in this area is well summarized in the master ref- erence list given in the papers by MiUiken [1],^ Gough [2], and KoUmann

[3]. Laboratory Tests The majority of tests conducted to measure tire characteristics

have been run in laboratories, where the tire rolls on a large drum.

However, internal drum tire test machines are also available [3-5].

These indoor tests are useful for comparing the characteristics of

different tires, but do not give exact information on the tire behavior on flat roads because significant differences in the tire behavior are induced by the curvature of the drum. KoUmann [3] and Krempel [4] have discussed the effect of the radius of curvature of external and

internal drums (fig. 7.2.1). They report that the advantage of an internal

drum lies mainly in the fact that it can be covered with any surface,

including an exactly defined thickness of water. To eliminate the effects caused by the curvature of the drum, a flat rotating disc has been used [3], but due to the finite width of the tire tread the velocity of the inner and outer tire tread edge will be different,

causing slip in the contact area (fig. 7.2.2). Another method of reproducing the traveling ground is the belt-type

tire tester, as shown in figure 7.2.3, consisting of an endless steel band or conveyor belt running on two drums. The flexible belt is supported in the

Figures in brackets indicate the literature references at the end of this section.
554 THE TIRE AS A VEHICLE COMPONENT

testwheel

a b R c Figure 7.2.1. Schematic representation of test tires on external and internal drums. testwheel

motor

Figure 7.2.2. Test wheel on flat rotating disc.

tire contact area by an extremely stiff and thin air bearing, in order to minimize the air gap variation due to the required variable vertical loading of the test tire. A drawback of such an air bearing is the large

air consumption [6^7].

These belt testers are sometimes equipped with an electro-hydraulic

vibrator in order to induce a variable load via the air bearing platform to the running belt (fig. 7.2.4). The static tire load in such cases is carried

by a special device mounted to the foundation block. Through the use of

such a shaker system, it is possible to excite in the laboratory the

various modes of vibration of an automobile, including the inputs due

to tire nonuniformities [8-9]. Another indoor test is the method using a movable platform, as

illustrated in figure 7.2.5.
STRAIGHT LINE ROLLING EXPERIMENTS 555

testwheel

Figure 7.2.3. Endless belt tire tester.

Figure 7.2.4. Belt tire tester equipped with an electro hydraulic vibrator to simulate road excited vibrations. The necessity to measure in a laboratory all forces and moments acting on a tire, without excessive wear which would change the tire characteristics, dictated the design of various movable table machines [10-14].

These flat surface machines have the limitation of low speed, but afford the easy control of test variables that can be only achieved in a laboratory.

Most tables provide for the possibility of bonding various surface materials, and others provide for a glass section for observing and photographing tire deformations. Often the road surface can be wet, dry or iced at a controlled temperature for additional frictional studies [15]. Most

machines measure the forces and moments of a loaded, steered, cambered or torqued tire. The results are often shown in the form of a carpet

plot [12], because this technique allows one variable to be plotted as a function of two variables, so that accurate interpolation with respect to both independent variables is possible.

The Delft movable platform machine as shown in figure 7.2.5 is equipped

with an air spring device, in order to maintain a constant normal tire load,

as required for tests as described in section 7.3.2 (fig. 7.3.21). The same machine can also maintain constant center height. The

machine is equipped with a slide mechanism and a turntable, both with

graduated scales [14].
556 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.5. Movable platform machine with glass section to observe tire deformation. Towed Trailer Road Tests The testing methods described above do not represent actual operating

conditions, and therefore devices were developed for measuring tire characteristics on road surfaces. The Air Force-Cornell tire tester [1] is well known, consisting of a single-wheeled trailer towed behind a truck

(fig. 7.2.6). The axle carrying the test tire is supported at its ends by units

which measure five forces, provision being made to include an instru- mented linkage for measuring the brake torque. The load cells are mounted on the free ends of an U-shaped frame whose normal position

is in a horizontal plane. The mechanism permits the test tire to be steered

and cambered 30 degrees and vertically loaded with a variable force up

to 3,000 lbs. Because the true slip angle applied to the tire consists of the steering

angle (tire relative to truck) minus the truck slip angle (truck relative to direction of motion at test wheel), a fifth wheel unit was added to measure

the truck slip angle (fig. 7.2.7). Another tire tester developed at the Delft Vehicle Research Laboratory avoided the difficulty in measuring the slip angle by using two test wheels in a trailer; the tire of the second wheel running at an equal

but opposed slip angle to the measuring wheel [10]. In this way the lateral forces induced by both wheels are now in equilibrium, resulting in zero slip angle for the test trailer. Of course the motion of the trailer without test tires must have zero slip angle, if accurately aligned and
STRAIGHT LINE ROLLING EXPERIMENTS 557

. ^

' Mechanical Cornpomnts

1. Bed-plate (tixed to truck). 7. Trunnion.

2. Steering axis. 8. Axle.

3. Hinge frame. 9. Test tyre.

4. Hinge box. 10. Right load cell.

5. Left load cell 11. Wheel frame,

o. i^'amber adjustment axis. 12. Pitch axis.

Figure 7.2.6. Schematic representation ofCornell tire tester.

fitted with tires having acceptable tire nonuniformity. A special air- sprung trailer (fig. 7.2.8) with a comfortable cabin for mounting auxiliary

equipment was developed, including a six-component tire tester. In

I
558 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.8. Towed trailer road tests with a six-component tire tester. The test wheel is mounted in the airsprung trailer.
STRAIGHT LINE ROLLING EXPERIMENTS 559

this trailer the tire load on the test wheel is kept constant by means of an air spring, so that vertical movements of the trailer cannot influence

the measuring results. The steering system for varying the slip angle

employs an extensive hydraulic regulating apparatus, in order to rapidly

adjust the various slip angles desired under constant vertical tire loads. The camber of the tire tester can also be adjusted. A water tank for spraying the road has been placed in the truck towing the trailer. Six-component Tire Testers The paper in 1956 by Close and Muzzey [1] described one of the most

modern means of measuring tire characteristics, since it was the first complete machine to measure tire parameters on flat road surfaces at different speeds and loads. Most available tire testing equipment was

capable of measuring the steady state characteristics of the tire, but

for steering system stability problems the time or distance behavior of the tire must be known.

The obvious advantage of a six-component tire tester is not only its ability to measure tire characteristics on flat roads, but also its utihty

as a precision laboratory test machine on a drum.

In this way it is possible to compare test results obtained from road

and drum tests with the same tire. The interpretation of forces acting in the tire road contact area has

been discussed by Fonda, Close and Muzzey [1]. In the contact area of a cambered tire there is a distribution of forces

normal to the road plane and a distribution of shearing forces in the road

plane. These forces can be resolved into three forces and moments acting on the tire at the so caUed center of tire contact C. Figure 7.2.9 shows a cambered wheel in contact with the ground, and

an enlarged view shows the forces at a point x, y in the area A. The equations are given by:

The vector of the pressure in the positive direction exerted by the tire upon the road is denoted px, Py, Pz. Thus Px is the force per unit area, and Px dA = — dFx, Py - dA = — dFy, pz ' dA = — dFz and therefore

>x-y-py'X PxdA ,Fy = —

J A( P,dA [40]

The choice of the location of the origin Cw of an additional set of axis

x'^y^z* through the center of the wheel reduces misinterpretation.
560 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.10 shows the new axis, and after transformation we obtain: = F, M% = - FyR COS y — FzR sin y

F* = My + F^R cos y

F* = F, M* = M,-h Fa:R sin y

Because the load cells make their measurements with respect to the

wheel axis, a third set of axis is shown in figure 7.2.11, and, after trans- formation, the equations read:

F% = F^^ M% = M^u:

Fy = Fyw cos y — Fzw sin y = Myw cos y — Mzw sin y

Ff = Fzw cos y + Fy«; sin y = Mzw cos y + Myw sin y

Figure 7.2.12 finally illustrates the relationship of wheel axis forces and

moments to load cell forces for the Cornell tire tester. The load cell readings can be used to compute the externally applied forces and moments

for the coordinate axes shown.

Figure 7.2.9. Forces and moments of a cambered wheel with enlarged view of the contact area.
Figure 7.2.11. The load cells make their measurements with respect to the wheel axis requiring the transformation offorces and moments to a third set of axes {Xw, Yw, Zw).
562 THE TIRE AS A VEHICLE COMPONENT

—fxl-^fxr bf.r- afzi TUy

=fzl+fzr afxi- bfxr

Figure 7.2.12. Relationship of wheel axis forces and moments to load cell forces for the Cornell tire tester. In figure 7.2.13, the axle arrangement for the tire tester is shown for the

Cornell tire tester. To provide for easier changing of the tire to be tested, a somewhat

different design was chosen for the Delft tire tester built in 1959. After some modifications it proved to be successful [14]. Using the same notations and symbols as figure 7.2.12, the reader will observe the different design approach in figure 7.2.14. The considerable

advantage obtained with this new design is the possibility of mounting

the tire at a free end of the measuring axle as shown in figure 7.2.15. Referring again to figure 7.2.14 it is observed that in this Delft tire tester the axle and axle bearing box are connected to the frame with the

measuring devices P, Q, R and a flexible coupling S. To calibrate the tire tester a special very stiff test rig was built (fig. 7.2.16) having three absolute perpendicular axis, the tire tester being
STRAIGHT LINE ROLLING EXPERIMENTS 563

/ \

QUICK DISCONN ECT UNIT

\ f6 V!9

^.r/^ Arrangement:, Plan View

Axle, non-rotating.

Hub.

Wheel and test tyre.

Brake drum.

Brake shoe.

Brake torque Unks.

Taper roller bearings.

Nut, bearing adjustment.

Shims, bearing adjust

ment.

10. V-band couplings.

11. Stub axle, left hand.

12. Stub axle, right hand.

13. Spherical bearings.

14. Linear ball bushings.

15. Spiders.

16. Flexures.

17. Load-cell frame, left hand.

18. Load-cell frame, right

hand.

19. Side force beam.

20. Links.

Figure 7.2.13. Axle arrangement for the Cornell tire tester. accurately fixed to an extremely stiff bed plate. With the aid of three

dead weight controlled dynamometers several combinations of forces

and moments can be induced on the tire tester, giving the required out- put corrections. With the test rig it is also possible to prove that, for instance, a static force in the Z-direction will not influence the output

of strain gage bridges in other directions.

Not only static calibration is required, but also assessment of the

dynamic behavior of the measuring unit is necessary, in order to define

its limitations.
564 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.15. The possibility of mounting the tire at a free end of the measuring axle

is shown.

The camber of the tire tester can be adjusted.
STRAIGHT LINE ROLLING EXPERIMENTS 565

Considering the automobile as a two degrees of freedom system, we

observe a low body frequency (0.8-2 Hz) and the high axle frequency

(10-16 Hz).

Influence of any of these vibrations on actual force variations of the

tire requires a tire tester with reasonably high limiting frequencies. After

mounting the tire tester in a rigid structure, excitation was impressed

with an excitor in aU three mutually perpendicular directions. Experiments were made with a loaded tire as well as with wheel and tire re- moved, the first resonant peak being observed 55 Hz, and flat response

curves were obtained up to 30 Hz, deviations beginning at 35 Hz. An

Figure 7.2.16. Test rig having three perpendicular axis to calibrate the Delft tire tester.

approximate method was employed to calculate the resonant frequencies

of the complex system, and yielded a frequency spectrum close to that determined experimentally. Filters with suitable cut off" frequencies were

available to ehminate disturbing resonant frequencies [17]. Road Platform Tire Tester An apparatus for measuring the total forces exerted on a road surface

by the wheels of a moving vehicle can be of considerable interest, be- cause it gives information about load transfer, cornering and traction or braking forces of a vehicle passing over the measuring platform [14].
566 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.17. Road platform tire tester having the possibility of measuring three per- pendicularforces of a tire moving over the platform.

wheelload

100 millisec.

lOOkm/h

mill isec.

7.5 25

25 km/h

output signal

Figure 7.2.18. Response curves of the road platform tire tester for vertical tire load at two speeds.
LOAD-DEFLECTION RELATIONSHIPS 567

Result Figure 7.2.19. Illustration of results obtained with the road platform tire tester by adding

a filter with an inverse transferfunction.

A photograph of such a platform with a steel surface table is shown in figure 7.2.17. There is the possibility of bonding various surface materials

to the table. The platform is supported by strain gaged bars operating in bending, the strain gages being bonded to the bars in such a way that only

shear and vertical forces are measured [18-19].

An approach such as this is satisfactory when measuring total tire forces at low car speeds, because the output signal obtained is not

strongly influenced by the excitation of the lowest natural frequency of

the platform at about 50 Hz. Figure 7.2.18 shows in dashed lines the sup- posed trapezoidal load of a tire moving onto, over and off the platform,

as well as the output signal obtained at speeds of 25 km/hr.. What happens when the tire passes over such a platform at 100 km/hr. is also

illustrated; it is seen that it is impossible to find the trapezoid from the

output signal obtained. However, using a filter with an inverse transfer

function of the measuring platform [20-21], satisfactory results could

be obtained as shown in figure 7.2.19. This example may illustrate that a platform originally intended to calibrate a mobile tire tester on the

proving ground, with certain limitations, could also be used as a tire tester. A better approach is of course to raise the stiffness of the measuring

system by replacing the bending elements supporting the table by bars

in compression or tension. The same output of the strain gage bridges

will be obtained with much lighter bars, having the advantage that the

lowest natural frequency will be raised by, for instance, a factor of six. The complicated inverse filter measuring system may be omitted [22]. Besides the possibility of measuring three perpendicular forces Fx,

Fy, Fz of a tire moving over a road platform, the position and path of the test wheel with respect to the platform axes have to be known, as

will be treated in the sections 7.3.3 and 7.3.5. 7.2.2. Load-Deflection Relationships

In this section the vertical load deflection characteristics will be

treated. In addition, some attention will be paid to the longitudinal (tangential) behavior, and to a lesser extent the lateral and torsional char-
568 THE TIRE AS A VEHICLE COMPONENT

acteristics of the tire will be discussed. The latter two types of deformation are directly related to cornering and yaw response of the tire treated

in sections 7.3 and 7.5. Vertical Load-Deflection Relationships

The lowest natural frequency of the vertical vibrations of an automobile is determined mainly by the sprung mass and the stiffness of the

wheel suspension springs. This frequency normally ranges from about

1 to 3 Hz. The second resonant frequency is determined by the unsprung mass and the radial tire stiffness. It usually Ues in between 8 and 20 Hz. The radial tire flexibility influences the vehicle vertical motion and the

time variation of the vertical tire load. The former aspect is important

for ride quality, whereas the second aspect indirectly influences the

directional stability of the vehicle. Besides, tire flexibility plays an important role in the life of vehicle components and road surface.

When a standing tire (nonrolling, V=0) is loaded vertically and after- wards unloaded, a loop in the load-deflection curve is observed, which is due to hysteresis and friction losses [14]. Investigation with a tire rolling on the inner surface of a rotating drum of 3.8 m. diameter [5], shows that

the loop area decreases with increasing rolling speed as in figure 7.2.20.

It has been generally accepted that the damping produced by the rolling

tire is very small, and can be neglected in studies dealing with vertical

axle motions of relatively low frequencies. Results obtained with the

Tire daflcction Zq

Figure 7.2.20. Vertical load deflection relationships at various speeds for a bias-ply tire, obtained on an internal drum machine.
LOAD-DEFLECTION RELATIONSHIPS 569

VzO km/h

V=50km/h

0 10 20 30 40 50 mm

Tire dtflection Zq

Figure 7.2.2 L Vertical load deflection relationships at two speeds for a radial ply tire obtained on an internal drum machine.

Delft tire tester are in conformity with the above statement [10]. The

curves of figure 7.2.20 furthermore indicate that for the bias ply tire under

consideration an increase of the vertical stiffness occurs with increasing

speed. The behavior of a radial ply tire shows a similar trend and has

been illustrated in figure 7.2.21. The shape of the curves, however, are

different [5]. Typical for bias ply tires is the double curved shape. The

radial ply belted tire usually shows a progressive load-deflection curve

as represented in figures 7.2.22-23. These results were obtained with

the Delft tire tester on a drum of 2.5 m. diameter [23-24].

The inflation pressure Pi has a marked effect on the radial stiffness. For a nonroUing bias ply tire the curves shown in figure 7.2.24 have been

obtained [5]. Similar results are shown for a 165-13 tire rolling at 5 km/hr. on a steel drum of 2.5 m. diameter [24]. The tire stiffness is composed of a more or less constant part originating

from the structural rigidity of the tire, and of a part dependent on the air pressure pi [25-26]. Under rated conditions, the latter part is predominant. It has been found that the structural rigidity of the aircraft tires may supply 3 to 8 percent of the total stiffness, whereas for automobile

tires this percentage may amount to about 15 percent [26-28]. These

relatively low percentages explain the considerable influence which the
570 THE TIRE AS A VEHICLE COMPONENT

0 10 20 30 40 mm

tire deflection Zq

0 10 20 30 40mm

tire deflection Zq

Figure 7.2.22. Vertical load deflection relationships at various speeds obtained on an

external drum machinefor a bias ply and radial ply tire.
LOAD-DEFLECTION RELATIONSHIPS 571

inflation pressure has upon the radial tire stifl'ness, as indicated in figure 7.2.24. According to measurements of Henker [5] the radial stiffness of the

tire under consideration becomes approximately 20 percent lower when

the tire rolls upon an external drum surface, as compared with an internal

drum surface of the same diameter. This means that when the tire rolls over a flat surface, the stiffness is expected to become approximately

10 percent lower than values obtainable from the internal drum machine.

It has been often observed that characteristics for both radial and

bias ply tires can be approximated very well by straight lines passing

through the origin. Only for relatively low values of vertical load does a noticeable deviation from linearity take place. The slope of these straight

lines determines the vertical stiffness of the tire, which consequently

is assumed to be independent of load in the practical range of interest. Figure 7.2.25 presents the results obtained for the spring stiffness of the tire as a function of speed for a number of values of inflation pressure

[29]. It may be noted that according to these experiments, the stiffness of the bias ply tire is very sensitive to the speed of travel. Similar results were obtained with the Delft tire tester as shown in figure 7.2.26. Due

to the increase in observed radial stiffness an increase in loaded tire radius (wheel center height) will occur. This increase with speed will become even larger due to the phenomenon of tire growth. Figure 7.2.27 shows the increase of the free tire radius R due to tire growth, and the

increase of the loaded radius Ri due to both tire growth and stiffening of the tire [29]. An alternative method for the assessment of the radial stiffness consists in determining the resonant frequency of a single mass spring sys- tem with known inertia and spring stiffness of applied load, in series

with the resilient tire rolling on a drum or belt [30]. A schematic of this
572 THE TIRE AS A VEHICLE COMPONENT

0 10 20 30 40 mm

Tire deflection Za

0 10 20 30 40 50 60 70 mm

Tire deflection Zo Figure 7.2.24. Variation of vertical load deflection relationships as a function of

inflation pressure pi.
LOAD-DEFLECTION RELATIONSHIPS 573

50 100 150

. speed of travel(km/h)

50 100 150 ^ speed of travel (km/h)

Figure 7.2.25. The vertical stiffness or spring rate as a function of speed at various

inflation pressures for a bias ply and radial ply tire.

X bias ply 175 S.U

kgf/mm radial ply 175 SRU

F„=335kgf

20

rPi=1.9kqf/6mi^

40 60 80 100 120 UO km/h

speed of travel Figure 7.2.26. The spring rate as a function of speed, demonstrating the difference in behavior of a bias ply and a radial ply tire.
574 THE TIRE AS A VEHICLE COMPONENT

mm

12 Tires

' X 5.60-15 155 SR15 Fn 0 kc Pj -1.5 k jf/cm» /

-//t AY

r

50 100 speed

150 km/h

t

FiM = 300

Pi -1.5

kgf <gf/cm2 >

50 100 150 km/h

speed Figure 7.2.27. TAe increase of the free tire radius at F.\ = 0 kgf as a function of speed,

and the increase of the loaded radius at F\ = 300 kgf as a function of speed for a bias ply and a radial ply tire.

drum type machine has been shown in figure 7.2.28 operating in the

frequency range of 10-15 Hz. The value of the spring rate obtained by

this method is called the dynamic tire stiffness, and it is shown that the

dynamic stiffness decreases sharply as soon as the tire is rolling. Beyond

a speed of about 20 km/hr. the influence of speed becomes less important,

as demonstrated in figure 7.2.28. These findings are in contrast to the

results as shown in figure 7.2.25 and figure 7.2.26, obtained from the

slopes of the load deflection curves at different speeds of rolling. For

amplitudes within the range investigated, up to 10 mm., the stiffness of

the rolling tire remains nearly constant, whereas a nonrolling tire shows

a non linear decrease of the stiffness with amplitude [31]. For further information on tire spring rate the reader is referred to the literature. The influence of preload, inflation pressure, rim width,

and cord angle has been discussed in reference [31], the influence of, among other things, speed and drum curvature in references [5.32] and

of tire size in references [33-34]. In section 7.2.1 of this chapter the

tire loading process and the enveloping properties of the tire have been

analyzed.

Horizontal Load-Deflection Relationships

The literature on the horizontal elastic characteristics of the nonrolling tire is scarce. Some of our own unpublished investigations

will be discussed below.

The same tires as in figure 7.2.23 and figure 7.2.26, have been examined

under rated conditions, and are of the radial and of the conventional bias

type.

The tire is now loaded upon a flat cast iron plate, in contrast to the tests described previously. The normal load has been kept constant during
LOAD-DEFLECTION RELATIONSHIPS 575

40x10*kgf/m

drum diameter 2.5 m

amplitud* 5-10 mm.

tirvs 165x400

0 5 10 20 30 40 50 60 70 100 km/h

Speed of travel Figure 7.2.28. The decrease of the vertical dynamic spring rate as a function of speed.

each test. The measured vertical stiffness of the nonrolHng tire amounts

to approximately = 17 and 20 kgf/mm. respectively. Except for the

longitudinal stiffness characteristics, all force and moment measure- ments have been executed with the six-component measuring hub.

The longitudinal stiffness Cx of a tire is important for the study of longitudinal isolation of disturbances caused by road irregularities. The

longitudinal stiffness may be determined in two ways which do not

i
576 THE TIRE AS A VEHICLE COMPONENT

necessarily give the same results. With the first method, the wheel is fixed in space and the surface upon which the tire is loaded is displaced

in the longitudinal direction as shown in figure 7.2.29. With the second

method the surface and the wheel axle remain fixed, while the wheel is rotated by a torque about the wheel axis. The resulting tire deformations

are different in these two cases. The results obtained with the second

method are presented in figure 7.2.30. The deformation rate of all tests was low. The rotation of the wheel has been increased until complete

sHding takes place. The longitudinal (tangential) stiffness of the radial

ply belted tire appears to be much lower than the stiffness of the bias

Figure 7.2.29. Measurement of the longitudinal stiffness. The left wheel fixed in space is loaded with an identical tire. The force required to displace the steelplate is measured

with a dynamometer.

ply tire. At zero longitudinal displacement of the wheel at road level, we

obtain for the stiffness approximately Cj- = 27 and 45 kgf/mm.

respectively [35]. The lateral stiffness Cy of a nonroUing tire is important for the relaxation properties of the rolling tire. In combination with the cornering stiff- ness C/Ta, to be discussed later, the so-called relaxation length can be

determined (eq 7.5.51)). The lateral comphance may furthermore give

rise to a lateral and yaw vibration of the vehicle with respect to the lower

tread portions of the tires.
LOAD-DEFLECTION RELATIONSHIPS 577

longitudinal displacement of wheel

at road level R^X (mm)

Figure 7.2.30. The tangential stiffness of the radial ply tire is much lower than the bias ply tire. For the same radial and bias ply tires figure 7.2.31 shows the measured lateral force-displacement characteristics. The flat cast iron plate on which the tire is loaded has been moved sideways, until complete

sliding takes place. Again the radial ply tire shows a lower stiffness than

the bias ply tire. We obtain at zero displacement approximately Cy = 9

and 12 kgf/mm. respectively.

Similar results were obtained by another method as reported by the

author [10]. The torsional stiffness C4, of a nonroUing tire about its vertical axis can be used in combination with the self-aligning torque rate Ca/o for the determination of the longitudinal tread stiffness parameter k* (eq

(7.5.120)). The torsional stiffness may furthermore be of value for the

assessment of the steering wheel torque required at zero or nearly zero forward speed (parking).

The characteristics obtained are presented in figure 7.2.32. The turning

of the wheel about the vertical axis has been continued until fuU sliding

with respect to the flat cast iron plate occurs. Similar to the tests dis- cussed above, the radial tire appears to be more compliant also in this respect. We find at zero angle of rotation approximately C,|, = 250 and

290 kgfm/rad respectively [35].
578 THE TIRE AS A VEHICLE COMPONENT
LOAD-DEFLECTION RELATIONSHIPS 579

In the past similar tests have been executed on the movable platform

machine, as illustrated in figure 7.2.5 of section 7.2.1. Actual Vertical Dynamic Tire Forces

The measurement of dynamic tire force has been required for various

purposes, such as car handling, optimizing shock absorber settings,

studies of what is decisive in respect to dynamic stress imposed upon

the wheel or road when passing over wavy short unevennesses, analysis

of enveloping forces of tires passing obstacles, etc. The dynamic tire force consists of a constant part Fzo and a variable

part Fz. Consequently, we have:

F F Fz

Considering a very simple two mass system such as shown in figure

7.2.33, it seems very easy to find the tire force variations Fz from:

Fz = mbZb-\- rUaZa The measurement of the accelerations of only the car body and the

axle are required. But with such a simplification it is very difficult to calculate the reduced masses involved accurately enough [36-37].

Even with a supposedly rigid car body the actual vibrations are much

more complicated, due to rolling, pitching, bouncing, etc. Acceleration

measurements usually fail to give results accurate enough to yield useful

data on actual tire forces. To measure the dynamic tire force Fz during road tests, the tire is considered to have no damping. We obtain Fz by measuring the tire deflection through the variable distance between axle center line and

the ground, and by assuming a known value of the radial stiffness C^.

It follows from figure 7.2.33 that: Fz=Cz{Zc — Za)

Fn(=-Fz)

Figure 7.2.33. Model oftwo mass spring system passing over an uneven road surface.
580 THE TIRE AS A VEHICLE COMPONENT

The radial tire stiffness Cz may be obtained as a function of speed from

laboratory drum tests. This test method can be refined by using an

electrohydraulic vibrator to shake the rolling tire at different amplitudes

and frequencies [23-31]. In other cases the frequency sensitivity is obtained by mounting the

test wheel in a stiff frame with an actuator plate moving the footprint

up and down. But it should be noted that results in this case are obtained

with a nonroUing tire, and the influence of the deformation of the cross

section of a rolling tire due to centrifugal forces cannot be taken into account. It will be very difficult to correct for an "effective" tire mass

in this case. Having discussed the radial tire stiffness, we observe several

methods of measuring the tire deflection.

The Miihlfeld antenna system [38] measures the variable capacity of a moving condensor plate above the ground, and requires a plate area of about 25 X 25 cm. As a consequence, dynamic tire loads due to road unevenness of half wavelength less than 25 cm. cannot be accurately

measured. There is thus some uncertainty of the value of the dynamic

tire stiffness when passing over a short obstacle.

Another system introduced by von Bombard [39, 90] measures the

variation of the width of a cross section of the tire. Later modifications

having sensing devices with a roller assembly measure the position of a

point on the left and right hand side of the tire wall as in figure 7.2.34.

Figure 7.2.34. Method of obtaining the vertical dynamic tire force by measuring the variation of the width ofa cross section ofthe tire.
LOAD-DEFLECTION RELATIONSHIPS 581 The buffing ribs of the side walls in contact with the rollers have to be

ground to correct for the lateral tire nonuniformity. The normal procedure

of averaging left and right hand displacements of the tire waU as a measure of the vertical load gives erroneous results when rolling under a slip

angle. It is seen from figure 7.2.35 that the lateral and radial displacements of a point on the tire wall are very much dependent on the value

of the sHp angle [40]. The method of making these measurements is illustrated in figure 7.2.36. As can be seen from the photograph, a point

Figure 7.2.35. Influence of the slip angle on the lateral and radial displacements of a

point on the tire wall.

on the tire wall is contacted by two spring loaded wires, which are perpendicular to each other, the wires being attached to potentiometers.

This method, shown in figure 7.2.34, can only be used in straight line rolling on relatively smooth roads when no lateral forces are present,

and corrections have to be made for the speed dependent character of the tire width.

The radial and lateral displacement of a point on the tire wall is also very much dependent on the position in degrees from the contact
582 THE TIRE AS A VEHICLE COMPONENT

center (fig. 7.2.37) due to the buckling effect in the center of the contact zone [40]. Attempts have been made to measure the tire deflection at a point in the wheel center plane at the inner liner of the tread when passing

through the contact zone (fig. 7.2.38). A spring loaded flexible cable was bonded to a piece of canvass attached to the inner liner of the tire, while the other end of the cable was connected to a potentiometer

mounted in the wheel rim of a tubeless tire. The restraining spring was relatively stiff in order to give a frequency response compatible

with the large accelerations of the tread when passing through the contact zone, resulting in a low life of the potentiometer bearing. A

photocell device was used to trigger the instant of passing through the center of the contact zone. Because the potentiometer reading had to be

corrected for the speed dependent character of the tire tread deflection,

and on rough roads the slider vibrated upon the potentiometer windings,

this method appeared to be too complicated and unreliable.

Various other sensing devices have been proposed to measure the tire deflection' of a point in the wheel center plane at the inner liner of the

tread when passing through the contact zone. Such devices are based

on the measurement of the variation of the resistance [40-42],

capacitance or inductance, on photocell counter devices for light beam

pulses, on the detection of the variation of the radioactivity of an emitter
LOAD-DEFLECTION RELATIONSHIPS 583

Radial

Displacement contact center

rotation angle

degrees

mnn -2Lateral Displacement 7.50-U

Load 300 kgf

2U psi

Figure 7.2.37. Variation of the lateral and radial displacements of a point on the tire wall

as a function of the wheel rotation through the contact center.

////////////

Figure 7.2.38. Measurement of the radial tire deflection with a spring loaded cable connected to a potentiometer.
584 THE TIRE AS A VEHICLE COMPONENT

plate mounted in the tread, etc., but all methods apparently are without success under conditions of a combination of traction and cornering [40]. Another method used the measurement of the variation of the signal

from a strain-gaged symmetrical rim, but due to the momentary position

of the strain gage bridge the load variations cannot be accurately measured w^hen passing over an obstacle [43—44].

The method of measuring inflation pressure changes has also been

reported for a tire rolling over a cleat on a test wheel [45]. The very small

change in inflation pressure of the tire was detected with a diff"erential manometer mounted outside the test tire, and this pressure change can

be related to the process of envelopment of the obstacle. A similar method

was adopted with a sensing device mounted at the rim inside a tubeless

tire. 7.2.3. Eflfective Rolling Radius

Introduction

An analysis of the deformations of a rolling tire shows that the eff'ective rolling radius Re is the ratio of the linear velocity V of the wheel center

in the Z-direction to the angular velocity fl of the wheel:

Figure 7.2.39. Measurement of free radius R, loaded radius Ri and effective radius Re

on the movable platform machine.
EFFECTIVE ROLLING RADIUS 585 The effective rolling radius is often erroneously assumed to be equal the center height h or so called loaded radius /?/, being the distance from

the center of tire contact to the wheel center. When R is the undeflected

free tire radius, it will be shown that: Rl<Re<R

that is, the rolling tire travels farther per revolution than determined by

using its center height as rolling radius, but less than described by the

free tire radius.

A rolling wheel may be considered as a wheel rolling along the sta- tionary ground, or the center of the wheel may be taken as fixed, so that the ground moves relative to it. Both cases will lead to correct results and the latter case is represented in the flat platform machine.

Consider figure 7.2.39, which represents a tire on a flat platform

machine with a number of equaUy spaced radii, drawn on the surface of the undeflected free tire, dividing the circumference into n equal ele- 2ttR

ments, each with length / =

n

It is observed from the photograph that tread elements in the zone immediately before contact are compressed, as well as elements in the area just after contact has been lost (fig. 7.2.40). Due to compression the

tread elements in the contact zone are shorter, and therefore in - of a n

Compression

V.Qh

Contact I

h = R,

V. Platform speed

Tire circumference Figure 7.2.40. Circumferential speed of a tread element when passing through the contact
586 THE TIRE AS A VEHICLE COMPONENT

revolution of the wheel, the platform moves a distance X/, which is less than length ' This is the same as saying that when the platform moves a distance 27tR, the wheel will turn more than one revolution.

From tests with a nominal loaded tire under straight line rolling conditions we also observe very little longitudinal slip in the contact area

(resulting in almost no wear), so that the compressed tread elements

will travel with the platform speed V. When the circumferential speed of the undeflected upper part of the

tire is called Vo, we find [24]

I A/ Vo=nR ] v=aRe\

Rl<Re<R

This was demonstrated experimentally by several authors [46-47] and

is in accordance with the situation shown in figure 7.2.41, where the

wheel center 0 is maintained fixed and the platform, in position 1, moves

to the right over the distance X. Then the point of the tire surface ini2<i

I Figure 7.2.41 Deformation of a tire on a movable platform when a radial load is applied.
EFFECTIVE ROLLING RADIUS 587

tially at Ci moves to Di. Simultaneously, the point of the tire at the

rim of the wheel A moves to B. The wheel thus rotates over the angle

AOB = ip. The distance between the center of the wheel and the platform

in the position 0 (no load) and position 1 (loaded) is OCo = R and OCi=Ri

respectively [47] C,D,=X = Re' <p

CiEi =Xi =Ri tg (f Xo = R sin (p

, Re X tg(f , Re X sin (f whence -s~=v ~d~~v K I Ai (f K Ao if

For (f not too large, tg (pl(p is slightly larger and sin ipl(p is slightly smaller

than unity.

Instead of an effective roUing radius, an effective deflection 6eff=/?—/?e can be introduced, which differs from the actual deflection 8 = R—Ri.

Methods ofMeasuring Effective Radius

A. Movable platform (fig. 7.2.39) Flat surface machines have the limitation of low speed, but good

control of variables can be achieved.

The radial load is applied and measured by means of a load cell, the

free radius R and loaded radius Ri are measured with a scale. When the platform is moved over a measurable distance X, the corre- sponding angle of rotation if of the wheel is determined by means of a

scale and pointer or other suitable instrument attached to the supporting

structure and wheel rim respectively.

The effective roUing radius is:

Re = 180 —7T(p when (f is measured in degrees.

B. Drum tests After conditioning the test tire at the required load, inflation pressure

and speed, the effective radius is determined by the ratio of the circumferential drum speed V and the angular velocity H of the test tire,
588 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.42. Photocell counting devicefor measurement of angular velocity.

The angular velocity is accurately measured with a photocell counting

device as in figure 7.2.42, which may be equipped with continuous

recording for the angular velocity [10, 24]. The variation in the effective rolling radius per wheel revolution due

to tire nonuniformities can also be measured with drum tests (sec. 7.2.6). Nonuniformities produce angular wheel accelerations that result in tangential force variations as measured with a tire tester or uniformity

machine.

The free radius R of the tire can be measured at a given speed through

skimming contact with the drum, by measuring the distance between

wheel center and the surface of the rotating drum. (See also tire uni- formity machine, sec. 7.2.6 (fig. 7.2.81)). C. Road tests Road tests have been reported [48] by driving an automobile at walking

speed over a measured distance and counting the number of revolutions

of the free rolling front wheel. Tests were run with several tires at various inflation pressures. Parameters Affecting the Effective Radius

A. Carcass construction Factors influencing the deformations of the rolling tire, such as carcass construction, tread thickness, number of plies, cord type and
EFFECTIVE ROLLING RADIUS 589

cord angle, rubber compound, etc., will also influence the eff"ective radius.

Experimental results, obtained with the Delft tire tester mounted in a rig with loading device above a drum 2.5 m. in diameter, show the

difference in behavior of a conventional bias compared with a radial- belted tire. Using Re = kR, the conventional bias tire shows a value of \ = 0.96

and h = Ri = OMR.

For the radial ply construction with an inextensible belt, having less compression of the tread elements when passing through the contact

zone, values oi k = 0.98 were obtained. The radial deflection of this tire

is more than in the former case, giving a value of Ri = 0.92 [24]

.

B. Effect of tire load and inflation pressure

An increase in load at constant pressure results in a larger deflection

of the tire. Due to an increase of the compression of the tread, a decrease

of the effective radius is observed shown in figure 7.2.43. A decrease of inflation pressure has the same effect as an increase of

tire load. C. Effect of speed

The effect of speed is shown in figure 7.2.43 for a conventional bias

tire, as measured on a drum of 2.5 m. in diameter. Due to larger centrifugal forces at higher speeds the tire wiU grow, resulting in an increase of

effective radius Re, increase of loaded radius Ri and an increase of free radius R. This effect will be almost negligible for the radial belted tire, due to its inextensible belt as shown in figure 7.2.44. Other measurements obtained from an internal drum of 3.8 m. in diameter on a conventional bias tire 5.60-15 and a radial belted tire 155 SR 15

also show the diff^erence in behavior of these tires [29].

7.60-15 Conventional bias tire

fr^fr^ DrufTi tTi cas u T 6 iTi 6 n t s Free radius R

Center height h

0 20 40 60 80 100 120 Speed km/h

Figure 7.2.43. Effect of tire load and inflation pressure on the effective radius as a function ofspeed.
590 THE TIRE AS A VEHICLE COMPONENT

= 350 kgf

Effective

rolling radius Rg

mm

340

330

320^0

Radial belted

Bias tire

Pj * 1,4 kgf/cm

7.60 -15 Drum measurements

tire

20 40 60 80 100 120 Speed km/h

Figure 7.2.44. Variation of the effective radius as a function of speed for a bias and

radial belted tire.

aR F„=Okgf

Pi =1,5 kgf/cm'

a y

v

^>

0 5 0 100 150 ^ speed ^

aRi FH=30d

Pi =1,5 kgf

<gf /cm

b

\.ed^j^

speed

Figure 7.2.45. (a) The increase A/? of the free radius R as a function of speed under zero load condition for a bias and radial belted tire. (b) The increase /\Ri of the loaded radius Ri as a function of speed for a bias and radial belted tire.

1.0 IS 2J0 25 3j0 kgf/cn,2 inflationprcssure

Figure 7.2.46. The influence of the inflation pressure on the increase A/?; of the loaded

radius for a bias and radial belted tire.
ROLLING RESISTANCE 591 The influence of increase of inflation pressure from 1.0-3.0 kgf/cm.^ on the unloaded radius R was neghgible (1 mm.). The increase of the

free radius A/? with speed for both tires under zero load conditions is shown in figure 7.2.45a while the increase A/? = AA of the loaded radius

/?/ at a load of 300 kgf. is illustrated in figure 7.2.456. The influence of

inflation pressure at constant load of 300 kgf. and constant speed of 50

km/hr. is illustrated in figure 7.2.46, showing the largest increase A/? / in loaded radius for the radial belted tire. The influence of different speeds and loads for the conventional bias

tire is shown in figure 7.2.47. The decrease A/?/ at "zero" speed was

actually measured at approximately 2 km/hr.

D. Effect of braking and traction

The effect of tangential loading on the effective radius through braking

or traction forces will be discussed in section 7.2.5.

! Bias tire

Pi =15 kgf/cm

AV=Okm/h

1^

"oV=50 km/h xV=100km/h

>^

0 100 200 300 AOO

vertical Load F^^kgf ^

Figure 7.2.47. The influence of the variation of the vertical load and speed on the decrease

A/?/ of the loaded radius for a bias tire.

1.2 A, Rolling Resistance

Introduction

The roUing resistance of a free rolling tire is mainly caused by the

internal friction in the rubber and cord, while the slip in the contact zone and the windage losses at moderate speeds are of less importance.

Slippage of tread surface on the road is about 5-10 percent of the total losses, while the drag due to air friction represents 1-3 percent of the

total loss. Other factors influencing the roUing resistance are bad road

conditions, involving large tire deflections, the presence of snow or the deformation of soft soil. At constant speed, a free roUing wheel requires a horizontal force Fr in the wheel center to overcome the rolling resistance. The balance

of moments around the wheel center is shown in figure 7.2.48 Fr-h^Fs'fr

where Fv = load carried by the wheel /i = axle height above the ground.
592 THE TIRE AS A VEHICLE COMPONENT

Direction of motion < «-

i

h

JfrL"

Offset

•^r = Rolling resistance

Figure 7.2.48. Rolling resistance of afree rolling tire.

The rolling resistance force Fr is the resultant of the longitudinal tan- gential stresses in the contact patch, while the resultant F.v of the normal

force distribution has an offset, ahead of the contact center as in fig- ure 7.2.49. The rolling resistance is commonly expressed per unit of load of the tire, thus Ib./lOOO lb. or kgf./lOOO kgf. The unit of load varies

in some cases, such as earthmover tires, where it is defined as the resist- ance per ton of load or for bicycle tires where it is given in Ib./lOO lb. or kgf./lOO kgf. The coefficient of rolling resistance /r is defined as

The rolling resistance of a tire is dependent on the load, inflation pressure, temperature, road conditions and speed. The level of the

rolling resistance ranges from 10 to 25 kgf./lOOO kgf. for passenger sizes,

whereas for commercial sizes these values vary from 5 to 15 kgf./lOOO

kgf. The power loss or power consumption is defined as the horsepower

used by the rolling tire at the indicated speed and under the stated

conditions of load and inflation. Methods of Measuring Rolling Resistance

A. On the road

Several methods are used to measure the rolling resistance on the

road under exact working conditions [49].

a. Towing a vehicle at constant speed.

b. Allowing a vehicle to coast and measure the rate of deceleration.

c. Measurement of the torque in the drive shaft of the vehicle re- quired to maintain its speed.

It has been found extremely difficult to find the rolling resistance ac- curately from these tests, because it is not easy to separate other friction from the measured results.

If 1 r fr = -pr or, expressed as a percentage, /r = t^- X 100%.
ROLLING RESISTANCE 593

TIRE 5.90-13 F„-300 kgf

p. = 1.5 kgf/cm2

^CONTACT BEGINS

10 15 i 20 CONTACT LENGTH cm

TIRE 5.90-13 FREE ROLLING

Figure 7.2.49. The vertical force and longitudinal force distribution over the contact

length ofa free rolling tire.

The principal method of towing a trailer has been refined by towing a shrouded passenger car over a range of speeds, and measurement of the

forces between the shroud and the enclosed vehicle (fig. 7.2.50a, h). This approach, due to the Motor Industries Research Association of

Great Britain, involved the isolation of a complete car from aerodynamic

forces by a plywood enclosure mounted on a two-wheeled light weight

trailer with sheet rubber skirting attached to the bottom of all the sides. A similar arrangement is the towing of a shrouded single wheeled

trailer at constant speed, and measuring the towing force by means of a dynamometer.

Because the horizontal force to be measured is only about 1 percent of the vertical load on the trailer, the determination of differences due to load changes, air pressure, or tire construction characteristics calls for an extremely high accuracy of measurement and carefully controlled

conditions of operation [50].
594 THE TIRE AS A VEHICLE COMPONENT

(a) Vehicle rolling resistance trailer

STRAIN GAUGED TOWING LINK TRAILING RUBBER SKIRT

(b) Vehicle rolling resistance trailer Figure 7.2.50. Method of measuring the rolling resistance by towing a shrouded passenger

car.

Difficulty has been experienced in obtaining a constant pull at constant speed, because of the inertia of the trailer, but results are also dependent

on the gradient of the test road surface, the friction of the wheel bearings,

climatic conditions, etc. It is almost impossible to control the tire temperature. In view of all these difficulties, it is generally accepted that in- door machine tests give more reliable comparative values between dif- ferent tires or operating conditions.

B. Indoor Machine Tests

Practically all methods measure the rolling resistance by loading the

tire against a smooth-faced steel drum, and by accepting that significant

differences in tire behavior are induced by the curvature of the drum so

that the values obtained are not those which would be obtained under

actual road conditions. The obvious advantages are that it is comparatively simple to maintain control over the variables speed, load, inflation pressure and temperature.

Four methods are used to determine the rolling resistance under free

rolling conditions:
ROLLING RESISTANCE 595

a. In the inertia method the tire is loaded against a drum of known mo- ment of inertia and the horsepower consumption of the tire is found

from the rate of deceleration of the drum through the chosen speed

or range of speed. The test is repeated with the tire in skimming

contact under zero load and the drum slows under bearing friction and the windage of the drum and tire. From both speed-time curves

the rolling resistance is determined. This simple method has the

disadvantage that it is impossible to determine the rolling resist- ance under stationary conditions at a given speed, and only limited

control of temperature is possible [49-51].

b. In the torque-shaft method the torque M can be determined on the

shaft in the drive to the drum by measuring the twist of a torsion

shaft. The twist is measured both with the tire in skimming contact and with the tire loaded against the drum. The difference between

the torques is used to determine the rolling resistance or the coeffi- cient of rolling resistance and can usually be measured with an

accuracy of 0.5-1.0 percent [52].

c. In the motor torque reaction method the motor driving the drum is trunnion mounted in order to measure the torque reaction in the

stator by a rigidly attached moment arm which engages with a spring

balance or a load cell [51]. The measuring technique is the same as

in test (b), and in both methods constant speed is very important to avoid errors due to acceleration and deceleration of high inertia

test drums.

d. The tire to be tested with the six-component tire tester is mounted in a rig with a loading device above the drum (fig. 7.2.51). This method

calls for an extremely high accuracy in placing the center of the tire in a vertical direction above the centerline of the drum. It is con- venient to average results obtained from tests with clockwise and

counterclockwise rotations of the drum.

Tire Conditioning

Since the rolling resistance of a new tire decreases during the first hours of running, the tires are preconditioned by running them 2-15

hours at about 50 km/hr. under 80-100 percent of their maximum scheduled load and inflation pressure. The tire will deform from its cured

state to an equilibrium condition by adjusting its localized stresses in- ternally. A distance of 150-300 km. is, however, often considered sufficient as a break-in period to bring the tire down to its ambient level of power loss [53].

After preconditioning, the load, speed, and inflation pressure are ad- justed to the desired values for test, and the tire is run until the temperature of the air contained within the tire remains constant for at least 10

minutes. Readings of rolling resistance and the temperatures of the con- tained air and the ambient air are recorded. Immediately after readings

are taken, the tire is lifted from the drum and stopped. Then the tire temperature is measured by a thermocouple needle at about four pre- selected points in the shoulder and in the tread.
596 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.51. Mounting of the six-comi>onent Delft tire tester in a rig with an air-spring

loading device above the drum. The sequence of measuring the various combinations of load, speed,

and inflation are varied in accordance with a statistical design, in order to eliminate systematic bias [54].

Parameters Affecting the Power Loss The most important factor influencing the power consumption in the

rolling tire is the hysteresis of the materials, representing 90-95 percent

of the total power loss. Decrease of the deformation of the tire during

rotation through the contact zone and decrease of vibrations after de- formation will help to minimize the power loss, as well as optimizing the

tire structure and rubber compound to lower hysteresis loss [53]. A. Effect of tire load, radial deformation and inflation pressure. An increase in load at constant pressure and speed results in larger

tire deflection and consequent increase in rolling resistance as illus- trated in figure 7.2.52. An increase in inflation pressure at constant load and speed results in less tire deflection, lower hysteresis loss and consequently, less rolling

resistance [55]. It also may reduce the amount of scrub because it reduces the size of the contact area. B. Effect of speed.

The effect of speed is shown in figure 7.2.53 with constant load and con-
ROLLING RESISTANCE 597

COEFFICIENT ROLLING RESISTANCE % OF F^

V - 30 km/h

165-13

"^^-.^.^t. Pressure DeFlectjoiv-"^^^

1.0 13 2J)

1 cm. 2.35 1.90 1.61 2 cm. 2.A0 1.91 1.61 3 cm. 2.A6 1.93 1.62 A cm. 2.59 2.12 1.72

100 200 300 /.OO 500 600 700 600 900 VERTICAL LOAD kgf Figure 7.2.52. Effect of vertical tire load and inflation pressure on the rolling resistance.

Tire 7.50-U tubeless g F,-A90kgf «

P; » 1.7 kgf/cm^ (constant) 30°

50 60 70 80 90 100 km/h

At constant temperaturg

At service temperature

50 75 100 125 150 » Speed km/h

Figure 7.2.53. Effect ofspeed on the rolling resistance.

Slant inflation pressure [51]. The data are plotted in the form of rolHng

resistance against speed as curves of constant temperature. At higher

speeds a progressive increase is observed as indicated in the lower fig- ure. At constant temperature curve A indicates the behavior, while curve B shows the effect of the higher "service" temperatures. Because

the inflation pressure was kept constant during these tests, there is an

additional stabilizing effect tending to further reduce the increase of
598 THE TIRE AS A VEHICLE COMPONENT

rolling resistance with speed, as a result of the rise in inflation pressure

as the tire gets hotter. There is another "fortunate" effect in that the

hysteresis loss decreases as the temperature increases. Also, slippage

losses in the contact area are reduced at higher tread temperature due

to a lower coefficient of friction. When starting at room temperature conditions, the rate at which the

temperature of a passenger tire rises and the rolling resistance diminishes is shown in figure 7.2.54. Both level off to constant values. The

curves represent constant load, speed and inflation pressure. The equilibrium temperature as measured with a thermocouple needle

in the shoulder after a run of 30 minutes is shown in the form of temperature against speed in figure 7.2.55. On flat roads these temperatures

wiU be lower due to less deformation. The temperature of the contained

air is about 25-30° C lower than the shoulder temperature at equilibrium conditions. C. The effect of rubber compound and cord.

The hysteresis losses within rubber and cord are well known. It folROLLING RESISTANCE kgf

ERATURE_ —^

/

s^^

RESIST/kNCE

TEMPERATURE IN SHOULDER

150

50

DRUM DIAMETER 2.5m

TIRE 7.25-13 NYLON

V «165 km/h

Fb-AOO kgf

Pj -1.5 kgf/cm2

0 10 20 30 TIME jm in) EFFECT OF TEMPERATURE ON ROLLING RESISTANCE Figure 7.2.54. Ejfect of temperature on rolling resistance. TEMPERATURE IN SHOULDER DRUM DIAMETER 2.5rr. TIRE 7.25-13 NYLON F^ -AOO kgf

P; - 1.5 kgf/cm 2

0 50 100 150 20 0 SPEE D km/h

EFFECT OF SPEED ON EQUILIBRIUM TEMPERATURE Figure 7.2.55. Effect ofspeed on equilibrium temperature in shoulder.
ROLLING RESISTANCE 599

lows that the higher the hysteresis losses the higher the rolling resist- ance. This subject has been treated in chapter 1. Readers are also referred to details given about energy losses due to tread, side wall, ply

rubber and cord in the literature [53, 55, 56]. D. The effect of carcass construction.

To show the influence of the carcass construction and tread thickness,

the difference in behavior of a conventional bias and a radial belted tire

is shown in figure 7.2.56. As can be seen, the radial ply construction,

having an inextensible belt, shows an advantage when new. To study the

effect of the tread, the tires were buffed to the bottom of the tread and

rerun for rolling resistance [53], showing a significant change in behavior. The rolling resistance of the conventional bias tire reduces to almost half with a decrease in rolling losses in the high speed range. The

radial-belted tire, although showing a loss in power consumption at the

lower speeds, showed an increase in rolling resistance at the higher

speeds.

30-] 1 1 1 1 1 1 1 1 1 1-

5

4-1 1 1 1 1 \ 1 \ 1 1 2 0 4 0 60 80 100 120 140 160 180 2 0 0 Km/^ SPEED Figure 7.2.56. Effect of tread wear on the rolling resistance — speed relationship for a

bias and radial belted tire.
600 THE TIRE AS A VEHICLE COMPONENT

In the bias construction, the inertia losses are associated with the

vibration or standing wave of the tread and its associated mass. When

the tread is removed, the weight reduction reduces the resulting inertial losses. In the radial belted tire the rigidity of the belt prevents the standing wave in the tread area of the tire, and the deformation is associated

basically with the flexural stiffness of the carcass side wall. Due to the loss of the tread mass the added tension component in the

sidewall is lost with the reduction of the tread-imposed centrifugal force. In radial belted tires the wave formation is associated mainly with the

flexural rigidity of the carcass side wall, and the amount of tread has a much smaller influence on the critical speed. Radial ply tires as high

speed tires therefore require a special approach because of their lower

critical speed [57].

Changes in the design parameters, such as the angle of the cords at the crown, the number of plies in the carcass and the rim width affect the deformation of the tire and consequently the rolling resistance. Truck

tires appear to have lower rolling resistance values (5-6 kgf./lOOO kgf.)

than passenger car tires, because of other tread rubber compounds,

higher inflation pressure and radial stiffness [54].

Lower cord angle increases the radial stiffness of the tire and conse- quently decreases the deformation at a given load and inflation. Because

the circumferential stiffness is also increased due to a lower cord angle

the interia losses are also decreased at high speed [53, 58].

Changing from four-ply to two-ply design gives a reduction in the

hysteresis losses which is related to the mass of the tire and to an in- crease in the deformation. The combined effect of these two factors leads to a decrease in the rolling resistance [58].

The effect of the rim width is that mounting of a conventional bias

tire on a wider rim gives a decreasing tire deflection, and therefore a reduction of the rolling resistance [51, 58].

The height-to-width ratio, known as the aspect ratio of the tire, has

recently received much attention. The trend towards lower aspect ratios

will continue, and low ratios of about 0.70 are already installed on cars, compared with 0.84-0.95 for bias tires of several years ago [59].

Lowering the aspect ratio is advantageous with respect to rolling

resistance because the radial stiffness is increased and the tread is flatter, which also decreases the deformation and consequently the

hysteresis loss. Due to an increase in circumferential stiffness the inertia losses at high speed are reduced [53].

At low speed the section height effect is greater for the bias tire than

the radial belted, but the high speed performance shows significant

improvements for the radial belted tire. Cornering

As the slip angle increases the rolling resistance increases due to a component of the cornering force Fy in the direction of motion, and also due to a slight increase of Fr as shown in figure 7.2.57.
ROLLING RESISTANCE 601

V

SLIP ANGLE a Figure 7.2.57. Effect ofslip angle on rolling resistance. Braking and Traction

The effects of braking and traction are shown in figure 7.2.58, where

the rolling resistance values are plotted against braking and traction

coefficients. These coefficients have been defined as the tangential braking or traction force Fb,t divided by the radial tire load Fn- Due to an in- crease in tractive or braking effort an increase in longitudinal slip is observed, resulting in a higher rolling resistance [51, 60].

The rolling losses under braking and traction conditions are equal to: Fb V -MbH and Mt^I-Ft V

respectively, where

Fb = — Fx= braking force

Ft = + Fx= traction force

Mb = braking torque

Mr = traction torque

n = angular velocity

V= speed.

ROLLING RESISTANCE rt - 10

-5—

Tire

Pi =

8.20-15

- 650 kgf

1.7 kgf/cm'

O.A 0.3 0.2 0.1 0 0.1 02 0.3 O.A ^

BRAKING COEFFICIENT DRIVING COEFFICIENT Figure 7.2.58. Effect of braking and traction on rolling resistance.
602 THE TIRE AS A VEHICLE COMPONENT

Temperature Measurements

Because tire temperatures are related to power and affect the service

life of tires, the measurement of the tire temperatures is considered as a

reliable indicator of the energy put into it and the durability of the tire. Due to deformation and hysteresis, heat is generated within the tire. Studying the heat build-up has always been an important factor in assess- ing the durability of the tire. This is because severe testing is often associated with tread separations from the cord body resulting from thermal

degradation and high centrifugal force. Operating temperatures have been measured while the tire is running

by means of thermocouples vulcanized or inserted into the tread or plies,

but the life of these thermocouples is short because of fatigue breakage

under load. More frequently, measurements have been made by inserting thermocouple needles under the tread or in the plies after stopping the tire. Measurements of the temperature of the air in a tire are also made, as well as determining the tire surface temperature.

Figure 7.2.59 shows typical values of tire temperatures taken on an

indoor test wheel as a function of speed, and with various test conditions. It appears that many normal passenger car tires operate in the

vicinity of 100° C at normal conditions, measured in the air cavity [62].

Several authors report the measurement of tire surface temperatures

using a contact thermometer or thermistor to relate the average skin or groove temperature to heat loss or the relative wear rating of tread

compounds [61,56].

Other methods were developed for determining the temperature of the

air in a tire [63].

The difficulties encountered using thermocouple needle measurements

are also reported in the literature [56, 64, 65].

Figure 7.2.59. (a and b) Equilibrium pressure and temperature rise as functions of tire load and speed.
BRAKING AND TRACTION 603

12

)o|40 50 60 70 80 90 100 SPEED MPH Figure 7.2.59. (c) Equilibrium pressure and temf)erature rise as functions of tire load and speed.

Refinements in measuring techniques made it possible to measure

internal tire shoulder temperatures under road testing conditions of

high loading and speeds, to study durability and relaibility of the tire [65, 66]. In endurance testing on indoor machines it has often been

found advantageous to trace the hot spots before severe tread separation occurs. The use of lead-sulphide photo-conductive cells for high

speed pyrometry has been reported [67, 68, 69].

7.2.5. Braking and Traction

Braking

A. Deformations

The difference in deformations between a free rolling and a braked

tire has already been discussed in chapter 6. However, to facilitate the

understanding of the following paragraph, another illustration of tire deformations shall be given [24].

To show this difference in behavior, a photograph was first taken of a

freely rolling tire and afterwards, using the same negative, a second

photo on top of the first one has been made of a braked tire. Triggering

of the camera has occurred at exactly the same position as in the free

rolling case. This is shown in figure 7.2.60. By measuring the distance between the radii which were originally

equally spaced on the surface of the undeflected tire, the deformations of tread and sidewalls can be determined for various braking conditions.

Figure 7.2.61 shows some results thus obtained on the movable platform machine, and it is seen when a braking force is applied that the

tread elements before contact are stretched. The alterations of the de- formation are also clearly shown on a photo of the contact area of a braking tire in figure 7.2.62.
604 THE TIRE AS A VEHICLE COMPONENT

Figure 7.2.60. Th e difference in deformation between a free rolling and a braked tire.

Tire 5.90-13 Bias carcass Bra kin g force 200 kgf

Load 300 kgf

I n fl pres £ ure 15 kgf/cm! f bra king un deflected

Figure 7.2.61. Deformations due to braking.
BRAKING AND TRACTION 605

Figure 7.2.62. Photo through glass plate of contact area of braking tire.

B. Force distributions

a. Longitudinal force distribution. The movable platform machine is also equipped with an apparatus in- corporating three measuring bars 2 cm. long and 20 cm. wide for measuring the distribution of the vertical, longitudinal, and lateral forces

[14,24].

With this modified "Gough" apparatus shown in figure 7.2.63, the

distribution of the longitudinal shear forces in the contact area is deter- mined. The distribution of the forces created by a free rolling tire is represented by line 1 in figure 7.2.64, and the additional shear force

created by the braking torque is represented by line 2. The resultant

shear force distribution along the length of contact is therefore repre- sented by line 3, as measured with the longitudinal force bar of the Gough

apparatus. The precise form of this curve depends very much on the

magnitude of the braking force for a given radial load, inflation pressure,

coefficient of friction, etc. as described in chapter 5, figure 5.52. The

reason for such a force distribution becomes clear by looking again at figure 7.2.61. An extended tread element adheres to the platform on first entering the contact zone (ch. 6, fig. 6.1c). As it moves further into the contact area, it produces a deflection which increases linearly with in- creasing distance (causing an increasing longitudinal force) until the

local value of limiting frictional force is reached and the tread element

begins to slide back, thus reducing again the longitudinal force as shown

by line 3 in figure 7.2.64.
606 THE TIRE AS A VEHICLE COMPONENT

- -

m

m

1

Figure 7.2.63. Modified Gough apparatus incorporating three bars measuring the lateral, longitudinal and vertical partial forces. The total longitudinal force Fb may be obtained using the six-component

tire tester, also mounted in the Delft movable platform machine of figure

7.2.5. The result obtained can be compared with the value obtained from

the longitudinal force measuring bar, by integrating the longitudinal

force function as the tire travels over it. b. Vertical force distribution.

The vertical force distribution is measured by a second bar of the

Gough apparatus. The difference in behavior of a free rolling and braked

tire is shown in figure 7.2.65. The total vertical force Fx may be obtained

either by integrating the vertical force function, or by using the tire tester, the latter method being preferable. The offset of Fv before the contact center may also be obtained. As already discussed in section

7.2.4, the coefficient of rolling resistance /, is higher than in the free

rolling case, mainly due to a rise in slip-induced rolling resistance. The

balance of moments around the wheel center 0 reads (fig. 7.2.64):

FB'h = MB-^Fsfr

Due to the braking torque, it is observed that the distance h is smaller

than in the free rolling case [24].

c. Longitudinal sliding

Attempts have been made to measure the longitudinal sliding with a

small cam wheel mounted in the longitudinal force bar of the Gough ap-
BRAKING AND TRACTION 607

Platform velocity

Vertical force distribution

Longitudinal force

distribution

Longitudinal

sliding distance

Vo=n.R

sliding velocity

Velocity

Tire circumference
608 THE TIRE AS A VEHICLE COMPONENT

Tire 5.90-13 bias carcass

Cor) tact length cm

Figure 7.2.65. Comparison of vertical force distributions over the contact length of a

free rolling and braking tire.

paratus. The cam wheel can rotate, but does not measure the longitudinal sliding of one single tread element because due to the sliding motion

the neighboring tread element subsequently comes into contact with the cam wheel. The total longitudinal sliding distance is measured with a potentiometer coupled to the cam wheel shown in figure 7.2.63. It is the sum of the longitudinal sliding of all different tread elements along a line parallel to the direction of travel of the platform. These measurements of

sliding in the contact patch were not accurate enough. Photography of the contact area of the tire, marked with a grid, during motion of the

glass surface of the platform machine, appeared to give better results

(fig. 7.2.62).

Due to stretching of the tread elements before contact the circumferential velocity will increase, and on the assumption of no sliding in the

front part of the contact zone for a moderate braking force, the tread

elements coming into contact with the platform will start to travel with

the platform speed V. The longitudinal shear force which increases

towards the rear of the contact zone, in combination with the decreasing

vertical force, will cause rearwards sliding of the tread elements in the

rear part. The resulting longitudinal sliding distance and sliding velocity curves, taken over the contact length, are shown in figure 7.2.64. In- creasing braking force at constant vertical load wiU result in increasing

sliding over the contact length as in figure 7.2.66. The slip ratio may be defined to be:

Ho
BRAKING AND TRACTION 609 where Clo = angular velocity at free rolling

CLb = angular velocity at braking

Both values can be measured at constant forward speed of the platform

or at constant drum speed [sec. 7.2.3, fig. 7.2.42]. As the braking force

Fb increases so does the percent slip. The resulting braking force co- Fb

efficient — is shown in figure 7.2.66 for constant speed as measured

r.v on a dry steel drum of 2.5 m. in diameter.

The difference in behavior of a radial ply and bias ply tire is clearly

demonstrated in figure 7.2.67 at a constant speed of 40 km/hr. [70]. It

is seen that the position of the peak coefficient /Lt/^ is often very difficult to determine since the curves are sometimes rather flat. Similar results are also often obtained on wet surfaces [71]. D. Effective radius

Returning to figures 7.2.61 and 7.2.62, the stretch of the tread ele- ments just before contact, as measured from the photograph, appears

to be approximately 10 percent. Assuming again no sliding in the front part of the contact zone, the same reasoning as in the free roUing case

(sec. 7.2.3) results in: V=l.lVo, that is Re=l.lR, showing an increase

in effective radius compared with the free rolling condition {Re = 0.96R)

and a decrease in angular velocity O, [70]

braking slip percent < = "° ~^ ^ ^lOO"/"

sliding

wheelslip- &5°/o 10°/o 15%

Figure 7.2.66. The relation between braking force coefficient and braking percent slip.
610 THE TIRE AS A VEHICLE COMPONENT

Drum me»surtm«nt

V = 40 km/h

F^=300 kgf

p^:l,5 kgf/cm^

10 20 30 40 50 60 70 80 90

Stip B no Figure 7.2.67. Comparison of the braking force coefficient-braking slip relationships of a radial ply and bias ply tire at constant drum speed.

The slip ratio may be defined to be: Hp Hfl ReB Reo Ho ReB

where /?eo = effective radius at free rolling

ReB = effective radius at braking

E. Wet road measurements

a. Distance method [72].

The locked wheel "sliding" friction coefficient is determined during

the speed interval under investigation.

The calculation of this friction value, sometimes called "braking

coefficient" is based on the accurate measurement of the skid distance

5 and the speed V. It follows that:

2gs

where Vi = initial velocity

^2=^ final velocity

^= gravitational constant. Due to the erroneous assumption in the above formula of a constant

deceleration over the speed interval (^t being speed dependent) it is
BRAKING AND TRACTION 611

rather difficult to compare results obtained with different tires on various

road surfaces. Because standard vehicles are used, a large number of

tests are required to arrive at an acceptable accuracy of the averaged

/jLis values. This is usually done at speed increments of about 10 km/hr.

in the required speed range of the test vehicle.

b. Deceleration method.

As discussed in chapter 6, the braking performance of tires can also be measured by the deceleration of the vehicle. In this method only the

front wheels of the test vehicle are braked in order to maintain directional

stability of the vehicle at all speeds [73] . Both the peak value of the braking force coefficient flip, and the locked

wheel slide value fJiis are determined from the film record of deceleration

(ch. 6, fig. 6.11).

Because these tests can be executed on a relatively short test sur- face, having a more or less constant frictional character and uniform

water depth, comparison of tire data thus obtained is usually preferred over the stopping distance method. The distance method with its long

skid distances has a poorer accuracy especially in the higher speed

range [72].

It has often been observed with the deceleration method that radial

ply tires gave higher peak braking force coefficients jJiip on all surfaces

than do bias ply tires. The differences between these peak coefficients were almost independent of speed and were least on coarse-textured

surfaces, and greatest on fine-textured surfaces, ranging from almost

zero to about 0.1. The radial ply tires gave higher locked braking force

coefficients fits on the fine textured surfaces but lower coefficients on the coarse textured surfaces than did the bias ply tires [73].

c. Force method.

Numerous braking force tests with different tire constructions and

tread rubber compounds have been executed on a variety of dry and wet

road surfaces and are reported in the literature. The advantages of the towed trailer road tests are that the towing

truck can maintain constant speed at the desired level, uniform wetting

immediately ahead of the tires is obtained, and a large variation in tire loading is possible without any load transfer effects (fig. 7.2.8). During these road tests it appears very difficult to obtain braking force

coefficient values in the range between the peak braking force coeffi- cient fJLijj and the locked braking force coefficient fJiis{KB = 100 percent),

because of almost immediately locking of the wheel after fJLip has been

reached. For this reason, usually only fxip and /jlis are determined to compare tread designs, rubber compounds, tire constructions etc. The nu- merous results obtained are very similar to data published in the litera- ture [73] and discussed in chapter 6. Another method to determine the braking force coefficient has been

published by the Road Research Laboratory [71]. Force measurements

are made on a fifth wheel mounted in a test vehicle. The angular velocity

of this test wheel can be held at any desired value, independent of the

vehicle speed.
612 THE TIRE AS A VEHICLE COMPONENT

At the beginning of a series of tests a record of wheel rotation is ob- tained with the free roHing tire in order to obtain its roUing radius Reo

for evaluation of the braking slip.

ReB ~ Reo ^B=—^

t\eB

Typical curves of braking force coefficient against braking slip kb are shown in figure 7.2.68 for a bias ply tire on three out of five of the test surfaces from table 7.2.1.

Tire bias ply 5.25_16

Figure 7.2.68. Typical curves of braking force coefficient against braking percent slip for a bias ply tire on three road surfaces at two speeds.

Surface 1 (rough, harsh), surface 4 (rough, poHshed), surface 5 (smooth, pohshed).

Table 7.2.1.

No. Description Texture

1 9.5 mm. quartzite macadam carpet Rough, harsh

2 Fine cold asphalt Smooth, harsh

3 9.5 mm. mixed aggregate macadam carpet Rough

4 9.5 mm. Bridport macadam

carpet Rough, polished

5 Mastic asphalt Smooth, polished
BRAKING AND TRACTION 613 The curves show in general a rapid initial rise, and then become

gradually less steep as they approach a peak at approach 7-25 percent

slip. Beyond the peak there may be a slowly falling region and finally a more rapid fall from 80 or 90 percent to the locked wheel values at 100

percent slip.

The rough and harsh surface No. 1 is capable of giving a high coefficient. The harsh microtexture has the largest effect on the coefficient

in comparison with the other surface No. 4. It has been shown that tires of radial ply construction have a more rapid initial rise of the braking

force coefficient curve than bias ply tires. On harsh surfaces the radial

ply tires gave higher peak coefficients but on polished macadam sur- faces lower values were observed. The same was true but less obvious

for the locked wheel case [71].

Curves of pure braking force coefficient against braking slip give

valuable data for development of anti-skid braking systems. These sys- tems are of particular value when braking and cornering forces act simultaneously [71, 74].

Traction

A. General observations

Having discussed the behavior of a braking tire at length in the pre- vious section, a short description of the action of a driving torque Mr

will be sufficient because the situation is analogous.

When a tractive force Ft is applied, the tread elements ahead of contact are compressed. The resultant shear force distribution along the

length of contact, as measured with the longitudinal force bar of the

Gough apparatus, is shown in figure 7.2.69. As discussed (ch. 5, fig. 5.53),

the form of this curve depends on the magnitude of the tractive force. The resultant shear force distribution (line 3) can be seen as the sum of the force distribution of a freely rolling tire (line 1) and the additional

shear force created by the tractive force (line 2). A compressed tread element adheres to the moving platform when

first entering the contact zone. As it moves further the increasing deflection of the tread element produces a linearly increasing longitudinal

force. Beginning in the rear part of the contact zone, forward sliding of the tread element will be observed [24]. The resulting longitudinal slid- ing distance and shding velocity are also shown in figure 7.2.69. The slip ratio may be defined to be: _ CIt — ^0 _ Reo ReT

fir Reo where Ho = angular velocity at free rolling

ilT = angular velocity with traction

As the tractive force Ft increases, so does the percent slip, and the

Ft

resulting traction force coefficient — as a function of percent sup may

r N be plotted, giving a curve similar to that obtained for the braking force

coefficient.
614 THE TIRE AS A VEHICLE COMPONENT

Tire circumference
BRAKING AND TRACTION 615 The decrease in effective radius Rer compared with the free roUing case may be obtained by measuring Q,t and ilo at constant platform speed

or drum speed.

B. Wet road measurements

a. Break-away method.

A light truck fitted with the test tire is attached to a large dynamometer truck. The dynamometer truck is used for braking and maintains

a low constant forward speed of about 5 km/hr. to ensure rolling of the

test tire. By measuring the maximum tractive draw-bar force developed

by the tire in the low slip range of less than 15 percent, the test deter- mines the maximum force available to accelerate a vehicle, as the torque

is gradually increased until the tire slides completely [72].

b. Dynamic sHp method.

This is a continuation of the break-away test, because dynamic test- ing is conducted at 75 to 200 percent slip.

Traction ratings for both break-away and dynamic tests are given as a

ratio of tractive force to tire load. Typical recordings are shown in figure 7.2.70 [72].

c. Maximum acceleration method.

The tire comparisons are based on the speed at wheel spin in the speed

range 40 to 100 km/hr. using a passenger car. The procedure is to drive

the car on the test surface at constant speed, then accelerate by means of

a full throttle down shift, or limiting the throttle position in order to re- duce the maximum available torque to a level compatible with the test conditions of surface and tires. The driving force vs. speed curves of the different gear ranges for the

test car are usually first obtained on a drum dynamometer or derived

50 100 150

Slip%

200

Figure 7.2.70. Relation between relative traction force percent and traction percent slip.
616 THE TIRE AS A VEHICLE COMPONENT

from engine torque curves. The theoretical point of sUp is shown in fig- ure 7.2.71 at the intersection of the maximum available driving force and

the tractive coefficient curves for two different tires on the same test surface. The speed at the point of wheel spin indicates relative tire traction. Tire comparisons are based on the maximum speed attained before

wheel spin, or the least time required to travel a given distance. For the

latter test, slip may occur throughout the test distance [72].

Several other methods are used to measure the tire resistance to wheel

spin under conditions of acceleration and with the vehicle travelling

straight ahead. Among these the single tire test technique is reported

[72, 75]. C. Factors affecting traction on wet roads.

The tire variables affecting passenger tire traction on wet road sur- faces include: tread design, tread compound and tire construction

(ch. 6). The tread design variables are bladed tread pattern versus plain design, groove pattern, tread width, tread radius and center shoulder effect. The tire construction variables such as crown angle, tread reinforcement, and radial construction versus bias construction have a smaller

effect on the wet tractive effort than the tread design variables.

Tread compounding is another complicated subject. For further infor- mation the reader is referred to the literature [59, 76].

7.2.6. Tire Nonuniformities

Introduction

As road surfaces became smoother their contribution to vibration deVehicLe speed km/h

Figure 7.2.71. Illustration of maximum acceleration method giving the relationship of

tire traction to driving force.
TIRE NONUNIFORMITIES 617

creased and increased attention has been focused on the tire-excited

vibrations introduced by the nonuniform properties of the tire.

It is impossible to manufacture perfectly uniform tires because each

item has its own manufacturing tolerances. Only the rigid control of all processes throughout the manufacture of materials and components for

tire building can minimize the unavoidable imperfections affecting

uniformity.

Lack of uniformity around the tire will produce variations in forces

applied by the tire to the vehicle, and repeat its influence with each

revolution of the tire. The resulting periodic vehicle vibrations are speed

dependent and often very annoying to the driver and passenger. Parts

of the vehicle vibrate and radiate energy as sound heard by the passenger, and vibrations of the steering wheel, floor and seats are felt directly

by the driver and passenger [77]

The transmission of vibrations by the tire caused by road irregularities, or by the natural frequencies inherent in the tire structure, are not con- sidered in this section, since these characteristics are not dependent

on the degree of nonuniformity. But it should be observed that chassis

tuning capable of attenuating all tire vibrations resulting from various

road irregularities is extremely difficult, even more so because the

various related vibratory systems should also avoid natural frequencies

that might synchronize with natural tire frequencies.

The Tire Manufacturing Process A passenger car tire may be composed of approximately 100 separate

ingredients. To insure quality and uniformity among these materials,

all have their own physical and chemical test qualifications. Tires are made of four basic materials as shown in figure 7.2.72, illustrating the

many steps in the manufacturing process that should receive inspection

and evaluation to insure conformity to the standards of quality and

uniformity [78].

I MATERIALS ORDERED TO SPECIFICATION . TESTED FOR QUALIFICATIOW | RUBBERS CHEMICALS FABRICS WIRE

i MATERIALS AND

i C0MP0UNDW6

PLASTISIZING

RECIPE. WEIGHING FORMULATION ADHESIVE COATING

I BUILDING

1 PREPARATON

CURING

HANDLING

AND

STOCK

AND

TREAD

CUT TO

SIZE.WglG

EXTRUSION INSUDSTE WIRE FORM TO BEAD SIZEAW WRAP

Figure 7.2.72. Representation of the many steps in the tire manufacturing process.
618 THE TIRE AS A VEHICLE COMPONENT

The phase of the stock preparation, being the fabrication of the rubber

compounds, fabrics, and wire into basic components of the tire, is very

critical of weight and dimensional tolerances.

Treads

The tread length, weight, and gage must be carefully controlled to in- sure uniform distribution of tread in the finished tire. Treads are generally applied to the building drum in such a way that one end butts the

other end, tending to concentrate any variation in length at the tread

splice. Exceeding the specified tolerances will result in an unacceptable

nonuniformity. Figure 7.2.73, obtained from special uniformity machines,

shows the effect of short tread length, but of course excessive length due

to stretching the tread during application on the building drum will also

affect the radial force, radial run out and tread shoulder thickness [78].

A thick tread joint causes unbalance, and this has long been known as a very important source of nonuniformity because the unbalanced force

is proportional to the square of the velocity. Balance can be compensated

for, either statically or dynamically. But a thick tread joint is also less

flexible than the rest of the tread and this difference in stiffness cannot

be corrected for or compensated by adding suitable balance weights to the wheel. The forces necessary to accelerate the tire tread in the contact area may change appreciably with a nonuniform mass distribution. The variation in inertial forces will also cause fluctuations of the effective

rolling radius. The effective rolling radius variations can be obtained by

measuring both wheel and drum angular velocities on uniformity machines. Rolling radius variations produce angular accelerations that re- sult in tangential force variations proportional to forward speed. Heavy

tread splices will also show fluctuations in the rolling resistance [79].

Poor tread centering on a portion of the tire will result in fluctuations

of the lateral force and shoulder gage thickness variation curves [78].

Figure 7.2.73. The influence of non uniform distribution of tread on the radial force and radial runout.
TIRE NONUNIFORMITIES 619 Cord Angle and Ply Fabric

The transformation from the cylindrical building drum shape into the

toroidal configuration within the tire mold is shown in figure 7.2.74 with

corresponding dimensional changes.

Any irregularities in the ply fabric are magnified due to the pantographing of the cords during the shaping of the flat ply stock into the final

tire. Since a one degree variation in bias angle can result in 2-3 degrees

variation in the cured tire, it will be evident that angle variation within a

tire can have significant effects on the dimensions as well as the related

forces. Figure 7.2.75 shows the effect of angle change on tire shape for several tires which have common contour lengths from bead to bead

[78]. Since the cord angle determines the ratio of height to width of the

tire carcass, the measurement of the width variation at the tire walls can

give an indication of the cord angle variation in the tire.

Shape relationship Parameters Building drum mold

Contour length L 19.6" 17.4" Tread width W 6.0" 4.6

" Cord angle ^ 66,5° 38* Cords per inch 25 21 Tread length Lt A7" 85" Figure 7.2.74. Transformation from the cylindrical building drum shape into the tire mold for a bias ply tire.

Figure 7.2.75. The effect of centerline angle change on tire shape.
620 THE TIRE AS A VEHICLE COMPONENT

Building and Caring

When the tire builder cuts the ply fabric too short, causing an open ply

splice, the effect can be an intolerable lateral and radial runout (fig. 7.2.76). Also, bulky ply stock splices or wrinkled and wavy pHes should

be avoided. Another cause of nonuniformity resulting in eccentric beads

may be due to improper alignment in the tire building machine.

1 La

t ,. .05^

teral r BS

unout

1 Lat cral r unout

*

Figure 7.2.76. The effect of an open ply splice on lateral and radial runout. A correctly specified drum width is important, since too wide a drum

results in irregular and loose cords, while too narrow a drum results in pulled or slipped cords in the cured tire [78]. The development of the

radial belted tire has progressed rapidly, but their adoption as original

equipment has been slow. This may be due to large investments in bias

conventional tire building equipment, but also to the difficulties en- countered in mass production of this type of tire to the required degree

of uniformity. The radial ply tire has two or three layers of cord at an

angle of 88-90 degrees to the crown and about three to six layers of

cords forming a circumferential belt. Although many possibilities exist,

the belt cords lie generally at an angle of 12-20 degrees to the crown

centerline. Accurate building and centering of the belt have received

concentrated effort in the area of precision assembly to minimize irreg- ularities affecting uniformity. Precision building and special handling

techniques are also required for bias-belted tires or tires with asymmetric carcasses.

Stability of Materials

Analyzing the influence of moisture on various cords (rayon, polyester,
TIRE NONUNIFORMITIES 621

nylon), it was found that in the region of cord tension due to inflation pressure within the tire, a large variation of cord elongation was ob- served for different relative humidity conditions. The uniformity of a tire may be adversely affected if the moisture content varies within a tire or during a production run of tires. Polyester cords have excellent

characteristics in this respect [78].

A localized difference in cord modulus in a ply will cause the tire to bulge slightly in the vincity of the more extensible cords, affecting the

radial force [79].

Another class of tire nonuniformity is the temperature induced or so

called heat-induced nonuniformity, found to exist in tires made with

thermoelastic cords such as nylon and polyester.

It was observed that uneven cooling of the tire during the conventional

postcure inflation process caused temperature differences across the

tire. Due to the temperature sensitivity of thermoelastic cords, these

temperature differences caused cord shrinkage, resulting in temperature

induced nonuniformities. Temperature measurements of tires on the

postcure inflators revealed differences of about 15° C around the circumference, with the hotter part commonly opposite the cooler part. To

obtain uniform temperatures across the tire a special tie cooling ap- paratus was developed [80].

Wheel Uniformity

Car wheels are subject to manufacturing fluctuations and contain some degree of nonuniformity. The maximum radial and lateral runout

at the beads should not exceed approximately 1.5 mm. (D.I.N. 7817).

Accurate wheel centering contributes greatly to a good degree of uniformity.

Tire uniformity measurements are made either on precision rims or car wheels.

Figure 7.2.77 shows car wheel radial and lateral runouts as measured

on the beads, together with the corresponding free tire runouts as measured on the side walls of a preconditioned tire. Lateral runout spectra of wheel and tire sometimes show a good correlation.

Effect of Tire Nonuniformities on Vehicles

The tire-excited vehicle vibrations introduced by the nonuniform properties of the tire are felt as vibrations of the parts of the car in contact

with the passenger, and heard as sound in the passenger compartment.

The automobile plays a significant part in the transmission of shake,

thump and roughness, as shown by the results of running the same set of tires on different cars. It appears that no car is insensitive to tire force

variations, but some are more sensitive than other cars. Shake problems can come from radial or lateral force variations and depend primarily on the amplitudes of the first and second harmonics of the force

variation [81]. Since the force variations repeat themselves with each

revolution of the tire, they constitute a periodic forcing input to the vari- our vibrating systems of the vehicle. It is a relatively simple matter with

today's computer capability to analyze the quantity of each harmonically
622 THE TIRE AS A VEHICLE COMPONENT

Radial Runout Figure 7.2.77. Car wheel radial and lateral runout together with the corresponding

free tire runouts. related sine wave which makes up the complex curve. The periodicity

allows a representation of the force variations by a series of sine waves, each at a frequency equal to an integral multiple of the wheel frequency.

Each sine wave component is called a harmonic of the wheel frequency,

and when all harmonics are summed with their proper phase relation- ships, the original variation is reconstructed [81]. This so-called Fourier

analysis of a runout curve is shown in figure 7.2.78. The amplitudes A,i of the harmonics are the magnitudes of the Fourier

coefficients of the periodic waveform expressed in the form:

A' F{t) =Ao-\- 2 sin i27mf^(fn)

ji=i with /= frequency and = phase angle.

With the aid of a mathematical shake model embodying beam effects,

it is possible to compute the response of the structure of a car to measured tire radial force variations. It was observed that the amplitude of the

first harmonic of tire radial force variation is an important parameter in vehicle beaming shake. This is a mode of vibration of the entire sprung mass involving predominantly bending deformation of the entire structure

[82].

Sometimes subjective ride ratings are used to demonstrate improvements in shake. For instance, tires having a certain level of nonuniformity are reduced to a lower value by removing a very small amount of rubber from certain regions of the shoulder ribs with special alteration

machines (fig. 7.2.79). These machines reduce the radial force variation

and proyide uniformity improvements on an automated basis. The altera- tion results in a demonstrable improvement in tire shake, but has no

consistent effect upon tire roughness [83].

The usual manner of demonstration is to select tire-wheel combinations

of known uniformity and evaluate the systems for shake and roughness on the given car. The tire-wheel combinations are then altered to better
TIRE NONUNIFORMITIES 623

90° 180° 270° 360°

Figure 7.2.78. Fourier analysis of a runout curve. uniformity levels, and the tests repeated. If tires are reduced to radial

force variations of 15 lbs. or less, tire shakes will usually be rated better

[83].

Subjective ride rating requires careful selection and thorough training

of the personnel involved in rating the vibration characteristics [84^86].

These subjective ratings, on the basis of studies of human tolerance to

FOtCE VAfhATION CYLMOER

Figure 7.2.79. Automatic reduction of radial force variation by removing a small amount

of rubber.
624 THE TIRE AS A VEHICLE COMPONENT

vibration, can be used to establish acceptable tire nonuniformity tol- erances as measured on tire uniformity grading machines.

Tire thump is a low pitched noise and is thought to be related to the

4th through 9th harmonic of tire rotation. Thump sound intensity is determined by the loudness of amplitude of modulation of the sound

pressure wave, and the sharpness of the increases and decreases of the

modulation envelope of the sound pressure wave [77].

Roughness intensity is also dependent on the modulation envelope,

but this envelope appears different at high and low speeds. There is an indication that of the harmonics present, the 4th through 15th are of consequence to roughness [77]. Another type of vibration observed is waddle, which is combination of nosing and rolling motion of the vehicle.

This motion is very sensitive to tire nonuniformities. Because the lateral and radial forces vary with each wheel revolution, and the effective roll- ing radii of left- and right-hand wheels are not exactly the same, the

resulting motions may be in phase or in opposite phase. The effect may

be felt through the steering wheel and experienced on flat roads as low

frequency oscillatory forward motion combined with slight rolling

motion. This low frequency nosing action of the vehicle can, for instance,

be recorded with a drift angle meter attached to a front wheel, as de- scribed in section 7.3.3 figure 7.3.43. Tire Uniformity Grading Machines

Numerous tire uniformity machines are described by various authors

and many machines are in use. Figure 7.2.80 is a schematic of a tire uniformity machine [83]. The test tire is mounted on a precision rim

with provision for rapid tire inflation. The tire is loaded against a drum

of the largest size that is practical, since the roll size affects the force

variations [82]. The tire is run at zero camber and zero sHp angle at speeds less than 60 rpm. This low speed of 60 rpm or 1 rps has been

chosen so that the amplitude of the tenth harmonic of the force varia- tions can be reliably measured. This requires the minimum machine

and tire resonant frequencies to be at least four times the frequency

of the tenth harmonic, or 40 times the rotational frequency. Since the

lowest tire resonance occurs at approximately 40 c.p.s., it is necessary

to restrict the maximum wheel speed to 1 rps [82].

Reproducible results for both forward and reverse rotations are only

obtained when the tires are properly conditioned.

The resulting force variations are recorded at the specified inflation and load, with a fixed distance between the tire and drum axle. This constant height method is employed since a constant load will cause axle

height variations for nonuniform tires with resulting inertial and fric- tional errors in load application.

Some machines can measure radial, lateral and tangential force

variations in tires, as well as free radial runout variations, free width

variations, and the crown thickness. Free radial runout variations, are measured either on the centerline or on both shoulder ribs. The free measurements can be executed when force variations are measured,

the displacement transducers being located a sufficient angular distance
TIRE NONUNIFORMITIES 625

TIR€ 1 r- SOUO OR r IK

WIVE MOTOR TORQUCMETER 1 60 RPM OPTIOIIAL f

SPLIT RM

ACTUATOR!

FORCE TRANSDUCERS

. TANCCNTIAL

F^. LATERAL

TIRE LOAD Figure 7.2.80. Scheme of a tire uniformity machine.

I

I U o

LATERAL FORCE VARIATION

RADIAL FORCE VARIATION

LOCATIONS OF SPLICES

LOADED RADIAL RUNOUT

FREE RADIAL RUNOUT

TIRE Nol

Figure 7.2.81. The relation between the location of splices and the corresponding uniformity measurements.
626 THE TIRE AS A VEHICLE COMPONENT

TREAD SPLICE LINER AREA SPLICE

SHOULDER

B. RIB HEEL

3 10 Y 30 40 SO 60 70' ' 8O 90 0

V ' V ' / \' ' <- X \ / \ \ ^, /

7^.

/

/ / ~y—^x^; x~ y

SERIAL ANo. 4 No. 1 No. 1 No. 3 No. 2 SERIAL No. 4 PLY ' PLY PLY PLY PLY PLY

TIRE No. 1 Figure 7.2.82. Chart showing the location of all ply splices and the liner splice. from the footprint area to avoid distortions from loading [83]. However,

the measurement of free radial runout is sometimes made on special

machines with a preconditioned unloaded tire. A displacement trans- ducer fixed in space contacts the crown of a slowly rotating tire. To illustrate some results obtained from uniformity grading machines,

figure 7.2.81 shows the effect of splices on uniformity measurements.

After the measurements the tires were cut circumferentially in three

places: the crown and the two buffing ribs on the sides of the tire. The

location of all ply-splices and the liner splice were determined and noted on a chart. A typical chart is shown in figure 7.2.82, and a replica of this chart is shown with the corresponding uniformity measurements of

figure 7.2.81. The measurements are "phased" so that all start from a common reference point on the tire, and the splice locations were deter- mined according to the same reference point. We see that radial force

variations occur at the liner splice and near the intersection of ply splices,

where such intersections occur near the crown. The lateral force, loaded

radius, and the free radius also show variation near the liner splice [88].

Another way to show that an automobile must be sensitive to uni- formity fluctuations is the Gough-plot of cornering force versus self- aligning torque. It shows in a single plot the cornering force and the

self-aligning torque generated at any position around the tire when the tire

is rolled in clockwise and counterclockwise directions. Figure 7.2.83 shows the characteristic data for a poorly constructed steel belt tire manufactured during the early stages of development of this tire [89].
TIRE NONUNIFORMITIES 627
628 THE TIRE AS A VEHICLE COMPONENT

References

This bibliography refers frequently to V.R.L.D. reports made by the Vehicle Research

Laboratory at Delft. These reports are in the Dutch language and can be obtained in microfilm or blueprint on payment.

[I] MiUiken, W. F., and Whitcomb, D. W., Close, W., and Muzzey, C. L. e.a., Symposium

on research in automobile stability and control and tire performance, IME, Auto,

Div., p. 107; SAE Trans., 344 (1956).

[2] Cough, V. E., and Roberts, G. B., Dunlop cornering force machine. Trans. I.R.I. 33(5), (1957).

[3] Kollmann, K., New testing machine for the study of tires. Revue Gen. du Caoutchouc

(10) (1959).

[4] Krempel, G., Experim. Beitrag zu Untersuchungen an Kraftfahrzeugreifen, Diss. Karlsruhe (1965).

[5] Henker, E., Dyn. Kennlinien von P. K. W.-Reifen, Wissenschaftl.-Techn. Veroffent- lichungen Automobielen H3 (1968).

[6] Bruinsma, F., Design and measurements of an airbearing for a running belt with an

electro-hydraulic vibrator system, V.R.L.D. Report No. 472^'^'^ (Sept. 1968).

[7] Groels, G. K., Measurements and improvements on an airbearing, V.R.L.D. Report

No. M080 (Aug. 1969).

[8] Elsinga, H., Design of an endless steel band road simulator, V.R.L.D. Report No.

511 (March 1969).

[9] Spaink, G. N., Friction and drainage measurements on abrasion paper of different

qualities, V.R.L.D. Report No. 391 (April 1967) (Translated by Cornell Aeronautical

Laboratory, Inc., Buffalo).

[10] Van Eldik Thieme, H. C. A., Experimental and theoretical research on mass spring systems, FISITA 1960 (Elsevier Publishing Company, Amsterdam) p. 386.

[II] Cough, V. E., and Whitehall, S. G., Universal tire test machine, FISITA 1962 (Inst. Mech. Eng., London).

[12] Nordeen, D. L., and Cortese, A. D., Force and moment characteristics of rolling tires, SAE Paper No. 713A (June 1963); SAE Trans., 32 5 (1964).

[13] Nothstine, J. R., and Beauvais, F. N., Laboratory determination of tire forces, SAE

Paper No. 713B (1963).

[14] Van Eldik Thieme, H. C. A., The measurement of tire characteristics, De Ingenieur, No. 28 and 30 (1964).

[15] Ginn, J. L., and Marlowe, R. L., Road contact forces of truck tires, SAE Paper No. 670793 (1967).

[16] Heesewijk, A. P. C. van, and Groeneweg, H. H., Description of six component tire tester, V.R.L.D. Report No. P055 (Aug. 1967).

[17] Savkoor, A. R., Dynamic behaviour of an orthogonal tyre force measuring unit, V.R.L.D. Report No. P029 (July 1963).

[18] Edema, L., Measuring platform, V.R.L.D. Report No. P017 (Nov. 1961); P017^ (Jan. 1962).

[19] Van der Zee, P., CaHbration of platform, V.R.L.D. Report No. 323 (Sept. 1964).

[20] Van Donkelaar, H., Dynamic response of platform, V.R.L.D. Report No. 358 (Sept.

1965).

[21] Yspeert, A. J., Inverse fiher for measuring platform, V.R.L.D. Report No. 377 and 377^

[22] Voermans, J. J., Results of pulsator excitations of measuring platform, V.R.L.D. Report No. 422 (Feb. 1968); Design of new measuring platform for truck tires, V.R.L.D. Report No. 150 (March 1968).

[23] Bakker, C. C, Normal force and longitudinal force response to vertical axle motions, V.R.L.D. Report No. 486 (Nov. 1968).

[24] Buis, P., Pneumatic tires, V.R.L.D. Report No. P084 (1967).

[25] Koolhof, F. J. W., Polman, J., and Olland, R. C, Static and dynamic testing of pneu- matic tires, V.R.L.D. Reports No. A032, P004-P005 (1958/59).

[26] Cooper, D. H., Radial stiffness of the pneumatic tire. Trans. I.R.I. 40, 58 (1965).

[27] Weber, G., Theorie des Reifens mit ihrer Auswirkung auf die Praxis bei hohen Bean- spruchungen, A.T.Z. 56(12), 325 (1956).

[28] Stulen, J. H. B., Comparison of road and drum measurements, V.R.L.D. Report No. 326 (Nov. 1964).
REFERENCES 629

[29] Gengenbach, W., and Weber, R., Messung der Einfederung von Diagonal-und giirtel- reifen, A.T.Z. 71, 196 (1969).

[30] Chiesa, A. , and Tangorra, G. , Tlie dynamic stiffness of tyres, Revue Gen. du Caoutchouc

36(10), 1321 (1959).

[31] Rasmussen, R. E., and Cortese, A. D., Dynamic spring rate performance of rolling

tires, SAE Paper No. 680408 (1968).

[32] Krempel, G., ExperimenteUer Beitrag zu Untersuchungen an Fahrzeugreifen, Diss.

(1965); A.T.Z. , 1 and 262 (1967).

[33] Davisson, J. A., Design and application of commercial type tyres, SAE Paper No. SP-344 (1969).

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[351 Timan, D. A., and Rooney, J. H. M., Static load deflection relationships of several tires, V.R.L.D. Report No. P108 (Jan. 1969).

[36] Essers, E., and Kotitschke, J., Ueber die dynamischen Radlasten von Lastkraftwagen, Techn. Hochschule Aachen (1957).

[37] Vegter, T., Measurements of dynamic wheel loads, V.R.L.D. Reports 214 (1959) and U020 (1960).

[38] Miihlfeld, A., Ein hochfrequenztechnisches Verfahrens fiir Reifen und Schwingungs- messung. Diss. T. H. Braunschweig (1949); A.T.Z., 147 (1953).

[39] Bombard, F. J. von, Verfahren zur Messung der dynamischen Radlast beim Kraft- wagen. Diss. T. H. Miincfien (1956); Deutsche Kraftfahrtforschung 131 (1959).

[40] Burgman, E. F. M., Dynamic tire deflection measurements, V.R.L.D. Report No. 390

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[41] Knight, S. J., and Green, A. J., Deflection of a moving tire on firm to soft surfaces, SAE Trans. 5, 116 (1962).

[42] Gengenbach, W., and Weber, R., Neues Verfahren zur gleichzeitigen Bestimmung der Einfederung und der Verformung eines Reifens in Umfangsrichtung wahrend des

Betriebes, Automobil Industrie 3, 93 (1969).

[43] Svenson, O., Untersuchung dynamische Krafte zwischen Rad und Fahrbahn, deutsche

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[44] Senger, G., Ueber dynamische Radlasten beim Ueberrollen kurzwelliger Unebenheiten

durch schwere luftreifen, Deutsche Kraftfahrtforschung 187 (1967).

[45] Lippmann, S. A., Nanny, J. D., Analysis of the enveloping forces of passenger tires, SAE Paper No. 670174 (Jan. 1967).

[46] Misset, H., Research on tire behaviour on a flat road. Diss. Techn. Univ. Delft (1932).

[47] Floor, W. K. G., The effective rolling radius of pneumatic tyred wheels, Nat. Aeron.

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[48] Vickers, H. H., and Robison, S. B., Measurement of tread motions and application

to tire performance, Proc. Int. Rubber Conf., 1959.

[49] Evans, R. D., Factors affecting the power consumption of pneumatic tyres, Proc. Second Rubber Tech. Conf., London, 1948, p. 438.

[50] Fogg, A., Measurement of aerodynamic drag and rolling resistance of vehicles, FISITA

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[51] Roberts, G. B., Power wastage in tires, Proc. Int. Rubber Conf., Nov. 1959.

[52] Frolov, L. B., and Khromov, M. K., Determination of the rolling resistance of tyres by an electric torque meter, Sov. Rubber Tech. (9) (1965).

[53] Curtiss, W. W., Low power loss tires, SAE Paper No. 690108 (Jan. 13-17, 1969).

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Proc. Int. Rubber Conf., Washington, Nov. 1959, p. 73.

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[56] Kainradl, P., Kaufmann, G., and Schmidt, F., Zusammenhang der Erwarmung von L.K.W. Reifen mit den visco-elastischen Eigenschaften der verwendeten Gummiquahtalen, Kautschuk und Gummi-Kunststoffe (1), 27-36 (1966).

[57] Yurkovski, B., Effect of design parameters on the rolhng resistance of tyres, Sov. Rubber Tech. (11), 32-54 (1966).

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[59] Anonymous, Tyres, a review of current constructions, developments, performance, Automobile Engineer, 274-288 (July 1969).
630 THE TIRE AS A VEHICLE COMPONENT

[60] Paish, M. G., Effects of tractive effort on rolling resistance and slip of pneumatic

tyres, Motor Industry Research Association M.I.R.A. Report 1965/16.

[61] Grosch, K. A., The effect of tyre surface temperature on the wear rating of tread compounds,!. I.R.I. 1(1) (Jan. 1967).

[62] Spelman, R. H., Determination of passenger tire performance levels — high speed,

SAE Paper No. 690508 (Chicago, IlUnois, May 1969).

[63] Richey, G. G., Hobbs, R. H., and Stiehler, R. D., Temperature studies of the air in a truck tire, Rubber Age 79, 273-276 (1956).

[64] Horning, V. J., The temperature measurement at the Government Tire Test Fleet, Rubber Age 74, 395-396 (1953).

[65] Coddington, D. M., Marsh, W. D., and Hodges, H. C, New approach to tire durability

testing, Rubber Chem. Tech. 38(4), 741-756 (1965).

[66] Ludwig, G., Rhodes, D., and Simson, B., Thermistor takes temperature of running

tire, Nat. Bur. Stand. (U.S.), Tech. News Bull. 52(6), 119-120 (1968).

[67] Parker, R. C, and Marshall, P. R., The measurement of the temperature of sliding surfaces, Proc. IME 148, 209 (1948).

[68] Minkes, S., Surface temperature measurements by an infrared radiation method with indium-antimonide cells, V.R.L.D. Report No. 538 ^' ^' ^ (1969).

[69] Hoogen, E. v.d.. Possibilities to trace hot spots in the tread of tires. V.R.L.D. Report P012 (1960).

[70] Buis, P., Braking measurements on pneumatic tires, V.R.L.D. Report No. 313 (1963).

[71] Holmes, K. E., and Stone, R. D., Tyre forces as functions of cornering and braking slip on wet road surfaces, R.R.L. Report LR254 (Road Research Laboratory, 1969).

[72] Davisson, J. A., Basic test methods for evaluating tire traction, SAE Paper No. 680136

(Jan. 1968).

[73] Meades, J. K., The effect of tyre construction on braking force coefficient, R.R.L. Report LR224 (Road Research Laboratory, 1969).

[74] Horz, E., Die Einfluss von Bremskraftreglern auf die Brems- und Fiihrungskraft eines gummibereiften Fahrzeugrads, Deutsche Kraftfahrtforschung 195 (1968).

[75] Bajer, J. J., Proposal for a procedure for evaluating wet skid resistance of a road-tire- vehicle, SAE Paper No. 690526 (May 1969).

[76] Kelley, J. D., Factors affecting passenger car tire traction on the wet road, SAE Paper

No. 680138 (Jan. 1968).

[77] Radt, H. S., The mechanism of tire thump and roughness, SAE Paper No. 322C (1961).

[78] Davisson, J. A., Factors of tire design and manufacture affecting uniformity, SAE

Paper No. 650521 (1965).

[79] Lippmann, S. A., The forces and torques associated with roughness in tires, SAE

Paper No. 322D (1961).

[80] Cimprich, F. J., Elimination of temperature induced non-uniformity in tires by air-ring control, SAE Paper No. 690075 (1969).

[81] Morrish, L. M., and Haist, R. R., The effect of loaded radial runout on the roughness and shake, SAE Paper No. 322E (1961).

[82] Nordeen, D. L., and Rasmussen, R. E., Factors influencing the measurement of tire uniformity, SAE Paper No. 650734 (1965).

[83] Hofeh, C, Uniformity control of cured tires, SAE Paper No. 690076 (1969).

[84] Van Eldik Thieme, H. C. A., Passenger riding comfort criteria and methods of analysing

ride and vibration data, SAE Paper No. 295A (1961).

[85] Hofkamp, H. J., Perception of shocks in vehicles, V.R.L.D. Report No. VI 19 (1969).

[86] I.S.O. Committee, Mechanical vibration and shock, I.S.O./T.C.-108-lOOE (June 1969).

[87] Bajer, J., The control of tire nonuniformity and a passenger car from manufacturer's

point of view, SAE Paper No. 667B (1963).

[88] Hofelt, C, Tarpinian, H. R., and Draves, C. Z., Measuring tire uniformity, SAE Paper

No. 65022 (1965).

[89] Gough, V. E., Barson, C. W., Gough, W. S., and Bennett, W. D., Tyre uniformity

grading machine, SAE Paper No. 322A (1961).

[90] Slager, K. M., Dynamic tire deflection measurements, V.R.L.D. Report No. 318 (1964).
CORNERING AND CAMBER EXPERIMENTS 631

7.3. Cornering and Camber Experiments

H. C. A. van Eldik Thieme

7.3.1. Introduction

When an externally applied side force F acts at the centerline of a nonrolhng wheel, the lateral frictional forces exerted by the road surface on the tire will cause a lateral tire deformation as shown in figure 7.3.1. The top part T of the tire, not in contact with the road, is seen to be almost

undeformed. Usually we are not so much interested in the deformation

of a standing tire as we are in the phenomena of the rolling tire. Therefore

consider a side force acting on a rolling wheel. The undeformed top part

T of the tire rolls gradually into the deformed condition by coming into contact with the road surface [1].^ A top view of the lateral deformation is given in figure 7.3.16, assuming

that these lateral tire deformations have attained steady state. At the

instant of contact with the road, we observe that the peripheral carcass

line already has a lateral deflection A'A. Passing through the contact

zone, the lateral carcass deflection increases until point E is reached,

after which side slip of the line reduces the lateral deflection. Due to some

extra lateral tread rubber deformation £"5, the tread elements first coming

in contact with the road adhere to the road surface over the distance ^fi. The equatorial line running along the middle of the tread surface shows

the lateral deformation of the tire tread, as represented by line ABD

where all deformations are exaggerated for clarity (fig. 7.3.1c). The direction of travel of the wheel is indicated by line AB, making an

angle with the plane of the wheel. The angle is called the slip angle. The

resultant lateral tire force Fy acts a distance t behind the geometric

center C of the contact area, causing the tire to generate a self-aligning

torque Mz. The distance t is called pneumatic trail. The diagram shown

in figure 7.3. Ic? is a schematic representation of figure 7.3.16. A slip angle a is formed between the direction of travel of the center of tire contact C and the jc-axis situated in the wheel plane.

Side forces may be due to centrifugal action when cornering, road unevenness, small movements of the wheels for directional control,

wind forces, and the lateral component of the weight on a cambered road

surface. For illustration, some typical cases of automobile response to side forces will be discussed below.

Nonsteered Vehicles

Consider the simplified case of a vehicle having constant forward

speed, with a driver holding the steering wheel in a fixed straight ahead

Figures in brackets indicate the literature references at the end of this section.
632 THE TIRE AS A VEHICLE COMPONENT

T

Figure 7.3.1. An externally applied side force causes a lateral tire deformation.

The slip angle a of the direction of motion with the plane of the wheel is shown together with the resultant lateral tire force Fy and the self- aligning torgue Mz.

position. The vehicle is allowed to change its direction of motion under

the action of lateral forces. This condition has been represented in figure 7.3.2. The externally applied lateral force must be in equilibrium with the

lateral forces Fy acting on the tires by the road. These forces result in

slip angles «/ and ar of the front and rear wheels. The relation between

lateral force and slip angle is governed by the tire characteristic. We

first consider the particular case in which, at equal front and rear slip

angle {af=ar), the resultant lateral tire force acts in the center of gravity Cc When the external lateral force is also applied at the Ca
CORNERING AND CAMBER EXPERIMENTS 633

Figure 7.3.2. The effect of an externally applied lateral force on the front and rear wheel

slip angles.

Force

Resultant

tire force

Path of vehicle

Figure 7.3.3. Path of the vehicle at equal front and rear slip angles {neutral steer).

(for instance on a cambered road), the vehicle drifts sidewards along a

linear path, with an angle between its longitudinal vehicle axis and the

direction of motion. When this translatory motion (without yaw velocity)

takes place as shown in figure 7.3.3, the automobile is said to have a

neutral steer character. A concept often employed is the neutral steerpoint, or the neutral steer-line in theX-Z plane, upon which an externally

applied lateral force will not produce a yaw velocity. In the particular case of neutral steer considered above, the neutral steer-point coincides

with the center of gravity. In figure 7.3.4 the response in case of understeer and oversteer has

also been shown. In these cases the neutral steer-point is located behind

and in front of the center of gravity, respectively. For an external lateral force acting at the center of gravity we have with an understeer vehicle

aj > a,- and with an oversteer vehicle aj < ar. As discussed before, the externally applied lateral force may also be

due to side winds. The resultant side wind force Fw is supposed to act

at the so-called "center of pressure" Cp, and this point may be located

in front of the center of gravity Ca. The location of Cp depends on the
634 THE TIRE AS A VEHICLE COMPONENT

Understeer

Neutral steer

Oversteer

Figure 7.3.4. Illustration of understeer, neutral steer and oversteer.
Figure 7.3.7. Path of the vehicle with the resultant side windforce Fw acting in the center of pressure Cp.
636 THE TIRE AS A VEHICLE COMPONENT

aerodynamic styling of the vehicle, on the forward vehicle speed, and

on the magnitude and direction of the wind velocity.

It is clear that depending upon vehicle inertia, tire characteristics and

magnitude and point of application of the external lateral forces, the slip

angles of the front and rear wheels may differ considerably.

A simplified explanation may be given using an idealized model,

where the center of gravity of the vehicle is assumed to lie in the road

surface. It is furthermore assumed that at front and rear, the left and right

wheels have equal slip angles, which means that the wheels may be

imagined as being compressed together along the axle center lines, and

consequently that the automobile is replaced by a two-wheeled model.

Consider a side force applied at the center of gravity when the vehicle

is moving forward along a straight path. Assume the front wheel to exhibit

larger slip angles than the rear wheels, as shown in figure 7.3.5. This

tends to turn the vehicle away from the applied lateral force F. The

rotation about the instantaneous center gives rise to a centrifugal

force Fc which opposes the applied side force F.

If the side force F produces slip angles a:/<ar, then the resulting

centrifugal force Fc tends to help the side force F. This condition has

been shown in figure 7.3.6. Consider again a nonsteering vehicle moving forward at a constant speed of 100 km/hr. in a straight path, but now suddenly subjected to a side wind with a lateral velocity 15 m/sec. resulting in a side

force Fw acting at the center of pressure Cp (fig. 7.3.7a). It may be of

interest to show some experimental results obtained [2]. It is observed that after a sudden wind force application the transient

motion may develop in a complicated manner. Figure 7.3.7 illustrates such a behavior, where first (fig. 7.3.7b) both tire forces oppose the wind

force Fw and afterwards the tire forces appear to change in sign. The

variation of the yaw velocity observed is also shown (fig. 7.3.7d).

Steered Vehicles

Consider a vehicle running through a curve with a transverse slope

of the road surface. The forces acting on the simplified vehicle with

rigid suspension system are shown in figure 7.3.8.

Figure 7.3.8. Forces acting on a vehicle running through a banked curve.
CORNERING AND CAMBER EXPERIMENTS 637

R

georn

geom

Figure 7.3.9. The reference steer angle 8ref and the geometric radius of turn R.

The equation of forces in the direction of the slope reads:

When the component of the inertial force Fc is in equihbrium with the

component of the weight (equiHbrium speed), there are no lateral tire forces Fy required in the road surface and consequently the slip angles

of the wheels are zero. A similar condition arises when going through a flat curve with a vehicle

speed approaching zero (fig. 7.3.9). In this simplified case of a vehicle

having an ideal Ackermann steer geometry, the wheel base angle /3 is equal to the reference steer angle 8ref- For R> I, where / is the

wheelbase, the geometric radius of turn Ry^eom — ^/P rn^Y be taken equal

to R. For 0 we consequently have:

The idealized two-wheeled model also shown in figure 7.3.9 can be

used effectively to interpret experimental tire data for application to a vehicle traveling at a speed V. The steer angle 8 required to keep a vehicle on a constant radius path in a so-called steady state turn is a function of the front and rear slip angles.

Assuming that all angles and traction forces are small, the equation for the required steer angle follows from figure 7.3.10

cos Jr — W sin Jr

«eom cj Oref
638 THE TIRE AS A VEHICLE COMPONENT

6

Figure 7.3.10. Idealized two wheeled model.

Sref-(«/-«r)

Consequently when:

af=ar then = /?geom (neutral steer)

(a/— ar) > 0 then R > /?geom (understeer)

{ar — ar) < 0 then R < R^eom (oversteer)

This simplified analysis of vehicle behavior illustrates the definition of

understeer and oversteer in the linear representation.

To obtain a better understanding of the behavior of vehicles at low and

high lateral accelerations, including the influence of suspension system,

it is necessary to consider a number of other important quantities such as moments of inertia, lateral and longitudinal load transfer, steering system

elasticity, roll steer and roll camber, self-aligning torques at the front and rear wheels, etc.

It is obvious that the study of directional vehicle control is very complicated, and beyond the scope of this book. The reader is referred to the

literature for a more detailed treatment of the subject of vehicles having

steered front wheels [3-6]. Readers are also referred to the reference

list of section 7.5 [29-43].
CORNERING EXPERIMENTS 639

It is hoped that the simplified introduction given above has shown the

importance of obtaining experimental tire data, such as slip angle,

cornering force and self-aligning torque relations, to be treated in the

following subsections.

7.3.2. Cornering Experiments

Photography ofContact Area Deformations

The difference in deformation between a vertically loaded free rolling

tire {a = Qi) and a cornering tire can be illustrated by photographing the

contact area of tires through a glass plate with grids during slow motion

[5] (fig. 7.3.65).

Measurements of the resulting deformations can be carried out in the same way as previously discussed in section 7.2.5 for the case of a braking tire. First a photograph is taken of the free rolling tire (q:=0). Then a second photograph can be taken of a standing laterally undeformed tire,

front

Direction movement

of platform

Figure 7.3.11. Cornering experiment showing deformation of equatorial line.
640 THE TIRE AS A VEHICLE COMPONENT

set of a predetermined angle to the direction of motion of the movable

platform, representing static condition.

Afterwards, using the same negative, a third photograph is made on

top of the second one, but now with the tire rolled in the laterally de- formed steady state cornering condition (fig. 7.3.11). The difficulty with

this method is the experience necessary to synchronize the photographs, but it is also very time consuming to analyze the photographic

data to define the form of the equatorial line. A glass plate device to study tire deformations is also used at several

proving grounds by driving a vehicle straight onto the glass plate with

the front wheels toed-in to a predetermined steer angle. Due to steering

system flexibility it is difficult to measure the exact slip angles, and it seems desirable to use a slip angle measuring device as described in section 7.3.3, figures 7.3.43-44.

The photographic technique is either to attach the camera onto the

vehicle and photograph through the glass plate, looking at the contact patch in a mirror, or to photograph from beneath as tires pass over the

glass surface [7]. Movable Platform Measurements with the Gough Apparatus

A. Lateral deformations

The deformation due to the lateral flexibility of a tire rolling at a slip

angle may also be obtained with a flat surface modified version of the

original Gough apparatus [6] mounted in the flat platform machine [5]

(fig. 7.3.65). The platform travels at a constant speed of about 0.05 m/sec.

When the platform has travelled such a distance that the lateral tire deformations are assumed to have reached steady state condition, the

tire is allowed to roll over the Gough apparatus [10]. The original version

of this apparatus, mounted behind the moving glass plate, had a steel upper surface. In order to avoid a difference in coefficients of friction on

steel and glass, both surfaces are now covered with a sheet of transparent

perspex. As discussed [5], the Gough apparatus has three measuring

bars. The bar measuring the lateral force distribution is equipped with a

steel fork which can move in a narrow slot, situated on the lateral centerline of this transverse bar of the Gough apparatus. Lateral sliding of the

tread element penetrated by the steel fork, with respect to the platform

(fig. 7.3.12), is recorded using a potentiometer coupled to the steel fork

[10]. We speak of sliding because a relative motion of a tread element

with respect to the road surface is observed.

Because the test tire is mounted on a structure with a turntable having

a transverse slide, the mechanism allows for a lateral movement of the

wheel center plane with respect to the platform. The initial lateral de- formation A'A of the tire where contact begins (fig. 7.3.16) can now be

taken into account by a suitable transverse movement CM of the slide, so that the steel fork penetrates the tire tread surface exactly at point A

of the equatorial line as marked on the tread (fig. 7.3.13). This is done by

observation in the longitudinal direction of the platform at the position

of the steel fork with respect to the equatorial line.
CORNERING EXPERIMENTS 641

Figure 7.3.13. Geometry of lateral slip and deformation of the equatorial line within the contact zone.
642 THE TIRE AS A VEHICLE COMPONENT

At low slip angles we observe that a tread element coming into contact with the slowly moving platform first adheres almost fully to the surface,

so that the assumption of no lateral sliding is reasonable iVsx= Vsy = 0) and the original undeformed equator line travels parallel to the direction

of motion. The steel fork does not move laterally.

As the tread element moves further in the contact zone, it produces a

deflection v with respect to the wheel plane which, depending on the

lateral carcass stiffness Cc and the rubber tread element stiffness C/>

[27], increases linearly with increasing distance, causing an increasing

lateral force until the local limiting frictional force is reached. The tread

element then begins to slide towards the wheel center plane. This lateral sliding of the tread element with respect to the platform is measured with

the movable steel fork. We are interested in the displacements u and v of a tread element with

respect to its position in the undeformed situation in the X and F direction

respectively.

The initial lateral displacement v of the tread element A, which first comes into contact with the platform (fig. 7.3.13) can be found to within

a small error, using the assumption that A'A" = 0. Hence, as represented

in figure 7.3.13,

V = CM cos a — (AM — CM sin a) tan a. The vertical force measuring bar gives a reference signal on a record

indicating the moment that contact begins with the platform. This is point A on the equator line, Mbeing the center of the turntable supporting

the test tire, and is indicated by a microswitch signal received from the

moving platform, so that the distance AM is known.

The distance CM is read from a scale attached to the support of the

lateral slide mechanism, and the rotation angle a of the turntable is also read from a scale. By transformation of the fork measurements, as indicated, the lateral deformation v can now be calculated.

B. Longitudinal deformation

As already discussed in section 7.2 (see [5] of sec. 7.2 and fig. 7.3.63),

the measurement of the longitudinal deformations with a camwheel

mounted in the longitudinal force bar is a very time consuming procedure

because several runs are necessary to obtain a reasonable accuracy.

Due to the lateral sliding, other neighboring tread elements will come into contact with the circumference of the camwheel. However, the track of one individual tread element of the equatorial line over the surface can be

constructed approximately because the lateral movement of one tread

element, as measured with the fork, is known.

By repeating the longitudinal deformation measurements several times,

by giving the slide a lateral movement of 5 mm. or less after each run.
CORNERING EXPERIMENTS 643

Figure 7.3.14. Longitudinal deformation measurements in the contact zone.

the real longitudinal slip line of one tread element can be reconstructed

[9]. As can be seen in figure 7.3.14, several lines are drawn parallel to the

direction of travel of the platform. The different points of intersection

of these lines with the deformed equator line give an indication of the

position along the contact length of the original tread element. The real longitudinal slip line, is now reconstructed with parts of the individual

longitudinal slip lines belonging to the points of intersection with the

lateral slip line as shown in figure 7.3.15.

C. Combination of lateral and longitudinal deformation

The lateral and longitudinal movement of a tread element of the

equatorial line can now be calculated and figure 7.3.16 shows the paths

of a tread element, parallel and perpendicular to the direction of motion,

for different slip angles a [24]. From these results it is seen that without

braking or driving forces longitudinal sliding occurs, and that the lateral sliding at the end of the contact zone is almost perpendicular to the direction of motion.

At increasing slip angles the deformation of the contact zone will increase, resulting in increasing lateral forces. It is seen that at larger

slip angles the adhesive zone decreases, resulting in an increasing

sliding zone at the rear of the contact area (fig. 7.3.17).
644 THE TIRE AS A VEHICLE COMPONENT

Contact Length

Lateral

sliding

Longitu- > dinal

sliding

signals

Longitudinal sliding

Approximated

Figure 7.3.15. Reconstruction of the longitudinal slip line of a tread element moving

through the contact zone.
CORNERING EXPERIMENTS 645

Sliding

parallel to direction

of travel i mm

Slipangle

Tire 5.90.13 Load 300 kgf

Infl. pressure 1.5 kgf /cm^

(mm.) Slidingx direction of travel Figure 7.3.16. Paths of a tread element for different slip angles a.

Direction of travel Sliding zone

Slipangle a-2°

Figure 7.3.17. The sliding zone at the rear of the contact area.

D. Sliding velocity

As can be seen from the lateral deformation curve of the equatorial

line, a tread element gradually rolls into its deformed situation. We

observe at the leading edge A where contact begins a lateral deflection r

v^ith respect to the undeformed situation (fig. 7.3.18). Assuming adhesion over the distance AB, the drifting tire shows an

equator line which is straight and parallel to the vector V, hence

a= constant. Because no lateral sliding occurs in this region,

the lateral sliding velocity Vsy = 0. Consequently, the value of the

deformation velocity Vd of the tread element will be equal to the lateral component Fsin a of the forward velocity F, so that we can write:

Vsy = -V sin a-hVd = 0

Travelling further in the contact zone over the distance BH, we observe

a decrease of dv/dx until point H, where we obtain the value dvldx = 0

and Vsy = — Vsin a.
646 THE TIRE AS A VEHICLE COMPONENT

Figure 7.3.18. Distribution of forces, lateral sliding distance and lateral velocity over

the contact length.

Due to the change in slope beyond H of the lateral deformation curve of the equatorial line, the lateral deformation velocity Vd now changes in sign resulting in further increase of the sliding velocity Vgy beyond

point H.

Having reached the rear end of the contact zone at point Z), the tread

element of the equatorial line gradually returns to the wheel center plane as indicated in the figure [24]. E. Lateral force distribution In discussing the lateral deformation of the tire, we observed for moderate slip angles a that a tread element on entering the contact zone
CORNERING EXPERIMENTS 647

at first adheres to the platform. As the tread element moves further in the contact zone, it produces a deflection which increases linearly with

increasing distance, causing an increasing lateral force. Dependent on

the vertical force distribution, which gradually drops to zero at the

rear of the contact zone, the local limiting friction is reached, and the

tread element begins to slide back towards the wheel center plane,

thus reducing the lateral force, as shown in figure 7.3.18. Note that the

peak value of lateral force distribution is not reached until after sliding

has started. When the tire travels over the lateral force measuring bar of the

Gough apparatus, we obtain the lateral force function. The resultant

lateral force Fy may be obtained by integrating the recorded force function, and can be compared with the total lateral force Fy, as measured

with the six-component tire tester, also mounted in the movable platform

machine.

For moderate slip angles a, as shown in figure 7.3.19, the resultant

lateral force Fy acts at a distance t behind the contact center C. The

distance tis known as the pneumatic trail. Neglecting the longitudinal force distribution in the contact area, we

find, in this simplified case of purely lateral slip, a value for the moment

about a vertical axis Mz = — Fy - 1.

Figure 7.3.19. The resultant lateral force Fy acts a distance t behind the contact center C (pneumatic trail).

In reality the longitudinal forces in the contact area influence the mo- ment about the contact center C (fig. 7.5.406), resulting in a total corner- ing moment Mz known as the self-ahgning torque. The force Fy is usually

called the cornering force. A more complete set of curves of the lateral or cornering force distribution over the contact area, obtained with the

Gough apparatus, are given for various sHp angles in Chapter 5 (fig. 5.58). The resuh of the cornering force-sHp angle relation is shown in Chapter

5 (fig. 5.59), also indicating the pneumatic trail. Bars for measuring the lateral force distribution have also been built into a drum of 3 m. in diameter, and the results obtained are similar to those just discussed [10].
648 THE TIRE AS A VEHICLE COMPONENT

F. Tread wear resistance

A nondestructive estimate of resistance to tread wear [11] of various

tire constructions is very important to the tire manufacturer, because

tread wear tests on the road require a large number of tires and are usually quite time consuming.

Although laboratory studies of movement of tread block elements on

a glass plate add to the knowledge of wear, they fail to solve problems

of varying rates of average wear. The original Gough apparatus [8] made it possible to give an estimate

of work done by the lateral frictional forces during passage through the contact patch when cornering.

The force as measured with the lateral force measuring bar is recorded

against the lateral movement of the central rib of the tread. The resulting

diagram gives, by measuring the area under the curve, the work done by

the lateral force when cornering. The tests are made at a range of slip

angles. From results obtained, it is found that the work done plotted

against the cornering force coefficient Fy/FN are straight lines for dif- ferent loads when shown on log-log paper.

Figure 7.3.20. Estimate of work done by the lateral frictional forces during passage

through the contact zone when cornering.
CORNERING EXPERIMENTS 649 Tread wear measurements obtained from road tests, made with vehicles

driven continuously around a constant radius at constant speed, showed

similar straight lines when tread wear was plotted against the cornering

coefficient on log-log paper. Neglecting longitudinal shpping, it follows

that work done by the lateral forces is strongly related to tread wear. The fact that the slope of the lines on the log (work done) versus log

(cornering coefficient) is influenced by the tire construction (fig. 7.3.20), means that some decision must be made as to what cornering coefficient

should be chosen for the basis of comparison of work done. The choice was a cornering coefficient of 0.3, which corresponds to a vehicle execut- ing a lateral acceleration of 0.3 g. [11]. The average value of 0.3 g. has also been obtained from statistical acceleration studies with 9 different makes of cars and 6 drivers, where

the lateral component of the gravitational force [12] due to the roll angle

has also been taken into account. The figure indicates that the work done, and consequently the wear

rate of a conventional bias tire is higher than a radial belted tire under

similar conditions.

Road and Drum Test Measurements

A. General observations

Although thousands of road and drum tests have been made it is felt to be outside the scope of this chapter to discuss the influence of parameters such as type of cord, cord angles, number of plies, rim width, aspect

ratio, stiffening of sidewalk and reinforcement of beads, bias, bias belted,

radial belted, or combinations of radial bias tires, etc. Due to its frictional character, the cornering force curve is also dependent on the texture of the road surface, the rubber compound and curing, the temperature and

the speed of sliding in the contact patch. It will be obvious that the data

mentioned above are not always available, and therefore parameters

which can be influenced by the vehicle engineer will be discussed, such

as slip angle, normal load, inflation pressure and speed.

As indicated, tire cornering force and self-aligning torque data depend

on so many factors that a general discussion is very difficult. However, some attention will be given in this section to the influence

of tire construction, tread pattern and tread resilience. For further infor- mation the reader is referred to chapter 6 and its literature references.

Further it should be noted that in order to obtain reproducible results, a standard test procedure which conditions the tire is required.

The data presented here for Fy and are obtained under steady state

rolling conditions. When a tire is made to roll at a slip angle a, it gradually

rolls into its deformed shape as in figure 7.3.1, thereby developing an

increasing lateral force with increasing distance travelled. After a dis- tance of about 4-5 times the contact length the lateral deformations

nearly attain their steady state values, and the lateral force Fy will remain constant (fig. 7.3.21).
650 THE TIRE AS A VEHICLE COMPONENT

Lateral ^

Force ^ F„-300kgf

Pi-^'5Kg!/crT^2

a, 40

Distance travelled (m)

Figure 7.3.21. A tire made to roll at a constant slip angle a develops an increasing

lateral force with increasing distance travelled and attains the steady state value.

B. Effect of slip angle a

a. Dry roads.

The influence of the slip angle a on the lateral force Fy for constant normal load F.v and constant inflation pressure pi is shown in figure 7.3.22

for a conventional bias tire and a radial belted tire. For small values of the slip angle of approximately 1-2 degrees, the

lateral force function Fy{a) is usually considered to be linear.

Limiting Friction

4 • 12 degrees a

Figure 7.3.22. Lateral force versus slip angle.
CORNERING EXPERIMENTS 651 The slope of the curve indicates the so-called cornering force stiffness CFa= (dFylda)a = o. This value is very much dependent on the tire construction due to different values of the lateral carcass stiffness Cc and

tread rubber stiffness Cp (fig. 7.5.33). Because it may be observed that at the larger slip angles the lateral force Fy increases, but the adhesive zone decreases, resulting in an increased sliding to the rear of the contact zone (fig. 7.3.17). the shape of the Fy(a) relation may be

understandable.

Although the laterally stiffer tire reduces tread motion in the contact

patch, and thus gives less wear, the variation in lateral force shown

in figure 7.3.22 may be too sudden for the capabilities of most drivers,

especially on wet roads. Some compromise has to be made, and the tire designer can influence the shape of these curves considerably by increasing or decreasing the cornering force stiffness, as shown in figure 7.3.23.

It should be observed that the recorded Fy{a) curves, as shown in the

figure, very often do not pass through the origin. It is a common practice

to shift all curves horizontally so as to pass through the origin, in order to compare results, thereby neglecting the well known existence of corner- ing forces at zero slip angle.

The same shifting applies for the self-aligning torque curves shown in the following figures.

As shown in figure 7.3.19, the resultant lateral force Fy acts at a distance t behind the contact center, partly explaining the value of the

cornering moment Mz. At increasing sUp angle a the lateral force Fy

increases, resulting in an increasing moment Mz. The slope of the Mz(a)

curve, as illustrated in figure 7.3.23, indicates the cornering moment

stiffness CMa = — {dMzlda)a=^o.

Due to the decrease of the pneumatic trail t with increasing slip

angle, the moment Mz reaches a maximum, and then drops to lower

values.

kgfm

Tires a and h are of different construction Figure 7.3.23. Effect of tire construction on lateral force and self aligning torque versus

slip angle.
652 THE TIRE AS A VEHICLE COMPONENI

The moment = }a{Px ' y — Py ' x)dA is influenced by the longitudinal and lateral force distributions, which are asymmetrical in shape

over the contact area^ (fig. 7.5.406).

We observe that the rolling resistance force is now acting out of the

wheel plane due to the lateral tire deformation. The local forces de- veloped are also dependent on the longitudinal and lateral carcass and

tread stiffnesses. Dependent on the vertical force distribution in the contact area, the local limiting frictional force may be reached and sliding occurs when Vp|+ Py= fJ^Pz- But as observed in section 7.3.5 (fig. 7.3.52), the coefficient of friction /x, depending very much on the speed

of sliding, reaches a maximum at a certain sliding velocity and then

drops to lower values. In the rear zone of the contact area the lateral sliding velocity Vsy increases (fig. 7.3.18), which may result in a de- crease of the coefficient of friction. Combining all these effects, the shape of the Mz{a) curve may now

be explained, and also the fact that a positive self-aligning torque Mz

can be obtained at high slip angles a. To indicate what happens when a dry road becomes wet, the Fy{a)

and Mz{cx) relations are shown for both conditions for another tire

(fig. 7.3.24).

0 5° 10° 15° 20° Slipangle

Selfaligning torque dry road

Figure 7.3.24. Effect of road condition on the cornering force coefficient and selfaligning

torque-slip angle relationships.
CORNERING EXPERIMENTS 653

b. Wet roads.

The difference in behavior on dry and wet roads may be best illustrated under extreme conditions, such as on a wet smooth polished asphalt test track. This road, as measured with the drainage meter (sec. 7.3.5, fig. 7.3.55) showed zero drainage, and a low skid resistance value of about 25, as measured with the Portable Skid Resistance Meter (sec. 7.3.5, fig. 7.3.57), indicating the dangerous situation of no micro- and no macroroughness. The wetted road temperature was about 18° C. The test results were obtained with the towed trailer (see ref. [1] of sec. 7.2, also fig. 7.2.8) by increasing the slip angle a very slowly, under constant

vertical load Fn and at a constant speed of 40 km/hr.

It is observed on wet roads that the lateral force Fy increases with

increasing angle a (fig. 7.3.25), but drops at larger slip angles. Returning

to lower slip angles, it is seen that the lateral force Fy cannot recover its initially higher values as shown by the closed loops at the indicated speed

of 40 km/hr. At a slightly lower speed (30 km/hr.), this loop has not been

observed in testing tire "a" (fig. 7.3.26). The variation in lateral force for

tire "6" is less than for tire "a," indicating a slight preference for tire "6" under these conditions. The self-afigning torque values Mz for tires "a" and "6" are also different as shown in figure 7.3.27. The footprint area of both tires has at equal loading conditions the same value, but

the value of the contact length of tire "a" is about 15 percent higher.

This may explain the higher maximum value of the self-aligning torque

of tire "a" compared with the lower value of tire "6."

The tread pattern, tread resilience, and tread hardness, obviously play

a significant role. Difference in skidding performance may be largely

attributed to these differences in tread pattern design and resilience. The center rib of tire "a" has, contrary to tire "6," no sipes or kerfs.

Road wet

kgf smooth, polished asphalt

2 4 6 8 10 12 U

a" Figure 7.3.25. The effect of tire construction on the corneringforce-slip angle relationships.
THE TIRE AS A VEHICLE COMPONENT

Road wet

smooth, polished asphalt

Tire = 5.60 -13 (a) F^=270 kgf

Pj =1.7kgf/cm^

2 4 6 8 10 12 U — a*

Figure 7.3.26. The effect ofspeed on the corneringforce-slip angle relationships.

kgf

2^0

Figure 7.3.27. The effect of tire construction on the self aligning torque-slij) angle

relationships.
CORNERING EXPERIMENTS 655 Road wet

smooth, polished asphalt

Tire = 5.60 - 13 (a) V =40km/h

kgf P =1.7kgf/cm^

))—

) /

30kgf

70kgf

\ —~A

//—

2 4 6 8 10 12 U ^ a'

2 A 6 8 10 12 U ^ a* Figure 7.3.28. The effect of normal load on the cornering force-slip angle relationships shown for tire a and tire b. Both tires have four drainage grooves, but the width of the drainage

grooves of tire "a" are smaller than those of tire "6," and the outer ribs of tire "6" have more sipes. Tire "6" has a lower tread rubber resihence

than tire "a," and the tread rubber hardness of tire "6," with a value of

61, is higher than tire "a," the latter having a hardness of 56. To judge which tire is better from this test along may be very difficult because the normal load (fig. 7.3.28a, h) also plays an important role when cornering, due to weight transfer. Therefore we shall now discuss

the lateral force and self-aligning torque relations versus the normal load.

t
656 THE TIRE AS A VEHICLE COMPONENT

C. Effect of normal load

The influence of the normal load W= Fn on the cornering force Fy is shown in figure 7.3.29. In discussing the vertical load-deflection curves

(sec. 7.2.4, fig. 7.2.49), we observed an increasing contact length with

increasing load at constant inflation pressure, resulting in almost no

increase of the vertical contact pressure Pz (ch. 5, fig. 5.32). The corner- ing force often shows at each slip angle a certain maximum value. The

observed drop of the lateral force Fy at increasing load Fn and at larger

slip angles may be explained by the influence of the increasing longitudinal forces and the decreasing lateral stiffness. The longitudinal force

distribution shows increasing forces with increasing load (fig. 7.2.49)

and therefore influences the lateral force Fy obtainable. Combining this effect and the decrease in lateral stiffness with decrease in normal load,

the shapes of the lateral force curves may be explained.

Due to the nonlinear shape of the Fy-FN curve (fig. 7.3.30), load

transfer from one wheel to another has an effect upon the values of the

Cornering Force

kgf

Dry asphalt

Tire 5.60 - 13 Speed 30 km/h

= 1.5 kgf/ cm^

Normal load kgf

Figure 7.3.29. Corneringforce versus normal load.

Figure 7.3.30. The loss in lateral force Fy due to load transfer.
CORNERING EXPERIMENTS 657

slip angles required for the generation of the total side force which

balances the centrifugal force Fc when cornering.

If theFy— F,v curve were linear no loss in lateral force Fy would be

observed at constant slip angle. Because the weight of the car remains

the same when cornering, load transfer results in increase of the required

slip angles ol to balance the centrifugal force Fc. A similar effect is ob- served when driving on bad, uneven roads with large variations in the
658 THE TIRE AS A VEHICLE COMPONENT

normal load, also causing a loss in cornering force due to the nonlinear

shape of the curve (cf., sec. 7.5.2). The self-ahgning torque Mz increases with increasing normal load due

to the increasing contact length, as shown in fig. 7.3.31 [15]. The Gough plot of the Fy-M^ relation for different sHp angles at constant normal load and inflation pressure is shown in figure 7.3.32.

Cornering Force

Fy kgf. Road dry asphalt

Tire 8.00-U

Speed 30 km/h

= ^00 kgf

10 kgf.m

Figure 7.3.32. Gough plot ofcorneringforce versus self aligning torque.

D. Effect of inflation pressure pi Increase of inflation pressure results in an increase of the lateral

stiffness, as observed from static load-deflection measurements (sec.

7.2.2, fig. 7.2.32) and therefore the lateral force Fy tends to increase at constant normal load and constant slip angle [16]. The self-aligning torque Mz decreases with increasing inflation pres- sure Pi because the contact length decreases, resulting in a decrease of the pneumatic trail t. Results obtained from measurements with a drum of 4.0 m. in diameter

[17] are illustrated in figure 7.3.33.

E. Effect of velocity

The infl,uence of speed on the lateral force Fy is considered to be rather

unimportant on dry roads. The effect of increased speed may be a slight

increase in lateral force. The added tension component in the carcass due to the centrifugal force tends to increase the tire stiffness, but the contact length decreases.
CORNERING EXPERIMENTS 659

ire 5.20-13 -„=250 kgf V = 40 km/h

kgf /cm'

1,1

1.7 o' 2 I 5 8 10 1*2

, a

100 200 300 ^00 500 ^gf

Normal load Figure 7.3.33. The effect of inflation pressure pi on self aligning torque and cornering

force.

— Mz kgfni

On wet roads, however, the decrease in lateral force Fy may be considerable at higher slip angles [14]. The tread pattern and state of the

road surface are extremely important parameters determining the directional stability. Although the speed dependence of the cornering force

of patterned and smooth tires has been shown in chapter 6, figure 6.30,

it may be of interest to show some cornering force coefficient (C.F.C.)—

slip angle relations in figure 7.3.34 as obtained on different wetted road

surfaces with patterned and smooth bias ply tires, the depth of water

being of the order of 0.5-0.75 mm.

The patterned tire was size 5.25-16 and had a Lupke pendulum resil- ience value of 31 at 20° C, compared with a smooth 5.00-16 tire having a
66f THE TIRE AS A VEHICLE COMPONENT

resilience value of 55. The hardness values were approximately the same,

namely 62 and 63, respectively.

The details of the test surfaces are given in the table below, and the

numbers of the test surfaces are the same as in figure 7.3.34 [14]. R,.230kgf

Pi.1.1 kgf/cm2 I"

Slipangle a Figure 7.3.34. The effect of velocity and tread pattern on the cornering force coefficient- slip angle relationships on various wet roads.

Surface 1 (rough, harsh), surface 4 (rough, polished), surface 5 (smooth, poUshed).

Detail of Test Surfaces

No. Description Texture

1 9.5 mm quarzite macadam surface rough, harsh

4 9.5 mm Bridport macadam surface rough, polished

5 mastic asphalt smooth, polished

The influence of speed and surface is clearly illustrated, and it is shown that the initial part of the curve is nearly linear and independent

of the surface, except at the higher speeds. The patterned tire gives
CORNERING EXPERIMENTS 661

greater C.F.C. values than the smooth. In general tires of low tread resil- fence give greater values than those of high resiUence tread rubber.

Therefore the smooth tire, having a high resihence tread rubber gives

extremely low C.F.C. values on the smooth pohshed mastic asphah

surface No. 5. In the extreme case of aquaplaning conditions, the cornermg torces

fall to such a low value that a vehicle may be directed from its straight

path by the influence of a sidewind or a cambered road. The feel at the

steering wheel should give sufficient early warning, but in certain cases

the self-aligning torque — cornering force relations can have a distorted

shape (fig. 7.3.35). suggesting that the steering feel may not always be a rehable indicator [18]. SPEED -0 mile/n SPEED 60 mile/h

n

A11 6°b:

4° \

'

2°

/^0°

001 002 001

Figure 7.3.35.

SELF ALIGNING TORQUE/ LOAD -ft

. tyre A _.x- - tyre B A Satisfactory tread pattern. B Unsatisfactory tread pattern characteristics. The effect of speed on the cornering force-self aligning torque relationships on a wet road.

Figure 7.3.36. The camber angle y produces a lateral tire deformation resulting in a camberforce Fyy.

1
662 THE TIRE AS A VEHICLE COMPONENT

Tire 5.20-13

0 100 200 300 ^00 Normal Load F^^
CAMBER AND CORNERING 663

7.3.3. Camber and Cornering

Camber

The inclination of the wheel plane to the vertical indicates the camber

angle y and is called positive when the wheel leans outward at the top

and negative when it leans inward (fig. 7.3.36).

The undeformed top part of the tire roUs gradually into the deformed

condition by coming in contact with the road surface, as illustrated in the figure. A lateral tire deflection is observed, resulting in a camber

force Fy^.

As demonstrated in figure 7.3.37 the lateral force Fyy developed by

a camber angle y is much smaller than the lateral force Fya developed by

an equal slip angle ol. The results shown are measured on a drum of 4.0 m. in diameter [17]. Similar relations obtained on a drum of 2.5 m.

in diameter are shown for another tire (fig. 7.3.38).

Y

200 400 600 kgf Normal load

Camber angle y

Figure 7.3.38. Camberforce versus normal load and camber angle.
664 THE TIRE AS A VEHICLE COMPONENT

For this conventional bias tire the camber force developed can be ap- proximately related to the normal load by:

with a constant camber force stiffness Cry, because of the linear character of the lateral force — camber angle relation. Radial ply tires show less camber force. The resultant lateral camber

force is said to act a certain distance ahead of the contact center, and this distance is called the pneumatic lead, resulting in a camber moment

Mzy. The camber moment is usually small and may be neglected. The

explanation of the camber moment is given by considering the rolling

tire to consist of two narrow tires, mounted rigidly a distance 2b apart on a spindle. As can be seen in figure 7.3.36, the rolling radius n > r-z. Because the distances travelled are equal, but the roUing radii are

different, an antisymmetric longitudinal slip must occur, producing two

equal longitudinal forces F^y, which act in opposite directions, resulting

in a moment = 2Fxy ' b. (See also fig. 7.5.21).

Combination ofCamber and Cornering

The combination of lateral forces due to a slip angle a and a camber

angle y is shown in figure 7.3.39, and according to the sign convention

it is shown that for positive values of a and y the lateral forces Fya and

Fyy act in the same directions. The influence of the camber angle y

on the total lateral force Fy decreases with increasing slip angle a, due

to sliding in the contact area. This is best illustrated (fig. 7.3.40) at low

normal load, for the case where Fya and Fyy both act in the same

direction.

Fy = F.vtan y= Cry, ' y

V X

V X

Figure 7.3.39. The combination of lateral forces due to camber and cornering.
CAMBER AND CORNERING 665

Tire = 5.20 - 13 V = ^Okm/h Y = const.

= 35 0 kgf =1.6kgf/cm^

Figure 7.3.40. The effect of normal load on the total lateral force Fy due to camber and

cornering.

A positive camber angle y develops a camber force at zero slip angle,

which can be counteracted by giving the wheels toe-in. Static toe-in of a pair of wheels is the difference in the transverse

distance between the wheel planes taken, respectively, at the extreme

rear and front points of the tire treads. When the distance at the rear is greater the wheels are said to be "toed-in" [3]. Camber angle variations due to change in wheel track of independent

suspension Unkages result in lateral sliding or "scrub" over the road

surface. The scrubbing action can result in unacceptable tire wear. The camber angle variation with respect to time causes the wheel to be

subjected to a gyroscopic moment Mz = Ix' ^ y, where Ix is the polar

moment of inertia. Actual Slip and Camber Angle Measurements

A well known method of obtaining sHp angles is by using a vehicle with

two symmetrically toed-in front wheels. However, elastic deformations

due to braking forces can influence the slip angles introduced into the

system. Because the tire forces of a vehicle under steady-state cornering

conditions are also of interest, another method is required to measure

the slip and camber angle of the test wheel.

Assuming zero camber angle y, the unknown slip angle a can be found

by tracing the path of two spring loaded ballpoint pens, situated a

fixed distance / apart on a subframe of the axlestub [19]. The pens will

Tire = 5.20-13

V = iiO km/h Y =const.

= 150 kgf

P: =1.6kgf/cm^
666 THE TIRE AS A VEHICLE COMPONENT

Figure 7.3.41. Slip angle measuring apparatus mounted on a wheel passing over the road

platform tire tester. trace two lines spearated by a distance d, depending on the slip angle a

introduced as in figure 7.3.41. It follows that sin a=d/L If the direction

of travel coincides with the Jc-axis of the platform (^=0), the cornering

force Fy is obtained from the road platform tire tester (sec. 7.2.1): Fy = — Fx sin a + Fy cos ol where Fx and Fy indicate the forces on the tire.
CAMBER AND CORNERING 667 A better method is using two electronic flashlamps triggered about

100 times per second with a short duration of the flash to ensure reasonable clear photographs at difl"erent vehicle speeds. The flashlamps are placed on a distance / on the axle stub in holders showing a small

rectangular slot, giving dashed lines on a photograph. The photo ap- paratus is placed in a structure well above the road (figs. 7.3.42a, b and 7.3.61).

Photoa p paratus

Dash No 1 2 3 4 5 tg 0< , 0,1785 0.1475 0ai83 0.0948 0.0837 tg « 5 0,1538 0J418 0/)964 0.081

6

O.O8O5 <^ , 10'7' 8° 24' 6''45' 5°25' 4° 47

' 8° 45' 8' 40' 5''30' 4" 40' 4° 36' Figure 7.3.42. Photographic slip angle measuring technique using two electronic flash lamps placed on the axle stub.
668 THE TIRE AS A VEHICLE COMPONENT

A small inclination of the axle stub arm due to cambering will hardly

influence the accuracy of the slip angle measurements. But obtaining the

slip angles from the dashed lines on the photograph is time consuming

work. Therefore this method has been used to control a special slip

and camber angle measuring device (S.C. meter) as shown in figure

7.3.43. The idea for such a device originated in 1964 with the Motor

Industry Research Association (M.I.R.A.) in England.

Figure 7.3.44. Principle ofthe slip and camber measurements.
INFLUENCE OF BRAKING AND TRACTION ON CORNERING 669 The principle of the sHp angle measurement is that a free trailing axle (1), gives the required direction of motion of the front wheel of a motorcar (fig. 7.3.44). Because two small spring-loaded trailing wheels

are used, the tubes (2) of the supporting frame are always perpendicular

to the road surface. The joint (3) between axle stub and frame allows

the measurement of the camber angle. The joint (3) is situated in the

vertical center line of the tubes (2). The slip angle measurements, however, require correction charts as a function of the lateral acceleration [19, 20]. The result is that the S.C. meter can be used on smooth and slippery

or even on bumpy roads as a tool to measure with reasonable accuracy the

required angles under cornering conditions.

Of course, this device can also be used without the road platform

(fig. 7.3.43). As an example, this device gives information about car "waddle'' when driving at low speeds on straight flat roads, due to tire nonuniformity. The slip angle varies in such a case from 0.1-0.0 degree

(sec. 7.2.6). 7.3.4. The Influence of Braking and Traction on Cornering

Dry Surface Measurements

The combination of a longitudinal force Fx and lateral force Fy, gives

a resultant horizontal force K as shown in figure 7.3.45 for the case of a braking tire. The difference in behavior of a bias ply and radial ply tire is illustrated. As can be expected, the lateral and longitudinal carcass and tread

rubber stiffnesses will greatly influence the results obtained.

The radial ply tire produces a more or less symmetrical shaped curve

for braking and traction. The bias ply tire produces a more pronounced

slope in the curve, as shown in dashed lines. The bias ply curve illustrates the effect that braking gives a higher

obtainable lateral force Fy than traction, for a given value of the longitudinal force Fx, at constant vertical load and constant inflation pressure.

radial ply bias ply y

Figure 7.3.45. The influence of braking and traction forces on the lateral force for a radial ply and bias ply tire.
670 THE TIRE AS A VEHICLE COMPONENT

Figure 7.3.46. The effect of traction forces on the lateral force.

The fact that larger traction forces often require larger slip angles a

to obtain the same lateral force Fy is illustrated in figure 7.3.46, thus

influencing the over- or under-steer character of a vehicleMeasurements taken on a dry steel drum of 4 m. in diameter are shown

in figure 7.3.47a, b for a number of slip angles [17]. The effect of braking and traction forces on cornering force Fy and self- aligning torque Mz is illustrated in figure 7.3.48 for two different loads

at a constant speed of 40 km/hr. and constant inflation pressure of

1.4 kgf/cm.2[171.

Wet Surface Measurements

Numerous tests have been carried out since 1960 with the Delft tire test trailer (fig. 7.2.8), but results have not been published in the litera- ture. The results obtained on different wet road surfaces with various

tires, however, are similar to those reported by the British Road Research

Laboratory [14], and because these results are readily available, they will be discussed. The force measurements are made on a fifth wheel which

can be set at a slip angle in the range of 0-20°, and whose angular velocity can be held at any desired value, independent of the vehicle speed, by means of a hydraulic transmission driven from the normal vehicle drive

[21].
INFLUENCE OF BRAKING AND TRACTION ON CORNERING 671

Tire 165-15 (radial ply) V=^0 km/h

= 250 kgf

Pj = 1,4 kgf/cm

kgf 200 160 120 80 O] AO 80 120 160 200 BRAKING TRACTION Figure 7.3.47. Lateral force-longitudinal force relationships at constant slip angles for a bias ply and radial ply tire.

In the following the results obtained with only one tire will be dis- cussed. For further information the reader is referred to the literature,

where detailed information is given on the test procedure and the evaluation of results, with further details of 10 test tires and 5 test surfaces

[14].
Figure 7.3.48. The effect of longitudinal forces on the cornerningforce-self aligning torque

relationships given for two normal loads.

The effect of the combined action of a longitudinal force Fx and a

lateral force Fy on a wet surface is illustrated in figure 7.3.49 for a patterned radial ply tire, with steel reinforcing belt. The test surface was

a 9.5 mm. Bridport macadam, having a rough, polished texture. The

depth of water was in the order of 0.5-0.75 mm.

Road wet

Tire 165.15 Speed ^8 km/h

Figure 7.3.49. The effect of the combined action of a longitudinal force and a lateral force on a wet surface is shown for constant braking force coefficients (BFC) and constant braking percent slip kb.
INFLUENCE OF BRAKING AND TRACTION ON CORNERING 673 A small braking force, generating little slip, does not affect the corner- ing force stiffness very much, but reduces the maximum cornering force

coefficient (C.F.C.) and the slip angle at which the peak is observed. It is shown that at increasing braking force coefficient B.F.C., the C.F.C. curve breaks away and reaches a lower maximum than before. At B.F.C.

values greater than the locked wheel value, the curves become closed

loops, as shown for a value of 0.35 B.F.C, illustrating the effect that a

large braking force cannot be satisfied at slip angles greater than a certain amount [14]. The 0.3 B.F.C. curves do not appear as loops, be- cause they extend outside the slip angle range covered by the figure.

The dashed lines, indicating constant braking slip kb, are quite

different from those of constant B.F.C. If a definite braking slip is imposed, the C.F.C. slip angle curve has a smaller initial slope, a lower

peak, and attains the peak at larger slip angles.

The effect of percent braking slip kb on the cornering force coefficient C.F.C. at a given slip angle is shown in figure 7.3.50 for three different

vertical loads. The well known fact that the cornering force falls off very

rapidly with braking slip is clearly demonstrated on this type of road

surface.

03 u

0.1

0

Road wet

Rough. polishtd macadam

Tirt 165-15 (Radial ply) Spctd ^8 km/h

Pi=1.4 kgt/cm2

) slipanglt

F^x3 60 kgf

20

F^=270kgf

0 10 20 Jo 40 50 60 70 80 90 100 Braking slip per cent tC

Figure 7.3.50. The effect of braking percent slip ,Kb on the cornering force coefficient CFC

at a given slip angle for three vertical loads.

The braking force coefficient (BFC) versus braking percent slip kb is also shown.
674 THE TIRE AS A VEHICLE COMPONENT

The effect of brake slip on the braking force coefficient B.F.C., is also shown in the figure for sHp angles of 0, 4 and 10 degrees. These effects are important when considering anti-locking devices.

The effect of traction, as observed with a special test vehicle, has also been reported [14], but discussion of resuhs obtained will be omitted.

However, because cornering traction methods, using conventional

vehicles, may be of interest from the standpoint of vehicle safety and

control, a short description will be given.

Conventional vehicle tests Because under maximum braking or driving conditions, the available

lateral force component is almost zero, there will be no force available for the vehicle control. In cornering traction tests utiUzing standard vehicles,

all tires on the vehicle are subjected simultaneously to their individual

complex conditions of load, inclination, torque, sHp angle, speed, etc. [22]. The nontethered cornering traction test is based on the combined

effects of all tires on the vehicle and their abiUty to generate sufficient forces to overcome the centrifugal effects on the vehicle.

For this test a car is driven in a circle at increasing speeds until breakaway occurs. The driver follows a procedure of increasing the speed in 0.5-1 km/hr. increments on alternate laps until a clearly distinguishable

level of the vehicle control is lost or rear breakaway is detected.

The maximum speed attained is usually the basis for tire comparison.

The momentary road surface condition, water depth, wind temperatures

of road and tire, etc. play an important role in the maximum attainable

speed.

Out of many procedures one test is to drive the vehicle in third or fourth gear on the test surface at a predetermined speed while trying

to follow a curved path. Arriving at the curve, the vehicle is accelerated

by means of a full throttle down shift. The driver follows again a pro- cedure of increasing the speed in 1 or 2 km/hr. increments on successive

laps, until at the so-called trace speed a distinguishable level of vehicle

control loss at the end of the curve is observed. The speed just before

this loss of control is called "just holding." When skidding already occurs at certain point, we speak of moderate slip.

Results obtained with a standard vehicle, on a smooth polished wet

asphalt surface, gave for the tires a and b of figures 7.3.25-28 the follow- ing speeds in km/hr.:

Tire a Tire b Just holding 37.5 40.9

Trace 37.5 42.8

Moderate slip 40.9 45

The large difference in cornering characteristics of tires a and b on

the same test surface have been discussed previously (figs. 7.3.25-28).
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 675 The combined effects of accelerating or braking while cornering are

also considered in the so-called slalom test, where the car is driven

alternately around a series of equally spaced markers positioned on the

test surface. Although a variety of test methods have been devised, all methods are subject to many factors which may adversely affect the repeatability

and accuracy of data. Even when A.S.T.M. E-17 control tires are used

for establishing the friction coefficient rating of the test surface, this practice does not provide the basis for the comparison of test data

among different tests. A program to determine the tractive performance

level for all modes of operation requires a large number of tests, and to reduce variations the individual tests have to be repeated several

times in order to average the data obtained [22]. 7.3.5. Difficulties in Measuring Forces and Moments

General Observations

The forces and moments acting between tire and road depend on the

distribution of the local normal pressure and the local tangential stresses

in the contact patch. The tangential stress occurring at a point x, y in the area A (fig. 7.2.9) is determined by the local normal pressure and the

local friction coefficient between tread and road surfaces. The normal

pressure is dependent upon the construction of the tire and the design of the tread pattern for a given load and inflation pressure. The coefficient of friction of a rubber compound against a clean dry road surface varies

with the speed of sliding and the temperature. Under isothermal conditions, the coefficient of friction tends to a constant value at very low

sliding speeds, approaching static conditions. As the speed of sliding

increases, the coefficient of friction rises and attains a maximum value

at a certain speed (fig. 7.3.51). The friction coefficient at constant speed

varies with the test temperature, as shown for a butadiene rubber com- pound on a glass surface (fig. 7.3.52). In the contact patch of a rolHng

tire, the temperature rises at higher speeds and the resulting friction is

2,0

0,01 0.1 1.0 10 100

Sliding speed, V, mph

Figure 7.3.51. The effect ofsliding: speed on the adhesion coefficient.
676 THE TIRE AS A VEHICLE COMPONENT

-5 -3 -2 -1

* ' « « . log V ^ m/sec

Figure 7.3.52. The effect of temperature on the coefficient of friction at constant speed

for a butadiene rubber on glass.

determined by the combined effect of speed and temperature. The nature

of the track surface, the rubber compound, the temperature and the speed

of sliding influence the local coefficient of friction in the contact patch.

The highest coefficients of friction are obtained on clean dry surfaces,

but contamination and/or water cause a decline in the forces developed.

The Nature ofthe Track Surface

In order to evaluate tire characteristics obtained from road tests, the

magnitude of the road surface influence should be assessed. When

comparing tire test data it seems advisable to include in the program test results obtained with a special standardized reference tire as a yardstick for the frictional rating of the road surfaces. If possible the road

surface characteristics should be obtained separately, in the form of a description of the macro- and microroughness of the test surface (fig. 7.3.53). Because the friction of rubber is temperature dependent, the

surface temperature plays an important role and should be given in test reports.
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 677

Surface typp

® Smooth iiiiiiiiiiiiiii (D Fine textured. rounded irimiiriiifn ® Fine textured, gritty llTlTIIIIlTnil ® Coarse textured, rounded rr>fTM>n1Tfl ® Coarse textured. gritty

Macro texture The effective friction on wet roads is controlled by the removal of the

fluid film throughout the contact patch of the tire. The road surface

drainage via macro texture (1-10 mm.) is thought to have an influence

upon the fluid displacement.

A photograph of the surface texture should be made available, in- dicating if required the size of coated chippings or quarzite macadam.

Stereo photography to determine the mean void width has been used,

giving a good picture of the form of the texture [26]. In other cases,

profile traces are obtained with an electromechanical roughness meter. Stereo photographs have been taken to record the surface profile,

from which the "profile ratio" was evaluated. The profile ratio is defined

as the ratio of the length of the surface profile along a line to the length of the base line. This method used to assess the profile bears some relation
678 THE TIRE AS A VEHICLE COMPONENT

to texture depth and takes into account the shape of the profile. Analysis

has indicated that the top 40 to 50 thousandths of an inch (1-1.25 mm.)

is the most significant part of the profile in determining the decrease in braking force coefficient with speed. It has been suggested that the

decrease in coefficient of friction fxis (k/}=100%) from 50 km/hr. to 125 km/hr. should not be greater than one quarter, with a minimum value

of /x/s=0.3, as measured with the small trailer apparatus of the Road

Research Laboratory [27]. In the sand patch method a known amount (p of fine dry sand or powder

is applied to the surface and is distributed in such a way that the sand

just fills the depressions and hollows in the area^ covered. The drainage

property is considered proportional to the ratio (flA. This method gives no information concerning the separation and shape of the texture height and the effectiveness of the surface drainage channels. We are interested in the channelling system and not in the hollows. We also want to take into account the draping behavior of tread rubber over

the individual asperities reducing the effectiveness of the surface

drainage channels.

With a drainage meter according to a design by Moore [28] the water

removing properties of a surface can be determined. A transparent bot- tomless cylinder is provided on its underside with a rubber ring upon

which the meter is placed on the surface to be investigated. The drainage

meter is loaded with the desired number of load rings so that the rubber

ring will drape itself over the asperities much as a tread element does.

Figure 7.3.54a. Road drainage meter.
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 679 The time required for the water level to fall from the top mark to the

bottom mark was originally measured and characterized the drainage

potential of the surface. Figure 7.3.54a, b shows the Delft drainage meter with an electro- magnetic operated valve to fill the outflow meter accurately to the desired

level. In order to measure the drainage on coarse roads it appeared necessary to compensate for stones or road unevennesses under the rubber

ring by placing foam rubber between the cylinder and the rubber ring.

The real measuring time was reduced to 10 seconds [29]. In view of the

large number of tests, the measurement results are automatically printed

with a digital recorder. Other data, such as water temperature and

number of load rings, are fed into a special computer program giving

the drainage number, defining the degree of drainage of the macro

texture of the road surface in question [30]. The criterion for the drainage

capacity of a road surface is expressed in the formula:

Figure 7.3.54b. Road drainage meter.
680 THE TIRE AS A VEHICLE COMPONENT

The factory indicates the influence of an increasing tire contact pressure

(number of load rings) on the drainage capacity of the road.

^eq 1-5 Heq 1.0 The equivalent height //gq 1.5 is the distance of the underside of the rubber

ring of the cylinder to a supposed flat plate, giving the same time of water flow as obtained from measurements on a road surface with a contact pressure of 1.5 kgf/cm.^. The results obtained from a large

variety of road s urfaces showed an extremely good correlation (0.98) for W= Vy • //gq 1.5 with results obtained by the Netherlands Road Research

Laboratory from the sand patch method, giving the texture depth

(TD) in mm.

(/)= volume of sand in cm^.

D= diameter of the area A covered with sand in cm. The results of an experimental program to establish a correlation be- tween drainage meter and skid resistance on difl'erent wet road surfaces

are shown in table 7.3.1. The high drainlage number //gq obtained on harsh and rough asphalt

with quartzite surface, indicates an excellent drainage capability. The

smooth polished asphalt shows the dangerous condition of no drainage,

having a zero value of the drainage number. The braking force coefficient

jjixs for K = 100 percent, also indicates lower values on the smooth asphalt

as compared with the asphalt and quartzite surface. The portable skid

resistance values (SRT) are also shown in the table. A new drainage meter measures the drainage as the total water flow

during 10 seconds at diff"erent water pressures (0.4-7 kgf/cm.^).

The drainage meter is shown in figure 7.3.546. After lowering frame

A from the test trailer of figure 7.2.8 to the road surface, the contact pressure between rubber ring and road can be varied with the air pres- sure in chamber 5 of part B. This part B is then fixed in its position with

the aid of a disk brake (7). The water pressure in chamber 2 is variable

between 0.4 and 7 kgf/cm.^.

Results obtained with this drainage meter are shown in figure 7.3.54c

for two diff"erent road surfaces. There is a strong indication, that for contact pressures in the range of truck tires (6-7 kgf/cm.^) no drainage

exists on the concrete road, whereas for the average passenger car tire pressure of 1.5 kgf/cm.^ a reasonable drainage is still available. This influence of the tire contact pressure has also been observed on towed truck trailer road tests, using the force method and giving the

locked braking force coefficient yns (fig. 7.3.556).
DIFFICULTIES IN MEASURING FORCES AND MOMENTS

o

deg

o

o ON CO

(N o o o o o

CO Csl CO

CO CO o o o o o

CO 1—CO o LO LO LO o" o o o o

-a
THE TIRE AS A VEHICLE COMPONENT

Waterf l(

12 3 4 5 6 7 8 kgf/cm^

Contact pressure rubber ring

Wate

Contact pressure rubber ring Figure 7.3.54c. Road drainage meter results.
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 683 Micro texture Small sharp points in the road surface can penetrate a thin fluid film,

but can also penetrate into the rubber tread surface of the tire. Because

the surfaces have thin films of oxide layer and water, they are far from

chemically clean and therefore it is not entirely a water film penetration.

A microroughness of 0.05-0.1 mm. on top of the macro texture produces

a high resistance to slip. The adhesive friction process consists of the

formation of adhesive bonds at the real area of contact, at the tips of the

hard asperities, and is caused by the normal load. The elastically stored

energy in shear, due to the tangential force, will try to overcome the

surface energy of the hard solid so as to free adhesive bonds [24]. Only

the rubber molecules forming the real area of contact may be considered

near enough to the field of forces of the hard solid and the deformation

is therefore concentrated in a very thin layer below the surface. Under

the action of the tangential force the adhesive bonds break and a fresh

cycle then begins with formation of new bonds elsewhere on the surface. In extreme cases, the stresses at the tips will be large enough to rupture

the rubber, causing abrasion, as has been proved in sliding on dry

quartzite [32]. The measurement and classification of the road surface

microroughness with a mechanical roughness meter is a difficult subject

because the microroughness is superimposed on the texture. The small-scale macroscopic roughness is sometimes measured

with a foil-piercing technique. In this technique a piece of aluminum

foil placed on the road surface is given an impact by a rubber tipped

rod released from a predetermined height [33]. The sharper tipped

particles pierce the foil and the number of piercings per square centi- meter are counted.

Another more reliable method using replicas has been developed by the

Dutch State Road Building Laboratory. A cast of the surface is made

with synthetic resin of silicon rubber. The cast is sectioned and the

section surface is projected on photographic paper. With a special

optical system the profile can then be measured and evaluated [25]. Skid testers The British Portable Skid Resistance Tester is a pendulum device

which measures the friction resistance of a wetted surface to the passage

of a rubber slider. Upon release from a horizontal position the pendulum

and pointer swing through an arc, the pendulum returns but the pointer

stays at the farthest point of the arc (fig. 7.3.55a), and the number read

from the scale gives the skid resistance value (S.R.T.). This instrument

gives as a first approximation a reasonable indication of the microroughness, but cannot measure the drainage properties of the road

surface. It is a low speed tester (2.8 m/sec.) and cannot sense the friction level at higher speeds. As a result, this tester rated the two surfaces

number 3 and 5 of figure 7.3.53 as very similar [23], even though the

friction levels of these surfaces were quite different at sliding speeds of 40 mph. The skid number SN is defined as the ratio of skid resistance to wheel load times 100. A skid number of 50 implies that the locked tire generates a skid resistance of 50 percent of the wheel load [23].
684 THE TIRE AS A VEHICLE COMPONENT
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 685

M Ls passenger tires 5.90 - 13 F^= 300 kgf

10.00 - 20.00 truck tires Fn = 2400 kgf

P. -5.5kgf/^^2

0,1 (b) 10 20 30 i.0 50 60 70 80 speed km/h Figure 7.3.55b. Friction coefficient vs. speedfor two types ofvehicle tires.

The shortcoming of all low-speed portable testers, their inability to sense the drainage properties of a surface and its friction level at higher

speeds, initiated the study and design of several Delft skid testers In wet countries with mixed traffic, it is vital to safety that there is no

dangerous discrepancy in stopping ability of trucks and busses with

respect to passenger cars. Since most truck and bus tires operate at high inflation pressures, when compared with passenger tires, it is in- teresting to evaluate truck tire tests versus passenger car tire tests on

surfaces having various macro- and microroughnesses.

With the Delft tire skid trailer [36, 37] an experimental program has

been initiated to study the behavior of truck tires having differences in tread design, compound and construction on a variety of road surfaces. Towed trailer road tests results obtained from passenger car tires and

truck tires are indicated in figure 7.3.55b. The large difference in the

locked braking force coefficient /ulis for the average passenger car tire and truck tire is clearly demonstrated.

Conditioning of the Tire

Due to surface conditions of both tire and road, and the temperature

dependent character of the friction, a precise description of the test conditions is absolutely necessary to compare test results of different

tires. The initial wear of the top layer of the tread rubber of a new tire, and

the warming-up effect during rolling and slipping, produce large varia- tions in the cornering force as shown in figure 7.3.56. These are typical

cornering forceslip angle relations as a function of the number of tests on a 6.40-13 bias angle tire with a natural tread rubber compound under

[34, 35].
a constant normal load of 350 kgf. and a constant inflation pressure of 1.50 kgf/cm^. In test number one, the new cold tire was pressed against

a drum of 250 cm. in diameter, rotating at a constant speed of 40 km/hr.

and having a smooth steel surface. In order to prevent unsymmetrical wear of the tread, the slip angle was varied in 30 seconds between

+ 15° and — 15°. The cornering force versus slip angle was registered as record number 1. After this first test the loaded tire was cycled 10 times

between -f 10° and —10° Now, with a cycle time of 3 seconds, record

number 2 was obtained after 30 seconds [38]. The same 10 cycles were repeated, giving test number 3, and so on, but in all tests the normal load and the inflation pressure were kept

constant. When after a number of tests reproducible results were ob- tained, with a more or less equilibrium temperature of the tire, say after

six tests, the average value of the cornering force for the sixth to tenth

test was considered to be representative for the test tire in question,

as in figure 7.3.57. From the above description it follows that to obtain

comparable results for both laboratory drum tests or road tests, it is necessary to "run in" test tires and to control the temperatures. A com- mon procedure, before drum test, road trailer tests, or vehicle tests are executed, is that all tires are run for break-in for a distance of about

300 km, at a speed of 80 km/hr. and with no hard cornering maneuvers.

Trailer Road Tests Versus Drum Tests Due to significant differences induced by the curvature of the drum,

the cornering force coefficient and self-aligning moment coefficient
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 687

1 2 2 i 5 6 7 8 9 10 testnumb«r Figure 7.3.57. The differences in cornering force-slip angle relationships of drum and

road tests.

Figure 7.3.58. Illustration of the self aligning torque-slip angle relationships obtained on drum and road tests.

obtained from drum tests can be about (10-20%) lower than the values

resulting from flat dry concrete road tests To illustrate these differences

in characteristics, the solid lines drawn in figure 7.3.57 show drum

test results, whereas the dashed lines give road test results obtained with

the same "run in" 165-15 tire. The self- aligning torque has been shown

in figure 7.3.58.
688 THE TIRE AS A VEHICLE COMPONENT

r->or-;r^^LOi-H\qoo

>OLOO^O^~-f--OOvOO^O^

o LO Lo lo CO ^ c^ 00

I I I I I I I I I

oOTfcoco-^-^ror-^coo

L0^O^-00t^CS|L0^0^

I I 7 I I I + I I I

OLO^Tf-TtOCOOOON

cOcOCMCMCslCSIcOrOCOfO OOfOO^CMOOfOOO- r—lONTtLOLOLOOr—Ii-HO

I I I I I I I

COCMt-COr-r^rfcO-^^O (^^^—lC^coooco•^o^^|—

i

fO CO -rf -rt S S
DIFFICULTIES IN MEASURING FORCES AND MOMENTS 689 To compare results of normal and experimental tires tested according

to the program described above under nominal constant load conditions,

and constant inflation pressure of 1.50 kgf/cm.^ at a speed of 40 km/hr.,

table 7.3.2 gives some results. The influence of the tire architecture

is clearly shown.

Trailer Road Tests Versus Vehicle Tests on the Road

Having discussed drum tests versus tire trailer tests, it may be of

interest to establish whether tire characteristics obtained from towed

trailer tests correlate with conventional vehicle tests on the road. The

problem encountered in the construction of a tire and the construction

of the vehicle suspension is, what are the actual operating conditions. It

is therefore essential to establish whether certain trailer and laboratory

tests are realistic or not. The tire characteristics obtained from the trailer tests as shown in figures 7.3.25-28 already account for road surface

irregularities, but it still remains a problem to decide which tire out of a number of test tires is the best from the standpoint of safety and vehicle

handling properties. The essence of the problem for the tire engineer is to know, what is the best compromise in tire construction regarding tread, carcass stiffness, tread compound, etc., in order to meet as far as possible

the conditions given by the vehicle suspension engineer.

Vehicle response tests determine vehicle handling properties by meas- urement of vehicle handling behavior in steady state cornering and

transient maneuvers, and by measurement of control modulation [39]. The effect of the driver on vehicle behavior has been practically elimi- nated in these tests. To correlate trailer test results with vehicle response tests for a given

set of tires, it is necessary to determine the actual conditions encountered

on different road surfaces. Therefore, measurements of vehicle response

tests should include the measurements of the tire forces on all wheels,

as well as the measurements of the path of the center of gravity of the

vehicle and the path of its individual wheels.

The measurement of the individual tire forces can be conducted with

the aid of the three-component road platform tester, as described in section 7.2.1 and shown in figure 7.2.17. The procedure of measuring

the actual tire forces with a road platform having the same surface as the test track may be preferred over the method of using force trans- ducers built into each individual axle. It avoids the necessity of designing

another axle force measuring device for each vehicle type.

Two road platforms already provide much information, but if required

four of these platforms could be used. At present the author has only

conducted experiments with one road platform, and load transfer could

be clearly demonstrated.

The measurement of the actual slip and camber angles when passing

over the road platform can be carried out with the S.C. meter as described

in section 7.3.3 and shown in figure 7.3.43. The path over the platform can be found by using the method of mounting flashHghts on the axle stub as shown in figure 7.3.42 and placing a photo apparatus in a steel structure well above the road. The dashed
690 THE TIRE AS A VEHICLE COMPONENT

Tire 5,60-15

Figure 7.3.59. The ejfect of tread depth and manufacturing fluctuation on the tire characteristics.

lines obtained on the photograph indicate the path over the road platform.

The path of the center of gravity of the vehicle can be traced by

filming or photographing from a high tower made of a steel structure,

placing flashlamps on the front and rear ends of the test vehicle.

Of course, if required, the well known techniques of using accurate accelerometers and gyros can be added to the instrumentation, but cor- rections for vehicle pitch and yaw motions have to be made.

The disadvantage of the combination of these methods is the complexity of the total instrumentation system. However, it is for obvious reasons of great importance to relate trailer test data to those obtained

from vehicle performance.

Tire Characteristic Variations Due to Manufacturing Fluctuations

Tires of the same manufacturer and having successive serial numbers

should have the same cornering characteristics. Due to problems of

tire processing and variation in tire components, substantial differences

in tire characteristics are often observed. To prove this, two tires 5.60-15,

both having 3 mm. tread depth, were compared at a speed of 0.2 m/sec. on a platform tire tester having a concrete surface [16]. Up to two degrees

slip angle the results obtained were similar, but above two degrees slip

angle it is seen that large variations can occur as in figure 7.3.59. The

influence of the tread depth is also shown, and the well known phe- nomena is observed that with 8 mm. tread depth, that is, with less wear,

a lower cornering force is observed on a dry concrete road.

Comparison of two tires of the same European manufacturer, but

according to serial numbers traced to come from different factories.
REFERENCES 691

gave still greater differences in cornering characteristics as shown

in figure 7.3.60. Such a variation in tire manufacturing is difficult for the car manufacturer, because both tires of figure 7.3.60 could be

mounted on the same car. It appears desirable that car manufacturers

specify allowable force variations.

References

This bibliography refers frequently to V.R.L.D. reports made by the Vehicle Research

Laboratory at Delft. These reports are in the Dutch language and can be obtained in microfilm or blueprint on payment.

[1] Freudenstein, G., Luftreifen bei Schrag- und Kurvenlauf, Deutsche Kraftfahrtfor- schung 152 (1961).
692 THE TIRE AS A VEHICLE COMPONENT

[2] Grotewohl, A., Seitenwinduntersuchungen an Personenwagen, A.T.Z., 11/12 (1967).

[3] Vehicle Dynamics Terminology, Handbook Supplement, SAE J 670a (1965).

[4] Nordeen, D. L., Analysis of tire lateral forces and interpretation of experimental tire data, SAE Paper No. 670173 (1967).

[5] Bundorf, R. T., The influence of vehicle design parameters on characteristic speed and understeer, SAE Paper No. 670078 (1967).

[6] Bergman, W., The basic nature of vehicle understeer-oversteer, SAE Paper No. 957B (1965).

[7] Sabey, B. E., and Lupton, G. N., Photography of the real contact area of tires during

motion. Road Research Laboratory Report LR64 (1967).

[8] Gough, V. E., Practical tire research, SAE Trans. 64 (1956); Tire to ground stresses. Wear 2(2) (Nov. 1958).

[9] Rooney, J. H. M., Measurements of the longitudinal deformation of a tread element, V.R.L.D. Report No. P090(1967).

[10] Iritani, S., and Baber, T., Forces on the contact patch of the tire. Paper B6, Proc. 10th FISITA Congress, Tokyo, 1964.

[11] Gough, V. E., Nondestructive estimation of resistance of tire construction of tread wear, SAE Paper No. 667A (1963).

[12] Gorp, H. A. van. Determination of lateral accelerations acceptable by motorists, V.R.L.D. Report No. 369 (1966).

[13] Dijks, A., Timan, D. A., and Ruyter, T. J., Trailer road tests versus vehicle tests on a smooth pohshed road surface, V.R.L.D. Report No. Pi 19 (June 1969).

[14] Holmes, K. E., and Stone, R. D., Tyre forces as functions of cornering and braking

slip on wet road surfaces. Road Research Laboratory Report LR254 (1969).

[15] Buis, P., Pneumatic tires, V.R.L.D. Report No. 084 (1967).

[16] Koeszler, P., and Senger, G., Vergleichende Untersuchungen der Seitenfiihrungseigenschaften von Personenwagen Reifen, Deutsche Kraftfahrtforschung 172 (1964).

[17] Henker, E., Dynamische kennlinien von P.K.W. reifen (V.E.B. Druckerei, D.D.R.); Wissenschaftlich-technische Verofifentlichungen aus dem Automobilbau. H3 (1968).

[18] Allbert, B. J., and Walker, J. C, Tyre to wet road friction at high speeds, Proc. IME

180, 105(1965/66).

[19] Koning. C. J. de. Design considerations of a device for measuring the camber- and slip angle, V.R.L.D. Reports No. 398^' ^' ^ (1967).

[20] Lems, F. L., The measurement of camber and slip angles of motorcars, V.R.L.D.

Reports No. 464^- (1968).

[21] Giles, C G., Lander, F. T. W., and Holmes, K. E., A test vehicle for studying the skid resisting properties of tyres and road surfaces under controlled cornering and

braking. Road Research Laboratory Report LR99.

[22] Davisson, J. A., Basic test methods for evaluating tire traction, SAE Paper No. 680136

(Jan. 1968).

[23] Kummer, H. W., and Meyer, W. E., Tentative skid-resistance requirements for main

rural highways, Nat. Corp. Highway Res. Program Report 37 (Highway Research

Board, Washington, D.C. 20418).

[24] Savkoor, A. E., The isothermal friction of rubber compounds on nominally smooth

hard tracks, V.R.L.D. Report No. P093 (1968).

[25] Spaink, G. N., Friction and drainage measurements on abrasion paper of different

qualities, V.R.L.D. Report No. 391 (Apr. 1967) (Translated by Cornell Aeronautical

Laboratory, Inc., Buffalo).

[26] Schulze, K. H., and Beckmann, L., Friction properties of pavements at different speeds, A.S.T.M. Spec. Tech. Publ, No. 326 (1962).

[27] Sabey, B. E., Wet road skidding resistance at high speeds on a variety of surfaces on A 1, Road Research Laboratory Report LR131 (1968). [28a] Moore, D. F., Drainage criteria for runway surface roughness, Cornell Aeronautical

Laboratory, Buffalo, J. Roy. Aero. Soc. (London) 69(653) (1965). [28b] Moore, D. F., A history of research on surface texture effects. Wear 13, 381-412

(1969).

[29] Edelman, A., Measurements with a refined drainage meter, V.R.L.D. Report No. 473 (Sept. 1968).

[30] Ernst, H. C, Correlation of results of measurements of coefficients of friction obtained

with a road trailer tire tester, a skid resistance tester (S.R.T.) and a modified Dr. Moore drainage meter, V.R.L.D. Report No. 4971 (June 1969).
REFERENCES 693

[31] Ernst. H. C. Development of a Delft high pressure drainage meter, V.R.L.D. Report

No. 497" (Aug. 1969).

[32] Grosch. K. A., and Maycock, G., Influence of test conditions on wet skid resistance of

of tyre tread compounds. Trans. I.R.I. 42(6)(1966).

[33] Gillespie. T. D.. Pavement surface characteristics and their correlation with skid

resistance. Penn. Dept. Highways Program Report No. 12 (1965).

[34] Gerritsen. R. R. V.. Description of portable pavement friction testers. Design of a pendulum tester. V.R.L.D. Reports No. 392-395 (1967).

[35] Koelewdjn. A. C.. Pavement friction testers. V.R.L.D. Report No. 516 (1969).

[36] Paar. H. G.. Skid trailer for testing truck tires, V.R.L.D. Report No. 376 (Nov. 1966).

[37] Schrier. J., and Groeneweg, H. H.. Design and calculation of a skid trailer for testing truck tires, V.R.L.D. Report No. M077 (May 1968).

[38] Timan, D. A., and Ruyter, T.. Cornering force and self-aligning torque measurements

on a drum and dry road with the tire tester, V.R.L.D. Report No. Pl-00 (Oct. 1968).

[39] Bergman. W.. Considerations in determining vehicle handling requirements, SAE

Paper No. 690234 (Jan. 1969).

NORMAL FORCE-DEFLECTION CHARACTERISTICS 695

7.4. Tire In-Plane Dynamics

H. B. Pacejka

In this part the symmetric behavior of tires will be discussed. Section

7.4.1 is devoted to relatively low-frequency phenomena with inertia

effects of the tire neglected. Attention will be paid to the normal and

longitudinal problem. Section 7.4.2 treats the high-frequency vibrational

behavior of tires, again mainly restricted to symmetric or in-plane aspects.

The frequency below which tire inertia effects may be neglected! must

lie well below the lowest natural frequency of the tire. Distinction may

be drawn here between radial and longitudinal motions and between

different types of tire construction. However, it is estimated that in general the low frequency range can be extended up to about 20 Hz.

It may be noted that for lateral tire movements this limit must be reduced

(cf. sec. 7.5.3 on gyroscopic effects). 7.4.1. Low Frequency Properties

Normal Force-Deflection Characteristics

It has been generally accepted that pneumatic tires transmit their

normal load mainly by the formation of a finite flat contact area,^, which

enables the internal air pressure, p/, to remain in equifibrium with the

external vertical contact pressure, Pz- We would therefore expect the

normal load, W, to be approximately equal to the inflation pressure

multipHed by the contact area. This would indeed apply for a thin

envelope or membrane. However, the tire tread-band in particular cannot

be considered to be thin. Automobile tires often show nearly rectangular

foot prints. This is due to the nearly flat profile of the undeflected tire cross section. The effective area, defined as Ae=Wlpi, then becomes

less than the gross foot print area^. This is because the contour of the

presumably thin side wall cross section does not show a horizontal tan- gent at the point where the contact area begins. The tension force in the

side waU, caused by the air pressure, has an upward component which

reduces the resulting pneumatic force. Later on in this paragraph an

analysis of this phenomenon will be given. Use will be made of a tire model showing thin flexible side walls and a rigid tread-band cross

section. In this analysis the internal pressure pi has been kept constant. For automobile tires this is an acceptable assumption. The effect of pres- sure rise may not be negligible for aircraft tires because of their relatively

large deflections [1].^

Figures in brackets indicate the literature references at the end of this section.
696 THE TIRE AS A VEHICLE COMPONENT

In the absence of internal air pressure the tire model proposed is not capable of transmitting a radial force if one neglects the flexural rigidity

of the tread-band. Consequently, the vertical stiffness of the tire model

is of a completely pneumatic nature. In reality, however, this is not entirely so. Experiments of various investigators [1, 2, 13] indicate that the

rigidity of the cover (side walls and tread-band) causes a noticeable contribution to the force transmission. According to these references, under

rated inflation pressure conditions the carcass carries about 15 percent

of the vertical load for cross ply automobile tires and 3 percent to 8

percent in case of aircraft tires. As a result of this cover rigidity the effective area A e increases and may even exceed the gross foot print area^.

The above observations hold for standing or nonrotating tires pressed

against a flat surface. On curved surfaces the tension force in the tread- band will take part in the force transmission. In addition, we observe

that with given radial deflection at the contact center, the contact length

changes with curvature of both tire and surface. It is therefore to be

expected that experiments indicate a dependence of normal tire stiffness upon road surface curvature. Both Marquard [3] and Chiesa [4] found a decrease in stiffness with decreasing radius of curvature of the surface. Marquard has shown that the influence of surface curvature can be

roughly approximated with the following equation:

In this equation, which originates from the Herzian theory of two parallel

cylinders pressed against each other, denotes the normal tire stiffness expressed as tire force per unit deflection, R the radius of the contact

surface and r the tire radius. Marquard found the formula to foUow the

experimental trend for ratios as small as Rjr = 0.25. When traversing

obstacles with radii of curvature much smaller than the tire contact

length, the tire clearly demonstrates its more than zero-dimensional

nature. A subsequent section treats the enveloping properties of tires. The remainder of this section will be confined to the contact with flat surfaces.

Until now, the tire has been considered as nonrotating. Once the tire roUs, fresh elements of cover are continuously entering the deformation

region. There is no a priori reason to believe that deformation of a rolling

tire follows the same rules which hold for a standing tire. Hysteresis,

which has been found to damp the vertical motion of the axle with a nonrotating wheel, appears to be practically absent with a rotating

tire once the rate of rolling becomes high as compared to the rate of

deflection. Instead, hysteresis produces rolling resistance. In addition,

it appears that vertical tire stiffness is affected by the rolling process. Rasmussen and Cortese [5] determined the effective tire stiffness by means of resonant tests. They show that the effective normal stiffness of

(7.4.1)
NORMAL FORCE-DEFLECTION CHARACTERISTICS 697

a rolling tire is virtually independent of hub amplitude whereas the

effective stiffness of a nonrolling tire varies with amplitude in a nonlinear fashion. At small amplitude the standing tire shows considerably

larger values for the stiffness than the roUing tire (50% higher at 1.5 mm. amplitude for a 6.50-14 tire). For increasing ampHtudes the stiffness of the standing tire decays gradually and tends to the constant value

of the rolling tire (15% higher at 10 mm.). Chiesa and Tangorra [4] found with their resonance tests that this level of stiffness is not very

much affected by the speed of rolling once the speed has exceeded a value of about 20 km/hr. (cf. fig. 7.3.32). The above observations are

of particular importance for the execution of laboratory ride simulator

tests (cf. Betz [61). At high roUing speeds the dynamic aspect, which among other things

is responsible for the formation of standing waves (sec. 7.4.2), must

be taken into account. Dodge [7] made an attempt to attack this problem.

With the aid of equations describing the dynamics of a rotating shell,

corresponding to eqs (7.4.90-91), the radial stiffness of the shell subjected

to a radial point load has been determined. The results are complicated

and difficult to interpret and are not directly appUcable to the tire pressed

upon a flat surface. An extension of the analysis is needed where the

following additional parameters are introduced: finite contact length,

internal pressure, tension force in tread-band and tangential stiffness of tread-band with respect to the wheel rim. Another theory related to the

work of Dodge has been proposed by Clark [43]. In this study the contact

length has been considered as finite. Before going into details of the static loading process, we consider

the following simple but interesting observation. For small normal tire deflections the contact length varies approximately as the square root

of the deflection. Consequently, for a foot print of constant shape the contact area varies as the deflection. This means that, with the assumption that the normal load equals internal pressure multiplied by a fixed

percentage of the contact area, the force-deflection curve is linear for small values of the deflection. For a more sophisticated analysis of the normal force-deflection

problem we make use of the membrane concept employed by Rotta [8] and Senger [9] among others for the analysis of the elastic properties of

a tire segment, and by Pacejka [10] for the investigation of air springs.

Senger used the model described earlier, consisting of flexible side walls

and a tread-band with rigid cross section. Senger replaced the circular

band by a linear elastically-supported beam under tension. The beam

is transversely deflected by the application of a normal force distributed over a finite length. The longitudinal tension force influences the re- sulting transverse stiffness due to the curvature caused by the application

of the normal forces. In the writers opinion the transformation to a linear

beam is not valid for analyses of radial deformations. In reahty, the contact area is not curved when pressed on a flat road. Therefore the

tension force, if present in this area, cannot influence the force distribution in the contact area. Flexural rigidity of tread band and side walls

produce additional normal forces but will be neglected henceforth.
698 THE TIRE AS A VEHICLE COMPONENT

Theoretical load-deflection relationship.

For the determination of the vertical force acting on a deflected tire we divide the tire into a large number of thin segments, and imagine

this force to be composed of the elementary radial forces acting on these

segments. For the calculation of the elementary forces we need to know

the load-deflection characteristic of one segment. This characteristic will be designated as the radial foundation characteristic.

We consider the model of the tire cross section shown in figure 7.4.1. The side wall is assumed to behave as an inextensible membrane. For

the segment having a nearly uniform thickness, the membrane assumes

a circular shape under the action of the internal air pressure pi. The

stresses in the membrane would not change if the membrane were ex- tended according to the dashed line shown in the figure. Evidently, the

external vertical force acting on the segment of unit thickness equals

p,dy=--2pibe (7.4.2)

with 2be denoting the effective width indicated in figure 7.4.1. With tire parameters / and hso introduced in the figure, we obtain the following

equations for be and the radial deflection —w in terms of the parameter

hso + w = l^-^^ (7.4.3)

Figure 7.4.1. Real and model representation of tire cross-section.
NORMAL FORCE-DEFLECTION CHARACTERISTICS 699

2(6-6,) =/^^- (7.4.4) The arc length / is assumed to be a given constant. At zero deflection,

i.e., w =0, the effective side wall height is hso. Due to the tensile rigidity of the tread band, the effective width be does not, in general, vanish in the undeflected case. The resulting radial force produces the hoop ten- sion force which is present in the undeflected portion of the tire tread

band.

These observations indicate that this characteristic will in general

not pass through the origin (^2 = 0, w=0). The foundation characteristic can be calculated from the equations above. For a segment with relative dimensions 1=1.75 6, hso=l-^ b, the dimensionless foundation characteristic is presented in figure 7.4.2. From the equations (7.4.3-4) the

stiffness of a segment in the radial direction (radial foundation stiffness per unit length) can be derived. We obtain:

Cr = -2p /dbe\ / cos (})s^4>s sin (/>, \ ^ \dw ) \ sin (/)s — (f)s cos 4>s )

Figure 7.4.2. Radial foundation characteristics with radial stiffness, Cr, and lateral stiff- ness, Cc, of tire segment of unit thickness shown in figure 7.4.1.
700 THE TIRE AS A VEHICLE COMPONENT

Together with eq (7.4.3) c, can be calculated as a function of w. In addition, we derive the lateral stiffness Cc of the tire segment. With

a small lateral displacement v of the tread band in lateral direction y, point A (fig. 7.4.1) moves upward and point B downward a distance

{b — be)vlhs. The total lateral force acting on a segment of unit thickness

becomes 2pi{b — be)vlhs. Hence, the lateral stiffness of the segment

(lateral foundation stiffness per unit length) reads:

c, = 2p,(^^)=J- (2p,-6-9.). (7.4.6) The lateral stiffness in an appropriate dimensionless form appears to bear a simple relationship to the dimensionless radial foundation stiff- ness, as shown in figure 7.4.2. From this figure it can be observed that both the radial and lateral stiffness decrease with increased deflection

{—w). The lateral stiffness even becomes negative for be > b. The lateral

stiffness is of particular importance in the analysis of the cornering

behavior of tires, to be dealt with in part 7.5. Rotta [8] and Clark [11]

employed alternative models for the determination of the lateral foundation stiffness. We shall continue now with the normal loading problem and determine

the load-deflection curve of the tire from the foundation characteristic

of figure 7.4.2. The radial deflection — w varies along the contact line. The contact length depends on the normal tire deflections, 8, and the

tire radius r. We use the approximate equations

a' = 2r8 (7.4.7) and

-w=8-Y^ (7.4.8) where x denotes the coordinate in the longitudinal direction. In figure

7.4.3 a comparison is made between calculated values of contact length

and experimental data from Senger [9]. Note that at large deflections

the calculated values are somewhat high.

The dimensionless normal load expressed in the integral form

(7.4.9)

can be calculated, in principle, with the aid of the foregoing equations. In the actual calculation the radial foundation characteristic of figure

7.4.2 has been approximated by a quadratic function of w/b, coinciding

with the original curve in — w/b = 0, 0.2, and 0.4. The final result for a

tire with r = 5.4 b has been shown in figure 7.4.4 in comparison with

measured data deduced from Senger's experimental results. Perhaps

owing to deviations between calculated and measured values of contact

length (fig. 7.4.3), the agreement is better than expected. As has been

mentioned before, around 15 percent of the normal force transmission
NORMAL FORCE-DEFLECTION CHARACTERISTICS 701

50

0 12 3 4 Deflection S(cm)

Figure 7.4.3. Experimental and calculated values of contact length [9]. A

cam

Ic u lated

ea s u red

0 .1 0 .2 0.3 . 0

b Figure 7.4.4. Calculated nondimensional load deflection curve compared with experi- mental results ofSenger [9] for a ll .00-20 cross-ply truck tire {pi = 4-7 bar).

is attributable to carcass rigidity. The shape of the measured and

calculated curves is representative for cross-ply tires. It suggests

that for most purposes a linear load-deflection relationship can be used

with sufficient accuracy. Vibration tests of Rasmussen and Cortese [5] and of Betz [6] with rolling tires substantiate this result. Tiemann [12] found great similarity between reduced load-deflection curves for a great variety of automobile and truck tires. Here, distinction

must be made between bias-ply and radial tires. For each of these cate-
702 THE TIRE AS A VEHICLE COMPONENT

gories of tires it turns out that the dimensionless-load versus dimensionless deflection can be reasonably well represented by a single curve

where B is section width, H is section height and r is rim radius. For

bias-ply tires the doubly-curved form of figure 7.4.4 applies, whereas

radial tires show a continuously concave characteristic. For aircraft

tires Smiley and Horne [13] propose a similar equation. To account for the influence of cover rigidity they replaced p, by p/ + 0.08 pr, with pr

denoting the rated inflation pressure. Since the measured curves are

so similar to those given by eq (7.4.10), it seems possible that vertical

load-deflection curves could be calculated for each type of tire construction. Enveloping Properties {Obstacles)

An important property of a pneumatic tire is its ability to cushion a vehicle against short road irregularities. A rigid wheel passing over an

obstacle would acquire a sudden vertical velocity which involves ex- tremely large vertical accelerations. Experiments never show such large

accelerations so that obviously the elastic enveloping properties enable

the pneumatic tire to partially "swallow" the obstacle while rolling over

it. The resulting vertical displacement of the axle is small relative to the

height of the obstacle.

Most of the published information on this subject concerns experimental data relating tire parameters to the response of tire forces or wheel axle motions as the tire slowly rolls over an obstacle. A number of investigators have examined the response of the vertical and longitudinal tire forces to a short prismatic obstacle extending over the whole

tire width, and possessing a rectangular, trapezoidal, sinusoidal or cy- lindrical cross section. In these tests the axle height above the road plane

has been kept constant. As an illustration we present in figure 7.4.5 (from Gough [14]) the response of a bias-ply and a radial tire. It is seen

that in the variation of the vertical load W= — Fz between two maxima,

a minimum arises when the wheel axle is located directly above the center of the obstacle. In this particular situation the minimum becomes

even lower than the initial normal load — F^o occurring on a flat surface. Extensive tests of Julien [15] show that the shape of the curves vary both

with size and shape of the obstacle and with the vertical deflection of

the tire. It appears that the minimum in vertical force variation only

arises when the static tire deflection is sufficiently large and the obstacle

size sufficiently small. The longitudinal force invariably shows one maximum and one minimum. These extreme values are virtually independent

of initial tire deflection [15]. From experiments conducted by Lippmann and his associates [16, 17]

it appears that with fixed axle height the vertical and longitudinal peak

forces vary nearly proportionally with internal pressure. For ordinary

inflation pressures the major part of the forces arise from internal

(7.4.10)
ENVELOPING PROPERTIES 703

MOO

Position of obstacle relative

Inches
704 THE TIRE AS A VEHICLE COMPONENT

pressure. This leads Lippmann to the conclusion that the core of the

process tests on some pneumatic mechanism.

Figure 7.4.5 furthermore shows the wheel rotation per unit of travel on the road. The angular acceleration of the wheel caused by obstacles

will generate additional longitudinal forces which may become quite

considerable in magnitude, particularly at high speed.

Theory ofenveloping capabilities.

An adequate theory explaining these measured force variations has

not been found in the existing literature. The following relatively simple

theory may furnish some insight into the problem. We employ a tire model consisting of a large number of radially directed springs. The

influence of tread band tension and bending stiffness will be neglected.

It has been pointed out in section 7.4.1 that the spring forces are mainly

due to pneumatic action. It is assumed furthermore that these forces

are directed perpendicular to the tire peripheral line. For the sake of simplicity the shape of the peripheral line is considered to remain

circular outside the contact zone, which extends from the leading to the

trailing contact points. When, in addition, the stiffness of the springs

is constant i.e., linear foundation characteristic, the vertical force

which acts on the tire can be obtained approximately by multiplying the

overlapping area of the tire circumscribed circle and the road profile

by the foundation stiffness per unit length of circumference. According

to the behavior of such a tire model, the vertical force increases when a short obstacle is encountered. The force remains constant as long as the

obstacle contour lies completely inside the circumscribed circle. The

experimental evidence of the occurrence of a minimum force cannot be

explained with this simple model unless a nonlinear softening foundation

characteristic is assumed. As has been indicated in section 7.4.1, such

a spring characteristic of a tire element will indeed exist (fig. 7.4.2). During the time in which the obstacle is completely swallowed by the tire,

it is obvious that the vertical force becomes a maximum when the

obstacle is in the foremost and rearmost positions since the slope of the load-deflection curve is then greatest. In the center position a minimum is expected as the obstacle now deforms the tire in the range of lowest stiffness. Experiments indicate that under particular conditions a minimum

vertical force can arise which is even lower than the initial force without an obstacle. Since the force-deflection characteristic of a tire element is not expected to be particularly nonlinear in the practical range of deflection, some other mechanism must be responsible for this phenomenon.

It is believed that a second possibly important effect on enveloping an

obstacle is the shrinkage of the circumference of the circumscribed

circle of the tread band, which is assumed inextensible in the ensuing

analysis. This circumference must become shorter in order to supply

length in the contact zone where the obstacle is partially surrounded by

the tread band. In figure 7.4.6 two circumscribed circles are shown.

The larger circle shows the tire pressed against a flat surface. Its size reduces to the smaller circle when the obstacle is in the contact center.
ENVELOPING PROPERTIES 705

TT

Figure 7.4.6. Tire peripheral line with and without the intrusion of an obstacle.

It has been assumed that the tread band deformations caused by the

intrusion of the obstacle vanish at the top of the tire. Consequently both

circles touch each other in this point. The difference in length of the two

circumferences equals the difference between obstacle contour length

and its base length. The relatively small variation of the difference

between arc length and chord length (contact length) of the circles in these two cases has been neglected. The possibility of partial loss of contact has been disregarded, an assumption which is admissable only

in case of relatively smooth obstacles. The reduction of the circumference of the circumscribed circle has

been designated as 2Kh, in which h denotes the obstacle height and \

a nondimensional form parameter of the obstacle. For a rectangular

obstacle shape, A.= 1. The value of A. decreases when the shape becomes

trapezoidal. The decrease in diameter of the circumscribed circle becomes IXhlir. For small values of /i the area of the section with chord

length 2a and width IKhjiT is approximately ^Xhalir. With c,- denoting

the radial (vertical) stiffness of the foundation per unit area, F^o the initial vertical load and A the area of the obstacle cross section, we obtain

for the vertical load acting on the tire when the obstacle has arrived

directly below the wheel axle:

It is seen that from this formula a decrease in vertical force can indeed

occur even using linear foundation characteristcs. The condition at

(7.4.11)
706 THE TIRE AS A VEHICLE COMPONENT

which this occurs is; A< (7.4.12)

It is of importance to introduce another form factor cr defined by the

relation: A = (jh\ (7.4.13)

For a square cross section cr= 1. When the length becomes larger than

the height cr increases. The condition (7.4.12) reads in dimensionless

form:

h ^ \ - <

a 77 a

(7.4.14)

Julien [15] has carried out experiments with an obstacle of square cross

section. We wish to compare his experimental results with our theory.

For this purpose we adopt a trapezoidal obstacle to the shape of which

the real tire deforms reasonable closely when it roUs over a square

obstacle. The symmetric trapezoid chosen has a height h, a base line 3h and a top line h. The area becomes A = 2h'^ so that cr = 2 and the

parameter k= V2 — 1 = 0.414. The condition (7.4.14) then becomes

hia < 0.263. From the results of Julien's experiments with a 5.0-15 tire

it can be deduced that the minimum vertical force becomes less than the

initial load at the same axle height when the ratio obstacle height to half contact length, h/a, becomes less than the values shown in the

table below, valid for three values of initial tire deflection 8. The agreement with the theoretical value 0.263 is very good considering

the simplicity of the model employed and its great sensitivity to the shape

of the obstacle, i.e., the shape of the actual tire deformation. It may

be noted that the critical ratio h/a increases with increasing initial deflection 8. This can be explained by the softening character of the

nonlinear foundation characteristic. This nonlinearity will in general

raise the critical value of h/a.

Table 7.4. 1 . Critical values ofh/a

h 8 a h_ (cm.) (cm.) (cm.) a

2 1.7 9.8 0.205

3 2.1 11.2 .267 4 2.6 12.4 .322

Using this theoretical tire model, the variation of the vertical and

longitudinal force has been calculated for a trapezoidal obstacle moving

from the leading edge to the center of the contact area. The results are
ENVELOPING PROPERTIES 707

Position of obstacle relative to contact centre Figure 7.4.7. Calculated variation of vertical and longitudinal force as a function of

position of obstacle relative to contact center.

shown in figure 7.4.7. The influence of foundation nonHnearity and of shrinkage of the tread-band are clearly demonstrated. The calculations

have been carried out for a tire with a cross section proportional to that

of the tire model used in section 7.4.1, of which the dimensionless foundation characteristic has been shown in figure 7.4.2. The tire model parameters are: r=27 cm., 8=1.5 cm., a — 9 cm., b = 5 cm., / = 8.25 cm., hso=8 cm., pi = 2 atm. The trapezoidal obstacle has the dimensions:

height h = 2 cm. , base line = 12 cm. , top line = 4 cm. This shape has been

chosen for simplifying the calculations during the first stage of contact. As the sloping side of the obstacle touches the peripheral line at the

instant of first contact, there will be practically no change in length of the

circumscribed circle during the first stage up to position 1. From this position on, reduction of the circumference will take place. The dotted

arc indicates the reduced circumference for the intermediate position 2. The circumference is minimum as soon as the obstacle lies completely

inside the circle. For position 2 the pressure distribution along the contact line has been

shown. The vertical components are found with the aid of the stiffness shown in figure 7.4.2. The integral of these components form the vertical
708 THE TIRE AS A VEHICLE COMPONENT

load —Fz. In the same way, the integral of the horizontal components

Qx form the drag force — FxIn figure 7.4.7 the calculated force variations are shown. The vertical

force has been calculated for four different combinations of the following

assumptions: linear (cr— 160 N/cm.^) and nonlinear foundation characteristic (fig. 7.4.2) and reduced and original circumferential length (curves

a, b, c, and d). Curve b, computed using the linear theory but with length

reduction taken into account, is close to exhibiting a minimum. According to condition (7.4.14) a minimum of curve b lower than the initial value — Fzo would arise when the obstacle is reduced in size so that h < 0.076 X

a =0.68 cm. This critical value increases when the nonlinear characteristic is taken into account. The variation of longitudinal force does not show great sensitivity to the model employed. Only curve e has been shown corresponding to the

combination of length reduction and nonlinearity.

The development ofa linear response theory.

One of the most important questions concerns the linearity of tire response to road irregularities. A linear behavior would simplify the

analytical treatment considerably. The tire has a finite contact length

and consequently is subjected to a large number of inputs at the same

time. The contact length varies with obstacle height, and negative

reaction forces cannot be transmitted so that partial loss of contact can

occur. In spite of these difficulties, which do not benefit linearity, Lippmann and his associates [16, 17] have experimentally shown that an

almost linear relationship exists between tire force variation and obstacle height. Tests have been carried out with constant axle height and

for the elementary step shape obstacle. Even the measured force

variation of a tire traversing a series of composite cleats (combinations

of blocks with rectangular cross sections and extending over the full tread width) could reasonably well be derived by linear combinations

of the response to a unit step in elevation of the road surface. A more

refined combination of these responses, where a distinction has been

drawn between positive and negative steps, gave very good correspond- ence to the measured total force variation. The applicability of this more

refined method is limited since linearity has been lost; for mathematical

details we refer to the original papers [16, 17]. The linear mathematical

representation of the tire force response to road irregularities, based

on the principle of superposition enunciated in the forementioned

papers, will be discussed hereafter. It has been stated that the assumption of linearity leads to sufficient accuracy in most practical cases. In figure 7.4.8 the positive directions are given for road coordinates

{xc, Zc), wheel axle position or distance traveled {s = x) , and reaction

forces (normal load — Fz and drag force — Fx)- Figure 7.4.9 shows the

force responses to positive and negative unit steps in road elevation.

The curves correspond in character to data given in [17]. Note that the curves differ slightly for positive and negative steps. The average of the absolute changes in force may be taken as the characteristic response

to a unit (upward) step. This function has been designated as for
ENVELOPING PROPERTIES 709

Figure 7.4.8. Positive directions of coordinates and forces.

Figure 7.4.9. Experimental force response to positive and negative steps in road

elevation. the longitudinal force response and as ^zii) for the vertical force response. The newly introduced coordinate f denotes the distance

traveled by the wheel axle from the instant of first contact with the step

irregularity.

We now decompose the road contour, zdxc) , into an infinite series of small positive and negative steps and use the principle of superposition

in order to obtain the total response. Expressed in the form of the

superposition integral of Duhamel, we obtain for the variation of the tire forces as a function of the distance traveled by the foremost contact

point Si = s -\- a with a denoting the assumedly constant half contact

length:

«1 dzc(xc)

dxc

^x,z{si — Xc)dXc (7.4.15)

At the starting position (^i = a) the forces have the value Fx^zo- This

general integral approach of expressing the response may be written in
710 THE TIRE AS A VEHICLE COMPONENT

the form of the more widely applicable frequency response of tire forces

to road irregularities. Following Lippmann's analysis we first rewrite

the above equation. With the conditions

Zc = 0 for Xc < a

ilJx,z = 0{or Xc > si

this equation may be written as follows

Fx,z{si)=Fjc,^o-^ j

^^^^^x,z{si—xc)dxc. (7.4.16)

The contour of the road may be expressed as a Fourier integral over

the spatial frequency co^:

zcixc) =^

I

^ D(o)s)e^^s^cd(Os. (7.4.17)

The Fourier transform becomes now with the above version of the

Fourier integral:

D{ojs) zc{xc)e-'^s^cdxc. (7.4.18)

The quantity D{o)s)do)sl2TT is the contribution of those harmonic components of the road contour whose spatial frequencies are included

in the interval between and + dois- Differentiation of (7.4. 17) yields

dzc{xc)

dXc 277^ f

" io}xD{cosW^s^c da)s. (7.4.19)

'TT J -y, Substitution of this expression into (7.4.16) yields with ^ = 5i —Xc'.

= ^ ^(^sDMe'^s^i

I"

e-'-/ • ilJx,zi^)d^ya)s. (7.4.20)

The quantity in brackets is the Fourier transform of iIjx,z{0 describing

the spectral content of the tire force response to a unit step variation

of the road level. This function is designated as Cx}z{(tis)' Hence eq.

(7.4.20) becomes:

Fx,zisi)-Fj,,,o = ^

I"

icosD(ojs)Cx,zMe'^shd(Os. (7.4.21)

Instead of measuring the response to a step, it may be more practical

to take a bump of short duration as the input. Such a bump may consist
LONGITUDINAL SLIP 711

of an upward step followed after a delay by a downward step of equal

height. The cross section of the unit bump has an area equal to unity

and a length tending to zero. The response to such a function simply

equals the distance derivative of i//^, ^C^). The Fourier transform of the response to a unit bump represents the frequency response function

of the tire force to road irregularities. This function, designated as

Hx,zi(^s), consequently equals icosCx, ^(ws).

In practive Hx,z{(Os) may be obtained by taking the Fourier transform

of the response to a bump which is short with respect to the shortest

wavelength of interest and dividing the result by the area of the bump

cross section.

Lippmann and Nanny [17] conducted a harmonic analysis of the

response of vertical and longitudinal forces to a cleat by feeding the

recorded signals into a tuneable resonator (tire 8.15-15 two-ply, Pi = 2

atm., deflection 2.54 cm.; cleat 1.27 cm. long, 0.63 cm. high; speed — 1 m/sec). The amplitude spectra exhibit two broad maxima. For the

vertical force response their maxima occur at the 2.4th and 14th order

of the tire revolution, and for the longitudinal response at the 8th and 21st

order. These figures correspond for the tire under consideration to the

following frequencies per unit of speed, reciprocal value of wavelength

expressed as (cycles/sec/meters/sec. ): vertical 1.1 and 6.3 c/m., longitudinal 3.6 and 9.4 c/m. At shorter wavelengths (higher frequencies) the

amplitude decays to low values. In the theories and experiments mentioned above the motions are as- sumed to develop slowly so that inertia forces can be neglected. The

traversing of obstacles at higher velocities involves high frequency

vibrations and dynamic forces in the tire tread band, the nature of which

will be dealt with in section 7.4.2. In the hterature, tests are described which have been carried out at higher speeds. In these tests the wheel axle has been free to move in its suspension. Care must be taken in the interpretation of the results as they do not solely refer to the response of the tire. We may mention here

the work of Barson [18] and Guslitzer [19] who measured the motion of vehicle components when passing over an obstacle. In the latter refer- ence attention has also been given to failure of tires due to obstacles.

The forces acting from the tire to the ground when rolling over an

obstacle have been measured by Hey [20] for a relatively long single

sinusoidal obstacle and by Senger [9] for a relatively short semicylindrical

obstacle. The velocities in these latter two tests did not exceed 50 km/hr.

Lonf^itudinal Slip

A tire which rolls freely at constant speed of travel, i.e., not subjected

to driving or braking torques, requires a thrust in the longitudinal direction acting on the wheel axle in order to overcome the rolling resistance.

We finaUy obtain for (7.4.21):

(7.4.22)
712 THE TIRE AS A VEHICLE COMPONENT

As has been shown in figure 7.2.56 the rolling resistance depends on the

normal load. For steady state motion the thrust P is in equilibrium with

the drag force F, and the rolling resistance moment Mr = riF,- where ri denotes the axle height above the road surface. At constant speed we may employ the linear relationship

Mr{=riFr)=DrW'=C,8 (7.4.23)

with W(= — Fz) denoting the normal load and 8 the normal tire deflection.

The coefficients of rolling resistance Dy and C, are, at least in the low

speed range, not influenced much by the speed V (cf. figs. 7.2.57, 7.2.60).

At free rolling the angular velocity fl and the speed V are related

through the eff"ective radius of rolling re V=are (7.4.24)

It has been found experimentally that ri<re<r (7.4.25)

According to Whitbread (cf. [1, 13 or 21]) the following approximate

relation exists

re=r-V3 8 (7.4.26)

This expression has been found by considering the compression of the

tread band in the contact area due to the normal deflection of the tire. Tests with aircraft tires roughly confirm this formula [13]. For a tire with an inextensible thin tread band and a worn-ofif tread pattern, so

that creep of the tread band with respect to the road does not take place,

the effective radius tends to the free radius r. Figure 7.2.48 provides more information about the variation of rolling radius with speed for different tire constructions.

We define now the longitudinal or tangential creep or slip velocity Vex as the tangential speed of a point C fixed to the wheel rim and

situated at road level (fig. 7.4.10). At the instant considered point C

coincides with the contact center C, which, according to its definition,

is located on a Une normal to the road surface and passing through the

wheel axle. Consequently, for a horizontal road surface, C lies directly

below the wheel axle. The longitudinal slip value relates the slip velocity to the speed of

of travel.

K = -VcJV=-^^-=f^. (7.4.27)

In section 7.5.1 a slightly different definition for the longitudinal slip

value has been adopted, viz K = — VcxlnO,. Their values are nearly the same at moderate longitudinal slip.
LONGITUDINAL SLIP 713

Assuming that relation (7.4.26) exists, the longitudinal slip speed

becomes at free rolling

Vcx=^/3 8CL (7.4.28)

or, expressed in more general terms,

Vcx=V^^ (7.4.29)

from which we obtain the longitudinal slip value at free rolHng:

(7.4.30)

The value of -q is approximately ^/s for bias ply tires and tends to unity

in case of an inextensible tread band with little tread rubber.

When applying additional driving or braking torques the slip value

changes in a manner analogous to that of the tire cornering characteristic relating lateral force to slip angle. Figure 7.4.11 gives an example

of measured longitudinal slip characteristics, taken from Horz [22]. Longitudinal slip is based on the following mechanisms. The first stems from the inextensible band theory, which states that since the same quantity of material passes at the top and at the bottom of the

tire per unit time, longitudinal slip must occur at the road level. Secondly, we have the influence of compressibility or extensibility of the tread

band. Compression in the contact area due to vertical deflection of the

tire diminishes the slip due to the first mechanism. Also under the action

of tangential force, the density in the neighborhood of the contact area
714 THE TIRE AS A VEHICLE COMPONENT

may change. As a result of this, the tangential velocity at the top and the

bottom do not necessarily have to be the same. Third, one may include

the influence of tread rubber longitudinal elasticity, which enables the

tread band to move with respect to the road without sliding. Finally, we have the possibility of partial or total sliding in the contact area. The first effect merely depends on geometry and does not involve

resultant longitudinal forces. The second and third effects give rise

to pure deformation slip characterized by the longitudinal or tangential creep, or slip stiffness Ck defined by the relation

C. = (7.4.31)

Since the tensile rigidity of the tread band is relatively large, the de- formation slip is expected to be mainly due to tread rubber elasticity.

The longitudinal slip stiffness Ck can then be expressed in terms of longitudinal tread profile rubber stiffness per unit area Cpx, half contact length a and half contact width b of the assumed rectangular contact

area: Ck = 4a26cp^ (7.4.32)

This expression suggests that Ck varies approximately as the square of the contact length. With the assumption that the contact length varies

^ hO

CJO

<-> 0,6

<uu

O 0,A c

0,2

1 V* 10,25 und 40 km/h

1 1 j t f

20 40 60 80 v. 100 brake slip value -k

Figure 7.4.11. Example of measured longitudinal slip characteristics [22].
LONGITUDINAL SLIP 715

quadratically with the normal deflection 6, we obtain for a linear tire load-deflection characteristic JF = C^S:

CK = -C,o=-i^ C,o (7.4.33)

with index o denoting the original situation. From test data [13, 21, 22, 23]

it can be deduced that the value of Ck ranges from 5 to 15 times

the vertical load W. This indicates that Ck is of the same order of magniI tude as the cornering force stiffness Cfa (sec. 7.5.1).

' For smaU deviations from the steady state conditions the longitudinal

force-slip characteristic may be linearized and written in the form

t\ = F,o + Fx (7.4.34)

where the index o indicates the original situation and the upper bar

indicates the deviation from this situation. The variable part Fx is a

linear function of the variation of vertical deflection 6 and of longitudinal

slip value R. With — Fxo = Fro denoting the original rolling resistance force we obtain in general:

Fx = - Fro +

^S +^K . (7.4.35) do dK

The values of the partial derivatives should be determined experimentally. From eq (7.4.31) the value of dFxIdK obviously equals C^o- With our knowledge of effective radii at free rolling we may derive dF.rld8

theoretically. From figure 7.4.12 we deduce for the longitudinal force at

load W

Figure 7.4.12. Longitudinal slip characteristics near the steady state conditions

{subscript o).
716 THE TIRE AS A VEHICLE COMPONENT

W=Wo-^W {or 8 = 80 + 8) and k = Ko + /<: F^ = -F,.-\-CAk + k') (7.4.36)

The quantity k' denotes the variation in longitudinal slip value of a freely

rolling tire due to a variation in normal deflection 8. According to eq

(7.4.30) the value of k' reads:

K=7)- (7.4.37)

fe The rolling resistance force F, becomes according to eq (7.4.23):

Fr = -Cr8 (7.4.38)

ri We finally obtain for eq (7.4.36), linear in the variations k and 8, and

with 8 neglected with respect to r:

Fa> = -Fro-\-C,ok-^~ {r}C,o-Cr)8. (7.4.39)

f0 Herewith, the coefficients of eq (7.4.35) have been determined. The

equation above is of great value for the theoretical assessment of the

longitudinal force response to road irregularities and tire nonuniformity.

The next section is devoted to this problem.

At relatively large values of longitudinal slip linearity is lost. For the

theoretical treatment of the behavior in the nonlinear range it is more

straight forward to use the alternative definition of longitudinal slip value K = —Vcxl^fi which results in a complete analogy with the side slip

phenomenon. For more information we refer to side slip theories enunciated in section 7.5.1. The latter part of this section treats the problem

of the interaction of longitudinal and lateral slip. We furthermore refer

to the early theories of Reynolds [24], Fromm [25], and Julien [26]. Experimental data are available in the publications of Yu Chen [21], Horz [22],

and a very extensive investigation of the behavior on wet roads by Holmes

and Stone [23]. For aircraft tires we refer to Smiley and Horne [13]. Theory of the Longitudinal Force Response to Road Wavyness, Tire

Nonuniformity and Vertical and Longitudinal Axle Motions

For the development of a theory of the longitudinal force response

to in-plane variations of wheel axle position, road contour, tire radius

and tire radial stiffness, a theoretical tire model is proposed, shown in figure 7.4.13, in which the following properties are represented:

1. radial tire elasticity with stiffness normal to road C^,

2. tangential carcass elasticity with stiffness tangential to the road

CcXi

3. tangential slip of carcass with respect to the road with shp stiffness
THEORY OF LONGITUDINAL FORCE RESPONSE 717

4. rolling resistance moment Mr with coefficient Cr,

5. moment of inertia of rotating mass including a great portion of the

tire, hThe system has two degrees of freedom, viz the angle x denoting the

deviation of the angle of revolution from the steady state situation characterized by the constant angular speed fl, and the torsion angle i// of the

wheel relative to the lower tread region (see fig. 7.4.13). The input varia- bles are:

t 1. horizontal and vertical wheel axle motions Xa and Za,

2. radial runout f and radial stiffness variations C^,

3. road level variations Zc.

I It may be noted that in reality the tire nonuniformity input quantities

I may vary with rotational speed H. This effect will not explicitly be taken

into account. The inputs are assumed to be of relatively long wavelength

so that first, the curvature of the road contour is much less than that of the tire; second, the wavelength is much longer than the contact length

between tire and road, which reduces the treatment of the longitudinal

slip behavior to the quasistationary problem; third, the frequency is small relative to the first natural frequency of the tire body (40-50 Hz)

permitting tire inertia effects to be neglected,

j We shall restrict ourselves to the investigation of the response to the

I input variables mentioned above. It may be noted, however, that other

input quantities may influence the longitudinal force. Among these,

the nonuniformly distributed mass along the tire circumference is also important.
718 THE TIRE AS A VEHICLE COMPONENT

One effect of nonuniformity in mass distribution may be the variation

in effective normal stiffness with speed of travel. By measuring the

periodic tire radial stiffness variations as a function of rotational speed

this effect will be included automatically. Also, the contribution of the mass nonuniformity to the change of radial runout with rotational speed

should be recognized.

The mass of the lower portion of the tire is subjected to the action of tangential Coriolis forces which arise when the lower tread-band portion moves in a radial direction with respect to the wheel axle. Similarly, a variation in tread-band mass causes a resultant tangential Coriolis force even in the case of fixed axle height. In both cases the variation in angular momentum of the tire-wheel combination about the wheel axle gives rise to the fore and aft force variations.

The influence of these effects deserves closer investigation. It may be

noted, however, that the whole phenomenon is restricted to a portion of the tire near the contact area. The tangential forces are in equilibrium

with the inertial forces acting on the particles present in this portion.

Therefore, the net force to be measured at the wheel axle is expected to be very small, if present at all. A rough calculation may provide some quantitative insight. Consider

a tire with radius r=0.3 m., contact length 2a = 0.15 m. and tread-band mass per unit length 1.5 kg/m. The mass which moves when deflecting

the tire becomes approximately 0.25 kg. When the tire rolls over a flat surface at a constant speed V=30 m/sec. and a constant angular speed

fl= 100 rad/sec, and moves vertically with a frequency of 10 Hz and an amplitude of 0.005 m., the lower tread band portion will be subjected to a Coriolis force which has an amplitude equal to

2 X 0.25 X 277 X 10 X 0.005 X 100 - 16 Newtons = 3.57 lbs. The problem of the influence of a gradient in mass distribution may be

illustrated with the following example. Consider the same tire in steady

state motion but possessing a heavy spot of 0.05 kg. This tire element

will be subjected to a Coriolis acceleration at the entrance of the contact area equal to 2a(F/r)^. The Coriolis force consequently becomes

0.15X100^X0.05 = 75 Newtons =16.8 lbs. Theoretically, these forces

are measurable at the contact surface and not at the wheel axle. We shall now turn to the restricted problem where inertia effects in the lower region of the tire are not considered. For the tire model of

figure 7.4.13 the equations of motion will be presented. Again, the

symbols provided with index o denote the stationary or average value

and the upper bar indicates the variable part. We restrict ourselves to small deviations from the rectilinear stationary motion, and can therefore

linearize the mathematical representation.

The vehicle has a constant speed of travel V. The distance traveled

becomes

s=Vt. (7.4.40)
THEORY OF LONGITUDINAL FORCE RESPONSE 719 The contact center has a coordinate in ^-direction somewhat deviating

from 5. We write

xc = s-^^x (7.4.41)

where

^x = xa-rl^ (7.4.42)

Taylor's formula yields for the slope at dzc dzc _|_ d^Zc dxr =-n+-d^+- ds ds

(7.4.43)

In the linear representation we may omit the terms with Ajc. For the

longitudinal force acting in the contact center we obtain

Xo = -Fro. (7.4.44)

X= F,+ Wo^^' (7.4.45) as According to eq (7.4.39) the tangential force variation equals

F:r = C,ok + - {y)C,o-Cr)l. (7.4.46)

^0 This equation, valid for steady state lontitudinal slip, can be used with

good approximation for nonstationary motions with wavelengths which

are long relative to the contact length. The problem is analogous to that

of the shimmy phenomenon treated in section 7.5.2. The tangential slip value defined by eq (7.4.27) becomes ~ ^ (7.4.47) V+Xa

with the steady state value

V r and variable part

Ko = —-^-7)— (7.4.48)

-y--Koy (7.4.49)
720 THE TIRE AS A VEHICLE COMPONENT

The tangential slip speed reads

Vcx^V-hxa- (n^X-^iij)n. (7.4.50)

with

ri = r—8 = rio-\-f — 8 we obtain for the variation of Vex

Vcx = ia- 0.ir-8)-r,o{x^ip). (7.4.51)

When neglecting 8 with respect to r we obtain for eq (7.4.49)

k = -y{xa-n{r-8)-ro{x^^)}. (7.4.52)

The equation of constraint for 8 reads:

8 = -Zc + Za+f. (7.4.53)

For the variation of the tangential force we have the additional relation

Fx = — Ccxro^. (7.4.54)

The normal load

r=C,8=(C,o + C.)(8o+8). (7.4.55)

Instead of using the variable stiffness Cz we prefer to introduce the

variable static deflection bo. This quantity can, in principle, be deter- mined by measuring the variation of the axle height of a perfectly round

tire which rolls under constant vertical load Wo over a smooth flat surface. The following relation applies

lo = -8oCzlCzo- (7.4.56)

Hence, the following relation holds for the variation in normal load

W=CzoC8-lo). (7.4.57)

According to relation (7.4.23) the variation in rolling resistance moment

becomes

Mr = DrW= Cr{8 - 80) (7.4.58)

with Cr approximately given by

C. = ^. (7.4.59) Oo
THEORY OF LONGITUDINAL FORCE RESPONSE 721 The dynamic equilibrium about the wheel axle is governed by the

differential equation

Fa:ri-Mr = Fro{r-b)-roF^-Mr. (7.4.60)

For the eight unknown quantities X, Fx, Mr, X, i//, 5, k and s we have

to our disposal the_ eight equations (7.4.40, 45, 46, 52, 53, 54, 58, 60). For the unknowns X, x ^ obtain by elimination of the remaining

unknown quantities the following set of equations

(7.4.61)

hX~Ccxr'd\\f= {Fro^Cr) {Zc — Za) —Cr{r —8o), (7.4.62)

ro(x+*)+^^r T7 + 7r^| — (Za—Zc)

CrW

(7.4.63)

These equations of motion can be used in the analysis of wheel suspension vibrations. We shall concern ourselves here with the derivation of the response functions of the longitudinal force to each of the input

quantities. It should be recognized that some of these input quantities

depend upon each other in a way which is a function of the suspension

properties.

A final elimination of the variables x ^nd i|/ and the introduction of the operator p=dldt yields the following expression for the response of the longitudinal force variations X to the road irregularities Zc, the

radial runout r, the variation in static deflection 5o and furthermore to the

longitudinal and vertical axle motions Xa and Za : X = - |/x(l-'r/-f^)^p + ro(F.o + C.)

I

(z«

— CrToho-^Ixp'^Xa^A

-Zc)

rro \ r Wo

pZc (7.4.64)

hich

+ Vp

From this expression of combined responses the single absolute fre- quency response functions can be derived.
722 THE TIRE AS A VEHICLE COMPONENT

Frequency response to vertical axle motions.

After replacement of p by icu, with oi denoting the frequency of the

motion, we find for the response to vertical axle motions the following

amplitude ratio.

Fro + Cr

(l-u2)2+4/fu2'

where have been introduced the nondimensional quantities

(7.4.65)

2/, = F^^foC KO

2/2 = 2/,

{\-y))C,

Fro~^ Cr

ay

col

(7.4.66)

(7.4.67)

(7.4.68)

with (Oxo denoting the rotational natural frequency of the standing

tire about the wheel axle

(7.4.69)

For illustration, some aspects of the response will be discussed for a tire-wheel combination with parameter values:

/x=-0.6 kgm.2, Ccx^SX 105 N*/m., ro = 0.3 m., Aixo-wxo/27r=43 Hz, v^^s, C,o = 40000 N, (7.4.70) Fro = 60N, (/o = 0.02m,, ro = 4000 N, Cr=900 N. The dimensionless parameters become, for this configuration,

/i = 0.023 F and /2 = 0.342 V. For CO = 0 we find

Fro~\~ Cr

3200 N/m. (7.4.71)

A' = Newton =.223 lbs. force.
THEORY OF LONGITUDINAL FORCE RESPONSE 723 For small w( <^ (oxo) we may simplify to Fro + C,

Fro + Cr

1 +

{1+ (l + 2/|-2/|)i;2}

Ix 1 F^/l f/(l -7^)C.o + C-

= 3200 {l+(l + 0.233F2)^j, (7.4.72)

where V denotes the speed of travel expressed in m/sec. and n denotes

the frequency of the axle motion in Hz. It may be observed that the

magnification of the zero frequency response (7.4.71) is in particular due

to the large tangential slip stiffness Cko and the moment of inertia Ix. The effect of changing Ccx is noticeable only at very small values of speed V. The factor 7) defined by eq (7.4.29) can exert considerable

influence. A value of 17 close to one is favorable, which can be explained

by the fact that if 17 = 1 the variation of the effective radius at free rolling

with deflection becomes very low and is in that case only due to the

change in rolling resistance.

Figure 7.4.14. Frequency response curves for longitudinal force X with respect to vertical axle motions Za, calculatedfor the tire-wheel parameter values (7.4.70).
724 THE TIRE AS A VEHICLE COMPONENT

At higher frequencies the ampHtude ratio deviates from the approximation (7.4.72). In figure 7.4.14 response curves are drawn for several

speeds of travel. Besides the point at zero frequency where the curves come together, two more invariant points appear to exist where the

response is independent of V. Since these points show the same amplitude ratio and are located near the natural frequency nxo, their height

gives a good indication of the maximum amplitude ratio, which occurs near the natural frequency and is virtually independent of V. The

curves show furthermore that for the higher range of speeds this maximum becomes very broad, which means that high amplitude ratios can be

attained at relatively low frequencies.

From eq (7.4.65) we obtain for the amplitude ratio in the invariant

pomts

^Fro±_Crf^^(l in)C.o^C,. _ ^^^^^ ^^^^ ^^^^^^^^

To J\ fo or a magnification of about 15 times the zero frequency response. From

the expression above it follows again that large Cko and low iq are unfavorable. The speed V and the parameters Ix and Ccx do not influence

the response in the higher frequency range, near the frequencies of the

invariant points, given by

Wfnvl,2 = colo(^l±jy (7.4.74)

Obviously, the invariant points lie close to and on either side of the

natural frequency.

Frequency response to longitudinal axle motions.

From eq (7.4.64) we find for the response to Xai CcrV~

(7.4.75)

V(l-v2)2 + 4/?v2

This is the well-known form for the acceleration response of a single

mass-spring system excited by a force. The dimensionless coefficient of damping become for our system:

2f,= V^^^^^' (7A76)

ro^KO Damping obviously increases with increasing speed V and decreases

with increasing slip stiffness C^o- Compared with the response to Za, the speed V has an opposite effect whereas a variation of Cko changes

the response in the same direction. The eff^ects of Ccx and Ix are more

difficult to evaluate since they influence the response also in other

respects. At relatively low frequencies Ix is the most important parameter. For the parameter values of (7.4.70) we find a dimensionless damping
THEORY OF LONGITUDINAL FORCE RESPONSE 725

coefficient /i = 0.023 V. Consequently, the damping becomes critical

at a speed approximately equal to V= 40 m/sec.

Frequency response to variations in static deflection. From the expression for the combined response (7.4.64) we obtain

(7.4.77)

V(l-v^)^ + 4/?v2

which corresponds to the displacement response of a single mass-spring

system excited by a force. The response to variations in radial stiffness or static deflection as formulated above is very small. However, the

indirect response with axle free to move vertically will be appreciable.

The variations in radial stiffness induce vertical axle motions which

involve a response of the longitudinal forces just discussed.

Frequency resjwnse to radial runout. The absolute frequency response function to radial runout f reads:

I with the dimensionless parameters

2/, = F^^S, (7,4,79)

^0^ KO

2/3 = 2/, (7.4.80)

It may be noted that great similarity exists with the response to vertical axle motions (7.4.65). The characteristics are similar to those shown in figure 7.4.14. The level of the response, however, can become considerably higher in the higher speed range. This is due to the fact that in the

dimensionless parameter fs the factor 1— 17 has been replaced by —r).

For tires with r) tending to one (inextensible tread band and little tread

rubber) which was favorable for the response to vertical axle motions, it turns out that their response to radial runout becomes worse. For the

values used in (7.4.70) we obtain for the amplitude ratio at the invariant

points

^Crl^^Cr l-^^KO + C.L 80500 N/m. (7.4.81)

or about 8 kgf. per mm. radial runout. The magnification with respect to the zero frequency response amounts approximately 27 times.
726 THE TIRE AS A VEHICLE COMPONENT

Frequency response to road irregularities.

According to eq (7.4.64) the response to road irregularities can be

formed by adding an extra term to the negative of the response to vertical axle motions

dX c)X 1

The solution for this combination of responses to variations in n and in slope dzclds is difficult, and is in fact impractical since no useful infor- mation can be obtained other than that v^hich can be drawn immediately

from (7.4.82). One might as well consider the slope dzdds as an isolated

input, the response to which must be regarded in combination with the

response to axle height to roadway variations. The response to slope

variations at constant axle distance from the road surface simply equals

the average normal load Wo.

7.4.2. High Frequency Properties

In the high frequency range, well above 30 Hz, tires exhibit a number

of natural frequencies in vertical (radial), longitudinal (tangential) and

lateral direction and will consequently show continuously distributed

vibrations of side walls and tread band.

Although the intensities of the high frequency vibrations of the un- sprung masses are lower than those of the lower frequency resonances, research on this class of high frequency vibrations is valuable since they

are lightly damped and in some cases are even amplified because of local | resonances occurring in the vehicle. They also play a fundamental role

in acoustical effects. The high frequency vibration behavior of rolling tires is a very complicated matter and has been dealt with only briefly in the literature. In the

following, important experimental and theoretical investigations on high

frequency tire properties will be discussed.

Vertical and Longitudinal Vibration Transmission

In recent years the transmission of high frequency vibrations from

contact patch to wheel hub has been experimentally investigated by

Chiesa, Oberto and Tamburini [27, 28] and by the team of Gough and

Barson [18,29].

Chiesa and his group mainly reported on investigations with the

nonrolling tire excited by a vertically vibrating platform on which

the tire stands. Barson and Gough, in addition, executed experiments

with tires which roll over a drum to which can be attached a great

variety of artificial road irregularities, ranging from sinusoidal shapes to random surfaces. In the case of a sinusoidal profile a wide frequency

range can be investigated by varying the drum speed. In their extensive

investigation the longitudinal aspect of vibration transmission has been

included.
VIBRATION TRANSMISSION 727

Figures 7.4.15a, b (from [18]) respectively show the vertical trans- mission ratio (amplitude ratio) of hub relative to platform for the nonrolling tire and the r.m.s. value of the vertical hub acceleration for the

tire rolling over a sinusoidal drum surface with wavelength 0.133 m. and

a

VERTICAL VIBRATOR

RADIAL- PLY TYRE

CROSS- PL Y TYR

>

>- _l

>- Q a

^ ^ ^

b

VERTICAL DRUM

CROSSPLY TYF >-

?E

\ /

RADIAL -PLY TYRE

^

1 V

1 Figure 7.4.15. Comparison of cross-ply and radial-ply tires in their vertical vibrational

behavior.

Test carried out on vertically vibrating platform (a) and on rotating drum {b} with sinusoidal surface (wavelength

0.133 m, amplitude 3 mm. speed 5-100 km/h) [18].
728 THE TIRE AS A VEHICLE COMPONENT

amplitude — 3 mm. Both tests have been carried out with a laboratory

strut-type wheel suspension showing vertical and longitudinal comphance, the latter resulting in a longitudinal natural frequency of about

11 Hz. In figures 7.4.16a, h the results of analogous tests for longitudinal

RADIAL- PLY 1 YRE

LONGITUDINAL

/ \ DRUM

•'-""7V

CROSS- PLY TYRE

ILiiiiifikJk li

1 1 1 1 I 1 Q< o

I- O IT < < O

0: a

40 60 80 100 FREQUENCY— HZ Figure 7.4.16. Comparison of cross-ply and radial-ply tyres in their longitudinal vibra- tional behavior.

Tests carried out on lon<:itudinally vibratint; platform (a) and on rotating; drum (b) under same conditions of fifiure 7.4.15 (b) [18|.
VIBRATION TRANSMISSION 729

vibration transmission have been shown. In this case the vibrator platform is excited in the longitudinal direction, while for the drum test the same sinusoidal surface is used as in figure 7.4.156. These experiments have been carried out with radial and bias ply

tires of the same dimensions. For clear comparison of the properties of these tires their response ratio has been shown in the lower diagrams.

It may be noted that for the vertical transmission of vibrations qualitative

similarity exists between results of both types of test, platform and drum.

The same resonance frequencies (attributable to both tire and suspension) arise, and from both tests the advantage of radial ply tires appears

above 100 Hz, the "road roar" range. Their disadvantage in the range of 60-100 Hz is also clear. In particular, for the longitudinal aspect of vibration transmission, the drum test appears to be essential. From this

test (fig. 7.4.166) it can be shown that the radial ply tire causes a much

greater longitudinal response below 100 Hz than the cross ply tire. The similarity of vertical response between rolling and standing tires led Chiesa to adopt the oscillating platform test for further detailed

investigation of the vertical vibration transmission. In order to over- come the problem of the presence of resonances other than those

attributable to the tire, the tire has been mounted on a massive heavy

wheel of approximately the same weight as the load sustained by a single

tire fitted to a car. This wheel was supported on the vibratory table by

an air spring system of negligible stiffness. In the experimental investigation described in [27] a comparison has

been made between the vibrational behavior of radial and cross ply tires. The oscillating platform produces vertical hub oscillations via tread band

and sidewall vibrations distributed along the tire circumference. The

Fi(;URE 7.4.17. Amplitude ratio between hub of heavy wheel and vibrating f>latform [27\.

0.06

0 I

I I I I 100 1

I

I 1 1 I

1 1 1 1 r

' » 50 150 200 Hz
730 THE TIRE AS A VEHICLE COMPONENT

question of whether a correlation exists between these distributed vibra- tions and the complex transfer function of platform motion to hub motion

forms the major part of that research.

The amplitude ratio of the hub oscillation and the sinusoidal platform

oscillation has been shown in figure 7.4.17 for both types of tire as a function of platform frequency. With the radial tire the system exhibits

a very pronounced resonance at about 90 Hz, followed by a number of

less important resonances. The cross ply tire, however, exhibits only one resonance near 150 Hz. In practice, this indicates that the radial

tire has more potential for developing resonances in the vehicl'^^ than the

cross ply tire. The tests have been carried out with an inflation pressure

Pi = 1.5 bar, (1 bar ~ 1 atm.). At higher inflation pressures the resonances appear to shift to slightly higher frequencies and the amplitude ratios mcrease a little. This is true for all resonances except the peak occurring

at about 230 Hz, being due to the resonance of the air column within the

tube. This appears to vary only with tire size and type of gas used for

inflating the tire. Experience shows that the whole picture of the trans- mission curve changes substantially when varying the tire size. When the

type of tire construction remains the same, the only effect is that with

larger sizes all resonance frequencies shift towards lower values. These

results are of importance for the tuning between tire and vehicle.

For determining the correlation between transmission curves and tire vibrations, Chiesa and his co-workers found the shape of the deformed

equatorial line by measuring the hub and tire tread band and sidewall

motions by means of accelerometers. The instantaneous displacement

of a point of the tread band is obtained by vectorial combination of the

radial and tangential displacements deduced from the signals of microaccelerometers placed between tube and tire. By connecting the positions of the points at several locations (azimuth angles) the instantaneous

deformation line is obtained. These displacements occur relative to the

stationary equatorial line, which is deformed due to the average vertical

load. For purposes of this research it has been considered acceptable to take as a reference equatorial line a perfect circle. In figure 7.4.18 de- formation patterns of a radial and a cross ply tire have been shown with

respect to the circular reference line. The deformation patterns which

are shown here for a series of excitation frequencies are obtained at that instant where the highest vertical resultant at the hub of the heavy wheel

occurs. The phase delay angle (/)r of the hub vertical motion relative to the shake table motion has been indicated for each frequency. The

vertical bold line segment indicates the amplitude of the platform motion.

At low frequencies the deformation is limited to the region near the contact area. By increasing the frequency the first mode of vibration

is attained. At this natural frequency (radial ply ~ 90 Hz, bias ply ~ 150

Hz) the deformation patterns are similar (two nodes). It can be seen from

the figure that with increasing frequency the number of wavelengths

along the circumference increases. At a frequency of 190 Hz the radial

ply tire already has seven wavelengths whereas the conventional bias

ply tire shows only two wavelengths at this frequency.

From observations over one complete period of excitation it has been

found that the wavy deformation as a whole does not vanish periodically.
VIBRATION TRANSMISSION 731

Fl(;URE 7.4.18. Deformation lines of radial and cross-ply tire for a series offrequencies taken at the instant of highest vertical resultant characterized by the phase delay angle

(f)r relative to the platform motion.

The nodes appear to move a little along the circumference, which is peculiar to dissipative distributed vibrations. Moreover it appeared that the number of nodes may vary during one period.

Measurements of the lateral displacements of the sidewall indicate

that a simple, generally valid relation with the radial tread displacements exists. It appears that these motions are always out of phase,

i.e., when the one contracts the other expands. This is true except in the direct vicinity of the contact area where both motions are in phase.

For investigating the correlation between the deformation line of the

tread band and the vertical force transmission, the following procedure was adopted. In increments of 10 degrees of azimuth angle the vertical

component of the displacement of the points of the equatorial line have

been determined, after which their algebraic sum has been calculated.

This sum has been determined for a number of instances over one excitation period and it appears that the variation with time is approximately

harmonic. In figure 7.4.19 the amplitude ratio and the phase relationship

of this calculated sum with respect to the platform motion has been

shown, together with the measured response of the hub motion. It appears that the agreement is very good between the calculated sum

of the vertical deformation, which may be considered to be proportional

to the force transmitted, and the experimental results of figure 7.4.17,

after the latter are cleared of low-frequency components. As Chiesa

states, the correlation which has been found to exist furnishes a broad
732 THE TIRE AS A VEHICLE COMPONENT

basis for the understanding of the high-frequency (vertical) behavior

of tires, to add to the well known low-frequency phenomena.

Similar research can in principle be carried out for the investigation of longitudinal and lateral force transmission. It may provide some insight,

but the practical application of the results obtained with a horizontally

shaking platform will be difficult, since the motions of the lower part of the tread-band running on a road are not known due to longitudinal and

lateral creep phenomena. Moreover, for longitudinal force transmission

the slope of a wavy road surface plays an important role, as has been

shown in the previous section 7.4.1. The natural frequencies and the mode shapes of the motions in these two directions however, are no doubt of importance for general

information and for purpose of tuning the tire-vehicle combination. The

information shown in figure 7.4.20 has been taken from reference [18].

It shows several modes of the tire-suspension combination. The tire is of the radial ply type and the suspension is of the strut type having

large longitudinal compliance. The resonances will change with mass,

polar moment of inertia and longitudinal compliance of the tire. Probably

the only peak which is due to tire natural vibrations is the one between

80 and 90 Hz, which corresponds to the mode found with the vertical

excitation. The resonance occurring at about 50 Hz does not correspond

to the natural frequency of the tire rotating as a whole with respect to the wheel fixed in space (cf. sec. 7.4.2, table 7.4.2: 45.5 Hz) but represents

the natural rotational vibration of the wheel with a great part of the tire (about a point approximately midway between hub and upper rim) with

respect to the contact patch. The peak at about 13 Hz will be due to the

natural frequency of the system, mainly owing to the longitudinal

compliance of the suspension.

The experiments discussed above are carried out on a tire mounted

on a hub which must be capable of moving in order to obtain the overall

transmissibility. The natural frequencies of the tire itself might be

measured more accurately on a tire mounted on a rim which is fixed in space. The results of some of these investigations will be given in the

subsequent paragraph as an illustration of the theory (cf. figs. 7.4.24-25).

Fl(;URE 7.4.19. Amplitude ratio and i>hase delay ofcalculated sum of vertical deformations

distributed along the circumference with respect to platform motion (adapted scale) compared with measured hub response of figure 7.4.17, after being cleared of the lower frequency comjwnents arising from the vibration of the system considered as a rigid

mass-spring system.
TREAD-BAND FREE VIBRATIONS 733

IN PHASE IN PHASE

Tx/2 OUT

OF PHASE

n OUT

OF PHASE

2-0 < 1 O

001

10

\ r

20 30 40

FREQUENCY — Hz 50 60 70 80 90 100 The magnitude and sense of the vibration are shown by the numbers and by the shading cf the

arrows. The phase with respect to the input for all the stations on any one diagram is shown above

the diagram. The unity transmission ratio line defines the magnitude of the input to the tyre. The transmission ratio curve itself refers to longitudinal vibrations at the axle.

Fi(,LRE 7.4.20. Modes of vibration of the suspension-ivheel-tire (radial-ply) combination

for longitudinal excitation by a vibrating platform.

Tread-Band Free Vibrations

For the development of a theory for the plane vibrations of a tire one may consider a circular membrane under tension or a circular beam or a combination of these two. In the early literature we encounter a number

of theories for the vibration of a two-dimensional elastically-supported

circular beam or shell [30, 31]. More recent studies by Tielking [32] and B(')hm [33] and a more simple theory by Fiala and Willumeit [34] lay

specific emphasis on the vibration of a pneumatic tire. The complicated

case of a torodial shell has been solved, after many simplifying assumptions, by Federhofer [35]. In the theory presented below we treat the

vibrations of circular beam and make use of the method employed by

Tielking, which is based on the princii)le of Hamilton. In an example we present the equations and results as given by Bcihm, which also cover

vibrations in the lateral direction. Unlike Tielking, we introduce a tension force acting in the beam which

may be due to the internal pressure and the rotational speed. We also account for the tangential stiffness between beam (tread-band) and wheel

rim. In figure 7.4.21 the pneumatic tire model is shown. Damping is not considered.

The application of Hamilton's principle requires the determination of the potential and the kinetic energy of the model as well as the work done

by internal pressure forces.
734 THE TIRE AS A VEHICLE COMPONENT

Figure 7.4.21. Cylindrical beam or shell model for a pneumatic tire.

Figure 7.4.22. Displacement and deformation of tread band element.

The potential energy is composed of a part due to elastic deformations

of the beam and a part due to radial and tangential displacements of the

tread band with respect to the wheel rim. For the determination of these

quantities we introduce the independent angular coordinate 6 and the

tire radius r, representing the position of a tread element at rest with

respect to a coordinate system fixed to the wheel. Furthermore we intro- duce the dependent coordinates u = np and w, denoting the tangential

and radial displacements, respectively, of the element with respect to the original position.

Figure' 7.4.22 shows an element in the original position (possibly in- fluenced by the centrifugal force) and in the deformed and deflected

situation. The tangential strain at the original radius r^^r + z is in- fluenced by the initial strain €o (due to internal pressure and centrifugal

force), by the additional "angular" strain dip/dO and by the radial dis-
TREAD-BAND FREE VIBRATIONS 735 placement w and its derivatives. The strain becomes: w \ d'-w

r

1 +

dijj_ do - 1

with

r + 2. The potential energy due to tread band deformations becomes, per unit

width:

1 r^eidhdO.

where the integration is over the tread band thickness h and in which E

denotes the elastic modulus of the beam material. With the introduction

of u = njj, I=y&bh^ (moment of inertia) and So = Cseo (tension force with

Cs denoting the tensile rigidity) we obtain, with the assumption that and (h/r)- are negligible with respect to unity:

+ w dd. (7.4.83)

in which only the second-order terms are shown.

The potential energy stored in the elastic foundation becomes, using

c, and Ct to denote the radial and tangential stiffness per unit length:

--if

The work done by the internal pressure pi is: W, = 2bpi(^^j^'\ldcf>-

d(f) = did-h 4j)

do. (7.4.84)

with

and

7Tr'

1 du

dO 7 M

rd= r w. When retaining only the second-order terms we obtain:

1 do

Wp=bpi j^^ (2w^^w-^ do. (7.4.85)

For calculation of the kinetic energy we consider figure 7.4.23. The

wheel rotates with an angular velocity Cl. In addition, a tread element

has a velocity with respect to the wheel. The velocity vector of such
736 THE TIRE AS A VEHICLE COMPONENT

Figure 7.4.23. Position of an element and its velocity. an element expressed in terms of the unit vectors in radial and tangential

direction Cr and et becomes:

--=w€r-\- iff) {r-\- w)et.

With the use of this expression we obtain the following formula for the kinetic energy of the shell per unit width with p denoting the mass

per unit area: T^ y2 pr j^^ \fd\-'dd = V2pr j'^ {w'+ {a-^iij)Hr+wy'}dd

which becomes when retaining only second-order terms and with u = njj:

f

277 T= V2 pr

J {w' + a'w' + u' + mwu}dO. (7.4.86)

According to Hamilton's principle the time integral of the Lagrangian

L must be minimized, so that:L=T-V-hWr (7.4.87)

The Lagrangian density dL/dd is easily formulated by combination of the

above expressions. We obtain: —= 72 pr{w--\- il^w^ + + Ulwu) — 72 — I -\-wj — 72 y I — I - (^^^wj - V2 r(crw-' + ctu') bpi(^2w w'^ (7.4.88)
TREAD-BAND FREE VIBRATIONS 737 The time integral of this quantity becomes stationary when the following

Euler-Lagrange differential equations for the dependent variables w and

w, which hold for our case of two independent variables d and t, are

satisfied [36]. With the abbreviations ( )' = a( )ldd and ( ) = d( )ldt we

obtain

dL' d /dL'\ d /dL'\ ('^^'\ = Q dw dt\dw) dO\dw') de-\dw")

du dt\duj de\du J

(7.4.89)

Application of these equations gives the following equations of motion

for the cylindrical beam of unit width.

FI S

+ (72-^) {u'^w)^crw=0, (7.4. 90)

p{u-h2aw) + {u"^w') 4- ^^w' = 0. (7.4.91)

Working along similar lines Tielking obtained almost the same equations.

However in the dynamic part of the second equation the additional term — pil'^u appears in his theory, while Tielking did not consider So and ct. The remaining analysis will be simplified by using the concept of

inextensibility of the cylindrical beam (c? 00). It is believed that this approximation is particularly good when the consideration is restricted to radial ply belted tires. Mathematically this may be accomplished by

putting:

u' =-w. (7.4.92)

We obtain by elimination of the terms with coefficient Cs in the equations

(7.4.90) and (7.4.91): EI

piib" -iv^ ^Clw - aV) + — (m;^' + 22^'^ + w") _ ^

^iv + + cr ) w" - ctw = 0. (7.4.93)

In an analogous way Bryan [31] derived essentially the same equation.

In his study of oscillatory motions it is assumed that the radial displacement w varies harmonically in the form:

w = A^m (sO^ (ot). (7.4.94)
738 THE TIRE AS A VEHICLE COMPONENT

Substitution in eq (7.4.93) yields

FJ

pis^o)' -co'- 405(0 + nV) + ^ (- 56 + - 52) - Q = 0 (7.4.95)

with the definition = - av +

^

5^5^ - 1)^ + 54 [s- +1 pr pr'

P we obtain

+ + (7.4.96)

^±0,. (7.4.97)

The possible solutions become herewith:

2^1 = ^1 Sin 56^+ , _, ? + Ct)^ 5^ 1 2m , . (^-^-^^^

W2 = A-z sin 50 H

„ , -, t — cot)

\ 5^ 4- i

In the particular case y4i = A2 = A we obtain for the solution w = W\ +1^2: w = 2A sin (^sO^ cos (7.4.99)

in which cb obviously represents the frequency of the vibration. The

mode number 5 denotes the number of periods along the tire circumference. The number of nodes is twice as much. The position of a node

e„ = ^

+ — {k=l,2, . . .) (7.4.100)

5- + 1 5 changes with respect to the wheel with an angular speed Of:

nf=-^0. (7.4.101)

5-^ + 1
TREAD-BAND FREE VIBRATIONS 739

For the observer stationary in space the nodes move with a different

velocity, O*, which obviously equals CLf + H:

n*=^f^a. (7.4.102)

For each number of periods s the frequency of the vibration may be

calculated with the aid of equation (7.4.96). The frequencies depend on

tire parameters and to a very small extent on the rotational speed Cl. The tension force So may be a function of Cl (cf. sec. 7.4.2 on standing

waves).

B(ihm, who did not consider a rotating tire, calculated the values of these parameters from the tire construction, geometry and material

properties. Bohm's equations differ in some respects from those derived

above [90, 91]. Bohm's equations, for the derivation of which we refer

to the original article [33], read:

piv-h^ {w'^^'^w") + (w' + m;) {w" + w)^crw = 0, (7.4.103)

pw + — {w"'-^w') --T, {u"^w')^ctu = 0. (7.4.104)

With the assumed solutions w = A sin ojt sin sO, u = B sin cot cos 5^ (7.4.105)

inserted in these equations the natural frequencies can be obtained for the various mode numbers s. The calculations were carried out by

Bohm for a 135-13 radial ply Michelin X tire with inflation pressure

p, = 1.25 bar. The following calculated values hold for the total width

26(= 0.075 m.): p = 1.25 kg/m., £7 = 0.7 Nm.\cs =59 X 10*N, So = 1920 N,

c,= 75 X 104 N/m.2, ct = 12.8 X 10^ N/m.^, r = 0.273 m.

In table 7.4.2 both the calculated and measured frequencies n = a)/27r (Hz) are shown for mode numbers 0—5. For comparison we have added

the results obtained with the equation earlier derived (7.4.96) for 11 = 0, so that dj — o). For each mode number a second natural frequency exists which, according to the theory, is 4 to 8 times higher than the lower one shown in table 7.4.2. The frequency 45.5 Hz at mode number 5 = 0

corresponds to the mode shape showing pure tangential torsion of the

tread band with respect to the wheel rim. The second frequency at

s = 0, which occurs at 376 Hz exhibits purely uniform radial displacements of the tread band. The experiments were carried out by applying

a radial excitation force so that the low frequency zero-mode natural

frequency was not detected.

In figure 7.4.24 the theoretical and measured mode shapes are pre- sented. The experimental frequency response characteristic for the

radial displacement amplitude A, with respect to the radially applied
740 THE TIRE AS A VEHICLE COMPONENT

THEORY

Figure 7.4.24. Theoretical and experimental mode shapes and measured frequency

response characteristic of the radial tread band deflections with respect to a radial excita- tion force for a radial-ply Michelin X 135-13 tire with pi = 1.25 bar (from Bohm [33]).
TREAD-BAND FREE VIBRATIONS 741

excitation force amplitude (constant) has also been indicated. The cor- respondence between theory and experiment is good enough for technical purposes. The differences between eqs (7.4.90-91) and (7.4.103-104) turn out to be of no importance in the range of mode numbers investigated.

Also, it appears that the concept of inextensibility used in eq (7.4.93)

is acceptable for this tire. From the calculations it appears that the

influence of the bending stiffness EI starts at higher values of 5. At

5 = 5 the influence of EI on co amounts approximately 2 percent. We shall see that in the study of standing waves with relatively small wave

lengths the influence of the bending stiffness is appreciable.

Table 7.4.2. Natural frequencies of symmetric {plane) tread band vibrations

5 = 0 5= 1 5 = 2 5 = 3 5 = 4 5 = 5 Measured n(Hz) 83 98.5 115 136 158 Calculated n(Hz)

(eqs (7.4.103-104)).. 45.5 83.7 105.5 119 134 150 Calculated n(Hz)

(eq (7.4.96)) 45.5 87 109 123 138 154

Lateral vibrations

For the same 135-15 Michelin X tire Bohm also studied the lateral vibrations. We shall restrict ourselves here to a short presentation

of his theoretical and experimental results. For the derivation of the

equations we refer to the original paper [33]. Again two degrees of freedom are considered for an element of the

tire tread-band. The variables are: the lateral deflection v and the torsion

angle /3 about a tangent to the peripheral line. These so-called anti- symmetric variables are not coupled with the symmetric variables u and

w. With the assumed solution

1; = ^ sin (t»^ sin 5^ and f3 — B sin cut cos sd (7.4.106)

the foUowing eigenvalue determinant is obtained:

in which So = 1920 N (total tension force), GIp = 3.2 Nm.^ (torsional stiff- ness of tread band), EIz = 150 Nm.^ (bending stiffness of tread-band about
742 THE TIRE AS A VEHICLE COMPONENT

wheel radius), q= 12.8X10'* N/m.^ (tangential foundation stiffness),

Cc= 10^ N/m.- (lateral foundation stiffness), Cr= 1050 N.m/rad.m. (tor- sional foundation stiffness), p/ = 1.25 X 105N/m.2,p = 1.56 kg/m., r = 0.273 m., 26 = 0.075 m. (width of carcass breaker), 6* = 0.056 m. (reduced

width), r= 0.003 m. (side wall thickness). In table 7.4.3 the calculated and measured natural frequencies are presented for mode numbers 5 = 0-4. Figure 7.4.25 shows the measured

mode shapes of the lateral deflection and in addition the frequency

response characteristics of the lateral tread deflection with respect to an excitation force acting upon the tread-band in purely lateral direction.

Table 7.4.3. Natural frequencies of anti-symmetric {lateral) tread band vibrations

5 = 0 5=1 5 = 2 5=3 5 = 4 Measured n(Hz) 39 44 76 114 149 Calculated n(Hz) 40.0 42.7 64.2 110 162
STANDING WAVES 743 Bohm found the mechanical characteristics of the tire by working

backwards from the measured spectra of natural frequencies, and

found good agreement with the above values which were calculated from

tire dimensions and material properties. B<">hm states that the backward

calculation of physical values is a powerful method, since the measured

natural frequencies describe the system more precisely than do static deformation experiments.

Standing Waves

When rolling at high speed, waves are formed on a tire behind the area

of contact with the road. The repeated deformation caused by the wave

process results in a considerable heat buildup which reduces the strength

of the tire and may lead to its ultimate destruction.

Figure 7.4.26. Standing wave formation at high speeds of travel (from B. Nylon S., Great Britain).

Because these waves present a stationary appearance to an observer

they have been called standing waves. Figure 7.4.26 gives a very good

illustration of such a stationary wave deformation. Gardner and Worswick

[37] published considerable experimental information on this phenomenon. Turner also [38] provides interesting theoretical and experimental

information on amongst other things the power consumption owing to standing waves. More recently Drozdov, et al. [39], Togo [40], Bohm

[33] and Fiala [34] presented theories on standing waves. Ames [41]

prepared a literature survey in which mention has been made of other

Russian and Japanese work. Ames proposes the introduction of nonlinear elements in the dynamics of the tread-band and side wall motion.

Akasaka and Yamagishi [42] studied the standing waves in the shell wall of a running tire by considering the tire as a cord reinforced toroidal
744 THE TIRE AS A VEHICLE COMPONENT

membrane shell with elliptical cross section. The cords have been as- sumed to be inextensible, the flexural rigidity has been neglected, and

tangential displacements have been neglected. Solutions have been

obtained by means of Galerkin's method. In the theories to be presented

below we shall restrict ourselves to shells of a circular cylindrical shape.

Tangential displacements will also be neglected here.

Membrane theory

The equation of the transverse motion of a membrance stretched

in longitudinal directions reads:

where p represents the mass per unit length, S the tension force, w the transverse displacement, x the longitudinal coordinate and t the time.

After a disturbance a wave is formed which propagates with a velocity

(7.4.109)

It is expected that when a normal load is applied which travels with re- spect to the membrane with a speed equal to the above propagation

velocity a critical situation occurs. Togo [40] has investigated this problem for an elastically supported membrane and found that for speeds

in excess of this critical value a standing wave is formed behind the

point of loading. In front of this point the displacement vanishes.

We shall examine the problem of a circular membrane or string

under tension, radially supported by an elastic foundation which may

show some viscous damping. The tension is supplied by the inflation pressure. This model tire rolls over a perfectly smooth horizontal surface

and is loaded vertically. We introduce the quantities: r=tire radius,

n rotational speed; F— Or^speed of travel; JF= normal load; pi=\wflation pressure; Cr = radial stiffness of foundation per unit length;

A:,- = coefficient of radial damping per unit length; Cs = tensile carcass

stiffness of unit length ( = average elastic modulus times tread-band

cross section); ^ = angular coordinate with respect to an axes-system

fixed to the wheel; it; = outward displacement of rotating membrane

due to W\ 17 = percentage of centrifugal force restored by radial forces;

1 — 17 = percentage of centrifugal force restored by tangential tension

forces; So = tension force in nonrotating tire; 26 = effective tread-band

width.

It is assumed that only radial displacements occur. In figure 7.4.27

forces acting on a membrane element of unit width are indicated. The

following partial differential equation applies for the radial displacements w {0, t) due to the load W in zones outside of the contact area.
STANDING WAVES

0

745 (C^W + k.dw) rde

2bpj (r4W)d9

(1-

. rpder(r+w)n'_^l ^ dt •' S.S.*(1-ii)r'pn^Csy

Figure 7.4.27. Forces acting on membrane element.

Consequently, the growth of tire radius due to centrifugal forces is not included in w.

where (7.4.110)

(7.4.111)

This equation corresponds to eq (7.4.90) for u = 0 and EI = 0 but with

the additional tension force due to Cl^. The solution of this equation

represents the tire deformation seen with respect to a rotating coordinate

system. For the description of the standing wave phenomenon we shall adopt coordinates with respect to a system fixed in space. The angle

(/) is introduced indicating the position of a tread element with respect

to the vertical through the wheel axis (cf. fig. 7.4.27),

cl) = 27r-nt-d

Equation (7.4.110) then assumes the following form:

(7.4.112)

d(t>'-

-kril |^-2pa f^+ {c'-pa^)w = 0. (7.4.113)

d(p dtdcp In case of stationary appearance of the radial deformations all deriva-
746 THE TIRE AS A VEHICLE COMPONENT

tives with respect to time vanish. We obtain the ordinary differential equation:

which could have been obtained immediately from (7.4.110) by putting

dl dt = — Cldldcf) and dl d6 = — d/dcj), which transformations are valid in case of deformations being stationary with respect to a coordinate sys- tem fixed in space. The general solution of (7.4.114) reads:

w = Cie^i'^-\-C2e^2'i> (7.4.115) in which the roots of the characteristic equation become:

^1,2 = T-^ ^-^

(7.4.116)

Comparison with existing theories of Bohm [33], Fiala [34], and Togo

[40], the latter two being reduced here to the case without bending

stiffness of the tread-band, reveals that their results deviate from (7.4.116)

in the following respects. Fiala and Togo neglect the effect of damping

(A:r = 0), Togo and Bohm assume that the centrifugal force does not

influence the tension force S (17 = !), Fiala assumes the opposite (17 = 0) and Togo neglects pO^ with respect to c'. For small values of k,- the solution (7.4.116) may be written as:

'So r.A~ So (7.4.117)

These roots become complex and consequently the solution (7.4.115)

becomes oscillatory when fi satisfies the following condition

ai<a<n2 (7.4. iis)

where Hi and 02 are critical values of rotational speed expressed by:

a?=^, ai=-- (7.4.119)

iqpr^ p

Before discussing the solutions in greater detail, we shall first examine

the order of magnitude of Hi and ^2 according to a number of known

theories. For a 135-13 Michelin radial ply tire Bohm obtains by experimental means the following physical values: So= 1920 N., p= 1.56 kg/m..
STANDING WAVES 747

Cs. = 59xl0^ N., c,. = 75Xl04 N/m.^, 2p,fe - W N/m. (p, = 1.25 bar), r= 0.273 m. We obtain with 17 = 1:

fli = 1285-1 ( Fi = 126 km/hr),

02 = 23505-1 (F2 = 2300 km/hr.).

For a Bridgestone 165^00 4 PR diagonal ply tire investigated by Togo

(to be treated later on in greater detail) the following theoretical values can be derived c'l2b = SX 10' Nfm.^ ( - 5 kgf/cm.-^), Sol2r^b= 136 X 10=^ N/m.-^ ( - 0.0136 kgf/cm.3), = 21.8 kg/m.^ ( - 2.22 X 10-^ kgf-s^/cm.-^), p/= 13X10^ N/m.- (=1.3 bar), r=0.34 m., which values resuh with

17 = 1 in.: a, =785-1 (Fi=96 km/hr.), n. = 15005-1 {V2 = 1830 km/hr.).

For a 165-400 radial ply tire we obtain, according to the formulation

of Fiala [34], c' = 35x 10^ N/m.^, p = 2.1 kg/m., r = 0.33 m. With Fiala's

assumption 17 = 0 the following results are obtained for Pi = l bar:

^2 = 4105-1 (^2 = 480 km/hr.).

The three theories deviate considerably from each other. Apart from

the differences in iq we have noted, amongst other things, that Fiala

took Cs = 0, which, in particular for a radial tire, is believed to be an

unacceptable assumption. It causes the low value of CI2 obtained with

Fiala's theory.

We continue now the discussion of the solution and write for (7.4.117),

with the aid of (7.4.119):

2r^p(n?-a2)- Vt? nj-a^' (7.4.120)

Two different classes of values wiU be considered. The first category

refers to the theories of Togo and Bohm which hold for Clz > Hi. We

obtain for the speed ranges:

a. n < Oi: Real roots Xi and k-z. Assume k,- small enough so that for the Cl considered one positive (Xi) and one negative root (X2) exist. With the requirements that for |(/)|^oo the deflection w remains finite
748 THE TIRE AS A VEHICLE COMPONENT

the solutions (7.4.115) become for the regions outside of the contact area with length 2a:

(f)< — j: z^ = Cie^i^

(7.4.121)

indicating an almost symmetrical monotonically damped deflection. In the neighborhood of fii, the deflection at the rear must vanish due

to the i)resence of kr{\i < Xi < 0, Ci = 0)

,

b. Hi < n < 112: Complex roots A.i,2 = a±i^ with a and P real positive

numbers. With the same conditions (|(/)| ^ oo, w finite) we obtain in this range of rotational speed:

(/)<--: z^^Cie^'^sin (j8(/) + i//i) ^

) (7.4.122)

(/)> ^: m; = C2«*^ sin (/3(/) + i//2)=0, (C2 = 0)

Consequently, in this speed range a standing wave will be formed

behind the contact area. The wave damps out due to the presence of the

radial damping coefficient /c,. In front of the contact area the solution

would indicate an exponentially increasing deflection w which is in disagreement with boundary conditions, so that the constant of integration

Cz must vanish, resulting in a front portion of the tire tread-band of which the deflection w vanishes. Figure 7.4.26 shows a beautiful picture

of the actual deformation when operating in excess of the critical speed.

The theoretical wave length of the standing wave varies with speed

according to the following formula:

2Trr I ft- — Or

^ = ^= 2-V.^- (7.4.123)

Figure 7.4.28 gives a graphical representation of the variation of / with n. It should be noted that the wavelength increases with ft from zero to

infinity, at which ft2 has been reached.

c. ft > ft2: This is a fictitious case as the tire will be destroyed long

before this very high speed range is reached. The theory predicts two

real roots for the stationary solution. A stationary situation, however,

will never be reached since with the unloaded undisturbed tire, where

derivatives with respect to 6 vanish, according to eq (7.4.110) an unstable

situation occurs. The last coefficient becomes negative in this speed

range and the tire explodes.
STANDING WAVES 749

0

Fi(;L RE 7.4.28. Wavelength I of the membrane type tire model as a function of speed V

for the case that 09 > O,.

In the second category of values Cl\ > Clz which refers to the theory

of Fiala, we distinguish:

a. 0 < < fli): Two real roots exist resulting again in a practically

symmetric damped deflection on both sides of the contact area.

b. flz < O, < Cti: When a stationary deflection would have been attained,

two complex roots A.i, 2 = — a' ± i/^' arise with a' and /3' representing

positive real numbers. We find now a damped standing wave in front

of the contact area (not recognized by Fiala) and vanishing deflections

at the rear. The wavelength appears to decrease with increasing fl until

Cli is attained, where the wavelength vanishes. This decrease corre- sponds to theoretical findings of Fiala who did not neglect the bending

stiffness of the carcass tread band. This, however, does not influence

the value of the critical rotational speed 112. Also when the bending

stiffness is introduced, the tire is theoretically expected to explode in excess of 02 so that Fiala's theory does not seem to have practical

value. The third speed range, Cl > Oi, is of no interest to our analysis.

Influence offlexural rigidity.

The introduction of bending stiffness gives rise to additional terms in the equation of motion. Consideration of equation (7.4.90) with it = 0,

derived in the preceding section, shows the additions to be introduced.

In case of deformations stationary with respect to an axes-system

fixed in space, we may write dldd = — d/dcj). Equation (7.4.114) now

becomes, when damping is omitted:

This equation differs in some respects (EI (w"" -\- w") instead of

EI {w"" + 2w" -\- w)) with respect to the equations of other authors
750 THE TIRE AS A VEHICLE COMPONENT

(Togo, Bohm, Drozdov, Fiala). We shall adopt the simplification in- troduced by Togo, viz omission of pO- relatively to c', and follow the

line of his theory [40]. The following quantities are introduced for the

sake of abbreviation:= 1, = a| = (7.4.125)

With Togo's simplification we obtain for the solutions it; = exp (kcj)) the

following characteristic equation:

v'^k^ + (2i/2 - + ) + ai = 0 (7.4. 126)

with four roots determined by: = ^ (7.4.127)

The discriminant vanishes at two speeds Ql and Clu:

ai=ci'i^2va2-2v\

(7.4.128)

These speeds represent the lower and upper critical rotational speeds

for the stretched beam type of model, whereas Hi is the critical rota- tional speed for the stretched string type model. The solution (7.4.42)

may be divided into three speed ranges 1, 2 and 3 also indicated in figure

7.4.29:

1. n < Cti/. Four real roots. The deflection curves are combinations of two damped exponential curves.

2. fl/ < n < fil l Four complex roots. The deflection curves are of a damped oscillatory character.

3. Of < O: Four imaginary roots. Standing waves are formed composed

of two different modes.

As an effect of the bending stiffness, a range of rotational speed

arises in which damped waves are formed in front of and behind the contact area. This speed range fi/ < ft < Clc is situated on both sides

of the value V(fl^ — v'^), which represents the critical rotational speed

arising when in eq (7.4.124) the fourth derivative is omitted.

Above the upper critical speed Cli the standing wave appears only

behind the loading point. At fl = Cii the standing wave already has a
STANDING WAVES 751

Q ranges

standing

waves

(two modes)

Im

Q ra nge 2

dam ped

standing wave

Q range 1 ex po - nentially

damped

fi=VrO

Fi(;l RE 7.4.29. Solutions of X- accordinfi to eg 7.4.27.

finite wavelength. Beyond this critical speed two wavelengths arise, one of which increases and the other decreases with increasing fl, as may be deduced from figure 7.4.29. Turner [38] reported that such a combination of wavelengths indeed can occur in practice. It is expected,

however, that in most cases the shorter waves are suppressed as a

result of material damping.

Example of calculation of the critical speed and wavelength for a bias

ply tire. The following equation which corresponds to eq (7.4.124) seems to have been used by Drozdov and others [39] in their calculations:

mo-B) ^+ (a-P^)w = 0. (7.4.129)

d^^ \p/n / d^' \ pirj

The expressions for the parameters A, B, C and D used by Drozdov may

be found in the original paper. The quantity n denotes the radius of the

tire cross section (half tire width) and ^ = xlr\ represents the dimensionless tangential coordinate {x = 4>r) . Togo [40] adopted the same equations but used different expressions

for A, B, C and D. We shall present here the theoretical and experi-
752 THE TIRE AS A VEHICLE COMPONENT

mental results of Togo for a bias ply Bridgestone tire with the following

data (1 kgf. - 10 N): Tire 165 X 400 4 Pr SAD; outer diameter 2r= 68 cm.;

tire width 2ri = l7 cm.; inflation pressure Pi = 1.3 kgf/cm.^, tire load

300 kgf.

The tire characteristic values to be used in the analysis are: crown

angle = average cord angle measured from the wheel center plane

q: = 40°; number of cord plies n = 4; number of cords at tire crown per

ply and per cm. width i = 9 cm.~^; average thickness of tread band

h = hi-\- h2 = l.9 cm.; distance from neutral ply to outer surface hi = 1.7 cm.; distance from neutral ply to inner surface h2 = 0.2 cm.; elasticity

modulus of rubber Eh = 111 kgf/cm.'-; Poisson's ratio of rubber fji = 0.2

(0.5 according to others); mass of tire per unit area p = 2.22XlO~^

kgf • s'-Zcm."^; tensile elasticity modulus of a cord £'c = 50kgf.; flexural

rigidity of the ply layers per unit width Ac=iEc ^ y'^ cos^ a =7.75

n kgf -cm. (y= distance of ply from neutral ply); flexural rigidity of rubber

tread-band ^« = V3 Eh (/i? + /if )/(1 - /^t^) = 200 kgf -cm.; it has been

assumed furthermore that the centrifugal force does not influence the

tension S of this bias ply tire so that 7] = !. From Togo's analysis it can be deduced that the dimensionless parameters A, B, C and D of eq (7.4.129) are expressed as follows (the rela- tions with parameters of eq (7.4.124) are given in between brackets):

. ( ^

c'r^^EI\ Enhr' ( 1 + tan^ a - 2/Lt tan^ a ) \

) (7.4.130)

\ 2bpir'^) Pi rf I D

rSubstitution of the solution w = e'^^, /3 real, yields relations which hold

in the speed range Cl > Clu where standing waves are formed behind

the contact area. The dimensionless reduced frequency (3 is inversely

proportional to the wavelength /,^ = ^. (7.4.131)

We obtain the equation:

Cf3' - D-B^ f3' +(^A- = 0 (7.4.132)
STANDING WAVES 753

with the solution:

=^D-B±J(^D-bY-^c(a-^). (7.4.133)

Again, two wavelengths occur theoretically. When the discriminant

vanishes, the critical value of speed Vr is attained, above which sinusoidal

solutions of ic exist. At this point of transition the values of F, (3 and / will be provided with the index U. We derive:

in which

(7.4.134)

(7.4.135)

Expression (7.4.135) is the same as the one used by Drozdov. With the

tire data listed above we obtain: Vr = 201 km/hr. and U = 16 cm. With

eqs (7.4.127-128) we find practically the same result for Vl {= rCli ) and

/r, the lower critical speed F/. becomes imaginary. For Fi = rHi, which

is the critical speed in case of the absence of bending stiffness, we

obtain: Fi = 96 km/hr. Considering equation (7.4.124) once again, it may be concluded that Togo's simplification (pO- <^ c') and the omission

in the coefficients of w" and w of El/r"^ (Bohm, Fiala, Togo, Drozdov) are permissible since their effects appear to be extremely small. Togo has carried out a number of test runs on a steel drum using the

tire described above. Wavelength and amplitude of the standing wave

have been measured from photographs. Figure 7.4.30 shows the meas0 ^0 80 120 160 200 V (km/h)

Figure 7.4.30. Experimentally obtained relation between wavelength I of standing wave and speed V for a Bridgestone cross-j>ly tire 165-400 {}^i = 1.3 bar) on a steel drum

(Togo [40\).
754 THE TIRE AS A VEHICLE COMPONENT

ured relation of wavelength versus speed. At the speed 140 km/hr., Togo

did not observe a wavy deformation but the endurance test was ended

after four minutes due to ply separation.

Comparison with theoretical results suggests that the theory only

provides an approximate insight into the problem. Drozdov reported

similar discrepancies between theory and experiment. Attempts have

been made to obtain a better agreement through the introduction of an

effective (smaller) tread thickness for the calculation of the flexural

rigidity. This may be supported by figure 7.4.26 (rear view) which clearly

shows the manner in which the tread rubber deforms. If we assume that the effective thickness is half of the geometric thickness h, then the

quantity Ac + Ar becomes approximately one-eighth of its originally

calculated value. The critical values become now: Vv = 150 km/hr. and

/t = 9.5 cm., which results are closer to the experimental values of

figure 7.4.30. From the analysis above and in particular from eq (7.4.134) with eq

(7.4.130) it may be concluded that the critical speed is shifted to higher

values when:

1. bending stiffness of tread EI is enlarged {n enlarged, ol reduced),

2. tension force in tread So is enlarged {pi enlarged),

3. radial tire stiffness Cr is enlarged (shape cross section, sec. 7.4.1).

4. tensile tread-band rigidity Cs is enlarged {a reduced),

5. percentage of the centrifugal force restored by radial forces

J) is reduced (a reduced),

6. mass density p is reduced.

The influences of inflation pressure pi, crown angle a and number of

plies n are in accordance with experiments of Gardner and Worswick

[37] (bias ply). Experiments of Curtiss show that unfavorable interaction

between parameters may occur. From measured rolling resistance curves

(cf. fig. 7.2.60) it is seen that radial and bias ply tires behave differently

when the mass of the tread-band is reduced through removal of the

tread. As expected, the bias ply tire then shows an increase in critical speed above which standing waves occur, causing a further progressive

rise of the roUing resistance with speed. For the radial tire, however,

it appears that the critical speed decreases, which, according to Curtiss,

may be explained by the loss in carcass sidewall rigidity due to a reduction in tension in the sidewall caused by the decrease in tread-imposed

centrifugal force. References

[1] Hadekel, R., The mechanical characteristics of pneumatic tires, S & T Memo, No.

10/52 (British Ministry of Supply, TPA 3/TIB, 1952).

[2] Weber^ G., Theorie des Reifens mit ihrer Auswirkung auf die Praxis bei hohen

Beanspruchungen. A.T.Z. 56 (12), 325 (1954).

[3] Marquard, E., Untersuchung iiber den Einfluss der Stossdampfer auf die Zwischen

Rad und Fahrbahn auftredenden senkrechten dynamischen Bodenkrafte, Deutsche

Kraftfahrtforschung 104, 32 (1957).

[4] Chiesa, A., and Tangorra,G.,The dynamic stiffness of tires. Revue Gen. du Caoutchouc

36(10), 1321 (1959).
REFERENCES 755

[5] Rasmussen, R. E., and Cortese, A. D., Dynamic spring rate performance of rolling

tires, SAE Paper No. 680408 (1968).

[6] Betz, E. R., Studying structure dynamics with the cadillac road simulator, SAE

Paper No. 660101 (1966).

[7] Dodge, R. N., Dynamic stiffness of a pneumatic tire model, SAE Paper No. 650491

(1965).

[8] Rotta, J., Zur Statik des Luftreifens, Ing. Archiv 1949, p. 129.

[9] Senger, G., Ueber dynamische Radlasten beim UeberroUen Kurzwelliger Unebenheiten

durch schwere Luftreifen, Deutsche Kraftfahrtforschung 187 (1967).

[10] Pacejka, H. B., Theoretische beschouwingen over het gedrag van luchtveren, De

Ingenieur 72(10), (maart 1960).

[llj Clark, S. K., An analytical model for lateral stiffness of a pneumatic tire. Paper 3-01,

FISITA, Barcelona, 1968.

[12] Tiemann, R., Aehnlichkeitsbeziehungen im Federungsverhalten von Luftreifen, A.T.Z.

65(4), 97 (1963).

[13] Smiley, R. F., and Horne, W. B., Mechanical properties of pneumatic tires with

special reference to modern aircraft tires, NASA Tech. Note 4110 (1958).

[14] Cough, V. E., Tires and air suspension. Advances in Automobile Engineering, Tidbury,

ed., (Pergamon Press, 1963).

[15] Julien, M. M. A., and Paulsen, M. J., Methode experimentale de mesure et definition du pouvoir absorbant du pneumatique, J. de la S.I. A., Janvier 1953, p. 33.

[16] Lippmann, S. A., Piccin, W. A., and Baker, T. P., Enveloping characteristics of truck

tires, SAE Trans. 74 (1966).

[17] Lippmann, S. A., and Nanny, J. D., A quantitative analysis of the enveloping forces

of passenger tires, SAE Paper No. 670174 (1967).

[18] Barson, C. W., James, D. H., and Morcombe, A. W., Some aspects of tire and vehicle

vibration testing, Proc. IMF 182,3B,32 (1967-68).

[19] Guslitzer, R. L., On the interaction of a motor-vehicle tire with an obstacle. Translation 817, Rubber Res. Ass. British Rubber Manufacturers, 1957.

[20] Hey, K. F., Untersuchungen von Langskraften zwischen Reifen und Fahrbahn beim

Ueberfahren von Hindernissen, Diss. T. H. Braunschweig (1963).

[21] Yu Chen, Studies of the interfacial phenomena during braking. Ford Motor Tech. Report A-788, 30 (1958).

[22] Hiirz, E., Der Einfluss von Bremskraftreglern auf die Brems- und Fiihrungskraft eines gummibereiften Fahrzeugrads, Deutsche Kraftfahrtforschung 195 (1968).

[23] Holmes, K. E., and Stone, R. D., Tire forces as functions of cornering and braking

slip on wet road surfaces, Proc. Symp. on Handling of Vehicles under Emergency

Conditions, IME, Auto. Div. (1969).

[24] Reynolds, O., On rolling friction, Phil. Trans. Roy. Soc. 166, 155 (1876).

[25] Fromm, H., Berechnung des Schlupfes beim RoUen deformierbarer Scheiben, ZAMM

7,27(1927).

[26] Julien, M., L'envirage et la tenue de route, S.LA. (April 1937).

[27] Chiesa, A., Obert, L., and Tamburini, L., High-frequency vibrations in tires under

vertical perturbation and their transmission to the wheels. Auto. Engr., 520 (1964).

[28] Chiesa, A., Vibrational performance differences between tires with cross-biased plies and radial pHes, SAE Paper No. 990B (1965).

[29] Barson, C. W., Cough, V. E., Hutchinson, J. C, and James, D. H., Tire and vehicle

vibration, Proc. IME, Auto. Div., 1964-65, p. 213.

[30] Strutt, J. W. (Lord Rayleigh), The Theory of Sound, Vol. 1 , London, 1878, p. 332.

[31] Bryan, J. W., On the beats in the vibrations of a revolving cylinder or bell, Proc. Cambr. Phil. Soc. 7, 101 (1890).

[32] Tielking, J. T., Plane vibration characteristics of a pneumatic tire model, SAE Paper

No. 650492 (1965).

[33] Biihm, F., Mechanik des Giirtelreifens, Ing. Archiv 35, 82 (1966).

[34] Fiala, E., and WiUomeit, H. P., Radiale Schwingungen von Giirtel-Radialreifen, A.T.Z. 68 (2), 33 (1966).

[35] Federhofer, K., Zur Schwingzahlberechnung des diinnwandigen Hohlenreifens, Ing. Archiv 10-11, 125 (1939-1940).

[36] Fliigge, W., Handbook of Engineering Mechanics (McGraw-Hill, 1962), pp. 16-2, 24-6.

[37] Gardner, E. R., and Worswick, T., Behaviour of tyres at high speed, Trans. I.R.I. 27, 127 (1951).
756 THE TIRE AS A VEHICLE COMPONENT

[38] Turner, D. M., Wave phenomena in tyres at high speed. Third Rubber Tech. Conf. London, Trans. I.R.I. (June 1954).

[39] Drozdov, V. K., e.a., Formation of stationary waves on pneumatic tires at high rolling speeds, Sov. Rubber Tech. 19(12), 36 (1960).

[40] Togo, K., Standing Wave on Pneumatic Tire at High Speed, Memoirs of the Defense Academy, Japan, Vol. IV, No. 1 (1964), p. 43.

[41] Ames, W. F., Wave phenomena in tires. University of Iowa, Tech. Report 1 (1967).

[42] Akasaka, T., and Yamagishi, K., On the standing waves in the shell wall of running

tyre. Trans. Japan Soc. Aer. Space Sc. 11(18), 12 (1968).

[43] Clark, S. K., The rolling tire under load, SAE Paper No. 650493 (1965).
YAW AND CAMBER ANALYSIS 757

7.5. Yaw and Camber Analysis

H. B. Pacejka

The so-called antisymmetric (cornering and camber) behavior of

tires is of importance for the investigation of the maneuverability and

stability of automobiles. In particular, the stationary (steady state)

characteristics are employed in such investigations. However, the

knowledge of nonsteady state tire properties is necessary for the in- vestigation of transient motions or of parasitic motions such as shimmy.

The Hterature provides a vast amount of experimental data, which

in particular covers the stationary cornering properties of tires. Cornering characteristics of automobile tires are published, amongst others,

by Joy and Hartley [1],^ Gauss and Wolff [2], Fonda [3], Freudenstein [4] (truck tires), Nordeen and Cortese [5], Henker [6], and Fonda and Radt

[7]. With special reference to aircraft tires. Smiley and Horne [8] have

presented a systematic survey of mechanical tire properties. Hadekel

[9] (1952) and Smiley [10] (1956) gave critical outlines and extensions

of existing theories for tire motions and wheel shimmy.

In the theory of tire mechanics relevant to the type of motion we are considering in this section, the road is assumed to be a smooth level boundary surface of an undeformable half space, while the tire is repre- sented by some elastic model. The literature provides tire models of various degrees of complexity. For all these models the following fundamental observations are applicable.

When the tire moves over the road, horizontal deformations will generally occur over and above the deformation due to static vertical

load. When the wheel moves in such a way that the contact points of an imaginary tire, which differs from the real tire only in that it does

not show horizontal deformations, do not move with respect to the road, we speak of pure rolling. When all the contact points of that imaginary

tire show the same relative velocity with respect to the road, we speak

of longitudinal (fore and aft) shp or creep when this relative velocity

and the rolling velocity have the same directions. We speak of lateral

(side) slip or drift when the relative velocity is directed perpendicular

to the rolling velocity. The angle between wheel center plane (direction

of rolling) and the vector of the velocity of the wheel center is called the

slip angle. When the wheel rotates about a vertical axis through the

wheel center without showing longitudinal or lateral slip, we speak of pure spin. When the wheel plane is tilted with respect to the vertical plane, the tire is said to show a camber or inclination angle.

A real tire will show additional horizontal deformation. In the case

of dry-frictional contact, the additional horizontal deformations may

' Fi(iures in brackets indicate the literature references at the end of this section.
758 THE TIRE AS A VEHICLE COMPONENT

cause regions of adhesion as well as regions of sliding. In the following,

the terms slip and sliding will always be used in the sense as expressed

above.

In this introduction mention must be made of the work which has

been performed on the problem of the steady state rolling and slipping

motion of two homogeneous bodies pressed to each other. In most of these theories equal elastic constants of the two bodies in contact are assumed (steel-on-steel problem). The two-dimensional theories of Carter and of Fromm were followed by three-dimensional theories of Johnson, De Pater, Kalker and Nayak and Paul (see [19, 21, and 22] for references).

General differential equations of a rolling and slipping body.

Consider a rotationally symmetric elastic body representing a wheel

and tire rolling over a smooth horizontal surface representing the road. Fixed to the road a coordinate system (0, x, y, z) is assumed, of which the

X- and y-axes lie in the road surface and the z-axis points downwards

(see fig. 7.5.1). Another coordinate system (C, x, y, z) is introduced of which the axes x and y lie in the {x, 0, y) plane and z points downwards.

The system moves with respect to the fixed system in such a way

that the x-axis lies in the wheel center plane and the y-axis forms the

vertical projection of the wheel axis. The body (tire) is deformed vertiFlGURE 7.5.1. Top-view of contact area showing position with respect to coordinate system

fixed in space (x,0,y) and deformations (u, v) with respect to moving system {x, C, y).
j YAW AND CAMBER ANALYSIS 759 cally so that a finite contact area is present. The center C travels with a constant speed V over the (Jc, 0, y) plane. The traveled distance s equals

s=Vt (7.5.1) where t denotes the time. The tangent to the orbit of C makes an angle /3 with the fixed Jc-axis. With respect to this tangent the jc-axis is rotated

with an angle a, defined as the slip angle. The angular deviation of the

wheel plane with respect to the x-axis (yaw angle) is denoted by

i// = /3 + q: (7.5.2) For small values of p the following relation with y, the lateral displacement of C, holds

/3 =f

(7.5.3) The horizontal displacements of a contact point with respect to its position in the horizontal undeformed situation with coordinates {x, y)

are indicated by u and v in x- and y- direction respectively. The dis- placements are functions of y and the independent variable s or t. The position in space of a material point of the rolling and slipping

body in contact with the road (cf. fig. 7.5.1) is indicated by the vector p = s^q

where s indicates the position of the contact center C in space and q the

position of the material point with respect to the moving system (jc, C, y) , q= {x-hu)ex+ {y-\-v)ey

with €x and €y representing the unit vectors. The vector of the sliding

velocity of the material point relative to the road becomes {x = dx/dt,

etc.):Vs =p = s-\-q= V-\- {x-\- u)ex-\- (y + i;)ey+ (jc + w)ey— {y-\-v)ex}

in which V denotes the vector of the speed of travel of the contact

center C. We introduce furthermore:

Vr = — {xej.-\-yey)

denoting the vector of the rolling velocity and

representing the vector of the slip or creep velocity of the body with

respect to the road. We reahze furthermore that
760 THE TIRE AS A VEHICLE COMPONENT

. _du _dii dx_^du dy _^du

dt dx dt by dt dt

'

. _dv__dv_dj^_^dv dy _^dv

dt dx dt dy dt dt The sliding velocity becomes herewith in vectorial form:

Henceforth we shall neglect the terms v • dip/dt and u • difj/dt, as these

are small of the second order. The components in x- and y-direction of the sliding velocity Vs of a point of a rolling body in the contact area with respect to the road are

in general:

du du du K.-y<o.-K.--V,,-^-,

(7.5.4)

17 ^ T/ T/ dv

ysy = Vcy + XOJz " Vrx T Vry T h —

dx dy dt where {Vex, Vcy) denotes the vector of the creep or slip velocity of the

tire, which is the sliding velocity of the point C of the horizontally un- deformed imaginary tire, which coincides with the center C at the in- stant considered; [Vrx, Vry) is the vector of the rolling velocity with

which point C moves relative to C. Moreover, oiz denotes the angular

velocity of the system (C, x, y, z) about the 2-axis (yaw velocity):

_dilj_d/B da

f7 r c^

0)z — —r — —r-r—r (7.5.5) dt dt dt ' ^ For better understanding, figure 7.5.1 illustrates the way in which

Vsy arises for the case Vry = 0. We will restrict ourselves to small values

of lateral slip and assume \a\ < 1. For the system under consideration,

i.e., the tire, where only rolling in the jc-direction occurs, the following

relations hold: Vrx=V and Vry = 0, (7.5.6)

when in addition the longitudinal creep velocity is taken equal to zero, which is approximately the case when no driving or braking forces are

applied. We obtain for the creep or slip velocities: Fca-==0 and Vcy = -Va (7.5.7)
YAW AND CAMBER ANALYSIS 761 We introduce the variable (/> denoting the spin:

(D,_d4f

(7.5.8)

The latter part of this relation holds owing to eqs (7.5.1) and (7.5.5). We finally obtain the following expressions for the sliding velocities of a point with coordinates {x, y):

dx ds VsylV^-a + x(\) dv ^ dv

dx ds

(7.5.9)

When the vector of the pressure exerted in the positive direction by

the tire upon the road is denoted by {px.Py, Pz)-> we obtain the following

relations for the case with finite friction coefficient ^t. In the adhesion

region, defined as the area where no sliding occurs (Vsx=Vsy = 0), the relations

du du

dx ds dv dvdx ds = — a-^ X(i),

%-\-pl< fxp.

(7.5.10)

hold, and in a sUding region the relations (7.5.9). For the pressure we

obtain in vectorial form:

(Px, Py)= lxpz(Vsx, Vsy)IVs

here

(7.5.11)

(7.5.12)

the velocity components Vsx, Vsy being determined by (7.5.9). For the case where only lateral slip occurs {(f) = 0), and. in addition,

Px = Vsx^ 0 throughout the contact area (which may occur with simplified systems to be treated later on), the relations (7.5.9-11) reduce to: dv

dx

dv

ds \Py\< f^Pz

dv dv — a —

dx ds

\Py\ = f^Pz Sgn* Vsy

V

'

in an adhesion region.

» in a sliding region.

(7.5.13)

(7.5.14)

"sgn signifies "sign of."
762 THE TIRE AS A VEHICLE COMPONENT

The equations above apply in general. Their solutions contain constants

of integration which depend on the construction of the tire of which an approximate physical description may be given. In case of a steady state motion, the partial derivatives with respect to the distance traveled s become zero {dvlds= du/ds = 0)

.

Tire models.

Many theories are known which describe the qualitative or quantitative behavior of the steady state or nonsteady state drifting tire. The

influence of camber has been studied only to a very limited extent. Due

to the immense complexity of the tire structure most theories are re- stricted to a qualitative description of tire behavior. Particularly in case

of the application of more advanced tire models, this qualitative picture can be adapted for quantitative use through fitting of the parameters.

This can be done by means of full scale tire experiments, either static or semi-static [4, 11, 23] or with the rolling tire [13]. The first method

leads to a greater insight into the problem, while the second achieves

higher accuracy since in that case the cornering characteristics which

are to be fitted are measured directly.

It should be pointed out that in order to avoid conceptual errors, which

may arise due to the use of oversimplified models, the development of advanced tire models firmly based on actual tire geometry and material

properties is of great importance. In this connection the work of Frank

[11, 12] should be mentioned.

A fundamental difference in structure is apparent between the tire models employed in steady state and nonsteady state tire theories. In most steady state theories, a model is used consisting of an elastic struc- ture (the carcass) provided with a great number of elastic tread elements

(see fig. 7.5.2). The tread elements contact the road surface in the contact area, where a region of sliding may occur when the adhesion limit

is locally exceeded. In most nonsteady state studies tread elements

have been omitted and, in addition, adhesion has been considered complete in the entire contact area. The use of an elastic continuous struc- ture representing the carcass is essential in nonsteady state tire theories.

0 X wheel plane

carcass

elastic foundation

tread rubber

y y Figure 7.5.2. Top-view of tire model in steady state rolling with constant slip angle a.
SIDE SLIP 763

7.5.1. Steady State Motion

Side Slip

Models of the carcass commonly encountered in the tire literature can either be of the beam type or of the stretched string type. The exact

representation of the carcass by a beam instead of a stretched string

is more difficult because of the fact that the differential equation for the

shape of the deformed peripheral line of the carcass becomes of the

fourth instead of the second order. For the study of steady state tire behavior, most authors therefore approximate the exact expressions for the lateral deformation of the beam.

As an extension of the model of Fromm and of Julien (cf. [9] for ref- erences) who did not consider carcass elasticity, Fiala [13] and Freudenstein [4] developed theories in which the carcass deformation has been

approximated with a symmetric parabola determined only by the lateral force. Bohm [15] and Borgmann [16] , the latter without tread elements,

use asymmetric approximate shapes determined by both the lateral force and the aHgning torque. Moreover, Bohm considers in his steady

state side sUp theory a two-dimensional contact area provided with a

finite number of tread elements which due to the width of the tread, will also deform in longitudinal direction under lateral carcass deformation.

As an additional complexity, Borgmann and Bohm both introduced a

coefficient of friction which is a function of the sliding velocity.

Frank [11] has carried out a thorough comparative investigation of the various one-dimensional models. He employed a general fourth order

differential equation with which stretched string, beam and stretched

beam tire models can be examined. He obtained the exact solution of the

stationary side sUp problem with the aid of a special analog computer

circuit. A correlation with Fourier components of the measured deformation of real tires revealed that the stretched string type of model is more suitable for the simulation of a bias ply tire, whereas the beam

model is more appropriate for the radial ply tire. It appears, furthermore,

that the use of a stretched string model requires a tension force which

is of the order of 25000 N, whereas measurements of Hinton [14] indi- cate that the tension force amounts to about 2000 N. This implies that apparently the stiffness of the carcass (shear and bending) is responsible

for the rest of the effective tension force. In reference [23] Savkoor enunciates his theory for the development

of a general mathematical model for the description of the lateral be- havior of the tire. He uses results of experiments carried out with a tire which roUs slowly at a constant slip angle.

Figure 7.5.3 (taken from [12]) presents the calculated characteristics

of several types of models. The parameters in cases a, b and c are chosen

in such a way as to give a best fit to experimental data for the cornering

force at small sHp angles. Curves d show the result when carcass elasticity has been neglected and only the flexibility of the tread elements

is taken into account (Fromm [9]). When the elastic constant of Fromm's

model is chosen in a similar way, no difference between Fromm's and

Fiala's results appear, since due to Fiala's approximations the coefficients
764 THE TIRE AS A VEHICLE COMPONENT

in the expressions for Fy{a) and Mz(oi) are equal to those obtained

directly by Fromm.

In the calculations for figure 7.5.3, Frank applied a vertical force dis- tribution Qzix), found from measurements, lying between a parabolic

and an elliptic shape. The positive aligning torque obtained at high

values of a (fig. 7.5.3) arose due to a slightly asymmetric shape of qz{x)

introduced in those calculations. The phenomenon that in practice

the aligning torque indeed varies in this way is probably due to a combination of several effects. Apart from the cause just mentioned above,

the rolling resistance force acting out of the wheel plane, due to the

lateral deformation, may contribute. Another important factor causing

the moment to become positive is the fact that the coefficient of fric- tion is not a constant but depends on the sliding velocity, the latter
SIDE SLIP 765

having its highest values in the real portion of the contact area. This

factor may also cause the slight drop in the Fy{a) curves as has sometimes been found experimentally at high values of a. The influence of

different but symmetric shapes for the vertical force distribution along

the jc-axis has been theoretically investigated by Borgmann [16]. He

finds that, especially for tires exhibiting a low carcass stiffness, the

influence of the pressure distribution is of importance and has, as may

be expected, particular effect on the aligning torque at higher values of the slip angle. Many authors adopt the parabolic distribution for pur- pose of mathematical simplicity (Fiala [13], Freudenstein [4], Bergman

Figure 7.5.3 shows that, when the model parameters are chosen

properly, the choice of the type of carcass model hardly influences

the results. For illustration, we shall present now the theory of steady

state side slip with the aid of the simple model of Fromm (cf. [9]), and

the more advanced model of the stretched string type with and without

tread elements (for details cf. Pacejka [19]). These two examples were

chosen for reasons of their connection with theories to be presented

later on concerning the influence of a driving or braking force and nonsteady state tire behavior, respectively.

Tire model with elastic tread elements and rigid carcass (Fromm).

The model to be treated first is shown in figure 7.5.4. The steady state

drifting tire shows a contact line which is straight and parallel to the

velocity vector V in the adhesion region, and curved in the sliding

region where the available side force becomes lower than the force

which would be required for the tips of the tread elements to follow the

straight line further. In the adhesion region the shape of the deformation is in accordance with the general equation (13). It is easy to prove

that at the leading edge the deformation of the tread elements vanishes.

Consequently the lateral deformation in the adhesion region reads,

when the drift angle a is assumed to be relatively small so that we can

[17]. Pacejka [18]).

Figure 7.5.4. Simple model of drifting tire (Fromm).
766 THE TIRE AS A VEHICLE COMPONENT

write tan a — a: v={a-x)a (7.5.15)

where a denotes half the contact length. In case of vanishing sliding, which will occur for 0 or for /x—> oo, expression (7.5.15) holds for the entire region of contact. Mter the

introduction of the total lateral stiffness Cp of all profile elements per unit

length of the assumedly rectangular contact area, the following expressions for the cornering force Fy and the aligning torque — are obtained:

Fy — Cp vdx = 2Cj,a^a

fa — Mz = — c,j vxdx = ^3 Cf,a^a

(7.5.16)

The cornering stiffnesses for the force and the moment consequently

become respectively:

CFa = {dFylda)a=() = 2cijaCMa = — idMzlda)a=o=ys c,,a^ (7.5.17)

We will consider now the case of a finite value of /x and a pressure

distribution which gradually drops to zero at both edges. For purpose

of simplicity we assume a parabolic distribution of the vertical force

per unit contact length as expressed by

(7.5.18)

where W represents the vertical wheel load (— — F^). The largest possible

side force distribution consequently reads:

9!,max = /xg. = =/4/xr(2^^) (7.5.19)

In figure 7.5.4 the maximum possible lateral deformation Vmax — Qy maJ^p

has been indicated. We introduce for the sake of abbreviation the fol- lowing tire model parameter

.^^/3(^) (7.5.20)

The distance from the leading edge to the point where the transition from

the adhesion to the sliding region occurs equals 2a\ and is determined
SIDE SLIP 767

by the nondimensional quantity A, which bears the following relation

to the slip angle a (assumed positive):

k=l-ea. (7.5.21)

From this equation the angle asi can be calculated, at which total sUding starts (X= 0): asi=ld. (7.5.22)

The force Fy and the moment M.z can now be derived easily as a function

of a. The results read:

Fy = ^JLW(l-k^)=fJilF{S^a-3{eay^{ear}

for a < asi for asi < a <

(7.5.23)

M, = ^JLWk^il-k)=fJiWa{ea-3(ear^S(ea

for a < asi

(Say}

M, = 0 for asi< a<

(7.5.24)

These relationships are shown graphically in figure 7.5.5. They cor- respond with curves c or in figure 7.5.3, but now for a symmetric

paraboUc pressure distribution. The pneumatic trail t, which indicates

the distance behind the contact center C where the resultant lateral force acts, becomes at vanishing slip angles:

(MA C.^,a y ^o = —{-Er] =73 a (7.5.25)

MnWa

mW

1

3

Figure 7.5.5 Cornering characteristics of the simple model with rigid carcass (Fromm).
768 THE TIRE AS A VEHICLE COMPONENT

This value is smaller than normally encountered. The introduction of an elastic carcass will, as we shall see, improve this quantitative aspect.

Another point in which the simple model deviates considerably from

experimental results is the effect of a variation of the vertical wheel load

W. With the assumption that a changes quadratically with W, it can

easily be shown the Fy and M will vary proportionally with W and W^l'^ respectively. As will appear later in this respect also the introduction of an elastic carcass improves agreement with experiment.

It should be noted that Fiala, who obtained the same expressions

(7.5.23-24) via approximation of his more complex relations derived

from a model with both beam and tread rubber, succeeded in finding

suitable values of his original model parameters (combinations of which

govern the parameter comparable with our quantity 6) so that reasonable

agreement with experimental curves are obtained including the varia- tion with wheel load W. This latter relation F^{W) even appears to show a maximum, after which Fy drops with increasing W. An investigation is needed as to whether this drop is due to the approximation of a symmetric parabolic shape of carcass. It should be noted that with Fiala's

model a drop of Fy versus W is accompanied by a region of negative

lateral force distribution in the forward portion of the contact line. This

force distribution does not appear to be possible for stretched string

tire models. Another theory explaining this drop, which has been ob- served in many experiments (Gauss and Wolff [2]), makes use of the

reduction of the lateral stiffness of the carcass as a function of vertical deflection, due to variations in geometry of the cross section of the tire in the contact region (cf. Rotta [20] for theory, Smiley and Horne [8] for experimental verification and Bergman [17] for application).

Curves obtained in the experiments of Freudenstein [4] for truck tires and Nordeen and Cortese [5] for passenger car tires do not show such a maximum in their ranges of measurement. As an illustration, some

experimental results are shown in figures 7.5.6-8. In these figures dif- ferent ways of plotting are shown, each of which have their specific

advantages. The curves of figure 7.5.6 may be more suitable for use in the analysis of automobile motions. Both the functions Fy{a) and Fy{W)

are directly obtainable from this kind of carpet plot. For further information we refer to part 7.3 which contains an extensive collection of experimentally determined cornering force and aligning torque characteristics

(figs. 7.3.22-35). In order to obtain a deeper insight into the problems

related to side slip, we shall turn now to the treatment of a more advanced

tire model.

Stretched string model with tread elements.

The analysis to be presented now has been taken in an abridged form

from Pacejka [19]. We shall start with a more general analysis which

will also cover the possibility of antisymmetric longitudinal deformations

occurring in the shimmy motion to be dealt with in section 7.5.3. In figure

7.5.9 a top view of the model is shown in an arbitrary position. The

carcass is represented by a number of elastically supported parallel

strings under tension, which are connected by cross cords. The points
SIDE SLIP 769

of connection can move only laterally, and their mutual distance remains

the same. When the strings are deformed laterally the rubber between

the strings will be sheared. Through the continuous elastic support, axial forces distributed over the length of the band can be transmitted to the

wheel plane. To this band under tension, several rows of an infinite number of elastic blocks are attached, representing tread elements. In contact area of length 2a and width 2b the ends of these elements are in contact with the road surface. The strings are assumed to be of infinite length.

The longitudinal deformation u is assumed to be proportional to the

longitudinal component of the contact pressure. The following relation holds: Px = — Cpxu, (7.5.26)

where Px is the force and Cpx the longitudinal stiffness of the profile

elements, both per unit area.
770 THE TIRE AS A VEHICLE COMPONENT

Normal

100 200 300 400 500 600 700 Aligning torque {Um)

Figure 7.5.7. Gough-plot of truck cross-ply tire 9.00-20 eHD, /»,- =5.5 bar.

Dry road measurements, V = 10 Km/h (from Freudenstein, Ref. [4]). pneumatic trail t (mm)

Aligning torque

Mz (Nm)

Figure 7.5.8. Gough-plot of 6.00-13 Dunlop tire, pi = lA bar, dry

internal drum with inner diameter 3.8 m, speed V=40 kmlh (from, Henker [6]). The lateral deflection v is made up of the lateral deflection of the string

(the carcass) Vc and the lateral deflection of the tread rubber Vpi v=vc + vp. (7.5.27)

We will consider only the case where Vp is constant along the width of
SIDE SLIP 771

Figure 7.5.9. Top-view of tire model considered.

the contact area, as will occur in cases to be investigated. We assume

Vp to vary proportionally with qy as expressed by:

'^bpy= qy = — CpVp (7.5.28)

where qy denotes the lateral force and Cp the lateral stiffness of the tread

elements, both per unit length. In order to obtain an expression for the deflection of the strings we

must consider the equilibrium of an element of the tire model as shown

in figure 7.5.10, where the longitudinal displacements w, resulting in a second-order effect, are neglected. In the lateral direction, the equilibrium of forces acting on the element of length dx and full tread width 2b

results in the following equation: — qvdx — CcVcdx-\- D — D — ^

dx — Si

dx dx

where Cc denotes the carcass stiffness ("pneumatic stiffness") per unit

length (cf. eq (7.4.6), fig. 7.4.2), Si the longitudinal component of the

total tension force in the strings and D the shear force in the cross

section of the tread-band. The shear force is assumed to be a linear

function of the shear angle, according to the formula:

D = -S,^-^. (7.5.30)

With the introduction of the constant S = Si-\-S2 we deduce from eq

(7.5.29):

Qy = S—--CcVc. (7.5.31)

dx^
772 THE TIRE AS A VEHICLE COMPONENT

X

Figure 7.5.10. Equilibrium of deflected tire element.

In the part of the tire not making contact with the road the contact pressure vanishes so that: S-^-CcVc = (^ for|jc|>a. (7.5.32)

For that part making contact with the road we obtain, with eqs (7.5.27-28,

7.5.31)

S -^— CcVc = — Cp{v — Vc) for|:c|<a (7.5.33)

We introduce the tire constants:

With increasing tread rubber stiffness the value of the parameter e decreases until it vanishes when Cp-^oo, which represents the case of no tread elements. When the quantities defined in eq (7.5.34) are intro- duced, one obtains the equations of equilibrium (7.5.31), (7.5.32), and

(7.5.33) as

d^Vc (T'-—-r — Vc=qy/cc, (7.5.35)

o--^-Vc=Ofor\x\>a, (7.5.36)
SIDE SLIP 773

cr-^—^-Vc^-{l-€')v for|x|<a. (7.5.37)

For large values of | jc | the deflection Vc tends to zero. Therefore the

solution to eq (7.5.36) reads:

Vc — Cie~-^"' for x> a

(7.5.38)

Vc=C2e-^^ foTx< — a At boundaries x = ±awe obtain:

Vc = — cr lim tt" for x= a

. dx

(7.5.39)

Vc=cr lim — foTx = — a

ox

X t —a Since for x = ±a the deflection Vc and its derivative dvddx vary con- tinuously with X, the latter due to the fact that no finite concentrated

forces can act on the strings, with finite deflection v,j and finite stiffness

Cfj (cf. also eqs (7.5.28) and (7.5.31)). in the expressions (7.5.39) the limit signs may be omitted, after which they can be used as boundary conditions for the solution of equation (7.5.37). For the determination of the

integration constants occurring in the solutions of the first-order partial

differential equations (7.5.9), the additional conditions are needed that the deflections v and u vary continuously at the leading edge, where

x = a. That this continuity does take place can be proved in the following

way.

For the real tire, where fji is finite and the vertical pressure gradually

tends to zero at the leading and trailing edges of the contact area, it will be obvious that the tread elements show no deflection just after entering

the contact area or just before leaving this area. Consequently the

deflections vary continuously in the neighborhood of both edges in this case. For the extreme case where finite shear stresses are available at the

leading and trailing edge of the contact area {/jl^ ^) a finite deflection

may occur at these edges. It can be shown, however, that if we consider

vanishing regions of sliding at both edges, at the leading edge sliding

velocities would occur which produce friction forces directed opposite

to the external forces required for maintaining the discontinuity, whereas

at the trailing edge this will not occur. Similar findings are obtained for the model without tread rubber when kinks are assumed in the string at both edges. The conclusion must be that only at the trailing edge may a

finite change in deflection of the tread rubber, or a kink in the string

without tread rubber, occur in the extreme case as /x-^^c (cf. [19] for detailed discussion).

The forces and the moment acting on the tire may be computed by
774 THE TIRE AS A VEHICLE COMPONENT

integration over the contact area^. The forces in the longitudinal and

lateral directions become respectively: = PxdA, Fy = - pydA. (7.5.40a)

The moment about the vertical axis reads:

M,=-|^ ip^y-pyx)dA. (7.5.406)

In case of purely lateral slip the tire model does not show longitudinal

deformations, so that px = Fx = 0. The lateral force and the moment

then, become:

Fy = ~

I ^ydx. M, = - \ Qjcxdx. (7.5.41) J—(I J—

a

The general analysis derived above will be applied here for the case of steady state rectilinear side slip. As before, the vertical force distribution

is assumed parabolic along the ir-axis. We obtain for the lateral force

distribution in the region of sliding:

^bpy = qy = fxq, sgn Vsy (7.5.42)

where

.. = 26p. =f{l-(fj). (7.5.43)

When we reduce the slip angle from a large value, where total sliding occurs, we find a point of first adhesion which is situated somewhat

behind the leading edge when the model parameter e is smaller than a certain critical value depending on cr, i.e., when the lateral stiffness of the tread rubber exceeds a certain value. This means that when reducing

the slip angle further, two regions of sliding occur: a small region in front of and a much larger one behind the region of adhesion. The critical value of € above which adhesion occurs immediately at the leading edge

is given by the following formula

(7.5.44)

We shall discuss here only the results for a model with relatively low

tread rubber stiffness exhibiting only one sliding region, and for a model

without tread rubber elements {c,,-^'^) showing two sliding regions. Figure

7.5.11 shows the deflected tire model of the former kind. Differential

equations (7.5.13) and (7.5.37) are applicable for the adhesion region

{a-? < X < a) and (7.5.35) and (7.5.43) for the sliding region {— a < x < a-z).
SIDE SLIP 775

a . a ^-^rT^ ^

-X

•

. ^ . 1 r

i * V

f

Vc JJl

+y Figure 7.5.11. Deflected tire model provided with tread elements, showing an adhesion and a sliding region. There are five constants of integration and one unknown a-z- We therefore

need six boundary conditions in order to find these. These conditions follow from (7.5.39) and the discussions thereafter. The slip angle a and the shape of the deformation are calculated as a function of the distance a-z for the special case e= 1/7.5 and (t= 3.74 a. Figure 7.5.12 shows the shape for a number of cases. The obliquely

shaded area indicates the sliding regions, which grow with increasing

slip angle until the whole contact line slides. Larger slip angles will not

alter the shape anymore.

The relaxation length o"* has been defined as the distance between

the leading edge of the contact area and the point of intersection of the

elongation of the straight portion of the contact line with the jc-axis. The

values of a and e mentioned above were chosen in such a way that cr* tends to the value 3a for a ^ 0. Note that the relaxation length decreases

from the value 3a to a value somewhat below 2a when total sliding starts. This property has been confirmed experimentally by Metcalf [55].
776 THE TIRE AS A VEHICLE COMPONENT

Once the deflection of the tire model is known, the force and moment

can be calculated by the use of eqs (7.5.14) and (7.5.28). Integration over

the contact length as indicated by eq (7.5.41) yields expressions for Fy and Mz in terms of a-i. Figure 7.5.13 shows the calculated tire characteristics. As with Fromm's model, the slope of both curves becomes

zero at the slip angle olsi where total sliding starts. We shall see that

this is not the case when the tread rubber is removed from the string

{cp^ 00, e = 0, 0-* = cr, Vp = 0,v = Vc).

Figure 7.5.13. Tire cornering characteristic for the tire model with treadrubber {a = 3.47a,e = ll7.5). String model without tread rubber elements.

Figure 7.5.14 shows this simple string model in a deflected situation. Two regions of sliding are expected to occur. In the region of adhesion

{a2<Xi<ai) the stationary version of eq (7.5.13) holds. The sliding

regions are governed by eqs (7.5.35) and (7.5.42). In the straight portion

of the contact line where adhesion occurs, the following inequality

holds according to eqs (7.5.35), (7.5.13), (7.5.42), and (7.5.43): v<y2j\l-[-] \ (7.5.45)

with the parameter

00 = %^. (7.5.45a)

This means that the straight portion of the contact line lies inside the

parabola as indicated in figure 7.5.14. The points of inflection of the contact line are located on this parabola when Vsu<0 in these points.

Near the edges x = ±a the available lateral force tends to zero. Since a

finite deflection z; > 0 is present in these places, the curvature of the
SIDE SLIP 777

Figure 7.5.14. Deflected tire model without tread rubber showing two

sliding regions.

String will be concave {d'^vldx^ > 0) according to (7.5.35). In the rear

portion of the contact line we have dvldx>0, so that according to

(7.5.14) Vsy < 0. Until the contact line intersects the parabola, the

shape remains concave. Inside the parabola, however, the curve becomes

convex. The boundary of the adhesion region is at x=a2. The point of

transition to the front sliding region is denoted by x = ai. When this

latter point lies inside the parabola, according to (7.5.35), the curve must

be convex just in front of that point when the shding velocity is negative.

This shape, however, would lead to an increase in slope, so that ac- cording to (7.5.14) the sliding direction becomes positive in that case, which is in contradiction with the assumption. In the same way a concave

shape can be shown to be impossible. The conclusion must be that this point of transition must lie on the parabola. The curve in the front sliding

region can only be concave, so that Vsy < 0. Its curvature tends to zero when the parabola is approached. This forms one of the seven boundary

conditions necessary for the calculation of ai and a-z and the five constants

Figure 7.5.15. The tire model drifting at various slip angles a{a = 3a,€ — 0)

.

The shaded areas indicate the regions of sliding.
778 THE TIRE AS A VEHICLE COMPONENT

of integration in the solutions for the three differential equations. By

means of iteration the contact line has been computed for a number of values of the slip angle a. Figure 7.5.15 shows the results. The variation

of the relaxation length with slip angle angle will be extremely small

for the model considered.

The cornering force and the aligning torque are found by integrating

the external lateral forces along the contact line. The results of the

computations are shown in figure 7.5.16. In contrast to the characteristics

of the model with tread elements shown in figure 7.5.13, the curves show

a discontinuity at the slip angle where total sliding starts. Behavior at vanishing sliding.

It is of interest to know the behavior of both models with and without

tread rubber in case of vanishing sliding, i.e., for coefficients of friction tending to infinity or for slip angles tending to zero (see fig. 7.5.17). The

model without tread rubber and with vanishing sliding was originally

treated by von Schlippe [9, 25, 26] and Temple [9]. Their theory is of particular importance for the study of nonsteady state motions as treated

in section 7.5.3. Omitting the detailed calculations, we obtain for the lateral deflection

at the foreword contact point:

i;i = o-*q: (7.5.46)

Figure 7.5.16. Tire cornering characteristics for tire model without tread elements

{a- =3a, e^O).
SIDE SLIP 779 where

o-{(l + e)e2"/^c+ (l-e)e--2 a I a, 2} -4a

1 + 6 1-e

+

1 (7.5.47)

-2«/cr, 1 + e which is the expression for the relaxation length at zero slip angle as previously defined (see figure 7.5.17a). The relaxation length equals cr when the tread stiffness tends to infinity (see fig. 7.5.176).

The cornering force and aligning torque becomes in this case: Fy=CFao: and— M^^CMaOi (7.5.48)

with the stiffnesses Cfa and Ca/q having the following values:

CFa = 2cc(l-e2)[(o-* + a)a-V40-o-*{(l + 6)e^''/^c+ (1 -e)e-^"/'^c-2}

+ V40-^ ( 1 - e^O (e-"/^^ + e-^"/°-c - 2) ]

,

(7.5.49)

Cv/a = 2cc(l-e^)[V3a3-V40-{o-*(l + e)-o-(l-e^)}{a(l + e=^"/«-c) + crc(l-e^"'^0}-V4O-{o-*(l-e) - o-(l - e^) }{a(l + e-^«/^c)

Figure 7.5.17. Drifting tire models at vanishing sliding; a. with tread rubber, b. without

tread rubber.
780 THE TIRE AS A VEHICLE COMPONENT

in which ctc and e are defined by (7.5.34). When the tread elements are omitted (ctc — € = 0) , these equations reduce to:

If the contact patch of a standing tire is moved sideways a lateral force

will arise, which for small deformations is proportional to this lateral

shift according to the relation

where Cy denotes the lateral stiffness of the tire and Vo represents the

lateral deformation of the contact patch relative to the wheel rim. In case of side slip, Fy assumes the same value when the center point of the contact hne shows a lateral deflection equal to Vo- When taking into

account that Vo = (a + cr'^)a the following relation appears to hold for the relaxation length of a tire

This expression may be used for the experimental determination of the relaxation length cr* of a real tire. For a number of values of cr and Cpjcc the resulting relaxation length

(T*, pneumatic trail toi=CMjCra). and cornering stiffness C^a are cal- culated and listed in table 7.5.1. The original value of the lateral car- cass stiffness per unit length Cc has been designated with Cco. It is remarkable that the inclusion of tread rubber of a relatively high stiffness

{Cfjlcc = S5) does reduce cr*, and so much. The model with tread

rubber may give results close to those obtained experimentally (cf. Fonda and Radt [7, figs. 25, 34]). In figure 7.5.18 for the same combinations of parameters (except

cr = 3a) the variation of cr*, C^a and C.ua with vertical wheel load W is shown when a parabolic variation of the contact length 2a with W is assumed. The original values are designated with ao and Wo - It is shown

that relative to the behavior of Fromm's model (Cc the variation of the cornering stiffness with wheel load has improved somewhat due to the introduction of carcass elasticity. The case without tread rubber

(cp^oo), in turn, differs qualitatively from reality since the cornering

stiffness Cra remains finite for the vertical load W equal to zero. The

drop sometimes observed at large values of IF cannot be simulated with

this model unless a Cc is varied. It is then necessary to introduce two

different functions of average lateral foundation stiffness Cci'{W) and

Ccm(W) , one for the force and the other for the moment. This is plausible

since for the force the lateral deformation of the center portion of the contact area is of primary importance, while for the moment the portions outside the contact area play the greatest role, and these show

less radial deflection. As has been shown in section 7.3.1 the stiffness Cc decreases with increasing radial deflection. The variation of the relaxaCFa = 2cc{cr+ a)~ and CMa = 2cca{(T{(j^a) + Vaa^} , (7.5.50)

Fy CyVo

(7.5.51)
SIDE SLIP 781

Figure 7.5.18. Influence of wheel load on theoretically obtained values of contact length

a, relaxation length cr - , and cornering force and moment stiffnesses Cya and C.ua for

three tire models.

a. carcass (strinji) + tread rubber (c,,/c, =55), b. carcass (string) only (c,,^3c). c. tread rubber only (c, —

tion length (t* as indicated in figure 7.5.18 is of importance for the

analysis of section 7.5.2, where the influence of a time-varying load

is treated.
782 THE TIRE AS A VEHICLE COMPONENT

Table 7.5.1. Influence of model parameters

Tire model

Parameters

Relaxation

length a*la Pneumatic

trail tola

Cornering

stiffness Ci.al{Cci)a-)

a/a C/, ICco

Stretched string with tread rubber 3.47 55 1 3 0.49 33.6 Stretched string 3.47 00 1 3.74 .80 45

without tread rubber 3 00 1 3 .77 32 Tread rubber only

(rigid carcass) 00 55 00 0 .33 110 Camber and Turning

For antisymmetric steady state motion of automobiles, one must consider not only side slip but also the influence of two other effects, which

in most cases are of much less importance than side slip. These two

variables which complete the description of the antisymmetric motion

are first the wheel camber, or tilt angle y between the wheel plane and

the normal to the road (cf. fig. 7.5.19), and second the spin or degree of turning. In the steady state case the spin equals the curvature IjR of a circular path with radius R. In idealized form the mechanisms of camber

and turning can be considered to be similar as has been enunciated in the following.

As indicated in figure 7.5.20, the wheel is considered to move tangentially to a circular horizontal path with radius R while the wheel plane

has a constant camber angle y. The wheel is assumed to be part of an imaginary ball. When lifted from the ground, the intersection of

wheel

axle wheel plane

^^imaginary ball

/path

Z "

- ^ ^

^^^——^iZl^ 0

Figure 7.5.19. Camber and turning without side sHjk
CAMBER AND TURNING 783

wheel plane and ball forms the peripheral line of the tire. When loaded

vertically, the ball and consequently the peripheral line are assumed

to show no horizontal deformations, which in reality will be the ap- proximate case for a homogeneous ball showing a relatively small

contact area (fig. 7.5.20). contact

area of imaginary

ball

X y' \^ -vertical projection of

horizontally undeformed

peripheral line TOP-VIEW

Figure 7.5.20. Toi>-view of peripheral line of nonrolling tire considered as a part of imaginary ball pressed on flat surface. We apply the theory of rolling and slipping and consider equations

(7.5.4). We assume: Vcx=Vcy=Vry = OandVrx=V.

Equations (7.5.4) become in the steady state case:

yo^z dx ' (7.5.52)

dv

dx

For the combination of turning and camber o)z reads:

V

(Oz = (Jizt + 0)zc = ^ — n sin y. (7.5.53) K

The total spin (f) consequently becomes (cf. eq (7.5.8):

, (x)z 11. (^ = —= --jsmy (7.5.54)

with re denoting the effective rolling radius. In case that complete

adhesion occurs in the contact area eqs (7.5.52) become:

^= — y(f), ^= x(f). (7.5.55) ax ax
784 THE TIRE AS A VEHICLE COMPONENT

Integration yields the following expressions for the horizontal deformations in the contact area:

i^ = — (/)y;c+ constant II (7.5.56) V = V2 c/)jc- + constant The constants of integrations follow from boundary conditions which

depend on the tire model employed. As an example, consider a simple

model corresponding to that of Fromm with horizontal deformations

through elastic tread elements only. The contact area is assumed to be

rectangular, of length 2a and width 26, and filled with an infinite number

of tread elements. In figure 7.5.21 three rows of tread elements are shown

in the deformed situation. For this model the following boundary conditions apply: x = a : v= u = 0. With the use of (7.5.56) the formulae for the deformations read: u = y{a — x)(f), I (7.5.57) v = -y2 {a'-x')(J), J After the introduction of Cpx and Cpy denoting the stiffness of the tread

rubber per unit area in s and y direction respectively, we obtain for the

lateral force and the moment about the vertical axis, by integration over the contact area:

(7.5.58)

Figure 7.5.21. Top-view of cambered tire model rolling in a curve with radius R.
CAMBER AND TURNING 785

or in terms of camber and path curvature (y small):

(7.5.59)

These expressions indicate that camber and path curvature have opposite

effects when their signs are equal, according to the sign convention. In case of pure turning, the force on the tire is directed away from the path

center and the moment acts opposite to the sense of turning. Consequently both the force and the moment try to reduce the curvature IjR. In case of pure camber, the force on the wheel is directed towards the

point of intersection of the wheel axis and the road plane, while the

moment tries to turn the rolling wheel towards this point of intersection.

No resulting force or torque is expected to occur when y=relR, which

is approximately the case when the point of intersection and the path

center are the same. A number of authors explain the presence of a moment by considering two wheels rigidly connected to each other on

one axle. In a curve the wheel centers travel different distances in a given time interval and when cambered, these distances are equal but

the effective rolling radii are different. In both situations antisymmetric

longitudinal creep must occur which produces the moment.

Lp to now we have dealt with the relatively simple case of complete

adhesion. When sHding is allowed by introducing a limited value of the

coefficient of friction jjl. the calculations become quite complicated. If, as before, a paraboHc pressure distribution is assumed, it is obvious

from eq (7.5.57). that for an infinitely thin tread (6—»0) no sliding will occur up to a certain critical value of spin (/).s/, where the adhesion limit

is reached throughout the contact length. Up to this point Fy varies

linearly with (f). According to eq (7.5.54). spin due to camber theoretically cannot

exceed the value l/r^. so that this discussion will be limited to the case

of turning when higher values of spin are considered. Beyond the critical value (i)si the situation becomes quite complex, as has been pointed out by Freudenstein [4]. In the front half of the contact line sliding will occur, whereas behind the contact center adhesion takes place up to a point

B (cf. fig. 7.5.22) where the deformation v is opposite in sign and reaches

a maximum, after which sliding occurs again. With increasing spin

(b{=\lR) this latter sliding zone grows while at the same time the side

force ~Fy decreases and the torque —Mz, which arises for (/)>(/)«/, increases until the situation has been reached where R and Fy vanish

and— becomes maximum.

When a finite width 2b is introduced, complete adhesion will only occur for vanishing values of spin. We expect that sliding will start at the left and right rear corners of the contact area, since in these points

the available horizontal contact forces reduce to zero and the longitudinal deformations u would become maximum for /x^ ^c. The zones
786 THE TIRE AS A VEHICLE COMPONENT

circle parabola

// (sliding )

Figure 7.5.22. Simple tire model in sharp curve. of sliding grow with increasing spin and will thereby cause a less than

proportional variation of — Fy and —M^ with (/). The maximum lateral force will be lower than in case of line contact. A more or less exact

theoretical treatment of the spin behavior of pneumatic tires with

sliding taken into account has not been found in the literature. This

behavior has been treated in great detail by Kalker [21] for the problem

of rolling contact between elastic balls. Freudenstein [4] presents a rough theory for the turning behavior up to an approximate value of

(j)si. He treats the longitudinal and lateral deformations as being un- coupled and introduces an effective coefficient of friction V2 /xV2.

In addition, Freudenstein gives an approximate value of the maximum

torque occuring at R = 0. For a parabolic pressure distribution in re- direction and a uniform distribution in y-direction, he obtains for contact areas with dimensions in the range b < a <2b:

Figure 7.5.23 (from [4]) gives the turning characteristics of a truck

tire. A comparison with experimentally obtained cornering characteristics (cf. fig. 7.5.7) indicates that the values of both the lateral force

and the moment reached at one degree sHp angle roughly equal the force

and moment obtained with the same tire when rolling in a curve with ^ = 3.5 m. (according to experiments with bias ply truck tires). Reference

[4] indicates, furthermore, that the cornering stiffnesses Cfa and Cma

are 20-30 percent higher for the radial ply (belted) tire. Analogously, the

radial ply tire is expected to show greater resistance against turning than

the diagonal ply tire. From the discussion above it follows that the effect of spin due to a stationary turning may only be of importance with slow city driving. In" theoretical investigations of automobile motions normally met in the

literature, this effect has been neglected. Camber, however, which is the other variable associated with the spin effect, will be of particular

importance for two-wheeled vehicles. The theoretical relationship be- tween turning and camber has been used by many authors (Fiala [13], Freudenstein [4], Bergman [17] and Maier [9]). In fact, however, this

\M,z max I - V4Mr(3/2a + 6) (7.5.60)
CAMBER AND TURNING 787

Normal wheel load W

0 ] 2 2 I 5 6 7 8 9 10

0 i 2 3A56789 10 Radius of curvature R ( m ) Figure 7.5.23. Measured turning characteristics for truck tire.

Lateral force Fy and moment about vertical axis as a function of radius of curvature R. Tire cross-ply 9.00-20 eHD:

|), = 5.5. bar. V = 1-3 km/h on dry road (from Freudenstein [4|).

relationship is only exact in the case of a rolling ball. A cambered tire will behave somewhat differently since there exists a combination of

radial and lateral stiffnesses of the tire which in general are not the same. We expect therefore that, unlike the ball, a cambered wheel loaded

1 vertically through a purely vertical motion of the wheel axle (in which

; position the vertical projection of the peripheral line remains approxii mately unchanged), will show an initial horizontal lateral force before

i the wheel starts to roll. The camber force of a steady rolling tire may be

obtained by superposition of the initial lateral force produced by the

I vertical displacement, and the change of this force during rolling due to

j the spinning action as treated in the theory above. In the final steady

j state stage, the cambered tire will show a straight contact line parallel

to the wheel plane if sliding can be neglected. The complete process is very complicated and a fundamental treatment of the behavior of a rolling

cambered wheel is needed.

In addition to the lateral (horizontal) force due to axial (i.e., normal to

I the wheel plane) deformation, we should take into account in such a theory the horizontal lateral force which arises when a standing tire is

! tilted about the line of intersection of wheel and road plane. From the

difference in curvature of the tire side walls in the tire cross section, Rotta [20] calculates the side force which is necessary to balance the

internal air pressure. Rotta states that this force is the main contribution

to the camber thrust. Once the tire rolls, axial deformation may occur

1

1
788 THE TIRE AS A VEHICLE COMPONENT

due to, amongst other things, the torsional stiffness of the carcass and

the finite width of the tread. These factors are responsible for the so- called overturning couple Mx about the jc-axis (intersection of wheel and

road plane).

For a tire model with line contact (for instance a single row of tread

elements or a single elastically supported stretched string) there is no need for the peripheral line to move out of the wheel plane when it is tilted. The camber force and moments then simply read:

These relationships appear to hold approximately for diagonal ply tires. Experiments indicate that radial ply tires produce less camber thrust than bias ply tires. This is in complete disagreement with the results deduced for turning, which reinforces the suspicion that the mechanism

of camber and turning are not completely equivalent. As Freudenstein did not give the camber characteristics of the truck tires on which the

turning behavior was measured, we are not able to compare the re- sponses to camber and turning. According to Hadekel, for aircraft

tires the lateral force due to turning is about four times higher than the

camber force at equal values of spin. Evaluation of Freudenstein's truck

tire data reveals that for bias ply tires (assuming that relation (7.5.61)

holds for these tires) a lower ratio of about 1 is expected; for radial ply

tires, however, Hadekel's value may be of the right order of magnitude.

In figure 7.5.24 some experimental camber characteristics are shown.

W e furthermore refer to figures 7.3.37 and 7.3.38. Influence of Braking and Traction Forces According to experiment large longitudinal forces have considerable

influence on tire cornering force and aligning torque. Theory on this subject is scarce. A significant paper is that of Bergman [17] which,

however, is restricted to the influence of traction. Bergman had to simplify the system considerably, due to the exceedingly complex

mechanism of the real tire. He employs the concept of interaction effect and effective lateral coefficient of friction. Bergman states that traction

reduces the lateral stiffness of the standing tire, resulting in a reduction

in cornering force. Similarly, one would expect that braking also would

soften the tire in the lateral direction. However, experiments indicate

that moderate braking increases the cornering force slightly. This contradicts Bergman's concept. The increasingly sharp reduction of the

cornering force at high values of traction has been explained by Bergman

by the introduction of the effective coefficient of friction in the lateral direction iJiy = V/x^ — jxi, where fXx represents the effective tractive coefficient FxIW. It is obvious that this is a crude approximation to reality, although the results appear to fit some observations. However,

for the description of the influence of the braking force, and especially

(7.5.61)
INFLUENCE OF BRAKING AND TRACTION FORCES 789

Y

Figure 7.5.24. Measured camber characteristics.

Lateral force Fy and moment as a function of camber angle y and wheel load W. Same tire as in figure 7.5.6 (a = 0) (from Nordeen and Cortese [5|).

for the influence of longitudinal forces on the aligning torque, Bergman's

theory is inadequate.

A theory which is probably more widely applicable will be presented

next. It is a further development of a first attempt of the author reported

in [18]. It describes the behavior of a relatively simple model, which

later on will be extended to a more advanced model.

Model without carcass elasticity.

The simple model is identical to Fromm's model, for which the steady

state side slip behavior has been treated in section 7.5.1. The carcass is considered to be rigid, and the tread elements provide the necessary elastic properties of the tire. For purposes of mathematical simplicity it is assumed that the horizontal stiffnesses of the tread elements, of which

several rows may be present, are equal in the lateral and longitudinal

direction. We introduce:

cjj = 2bcpj: = 2bc„y (7.5.62)

with Cf,x and Cpy denoting stiffnesses per unit area, and c,, the horizontal

stiffness per unit of length.
790 THE TIRE AS A VEHICLE COMPONENT

^-FX\f\\\ \ \ \ \

Figure 7.5.25. Deformations of the simple model due to side slip (a) and longitudinal

slip {k).

In figure 7.5.25 the deformations of the tread elements are shown for both the cases of traction {Fx > 0) and braking {Fx < 0). The longitudinal creep velocity Vex obviously becomes:

Vex = V COS a — Clri. The longitudinal slip ratio has been defined as _ Vex _ ^ V cos a

Clri Clri

(7.5.63)

(7.5.64)

The deformations due to longitudinal slip and side slip become, in case

of full adhesion {fx^

u= {a — x)K

v= {a — x) (1 — K)a

(7.5.65)
INFLUENCE OF BRAKING AND TRACTION FORCES 791 where for simplicity a is assumed to be small. We introduce the angle 8,

indicating the angular deviation of the total horizontal force from the

lateral direction (cf. fig. 7.5.25). The following relations hold:

sin b = Fj: IK, cos 6 = Fy IK. (7.5.66)

The relation between k and a reads:

tanS = -- — • (7.5.67) {i — K)a

Analogous to eq (7.5.21), we solve with (7.5.20) for the point of initial sliding, but now with the presence of longitudinal slip (a > 0)

0{\ — k)ol , ^ /< , OoL ^ ^ .r.. A= 1 ^ 1 - Q

—=1 z • (7.5.68 cos 0 sm o a sm 6+ cos o For a given slip angle a, \ will be greater at positive 6 (traction) than at negative 8 (braking) for same absolute values of 6. When for a combination of Fj- and Fy total sliding begins (X = 0), the

slip angle and the longitudinal slip ratio are: cos § 1 • s c /:m asi=7i :—r; Ks/^- sm 6 (7.5.69) ^ — sm o d Integration over the contact length yields the total horizontal force: K=ixW{\->?) (7.5.70)

and for the aligning torque:

Mz = -iiWah?{\-\) cos8. (7.5.71)

From these formulae, for given a and k, the tire forces and moment

can be calculated with the use of eqs (7.5.66-68). In figure 7.5.26 a number of curves are shown for the parameter value ^ = 5. For both

the moment Mz and the force Fy plotted against Fx a slight asymmetry

appears to occur.

In the (Fy, Fx) diagram curves for constant longitudinal slip values K are also shown. From eqs (7.5.66, 68, 70) we obtain the following

formula:

For K < fjiW, one finds that 0 < OkI sin 8 < 1 and formula (7.5.72) indi- cates that for a given k the absolute value of the longitudinal force

\Fx \ decreases with increasing cos 6, and consequently with increasing

slip angle a.
792 THE TIRE AS A VEHICLE COMPONENT

Figure 7.5.26. Variation of cornering force Fy and aligning torque as a function of

driving or brakingforce Fxfor the tire model skown in figure 7.5.25.

These theoretical observations correspond to experiments carried out by, among others, Henker [6] on a rotating internal drum (dry, cf.

fig. 7.5.27) and by Holmes and Stone [27] on wet pavement. The phe- nomenon that the curves on wet surfaces tend to end inwards may be

due to the fact that at higher sliding velocities the coefficient of friction decreases (cf. fig. 7.3.49).

However, the variation of the aligning torque with the longitudinal

force Fx does deviate markedly from experimental results presented

by Nordeen and Cortese [5] (cf. fig. 7.5.28). As shown, the force Fy

varies as expected but the moment does not agree with the theory. > It appears the changes its sign during heavy braking. This phe-

| nomenon cannot be explained with the simple model employed so far. Before we adopt a more advanced model, the cornering stiffnesses of the simple model will be derived as a function of F^-. After- elimination of X from eqs (7.5.68) and (7.5.70) we obtain, for F,-^OAa-^0,K^\F^\):
INFLUENCE OF BRAKING AND TRACTION FORCES 793

radiuS:

Fx(N) W 2000 2000 W -Fx(N) TRACTION BRAKING Figure 7.5.27. Measured side force {Fy)-longitudinalforce {F.r) relationship for constant

slip angle a {also shown for constant longitudinal slip value k). Same tire and conditions as in figure 7.5.8. (W=2500 N) (from Henker [6|).

+ Mz *Fy (Nmy^ (n;

Figure 7.5.28. Measured variation of both Fy and M^ with Fx for constant slip angle a. Same tire as in figure 7.5.6 (W=4500 N, y = 0). (from Nordeen and Cortese [5|). which yields:

0\F.\

(7.5.74)
794 THE TIRE AS A VEHICLE COMPONENT

which reduces, for /x ^ oo, to

C,.a-^2c„a--Fjc. (7.5.75)

The pneumatic trail t = — MJFy becomes (using eqs (7.5.66. 68, 70)):

' = ^ 1_;,3 (7.5.76)

or when a ^ 0:

to = ad '^ ' (7.5.77)

from which is obtained;

and for jjl—^^:

CMa=to'Cra=a6Cra^^^—^ (7.5.78)

C.Ma-^ V3(2cpa3-F^a). (7.5.79)

These resuhs are not of direct importance since they deviate too much

from experiment. They are, however, needed for the development of the theory of the more advanced model with carcass elasticity.

Model with carcass elasticity.

The simple model will now be extended by the introduction of lateral,

longitudinal and torsional elasticity of the carcass. For simpUcity we

shall assume the carcass equatorial line to be straight in the contact

region (see fig. 7.5.29). For the sake of distinction the quantities referring

to the simple model of figure 7.5.25, which can be recognized as a part of the^more advanced model, are indicated by upper bars. The quantities

Fy, Mz, i and Cra have been determined as a function of Fx and a in the preceding portion of this section.

The angles a and fS (fig. 7.5.29) are assumed to be small. With the carcass stiffnesses Ccx, Ccy and Cc^ we obtain for the displacements of the straight carcass section: Uo FX ICcx

Vo = Fy/Ccy (7.5.80)
INFLUENCE OF BRAKING AND TRACTION FORCES 795

Figure 7.5.29. Deformations of the more advanced tire model showing carcass and tread rubber elasticity.

Furthermore the following relations hold:

F^ = F^^Fy(3

Mz = Mz — Fj:Vo-^FyUo

and in case of complete sliding:

Mz = -Fj:Vo FyUo — FxFy ( j;^ ^ cx ^ c

(7.5.81)

(7.5.82)

With the above equations. Fx, Fy, Mg, and a can be calculated without

difficulty for a given combination of K and 6. By means of interpolation, curves for constant a have been obtained, which are shown in figure

7.5.30. The calculations have been carried out using the values:

^ = 2/3Cpa^/)Ltr=5. Ccx^^, CcyalfjiW=3, CcfilyiWa=l. (7.5.83)

For investigation of the influence of longitudinal elasticity, some finite values of Ccx are considered as well. The Fy vs. a and Mz vs. a curves obtained for Fj. = 0, figure 7.5.30c, d,

appear to be qualitatively equal to those shown in figure 7.5.5 for the

simple model. As expected, the slopes at o: — 0 are less steep but the slip

angle at which total sliding starts remains the same. Therefore, the curves in the Fy vs. Fx diagram (a) become somewhat flatter relative to the curves of figure 7.5.26. Owing to the lateral deformation Vo the

longitudinal force Fx produces a moment about the contact center C
796 THE TIRE AS A VEHICLE COMPONENT

which opposes the aUgning torque —Mz in case of braking, and thereby causes a change in sign of the ahgning torque (diagram b) as has also been

observed experimentally (cf. fig. 7.5.28). In the same figure 7.5.306, the

influence of equal stiffnesses Ccx = Ccy has been indicated. In this case

the torques produced by Fx and Fy about C balance each other so that no change in sign occurs. The influence of carcass elasticity may be

smaller than predicted here because of the expected property that the

displacements of the lines of action of Fx and Fy are smaller than the

deflection Vo and Uo respectively. Smiley and Horne [8] give some in- formation on the shift of the normal force relative to the horizontal defor- mations of a standing tire. They give figures of 80 percent and 25 percent

respectively for aircraft tires. The cornering stiffnesses valid for the more advanced model are derived as follows. With the relations (81, 82) we obtain for a —> 0:
INFLUENCE OF BRAKING AND TRACTION FORCES 797

^Fa —

a

Cfa

1 —Fxto/Ccp

Cxia —

a-\-l3 l + CfatolCc^'

-M,_-M,^ Fg-Vo - FyUo

1

(7.5.84)

1 + CfatolCcli \Ccy Ccx

in which Cfa and to are given by eqs (7.5.74) and (7.5.77). In figure 7.5.31 the functions (7.5.84) are plotted for the values given in eq (7.5.83), and in addition for two finite values of Ccx- It has been found that through

the introduction of carcass elasticity considerable qualitative changes

in characteristics can arise. When braking, the cornering stiffness Cfu initially increases slightly,

it then passes a maximum after which a sharp drop occurs, and finally

complete sliding takes place. From this point Cfa decreases with in- creasing longitudinal slip velocity up to the point of a locked wheel.

In case of traction Cfu decreases continuously and drops to zero in the

range of complete sliding, when the slip velocity — Vex tends to infinity

(wheel spin-up).

As indicated, the variation of C.Ma depends on the ratio of longitudinal

stiffness. For Ccx ~^ ^, the aligning torque stiffness Cv/a decreases with

increasing braking force, becomes negative, passes a minimum and

finally shows a small negative value when the wheel is locked. In case of

traction C\ia first shows a slight rise after which Cv/a tends to zero for Fx^fx^f"-

e=5

7MWa

Figure 7.5.31. Corneringforce and moment stiffness {ata = 0)asafunction of longitudinal

force.
798 THE TIRE AS A VEHICLE COMPONENT

The influence of the longitudinal stiffness is of great importance as shown in the figure. In case of equal stiffnesses Ccx=Ccy, the variation of

Cxia (and of the moment M^, cf. fig. 7.5.306) becomes more symmetrical,

which corresponds to the experimental curves for radial ply tires. Bias

ply tires, however, behave according to the theoretical curves for rela- tively large longitudinal carcass stiffness. The much lower longitudinal

stiffness of radial tires might indeed be responsible for the great qualitative differences observed experimentally between diagonal and radial

ply tires [28].

Limitations of theory.

The theory just developed is limited due to simplifications in the tire model. Probably the most questionable simplifications are the equal

stiffnesses of the tread rubber elements in the lateral and longitudinal

directions, and the assumption that the carcass remains straight in the contact zone. Furthermore, the theory assumes a constant coefficient of

friction. As has been pointed out previously, the force and moment characteristics will change in shape due to the drop of the coefficient of fric- tion with sliding velocity. Although a number of aspects may be clarified with the aid of this model, it is certain that important factors are not yet

taken into account. One of these will be mentioned briefly. It is the in- fluence of the change in tension force or effective tension force S in the carcass tread band due to braking or driving forces. Influence of change in tension force, front and rear. In order to investigate this influence, an alternative tire model must be

adopted. Obviously, the simple stretched string model is most suitable. The slopes of the Cpa {Fx) and Cv/a {Fx) curves at Fj. = 0 will be calcu- lated while we restrict ourselves to the case of vanishing sliding (^t oo). The slopes to be calculated, consequently, are purely due to the elastic properties of the tire and may be compared with the slopes obtained

with the simple model for ^ (cf. eqs (7.5.75) and (7.5.79)). In figure 7.5.32 the stretched string model is shown subjected to lateral and longitudinal forces and the aligning torque. Like in the theory of steady state side slip (eq 7.5.31), the effective longitudinal tension force S is introFlGURE 7.5.32. Stretched string model deformed due to side slip and traction {no sliding, UL-^ 00).
APPROXIMATE MATHEMATICAL REPRESENTATION 799

duced. which, due to the longitudinal force Fx, will differ now at the

trailing and leading edges. We assume the following relations to hold:

S2 = S + V2 Fx.

(7.5.85)

According to the definition of the relaxation length when tread elements

are absent (cr. eq (7.5.34)) we obtain the following formulae for the relax- ation length front and rear respectively:

af={S-y2Fx)lcc. (Tl={S + V2Fx)lcc. (7.5.86)

Analogous to eq (7.5.50) the cornering stiffnesses read:

CFa = Cc {(o-i +cr2 + 2a) (cTi + a) +a(cr2 — (Ti)}

C.\ia = Cca {(J2(cr2 + a) +C7i(cri + a)

+ 2/3 ^2+ (^,_cri)(o-i + a)}, (7.5.87)

which with the aid of (7.5.68) and with the assumption Fx ^ S reduce to:

Cf-a — Cfao 'kFx , Cuq = C.i/ao + iFxa ( cTo + a ) /o-o

(7.5.88)

where index o refers to the situation without external longitudinal

forces. The slope at Fx = 0 becomes:

-7Er=—h ;r — (cTo + a). (7.5.89) dFx dFx (To

It may be noted that the slopes of C^a and Cuq differ in sign. The change

in tension force due to traction or braking, consequently, intensifies the effect of lateral carcass elasticity.

The influence of a longitudinal force on camber force and moment

has not been investigated properly as far as the author knows. Since the

influence of camber itself is small relative to the influence of side

slip, we may roughly approximate the effect of longitudinal forces on camber force and moment, as will be done in the next section.

Approximate Mathematical Representation of Combined Effects In theories of steady state or quasi-steady state automobile motions,

tire forces and moments must be introduced in an appropriate mathematical form. The complexity of this representation depends upon the

object of the investigation. Many authors [29. 30. 31. 32. 33] restrict the

motions to be investigated to relatively small deviations from the recti- linear path. In that case the equations of motion can be linearized and

only coefficients like cornering stiffnesses are of importance. When,
800 THE TIRE AS A VEHICLE COMPONENT

in addition, longitudinal forces are not taken into account these coeffi- cients depend only on the vertical wheel load, which may vary due

to lateral load transfer. In the hnear representation the latter effect is only of importance when initial steer or camber angles of the wheels

front or rear are present. The change in roUing resistance (cf. part 7.2,

fig. 7.2.56 and sec. 7.4.1) with wheel load will always enter the problem

as soon as a finite height of the vehicle center of gravity is considered

[31]. The order of magnitude of the cornering stiffness Cya (force per

radian), expressed in terms of the nominal vertical load, Hes between

bWo and lO^To; the pneumatic trail to expressed in terms of half the contact length hes in the range 0.4a to 0.7a. Theories which consider high lateral (cornering) accelerations (refs. [34] to [42]) need a more or less complete mathematical description of

tire behavior. In these investigations most authors describe tire characteristics by means of simple mathematical expressions (parabola,

sine, exponential) which correspond more or less to actual characteristics known from the literature. More sophisticated theories [38, 42]

show the employment of actual tire characteristics in which the influence

of vertical and longitudinal forces are also taken into account. A complete, more or less exact, mathematical representation of measured data is difficult to accomplish. Fiala has combined his theories for side slip (with influence of W and /x) and camber. He presents explicit

expressions for Fy and Mz for which we refer to the original paper [13].

We shall present a procedure with which the combination of most

effects treated in this part 7.5 can be approximately represented. The

principle of this method has been given in [18]. The philosophy is as follows. Consider the tire characteristics {Fy vs. a, Mz vs. a) measured

at nominal wheel load Wo and zero camber and longitudinal force (ex- cept small roUing resistance force). From these basic tire characteristics we attempt, using the theoretical and experimental experience

obtained, to derive Fy and Mz in cases where the conditions differ from

basic conditions. These basic conditions are: W=Wo. ix=iJio.F:, = 0,y = 0. Under these circumstances we obtain

Fy = Fyoioi) , Mz = Mzo{oc) , Cpa^ Cfao, Ckia^ CMao- The functions Fyo{a) and Mzoioi-) may be approximated by antisymmetrical mathematical functions.

The influence of a variation of the vertical load W has two effects. First the. cornering stiffnesses vary according to functions which may be

measured (cf. fig. 7.5.6, low values of a). Secondly the maximum Fy

changes proportionally with yiW (experiments of Borgmann [16] show

that in fact this variation is somewhat less than proportional). Through

this effect the influence of /x has been taken into account. It is assumed

that /X effects only the more or less horizontal level of the cornering force
APPROXIMATE MATHEMATICAL REPRESENTATION 801

characteristic. On slippery roads the shape of the curves may change

considerably (fig. 7.3.26) and a different approach is needed.

The following equations hold approximately when only W and ijl differ from the basic conditions:

Fyoiafeci),

Mz =-^^ —p; ^ .jjr.. M,o (a.v/eq)

with the equivalent slip angles _ fJioWoCyaiW)

(7.5.90)

(7.5.91)

The cornering stiffnesses C^a and Ca/o are assumed to depend only on and are independent of fx. The functions CpaiW) and CnaiW) may

be approximated by polynomials. In figure 7.5.33 the transformation of curves (0) (= basic) to curves (2), obtained by means of multiplication

as indicated, result in the characteristics relevant for the new situation

(M, n

The influence of camber can be approximated by shifting the curves

(2) horizontally, so that for (x = 0 the force Fy and the moment Mz equal

Cfyy and C.uyy respectively. For experimental verification cf. Fonda

[3, fig. 73], or Henker [6, fig. 98] and figure 7.3.40. Equations (7.5.94) can be applied but with different arguments.

OLFeq —

/xr cFao Cram ^

(7.5.92)

OiMeq Fao \

The functions CyyiW) and CMy{JV) may be approximated by linear

functions of W. The quantity Ca/y is relatively small and might very

well be neglected. The new curves (3) are shown in figure 7.5.33. The influence of Fx is complicated. We propose the application of the following formulations which are expected to give reasonable

results. With consideration of the theory of section 7.5.1 the cornering

stiffnesses may be represented by:

cv.(/x, r, Fx) = (i

CA/a(/X, r, Fx) = (l

Fx

(7.5.93)

]CMcm-^{^-^]FxCFa{lX. r, Fx)
802 THE TIRE AS A VEHICLE COMPONENT

Figure 7.5.33. Illustration of characteristics obtained from the basic curves [0) according

to formulae (7.5.90- 95).

Diagram a:

The following successive steps are carried out: Diagram b: 00- 1 1-2

2

basic (/i.0, r,,, y=Fjc = 0) horizontal mult, with Ci.-,JC radial mult, with ixWIixqWq new conditions (/a, W)

2-3 : horizontal shift 3 : new conditions (/i,, W, y) 3-4 : hor. muh. C^a(W)|C ygiyi, W. Fs) 4-5 : rad. muh. Vi-CFx/m'^)'

5 : new conditions [(x, W, y, Fx) Diagram d:

Diagram c: 0 : basic (Mo, ro, y=F^ = 0) 0-1 : rad. mult, with ^ ^fa" 1-2 : vertical muh. C,„a(>F)/C,v/ao 2 : new conditions (/x, W)

2-3 : horizontal shift 3 : new conditions (pi, y)

^ ^ ^ ^\-{FJ,xWV-C,AW) 3-4 : rad. m. —

c^a(^Ji, w, F,) 4-5 : vert. mult. ( 1 - | F^m^^ I " ) 5-6 : vert. add.^^-^^F^F,

6 ; new conditions (/i,, W. y. Fx)

in which Ccx and Ccy denote the lateral and longitudinal stiffnesses of the carcass relative to the wheel rim. The constant t^y and t,x represent

the percentages of effectiveness of the carcass elasticity. They 'may be

found experimentally or by curve fitting of the whole function Cua

The approximate formulations (7.5.93) in which the exponents are of the order oi n— 10-20 and m — 1-2, do not take into account the

theoretically exact vertical portions (dotted in fig. 7.5.31) where com-
APPROXIMATE MATHEMATICAL REPRESENTATION 803

plete sliding occurs at \Fx\= fJ^^- In order to describe the variation of Cpa and C.Ma in these portions, the longitudinal slip ratio actually should

be introduced. We approximate this with a steep approach of the endpoints (1,0) and (— 1, 1) in the diagram {CfJtJ^W y?>. F^//>t^F) . Most authors

[43, 41] approximate Cpa {fJi, Fx) by an ellipse. As a result of this, the curves {Fy vs. a) are merely multiplied in the vertical direction. This

does not agree with theoretical findings (fig. 7.5.30c). A multiplication

of the basic curve Fyo{oL), mainly in the radial direction from the origin,

is expected to occur. We obtain the following formulae:

1^ c,/a(r) 'Fao CFai^l, r, Fx)

f \ F F

\Ccy Ccx / in which the equivalent slip angles read:

(7.5.94)

_ IXqWq CfaifJi, W, Fx)

VfjLW'-Fl Cf. l^oWo CFai^^.W.Fx)

CFao

CFaiW)

C.v/y(r)

Cmam (7.5.95)

Figure 7.5.33 shows the change of the curves due to the introduction of the longitudinal force Fx. For Fy, curve (5) is the final curve and for M^,

curve (6) is the final curve, covering all effects of deviations from the

basic conditions.

It should be pointed out finally that for the description of the unrestricted motion of a vehicle, tire characteristics must be used with the

abscissa sin a, instead of a, extending from — 1 to + 1 (cf. [40]).

These final characteristics are ready for use in the theory of vehicle

dynamics. It should be noted that the formulations above are but approximations of measured behavior. The basic curve measured under nominal

conditions (average vertical load, zero camber, free rolling, actual road

surface) depends on tire construction and inflation pressure. Quantitative experimental data is needed for the execution of each intermediate

step. The simulation of wet traction behavior (fig. 7.3.49) is expected

to be more complicated. In that case the influence of the speed of travel

is appreciable.

7.5.2. Nonsteady State Motions

In this section two kinds of nonsteady state motion will be discussed.

First, the response to nonsteady state horizontal motions of the wheel
804 THE TIRE AS A VEHICLE COMPONENT

axle and secondly, the influence of vertical oscillations of the wheel axle upon cornering force. Horizontal Motions {Transient and Shimmy)

In the theories describing the horizontal nonsteady state behavior of

tires one can identify two trends of theoretical development. One group

of authors assumes a bending stiffness of the carcass and the other bases

its theory on the string concept.

In principle, the string theory is simpler than the beam theory, since

with the string model the deflection of the foremost point alone deter- mines the path of the tread for certain wheel movements, whereas with

the beam model the slope at the foremost point also has to be taken into account as an additional variable. The latter leads to an increase in order

of the system by one. Probably the first investigator who tried to describe tire behavior

mathematically in the study of shimmy is Kantrowitz [44] (1937). In

spite of his rough and theoretically unsatisfactory assumptions, the

theory developed gave a fair correspondence with measured values of divergence of wheel deflections and frequency of the shimmy motion.

Kantrowitz studied the damping effect of the gyroscopic couple due to lateral distortion of the rotating tire. Another theory, apparently

inspired by Kantrowitz' work, was developed in 1942 by Greidanus [45]. Where Knatrowitz' work shows features of both the beam and the

string, Greidanus is consistent in applying the bending concept in his interesting study. Besides the slope, the curvature of the peripheral

line just in front of the contact point is also important for the further

development of the motion. In Greidanus' model a vanishing area of contact was considered as may be deduced from his approach. In a discussion on Saito's paper [46] Pacejka has given the differential equations which govern the kinematical variations in lateral tire distortion

for the beam type model with finite contact length. These equations

appear to be identical to those given by Greidanus when the influence of camber is not considered, and when the contact length is taken equal

to zero. In 1962 Saito [46] presented a theory using a tire model consisting of an elastic beam of which a finite length makes contact with the road. The theory is based on an approximate treatment of the kinematic

behavior of the contact line. Frequency response curves are given for the force and moment with respect to lateral and angular motions of the wheel plane. In order to obtain better agreement with experimental

results, Saito introduced theoretically unjustifiable empirical corrections.

Besides this group of investigators which were inspired by the work

of Kantrowitz, another group exists which has studied the problem with

the aid of tire models more or less based on the string concept.

In 1941 Fromm [47] gave a simple theory where this model (although

not mentioned by name) is investigated for the case of point contact. A

similar theory was developed by Bourcier de Carbon [48] in 1948 together

with an extension, somewhat unclear, which increases the order of the system by one. A similar simple theory originates from Bohm [49]. He
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 805

uses the nonlinear steady state cornering characteristics in order to find the amplitude of the periodic shimmy motion.

In 1941 von Schlippe [25] presented his well-known theory of the

kinematics of a roUing tire, and introduced the concept of the stretched

string model. For the first time a finite contact length was considered.

In the same paper Dietrich applied this theory to the shimmy problem.

Mathematical difficulties arose in the form of transcendental equations,

due to the retardation effect of the assumption of a finite contact length. Later on, two papers of von Schlippe and Dietrich [50, 51] were published

in which the effect of the width of the contact area is also considered.

Two rigidly connected coaxial wheels, both approximated by a one- dimensional string model, are considered. The strings and their elastic supports are also supposed to be elastic in the circumferential direction. Segel [52] derived the frequency response characteristics for the

one-dimensional string model, and these are similar to response curves which arise in Saito's approximate theory for the beam model.

Smiley [10] gave a summary theory resembling the one-dimensional

theory of von Schlippe [25]. He has correlated various known theories

with several systematic approximations to his summary theory. In [19] Pacejka gave the nonsteady state response of the string model

of finite width provided with tread elements. The important gyroscopic

effect has been introduced and the nonlinear behavior of the tire due to partial sliding has been discussed. Applications of the tire theory to the

shimmy motion of automobiles have been presented.

Transfer functions.

We shaU discuss here some theories of nonsteady state tire motion,

based on the stretched string concept, starting with the relatively simple

case where sliding in the contact area does not occur (/u,—» oo or ct^O

and </) 0). The response of the force Fy and the moment Mz with respect

to arbitrary variations of the slip angle a and the spin </> will be deter- mined for models which are successive approximations of the stretched

string model with tread rubber elements shown in figure 7.5.9, of which

the steady state behavior has been treated in section 7.5.1. The contact equations (7.5.9) apply when the velocities of sliding

F.sj- and Vsy are taken equal to zero. They read then:

du du , -——=-y(l),

dx ds

dv dv . = — , a-\- x(p. dx ds

(7.5.96)

These partial differential equations will be solved by using Laplace

transformation. The Laplace transforms have been written in capitals.

We will not transform with respect to time, as is done usually, but with

respect to the distance travelled s=Vt, where F is a constant. The
806 THE TIRE AS A VEHICLE COMPONENT

Laplace transform of a variable quantity, generally indicated by q, is defined through: ^ {q{s)} = Q(ps)= j^e-plq{s)ds. (7.5.97)

With the initial condition u{x, 0) = v{x, 0) = 0 at 5 = 0 we obtain:

^-PsU= -y^, (7.5.98a)

dV-^-PsV=-A+x^. (7.5.986)

The solutions of these ordinary first-order differential equations read:

t/=C„e.». +i-34., ^^^^^^ V= C^ePs^^ —A —^f —+ (7.5.996) Ps Ps\Ps J The terms CuC^s^ and CvC^s^ point to a retardational behavior. The

coefficients Cu and Cv are constants of integration. They are functions

of Ps and depend on the tire construction, expressed for example by

equation (7.5.37) and the boundary conditions (eqs. (7.5.39) and further).

The conditions at the leading edge are: x=a: u = OorU=0 (7.5.100)

leads to the following expression for Cu : Cu = -- y^e-Pl- (7.5.101) Ps For the determination of Cv we turn to eq. (7.5.37) whose transform is <Ti^-V, = -(l-e-^)V. (7.5.102)

With eq. (7.5.996) the following solution is obtained:

Fe=C+e^^-c + C_e--/-c+(l-62) |

-h —A-— ( —-fjt W|- (7.5.103) Ps Ps \Ps I J
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 807

The three constants of integration Cv, C+ and C_ may be solved with

the aid of boundary conditions discussed before (cf. eq (7.5.39) and

further).

With the use of eqs (7.5.26-28) and (7.5.40) the Laplace transform of

lateral force and moment can be obtained. For more details we refer

to reference [19]. The discussion will be continued for the simpler model

without lateral flexibihty of the tread rubber elements (cp^Qo, v=Vc). The

longitudinal flexibility of the tread elements will be maintained. The

constant of integration Cv appearing in (7.5.996) can now be found with

the aid of the condition at x=a expressed by (7.5.39), with v^Vc- Expression (7.5.996) then becomes:

Psl 1 + O-ps \ PsJ

At the leading edge the deflection becomes:

(7.5.104)

^^ = TT^^^^-"^^ (7.5.105)

or transformed back:

-Y-^ — =oL-ci<^ = ^--T-a-^, (7.5.106) as (J as as where use has been made of the relations (7.5.2-4).

The first-order differential equation may be found immediately from

the original differential equation (7.5.10), when the condition that for x = a the slope becomes dvldx = — Vilcr is taken into account. For the calculation of the lateral force Fy and the moment acting on the simple string model, two methods are encountered in the litera- ture. The first method, employed by von Schlippe, makes use of inte- gration of the internal lateral forces along the length of the string ex- tending from minus to plus infinity. For the calculation of the moment,

von Schlippe introduced a correction factor (p in [51, eq (77)]) with

which the influence of the circular tire of radius r is meant to be ex- pressed. It turns out that the effect of the circular tire upon the moment

about the vertical z-axis through the contact center generated by the

internal lateral forces CcV is completely cancelled out by the torque

exerted by radial reactive forces Sir produced by tension in the string

which is stretched around an imaginary cylinder and shifted laterally a distance v. Smiley [10] and Hadekel [9] adopted the same erroneous

correction factor. Consequently, the circular shape of the tire (string)

has no effect (except a smaU influence due to the finite length of the cir- cular string (cf. Frank [12]) and the string can be considered to be

developed in a plane (i.e., the road surface).

h
808 THE TIRE AS A VEHICLE COMPONENT

The other method which leads to the same results is due to Temple

[9] who integrated the internal lateral forces only over the contact length

and added the influence of the internal tension forces in the string just

outside the contact region.

According to Temple's method we obtain for the lateral force: Fy = Cc\ vdx-\-S{vi-hv2)l(r (7.5.107)

J -a and for the moment due to lateral deformations denoted by M'^

M', = Cc \ vxdx + S(a + cr) (vx V2)/(t (7.5.108)

J -a where Vi and V2 are the deflections at x—a and — a respectively, and

S = cr^Cc according to eq (7.5.34). The moment due to longitudinal deformations of the tread elements u denoted by Mt becomes with eqs (7.5.26) and (7.5.40):

Mf=-cpx uydxdy. (7.5.109)

By adding up the contributions (7.5.108) and (7.5.109) the total moment

about the z-axis is obtained

(7.5.110)

The Laplace transforms of Fy, M'^ and Mf can readily be obtained now

with tl^e aid of eqs (7.5.34), (7.5.104), and (7.5.99a). In general, these

transforms can be written in the form:

Jf{Fy} = FaA-hF^^

^{M',} = M'aA+M^^ (7.5.111)

in which Fa, F^, Ma, Mi, and M$ represent the transfer functions

of Fy, Mz and with respect to the slip angle a and the spin (/) {=d4flds). In order to avoid double subscripts, the subscripts y and z are omitted.

We find the following transfer functions in vector form for the tire model

considered

(Fa,F^)=^

Ps 2{a + a) (l,-l)+lfl+^^e-/.«)

\ Ps/ Ps \ (TPs +1 /

PsJ. 1,(7 + (7.5.112)
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 809

2a{cr(cr+a) + Vsa^} (0,-1)

a(l + e-2/'s«) + ps{(T{(r+a) - l/p|} (1 - e-'"'s") {o-ps+l)ps

and furthermore

MS = 0

aps 1-2aps

in which the quantity has been introduced:

(7.5.113)

(7.5.114)

(7.5.115)

By transforming back the above expressions, the deflection, the force

and the moment can be found as a function of distance travelled for given variation of a and (j).

Response to step function of the slip angle.

An important characteristic aspect of tire behavior is the response of the lateral force to a stepwise variation of the slip angle a. The initial conditions at 5 = 0 read: v{x) =0; for 5 > 0 the slip angle becomes a = ao. From (7.5.104) we obtain for the lateral deflection of the string in the contact region: —=a-x + o-{l-e-(«+-^-«>/^} "(forjc >a-s) (7.5.116)

while for the original points the following simple relation holds: —=5 (for:^^a-5) (7.5.117) do

With (7.5.107), finally, the force has been calculated for the two

intervals, with and without original contact points.

Fy = Cc{2(a^a)s-y2s''}ao {s ^ 2a) (7.5.118a)

Fy = Cc{2{(T^a)^-2(T''e-^'-^''^'^}ao {s > 2a) (7.5.1186)
810 THE TIRE AS A VEHICLE COMPONENT

The latter part (7.5.1186) could have been obtained immediately from

(7.5.112). The variation of Fy, graphically shown in figure 7.5.34, may be

used for an experimental determination of the relaxation length cr of the tire. For this purpose the ratio of the force attained at 5 = 2a and the

steady state value CfaOi-o may serve. Another method for the determination of the relaxation length has been given in section 7.5.1, eq (7.5.51).

Figure 7.5.34. The response of the lateral force Fy on a step input of the slip angle a, calculatedfor the relaxation length cr = 3a.

Response to sinusoidal inputs (shimmy).

The frequency response functions for Fy and Mz can easily be found

by replacing ps by io)s in eqs (7.5.112-114). The path frequency (Os equals

27r/\, wh^re k denotes the wavelength of the motion. When we are dealing with sinusoidal motions with the x-axis deviating only slightly

from the Jc-axis fixed to the road (cf. fig. 7.5.1), it is convenient to replace a and cf) by the variables i// and y or )8(== dy/ds for /3 <^ 1). With the aid of the relations (7.5.2), (7.5.3), and (7.5.8) we find foi; the transfer functions

I
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 811

with respect to i// and ^ (or y), expressed in terms of the transfer functions found before:

(=-^;)=o.

)8

(7.5.119)

The frequency response functions, F^iiojs) etc., are the complex ratios between output, Fy etc., and the input ip etc. For the tire model with

and without tread elements their absolute and phase relationship has

been calculated. For shimmy analysis, the response of to ip is of greatest importance.

In figures 7.5.35a, b, c, d the various responses are shown, as a ratio

to their steady state values as a function of the nondimensional path

frequency, ajsO. Figure 7.5.35e shows the response divided by the constant K:*/a, the value of which approaches when ajs^°°. The

approximate responses treated below are also shown in these figures.

The phase angles (j) are taken as positive when the output lags behind

the input, which appears to be the case with the force and moment due

to lateral deformation. The moment M* due to longitudinal deformation

however, appears to lead in phase. The phase lead of M* causes a reduction in phase lag of the total moment with respect to i//, as has been

illustrated in the diagram of figure 7.5.36. This is a favorable effect for the suppression of shimmy.

In the complex plane shown in figure 7.5.36 the response curves are drawn for the moment M'^ , which applies for an infinitely thin tire, and

for the moment = M'^-\- for a tire of finite width with /<:*=CMaa.

The moments are made nondimensional by division by the steady state value Mzo-, which occurs at a=ao=ilJo- The curve for a tire with K*—CMaa

is obtained by vector addition of M'z and M^. Curves for other values of may be obtained by multiplying the vector oi by the factor K*/CMaa.

The calculated behavior of the linear tire model has unmistakeable points

of agreement with motions found experimentally at low values of the

swivel frequency. At higher values of the frequency, the influence of the

gyroscopic couple due to tire deformation, dealt with later on, is no

longer negligible.

We may note, furthermore, that above a certain value of k* the curve

for the total couple MzlM^o will not encircle the origin but will remain

on the right-hand side of this point. This appears to be a typical property

of curves obtained experimentally, which has not been explained before. The point of intersection of the curve for and the real axis (fig. 7.5.36)
812 THE TIRE AS A VEHICLE COMPONENT

Figure 7.5.35. Exact and approximate response characteristics of force and moment

with respect to \jj and ^for string type models (a* ^ 3a, a = 3.74a, e = 117.5; without profile elements: cr= 3a).

finite

Figure 7.5.36. Response curves in the complex plane for the moment Mz with respect to yaw angle i//.

represents the point of "kinematic" shimmy (cf. Kantrowitz [44] and

Saito [46]). This sort of shimmy may occur at very low values of speed

of travel, where the frequency and consequently the moments due to viscous damping and inertia acting about the king-pin axis become very

small.
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 813 Dynamic tire tests at low values of the swivel frequency show good

correspondence with the theory as will be shown after the introduction

of the gyroscopic effect. Experiments with tires of different tread shapes

indicate that tires with longitudinal ribs have a greater k* than tires with

a block profile. Tests indeed show that ribbed tires are less sensitive to shimmy [19]. The response for cog-^ ^ (X = 0, standing tire) might be used for experimental determination of parameter k*. Torsion of a standing tire about the vertical axis over a small angle i//o produces a moment:

-M,= Qijjo = {C.Ma + K*/a)i//o (7.5.120)

from which the quantity k* can be obtained after the determination of the aligning torque stiffness. Figure 7.5.35 shows that the exact responses of the models with and

without tread elements are qualitatively the same. Quantitatively, the

deviation from the response of the model with tread rubber becomes

larger for shorter wavelengths (larger o)s). In the region important for shimmy analysis, at the left-hand side of the hatched band, very good

correspondence exists. The exact treatment of the simpler model, however, is stiU too complicated to be used in the actual shimmy analysis.

We shall therefore consider three approximations for the response of the

model without lateral tread rubber flexibiHty.

Approximations.

A first approximate description of the behavior of this model originates

from von Schlippe [25]. The contact line is considered to be a straight

line connecting the two endpoints of the real contact line (see fig. 7.5.37). Only the deflections Vi and v-? of these points are of importance now. For

the transformed force and moment we obtain, after some manipulation

and with the aid of expressions (7.5.50): ^{Fy} = Cc(a+a){V, + V-,) = ^Fa ^^^^'^y

^{M;} = Cc{(t{(t + a) + Vsa^KFi - V2) = Cxia v,-v..

(7.5.121)

2a

With the aid of (7.5.104) the transfer functions become:

iFa.F^) = ^^ 2 1,-- + '^\, (-l,o-+a+l/p.)

2{(T-\-a)ps[ V psj o-p,+ l (7.5.122)

(Mi, Mi)

2aps

1 — e-2Ps« 2« (0, - 1 ) +

^^^^^

(- 1 , o- + a + 1/p,) [. (7.5.123)
Figure 7.5.37. The exact deflection of the simple stretched string model and three ap- proximations.

The responses with respect to i// and j8, obtained with the aid of formulae

(7.5.119), appear to coincide practically with those obtained from formulae

(7.5.112) and (7.5.113) in the range of wavelengths investigated for

cr = 3a (see curves "exact/v. Schlippe" in figs. 7.5.35-36).

Simulation of the von Schlippe representation by means of an analog

computer appeared feasible although complicated (cf. [19]). In this simulation use has to be made of equation (7.5.106) for obtaining V\, and of the

retardational behavior in order to generate v-i- The latter may be carried out with the aid of a memory device (magnetic tape recorder or otherwise).

Simpler directly applicable approximations are obtained by expanding

the exact response functions (7.5.112), (7.5.113) and also (7.5.114) into powers of ps. With the use of relations (7.5.119) we find for the power

series of the responses to the angular displacement \\f and the lateral displacement y of the wheel plane:

F^{crps-\- 1) = CFa{l — aps)-\-2cca^((T+a){{o-\-y3a)pi

-V3a2pf}+ . . . Fyi(Tps^l)=- C.aPs + 2ceaH ( o- + a ) Pl

-V3a2p3} + . . . ^i(o-ps+ 1) = — CMa(l — aps) — (cr+ a)aCMaPl

+ 2/3 (o-+a)a2(C'Ma-0.0667cca3)p|+. . .

(7.5.124)

Myicrps + 1 ) = CMaPs ( 1 " aps)

+ 2/3 aHCMa-0.0661cca^)pl-^
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 815

(7.5.125)

In the periodic case we have: ps= i(Os= io)/V=2Trilk. When in the

power series (7.5.124) truncation is made to the second power of ps, the shape of the contact Hne is approximated by a parabola and when the

second and higher power are neglected, the contact hne is approximated

by a straight hne, both touching the real contact line at the leading edge

(cf. fig. 7.5.37). The larger the wavelength becomes relative to the contact

length, the better the approximation will be. In the practical range of wavelengths, especially for the amplitude responses of to xfj, the

paraboHc approximation furnishes a great improvement with respect to the straight tangent approximation. The responses gotten from these ap- proximation are shown in figure 7.5.35. Also, the response due to longitudinal deformations (7.5.125) has been approximated according to a quadratic and a linear variation of these deflections along the ;c-axis (see fig. 7.5.35e). The linear representation (up to ps) corresponds to a viscous damping. The approximate differential equations for the force

and the moment, directly applicable for shimmy analyses, which cor- respond to the quadratic approximation of the response functions

(paraboUc lateral deflection ) read:

ds

dM'

cr — \- 1 ds

+ 2c.a2(o-+2/3^)|(o-+a)

^, ^ r dx\j dy\ r \i ^ d'y]

dxjj^ ds

d^f

ds''

(7.5.126)

When omitting the underlined portions of eqs (7.5.126), the linear ap- proximation (lateral deflection according to straight tangent) is obtained.

In that case Fy and M'^ only depend on v\ (cf. eqs (7.5.106), (7.5.46)

cr'^ = cr, (7.5.49)), and M* corresponds to a moment due to viscous damping (damping coefficient = /<*/F)

.

A final approximation can be obtained by neglecting the dimensions

of the contact area {a=b = ^). This leads to a representation of the

nonsteady state tire behavior which has been used by various authors.

Although the moment should vanish for the string model, some authors

maintain M' and use the measured aligning torque stiffness C.v/a.
816 THE TIRE AS A VEHICLE COMPONENT

Nonlinear behavior due to partial sliding.

The general treatment of the combination of nonsteady state drift and turning is extremely difficult. We shall therefore restrict ourselves

to the simple string type tire model without lateral tread rubber elasticity. The effect of tread width (M*) may be treated separately when

longitudinal sliding velocities are assumed to be small. From the above

analysis it appears that for a wheel swivelling about a vertical axis through

the wheel center, the two moments M* and M' do not attain their maximum at the same time. A phase difference ranging from V47r to about

V27r radians will occur in the practical range of wavelengths. In this range of wavelengths, many times larger than the contact length (order

of 10 times), the shape of the deflection of the tire approaches the sta- tionary shape, i.e., shows an almost straight contact line in case of complete adhesion (linear approximation mentioned above). If, further- more, in figure 7.5.15 the front sUding region is neglected, the lateral deflection at the leading edge V\ is the only quantity which governs the

lateral tire deformation. As long as no total sliding occurs, Vi varies ac- cording to the differential equation (7.5.106). Finally, for Fy and M^, measured or calculated characteristics as a function of slip angle a=V\l(j may

be employed. According to the steady state side slip theory, the relaxation length of a tire with tread elements decreases somewhat with

increasing slip angle (fig. 7.5.11). The introduction of a nonconstant cr, however, would lead to great mathematical complexity.

In [19] Pacejka uses an analog computer in connection with a tape

recorder for the simulation of a more exact representation of the nonstationary behavior of the partially sliding tire, which can be seen as a nonlinear extension of von Schlippe's approximation of the behavior at complete adhesion.

Gyroscopic couple due to lateral tire deformation.

In [19] it has been shown that inertial effects due to lateral vibrations

of the peripheral line of the nonrolling tire may be neglected for frequencies lower than about 8 Hz. At higher frequencies tire inertial forces are expected to have considerable influence upon the tire deformation and

consequently upon tire force and moment response. Approximate cal- culations indicate that the ratio of inertial and elastic forces amounts

to 6 percent and 24 percent for a frequency of 7 c.p.s. and 14 c.p.s. respectively. In case of a rolling tire we have to deal with the substantial acceleration of a material point of the tire. For a point P (fig. 7.5.38) of a tire with

angular speed Cl and radius r we obtain:

d^v _d^v

d?~^ dxdt

ra^'-^r^a^. (7.5.127)

•The first term represents the inertial force discussed above. We shall neglect this term in the following. When calculating the moment about

the vertical axis, taking into account the last two terms of (7.5.127) to-
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 817

Figure 7.5.38. The deflected tire peripheral line.

gether with the centrifugal force acting on a tire element in radial direction, and furthermore a lateral deflection v{x) of a massless tire model

according to eqs (7.5.38) and (7.5.104). we find that only one term remains,

which represents the gyroscopic couple. This couple corresponds to the

g>Toscopic couple which would arise when the circular peripheral line of the tire is tilted about the horizontal fine which lies in the wheel plane

and passes through the wheel axis. The lateral deflection of the tilted peripheral line represents the first odd harmonic of the Fourier expansion

of the lateral tire deflection. This imaginary angle of tilt {—yt) wiU be

approximately proportional to the lateral tire force Fy. We obtain, then,

for the gyroscopic couple:

M,„ = /;ny, = -C,,,F^, (7.5.128)

in which represents the effective fraction of the tire polar moment of

inertia. Cgyr is a tire constant which is proportional to tire mass, m?, and

tire lateral compliance, 1/Cy, and is furthermore influenced by the type

of tire construction expressed by the dimensionless parameter Cgyj.. (7.5.129)
818 THE TIRE AS A VEHICLE COMPONENT

For a conventional tire we found that Cgyi.= 0.12 is a value vs^hich pro- duces a good approximation to experimentally obtained responses.

From (7.5.128) we obtain the following expressions for the transfer

functions oi M^.yj.:

(7.5.130)

from which we see that for a certain wavelength X of the swivel motion,

where ps= io)s — '27rilX is a constant, the response increases quadratically with the speed of travel V. In the complex plane, the vector of M^yj. will be directed perpendicular to the vector of Fy. In figure 7.5.39 theoretically obtained response curves of the moment with respect to i// are shown for the model with lateral tread rubber elasticity (cr*=2.4 a, 6=1/7.5). The value of cr* has been experimentally determined with

the aid of eq (7.5.51) for a tire whose experimental response curves are shown in figure 7.5.40. The value of e has been estimated. Figures 7.5.35-

36 show that the use of the simpler model without lateral tread rubber

flexibility (exact or parabolic approximation) is expected to yield results which are close to the theoretical curves of figure 7.5.39 in the practical

range of the dimensionless path frequency w^a < 0.35. The straight

tangent approximation needs longer wavelengths for acceptable results.
HORIZONTAL MOTIONS (TRANSIENT AND SHIMMY) 819

Figure 7.5.40. Influence offrequency of motion upon response (experimental).

Tests carried out on drum test stand (cf. figure 7.5.40a). Tire 9.00-16 block tread pt= 1.75 bar. A reasonable correspondence between theory and experiment appears

to exist. The figures clearly show that an important time influence exists. With increasing swivel frequency the curves of the moment rise, which means that the phase lag decreases and thus the degree of self-excitation of the system becomes less. The points of the experimental curves obtained at V= 0 (standing tire) are situated above the real axis and become lower and further to the

right at increasing frequencies. This may be due to the viscoelastic

properties of the tire rubber, which shows larger stiffness and less damping at higher frequencies. The amplitude of the force Fy appears to

Figure 7.5.40a. Shimmy excitation test standfor measuring tire frequency response.
820 THE TIRE AS A VEHICLE COMPONENT

increase with increasing frequency. This property has not yet been ex- plained theoretically. For shimmy analysis the effect of such an increase

in amplitude appears to be of less importance.

The machine with which the shimmy response tests have been carried out is shown in figure 7.5.40a. The wheel is swivelled as it rolls over the

drum, together with the whole structure in which the wheel axle is mounted. The structure is excited against four coil springs in the reso- nance frequency so that only a small force of excitation is needed. A

special measuring hub (described in sec 7.2) has been used for the

measurement of forces and moments. Because of the fact that the wheel

inertia distorts the signal, a correction is needed in order to obtain the

torque Mz acting from drum to tire. Results of other shimmy response

tests can be found in references [46] and [57].

Effect of a Time-Varying Load

Due to road roughness, variations in vertical tire load occur which

cause unfavorable changes in the tire cornering behavior. This has been

found experimentally by, amongst others, Kurz [53]. The long wavelength

phenomenon can be explained sufficiently by consideration of the steady

state tire characteristics. As has been indicated by Kurz, the static loss of the average cornering force is due to the typical nonlinear variation

of cornering force with normal load (cf. fig. 7.5.41). Tests carried out by

Metcalf [55] (cf. also Endres [54]) reveal that short wave length motions

show an additional loss in side force, which Metcalf has termed the

dynamic loss, although the cause may be entirely of a kinematic nature. In [56] Pacejka made mention of a theoretical study dealing with this problem. The essence of this theory, which has been restricted to small

slip angles, will be enunciated below. After that, a semi-empirical theory

Figure 7.5.41. Effect ofslip angle and normal load on tire side force {from Kurz [53\).
EFFECT OF A TIME-VARYING LOAD 821 due to Metcalf will be given, and finally the direct interaction of vertical and horizontal tire motions will be treated approximately according to a theory of Bohm.

Vanishing values of slip angle.

For the theoretical explanation of the behavior of a tire drifting over

a rough road, theoretical tire models will be employed of which the steady

state and shimmy properties have been dealt with in sections 7.5.1 and

7.5.2. The more advanced string tire model with tread rubber elements

appears to be the most promising in predicting the real response. The

analysis is restricted to the case of complete adhesion in the contact

zone, so that in fact the variation of cornering stiffness at zero slip angle

will be determined for the model considered.

In the theory below, the slip angle a and the direction of motion of the

wheel axle remain constant. Camber and longitudinal forces are not considered. Figure 7.5.42 shows the model in an arbitrary situation. When

the contact center is moved a distance As, the following changes occur

simultaneously: lateral shift of wheel plane, forward rolling and change

in contact length. These changes influence the position of the leading and

traihng edges of the string.

The x-coordinates of these points read:

(7.5.131)

of which the changes become

Axi = A5 + Aa I (7.5.132)

Ax2 = As — Aa\

Figure 7.5.42. Two successive positions of tire model with varying contact length rolling

at constant slip angle a (sliding not considered).
822 THE TIRE AS A VEHICLE COMPONENT

The changes in front and rear lateral deflection Vci,2 are composed of contributions due to various causes. For the deflection at the front we

obtain:

1. due to lateral wheel displacement (aAs)

Avci^A(a) • aAs

2. due to loss of contact rear (Ax-? > 0):

ilVci= — ti{a) X2

3. due to loss of contact front {Axi < 0):

Az;ci = 5*(a)

Vi—Vc

\Xi

(7.5.133)

4. due to longitudinal displacement of leading edge (A^i):

Avci = -'^'Ax,

Similar expressions are obtained for the contributions to the change of the lateral deflection at the rear {vci). The contact-length-dependent

coefficients appearing in the expressions are: ^ ( G ) = { -2e + ( 1 + e ) e2«/o, - ( 1 _ e ) e-2«/o, }IP(a)

B{a)=2IP{a)

B*{a)= {(l + e)e2«/^c+ [l - €)e-^^l^c}IP(a)

1-e

1 —

e

a/cr„ 1 + e

(7.5.134)

They are derived from solutions of the differential equation for the shape

of the string (7.5.35). For the parameters employed we refer to expressions (7.5.34). From the differences shown above (7.5.132-133) the fol- lowing differential equations for the unknowns Vci and Vc2 are formed:

dvci ds

dVc2 ds

da\ V2 — Vc

+ 5*(a) fl + ^) \ as J (Tc

0"c da\ vi — Vc\ + A{a)a,

+ B{ 1 +

(Tc da\ vi—vc\

ds

+ A{a)a.

(7.5.135)
EFFECT OF A TIME-VARYING LOAD 823 The two remaining unknowns Vi and v-z^ denoting the total lateral deflec- tions and consequently the distance between wheel plane and contact

line at the leading and trailing edges, are found by use of the following

considerations. In the rolling process of a drifting tire with a continuously

changing contact length, in general, three intervals can be distinguished

during the contact phase of the loading cycle. In figure 7.5.43 a possible

variation of the contact length 2a has been shown in the road plane {x, y). The tire touches the ground over a certain distance of travel. Immediately

after the first point touches the road, the contact fine will grow in two

directions. This will continue until the second interval is reached, where

growth of contact takes place only at the front, and at the rear loss of contact occurs. In the third interval, finally, loss of contact at both ends

takes place until the tire leaves the road. When the tire does not leave

the road, an additional interval II occurs before interval I is reached

again and the cycle has started anew. In less severe cases, intervals I and III may not occur. We then have the relatively simple situation of continuous growth of contact at the front and loss of contact at the rear.

Figure 7.5.43. The development of the contact line of the tire model which periodically

loses contact with the road.

Arrows indicate the directions in which the positions of the leading and traiHng contact points change.
824 THE TIRE AS A VEHICLE COMPONENT

The unknowns V\ and V2 and the y-coordinate of the contact points in the (Jc, y) plane are obtained as follows (cf. fig. 7.5.43).

Interval I: 5;i > 0, ^2 < 0 > 1

Vi = Vcl , V2 = Vc2

Interval II:

— yci = sa — Vcu— yc2 = sa — Vc2

JCi>0, ^2>0 (|-1<^<1

Vi = Vci , V2 = sa-\- yc2

Yd =sa — vc

Interval III: JCl < 0, X2 > 0

Vi=sa-\- jc V2

(7.5.136)

By means of numerical integration equations (7.5.135) have been solved

with the aid of the equations (7.5.136) and the expressions (7.5.134) for the case formulated below in which the vertical tire deflection 6 has been

given a periodic variation with amplitude h„ greater than the static deflection So, so that periodic loss of contact between tire and road occurs. The

rated or static situation will be indicated with subscript o. The tire radius

is denoted by r. The vertical deflection and the contact length are gov- erned by the equations

8 = 80 — ha sin a)s5, (7.5.137)

(7.5.138)

The numerical values of the system under consideration are as follows:

ao = 0.25r, (80 = 0.03175r) ,da = 0. Ir, =

a = 3ao, 6 = 0.25, {(Tc = 0.7Sao)

.

(7.5.139)

In figure 7.5.44 the calculated variation of contact length 2a (oval curve),

the path of the contact points (lower curve AB) and the course of the

points of the string at the leading and trailing edge {AC and BC) are indi- cated. Two positions of the tire are shown, one in interval II and one in interval III.

It is necessary that the calculations in the numerical integration proc- ess be carried out extremely accurately. A small error gives rise to a
EFFECT OF A TIME-VARYING LOAD 825

Figure 7.5.44. The deflected tire model in two positions during the interval ofcontact with

the road. rapid build up of deviation. In the case considered above, the integration time is Hmited and accurate resuhs can be obtained. In cases, however, where continuous contact between tire and road exists, a very

long integration time is needed before a steady state situation has been

attained, which is due to the fact that the exact initial conditions of one

loading cycle are not known. For this sort of situation, the exact method

described above will be difficult to apply due to strong drift of Vc-i in particular.

For further investigation of the effect of a time varying load we will turn to an approximate method based on the behavior of a string model

without tread elements.

Simple string model.

The treatment of this model is much simpler since the deflections at both ends are independent of each other. Drift does not occur in the

calculation process. In figure 7.5.45 the deflected model has been shown

in interval II. In the three intervals the following sets of equations apply.

Growth of contact at leading edge:

(7.5.140)

— Yd =sa — vi.
826 THE TIRE AS A VEHICLE COMPONENT

Growth of contact at trailing edge:

X2

ds

(i>')-

. cr \ds / X2 = s — a, — yc2 = sa — V2. Loss of contact at leading edge:

fcl =yc,s+n,

Vi=sa-\- fcl

.

(7.5.141)

(7.5.142)

Loss of contact at trailing edge:

X2 >

yc2 = yc,s-a,

V2 = sa-\-yc2.

(7.5.143)

Solutions of the above equations show considerable differences from the

results obtained using the more advanced model with tread elements.

The most important difference is the fact that with the simple model the

lateral force does not gradually drop to zero as the tire leaves the ground.

In order to get better agreement we introduce a relaxation length

cr = cr* (a) which is a function of the contact length 2a. We will take cr* equal to the relaxation length of the more advanced model according to eq (7.5.47) and figure 7.5.18. In figure 7.5.46 a comparison is made of the results for the three cases: without tread elements, with tread elements (exact) and according to the

approximation with varying (t = ct* (a). The calculations are carried out

for the values (7.5.139). The approximate path shows good agreement

with the path of the contact points for the model with tread elements.

When the tread elements are omitted the path becomes wider and the

lateral deflections become greater.

The lateral force Fy and the moment Mz which act on the tire can be

determined with good approximation by the following simplified for-
EFFECT OF A TIME-VARYING LOAD 827

Figure 7.5.45. The development of the contact line for the simple tire model without

tread rubber.

Figure 7.5.46. Paths of contact points and variations of lateral force Fy according

to three theories.

mulae. In their derivations we have replaced the contact line by the

straight line connecting the begin and end points of contact. For the

model with tread rubber elements we obtain:
828 THE TIRE AS A VEHICLE COMPONENT

2ccCp {a-h (TA{a)} V( Cc + c,j — 2ccCpa

(7.5.144)

Cc + Cfj {Vs a'^ + a (crH-a) C (a)} ao.

and for the model without tread elements:

Fy = 2cc (a + cr) Vo, M,'-=-2cca {Vs a^ + cr (cr + a)} ao. (7.5.145)

Here we have introduced the quantities

and Vo=y2 {Vi-\-V2), Oio=y2 {Vi—V2)la (7.5.146)

C{a) =A{a) -- eB^'ia) + (2 + -) €B{a) (7.5.147) a \ a/

in which A, B and B* are given by expressions (7.5.134).

For the sake of completeness the formulae for the moment have been

given. We shall restrict ourselves to discussion of the variation of the

force. For the three cases considered, the variation of the cornering

force has been shown in the same figure 7.5.46. As expected, the approximate and exact theories drop to zero at the point of lift off, whereas the

force acting on the simple string model remains finite, under the assumption of no sliding. The correspondence between exact and approximate

solutions of the path and of the force are satisfactory, and we will henceforth use the approximate method exclusively for the investigation of the

model with tread elements.

For a series of amplitudes 8a and path frequencies cos the variation of the cornering force, or rather of the cornering stiffness, has been calculated. For illustration figure 7.5.47 gives the time histories of the corner- ing force acting upon the model without tread rubber elements. Four

cases are considered. The upper figure shows the effect of a large amplitude, and it is clearly seen that oscillations with the larger wavelength

(\) attain a higher average side force. The effect of an amplitude equal

to the static deflection is presented in the second figure. Two possibilities are considered. When contact is not lost, the tire lateral deflection

does not need to be developed anew each cycle from the undeformed

state, which happens when the tire leaves the ground. This causes the

much lower level of the cornering force shown in the latter case. The

lower figure gives the force variation which occurs at moderate values of the vertical amplitude.

Since during loss of contact negative vertical forces cannot be trans- mitted, the period of the total loading cycle must become greater in order to keep the average vertical load unchanged. This change in period

has been taken into account in the calculation of the average side force

or cornering stiffness Cpa.
EFFECT OF A TIME-VARYING LOAD 829

V(orW)

Figure 7.5.47. Calculated lateral force variation of the drifting simple string model

(sliding not considered) due to a periodically changing vertical load W.

The lower figure represents the case of constant contact, the center figure the cases of just maintaining and losing contact and the upper figure the case of periodic loss of contact. Figure 7.5.48 shows the final result of this investigation, viz the corner- ing stiffness averaged over one complete loading cycle as a function of

vertical deflection amplitude b,, and path frequency oig for both models

with and without tread elements. With the values cr = 3ao and e = 0.25
830 THE TIRE AS A VEHICLE COMPONENT

STRING MODEL WITH TREAD ELEMENTS

Figure 7.5.48. Calculated variation of average cornering stiffness Ci-a with amplitude of

tire normal deflection d„ and path frequency.

the relaxation length of the more advanced model becomes ao=l.7ao.

For the model without tread rubber a relaxation length has been con- sidered equal to (t = 2ao. The figure clearly illustrates the unfavorable effect of increasing the

amplitude and the path frequency. The curve at zero frequency is purely

due to the nonlinear variation of the cornering stiffness as a function of

vertical load shown in figures 7.5.18 and 7.5.41. A pronounced difference

between the response of both types of models is the discontinuity which

takes place in the curves for the simple string model. This is in contrast

to the gradual variation in average cornering stiffness which occurs with

the more advanced model, and no doubt also with the real tire. The more

advanced model, furthermore, already shows a noticeable decrease in average cornering stiffness before the tire periodically leaves the ground.

The very low level to which the average stiffness has been reduced, once

loss of contact occurs, is of the same order of magnitude for both tire models.
EFFECT OF A TIME-VARYING LOAD 831

Finite values of slip angle.

Thus far we have only discussed the behavior at small slip angles,

where no sliding is assumed to occur. Larger slip angles will induce

zones of sliding and cause a much more complicated situation. We have

not attempted to extend this investigation into the nonlinear range of the

problem.

For the analysis of automobile motions we might use the experimentally or theoretically determined average cornering stiffness function

Cfa(8a, ois) for the construction of the average tire characteristic Fy{a)

from the basic characteristic Fyo(«eq) (cf. sec. 7.5.1) through the intro- duction of the equivalent slip angle.

«eq = 7; (7.5.148)

A semi-empirical theory of the effect of a time varying load has been

developed by Metcalf [55]. In this theory large slip angles and the

possibility of sliding have been included. Metcalf carried out a number of experiments with a 4.00-7 tire which was loaded against a rotating drum.

During each loading cycle, induced by a sinusoidally changing axle dis- placement, two distinctly different phenomena were observed. First, as the normal load increases from a minimum value, the lateral force lags

behind the applied vertical load. Secondly, after the vertical force has

reached a maximum and has begun to decrease, a point is reached where

the lateral force becomes equal to the instantaneous load on the tire multiplied by the friction coefficient between tire and drum.

This behavior has been simulated with the aid of an analog computer.

A first-order filter has been employed for the generation of the phase lag.

The response to a step change of the input corresponds to the lateral force response of the tire to a step change in vertical load (equal to the

response shown in fig. 7.5.34) approximated by:

Fy = Fyo{l-e-^l-). (7.5.149)

Although Metcalf recognized that the relaxation length a changes with

vertical tire deflection, in the computations cr has been held constant for the sake of simpHcity. By means of a "least selector," the lateral force

has been limited to the friction coefficient multiplied by the normal load

In figures 7.5.49-54 Metcalfs computer and experimental results are presented. In the simulation a number of constant values of cr have been

tried. In figure 7.5.49 the dynamic loss, which together with the static loss indicated in figure 7.5.41 forms the reduction in lateral force due to load variations, has been plotted against reduced or path frequency o)s for three drum velocities. The test data from this figure support the hypothesis that the phenomenon is distance rather than time dependent. In

figure 7.5.50 the average lateral force Fy has been plotted against re- duced frequency ojs for two values of slip angle. The amplitude of the
832 THE TIRE AS A VEHICLE COMPONENT

'200

100

400 _ 7 tyre

Pi r1.^ bar

Av. load Wo =13A0 N

Osc. ampLfa =1.27cm

Steer angle o(,5** O V .175km/h

Speed A V =43km/h

V , 78 km/h

10 20 Reduced Loading frequency, OJs (rad/m)

Figure 7.5.49. Dynamic side force loss versus reduced loading frequency for three drum

velocities.

(Figs. 7.5.49-54 are taken from Metcalf [55]).
833 EFFECT OF A TIME-VARYING LOAD

tire vertical deflection is lower than the static deflection so that contact between road and tire is maintained. In figure 7.5.51 results are plotted

for the same tire but with higher inflation pressure and an amplitude

greater than the static deflection, which results in a periodic loss of contact. Completely in accordance with the theory (cf. also fig. 7.5.48), the

reduction with increasing reduced frequency becomes stronger for amplitudes in excess of the static tire deflection.

(UD)

n)

i_

(U><

O measured compu ted

4.00 - 7 tyre

P. : 2.8 bar

Av. load = 13A0 N

Av.defl. So : 0.9 cm

Osc. a mpl. 8a : 1.3 cm

Steer angle a = 5° Speid V : 17.5 km /h

0 = 18cm

0 20 40 Reduced Loading frequency , (rad /m)

Figure 7.5.51. Average side force versus reduced loading frequency.

In figures 7.5.52-54 the time histories are shown for this latter configuration. These plots show agreement between test data and computed

results. The lateral resonance of the measuring axle accounts for the high

frequency osciUations present in the measured time histories. The decrease of Fy with ojs has been reasonably well predicted by the

analog computer results for cr= 18 cm. except at high values of slip angle

where tests show that a sharper decrease of Fy with Ws occurs. This

value (7=18 cm. is approximately 20 percent larger than the value ob- tained experimentally at rated constant deflection Wo and at small slip

angles.

Interaction of vertical and horizontal motions.

Hitherto the slip angle of the tire has been considered as constant. In

reality, however, the mass of the automobile is not infinitely large. Due

to the variation of the lateral force being caused by a time-varying load,

the wheel plane will move laterally. This brings us to the problem of combined nonstationary tire behavior influenced by both the vertical and

horizontal wheel motions.
834 THE TIRE AS A VEHICLE COMPONENT

0 04 08 ^2 Te 20 tl> Time (sec)

Figure 7.5.52. Time history, configuration offigure 7.5.51, low frequency.

P

r —0--Test data

y I. Analogue results 0*= 18 cm

3- 2a(=S) Axle displacement

(jj r A2.8 rad/s , 0)^= 8.2 rad/m

0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 Time (sec) Figure 7.5.53. Time history, configuration offigure 7.5.51, moderate frequency.

In the literature, this problem has been touched at by Bohm [49].

He presents an approximate theory of the lateral motion of the mass m

rigidly connected to the king-pin of the wheel which shows a constant

steer angle i|/ (cf. fig. 7.5.55). The mass is subjected to a constant lateral force K representing, for instance, the centrifugal force. Bohm considers
EFFECT OF A TIME-VARYING LOAD 835

^(.8) —-

Test data Analog results. C. 18cm

Axle displacement

Time (sec.

Figure 7.5.54. Time history, configuration offigure 7.5.51, high frequency. a periodically changing vertical load and cornering stiffness, from which

follows, according to eq (7.5.51), that for a constant lateral stiffness Cy

the relaxation length cr* also changes periodically. The following for- mulae are assumed to hold: r=ro/(i + ^ sin ojt). Cya = Cyaol ( 1 + ^ slu (i)t ) , o-* = o-*/(l + ^ sin cot).

(7.5.150)

From these assumptions it may be noted that the average values are not constant but increase with ^!

Figure 7.5.55. The simple tire model used for the analysis of interaction between vertical and lateral wheel motions.
836 THE TIRE AS A VEHICLE COMPONENT

The following simple differential equation for the nonsteady state variation of the lateral deflection v of the tire, valid for vanishing size of the contact area, has been applied:

1 . , 1 + r sin , 1 . yy^ —^ v-4^-yy- (7.5.151)

The equation of motion for the mass reads:

my = CyV-K. (7.5.152)

Elimination of y yields:

V /Cy ojV \ K

v-\- C sin (jjt) v-i- '\- C —r cos o)t v = — (7.5.153)

This equation with periodically changing coefficients may be solved with

the aid of the perturbation method. We consider f small and put

v = Vo^^v,{t) + C'v2{t)^. . . (7.5.154)

After insertion of this power series in (7.5.153) another series arises

which must vanish identically in f

, hence the coefficients of the successive powers of ^ must vanish.

When we restrict ourselves to the second power of ^ the following

form is obtained for the lateral tire deflection:

v = Vo+CAi cos (w^-(/)i) + ^2^2 COS (2wf-(/)2)+ . . . (7.5.155)

For the factor of the first harmonic we obtain:

ojV ojV

(7.5.156)

It may be noted from these results that the maximum variations of lateral deflection are obtained at a frequency of the load variation equal to the

natural frequency of the horizontal motion (o)'^ = Cy/m). Insertion of this solution in equation (7.5.151) yields for the velocity of the mass in the

lateral direction (assumed equal to zero for ^ = 0): V — y=^^i sin (wi — (//i) + ^2—^1 sin^w^- sin (/) 1 + ^-52 sin (2w^ — i//2)+ . . CTo (7.5.157)
REFERENCES 837 The average of this expression does not vanish. As Ai is negative (7.5.156)

the average cornering stiffness apparently increases with ^ (this is in contradiction with the result of Bohm, presumably due to sign error). The maximum average lateral velocity

jmax, ave = V2 C' —Vo (7.5.158) ^0

exactly equals the decrease in side slip velocity resulting from the rise

in average cornering stiffness due to the assumed variations (7.5.150).

A better assumption keeping the average load constant no doubt wiU

lead to different results.

References

[1] Joy. T. J. P., and Hartley, D. C, Tire characteristics as applicable to vehicle stability problems. Proc. IME, Auto. Div. (1953-1954), p. 113.

[2] Gausz, F., and Wolff, H., Ueber die Seitenfiihrungskraft von Personen-wagen-Reifen, Deutsche Kraftf. forsch. und Str. verk. techn.. Heft 133 (1959).

[3] Fonda. A. G., Tire tests and interpretation of experimental data, Papers on research

in automobile stability and control and in tire performance. Auto. Div. of IME, No. 7

(1956/57).

[4] Freudenstein, G., Luftreifen bei Schrag- und Kurvenlauf, Deutsche Kraftf.forsch. und

Str. verk. techn.. Heft 152 (1961).

[5] Nordeen, D. L., and Cortese, A. D., Force and moment characteristics of rolling

tires. SAE Paper No. 713A (June 1963); SAE Trans., 325 (1964).

[6] Henker, E., Dynamische Kennlinien von PKW-Reifen, Wissenschaftlich-Technische

Veriiffenthchungen aus dem Automobilbau (IFA-DDR) Heft 3 (1968).

[7j Fonda. A. G., and Radt. H., Summary of Cornell Aeronautical Laboratory tire test data. CAL Report YD-1059-F-2 ( 1958)

.

[81 Smiley. R. F.. and Horne. W. B., Mechanical properties of pneumatic tires with

special reference to modern aircraft tires, NACA (NASA) Tech. Note 4110 (1958).

[91 Hadekel, R., The mechanical characteristics of pneumatic tires, S & T Memo No.

10/52 (British Ministry of Supply, TPA 3/TIB, 1952).

[10] Smiley, R. F., Correlation, evaluation, and extension of linearized theories for tire motion and wheel shimmy, NACA (NASA) Tech. Note 3632 (June 1956).

[11] Frank, F., Grundlagen zur Berechnung der Seitenfiihrungskennlinien von Reifen, Kaut. Gummi 8 (18), 515 (1965).

[12] Frank, F., Theorie des Reifenschraglaufs, Diss. T. H. Darmstadt (1965).

[13] Fiala, E., Seitenkrafte am roUenden Luftreifen, VDI-Zeitschrift 96, 973 (1964).

[14] Ellis. J. R., Frank. F., and Hinton, B. J., The experimental determination of tire model

parameters. A.S.A.E. Report No. 2 (Sept. 1966).

[15] Biihm, F., Der RoUvorgang des Automobil-Rades, ZAMM 43, T56-T60 (1963).

[l6] Borgmann, W., Theoretische and ExperimenteUe Untersuchungen an Luftreifen bei Schraglauf, Diss, Braunschweig (1963).

[17] Bergman, W.. Theoretical prediction of the effect of traction on cornering force, SAE Trans., 614 (1965).

[18J Pacejka. H. B., Study of the lateral behaviour of an automobile moving upon a flat level road, Cornell Aeronautical Laboratory Report YC-857-F-23 (1958).

[19| Pacejka, H. B., The wheel shimmy phenomenon. Diss., Tech. University of Delft (1966).
838 THE TIRE AS A VEHICLE COMPONENT

[20] Rotta, J., Zur Statik des Luftreifens, Ing. Archiv, 129 (1949).

[21] Kalker, J. J., On the rolling contact of two elastic bodies in the presence of dry friction. Diss., Delft (1967).

[22] Nayak, P. R., and Paul, I. L., A new theory of roHing contact, Engrg. Proj. Lab., Dept. of Mech. Engrg., M.I.T. (1968).

[23] Savkoor, A. R., The lateral flexibility of pneumatic tire and its application to the

lateral rolHng contact problem, SAE Paper No. 700378 (1970); FISITA Int. Auto. Safety Conf. Compendium, p. 367, New York (1970).

[24] Savkoor, A. R., The relation of the adhesional friction of rubber to the friction between

tire and ground, Paper B-12, FISITA, Munchen (1966).

[25] Schlippe, B. von, and Dietrich, R., Das Pattern eines bepneuten Rades, Ber. Lilienthal Ges. 140, p. 35 (1941).

[26] Schlippe, B. von, and Dietrich, R., Shimmying of a pneumatic wheel (Translation of

[25]), NACA (NASA) TM 1365, p. 125 (1954).

[27] Holmes, K. E., and Stone, R. D., Tire forces as functions of cornering and braking slip on wet road surfaces. Paper 6 (IME Auto. Div. Symp., Handling of Vehicles under Emergency Conditions, Univ. of Technology, Loughborough, England, Jan. 8, 1969), Proc. IME 1968-69 1 83 (Pt 3H), 35 (1969).

[28] Gengenbach, W. (Private Communication).

[29] Riekert, P., und Schunch, T. E., Zur Fahrmechanik des Gummi-bereiften Kraft- fahrzeugs, Ing. Archiv 1 1, 210 (1940).

[30] Whitcomb, D. W., and Milliken, W. F., Design implications of a general theory of automobile stability and control. Papers on research in automobile stability, Proc. Auto. Div. of IME, No. 7 (1956-57), with many references.

[31] Segel, L., Theoretical prediction and experimental substantiation of the response

of the automobile to steering control, Proc. Auto. Div. of IME, No. 7 (1956-57).

[32] Mitschke, M., Fahrtrichtungshaltung, Analyse der Theorien, A.T.Z. 70(5), 157 (1968), with many references.

[33] Fiala, E., Zur Fahrdynamik des Strassenfahrzeuges unter Beriicksichtigung der

Lenkungselastizitat, A.T.Z. , 71 (1960).

[34] Pevsner, Ja. M., Theory of the stability of automobile motions (in Russian), Masjgiz,

Leningrad (1947).

[35] Hoffmann, E. R., Note on vehicle stability. Austral. Road Research 1, (1964) III, p. 15. [36] Antonov, D. A., Stability calculation of automobiles (in Russian), Avtomobiljnaja

promisjlennost. No. 9 (1963).

[37] Apetaur, M., Beurteilung der Fahreigenschaften von Fahrzeugen nach der Ergebnissen

der Priifungen durch gleichmassige Fahrt auf einer Kreisbahn, Conference on

Testing of Automobiles, Prague (Oct. 1965).

[38] Radt, H. S., and Pacejka, H. B., Analysis of the steady-state turning behavior of an automobile. Proc. Symp. Control of Vehicles 1963, p. 66 (IME, London).

[39] Bojm, F., Ueber den Fahrzustand des Kraftwagens auf einer ebenen Kreisbahn ohne Ueberhohung, Ing. Archiv. 32, 112 (1963).

[40] Koiter, W. T., and Pacejka, H. B., On the skidding of vehicles due to locked wheels. Paper 1 (IME Auto. Div. Symp., HandUng of Vehicles under Emergency Conditions, Univ. of Technology, Loughborough, England, Jan. 8, 1969), Proc. IME 1968-69, 183 (Pt3H),3(1969).

[41] Chiesa, A., Rinonapoli, L., and Bergoni, P. I. R., A new loose inverse procedure for matching tires and car using a mathematical model. Paper 4 (IME Auto. Div. Symp.,

Handling of Vehicles under Emergency Conditions, Univ. of Technology, Loughborough, England, Jan. 8, 1969), Proc. IME 1968-69, 183 (Pt 3H), 93 (1969).

[42] Harris, A. J., and Riley, B. S., Vehicle behavior in combined cornering and braking.

Paper 5 (IME Auto. Div. Symp., Handling of Vehicles under Emergency Conditions, Univ. of Technology, Loughborough, England, Jan. 8, 1969), Proc. IME 1968-69 183 (Pt3H), 19(1969).

[43] EUis, J. R., Understeer and oversteer. Auto. Engr. (London) (May 1963).

[44] Kantrowitz, A., Stability of castering wheels for aircraft landing gears, NACA (NASA)

Report 686 (1937).

[45] Greidanus, J. H., Besturing en stabiliteit van het neuswielonderstel, NLL Report

V 1038, Amsterdam (1942).

[46] Saito, Y., A study of the dynamic steering properties of tires, FISITA London (1962), pp. 101,246,282.
REFERENCES 839

[47] Fromm. H., Kurzer Bericht ueber die Geschichte der Theorie des Radflatterns, Ber.

Lilienthal Ges. 140, 53 (1941); or NAGA (NASA) TM 1365, p. 181 (1954).

[48j Bourcier de Carbon. C Etude theorique due shimmy des roues d'avion. Off. Nat. d'Etude et de Rech. Aer. 7 (1948); or NACA (NASA) TM 1337 (1952).

[49] Biihm. F.. Reifenmechanik und fahreigenschaften des automobils. Paper B-3, FISITA

Miinchen (1966).

[50] Schlippe, B. von, and Dietrich, R., Zur Mechanik des Luftreifens, Zentral fiir Wiss.

Ber.. BerHn (1942).

[51] Schlippe, B. von, and Dietrich, R., Das Flattem eines mit Luftreifen versehenes Rades, Jahrbuch Deutsche Luftfahrtforschung (1943).

[52] Segel. L.. Force and moment response of pneumatic tires to lateral motion inputs, Trans. ASME. J. Engr. for Ind. 88B(1) (1966).

[53] Kurz, H., Seitenfiihrungskraft des Draftwagenrades bei wechselnder Radlast, A.T.Z. 60(5) (1958).

[54] Endres. W.. \ ersuche liber das Verhalten des Autorades in der Kurve, VDI-Zeitschrift 106(4) (1964).

[55] Metcalf. W. H.. Effect of a time-varying load on side force generated by a tire operating

at constant slip angle. SAE Paper No. 713C (1963).

[56] Pacejka. H. B.. Discussion. Proc. Symp. Control of Vehicles during Braking and

Cornering 1963, p. 116 (IME, London).

[57] Ginn. J. L.. Miller. R. F.. Marlow . R. L., and Heimovics, J. F., The B. F. Goodrich Tire Dynamics Machine. SAE preprint 490 B (March 1962).

Index

A

Abrasion, rubber, tests for, 12 Accelerating force, interaction with cornering, 507 Adhesion

cord-to-rubber, RFL, 269

cord-stripping, 294

fabric stripping test, 295

H-test, 291

I-test, 293

polyester, 279

pop-test, 294

static tests of cord, 291

steel wire, 290

T-test, 291

testing of, 291

U-test, 291 Adhesion tests, dynamic, 295 Adhesive treatment

glass fiber, 291

polypropylene, 291

vinylon, 291 Aging, rubber, 9 Angles, helix yarn, 96 Aquaplaning of tires, 534 B

Bead, tire, 362

multiple, 366 Behed tires, 373

Bias ply tire, 372

Braking, 515 Braking force, interaction with cornering, 507, 669

Breakers, tire, 360 c

Camber, 663 Camber and cornering, 664, 782 Camber measurements, 665 Carbon black

description, 4, 33 reinforcement of rubber with, 32 Carcass elasticity, influence on string models of, 794 Carcass construction, effect on friction of, 526

Centrifugal loads on tire, 421

Chafer, 366

Chains, effect on friction of tire, 539

Circumferential slip, 515 Compression of cord, 172 Compression in contact area, 375 Compression test for adhesion, 295

Composition, tire rubber compounds, 3 Conditioning, tire, 595, 685

Construction, tire, 360, 379 Contact area

actual net, 457 membranes, 456

solid bodies, 454

tire, 447

cornering, influence of, 451 curved surfaces, influence of, 449

pressure, influence of, 449

velocity, influence of, 450

Contact pressures

in soils, 480

influence or structure, 472 normal or vertical, 472 on hard surfaced roads, 480

tangential

in cornering, 494

lateral, 494

longitudinal, 487, 605

longitudinal during braking, 491

Contraction, yarn, 71 Cord

adhesion, static tests, 291 behavior in compression, 172, 186 bending, 191 bending rigidity, 196 bonding, 265

buckling, 172

efficiency, 177

fatigue resistance, 253

fatigue tests, 257

fiber glass, 241

friction, 190 geometric models, 67, 73 geometry, 65

heat-treating, 240

heat treatment after dipping

for adhesion, 277

841
842 INDEX

Cord — Continued

impact resistance, 246

lubricants, 190, 222

nylon, 220

physical properties, 245

polyester, 238

radial tires, 244

rayon, 220

steel wire, 243

strain rate, effect of, 183

strength characteristics, 251

stress-strain curves, 186

stripping test for adhesion, 294

temperature, effect of, 183

tensile behavior, 111, 175 Cord path, 385

tire, 414

Cornering, definitions, 631 Cornering force, 506, 649

braking and traction, interaction with, 508, 669, 788 measurements, 639

with vehicles, 674

Cosine law, 415 Creep of rubber, 23

Critical speed, tire, 424, 746 D

Damping, cord-rubber laminates, 337 Deformation combined, 643

of laminates, 377 under braking, 603 under cornering, 639

Degradation, cord-rubber laminates, 346 Dynamic tire forces, vertical, 579 Dynamic modulus, rubber, 26, 29 E

Elastic constants calendered cord-rubber, 319 composite bias laminates, 324

Elasticity, theory of rubber, 14 Energy loss, rubber, 16 Energy loss, in tires, 31 Enveloping properties, 702

Equilibrium equations, shell, 408 F

Fabric stripping test for adhesion, 295

Fatigue, cord-rubber laminates, 341 Fatigue of cords, 253

Fatigue tests of cords, 257

Failure, cord-rubber laminates, 341

Fiber glass cord, 241

Fibers, merged, 237

Flange, rim, 367

Flatspot index, 234

Flatspotting, 233

Flex cracking, rubber, 12

Flexure tests for adhesion, 297

Flipper, tire, 365 Force response, longitudinal, 716

Fracture, cord, 347 Frequency method for tire stiffness, 580 Frequency response, tire, 722

Friction

definitions of, 44

influence of tread pattern and

carcass construction, 526 measuring techniques, 611 on ice, 539 rubber

adhesive, 46

classification, 45

deformation, 49

dry, 54

tearing, 54

viscous, 52

wet, 56 speed dependence, 46, 518

tires, 525 G

Geometry, cord, 65, 67, 73 Glass fiber, adhesive treatment, 291 Glass formulation, 241 Groove depth, effect on braking, 470 Gyroscopic effects, 816 H

H-test for adhesion, 291 Hardness, rubber, 11 High frequency properties, tire, 726 Hooke's Law, plane orthotropic

materials, 313 Hydrodynamic effects in skid, 468, 520

Hydroplaning, 468, 534

Hysteresis, effect on friction, 533

I

I-test for adhesion, 293

Ice, tire friction on, 539 Impact resistance, cords, 246

Inflation pressure

contact area, influence on, 449

cornering, influence on, 658

Irregularity in tires, see Nonuniformity

K

Kerfs, influence on tire friction, 528

Lateral stiffness of tires, 576

Lateral force distribution

in cornering, 646
INDEX 843

Load-vertical deflection, tire, 567 Load carrying, tire, 392, 398

Longitudinal deformation of tire, 642

Longitudinal slip, 711

Longitudinal stiffness of tire, 575

Lubricant, cord, 190M

Membrane stress resultants in a tire, 408

Migration, filament, 101 Modulus, rubber, 6 Molding, tire, 384 Mooney material, model for rubber, 22 N

Nonsteady state motion, 803 Nonuniformity

effect on vehicles, 621

grading machines, 624

tire, 616, 690

wheel, 621 Normal load, see Vertical load Nylon cord, 220

Nylon, high tenacity, 224 Nylon 6 tire cord, 225 Nylon 66, processing conditions, 228 o

Obstacles, 702

Oversteer, 512, 638P

Physical properties, fiber and

filament, 241

Polyester, adhesion, 279

Polyester cord, 238

Polyester, heat-treating, 240

Polypropylene, adhesive treatment, 291

Pop-test for adhesion, 294 Power loss, tire, see Rolling resistance Pressure districution, see Contact

pressures R

RFL, 269

Radial tires, 244

cord, 373 Radius of curvature, tire, 414, 417 Rayon

extra high modulus, 224

high tenacity, 221

high tenacity, production processes, 222 Rayon cord, 220 Reduced flatspot materials, 236

Relaxation length, cord in rubber, 336

Retraction, yarn, 88, 93 Rims, tire, 366 Road platform, 565 Road surface roughness, 521 measurement techniques, 676

Rolling radius

influence of carcass, 588

influence of load and inflation, 589

influence of speed, 589 measurement of, 587

tire, 584

Rolling resistance

effect of construction, 599

effect of material, 596

effect of operating conditions, 600

effect of speed, 591 measurement of, 592

tire, 591 Running-in of tires, 595 s

Self-aligning torque, 513 Shear stress, cord-rubber in tire, 419 Shear test for adhesion, 296 Shimmy, 810

Sipes, influence on tire friction, 528 Skid testers, 683

Slip angle, 631

Slip

definitions, 463, 503, 608, 757

longitudinal, 711 measurement of, 467

Slip regions in cornering, 463 Snow, tire friction on, 539 Speed, influence on tire friction, 46, 518, 523

Standing waves, see Critical speed

Steering of vehicles, 636

Strain energy functions, rubber, 21

Strains, measurement of, 430, 437

Strength loss, nylon cord, 230

Stress relaxation, rubber, 26

Stress resultants, tire, 417

Stress-strain properties

cord, 309

cord-rubber laminates, 316

rubber, 5, 309

String models combined effects, 799

generalized, 763

limitations, 798

Strip tests for adhesion, 296

Studs, effect on friction, 539

Superposition principle, rubber, 26 T

T-test for adhesion, 291 Tear tests, rubber, 10 Temperature, measurements, tire, 602

Tenacity, 221
844 INDEX

Tension

cord, 418

bead, 419

Testing equipment, tire 553

Texture, road surface, 521 Thermal shrinkage, nylon 6 tire cord, 228

Tire cords, see Cord

Torsional stiffness of tire, 577

Traction, 613

interaction with cornering, 669

Trailers, test, 686

Transfer functions, cornering, 800

Transient motion, see Nonsteady

state motion

Traveling waves, tire, 424 Tread compound, effect on friction, 531 Tread, function of, 369 Tread pattern, effect on friction, 471, 526 Tread wear, 648

Turn-ups, tire, 364 u

U-test for adhesion, 291 Understeering, 512, 638

Uniformity, fiber, 129V

Variable vertical load, influence of, 820

Variability in singles yarn, 135

Velocity effects on cornering force, 658

Velocity effects on rolling radius, 589

Velocity effects on

tire friction, 46, 518, 523

Vertical load

influence on cornering, 656

variable, influence on cornering, 820

Vertical stiffness, tire, 568, 695

Vibration transmission of tire, 726

Vibrations, tire tread normal, 733

lateral, 741

Vinylon, adhesive treatment, 291

Viscoelasticity, rubber, 23

Vulcanizing, tire, 385 w

Water films, 52, 521 Wear, see Tread wear Wet roads, influence on cornering, 653 Winter tires, 539 Y

Yarn

analysis by energy, 127

contraction, 70

cross-sections, 100

effect of transverse force, 121

helix angles, 96

large extension, 125

migration of filaments, 101

retraction, 88, 93

structure, 87
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. 2. Gov't Accession BIBLIOGRAPHIC DATA NBS-MN-L22

3. Recipient's Accession No.

4. TITLE AND SUBTITLE MECHANICS OF PNEUMATIC TIRES

5. Publication Date November 1971

6. Performing Organization Code la»r Clark, Editor

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS University of Michigan

Ann Arbor, Michigan 48104

10. Project/Task/Work Unit No. 4080401

11. Contract/Grant No. CST-386

12. Sponsoring Organization Name and Address National Bureau of Standards Department of Commerce Washington, D. C. 20234

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

This publication was funded by the National Highway Traffic Safety Administration, Department of Transportation, through the NBS, Interagency Agreement FH--1 1-6090.

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) The pneumatic tire has been an integral part of automotive transportation almost

since its inception, yet it remains a product whose characteristics are not easily

predictable or comprehensible by conventional engineering techniques. This treatise

is an attempt to provide a rational descriptive and analytical basis for tire mechanics.

Chapters of this book are contributed by active research workers in the fields of rubber and textile properties, friction, material properties, tire stress problems,

tire design and construction, vehicle skid and handling, and tire mechanical properties.

17. KEY WORDS (Alphabetical order, separated by semicolons) Friction; rubber; skid; tires; tire cord; tire contact; tire stress; tire structure.

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