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NASA CR-168171

NAS

National Aeronautics

A

 and

Space Administration

NONLINEAR DAMAGE ANALYSIS —

POSTULATE AND EVALUATION

March 1983

by

B.N. Leis and T.P. Forte

BATTELLE-COLUMBUS LABORATORIES

(NASA-CB-168171) NONLINEAR DAMAGE ANALYSIS: N86-26652

POSTULATE AND EVALUATION Final Report, 30

Sep. 1981 - 6 Apr, 1983 (Eattelle Columbus

Labs., Ohio.) 101 p HC A06/MF A0 1 CSCL 20K

Prepared for

National Aeronautics and Space Administration

LEWIS RESEARCH CENTER

Contract NAS3-22825

https://ntrs.nasa.gov/search.jsp?R=19860017180 2019-01-16T08:29:51+00:00Z
1 Report No

CR 168171

2 Government Accession No

4 Title and Subtitle

Nonlinear Damage Analysis — Postulate and Evaluation

7. Author(s)

B. N. Leis and T. P. Forte

9. Performing Organization Name and Address

Battell e Columbus Laboratories

505 King Avenue

Columbus, Ohio 43201

12 Sponsoring Agency Name and Address

NASA Lewis Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

3

5

6

a

10

11

13.

14

Recipient's Catalog No

Report Date

April 6, 1983

Performing Organization Code

Performing Organization Report No

Work Unit No

Contract or Grant No.

MAS 3-22825

Type of Repon and Period Covered

Final 9/30/81-4/6/83

Sponsoring Agency Code

15 Supplementary Notes

16 Abstract

The objective of this program was to assess the viability of a damage postulate which

asserted that the fatigue resistance curve of a metal is history dependent due to

inelastic action. The study focussed on OFE copper because this simple model material

accentuated the inelastic action central to the damage postulate. Data relevant to

damage evolution and crack initiation were developed via a study of surface topography.

The effectrof surface layer residual stresses were explored via comparative testing

as were the effects in initial prestraining.

The results of the study very clearly indicated the deformation history dependence of

the fatigue resistance of OFE copper. Furthermore the concept of deformation history

dependence was shown to qualitatively explain the fatigue resistance of all histories

considered. Likewise quantitative predictions for block cycle histories were found

to accurately track the observed results. In this respect the assertion that damage

per cycle for a given level of the damage parameter is deformation history dependent

appears to physically justified. Also, use of a history dependent nonlinear assessment

quantitative predictions of life to form a small crack when damage was linearly

accumulated.

17. Key Words (Suggested by Author(sl)

Fatigue

Crack initiation

Life Prediction

Damage Assessment

Damage Accumulation

OFE Coooer

19. Security Qassif (of this report)

Unclassified

18. Distribution Statement

20 Security Classif. (of this pagel 21 No of Pages 22 *'«'

Unclassified

" For sale by the National Technical Information Service Sprmgneld Virginia 22151

NASA-C-168 (Rev 10-73)
TABLE OF CONTENTS

Page

FORWARD 1

INTRODUCTION 1

APPROACH AND SCOPE 3

DAMAGE ANALYSIS 6

Concepts Involved in Damage Analysis 6

Damage 6

Damage Parameters 6

Similitude 8

Operational Definition of Nucleation 10

Phenomenology Relevant to Damage Analysis 12

Assumptions 13

Important Questions 14

THE DAMAGE POSTULATE 15

EXPERIMENTAL ASPECTS -. 17

Test Matrix 17

Data Requirements 17

Approach to Meet Objectives 18

Test Plan 20

Experimental Details 20

Material and Specimen 20

Apparatus and Procedure 24

RESULTS AND DISCUSSION 30

Material Characterization 30

Damage Evolution and Crack Detection 33

Stress Response 33

Ultrasonics 37

Surface Topography 38

Microhardness 52

Cyclic Deformation and Fatigue Behavior 54
TABLE OF CONTENTS

(Continued)

Page

Surface Topography 58

Surface Stress States 59

Measures of Damage 60

History Dependence of the Fatigue Resistance 63

EXAMINATION OF THE DAMAGE POSTULATE 65

History Dependence in a Strain Based Framework 65

History Dependence in the Absence of Transient Effects 70

Damage Assessment and Accumulation Predictions

for Block Cycling 78

Commentary 83

SUMMARY 84

CONCLUSIONS 85

REFERENCES 87

LIST OF TABLES

Table 1. > Test Plan 21

Table 2 Area and Crack Depth as a Function of Load Drop 36

Table 3. Surface Topography Study 40

Table 4. Change in Bulk Microhardness in OFE Copper

Under CA Cycling to Initiation 53

Table 5. Fatigue Resistance Data Developed Following

the Test Plan of Table 1 56

Table 6. Parameters Related to the Fatigue Resistance

Data Presented in Table 5 57

LIST OF FIGURES

Figure 1. Typical Application of a Damage Parameter Used to

Consolidate Multiaxial Fatigue Resistance of 1015

Mild Steel - As and Aet Are Equivalent Unixia l

Measures of Stress Range and Total Strain Range 9
LIST OF FIGURES

(Continued)

Page

Figure 2. Schematic Illustrating the Back Extrapolation

Procedure to Define Nucleation (OFE Copper,

Nf = 853 Cycles) ...................... 11

Figure 3. Essential Features of the Damage Analysis Postulate ..... 16

Figure 4. Axial Fatigue Test Specimen ................. 23

Figure 5. Evidence of CU20 Formation ................. 26

Figure 6. Evidence of CU20 and Its Apparent Influence

on Initiation ........................ 27

Figure 7. Overview of Test Cell and Gripping Arrangement ....... 29

Figure 8. Microstructures for Various Heat Treatments of

OFE Copper ......................... 31

Figure 9. Stress Response as a Function of Cycles for the

OFE Copper: AR, AR/IPS, HTA, and HTB Conditions ...... 34

Figure 10. Examples of Ultrasonic Records and Load Traces

As Indicators of Initiation ................. 39

Figure 11. Evaluation of Surface Topography in OFE Copper

Under Cyclic Loading .................... 43

Figure 12. Comparison of Monotonic and Half Life Stress

Response for OFE Copper from Selected Tests:

AR, HTA and HTB Conditions ................. 55

Figure 13. Trends in Microstructural Measures of

Transient Behavior ..................... 61

Figure 14. Fatigue Resistance of OFE Copper at 20 C

in Flowing Argon ...................... 67

Figure 15. Strain-Life Curves for a Variety of Steels ......... 71

Figure 16. Transients Evident in Dissipated Energy

As a Function of Cycles ................... 72

Figure 17. Fatigue Resistance As a Function of

Dissipated Energy ...................... 72

Figure 18. Fatigue Resistance As a Function of

80
FOREWORD

I

This program was developed as a result of NASA's support of

Battelle's unsolicited research proposal titled Nonlinear Damage Accumulation

Criterion, submitted April 14, 1981. The proposal followed the concepts and

approach outlined in a suggested research program of the same title submitted

by Battelle on March 13, 1981, following discussion of the subject program

with Dr. Gary Halford. The objective of the suggested program was to explore

the viability of a nonlinear damage postulate that differs from the literature

in the following significant way. Instead of assessing damage using the

conventional resistance curve, the postulate asserted that the resistance

curve is itself history dependent.

INTRODUCTION

Tracking or predicting the damage state of structural components is

essential to maximize utilization of materials and minimize losses associated

with unscheduled downtime for repair or inadvertent failures. Some components

are loaded in such a fashion that the maximum load encountered in service can

be apriority estimated within a small margin of error. In such cases design

against fatigue failure can be achieved by sizing components and selecting

materials such that the design stress is less than the endurance limit using

traditional concepts. That is "constant amplitude" cycling at the design

stress would not cause failure. However, most structures and their components

are not subjected to peak load limited situations which admit or ensure

"infinite life" design. Rather the majority of the loadings are variable in

amplitude (and, in general, involve a time varying direction of application).

Many times the maximum loading cannot be predicted within a small margin of

uncertainty, nor can the frequency of occurrence of very large loads be

predicted. In such cases a variable amplitude damage analysis leading to

"finite life" must be available to explore the extent to which such events

limit life and serviceability.
Damage analysis under variable amplitude cycling requires three

steps to be taken. First, ascending and decending segments of'the stress

strain history must l>e identified and matched, to construct closed loops

similar to that formed under constant amplitude cycling. Second, damage done

by a given event of loading must be assessed ±>y comparing the number of times

that event-occurs with the resistance of the material to such events. This

assessment thus involves a reference data base that characterizes this

resistance. In the simplest framework - uniaxial stressing - this damage

assessment requires a parameter that acccounts for the effect of the mean

stresses developed under variable amplitude cycling as compared to that of the

constant amplitude reference data,used in damage calculations. Finally,

damage done must be accumulated in accordance with a criterion .which specifies

how the damage is integrated, and a running damage sum must be maintained and

compared with a prescribed value that indicates when failure is predicted.

For more than 10 years the Japanese-developed rainflow cycle counting procedure [1] has found acceptance in damage analysis schemes, based on its

successful use in such applications 12]. Subsequently, a-number-of-numerical

procedures have evolved for the purpose of forming cycles [3-7]. These

procedures identify segments of hysteFesis consistent with the material's

plastic flow process, and have tended to displace rainflow procedures in

damage analysis. In any event, a number of experience proven schemes are

available for.purposes of matching hysteresis segments to form cycles—the

first step in damage analysis.

Numerous schemes are also available and continue to be developed to

account for mean stress effects. The list of mean stress parameters is long.

It includes one due to Morrow [8], one due to Smith et al [9] and variants or

alternately developed forms similar to that of Smith et al [10-11] and others

[12]. While successes are claimed in the literature for all mean stress

parameters, the fart that such parameters continue to evolve indicates

shortcomings exist in this aspect of damage analysis. Regarding the last

aspects of damage analysis—damage assessment and accumulation—numerous

studies -illustrate the controversy and uncertainties involved 'with linear

damage accumulation (e.g.[13]) whereas the subject of damage .assessment has

laid virtually untouched.
The mainstream of current technology continues to track the damage

stats by assessing and accumulating increments of fatigue damage on a cycleby-cycle basis according to the Palmgren-Miner linear damage hypothesis.

Studies have consistently shown the shortcomings of this hypothesis in the

context of comparisons of observed life and life based on a linear damage

calculation. Indicated errors are either conservative or nonconservative,

sometimes by as much as a factor of 10 [14]. While studies have demonstrated

errors as a consequence of the hypothesis, few if any have attempted to

isolate the cause of these errors. Pertinent data indicate that, even when

crack nucleation and crack growth damage are dealt with independently and

cyclic, history dependent, mean stresses are accounted for, the linear

accumulation hypothesis is still significantly in error when implemented with

a linear assessment scheme.

The objective of this research was to develop an understanding of,

and initiate formulation of an analytical model for damage analysis that

reflects .observed nonlinearities. It was desired that this model be sufficiently-simple, so as to permit its use in tracking the damage state in

metallic components subjected to a wid_e range of service histories, eventually

including elevated temperature. ~~

APPROACH AND SCOPE

While much data exist that indicate the inadequacies of the linear

hypothesis, none of these data were developed as a result of an experimental

program designed to fully explore and isolate the causes for the inadequacies.

For this reason, the damage accumulation process is ill-understood at both the

phenomenological and microstructural levels. The approach adopted for this

program has been to perform a series of experiments designed to develop

phenomenological and microstructural information regarding damage accumulation

that facilitate evaluation of the utility of the nonlinear postulate. Given

the fact that a new approach to damage analysis is envisioned, the scope of

this research has been restricted to a preliminary study designed to explore

the potential utility of the nonlinear analysis postulate.
Because little is known about damage accumulation, every effort was

made to isolate the process from the various factors known to influence the

damage rate - except for those being directly considered. For this reason,

uniaxial specimens were used to avoid multiaxial stress effects. To avoid the

complex interplay between strengthening mechanisms, a relatively simple

material was preferred. For this reason, OFE* copper was used; the material

was taken from one bar from one heat.

To avoid the uncertainty in dealing with both nucleation and growth

damage, this study focused on nucleation. ("Nucleation" is defined on page

10.) This has been achieved by using a smooth axial specimen designed such

that most of its life is spent developing cracks of the size defined as

nucleation. Finally, to avoid cycle dependent mean stress relaxation and the

influence of mean stresses on the damage rate, the experiments have been

designed such that unloading from tension or reloading from compression

following sequenced blocks and overloads induced a small mean strain, rather

than a mean stress. Small mean strains are considered not to be of much

consequence for the type of study reported herein.

After isolating the problem_as just detailed, one is left with the

conclusion that the shortcomings in tfie linear hypothesis are due to the

history dependence of the damage rate process. It is asserted that this

dependence is manifest in terms of changes in (1) the bulk microstructure and

(2) the surface topography and near surface residual stresses.

Changes in the microstructure are associated with hardening or

softening, and occur in the bulk until decohesion in persistent slip bands and

crack nucleation occurs. Thereafter, they are confined to the crack tip. It

is postulated that a particular microstructure forms in the reference constant

amplitude data base used in conjunction with the linear damage hypothesis in

predicting damage under variable amplitude loading. It is further postulated

that this microstructure may differ at a given damage (life) fraction from

that developed under variable amplitude cycling. It is known that the fatigue

*QFE denotes oxygen free electronic grade copper—the material is discussed

later in the Experimental Details section.
resistance of a material changes with microstructural changes that are manifest as changes in hardness/ductility. For this reason, it is asserted that

the baseline used in damage analysis should change to reflect history dependent changes in hardness/ductility associated with cyclic hardening or

softening due to large or small load levels encountered in a given strain

history. Because cumulative plastic strain is a measure of both hardening/

softening and the bulk micro-structural state, it is asserted that this

parameter may provide a very convenient basis to track the history dependence

of the damage process. Cumulative plastic strain is already tracked in

computer-aided damage analysis procedures. Therefore, accounting for the

history dependence of the damage process to such procedures should not be too

difficult.

With regard to surface, the character of the surface topography and

surface residual stresses also reflects the history. Consequently, the

evolution of surface residuals and roughening associated with slip step

formation through cyclic plastic flow, can also be tracked in terms of

cumulative plastic strain. Based on prior work, there appears to be a minimum

level or threshold in the development_of surface slip steps, after which slip

band decohesion occurs and crack nuclTation and growth ensues. Often the role

of surface roughening is thought of in the context of mechanical notch

effects. Consequently, until a notch forms, there is no effect - thus

tentatively explaining the expected apparent threshold.

It is asserted that the just detailed history dependence of the

damage process leads to the observed inadequacies in the linear history

independent hypothesis. An experimental program designed to pursue bulk and

surface manifestations of this history dependence in a preliminary fashion has

been developed and performed. The results have been analyzed to facilitate

evaluation of this postulate, and identification of controlling phenomenological parameters for the data developed. The approach adopted in this study to

develop a history dependent damage theory, therefore, has involved a series of

key experiments and analysis to identify the character of physically based

damage theory. Results of this preliminary study follow in sections dealing

with Damage Analysis, The Damage Postulate, Experimental Aspects, Results and

Discussion, Examination of the Damage Postulate, Summary, and Conclusions.
DAMAGE ANALYSIS

Concepts Involved in Damage Analysis

Damage

Damage is herein taken as the progressive change in a material's

microstructure, that eventually culminates in the nucleation and growth of

cracks, due to the localized action of reversed microplastic strain and

tensile stress. Thus the damage process involves the nucleation and growth of

cracks. During the nucleation stage the damage process occurs at a scale

strongly affected by the random nature of the microstructure, suggesting that

damage models at the micromechanics scale should be probabilistic. Furthermore, because the process occurs at a microlevel it is difficult to track.

Fortunately changes that lead to nucleation apparently also result in changes

in the bulk deformation behavior and in the character of the material's

surface. This coupled with the fact that bulk deformation parameters are

easily measured has led to their popular use as measures of the damage per

cycle. ~"

After nucleation fundamental studies indicate that the advance of a

crack is a simple geometric consequence of the slip at the crack tip.

Measures of damage related to crack advance therefore may be provided in terms

of the propensity for slip at the tip (local strain, etc., related to far

field stress and crack geometry) or geometric crack variables (CTOD, crack

length, etc.). Damage can be characterized by continuum mechanics parameters

before nucleation, whereas after nucleation the mechanics of cracked geometries is required. Clearly then when initiation and propagation are being

considered in a fatigue life prediction there is a "grey" area wherein a

transition must be made, based on crack size, from the initiation analysis

scheme to the propagation analysis scheme. The crack size at which the

transition is made is based on ones assessment of the size of crack at

nucleation. Fortunately, with an appropriate propagation analysis scheme the

error made as a result of the selection of the crack size at the "just

initiated" crack is small.
Damage Parameters

The concept of a damage parameter is introduced into damage analyses

because the reference data used to assess the damage done by a variable

amplitude event may not be developed under identical conditions. In general,

stress-strain path/multiaxiality, temperature, environment, loading rate, etc.

may differ between the component being analyzed and the reference data. Thus

a damage parameter, defined as an analytical (or empirical) relationship which

establishes the link between the complex damage process in the component and

the simpler situation for the laboratory reference data, is required.

For the present case the only potential difference between constant

amplitude reference condition and variable amplitude loading is mean stress.*

Thus only a "mean stress" damage parameter is required and discussed. Damage

parameters used in this study have the form [12]:

D = sm Aet + AsAet, (1)

where sm is the mean stress and As and_Aet are the stress and total strain

ranges respectively. This parameter was selected because it has been shown

useful in correlating data developed under both complex multiaxial proportional loading [12,16] and uniaxial mean stress conditions [17].

Simpler uniaxial forms for the parameter include the product As-Aet

in the absence of mean stress effects. This product is equal within a scalar

multiplier to the denominator of the Neuber parameter, which is popular in

notch analyses [18-19]. It can likewise be manipulated to yield

*While it was sought to avoid mean stresses through the use of specific

strain histories in the variable amplitude experiments, small mean stresses

did develop in some instances. Thus a mean stress parameter was required.

In cases where mean stresses have been avoided, this has been accomplished

with the introduction of small mean strains. Mean strain may be a factor

under certain conditions - for example forming prestrains [15]. However,

the small mean strains introduced to avoid mean stresses in the present

study were very small and thus their effect was ignored.
8

in which the right hand side is equal to the sum of elastic and plastic strain

ranges used by Morrow [20] to characterize fully reversed (zero mean stress)

fatigue data. For linear elastic conditions the parameter can be manipulated

to yield a parametric form which has units of stress:

D. = (DE)*5 = (S + As)As * (3)

i m

or

_ . (3-RH1-R) * (3a)

where smx is the maximum stress in a cycle and R is the stress ratio. This

form is similar to one variant of the parameter proposed by Smith et al [9].

For linear elastic fully reversed cases one obtains DI = As2, a form similar

to the Basquin relationship [21] when correlated with fatigue data. Finally

for situations where the inelastic strain dominates (Aet = AeP and sm ~ o),

one obtains D = (As) (AeP) or D = C (AeP)n'

+1 which, when correlated with

fatigue data, is a form similar to the Coffin-Manson relationship [22,23]. In

these respects the parameter given in-Equation (1) has a form operationally

similar to several of the experience proven parameters, when applied to specific simplified situations that these previously developed parameters had

been developed for. Its utility has been demonstrated elsewhere for a broader

more general set of situations [12,17]. The concept of the damage parameter is

illustrated by its application to correlating fatigue data for a typical case

in Figure 1.

Similitude

For a damage parameter to provide the link between the situation at

a critical area in a component, all of the factors which impact on the damage

per cycle at that location not embedded in the reference data must beaccounted

for by the damage parameter. Similitude means that equal values of the damage

parameter applied at two critical locations* results in equal lives

*Also, there must be similitude between the materials at these locations. In

a statistical sense, chemistry, phase(s), orientation, and other microstructural variables must be equal. It follows that there is a minimum volume of

material needed to develop this statistical homogeneity.
o>

«

u

Uniaxial

D Biaxial,

• Biaxial,

X Biaxial,

Biaxial,

Biaxial,

Biaxial,

0.01

o

a.

u

Cycles 1o Crack Initiallon

FIGURE 1. TYPICAL APPLICATION OF A DAMAGE PARAMETER USED TO

CONSOLIDATE MULTIAXIALfFATIGUE RESISTANCE OF 1015

MILD STEEL - As AND Ae ARE EQUIVALENT UNIAXIAL

MEASURES OF STRESS RANGE AND TOTAL STRAIN RANGE

(AFTER (12))
10

to crack nucleation within statistical scatter at both locations. Obviously

similitude between the damage processes at the two locations must be ensured

if accurate predictions are to be expected. Similitude can be achieved most

easily by demanding less of the damage parameter. This can be accomplished by

using constant amplitude reference data that reflect the important component

loadings - thermal, multiaxial mechanical, time and path dependent considerations being among the more difficult aspects to analytically account for.

Clearly too, different damage parameters and reference data are required if a

significant macrocrack growth stage exists in the component for a period

different than in the reference data.

Similitude is a key aspect of any damage analysis, be it for nucleation or for macrocrack growth as detailed in Reference [24] and [25] which deal

with crack growth and nucleation respectively.

Operational Definition of Nucleation

With respect to the above discussion, the focus here is on the

nucleation stage established in terms of when bulk parameters indicate cracking. The operational definition of nucleation* adopted herein is the life for

which the back extrapolated asymmetric drop in the tensile load intersects the

stable deformation response, as shown schematically in Figure 2. Calculations

of "area lost" due to cracking indicate this definition is associated with

surface cracks with a shallow aspect ratio whose depth is on the order of 50

to 200 urn. Similar calculations indicate that a one percent load drop relates

to a depth of 400 ym and the often reported five percent load drop related to

a depth of >1200 urn.

Experience with the above definition suggests that it typically

indicates "nucleation" before continuous monitoring with a reflection ultrasonic scheme using a specimen geometry and transducer optimized for crack

detection as detailed later in the section dealing with experimental results.

The threshold for detection using that scheme is a depth of about 75 urn for

calibration specimens. Nucleation based on the above operational definition

*Hereafter, the terms nucleation and initiation are used interchangeably.
11

Locus of stable

loop tips

Nucleation

Back

extrapolation

Time

FIG'JRE 2 SCHEMATIC ILLUSTRATING THE BACK EXTRAPOLATION

' PROCEDURE TO DEFINE NUCLEATION (OFE COPPER,

Nf = 853 CYCLES)
12

also has been compared with results of surface studies made using single stage

gold coated replicas interpreted a. up to 5000X magnification in a SEM. This

comparison detailed later in the section dealing with experimental results

indicates nucleation defined as above using bulk parameters correlates well

with the onset of growth of Mode I cracks from the much earlier formed and

subsequently intensified Mode II and Mode II/Mode I slip band extrusions and

long-shallow shear oriented surface cracks.

Phenomenology Relevant to Damage Analysis

The literature on damage analysis indicates either linear damage

accumulation, as suggested by Palmgren [26] and by Miner [27], or some nonlinear scheme (for a summary see [13]) is employed. In virtually all cases a

single curve is used to characterize the materials damage resistance—as for

example either a stress-life or a strain-life plot. The essence of all procedures is to compare cycles applied in a variable amplitude history to the

damage resistance observed under constant amplitude cycling. It is therefore

instructive to examine the constant amplitude situation to see if some insight

as to the character of a damage theory can be extracted.

First, under constant amplitude control, the response parameter

changes as a function of cycles and the driving force for transient action—

inelastic strain. Under strain control, hardening or softening occur as does

mean stress relaxation. Under load control hardening and softening also occur

but given the unstable nature of the control condition, cyclic creep may

intervene to cause premature failure. So long as one deals with constant

amplitude cycling, knowledge of the magnitude of the control parameter is all

that is required to specify the life within statistical bounds. This is

because changes in the response parameter which occur do so in similar fashion

from test to test. For strain control, Aee and AeP change so that the

ratio AeP/Aee varies on a cycle by cycle basis even though Ae^ remains

constant. Thus, as Equation (2) indicates stress, a factor for consideration

in more general situations, can be factored out to yield elastic and plastic

strain as the sole basis to characterize fatigue resistance for a fully

reversed situation.
13

Assumptions

For constant amplitude cycling it is reasonable to assume that up to

nucleation each cycle in the history does an amount of damage that is proportional to the value of the bulk damage parameter on the cycle. Because these

data serve as a basis for variable amplitude damage analysis, transients that

could be ignored in selecting the damage parameter for constant amplitude

cycling no longer can be ignored. The implication is that Ae^ may be an

adequate basis for comparing constant amplitude data but in general is an

inadequate basis for characterizing the driving force for damage under block

sequence or variable amplitude cycling.

The just stated assumption means that each cycle of a history for

which the value of the damage parameter is the same (damage parameter control)

causes an equal amount of damage. When integrated over the history, one

obtains a measure of the material's resistance to damage. Clearly one could

integrate the damage per cycle over the history using any weighting per cycle

he chose. However, the phenomono-logy does not indicate any one cycle is more

damaging than another. Thus it is assumed that damage per cycle is equally

weighted over the life to nucleation. This equal weighting leads to a linear

accumulation criterion. Applying this same logic to a constant amplitude test

done in strain control leads to a nonlinear weighting per cycle if strain

range is used as the damage parameter. This weighting follows from a comparison of the values of the damage parameter given respectively by Equations (1)

and (2). For materials which stabilize quickly the weighting affects only a

few cycles, whereas for materials such as RQC-100, which exhibits exponential

softening [5], the weighting affects much more of the life.

One can derive empirical weightings to modify the linear accumulation criterion to suit the damage parameter employed such as just illustrated.

However, it is much simpler to couple a more general parameter with the linear

accumulation assumption and assess and integrate damage on a cycle by cycle

basis for constant amplitude loading. Extending this approach to variable

amplitude loading requires a second assumption—that closed loops* under

*While closed loops are discussed this assumption and the one just prior

could have been equally stated in terms of reversals.
14

constant amplitude loading do the same amount of damage as their variable

amplitude counterparts. It is instructive to look briefly now at phenomenology related to damage assessment.

Important Questions

Some now consider it well established that a material's long life

endurance relates to hardness (strength) while its short life resistance

relates to its ductility in view of published correlations (e.g., [28]).

Transient action develops under cyclic loading (e.g., [20]) for which it seems

rational to postulate that the bulk hardness and ductility depend on.the

cyclic history. Also there tends not to be a saturation or steady state under

variable amplitude loading [29]. Furthermore variable amplitude stress

histories are not the same as that of the constant amplitude results used as

reference data in damage analysis. For example, in the long life regime

cyclic hardening would in this context enhance the material's resistance to

damage and require a nonlinear assessment of the damage done based on the

imposed history. Therefore it seems reasonab-le to ask: should the reference

data used for damage assessment reflect the history dependent hardness/

ductility state of the bulk component material in the vicinity of nucleation?

Second, it is reasonable to postulate in view of published data

(e.g., [30]) that the surface of a material roughens as damage localizes and

eventually tends to concentrate in areas from which cracks nucleate. Clearly

the surface character may evolve differently under variable amplitude loading

as compared to the constant amplitude results used as reference data in damage

analysis. Given that surface roughening acts as a notch one might expect

based on the literature (e.g., [31]) that in the long life regime the fatigue

resistance is strongly reduced once a notch of a threshold size is developed.

Again damage assessment is nonlinear, depending on the history imposed.

Therefore it also seems reasonable to ask: should history dependent surface

topography be embedded in damage assessment and if so how?

Third and finally, surface grains are free to deform under plane

stress at strain levels below their constrained interior counterparts*.

*In this context, surface means the plane stress surface layer.
15

Therefore it is conceivable that, for a narrow window of cyclic strains within

the low strain, long life regime, the surface could yield in tension or

compression and be constrained by the elastic interior. In turn this would

set-up surface residual stresses that could either retard or enhance the

damage process depending on the sense of the reversal that causes yielding.

(Pangborn [32] discusses this aspect in the literature.) Related to this are

grains most favorably oriented for slip which may also yield and be constrained by their unyielded counterparts in a very narrow window in strains,

biased again to the low strain long life regime. While the sign of the

residual stress that develops depends on the sense of the strain that first

caused yielding, fatigue is a selective process and will seek out grains in

which damage will accumulate most rapidly. Under constant as well as under

variable amplitude loading grains may be expected to yield under both tensile

and compression going cycles, with a few exceptions. Thus the effect of

compression yielding creating tensile residual stress will always be observed

in that it will reduce the life by locally enhancing the damage per cycle.

But the effect of local compression residuals due to tension yielding will

seldom be seen because cracking in other grains will intervene. In view of

this it is reasonable to ask: should the character of surface residual

stresses due to the loading history be imbedded in damage assessment.

THE DAMAGE POSTULATE

It is postulated that the nonlinear nature of the damage process

arises through a history dependence of the material's fatigue resistance which

depends on the bulk hardness/ductility, and the character of the surface

topography and residual stress field. In this respect the nonlinear nature of

the damage process involves damage assessment but not damage accumulation.

The essential features of the postulate are presented in Figure 3.

Experiments performed to evaluate the potential utility of this analysis

framework are now presented.
16

A Schematic of the General Situation

t

&o

I

Effect of hardness/ductility causes

layering due to bulk condition

CT^_ \ Effect of surface topography

~""-"^. J and residual stress state

The deformation history controls which

curve defines the materials resistance at

any instant in time

Cycles to Nucleation, N:

t

Damage Analysis at Specific Instants in Time

assessment:

Instant

Damage assessed for

N cycles at D|

Reference

N/NjR

A

N/NjA

B

N/NjB

0)

o>

o

o

a

Damage level I, D

Damage accumulation: linear throughout history to

nucleation

Initial resistance (Reference)

IA ^ INB

 cvc'es

N.B 'A r*^!:;—-"-Instant A after NA cycles

'

 Ni NjR

 ^-Instant "B" after NA-»-NB

t>

Cycles to Nucleation,

FIGURE 3. ESSENTIAL FEATURES OF THE DAMAGE ANALYSIS POSTULATE
17

EXPERIMENTAL ASPECTS

Test Matrix

Data Requirements

Meeting the objectives of this preliminary study required data be

developed to explore each of the key features of the damage postulate. To

this end the experimental program developed data to determine if (1) hardness/

ductility, (2) surface residuals and (3) surface topography develop in a

characterizable fashion as a function of history. Thereafter, data were

required to determine if there was a history dependence of the fatigue resistance and whether such a dependence could explain the apparent nonlinearity in

damage analysis under variable amplitude loading.

Detailed evaluation of the postulate required data to explore each

of the bulk and the surface contributions in an uncoupled fashion over a range

of lives that embraced a range of lives both well above, through, and well

below the transition life. Studies of the bulk needed to introduce bulk

changes, and then isolate their effect by removing the surface after pretreatment. Studies of the surface topography needed to develop the surface

morphology desired using various prestrain histories and then relieve all mean

stresses and hardening either through a decremental step loading and/or heat

treatment. Surface residual stresses needed to be studied either directly via

measurement or indirectly via their effect on life for comparable bulk states

and surface topographies.

Detailed study of damage evolution and the postulate advanced, while

useful and warranted in a fundamental sense, would involve far more data

development than that needed to explore the viability of the concepts that

underlie the postulate. This coupled with the novel history dependent

resistance curve central to the postulate suggests that the viability of the

postulate should be explored in a much reduced study. Finally it is often

easier to identify trends which if developed in the data for critical experiments indicate a postulate is either viable or questionable on empirical or

physical grounds, than it is to prove a postulates value. For this reason the
18

simpler approach was adopted. Consequently, if the postulate would be shown

to be consistent with the phenomenology, it will be argued that the postulate

is viable, and warrants further study as a basis to characterize the nonlinearity in damage analysis.

i

Approach to Meet Objectives

The postulate advanced followed from consideration of the phenomenology of damage which led to three questions. The purpose of the experiments performed was to develop answers to these questions through study of

three key aspects namely, the existance of the implied correlation of microstructural changes with bulk deformation parameters, and the evolution of

surface topography and stress-strain history. Thereafter the existance of

history dependent fatigue resistance curves is pursued.

The correlation between microstructural changes and changes in bulk

deformation parameters can and has been explored at many levels. Studies at a

very microscale address dislocation structures, as presented in the notable

works of Mughrabi [33], Laird [34] and others, and as summarized for example by

Kocanda [35]. It is changes at that level which are shown to relate to

eventual cracking and in turn are postulated to relate to bulk measures such

as stress. Changes in microhardness reflect the integrated changes in microstructure over a small volume of material and in turn these relate to changes

in dislocation behavior. Consequently, changes in microhardness measurements

reflect changes at the dislocation level. Therefore, a correlation of changes

in bulk measures with changes at the dislocation level can be demonstrated by

directly comparing the measured change in microhardness with the corresponding

change in stress under strain control. For the postulate to be conveniently

implemented, these changes must be a function of cumulative plastic strain

ZeP, and be independent of how ZeP is introduced.

Correlation between surface topography and stress-strain history can

be explored by tracking the surface via replication under the action of various histories. In the present study this was conveniently accomplished using

single stage replication performed on samples tested under selected combinations of strains and cycles. The procedure involved replication after each
19

major sequence event as well as at intervals equal to one tenth of the

expected life. The replicas were then examined in an SEM and the results

documented at SOX, 250X, 500X, 1000X and 5000X magnification. Again for the

results of this study to be useful they must correlate uniquely with some

parameter such as ZeP or some other related sum that includes stress.

Surface stress state effects can be examined in terms of the life

obtained under selected sequences over a range of strains. This study

explored the possible effects of surface stress states by starting tests in

opposing directions - one in tension and one in compression. Because the

extensive bulk inelastic action that develops at higher strains leads to a

homogeneous deformation state that washes out any microresiduals should they

develop, consideration of this possible effect began at the lowest strain

being considered. If there was an effect, the test started in tension should

survive longer than that started in compression for the same fully reversed

cyclic strain. If this result developed, testing would explore higher

strains, until the effect washed out. In all cases, the behavior of the

surface residual stress (if it is a factor) should be characterized as a

function of the imposed history.

The final aspect of the postulate examined was the implication that

a material's resistance to damage is history dependent, beyond the sequence

related effects of surface roughening. Recall, as hypothesized earlier, that

bulk changes in stress relate to changes in microstructure which in turn

relate to damage. This means that the material evolves in its hardness or

ductility so long as microstructural changes, evident in terms of changes in

bulk parameters, are occurring. Also, it means that if different stress

levels develop for the same strain range after different deformation histories

in a given material, then differing amounts of damage per cycle develop even

though the ensuing strain history is the same. Consequently if different

fatigue resistance curves can be developed for the same material in differing

initial states of hardness/ductility one can conclude that a sample of

material exhibiting transient response wil l also exhibit differing fatigue

resistances over its life. Because transient behavior is history dependent in

general, one could also conclude that the fatigue resistance is history

dependent, consistent with the postulate.
20

History dependence of a material's fatigue resistance was examined

as just suggested by comparing the fatigue resistances of a material in

several different initial hardnesses/ductilities. Results were developed by

testing samples prepared in one of four hardnesses (developed by heat treating

or mechanically working the as-received material). Testing was done at one of

three identical total strain levels, designated as Aej, Ae£ and £63, selected

to cause nucleation at about 10^, 10^, and 10^ reversals. These same levels

were used throughout the study.

Test Plan

The test plan followed is detailed in Table 1. It embraced a total

of 26 tests. Of these, 10 were targeted for constant amplitude cycling while

the remainder were subjected to some form of strain history. Additional

samples were committed to establish optimum conditions for ultrasonic detection of nuclea-tion, as well as for calibration of the ultrasonic output with

crack size.

Experimental Details

Material and Specimen

The material used in this study was specified to be oxygen free

electronic (OFE) grade copper. Copper was used because it provides a single

phase model material for which the inelastic aspects, indicated by the

postulate to control nonlinearities in damage analysis, will be accentuated.

The material was obtained in the form of 25.4 mm diameter bar, a total of 20

kg being purchased. The material's reported composition was verified via

spectrographic analysis as (in ppm) Pb:3, Zn<l, Te<l, Se<l, P:l, 0:2, 5:12,

Hg<l, Cd<l, Bi<l, Cu: balance. Some samples also indicated traces of silver.

In view of the chemistry, the material met the OFE copper specification ASTM:

F-68(77). Trace elements account for ~13 ppm so that the sample was 99.9987

percent pure copper. Mechanical properties determined by the supplier were a

297 MPa yield strength (0.5 percent offset) with an average elongation of 24
21

TABLE 1. TEST PLAN

No. of Tests

9

1

6

History

Constant

Amplitude (CA)*

Constant

Amplitudeb

Incremental

Purpose

Develop data for differing

hardness/ductility states

Examine effects of surface

residual stress

Develop a different initial

initial

hardnes

Prestrain (IPS)C

Initial

Overstrain (IOS)d

with Decremental

Prestrain (DPS)

Block

Cycling (BC)e

ductility state by mechanically

action

Develop differing initial surface

topographies

Develop data to test viability of

postulate

(a) CA fully reversed strain cycling starting in tension

(b) CA fully reversed strain cycling starting compression

(c) IPS consisted of a gradually decrementing block of 30 reversals to zero

strain from Aei, followed by a gradually incrementing block to the

desired peak strain for the test

(d) IDS/DPS consisted of an initial CA overstrain at Ae = 0.0204 for N

cycles as indicated case by case, followed by a reversal decremental

sequence to zero strain, followed by CA cycling

(e) BC consisted of alternating blocks of low high or high low CA straining

as follows: 104 @ Ae3 / 10 @ Ae^, repeated to failure or

10 @ Aei / 104 (3 Ae3, repeated to failure
22

percent. These properties indicated the material provided was cold worked

between 20 and 40 percent. The grain size was ASTM 2% to 3.

The specimen design involved a trade-off between the shape and size

desired for low cycle fatigue testing and the requirements of ultrasonic crack

detection.

An optimum design from an ultrasonic viewpoint involved consideration of limiting beam spread, the desire to achieve plane waves, and the

desire to locate the transducer as close to the gage section as possible to

enhance crack detection sensitivity. One final consideration involved

choosing a geometry to minimize noise from reflections outside the gage

section. To this end a transducer optimized to the material and geometry was

designed and fabricated by Search Unit Systems Incorporated.* The transducer

was located nearer the gage length by burying the transducer in an access hole

in one and of the specimen. The distance from the gage section, the diameter

of the transducer, and the shape of the specimen were chosen to limit the

amount of signal reflected from outside the gage section that would be

returned to the transducer during the time period the signal would return from

the gage section. These considerations led to the gage section and transition

shoulder geometry shown in Figure 4.

A11 specimens were machined in a tracer lathe using standard shop

procedures for copper. Samples were turned in the hardened as-received

conditions using progressively finer cuts. The last 0.5 mm of radius was

removed using two cuts of .1 mm, two cuts of 0.05 mm and eight approximately

equal cuts of about 0.012 mm. This procedure was used to minimize any

residual stresses induced by machining, even though enough inelastic action

was expected in the tests to quickly wash out their effects. A total of 36

specimens were machined for the program. Of these 4 were used in various

facets of testing the ultrasonic procedure. The remainder were committed to

mechanical testing.

*Search Unit Systems, Inc.

San Antonio Texas, 78228

Attn: Mr. K. Briers.
23

0.375" D-ljJ'eo.rX 07375"D-I

is I-UNFI4

5.00"-

0.25"

:: Do not undercut blend radii

at transition regions

0.750"

J."

4

- 0.3125" r, blend radii

(4 places)

FIGURE 4. AXIAL FATIGUE TEST SPECIMEN (DIMENSIONS IN INCHES)
24

Apparatus and Procedure

All mechanical testing was performed in one commercially available

closed loop servocontrolled electrohydraulic test system. All testing was

performed in strain control at frequencies ranging from 2 Hz for the smaller

strains to 0.2 Hz for the highest strain. Temperature rise measured in a

benchmark test was less than 4 F under these conditions. Specimens were

subjected to a variety of essentially fully reversed axial straining sequences

primarily using a sinusoidal pacing function. Strain was controlled in the

12.52 mm long gage section using a clip-on extensometer calibrated at the

start of and three times during the program to ASTM Class B]_. Load was

monitored in all tests using a load cell mounted in series with the specimen.

Calibration of the load cell was performed prior to and verified once during

the test program. The load cell was observed to be accurate and linear within

0.1 percent of the operating range used in the present program. All recording

devices were calibrated at the start of the testing.

All specimens were gripped in a fixture that featured a liquid/solid

Cerrotru alloy grip which served to minimize stresses when mounting the

specimens. Prior to starting the test program, the alignment was adjusted to

minimize bending strains to less than + 5 percent of the imposed strain, as

required by ASTM E606. Before commencing with the experimental program, the

closed loop system was tuned to allow testing at the optimum system gain. The

procedure used for each test follows: after mounting the specimens in the

upper grip, the load train was closed by freezing the liquid-solid grip under

load control conditions. Maintaining the system at zero load, the extensometer was mounted and zero suppressed and'all recording devices were activated

and zero suppressed. The system was then shut down and strain control

selected. The system was then reenergized and allowed to stabilize before

strain cycling. In tests where the system had to be periodically shut down

and restarted (such as for surface damage replication), care was taken to

unload and thereafter reload without inducing spurious mean stresses. Monotonic and cyclic deformation response was recorded continuously during the

first ten cycles and at logarithmic intervals thereafter on an X-Y recorder.

Both load and ultrasonic output were continuously recorded on a time-based,

high-speed strip chart recorder.
25

Initial tests were conducted in an ambient air environment at 40 percent

relative humidity (+ 5 percent) and 20 C (+ 1 C). But some uncertainty as to

the value of these initial tests existed in that intergranular (IG) cracking

occurred whereas a ductile initiation was anticipated as well as required for

this study. Examination of the failed specimens showed evidence of CU20, as

is apparent in the randomly located dark areas shown in the micrograph in

Figure 5(a). Furthermore, the near surface grain boundaries were preferentially decorated by Cu20, as shown for example in Figure 5(b). Concern for

the presence of the CU20 developed when further study showed crack initiation

in these tests occured preferentially at grain boundaries, and continued along

grain boundaries. Examples of two such cracks are shown in Figures 6(a)

through 6(d) at 350X and 1500X magnification. Such brittle initiations

jeopardize meeting the objective of these tests—study of nonlinear history

effects which according to the postulate arise due to inelastic action

culminating in ductile initiation. After identifying that IG cracking

associated with CU20 observed in grain boundaries near the surface was

occurring, sample specimens of the copper bar were tested against the standard

acceptance criteria for oxygen free copper*. The tests indicated that the

samples met the standard for OFE copper. Consequently the Cu20 evident near

the surface of specimens was believed to be forming during the cycling. To

circumvent the IG initiation (without a detailed study of its cause), all

subsequent tests were performed in a 99.999 percent flowing argon atmosphere

at a back pressure of one inch of mineral oil, with testing commencing after a

15 minute chamber purge. Whether or not oxygen entry during testing caused

the observed IG cracking, the incidence of IG initiation and growth ceased

once testing in Argon began. All results developed in this fashion showed

uniform deformation in the gage section and ductile initiation, in accordance

with the requirements of this study. A photograph of the test chamber and

gripping arrangement used is shown in Figure 7.

*The test consisted of examining a polished sample of OFE copper after etching

to decorate the Cu20 (red). Evidence of decoration below 75X magnification

is defined as excessive Cu20 and the material fails to qualify as OFE.
26

ORIGINAL PA3E t3

OF POOR QUALITY

(a) View in bulk at 100X (7K776)

• .

" •

.

.

•

.

.

(b) View showing decoration near surface

at 100X (7K777)

FIGURE 5. EVIDENCE OF Cu20 FORMATION
OF POOR QUALfTY

27

(a) IG initiation at surface - view A

at 350X (SEM-34566)

(b) IG initiation at surface - view B

at 350X (SEM-34568)

FIGURE 6. EVIDENCE OF Cu?0 AND ITS APPARENT INFLUENCE

ON INITIATION (SPECIMEN 2)
ORIGINAL

28

(c) 16 initiation - View A at 1500X (SEM-34567)

(d) IG initiation - View B at 1500X (SEM-34569)

FIGURE 6. Continued
29

ORIGINAL PAGE-is

0* POOR QUALITY

FIGURE 7. OVERVIEW OF TEST CELL AND GRIPPING ARRANGEMENT
30

RESULTS AND DISCUSSION

Material Characterization

The OFE copper used in this study was cold worked between 20-40

percent and according to the suppliers certification had a nominal hardness of

87.5 on the Rockwell hardness F scale (HRF). A value of 87.5 HRF corresponds

to a diamond pyramid hardness (DPH) value of 98. Hereafter, all hardnesses

will be reported in terms of DPH values.

The microstructure of the as-received material was found to contain

extensive deformation induced twins, as shown in Figure 8(a), and had a grain

size of ASTM 24 to 3. Because initial hardness/ductility of the material was

a program parameter, samples of the as-received material were heat treated

under a variety of conditions chosen with a view to decreasing hardness without radically changing the microstructure. Extensive recrystallization and in

particular grain growth thus were undesirable. Study of the literature for

copper base alloys (e.g., [36]) indicated major reductions in hardness could be

achieved using anneals at 270 C, and that this would promote limited recrystallization and produce only nominal increases in grain size. To this end

samples of the as-received (AR) material were annealed at 270 C* for periods

of 1,5, 10, and 24 minutes denoted as HT1, HT5, HT10, and HTA. Microstructures that developed showed nominally no changes in grain size or twin

morphology, as evident in Figure 8 parts (b)-(e). Grain sizes remained at

ASTM 2*s to 3. Hardnesses measured 8 or more times developed the following

mean/standard deviation DPH values for the condition noted: AR, 101.1/5.67;

HT1, 102.64/2.06; HT5, 101.93/1.28; HT10, 100.68/1.08; and HTA, 103.58/4.27.

In view of these data, there was no significant effect apparent in the

microstructure or hardness due to these short term anneals.

Mechanical properties testing indicated 0.2 percent offset yield

stresses of 305.3 MPa for the AR condition and 277.7 MPa for the HTA condition, indicating that the HTA anneal actually had softened the material

*A11 heat treatment was done with samples sealed in pyrex vial s evacuated and

sealed in presence of 99.999 percent flowing Argon. Each via l contained a

few grams of clean titanium chips to serve as a getter for any impurities.
31

ORIGINAL PAGE IS

OF POOR QUALITY

(b) AR1 - AR + 1 minute at 250 C

<>// ' % V ^^- - /——^n

'• H '•

 x ^ / Y«t \. V \ ^\ \ i

\- _/;x' 1^-,A ' i }

/'• "'' •" \ '=' \

Wr, A > V\/

'' ^ :

(c) AR1 - AR + 5 minutes at 250 C

FIGURE 8. MICROSTRUCTURES FOR VARIOUS HEAT TREATMENTS OF

OFE COPPER (100X)
(d) AR10 - AR + 10 minutes at 250 C

/v

(

(e) HTA - AR + 24 minutes at 250 C

(f) HTB - AR + 50 minutes at 270 C

FIGURE 8. Continued
33

somewhat. But this was not enough for present purposes. For this reason

samples of the AR condition were annealed at 285 C for 50 minutes, an anneal

hereafter termed HTB. Heat treatment B produced some recyrstalization, as

evident in Figure 8(f). The grain size was measured as ASTM 24 to 3% and

included some finer grains as compared to the other structures, apparently due

to the recrystalization. The hardness was 84.17 with a standard deviation of

16.96 (12 observations). The microstructural modifications produced a

material with an 0.2 percent offset yield of 246.7 MPa, a value substantially

lower than that for the AR and HTA materials.

Damage Evolution and Crack Detection

Four methods were used to track the development of damage under

strain control testing in this study. Stress response was measured continuously and served as a bulk indicator of microstructural changes and the

difference in compliance due to cracking. Changes in the ultrasonic impedance, manifest as a mean shift in an analogue signal, also served as a bulk

measure of damage. In addition, the ultrasonic impedance also served to

define the formation and development of cracking, as evident in changes in the

magnitude of reflected energy. Changes in surface topography, mapped via

replication, served as localized measures of near surface microstructural

changes, and also provided direct characterization of cracking. Lastly,

changes in microhardness served as a measure of bulk microstructural changes.

Stress Response

Results of studies directed at the evolution of damage under baseline histories in terms of bulk stress response are presented in Figure 9.

The AR and HTA conditions produced cyclic softening. However, at lower

strains HTB cyclically hardened while at large strains HTB cyclically softened. This hardening and softening was viewed as a macroscopic manifestation

of microstructural reorganization prior to cracking (e.g., [37,38]). This

involves a breakdown of the initial dislocation morphology into cells, at

least for the range of strains studied herein [34]. Further discussion of the
34

TOO

soo

400 -

OFE Cooo«r

Fully R«v«rs«d Strom Cycling

Cycm

FIGURE 9. STRESS RESPONSE AS A FUNCTION OF CYCLES FOR THE

OFE COPPER: AR, AR/IPS, HTA, AND HTB CONDITIONS
35

correlation between microstructural changes and changes in stress response

follows later in the section titled Cyclic Deformation and Fatigue Behavior.

Stress response also served as a macroscopic indicator of the

cracking process in that the specimen stiffness changed in the tensile

direction with cracking, but remained more or less the same in compression

once the crack faces close. This change was evident as an asymmetric decrease

in tension load, as shown for example in Figure 2. If hardening or softening

did not occur, and a single dominant crack formed, this asymmetric decrease in

tensile stress response would define the net section area lost due to cracking. Provided that fractography permitted determination of the crack aspect

ratio as a function of crack length, load drop also would define crack depth.

Table 2 indicates that load drop* provided a very sensitive indicator of the

formation of small crack depths (under this set of conditions) for the segment

configuration. However, for a semicircular crack configuration a one percent

load drop equated with a quite large crack depth (650 urn). For the present

study, macrofractography of failed specimens suggested that the initial aspect

ratio was between 0.143 and 0.5, and that the depth of the crack was about

twice that of the segment crack, as shown for example by the photomacrographs

shown below the table. The cases shown illustrate double segment cracks for

which a one percent load drop equated to a crack about 310 urn deep.

Crack depths listed in Table 2 were upper bounds on the actual crack

depth if multiple cracks formed. They were also upper bounds in materials

which soften since such softening at the crack tip would accelerate the load

drop. Conversely, in materials which harden and form a dominant crack, these

crack depths may not be an upper bound in that hardening at the crack tip

would tend to offset the load drop due to lost cross-section**. Results

developed in this study indicated that the number of cycles associated with a

1 percent load drop (or the formation of a crack on the order of 1 percent

lost area or upper bound depth of 310 ym*), was very large compared to the

*Load drop was assumed to be 1:1 related to crack area in the calculations

presented in the table.

**Engineering estimates indicated that hardening did not significantly offset

the load drop due to cracking for the present study.
36

TABLE 2. AREA AND CRACK DEPTH AS A

FUNCTION OF LOAD DROP

Load

Drop,

Percent

1

5

Crack Depth, a, mm

Crack Shape and Aspect Ratio, ' a/2c

Segm

V"-•-_

0.101

0.310

ent

" /

_- •"•

0.164

0.925

Double

Segment

0.07^2

0.192

0.155

0.570

Semi

Circular

V

i

0

0

1

^y

.5

.65

.5

} a/2c

(a) Crack shapes were mostly double segment with aspect ratios ranging

from .14 up to about 0.5. Typical shapes such as that shown below

indicated aspect ratios of about 0.3, Note that 2c is the distance

along the surface while a is the crack depth.
37

number of cycles to grow that crack to separation. Furthermore, the load drop

records indicated that the 1 percent load drop followed very soon after both

the first discernable load drop and the life defined as initiation based on

back extrapolated load drop (cf Figure 2). Note that these observations were

contrary to the popular opinion that low cycle fatigue in ductile metals

involves macrocrack propagation following almost immediate initiation of

macrocracks. Whether or not this contrary observation was due to the material

studied and the macroscopic nature of the definition of initiation is pursued

in a few paragraphs.

Ultrasonics

Ultrasonic tracking of damage evolution works on the principle that

changes in the microstructure tend to alter the attenuation of sound waves as

they pass through a material. This change in attenuation can be amplified and

recorded in comparison to the initial (reference) condition. In turn this

change can be correlated with physical manifestations of damage such as

changes in hardness, surface topography, and stress response. Clearly, for

this scheme to work the reference must remain invariant in time, and driftfree electronics must be available that match amplification needs. Furthermore, factors such as probe/substrate coupling must remain invariant with

time, and be small compared to the amplified attenuation.

Calibration of the ultrasonic equipment indicated that the position

of segment shaped electrical discharge machined (EDM) flaws 1 percent of the

cross-sectional area (310 urn deep)* could be consistently located within the

gage length. Thus the threshold for detection via ultrasonics as implemented

in this study was a reflecting area on the order of one percent of the net

cross sectional area. Studies with multiple flawed EDM samples indicated, as

expected, that multiple small flaws on the same crack plane or in each others

sound shadow tended to make signal interpretation difficult. These studies

suggested that indications of cracking tended to provide upper bound estimates

*Smaller depths were detectable, but not consistently.
38

of the life to form cracks of the threshold size, whose sound reflective

character was comparable to that of EDM flaws. Comparison of responses from

actual cracks on the order of the threshold area suggested that for the OFE

copper studied EDM flaws and cracks were indeed similar reflectors. Calibration of the ultrasonic analogue output with EDM flaw area indicated a

correspondence could be developed with actual cracks. However, difficulties

with the probe-specimen coupling introduced due to mechanical effects during

test start-up/shut-down for purposes of making replicas caused the reference

zero to shift. For this reason ultrasonic results were useful only as an

indicator of cracking once the threshold crack area was developed.

Data developed indicated attenuation occured during the period of

transient hardening or softening. Thereafter changes were more subtle until

indications of cracking were noted. Cracking produced analogue indications

comparable to the attenuation observed during the transient period. However,

the load dependent length of the specimen caused a cycle dependent variation

to be superimposed on the attenuated response. The amplitude of this variation tended to increase with the increase in specimen length that developed

with cracking. Generally, asymmetric load drop as defined in Figure 2

indicated initiation before the ultrasonic technique. Examples of this are

apparent in the two typical strip chart recordings shown in Figure 10. Note

that differences existed between the cycle identified as "initiation" using

these methods of tracking damage. But, this difference was not of much consequence in a relative sense when compared with the number of cycles involved.

The data developed indicated that lives defined as Initiation by the load drop

and ultrasonic schemes were equal within 1 percent, with only one exception.

Surface Topography

Single stage replication provided a more direct but also more

tedious and expensive method of tracking damage as compared to stress response

and ultrasonic measures. Acetate replicas were made at various stages of the

life for every sample tested. Sufficient replicas were made to characterize

the reference as machined/polished surface, and the surface after each major

straining sequence, as well as at tenth points of the expected life. Funding
39

Test is stopped for replication and then restarted

(-) Load Analogue (+) Ultrasonic Analogue

ORIGINAL PAGE is

OF POOR QUAUTY

Consistent indications

of initiation

Load reference (zero)

Time

Reference after

stabilization

(a) Specimen 16, Nf = 12,445 cycles at Ae2 in HTB

Test is stopped for replication and then restarted

(-) Load Analogue (+) Ultrasonic. Analogue

Consistent indications

of initiation

=*** Reference after

stabilization

Load reference (zero)

Time

(b) Specimen 17, Nf = 853 cycles at Ae: in HTB

FIGURE 10. EXAMPLES OF ULTRASONIC RECORDS AND LOAD TRACES

AS INDICATORS OF INITIATION
40

limitations precluded gold coating and detailed examination of all replicas

for all specimens. For this reason results have been developed to characterize general trends and establish the relative effects of strain range,

material condition, mechanical hardening and initial overstraining. Study of

the seven specimens noted in Table 3 provided some insight into the effects of

surface topography.

TABLE 3. SURFACE TOPOGRAPHY STUDY

Specimen

35

14

4

36

29

15

24

Condition

AR

AR

AR

AR

HTA

HTB

HTB

Strain History

CA @ Ae]_

CA @ Ae2

IPS followed by CA Ae^

IPS followed by CA Ae3

CA @ Ae3 started in compression

CA @ Ae3

IDS for 100 cycles, followed

by a DPS, followed by

CA @ Ae3

Results developed from the examination of surface topography for

these seven samples are presented in Figure 11, parts (a) through (g). Each

figure includes photographs at low (SOX) magnification and selected photographs at one of 250, 1,000 or 5,OOOX magnification* as needed to characterize

the topograpy and illustrate trends. In studying these photographs, bear in

mind that photographs of single stage replicas depict a negative impression of

the surface.

A few general observations are in order before results are discussed

on a specimen to specimen basis. At high strains the surface topography was

rather uniform and began to develop early in the life. Continued cycling

intensified surface features which tended to be aligned with the macroshear

orientations for the specimen. The formation of what appeared to be long and

*The study was made using the magnifications noted. However, to facilitate

reporting, these magnifications have been reduced by about 0.5, resulting in

magnifications reported in Figure 11 as 25X, 125X, 500X, and 2500X. As well

some detail is lost using the reproduction procedure employed.
41

shallow Mode II cracks developed early in the life. Eventually Mode I cracks

grew from areas containing Mode 11/Mode I cracks, which evolved earlier from

the long shallow shear oriented (Mode II) surface cracks. Lower intensities

of cyclic strain led to trends comparable to those at higher strains, except

the deformation was not homogeneous and consequently resulted in a reduction

in the extent of surface area involved in Mode II cracking. This in turn

produced fewer regions from which Mode I cracks could evolve. Initially

softer copper exhibited more extensive and well developed Mode II cracking

compared to its harder counterparts at equal cycle numbers or comparable life

fractions. Initial prestrains served to create Mode II cracks earlier in the

life. However, this topography was very limited in extent and intensity

compared to that developed by continued cycling until the prestrain/overstrain

history constituted a linear damage fraction of 0.4 to 0.5.

Study of all specimens indicated that the onset of Mode I cracking

occurs very much later in life than did the formation of Mode II crack like

features. Lives to form Mode I cracks correlated reasonably with the earlier

discussed bulk measures of initiation. This suggested the popular view that

the life of low cycle fatigue specimens made of ductile metals was spent in

large part growing cracks, did not hold for the copper specimens examined.

Furthermore, it suggested that load drop and other such bulk measures of

"initiation" provided a reasonable characterization of the life to form Mode I

microcracks,

Whether or not the copper behaved differently from engineering

alloys because it formed shear cracks in the absence of inclusions or second

phase particles which may tend to directly form Mode I cracks in engineering

alloys is open to speculation. Nevertheless, it was clear that long and

shallow Mode II cracks which form early in the life were only of consequence

when they became life-limiting. Mode II cracks became life threatening when

they ceased to grow as shallow surface shear cracks by turning into combined

Mode I I/Mode I (perhaps as the Mode II crack grew into adjacent grains).

Thereafter the Mode I component of the crack grew across the transverse netsection leading to failure. It was these Mode I cracks that most strongly

reflected ultrasonic energy and resulted in the significant reduction in load

used to define nucleation by back extrapolation (cf Figure 2).
42

Figure ll(a) presents results which show the evolution of surface

topography in specimen number 35 tested under CA cycling at Ae]_ in AR

material. The reference surface showed some evidence of machining marks, but

did not show evidence of preferential slip due to machining and polishing.

After 50 cycles or O.lN-j, the surface began to show rumpling on axes inclined

to the tensile loading direction. Slip traces were somewhat randomly oriented

but tended to be aligned with the rumpling. The majority of the slip appeared

to be oriented at or near the macroshear planes for the specimen. These same

features were evident at 300 cycles or 0.6 N-j, however, they were intensified,

i.e., they occured more frequently and were better developed. Slip traces

perpendicular to the tensile axis were also evident as are Mode II crack like

features. There was also a concentration of damage perpendicular to the

tensile axis that developed within as well as across the checkerboard of shear

oriented rumpling. This damage, which could lead to Mode I cracks, appeared

to be best developed in regions of intense shear oriented rumpling. Some long

and shallow intrusion like features were evident growing from these areas.

Finally, results are shown at 500 cycles or 0.98 N-j obtained near the site

forming the fatal crack. A still more intensified development of rumpling and

shear oriented cracking was evident along with pucker like features that

formed in the heavily deformed regions. These pucker like features had a

Mode II/Mode I character, as did many of the earlier pure Mode II features.

Cracking appeared to occur by decohesion of damaged areas that had a

Modell/Mode I character, with the Mode I character evolving late in the life.

Results for cyclic conditions comparable to those for specimen 35,

except for the introduction of the IPS, are shown in Figure ll(b) for specimen

4. Note that features of damage evolution and cracking present in part (b)

compare closely with those in part (a). Discussion of these therefore will

not be repeated. Given these similarities, the prestrain apparently had

little relative effect on the surface topography. But this may be expected

given the extensive flow that occurred with each cycle of Ae^* following the

*Study of the surface after the IPS at 5,OOOX indicated no change in surface

topography had occurred, aside from formation of a few sli p traces, as

detailed next in discussing specimen 36. Thus the IPS altered the hardness

mechanically without perceptably influencing the surface topography.
Cycle

Cycle Fraction (Nj)

-25X

-125X

~500X

-2500X

t

500

0.98

(a) Specimen 35: CA at Ae-j in AR material; Nj = 508, Nf = 572

FIGURE 11. EVALUATION OF SURFACE TOPOGRAPHY IN OFE COPPER UNDER CYCLIC LOADING
Cycle

Cycle Fraction (Nj)

~25X

~125X

-500X

-2500X

b

?

!

o

0

(b) Specimen 4: CA at Ae-j in AR material following the IPS; Nj = 500, Nf = 502

FIGURE 11. (CONTINUED)
45

IPS. As with specimen 35, there was a gradual intensification in the topography as cycling continued. Note that the lives of specimens 4 and 35 were

virtually identical, and in both cases a very few cycles of straining were

required to grow (not nucleate) detectable microcracks (on the order of one

percent of the specimen area) to their critical size.

Specimen 36 represents test conditions identical to those of

specimen 4 except that the IPS was followed by a lower strain range (£63) in

the case of specimen 36. Examination of the topography for this specimen,

shown in Figure ll(c), indicated that features similar to those for the

previous specimens developed. However, here the damage was very localized and

was not as well developed. When crack like features were apparent they did

not arise from the extensive network of shear flow evident at high strains.

Such localization of damage and Mode II cracking was expected in that, as the

strain range decreases, slip occurs selectively in grains where the crystallographic orientation favors slip. Cracking from well developed slip via

decohesion was also expected to occur selectively in grains where the adjacent

grains admit continued easy slip. Mode II cracking was again a precursor for

Mode I I/Mode I and Mode I cracking. Finally, Mode I cracking was again

observed late in the life. Note that the results for 5,000 cycles indicate

the surface topography was virtually identical to that of the virgin surface.

In this respect the IPS did not alter the surface, but did mechanically change

the bulk hardness. In turn this meant that topographic results for IPS and

non IPS histories can be directly compared as though the IPS had not been

imposed.

Consider now results for specimen 14, developed in AR material under

CA cycling at Ae2, shown in Figure ll(d). Observe that these conditions,

which were bounded above by specimen 35 and below by specimen 36, developed a

topography which was a cross between these bounds. At low cycle fractions the

topography showed the very localized and light damage evident late in the life

of specimen 36. However, by midlife this damage intensified somewhat developing evidence of the puckers and Mode II, and Mode I I/Mode I, cracking

reminescant of specimen 35. The essential topographical features of specimens

36 and 35 were thus evident in specimen 14. The only major difference between

these features as a function of Ae]_, Ae£ and Ae3 was the intensity and
Cycle

Cycle Fraction (N;

5,000

0.082

10,000

0.164

25,000

0.61

40,000

0.98

~25X

125X

Q

?

1

~500X

-2500X

I

00

^1 »

o -u

ss

(c) Specimen 36: CA at Ae3 in AR material following the IPS; Nj = 41,000, Nf = 44,878

FIGURE 11. (CONTINUED)
Cycle

Cycle Fraction (Nj

0

0

500

0.114

25X

125X

I

is

o

-500X

~2500X

•gj

f!

as

(d) Specimen 14: CA at Ae2 in AR material; Nj = 4376, Nf = 5,836

FIGURE 11. (CONTINUED)
48

distribution of damage. For all strain ranges the damage process was the

same—after much shear mode cracking, life limiting Mode I cracks formed and

grew across the transverse net section.

Differences in the initial hardness of the material were studied by

comparing results for topographies characteristic of CA cycling at Ae3- The

AR condition is represented by specimen 36, shown earlier in Figure ll(c).

The HTA and HTB conditions are represented by specimens 29 and 15, results for

which are reported in Figures ll(e) and ll(f), respectively. Note in comparing these figures that the stages in life for a given cycle number represent

increasing cycle fractions for the harder materials due to the increased

fatigue resistance for HTB vs HTA vs AR conditions. These results showed the

surface mottling to be initially sparse, but that it increased with further

cycling. The mottling tended to be oriented along the macroshear planes, and

developed earlier and more extensively in the softer materials. Toward the

end of the life, puckers formed and Mode II/Mode I and Mode I cracking

followed, just as had been previously discussed. Comparison of the topography

indicated the damage became increasingly widespread and better developed as

the material became more ductile, just as expected. Softer materials will

have more grains favorably oriented for slip at a given strain range because,

as reported earlier, their yield strength is reduced. Likewise flow within

surface grains wil l be easier as compared to harder materials. In this

respect, crystallographic notches at intrusions develop earlier in the softer

materials and may serve to accelerate decohesion and initiation of all

cracking modes. For this reason the life of softer materials may be limited

at long lives by their earlier crystallographic notching as compared to their

harder counterparts.

The effects of 100 cycles of IOS on damage topography are evident in

the first of the results for specimen 24 shown in Figure ll(g). Note that the

material studied was in the HTB condition. This softer material was expected

to quickly develop extensive surface damage, particularly since the 100 cycles

imposed represented a cycle fraction of about 0.143 (N^) under continued CA

cycling at the IDS level. Such damage indeed developed with evidence of

surface rumpling spread across the gage section. Examination up to 5,OOOX

showed the presence of extensive shear oriented slip. But as expected given
Cycle

Cycle Fraction (Nj)

~25X

~125X

.2

I

Q

f

-500X

-2500X

i

5,000

0.084

10,000

0.168

30,000

0.505

50,000

0.842

»»*...-s»«- £"'_—•"""^GSi

—~ JrJ-M

--

J

*>_. it^^"^^/!* «•"'* *^u

~.» •"**• •'

(e) Specimen 29: CA at Ae3 in HTA material; Nj = 59,375, Nf = 63,404

FIGURE 11. (CONTINUED)
Cycle

Cycle Fraction (Nj)

5,000

0.075

10,000

0.15

30,000

0.449

50,000

0.748

-25X

125X

-500X

-2500X

I

O O

•

O TJ

c *»

•> o

(f) Specimen 15: CA at Ae3 in HTB material; Nj = 66,840, Nf = 72,676

FIGURE 11. (CONTINUED)
Cycle

Cycle Fraction (Nj)

IOS -*

100

0.143

-25X

~125X

o

g-

-500X

-2500X

-*- CACyling

10,000

0.215

25,000

0.539

50,000

(g) Specimen 24: 100 cycles (OS followed by CA at Ae3; Nj = 46,428, Nf = 52,625

FIGURE 11. (CONTINUED)
52

the earlier noted gradual evolution of Mode II, then Mode II/Mode I, and

finally Mode I cracking late in life, there was no evidence of cracking, or

even puckering. With continued cycling at Ae3 following the IDS, the damage

topography tended not to change in an obvious fashion* until late in the life,

When changes became apparent, there was evidence of puckering and Mode

II/Mode I and Mode I cracking in areas adjacent to the site of the dominant

crack.

Somewhat surprising was the fact that the IOS did not appear to

significantly reduce the life as compared to pure CA cycling when the damage

due to the IOS was linearly accounted for. In this case, the IOS reduced the

life by about 12,700 cycles or 19 percent where an order of magnitude or

greater life reductions are reported in the literature [5,39]. Whether or not

this was due to differences in cracking processes between OFE copper and

engineering alloys (Mode II to Mode II/Mode I to Mode I vs Mode I at inclusions and intermetallics, respectively) remains to be seen. Such considerations are deferred until all results are presented.

Microhardness -

Detailed microhardness surveys were made on the 12 specimens that

characterized the baseline fatigue behavior of the HTB, HTA, AR, and AR plus

IPS materials. Results for specimen groups 15, 16 and 17—26, 27 and 28--14,

21 and 35—and 4, 32, and 36, are listed in Table 4. Included in the table

are mean values of the initial and final bulk hardnesses denotated as xg and

xg. These hardness values were obtained by first sectioning and polishing

failed specimens and then measuring the hardnesses in the gage section and at

a low stressed location in the grip area. In most cases the results reported

represent not less than six microhardness measurements. However, in some

cases where scatter was large, more than 10 readings were made. Table 4 also

lists the change in hardness, i.e., the difference between the initial and

*Subtle changes like those observed for specimen 15 may have occurred but

were not easily resolved because the surface rumpling tended to disguise

such localized changes.
53

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54

final mean bulk hardnesses. The change in hardness normalized by the initial

hardness is also given in Table 4 for each specimen. In cases where the

hardness increased these differences are noted as (+) whereas in cases where

it decreased, the difference is noted as (-). Observe from the table that

initially harder material showed negative changes while initially soft material showed positive changes at smaller strains and negative changes at larger

strains.

Cyclic Deformation and Fatigue Behavior

Under CA cycling the locii of the half life loop tips, shown in

comparison to the monotonic responses in Figure 12, indicated each of the AR,

HTA, and HTB conditions studied cyclically softens. However, this is an over

simplification in that hardening was observed initially followed by softening,

and in one case pure hardening was observed, as evident earlier in Figure 9.

The implications of this transient behavior will be pursued shortly.

Fatigue resistance data developed in this program are reported in

Table 5. Included in the table are total, plastic, and elastic strain

ranges Aet, AeP, and Aee, stress range, As, cycles to initiation*, Ni, and

failure**, Nf, and a description of the material heat treatment and strain

history. Information extracted from the raw data of Table 5 or the mechanical

response of the specimens is reported in Table 6. Included in this table are

parameters such as smxAet, the ratio N-j/Nf, cumulative energy, £E-J, energy per

cycle, E-j, cummulative plastic strain, £AeP, the product smx£eP and the mean

change in flow resistance based on bulk and microstructural measures, As/As0

and AHB/H0.

As the data in Tables 5 and 6 are used to further explore fundamental aspects of damage evolution and accumulation, and assess the viability of

the postulate, these and other data will be introduced in graphical form.

Recall that these data have been developed with a view to answering three

earlier posed fundamental questions regarding the phenomenology of damage, for

initiation was defined here by the back extrapolation technique defined

earlier, cf. Figure 2.

**Failure was defined here as separation of the specimen into two pieces.
55

350r

300 —

250 -

200 -

S

150 hOFE Copper

— >Monoton|c_

100 -

50

As received

• —— 500 F for 24 minutes

550 F for 50 minutes

I

02 0.4 0.6 0.8

Strom, percent

10

50

40

30

20

10

12

FIGURE 12. COMPARISON OF MONOTONIC AND HALF LIFE STRESS

RESPONSE FOR OFE COPPER FROM SELECTED TESTS:

AR, HTA AND HTB CONDITIONS
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57

TABLE 6. PARAMETERS RELATED TO THE FATIGUE RESISTANCE

DATA PRESENTED IN TABLE 5

Specimen

Number

35

14

21

26

27

28

29

17

16

15

4

32

36

37

37R

12

7

19

11

18

9

24

20

25

23

22

Ni/Nf

0.89

0.75

0.91

0.93

0.84

0.85

0.94

0.96

0.97

0.92

~1.0

0.72

0.12

--

0.88

0.67

0.89

0.96

0.84

0.57

0.66

0.88

0.93

0.77

0.82

0.91

TSpa

4.43

1.25

0.49

4.24

1.21

0.50

0.50

4.20

1.11

0.41

4.22

1.20

0.46

0.18

0.48

0.52

0.51

0.52

0.50

0.44

0.49

0.48

0.52

0.48

--

--

y E '

K Joules

3.51

4.75

5.05

3.68

7.11

11.93

—

4.67

11.25

15.00

3.04

3.85

5.46

--

--

—

9.02

—

8.13

13.61

8.94

10.22

—

--

--

--

Ei/cycle,

Joules

6.91

1.08

0.16

5.90

1.09

0.22

--

5.71

0.94

0.22

6.08

1.06

0.13

—

—

~

0.22

—

0.23

0.21

0.21

0.22

—

--

—

--

ZeP

7.75

16.30

34.07

9.47

24.75

60.81

57.07

12.33

44.00

89.75

7.61

13.17

40.70

--

—

—

44.90

—

40.30

77.30

45.80

52.40

--

--

53.30

78.20

Smx 2sP,

MPa

1849

3126

5192

2167

4595

9253

8642

2798

7500

11054

1760

2453

6065

«

—

--

6568

--

5895

10107

6573

7430

--

--

--

—

*S/*S0

-0.286

-0.462

-0.366

-0.26

-0.46

-0.44

—

0.135

-0.195

+0.428

-0.281

-0.466

-0.347

--

--

—

--

--

--

--

—

--

--

--

«

—

iHg/ HO

-0.14

-0.23

-0.20

-0.185

-0.271

-0.163

—

-0.18

+0.012

+0.095

-0.09

-0.19

-0.16

—

--

~

—

—

—

--

—

--

--

--

--

--
58

OFE copper over a limited range of lives. If the expected behavior was not

observed or the results were inconsistent with the expected trends it would be

concluded that the postulate was not viable. On the other hand, if the

results were consistent with the expected trends and predicted behavior, it

would be concluded that the postulate was viable. The Approach to implement

the study focussed on whether or not (1) hardness/ductility, (2) surface

residual stress, and (3) surface topography developed in a characterizable

fashion as a function of history. Further the approach involved exploring the

history dependence of the material's fatigue resistance.

Surface Topography

To this point it has been shown in some detail that the surface

damage process was similar in all material conditions studied. These same

results suggest that cracking at crystallographic notches due to well developed Mode II/Mode I microcracking would serve to limit life due to IDS or IPS

under certain conditions. Results indicated that this effect would be particularly marked at longer lives and lower strains thus requiring a nonlinear

damage assessment using a reference resistance curve that was a function of

the strain history. Furthermore, the data clearly showed that there was a

threshold damage state that must be developed before IPS or IDS would have a

significant impact. Since this threshold evolves as a result of inelastic

action, it is reasonable to speculate that this damage threshold wil l correlate uniquely with £AeP or smx£AeP. Available data indeed indicated a

correlation was possible using such parameters, but the sparse data did not

permit quantitative evaluation at this time. Thus it remains to explore

hardness and surface residuals, and further demonstrate the history dependence

of the resistance curve.

Correlation between surface topography and stress-strain history was

explored by tracking the surface via replication under the action of various

histories. Generally speaking, surface replication results indicated surface

roughening continued throghout the life to initiation. The effect of initial

prestrains was to roughen the surface creating a network of slip bands oriented at 45 degrees to the tensile axis (shear oriented). Initially soft OFE
59

copper samples exhibited a greater roughening than did samples of initially

hard (drawn) OFE copper. The initial roughness developed, however, was very

limited compared to that developed by continued cycling until the initial

prestrain constituted a block of cycles with a damage fraction of about 0.4 to

0.5. In this respect the effect, of initial prestrain certainly was seen to

roughen the surface although the extent was small compared to the character of

the surface developed at saturation. When prestrained lives (specimens 7, 9,

11, 18, 19, 20, 24, 25 in Table 5) were compared to non prestrained lives

(specimens 15, 21 in Table 5), very little difference was observed except in

cases where the prestrain constituted a significant damage fraction. As

emphasized earlier, this trend was not expected given the apparent degrading

effects of prestrains published for other materials (e.g.[5,36]). However this

behavior was not inconsistent with the postulate, and indeed was easily

rationalized in terms of a history dependent fatigue resistance curve, as

demonstrated in a subsequent section.

Surface Stress States

In this study surface stress state effects have been examined in

terms of the life obtained under selected sequences over a range of strains.

The possible effects of surface stress states have been studied here by starting tests in opposing directions - one in tension and one in compression.

Because bulk inelastic action at higher strains typical of the OFE copper

being studied lead to a homogeneous deformation state that washed out any

microresiduals, consideration of this possible effect was limited to Ae3. If

there was an effect, this comparison would show that the test started in

tension survived longer than its counterpart started in compression.

The results generated (specimens 28 and 29 of Table 5) showed virtually identical lives, differing by less than 2 percent at failure. In view

of this limited comparison, it was tentatively concluded that surface stress

state had nominally no influence on the damage process under the circumstances

examined. This result was not surprising for the OFE copper which showed extensive plasticity, even at the lowest strain examined. It thus remains to be

shown whether surface residual stresses develop or are of consequence at stil l

lower strains in the OFE copper, or in engineering alloys.
60

Measures of Damage

Consider now results developed to explore correlations of bulk and

surface measures of damage with each other, and with other characteristic

damage parameters. Tables 5 and 6 present data needed for this study.

Observe from Figure 9 that the transient changes in stress response typically

involved a period of initially stable to slightly hardening response followed

by almost exponential softening. Higher strains and thus higher stable

stresses tended to be associated with greater changes in stress whereas

smaller changes occur at smaller strains since the driving force for transient

action - plastic strain range - was reduced. In the context of cumulative

plastic strain, this tendency was countered by the fact that at larger strain

ranges less plastic strain accumulates before initiation whereas in the case

of smaller strain ranges, the large number of cycles to initiation resulted in

an extensive accumulation of plastic strain.

The net result of the counteracting factors controlling transient

action is evident in Figures 13(a) and 13(b). In both figures changes in

microhardness, postulated as being related to damage, first increase and then

decrease. In this respect there were two distinct domains of response. Such

a situation developed for OFE copper because of competing factors operative in

all metals, as noted earlier. For this reason it may be anticipated to occur

to some extent in all engineering alloys. It should be noted that the nonunique nature of this microstructural measure of damage does not preclude its

being of value in a fundamental sense. This behavior is not inconsistent with

the damage postulate, but it does lead to complications in tracking the damage

process.

If changes in stress response are indeed measures of the microstructural process of damage, these changes in stress response should correlate with the just discussed changes in microhardness. As evident in Figure

13(c) the results introduced above do indeed correlate with changes in stress

response reported in Table 6. Such a correlation suggests that bulk measures

like stress response can indeed be used to characterize the microstructural

changes leading to initiation, consistent with the postulate.
ORIGINAL PAGE-5S

OF POOR QUA4JTY

61

10

i

2

I

S

10

10"

_ 0 AR-CA OFE Copper

• AR-IPS/CA Fully Reversed Strain Cycling

~ O HTA-CA

& HTB-CA, softens

- 7 HTB-CA, hardens

"~

— I

1 1 1

T

-J

1

|

L

 2

> < >

(

i

"

10° I01

Increment in £A«P to Initiation

a. As a function of the increment in £Ae to initiation

to initiation, ksi

ra4

II

1 1 1 1 1

O AR-CA

• AR-IPS/CA

D HTA-CA

V HTB-CA, softens

A HTB-CA, Ixwenj

T

~ i

i

T ^

|

,1

' ' 1

1

Fulty

i

< \

i

iii ,

OFE Coopw

ftevened Slran Cycling

( 1

t

1 I I I

10s 0"

2Sm, I^e" to Initiation, MPa

b. As a function of 2s £Aep to initiation

FIGURE 13. TRENDS IN MICROSTRUCTURAL MEASURES OF TRANSIENT BEHAVIOR
62

ue

as

Q4

0.3

O2

ai

OFE Copper

Fully Reversed Strain Cycling

-

Constant Amplitude

O AR, hardens

D HTA, hardens

A HTB, hardens

V HT8, softens

Incremental Prestram

~ • AR, hara

T '

- >"

^

ens

v t

L^

i

i

^

•r

1

\

i i •

I

'\

\ "

01 02 03 04

Normalized Change in Sulk Stress Response

05

c. As a function of changes in bulk stress

FIGURE 13. (Continued)
63

History Dependence of the Fatigue Resistance

One key feature of the postulate arises with a view to answering the

first of the earlier posed questions. "Should the reference data used in

damage assessment reflect the history dependent hardness/ductility state of

the material?" A related question is: can a test control condition such

as Aet serve as a measure of the propensity for damage per cycle when the bulk

stress, whose transient behavior is history dependent, does not vary in the

same manner in both the reference data and the variable amplitude history

being analyzed? The first question is of consequence to the postulate while

the second addresses errors that may creep into damage analysis when a

nonconstant damage per cycle is assumed to be constant.

The second question is more general and is addressed first. Data

useful for this study may be taken for example from specimen 14 (Table 5).

Analyses of the transient stress response (Figure 9) taken in conjunction with

these data indicated that the dissipated energy per cycle was essentially

constant for this test. In this sense dissipated energy per cycle reasonably

matched the often made linear assumption that a given level of damage

parameter produces the same amount of damage regardless of where it occurs in

the history. In contrast, if Aet (or any purely strain or stress based

measure) is used, the damage done in the first 0.25 percent of the life

occured at a level 38 percent above that in the last 2.5 percent of the life.

During the first 2.5 percent, 12.5 percent, 37.5 percent, 62.5 percent and

87.5 percent of the life, the level of damage done compared to the last 2.5

percent of the life was, respectively, 23.4 percent, 10.5 percent, 4.97

percent, 3.13 percent, and 0.37 percent greater. The cumulative effect of

ignoring this transient action amounted to a 2.13 percent error in the damage

parameter, for the CA cycling imposed. For the AR material this translated

into about 50 cycles to initiation at a life of about 1000 cycles - about 5

percent. In past, errors on this order have been accepted in exchange for the

simplicity gained by ignoring the transient - thus steady state or half life

values tend now to be the popular basis for developing reference data for

damage analysis. However, greater errors may develop under BC histories and

variable amplitude sequences as indicated in the following.
64

In situations such as that just examined, potential errors due to

transient effects tended to be underestimated. For example, consider a

history for which initial hardening increased the stress response, followed by

CA cycling at a strain which did not promote softening. Calculations indicated that linear analysis based on total strain range as a damage parameter

can underestimate the "effectiveness" of strain based damage as compared to

that evaluated via a dissipated energy parameter by more than 60 percent, for

the case of HTB material hardened and then cycled at a level on the order of

Ae3- This error arose because the hardening developed a stress level in

excess of that developed in the CA test at the same strain range. This

difference translated into an error in life of more than an order of

magnitude, if CA cycling that did not cause softening followed the IPS that

caused the hardening. In this respect, care must be taken to accommodate the

shortcomings of the damage parameter. Note that the use of such a parameter

essentially precludes successful implementation of the commonly used linear

damage accumulation assumption, even before one starts damage analysis. It

follows that complicated histories can produce a variety of counteracting

transient effects that may influence life in a confounding fashion. Likewise

materials which harden at high strains and soften at low strains can also

produce confounding trends. Finally, note that softening would produce trends

opposite to hardening.

In all of the just noted cases, the resistance curve used in damage

analysis must be history dependent to account for the inadequacies of the

damage parameter in regard to the assumption that each cycle causing a given

value of that parameter does an equal amount of damage. The above discussion

suggests that the result of violating this assumption will bias the error

depending on the material and the history. Hardening materials can suffer

more damage than expected, whereas softening materials may suffer less damage

than expected using a pure strain based damage parameter. Histories which

impose and sustain the effects of hardening or softening therefore are more

discriminating than are those histories that tend to wash out the effects of

the hardening or softening. These transient effects also are coupled with the

possible influence of the larger cycles to significantly alter the evolution

of the surface topography as compared to the CA case. This effect tends to
65

reduce the material's fatigue resistance, perhaps due to the early initiation

of cracks which grow at higher rates and lower stresses as compared to the CA

reference condition. Such coupling of transient effects and premature

initiation is the situation commonly encountered in making damage (life)

predictions for BC or variable amplitude (VA) cycling using CA reference data.

In view of the above discussion it appears that the fatigue resistance curve used in VA cycling or BC situations should be history dependent.

(Clear evidence of this lies in the different resistance curves that develop

in the absence of surface effects for AR material versus AR hardened by the

IPS, results for which are shown in Table 5.) But, in view of the significance of this conclusion, the question merits further study herein for OFE

copper - and still further study beyond this preliminary investigation for

both OFE copper and engineering alloys. The next section begins with an

examination of the common situation where transient effects and sequence

induced premature initiation are combined through the use of Ae^ as a damage

parameter. Thereafter transient effects are isolated from sequence induced

premature initiation.

EXAMINATION OF THE DAMAGE POSTULATE

History Dependence In a Strain Based Framework

Recall as indicated earlier that bulk changes in stress related to

changes in microstructure which in turn related to damage. This meant that

the material evolved in its hardness or ductility so long as microstructural

changes evident in terms of changes in bulk parameters were occurring. Also,

as just detailed it meant that if different stress levels develop for the same

strain led to different values of the damage parameter in a given material,

then differing amounts of damage per cycle develop even though the strain was

the same. Consequently if different fatigue resistance curves can be

developed for the same material in differing initial states of hardness/

ductility one could conclude that a sample of material exhibiting transient

response also exhibits differing fatigue resistances over its life. Because
66

transient behavior is history dependent in general, one could conclude that

the fatigue resistance is also history dependent, consistent with the

postulate.

The possible history dependence of a material's fatigue resistance

can be examined by comparing the fatigue resistances of a material in several

i

different initial hardnesses/ductilities, based on the data in Table 5. These

results represent OFE copper prepared in one of four hardnesses developed

either by the IPS or by heat treating the half hard (drawn) material as

detailed earlier. Testing was done at one of three identical strain levels -

Aei, Ae2, and Ae3 - selected to cause nucleation at about 103, 104, and 105

reversals. Raw data developed are presented in Figure 14(a). Observe that

there is a layering of the data at high strains—longer lives being associated

with increased ductility as expected. However, at the lowest strain this

layering begins to wash out. This observation is less than surprising when

interpreted within conventional wisdom which suggests curves for materials

with different initial hardness crossover as life increases, in that long life

is considered to be controlled by strength, not ductility. Regardless of why

the layering begins to wash out, the data does layer indicating differing

fatigue resistance curves for mechanically and thermally induced hardness

changes such as develop under cyclic loading. This layering thus implies that

different resistance curves exist at different stages of the life. This in

turn implies a history dependence of the damage process consistent with the

postulate. When coupled with the indications that transient effects cause a

history dependence of damage as discussed previously, these trends are

considered to constitute reasonable proof that the material's resistance curve

under VA cycling is history dependent. Furthermore, unless transient effects

can be accounted for by a damage parameter (which yields an equal increment of

damage for a given level of damage parameter anywhere in the history up to

nucleation), damage analysis will in general require a history dependent

resistance. This last aspect is deferred for later consideration.

Another interpretation of the trends in layering also exists which

is consistent with the postulate and these data, as well as the published data

that lead to the conventional wisdom, as follows. Figure 14(b) presents the

elastic and plastic components [20] of the results shown in part (a). Note
67

100

10 0>

o>

in

I '

0.1

OFE Copper

Fully Reversed Strom Control

Baseline Results

• As received, half hard

A Heat treat A (HTA)

• Heat treat B (HTB)

? HTA, started in compression

I01 I02 I03 I04 10s 10s

Reversals to Nucleation

(a) Total strain-life behavior

10

c

'

O.I

OFE Copper

Fully Reversed Strain Control

-Plastic strain response

Heat B

Heat A

As received

Heat treat B

/Heat treat A

As received

Elastic strain response

(b)

I05 10"

Cycles to Nucleation

Elastic and plastic strain-life behavior

10=

FIGURE 14. FATIGUE RESISTANCE OF OFE COPPER AT

20°C IN FLOWING ARGON
68

that the plastic components of the total strain layer as did the data based on

the total strain. Note too that the stress response embodied in the elastic

component is essentially independent of the initial hardness of the material.

Figure 14(c) indicates this by a strong correlation of the stresses, with the

exception of the one result for the softest material and longest life. Recall

that Basquin's law [21] indicates the elastic strain (stress/E) life trend

follows a power law out to the endurance limit. Extrapolating the elastic

strain life line (excluding the just noted point for the time being) accordingly and adding that result to the plastic components in Figure 14(b)

produces strain life trends which now layer at low strains as they do at high

strains. The results of this construction are shown in Figure 14(d) with

regard to the long life trend for HTB.

Clearly the construction of a small portion of Figure 14(d) is

artificial and at odds with conventional wisdom that suggests these layered

curves should cross. Note however that the constructed portion of the curve

differs from conventional life data in that material resistance in the absence

of surface effects is sought. Recall that the softest material has been noted

earlier to develop a more intensified surface roughness more extensively than

its harder counterparts, and that this process occurred much earlier in life.

At shorter lives higher plasticity tends to wash out notch effects{!9,40]. Also

surface roughening tends to develop in all materials. At high strains the

effects of surface are more or less equally present in the material conditions

presented here, based on replication results. This means that at high strains

the relative effect of surface tends not to be a factor while at lower strains

it becomes important first for softer materials. The difference between the

constructed portion of Figure 14(d) as compared to the raw data of Figures

14(a) and (b) therefore is postulated as being due to intervening surface

effects. That is, it is suggested that instead of the fatigue resistance

layering based on hardness/ductility throughout life, more ductile materials

fail to achieve their potential resistance because intensified surface

degradation intervenes by creating notches which serve to reduce fatigue

resistance.

The interpretation of fatigue resistance as just detailed suggested

that the so called "rocking chair effect" has a physical interpretation other
69

100

OFE Copper

Fully Reversed Strain Control

v>

in

V)

• As received

A HTA

• HTB

? HTA, started in compression

10

I02

10

a

o>

I

i

0.1

10'

10"

Cycles to Nucleation

(c) Stress-life behavior

Trends Anticipated in Absence of

Surface Effects .

Heat treat B

Heat treat A

As received

(half hard)

I I

IOZ 10" 10= 10"

Reversals to Nucleation

(d) Constructed total strain-life behavior

10=

FIGURE 14. Continued
70

than balancing ductility for short life resistance against strength for long

life resistance. Clearly strain-life data do exhibit this rocking chair

effect, shown schematically in Figure 15. The postulate suggests these data

trends develop as a consequence of layering bulk resistances based on

hardness/ductility coupled with a loss of resistance at the surface by

intervening notch enhanced initiation at longer lives.

In passing it is worth noting that the constant amplitude fatigue

resistance at the lowest strain shown in the conventional plot (Figure 14(a))

is similar to that developed under various significant prestrains followed by

cycling to failure at that strain level. This situation is at odds with other

prestrain data which show a much reduced fatigue resistance attributed to the

effect of the prestrain. However, if the plot of Figure 14(d) is used as the

reference for constant amplitude cycling, the anticipated marked effect of

prestrain is indicated. While this can't be taken as evidence supporting the

postulate and the related reinterpretation of the rocking chair effect, it

does indicate a further look at basic damage mechanics is warranted.

History Dependence in the Absence of Transient Effects

Conside'r now analysis of the history dependence of the resistance

curve in terms of dissipated energy as a damage parameter. Note in this

respect that the materials resistance is being correlated with dissipated

energy—nothing fundamental is being assumed from which life is predicted from

dissipated energy. The potential advantage of using dissipated energy as a

damage parameter lies in its being nearly constant over life, as shown for the

OFE copper in Figure 16.* This tends to negate the influence of transients on

the CA reference resistance curve, at least for the lives and material of

interest in the present study. Following this scheme the value of the damage

parameter per cycle is equal to the integrated dissipated energy divided by

the number of cycles to initiation.

*This nearly constant behavior is not typical (e.g. see [20]), nor is dissipated energy useful at long life where AeP-»0 as Aefc decreases. Other

parameters which circumvent these difficulties therefore should be sought.
71

10

TJ

3 -2

10

<

10

Aus H-ll

18 Ni Mar

SAE 1045 (4IOBHN)

SAE 5132 (290BHN)

SAE 950X

SAE 1010

_ I Transition fatigue life

I Mil I I III I I I il i i I il I i IH i

I01

I03 KT

Reversals to Failure

10= I0610'

FIGURE 15. STRAIN-LIFE CURVES FOR A VARIETY OF STEELS:

NOTE THE CROSSOVER TENDENCIES AND THE FACT

THAT STRONG MATERIALS ARE BEST AT LONG LIVES

WHILE DUCTILE MATERIALS ARE SUPERIOR AT SHORT

LIVES (AFTER (41))
72

I04

10°

OFECoccet

Fully Reversed Slron Cycling

-^^—^^— As received

As received (IPS)

HTA

HTB

-I 8%

-Q65%

Af-032%

Ml I I I I I I I I I I I I I i I I i ' I

10' IO2 >05 10* I0:

Cycles

FIGURE 16. TRANSIENTS EVIDENT IN DISSIPATED ENERGY

AS A FUNCTION OF CYCLES

\> OFE Copper

Fully Reversed Strom Cycling

AR-CA

-•- AR-IPS/CA _

10'

i

LU

I

10° S

Q

10' I02 10* 10s

 I06 I07

Cycles to Initiation

FIGURE 17. FATIGUE RESISTANCE AS A FUNCTION OF DISSIPATED ENERGY
73

Figure 17 plots dissipated energy versus cycles to initiation as

just detailed, and it shows the same layering evident in Figure 14.* Because

transients occurring during the life have been negated in developing these

histories such as IPS early in the life are thus expected to have a resistance

curve shifted toward the left (decreased life). Indeed, although the difference is not too great, this is exactly the result observed for the AR material

when the CA cycling is preceded by IPS.

As noted in dealing with the topography for the IPS, its effect is

observed to be inconsequential for the AR material. For this reason, the

entire shift to decreased resistance for the AR material must be ascribed to a

reduction in the innate resistance to cracking of the bulk material. In this

respect it is noted that the IPS cyclically hardens this material, and the

following cycles do not cause softening to the level in CA tests of AR material. Given the trend of reduced life with increasing hardness (decreasing

ductility) evident in Figures 14 and 17, the results for the IPS hardened OFE

copper match exactly the trend predicted from the damage postulate. That is

the fatigue resistance of the bulk material is controlled by hardness (ductility) changes due to the deformation history. It follows then (in the

absence of history related differences in the surface topography) that differences in the bulk resistance of the material can be characterized in terms of

the current histeresis loop shape and deformation resistance normalized to

some reference condition. The best material reference condition .is one for

which transient effects are minimal.

Conveniently, deformation resistance and loop shape vary in a unique

history dependent fashion which can be characterized as functions of £AeP,

with respect to some reference deformation condition - say that characterized

by the monotonic stress-strain curve. Differences in fatigue resistance

*Note that in this figure the long life behavior for HTB material has been

developed following the construction used in Figure 14(d). Recall that this

construction attempts to develop the fatigue resistance of the bulk material

from the actual behavior which represents an increased bulk resistance offset

by the well developed surface which acts as a distribution of notches to

increase the surface strain above the bulk strain. In this sense, differences in the fatigue resistances of the material conditions shown in Figure

17 does not depend strongly on surface topography. Instead bulk properties

control.
74

evident here are due solely to innate differences in the bulk resistance of

these materials to crack initiation. According to the postulate, this is due

to the history dependence of the deformation response. Soft material, such as

HTB, harden under cyclic loading whereas hard material, such as AR/IOS,

soften. Certainly these contrasting transients complicate the picture. But,

fortunately for present purposes, such transients can be simply tracked and

accurately predicted via computer models (e.g., see [5]). Consequently, this

added degree of complexity does not adversely complicate damage analysis under

variable amplitude (VA) cycling. However, it will enhance the accuracy by

accounting for history dependent nonlinearities in damage assessment, while

retaining the simple nonarbitrary assumption of linear accumulation.

It is interesting to explore the implications of the above suggestion that the bulk state of the material controls fatigue resistance in the

absence of strongly different surface topographies. Consider first the

results of specimens 7, 9, 11, and 19 subjected to various amounts of the IDS*

(20, 100, 50 and 20 cycles, respectively). Observe from Table 5 that the

lives to initiation for these specimens do not show an ordered pattern in life

reduction with increased number of cycles of IDS. The mean life for the group

is 42,394 cycles with a standard deviation** of 6046 cycles. All of these

lives are in excess of the life to initiation observed for the non IOS material represented by specimen 21 (32520 cycles).

The fatigue resistances for IOS and non IOS specimens imply that the

IOS (1) did not alter the surface to enhance initiation as compared to the non

IOS case and (2) either the IOS increased the innate resistance to cracking of

the bulk material, or reduced the effective driving force for damage, or

induced beneficial surface residual stresses. Both (1) and the last item of

(2) have been shown earlier not to be factors in this study. Regarding the

driving force, the CA value of Ae is the same in all cases. However, the

*Note that the IOS was followed by an IPS to wipe out any mean stress due to

the stepdown in strains.

**The significance of this number is questionable in a rigorous statistical

sense - its use here is as a relative measure of scatter. Its usual

statistical interpretation is not invoked.
75

magnitudes of the dissipated energies per cycle are higher in the case of the

IDS plus CA, suggesting the lives for these histories should be shorter, but

certainly not longer as was observed.

One possible explanation for this inversion of expected trends

follows from the postulate: differing innate resistances to cracking develop

due to the differing deformation histories imposed." Study of deformation

responses shows the IDS data soften very rapidly to levels somewhat below

their non IDS (CA) counterpart. Based on Figure 17, this means that the

fatigue resistance curve for the IDS specimens actually lies to the right of

AR material trend. This of course means longer lives for the IDS specimens,

just as observed.

A test involving a slightly more complicated but less severe history

as compared to the just discussed IDS cases was also performed, as indicated

in Table 5 for specimen 12 made from AR material. This specimen was initially

overstrained in compression and then partially unloaded so that a compressive

mean strain remained. Following this the specimen was subjected to 8 cycles

of incrementally decreasing amplitude to reach Ae3. The slight compression

mean stress was removed by a corresponding reduction in the mean strain.

Following this, CA cycling at Ae3 was applied until failure occurred. The

result after about 30 reversals of this CA history was a hysteresis loop

bounded in energy content by specimens 21 and 36, but involved a somewhat

increased stress for the same total strain range (Ae3) as compared to either

specimen. The postulate indicates that the life of specimen 12 should be

bounded above by the AR curve (specimen 21) and below by the curve for the AR

material after the IPS (specimen 36), for the same level of damage. As can be

found by study of Figure 17 at the damage level of interest, the life to

initiation for specimen 12 was indeed so bounded. While this in itself does

not prove the viability of the postulate it is yet another result

qualitatively consistent with it.

The result for specimen 12, like most of the others, is not exactly

what is expected based on trends in the literature. Yet it, as well as the

other results discussed to date, qualitatively fit predictions of the postulate. Significantly the postulate can qualitatively explain the more typically observed literature trends as well - but further discussion of this is

deferred until completing this section dealing with this investigation.
76

Results were also developed for IDS of HTB material using specimens

18 and 24. Specimen 18 had a 50 cycle IDS, while specimen 24 had a 100 cycle

IDS history. For these specimens hardening was greatest in specimen 24, and

the rate of softening following the IDS was slowest and occured to the least

extent in specimen 24 as compared to specimen 18. The postulate indicates

that specimen 24 should not last as long as specimen 18 given the same damage

level is imposed in that the innate bulk resistance of specimen 24 is less

than that of specimen 18. Such a result indeed developed. Table 5 shows

their lives are 64,290 and 46,428 cycles to initiation, respectively.

However, Table 5 also shows that the damage levels imposed on specimens 18 and

24 actually differed. The energy per cycle introduced was greater for

specimen 24 as compared to specimen 18 (17.33 joules vs 16.45 joules; see

Table 6), even though the CA strain was the same. This difference in energy

levels can be accounted for using the slope of the HTB data of Figure 17 to

define the rate of change of resistance as a function of dissipated energy

level. Normalized against specimen 18 with its observed life of 64,290 cycles

at an energy level of 16.45 joules/cycle, the expected life of specimen 24 at

17.33 joules/cycle is in excess of 60,000 cycles in the absence of a deformation history dependent resistance curve. The observed life of 46,428 cycles

therefore is reduced compared to specimen 18 at the same energy level as

expected based on the postulate.

Note too that both IDS specimens 24 and 18 softened at a slower rate

and to a lesser extent the non IDS specimen (number 15 in Table 5). Therefore, according to the postulate, specimen 15 should survive longer than

either specimen 24 or 18. Such was indeed observed, a result which again

points to a dependence of the innate fatigue resistance of a material on the

deformation history.

Consider now the IDS results developed for specimen 20, which

started as AR but following the IDS was heat treated to the HTB condition.

For this specimen the first 100 cycles did damage in the AR condition which,

based on Figure 17, occurs at a just slightly faster rate than it would in the

HTB condition. Following the IOS, the deformation history was wiped out by

the HTB creating essentially virgin bulk material with some initia l surface

damage. Cycling of this specimen in the HTB condition produced a gradual
77

softening to a condition almost identical to specimen 24, but somewhat harder

than the CA reference case (specimen 15). The postulate suggests that the

life of specimen 20 should be reduced from that of specimen 15. Furthermore,

given its deformation history is similar to that of specimen 24, comparable

lives are expected. Interestingly, both qualitative predictions of the

postulate are borne out in the lives observed, as evident in Tables 5 and 6.

Specimen 25, which starts as HTB but after the IDS is again subject

to HTB, also constitutes an interesting qualitative test. The first 100

cycles in this test are identical to the first hundred cycles in specimen 15.

After reheat treatment to HTB some of the prior mechanical history remains, as

it did for specimen 25 because the deformation response upon reloading showed

higher yield stresses than did AR material subjected to HTB. This difference

means that specimen 20 gradually softens, just as specimen 25 did, except the

rate of softening is somewhat slower and the extent somewhat greater in

specimen 20 as compared to specimen 15. Qualitatively the harder specimen

(number 20) is expected to fail before the HTB OFE copper reheat treated after

further cold working (specimen 25). This further cold work enhances

recrystallization and would develop a material that is softer (more ductile)

than that for HTB. Given the trend of resistance with hardness evident in

Figure 17, this material's fatigue resistance curve would lie to the right of

that for HTB. This new trend is the upper bound to the life for specimen 25,

because some life is exhausted by the first 100 cycles of IDS. Unfortunately

in the absence of the reference curve for this double heat treated and cold

worked material little can be said quantitatively. But the observed life of

74,600 cycles is well in excess of the life for specimen 15 (66,840 cycles).

It should be noted that concern must in general also be given to the

effect of specimen surface topography that develops differently from the

reference CA condition. The fact that the trends in Figures 14 and 17 lie

well beyond the observed life for specimen 15 (HTB) has been ascribed to the

fact that the surface which develops in this sample creates a crystallographic

notching which is "effective" in reducing the life. This is not to say that

notches do not develop in the other samples. Rather it means that specimen 15

behaves differently than the other samples because (1) the strain level is low

so that damage is localized and notches are not blunted by the extensive
78

plastic flow, (2) crystallographic notches are tending to become the dominant

surface feature at lower strains, and (3) the softer material develops more

extensive localized plasticity in fewer cycles, much earlier in the life based

on the bulk resistance. As suggested earlier in regard to Figure 14(d), this

tendency (exhibited here at £63 for HTB and expected for HTA and AR at still

longer lives and lower strains) would lead to the so called rocking chair

effect evident in Figure 15. But further study is needed before this can be

discussed in other than a speculative fashion.

Damage Assessment and Accumulation Predictions for Block Cycling

To this point, the discussion of the history dependence of fatigue

resistance has focussed on IPS and IDS histories followed by CA cycling. One

interesting implication of the application of the postulate to these results

is that the current hysteresis loop shape and deformation resistance dictates

which fatigue resistance curve should be used to assess damage per cycle.

Differences in deformation resistance are due to transient hardening or

softening, or in general mean stress relaxation and cyclic creep as well.

They may also be due to hardening in tensile or compressive excursions along

the skeleton (monotonic) deformation curve.

The postulate suggests that nonlinearities in damage analysis for

block cycling arise due to transients taking place within the blocks, and to

differences in the resistance curves associated with the stable deformation

response that may develop if a block is sufficiently long. For example,

consider a history composed of two constant amplitude blocks, with each of

these blocks being repeated alternately until initiation. For tests in which

all transients occur in the first pair of blocks, the postulate suggests that

a range of resistance curves are needed during the first pair, and thereafter

a single pair of resistance curves are sufficient for all subsequent blocks.

In situations where transient action persists for several blocks, nonlinearities continue over those blocks. In this case the postulate requires assessment of damage using distinctly different resistance curves depending on the

transient behavior in each block. Ultimately, if the transients washout,

different resistance curves (one for each block) define the materials fatigue
79

resistance until initiation. In this respect, block cycling provides an

interesting test of the viability of the postulate. For this reason tests

have been done using low-high (lo-hi) strain and hi-lo strain block cycling.

Test conditions are detailed in Table 1 while the results are presented in

Tables 5 and 6.

Analysis of the block cycling results would in general require

presenting the fatigue resistance trends presented in Figures 14 and 17 as a

function of the material's deformation response. For the sake of convenience

in analysis, these resistance curves should be expressed as continuous

functions normalized with respect to one of the material conditions. This

would premit simple yet convenient interpolation over the range of deformation

behaviors observed. Furthermore the analysis would best be done numerically,

assessing damage per cycle and intergrating this subject to the failure

criterion that a sum of unity defines initiation in the spirit of Figure 2.

Such a numerical formulation is desirable, but is not warranted for this

preliminary investigation involving two rather simple tests. For this reason,

bounding resistance curves have been used to simplify this preliminary

analysis, and a variant of the dissipated energy formulation—the product

smxAet—is used as a damage paramter. The results therefore, are approximate

compared to what could be achieved with a more detailed numerical formulation.

Consider first the results for the lo-hi history—specimen 23. The

first 104 cycles of this test are identical to that for specimen 15 (HTB

material). For this reason the correct initial resistance curve is that for

HTB material, results for which are plotted in Figure 18. At Ae3 = 0.00337

and smx = As/2 = 180.65 MPa for the block, the damage level/cycle D = s^Aet =

0.606 MPa so that the expected life is 34,000 cycles. Therefore, the damage/

block DB = 0.294 and, the residual damage capacity, Dr = l-£Dg = 0.706. This

history does little to harden the HTB and so the HTB resistance curve also

reasonably represents the resistance at the end of the first block. The

second block of cycles is the high block, which involves 10 cycles identical

to the IOS. Based on results for the HTB material subjected to 20 cycles of

this IPS (specimen 18) it is expected that the material hardens and then

softens during this block. Such hardening and softening is observed, the

stress response approaching 321.1 MPa, indicating according to the postulate
80

I01

OFE Copper

Fully Reversed Strom Cycling

—O—— AR-CA

-•- AR-IPS/CA

O HTA-CA

•— -£f HT8-CA

•Trend following

Figure 14 d

ll I I III

tf) Jt

"o><

10°

10' I03 10* I05

Cycles to Initiation

FIGURE 18. FATIGUE RESISTANCE AS A FUNCTION OF
81

that a resistance curve comparable to that for AR material is appropriate.

Using Ae* = 0.0204 and smx = 452/2 MPa, D = 4.61 MPa so that N-j = 490 cycles,

and DB = 0.0204 which leaves Dr = 0.706 - 0.0204 = 0.686.

The next and all subsequent lo blocks show that the material softens

quickly to a nearly stable stress range As = 283.86 MPa. At Ae3 = 0.00337,

D = 0.47 MPa. The resistance curve for this case corresponds to HTB material

softened by an IDS to As = 280 MPa cycling at Ae = 0.00337. Examination of

the deformation responses for the various material conditions listed in Table

5 indicates that a resistance curve between HTA and HTB, but closer to HTA

would be associated with the observed combination of stable stress and strain.

Using the lower bound curve (the HTA resistance curve), this value of D = 0.47

MPa corresponds to a life of about 74,000 cycle so that Dg = 0.1351. The next

and all subsequent hi blocks are reasonably represented by a stress range

As = 463 MPa at Ae = 0.0204, so that D = 4.72 MPa. Examining the deformaion

responses listed in Table 5 indicates this behavior is also bounded by the HTB

and HTA conditions, and again the HTA (lower bound) most closely corresponds

to the observed behavior. AT D = 4.72 MPa, the life expected based on HTA is

602 cycles, so that the damage per lo-hi pair, D|_H = 0.0166 + 0.1351 = 0.1517

for each pair of blocks. Thus the number of blocks remaining is 4.52, or

45,252 cycles. This number of cycles, when added to the earlier analyzed pair

of 10,010 cycles, indicates a predicted life of 55,262 cycles.

The predicted life is essentially identical to the observed life of

55,084 cycles to initiation. It should be noted, because of the preliminary

nature of the study, that approximations were made to simplify the calculations. But in performing these calculations an attempt has been made to make

offsetting simplifications. For this reason, transients have been ignored,

which tends to reduce damage somewhat, whereas lower bound resistance curves

have been used which tends to overestimate damage. While these approximations

offset, the exact extent of this approximation and the related error wil l

remain uncertain until the damage algorithm has been coded for numerical

evaluation. Bear in mind that these same approximations carry over into the

analysis of the hi-lo BC history which follows.

Specimen 22 has been subjected to the hi-lo history detailed in

Table 1, results for which are reported in Table 5. Study of the stable
82

stress response for the high blocks indicates the hardening during the first

block is similar to that for the AR condition. The second block stabilizes at

stresses similar to that for the HTA condition, whereas in subsequent blocks

the deformation response is comparable to the HTB condition. Corresponding

results for the low blocks indicate the deformation response in the first

block is like that of the HTA condition, while that of the second and third

blocks lies between the HTA and HTB conditions (but is close to the HTA lower

bound). Subsequent low blocks soften to a state for which the lower bound is

reasonably represented by a trend midway between the HTA and HTB conditions.

Analysis of the first 10 cycles indicates D = 0.0204 x 526/2

MPa = 5.37 MPa, for which N-j = 375 cycles and DB = 0.0267. The next 104

cycles occur at D = 0.501 for which N-j = 60,000 and DB = 0.1667 leaving

Dr = 0.807 after the first hi-lo pair. The next pair does less damage, the

values being D = 4.63 MPa and D = 0.478 MPa for the hi and lo blocks,

respectively. Corresponding lives to initiation are respectively 535 and

75,000 cycles, which yields Dg = 0.0187 and DB = 0.133 for a sum, D|_H, of

0.152 leaving Dr = 0.655. The third pair of hi-lo blocks involves D = 4.63

MPa and D = 0.478 respectively. Corresponding lives to initiation are 700

cycles and 75,000 cycles, which yields DB = 0.0143 and DB = 0.133 for a sum,

D[_n» of 0-148 leaving Dr = 0.5074. The fourth and all subsequent blocks are

similar in terms of damage per cycle. All high blocks share the HTB resistance

curve while all low blocks share a resistance curve between HTA and HTB. The

values of D = 4.47 MPa and D = 0.468 MPa are associated with lives to initiation of 740 cycles and 77,500. (The last result is based on HTA, a lower

bound life which overestimates damage.) Damage per block, DB = 0.0135 and

DB - 0.129, yields a sum D|_H = 0.1425, so that the remaining number of blocks

to failure is 0.5074/0.1425 = 3.56, or 35,631 cycles. The predicted total

life thus is 35,631 + 30,030 = 65,661 cycles, as compared to an observed life

of 73,260 cycles.

While the prediction for the hi-lo case does not match that observed

as closely as did the lo-hi case, the result does match the qualitative trend

to much longer life. Indeed, if, (as the deformation response suggests) a

resistance curve biased away from the HTA toward the HTB condition is used,

much of the discrepancy between predicted and observed lives disappears.
83

Results of the reanalysis indicates a predicted life between 70,000 to 85,000

cycles, depending on where the resistance curve is placed between the HTA

lower bound and the HTB upper bound.

Commentary

Damage analysis of the BC results as well as the IPS and IDS

histories using history dependent resistance curves consistent with the

postulate correctly predicts trends observed in all cases, including the lo-hi

and hi-lo tests. On the other hand the usual strain based linear analysis

fails to account for the IPS/IOS trends, and predicts the same life for both

BC histories—about 60,919 cycles.

The successes with the IPS/IOS results aside, one could still argue

that the above differences in observed lives for lo-hi and hi-lo BC histories

develop as a result of scatter, so that successful prediction of any observed

life is fortuitous. While such an argument cannot be proven incorrect, the

trends for BC histories in the literature, coupled with the nearly 20,000

cycles difference in lives observed herein suggest this arguement is vacuous.

It could also be argued that statistical scatter in subsequent similar tests

would cover a sufficiently broad range of lives such that predictions close to

any of the one shot test results reported herein are in general of little

consequence. Again such an arguement is difficult to prove wrong. But, one

must bear in mind that the framework advance herein does not predict a fixed

life for a given strain controlled test history. That is, different lives are

predicted by this damage postulate to the extent that differences in deformation response develop. For this reason arguements related to scatter are

equally vacuous in the present context.

While the postulate advanced holds promise, it has only been applied

in situations for which differences in surface topography were a minor factor.

This was not by design, as evident in attempts to experimentally develop surface effects by tests such as 100 IOS cycles at Ae = 0.02 followd by Ae3 to

failure. For this reason, it remains to explore the viability of the postulate at longer lives (lower strains) where surface wil l be a factor. It likewise remains to examine the postulate in engineering alloys, as well as in

combination with other damage mechanisms.
84

It should be noted in closing that the postulate advanced is rather

general. For example, in the absence of surface effects one has:

_s = §(e) (4)

i

which defines the current (history dependent) deformation state, and it is

postulated that damage per cycle, D, and the damage resistance, R, are also

history dependent

D = D(s.,e4 ...) , and , (5)

R = R(s,e_, ...) .

Equations of the form of Eq. (5) have been shown by the data developed herein

to hold for a variety of conditions. In general the resistance wil also have

to be related to he surface topography, S, and residual stress state, g, which

may be step funcions of R, once a threshold is reached. That is, more

generally,

B = _

Rl(s»e,IeP,S,g, . . .) (5a)

By analogy to the results and postulate for fatigue damage, a similar more

general postulate and, similar equations could be written for creep fatigue,

or thermal-mechanical fatigue, but care must be taken to include the appropriate rate dependence in Eq (4) and differences in surface effects (wedging,

area loss, etc.) in Eqs. (5) and (5a). Simple equations of the form of Eqs.

(4) and (5) have already been proposed (e.g., see [12]), and shown to be useful

on a limited basis.

SUMMARY

The objective of this progam was to assess the viability of a damage

postulate which asserted that the fatigue resistance curve of a metal is

history dependent due to inelastic action. The study focussed on OFE copper

because this simple model material accentuated the inelastic action central to
85

the damage postulate. Given the novel nature of the postulate, the scope of

the study was limited to that of a preliminary evaluation.- The study carried

out was purely phenomenological. Data relevant to damage evolution and crack

initiation were developed via a study of surface topography. The effect of

surface layer residual stresses were explored via comparative testing as were

i

the effects in initial prestraining.

The results of the study very clearly indicated the deformation

history dependence of the fatigue resistance of OFE copper. Furthermore the

concept of deformation history dependence was shown to qualitatively explain

the fatigue resistance of all histories considered. Likewise quantitative

predictions for block cycle histories were found to accurately track the

observed results. In this respect the assertion that damage per cycle for a

given level of the damage parameter is deformation history dependent appears

to be physically justified. Also, use of a history dependent nonlinear

assessment procedure consistent with the postulate provided qualitative and

accurate quantitative predictions of life to form a small crack when damage

was linearly accumulated. For this reason the nonarbitrary linear accumulation assumption appears to be valid when nonlinearities in assessment were

accounted for. Double linear accumulation criteria [42,43] which deal

separately with initiation and propagation would properly deal with initiation

provided some nonlinear assessment scheme were adopted.

CONCLUSIONS

Many conclusions can be drawn as a result of this study. Among the

more significant are the following.

• The use of simple damage parameters such as strain range may lead

to errors in adapting constant amplitude data to damage analysis

of variable amplitude histories.

• Transients driven by cyclic inelastic action may violate the

linear damage accumulation assumption in the reference constant

amplitude data base. This coupled with the use of simple damage

parameters may effectively preclude adapting constant amplitude

data for damage analysis of complex histories in terms of this

linear assumption.
86

The fatigue resistance curve of a material is deformation history

dependent.

When a deformation history dependent resistance curve is used in

damage analysis, accurate qualitative and quantitative

predictions for variable amplitude histories can be made in terms

of a linear accumulation criterion, up to the initiation of a

small crack.
87

REFERENCES

1. Matsuishi, M., and Endo, T., "Fatigue of Metals Subjected to Varying

• Stress", Paper presented to Japan Society of Mechanical Engineers,

Fukouka, Japan, March 1968; see also Endo, T., Mitsunaga, K.,

Takahashi, K., Kobayashi, K., and Matsuishi, M., "Damage Evaluation

of Metals for Random or Varying Load", Paper presented at 1974

Symposium on Mechanical Behavior of Materials, Kyoto, August, 1974.

2. Dowling, N. E., "Fatigue Life and Inelastic Strain Response Under

Complex Histories for an Alloy Steel", J. of Testing and Evaluation,

Vol. 1, (4), July, 1973.

3. Wetzel, R. M., "A Method of Fatigue Damage Analysis", Ph.D. Thesis,

University of Waterloo, 1971.

4. Jhansale, H. R., "Inelastic Deformation and Fatigue Response of

Spectrum Loaded Axial and Flexural Members", Ph.D. Thesis, University

of Waterloo, March, 1971.

5. Conle, F. A., "A Computer Simulation Assisted Statistical Approach to

the Problem of Random Fatigue", M.A.Sc. Thesis, University of

Waterloo, March, 1974.

6. Landgraf, R. W., Richards, F. D., and LaPointe, N. R., "Fatigue Life

Predictions for a Notched Member Under Complex Loading Histories",

SAE AE-6, SAE, 1977, pp 95-106.

7. Leis, B. N., "Fatigue Analysis to Assess Crack Initiation Life for

Notched Coupons and Complex Components", Ph.D. Thesis, University of

Waterloo, September, 1976.

8. Morrow, J. D., in Fatigue Design Handbook, SAE, Chapter 3.2, 1968.

9. Smith, K. N., Watson, P., and Topper, T. H., "A Stress-Strain

Function for the Fatigue of Metals", SMD Report 21, University of

Waterloo, Ocotber, 1969; see also, Smith, K. N., Watson, P., and

Topper, T. H., "A Stress-Strain Function for the Fatigue of Metals",

J. of MatIs.. JMLSA, Vol. 5 (4), pp 767-778, December, 1970.

10. Haibach, E., and Lehrke, H. P., Das Verfahren der AmplitudenTransformation, LBF Report No. FB-125, LBF Darmstadt, 1975.

11. Ostergren, W. J., "A Damage Function and Associated Failure Equations

for Predicting Hold Time and Frequency Effects in Elevated

Temperature Low Cycle Fatigue", J. Testing and Evaluation, Vol. 4,

No. 5, pp 327-339, 1976.
12. Leis, B. N., "An Energy-Based Fatigue and Creep-Fatigue Damage

Parameter", Journal of Pressure Vessel Technology, Trans. ASME, Vol.

99, No. 4, pp 524-533, November, 1977.

13. Leve, H. L., "Cumulative Damage Theories", in Metal Fatigue: Theory

and Design, Wiley, 1969.

14. Schijve, J. "Estimation of Fatigue Performance of Aircraft

Structures", in Fatigue Testing of Aircraft Structures, ASTM STP 338,

1961, pp 193-201.

15. Libertiny, G. Z., Topper, T. H., and Leis, 8. N., "The Effect of

Large Pre-Strains on Fatigue", Experimental Mechanics, Vol. 17, No.

2, pp 64-68, February, 1977.

16. Leis, B. N., and Laflen, J. H., "An Energy Based Postulate for Damage

Assessment of Cyclic Nonproportional Loadings with Fixed Principal

Directions", Proceedings ASME Symposium on Ductility and Toughness in

Elevated Temperature Service, San Francisco, MPC-8, pp 371-389,

November, 1978.

17. Leis, 8. N., and Forte, T. P., "Fatigue Damage Analysis Under

Variable Amplitude Cycling", in Random Fatigue Life Prediction, ASME,

to appear, 1983.

18. Morrow, J. D., Wetzel, R. M., and Topper, T. H., "Laboratory

Simulation of Structural Fatigue Behavior", Effects of Environment

and Complex Load History on Fatigue Life, ASTM STP 462, pp 74-91,

1970.

19. Leis, B. N., Gowda, C.V.B., and Topper, T. H., "Cyclic Inelastic

Deformation and the Fatigue Notch Factor", Cyclic Stress-Strain

Behavior Analysis, Experimentation and Failure Prediction, ASTM STP

519, American Society for Testing and Materials, pp 133-150, 1973.

20. Morrow, J., "Cyclic Plastic Strain Energy and Fatigue of Metals",

Internal Friction, Damping, and Cyclic Plasticity, ASTM STP 378,

American Society for Testing and Materials, 1965, pp 45-84.

21. Basquin, 0. H., "The Exponential Law of Endurance Tests", Proc. ASTM,

Vol. 10, pp 625-630, 1910.

22. Coffin, L. F., "A Study of Cyclic Thermal Stresses in a Ductile

Metal", Trans. ASME, Vol. 76, pp 931-950, 1954.

23. Manson, S. S., "Behavior of Materials Under Conditions of Thermal

Stress; Heat Transfer", Proc. Symposium, U. of Michigan Engr.

Research Inst., pp 9-75, 1953.

24. Broek, D., and Leis, B. N., "Similitude and Anamolies in Crack Growth

Rates", in Materials, Experimentation, and Design in Fatigue,

Westbury House, IPC Science Press, UK, pp 129-146, March, 1981.
89

25. Leis, B. N., "Predicting Crack Initiation Fatigue Life in Structural

Components", in Methods of Predicting Fatigue Life, ASME, pp 57-76,

1979.

26. Palmgren, A., "Die Lebensdauer von Kugellagern", VDI-Z, Vol. 69, pp

337-341, 1929.

27. Miner, M. A., "Cumulative Damage in Fatigue", J. of App. Mech. , Vol.

12, pp A-159, 1945.

28. Landgraf, R. W., "The Resistance of Metals to Cyclic Deformation", in

Achievement of High Fatigue Resistance in Metals and Alloys, ASTM STP

467, 1970, pp 3-36.

29. Jhansale, H. R., "Inelastic Deformation and Fatigue Response of

Spectrum Loaded Axial and Flexural Members", Ph.D. Thesis, University

of Waterloo, March, 1971.

30. Forsyth, P.J.E., "The Physical Basis of Metal Fatigue", Blackie and

Son Ltd., London, 1969.

31. Allery, M.B.P. and Birkbeck, G., "Effect of Notch Root Radius on the

Initiation and Propagation of Fatigue Cracks", Enqr. Fract. Mech.,

Vol. 4, No. 2, June 1972, pp. 325-331.

32. Pangborn, R. N., Weissmann, S., and Kramer, I. R., "Dislocation

Distribution and Prediction of Fatigue Damage", Metallurgical

Transactions A, Vo. 12A, 1981, pp 109-119.

33. Mughrabi, H., Ackermann, F., and Herz, K., "Persistent Slip Bands in

Fatigued Face Centered and Body Centered Cubic Metals", Fatigue

Mechanisms, ASTM STP 675, pp 69-105, 1979.

34. Laird, C., "Recent Advances in Understanding the Cyclic Deformation

of Metals and Solid Solutions", in Work Hardening in Tension and

Fatigue, AIME, pp 150-176, 1976.

35. Kocanda, K., Fatigue Failure of Metals, Noordhoff, 1978.

36. Brick, R. M., Gorden, R. E., and Phillips, A., "Structure and

Properties of Alloys", McGraw-Hill, Third Edition.

37. Katagiri, K., Omura, A., Koyanagi, K., Awatani, J., Shiraishi, T.,

and Kaneshiro, H., "Early Stage Crack Tip Dislocation Morphology in

Fatigued Copper", Metallurgical Transactions A, Vol. 8A, 1977, pp

1769-1773; see also, Katagiri, K., Awatani, J., Omura, A., Koyanagi,

K., and Shiraishi, T., "Dislocation Structures Around the Crack Tips

in Early Stage in Fatigue of Iron", Fatigue Mechanisms, ASTM STP 675,

1979, pp. 106-128.

38. Pratt, J. E., "Dislocation Subfracture in Strain-Cycled Copper,

Journal of Materials, Vol. 1, No. 1, March, 1966, pp. 77-88.
90

39. Watson, P., "The Effect of Mean Stress and Overstrain on the Fatigue

Behavior of Structural Steels", Ph.D. Thesis, University of Waterloo,

1971.

40. Gowda, C.V.B., Leis, B. N., and Smith, K. N., "Dependence of Notch

Strength Reduction Factor on Plasticity and Duration of Crack

Growth", Journal of Testing and Evaluation, ASTM, Vol. 2, No. 1,

pp 57-61, January, 1974.

41. Landgraf, R. W., and La Pointe, N. R., "Cyclic Stres-Strain Concepts

Applied to Component Fatigue Life Prediction", SAE Transactios, Vol.

83, Sect. 2, 1974, pp. 1198-1207.

42. Grover, H. J., "An Observation Concerning the Cycle Ratio in

Cumulative Damage", Fatigue in Aircraft Structures, ASTM STP 274,

1960, pp. 120-124.

43. Manson, S. S., Freche, J. C., and Ensign, C. R., "Application of a

Double Linear Damage Rule to Cumulative Fatigue", Fatigue Crack

Propagation, ASTM STP 415, 1967, pp. 384.