NBSIR 76-1 062 (R)

A Micro- Degradation Model

for Paper

Jeffrey T. Fong and Ronald G. Rehm

Mathematical Analysis Section

Applied Mathematics Division

E. L. Graminski

Polymer Stability and Reactivity Section

Polymers Division

April 15, 1976

Progress Report Covering the Period

June 1, 1975 - February 29, 1976

Note: This Document has been prepared for the use of the Bureau of

Engraving and Printing. Responsibility for its further use

rests with that agency.

Prepared for Bureau of Engraving and Printing

U. S. Department of the Treasury

Washington, D. C. 20234

l^4-o

I

NBSIR 76-1062

A MICRO-DEGRADATION MODEL

FOR PAPER

Jeffrey T. Fong and Ronald G. Rhem

Mathematical Analysis Section

Applied Mathematics Division

E. L Graminski

Polymer Stability and Reactivity Section

Polymers Division

April 15, 1976

Progress Report Covering the Period

June 1, 1975 - February 29, 1976

Note: This Document has been prepared for the use of the Bureau of

Engraving and Printing. Responsibility for its further use

rests with that agency.

Prepared for Bureau of Engraving and Printing

U. S. Department of the Treasury

Washington, D. C. 20234

U.S. DEPARTMENT OF COMMERCE, Elliot L Richardson. Sacntary

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Seianea and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director
I
CONTENTS

Page

1. Siimmary 1

2. Introduction 3

3. Analysis of Flexing Data . . . , 4

4. Weibull Statistics 6

5. Analysis of Specific Structural Parameters 8

5.1 Fiber Length Distribution 8

5.2 Microscopic Strength Data 8

5.3 Porosity of Paper 9

6. Proposed Mechanisms for Microscopic Degradation

of Paper 11

6.1 Failure Mechanism Type "A" 11

6.2 Failure Mechanism Type "B" 12

7. Discussion 14

8. Acknowledgement 15

9. Bibliography 16

1. SUMMARY

Generally, the approaches to solving problems associated with

paper are empirical in nature. Hypotheses are formulated on an

intuitive basis and experiments are designed to verify the hypotheses.

All too frequently, however, the experimental results can be used

to verify conflicting hypotheses. Frequently the empirical approach

results in an illusory solution to a specific problem. The

system functions smoothly until changes are made. Problems surface

because the solution to the original problem was not real. At

that time there are three courses to take: 1) determine the real

solution to the original problem, or 2) revert back to the original

scheme of things and lose all flexibility in the system, or 3)

obtain another illusory solution which will suffice until a desired , or necessary change is made on the process at some future date.

One of the most powerful tools that can be used for solving

problems involving the mechanical properties of paper is a mathematical model. A practicable mathematical model would further a better understanding of the relationship between the microstructure

of paper and its mechanical properties. Ultimately the model would

greatly assist in selecting the structural parameters important

to the circulation life of currency. With this in mind an attempt

was made to formulate a mathematical model for paper.

In order to interpret macroscopic data with microscopic

observations, a decision must be made as to what should be observed

and how the massive amount of data generated for each microvariable

is to be treated. After all, the viability of any microvariable

is contingent on the ability to analyze and interpret the data

corresponding to that variable. It is important to be able to

analyze and interpret microscopic data as a function of time. The

two most popular approaches to analyze these data are through the

well-known log-normal and gamma distributions. However, they are

unsatisfactory as both distributions are always asymmetric to the

left. There is no a priori reason to assume a microvariable has

only one way to change its distribution with time.

The statistical theory proposed by Weibull was selected to

formulate a mathematical model for paper. Three parameters

describe a Weibull distribution and the curve can skew to the left

or right. When the shape parameter of the distribution curve is

3.6, the curve is almost indistinguishable from the well known

Gaussian distribution.

An analysis of the extensive data obtained on the flexing

of currency paper was made. Important parameters such as fiber

length distribution, microscopic strength data, pore size distribution

and the elastic and sonic moduli were analyzed rigorously. Two

mechanisms for the microscopic degradation of paper were proposed.

One of the mechanisms involved fiber failure; the second involved

the distribution of "equivalent" void diameter in paper. Both

mechanisms account for a rapid loss of modulus at the early stages

of flexing of paper and a gradual leveling of the modulus during
2

subsequent stages. The actual problem is probably much more

complicated than postulated by either mechanism. All processes

causing micro- structure degradation are probably occurring simultaneously albeit at substantially different rates.

Additional data, such as the shear modulus and Poisson's

ratio of currency paper as well as the elastic properties of cotton

and linen fibers was needed^. Since this data had not been determined

previously, a thorough search of the literature was made in an

effort to get useful data which would allow completion of even a crude mathematical model. Only meager data was found in the

literature and much of the data that was available appeared to be

unsound.

In conclusion, a novel approach to the problem of modelling

the degradation of paper has been found. The 3-parameter Weibull

distribution is adequate to characterize paper micro-structure as

well as the change in micro-structure with time or number of cycles.

Two mechanisms for the loss of modulus due to repeated flexing

have been proposed. However, additional data is needed for the

eventual completion of the mathematical model.

The technique required to obtain data on the shear modulus and

Poisson's ratio for paper are complex, while the single fiber

techniques are tedious and time consuming. This could account

for the limited and incomplete data of this kind in the literature.

A significant effort should be made to obtain the necessary data

required for formulating a mathematical model for paper. Development of a mathematical model would be beneficial to the solution

of a large number of paper problems related to its mechanical

properties. The mathematical model could be especially useful

for the design of a durable currency paper containing all or part

wood pulp fibers.

I
3

2. INTRODUCTION

The statistical nature of microscopic parameters such as the

fiber length [1] , the fiber breaking load [2] , the pore size [3]

,

etc., has been recognized by paper research workers for quite

some time. However, mathematical modelling of the mechanical

behavior of paper as related to its micro-structure has been

limited to the case of a "quasi-^^atic" process of deformation,

where the microvariables were either endowed with a "random"

statistical distribution [4,5,6] or stripped of their statistical

content in formulating a variety of empirical relations [7] , To

account for the physical phenomena such as anisotropy [8] , fatigue

[8], yield, and rate-dependent modulus [9], it is necessary to

introduce either the time variable or the number of cycles of

deformation explicitly into the statistical model. Clearly, any

such modelling effort must be coupled with an experimental program

where the variation of the micro-structure is monitored in conjunction

with the change of macro-strength characteristics.

Recent advances in automatic scanning devices and their

application in metallurgy [10] have convinced us that the same

devices may be applied to the study of the micro- structure of

paper. As more experimental data become available, a new mathematical framework other than random statistics is needed to interpret the data and guide the development of a time-dependent model.

The purpose of this report is to provide a preliminary analysis

of the data developed on currency paper at NBS over the past

several years.
4

3. ANALYSIS OF FLEXING DATA

In Fig. 1 a sheet of currency paper, 15.2 cm wide, 30.5 cm

long, and 0.13 mm thick is shown undergoing a flexing test. The

rollers used for testing the strength properties of the paper in

the cross direction are 3.18 mm in diameter. Assuming plane

sections to remain plane when the paper is bent around a roller,

and assuming further that the neutral axis is found half-way

between the thickness of the paper, the radius of curvature is

estimated to be 1.655 mm, and the maximum strain at extreme fibers

is about ± 3.9%. It is, of course, understood that the estimated

strain, whether tensile or compressive, is a macroscopic quantity.

With the constraints supplied by the roller and the specimen

geometry, macroscopic buckling simply cannot take place. In Fig. 2,

is shown how the motion of the pair of rollers with a period of

2.0 sec. induces four bursts of bending strain at a typical point

"E" along the flexing specimen D-F as shown in Fig. 1. Theoretically,

as the paper is brought to contact with a roller, the rate of strain

is infinite, i.e., the loading and unloading processes are both

instantaneous. The duration of the maximum strain (± 3.9%) for

the extreme fibers is estimated to be of the order of 0.02 s. The load-elongation curves for a tensile test for currency

paper in the cross direction before and after 80,000 flexes is

shown in Fig. 3. The initial modulus of the quasi-static loading

test (1.67 X 10~3 s"^) dropped from a mean value of 246 kgf/mm^

(2,410 N/mm2) before flexing to 167 kgf/mm^ (1,640 N/mm^) after

80,000 flexes. Since the flexing test involves both loading and

unloading, some previous data [12] is plotted in Fig. 3 on currency

paper when it underwent a cyclic strain of tension in the cross

direction. The initial portion of the unloading curve is seen to

be elastic with an estimated modulus of 400 kgf/mm^ (3,900 N/mm ).

To complete the picture, the dynamic modulus (= 850 kgf/mm2 or

8,300 N/mm ) , based on wave propagation measurement is also plotted

[11] . In Fig. 4, the dynamic modulus and the quasi-static modulus

of currency paper in the cross direction is plotted as a function

of the number of flexes. The dramatic drop in both moduli during

the first 5,000 flexes with the subsequent flat lines in Fig. 4

characterizes the macroscopic phenomenon of fatigue of paper and

will be the central problem in constructing a microscopic degradation

model.

To complete this analysis, two additional points must be

emphasized. In the first place, the load-elongation curve in

Fig. 3 is given only for a slow rate of tensile loading. Unfortunately

the strain induced by the flexing machine is not only alternating

between tension and compression, but also at a very high rate.

This implies that this is insufficient experimental data to launch

a rigorous stress analysis which is a prerequisite to a rational

assessment of fatigue damage. The second point is related to the

fact that, while the flexing test imposes a non-homogeneous strain

distribution in the paper sample, the loss of modulus phenomenon

as shown in Fig. 4 was observed on samples with a homogeneous strain
5

distribution in the paper sample, the loss of modulus phenomenon

as shown in Fig. 4 was observed on samples with a homogeneous

strain distribution. Caution must be exercised in proceeding with

the formulation of a degradation model where there is not a oneto-one correspondence between loadings that cause damage and

i strength tests that assess damage. Direct fatigue tests will have

to be conducted in the future.
6

4. WEIBULL STATISTICS

To undertake a task of interpreting macroscopic data with

microscopic observations, one is immediately confronted with two

difficult problems, namely, what to observe and how to treat the

massive amount of data that is generated for each microvariable.

The two problems are not completely independent of each other,

since the test of the viability of any microvariable includes the

requirement that the data corresponding to that variable can be

analyzed and interpreted. However, a three-parameter probability

density function due to Weibull [13] is a natural tool for

analyzing and interpreting microscopic data as a function of time.

To begin with, it is assumed the reader is familar with the

notion that a probability density function known as Gaussian or

"normal" is given by two parameters, namely, the mean and the

variance, and is symmetric al?out the mean. If any microvariable is

describable by a normal distribution at a specific time, say, t^,

and if that variable changes with time explicitly due to physical

reasons, then it is conceivable that the distribution may become

non-Gaussian at any time before or after tg. This creates a need

to look for a distribution which in general is asymmetric and has

an extra parameter that measures the asymmetry.

V- Two obvious candidates, which seem most popular in the current

literature (the well-known log-normal and gamma distributions)

,

have been rejected. It is a mathematical fact that both distributions

are asymmetric always to the left, i.e., the vertex of the density

curve lies to the left of the mean. For the specific purpose in

mind, a distribution that could skew either to the left or to

the right is needed since there is no a priori reason to demand

that a microvariable has only one way to change its distribution

in time.

In his pioneering work on a statistical theory of the breaking

strength of materials, Weibull [13] proposed a distribution which

appears to be ideally suited for paper. The three parameters, m,

Xq, x^, can be given a physical meaning in every application

involving some ultimate property of a material. Since the mathematical properties of a Weibull distribution are nothing new [13,

14] its graphical behavior as we change one of its three parameters

will be discussed. In Fig. 5 is shown not only the exact mathematical expression of the distribution but also its shape as the

shape parameter m is varied. In Fig. 6, the scale parameter x^

is varied while m remains fixed. There is an apparent change in

the shape of the curve. In both figures, the location parameter

x^ has been ignored, simply because its variation merely corresponds

to a horizontal shift of any curve for specific values of m and Xq.

It is worth mentioning that for m = 3.6 (approx.), the Weibull

distribution is almost indistinguishable from the Gaussian (normal)

distribution. For m less than 3.6, the distribution is asymmetric

to the left, and for m greater than 3.6, the distribution skews
I

7

to the right, as shown in Fig. 5. When m approaches infinity,

the Weibull distribution approaches that of a Dirac-delta function

with the spike located at x = + x^ , This is a useful property

particularly when we know that in applications we are constantly

faced with a judgmental question in choosing either a deterministic

model or a probabilistic model. With the use of a Weibull distribution,

both models can be formulated without asking the question whether

m is finite or infinite.
8

5. ANALYSIS OF SPECIFIC STRUCTURAL PARAMETERS

5 . 1 Fiber Length Distribution

Having settled on a specific tool for the statistical treatment of microscopic data, attention was turned to an obvious microvariable, namely, the fiber length.

The data was obtained on 600 fibers from currency paper before

and after 80,000 flexes in the cross direction [21], The histograms

and the appropriate Weibull curves are plotted in Fig. 7. Physically,

the fibers in the neighborhood of 2 to 3 mm lengths disappear and

those around 1 mm in length increase after 80,000 flexes. Mathematically, through the Weibull parameters, the shape parameter m

hardly changes, while both the scale parameter Xq and the location

parameter x^ suffer considerable changes. This coincides with

the discussion associated with Fig. 6 of the previous section,

where a change in scale parameter can change the apparent shape

of a distribution curve when the shape parameter remains fixed.

5 . 2 Microscopic Strength Data

Since Weibull [13] intended his distribution for application

to the breaking strength of material, our next illustration is

motivated by the curiosity whether the microscopic strength parameters

of wood and cotton fibers can also be analyzed with Weibull

statistics. Two quantities are of direct interest, the fiber

breaking load, and the fiber- to- fiber bond shear strength. In Fig. 8

is shown some typical data on those two quantities. Even though

we can always fit some 3-parameter distribution to a given histogram

by calculating its first three moments, in the case of the two

examples listed in Fig. 8, the fit is not necessarily "smooth"

.

Nevertheless, the availability of an extra parameter in our approach

speaks well of the analysis as compared with the conventional meanand-variance analysis given in both references 15 and 16. For

example, with m = 2.0, the breaking load distribution given in Fig.

8(b) is asymmetric to the left with the implication that any

degradation model based on fiber breaking as a mechanism should

consider x^, the minimum breaking load as a critical microvariable.

On the other hand, if we extrapolate Weibull 's result [13] on

cotton yearn and assume that for cotton fibers, the shape parameter

m is about 5.5, we conclude that the breaking load distribution

for cotton fibers may be asymmetric to the right (as illustrated

in Fig. 5 for m = 5.5) , and the critical microvariable may be x^ + Xq rather than x^.

In conclusion, the Weibull distribution is useful for representing micro- strength data. Unfortunately, for currency paper,

no data on fiber breaking load or fiber-to- fiber bond is available

for analysis and model-building.
9

5. 3 Poyosity of Paper

So far, only one micro-geometric variable has been discussed,

i.e., the fiber length, and two micro-strength parameters, the

fiber breaking load, and the fiber-to-fiber bond. To complete

the crude formulation, we need to introduce a second micro-geometric

variable, namely, the concept of a pore radius distribution as

expounded by Corte [3] . Once again no pore size data on currency

paper, before or after flexing, is available. Presumably, the data

are not difficult to get, and will be available in time to guide

the modelling effort in the near future. For the time being, an

extrapolation based on air permeability measurements will have to

be used.

The porous structure of paper is one of its fundamental

characteristics. Microscopic examination suggests that an adequate

description of this complicated structure is not presently obtainable. However, a simple picture of the structure which is acceptable

for many purposes is one in which the voids are described by

circular pipes, or spherical cavities. Specification of statistical

distribution of these voids as a function of their diameter then

defines the porous structure.

Air permeability can be used to provide a measure of the

paper porosity. The permeability relates the rate at which air

flows through a sample to the small pressure drop imposed across

it. At normal pressures the volume flow of air is found to follow

the theoretical predictions for Poiseuille flow through long

capillary tubes. When a Poiseuille law is followed and the simple

model of circular holes is used to describe porosity, the

permeability data specifies the product of the mean number of voids

and the mean area of a void over the paper sample. In Fig. 9,

air permeability data [8] is plotted as a function of the number

of flexes in the cross direction for currency paper. The air

permeability of paper is measured in units of cubic centimeters

of air per second through an area of one square meter when the

air pressure difference is 98.08 N/m2 . The commonly accepted

notion of permeability of a porous medium differs from that given

in Fig. 9 by merely a multiplicative constant.

To interpret Fig. 9, we note that there is a dramatic increase

of air permeability from the initial state to the intermediate

state corresponding to either 5,000 or 10,000 flexes. This

corresponds to the sharp drop in modulus as shown in Fig. 4. The

increase in permeability can be interpreted as an increase in a

single parameter called the equivalent void diameter.. If the

initial void diameter is assumed to be known, then the air

permeability data and this simple model enable us to estimate the

new void diameter by multiplying the initial one with the square

root of the ratio of the new permeability to the initial permeability.

For example, there is a five-fold increase in permeability after

5,000 flexes. This indicates an increase in the equivalent void

diameter by a factor of 2.2, There is a less dramatic increase

in permeability between 10,000 and 80,000 flexes. With the square
10

root relationship, the increase in equivalent void diameter in

the same cycle interval is even more gradual. The fact that the

modulus of a flexed paper also changes very slowly during that

cycle interval as shown in Fig. 4 provides us with a reasonable

argument to adopt the equivalent void diameter as a second microgeometric variable.
11

6. TWO MECHANISjyiS FOR MICROSCOPIC DEGRADATION OF PAPER

Two micro- geometric variables, the fiber length and the

"equivalent" void diameter, and two micro-strength parameters, the

fiber breaking load and the fiber-to-fiber bond have been singled out

as the basic elements for a microscopic degradation model of paper.

Since the only degradation data available are based on uniaxial

tensile testing, the modelling effort can only be as primitive as

a one-dimensional balance of forces and strains. Unfortunately,

even that is too much to ask for, because both fiber length and

void diameter (through air permeability) measurements yield no

information about the anisotropic character of the mechanical

behavior of paper. To resolve this dilemma, the new notion of an

"effective" fiber length equal to the length of a fiber as seen

under a microscope and projected along an axis that is parallel

to the direction of loading and unloading of a given specimen is

introduced. The "effective" fiber length distribution is obtainable

by taking advantage of the capability of an automatic scanning

device as discussed by Moore [10]. For our purposes here, it shall

be ^£5sumed that all four microvariables , namely, the "effective"

fib^jf length, the "equivalent" void diameter, the fiber breaking

load distribution for some critical fiber length, and the fiberto-fiber bond for some typical fiber width, have all been estimated

as a function of time or number of cycles. Within the fram.ework

of a: one-dimensional degradation model, two mechanisms are proposed

to relate the change of microvariables to the loss of modulus

as shown in Fig. 4.

6. 1 Failure Mechanism Type "A"

In Fig. 10, is shown a sequence of events leading to a loss

of load-carrying capacity of a single fiber in tension, if there

is experimental evidence that the fiber-breaking load distribution,

as a material property, changes with time or number of cycles more

rapidly than the fiber-to-fiber bond distribution. In that case,

the fiber- to- fiber bond strength, as a first approximation could

be represented by the mean of the observed strength distribution

mentioned earlier. With this simplification, the true tensile

load distribution curve in a typical single fiber could be represented

with an idealized one as shown in Fig. 10(a).

To illustrate this concept, let us construct a numerical

example based on the bond shear strength data of white fir summerwood as shown in Fig. 3(a). The mean shear strength of the fiberto-fiber bond is given as 2.516 x 10^ N/m^ (or 0.26 kgf/mm^ which

is alsoO. 00026 g^/ym^) . Assumming a typical fiber having an oblong

cross-section of 20 ym by 5 ym, the perimeter is approximately 50

ym. The estimated bond strength per line dimension of the fiber

may be given by 0.00026 x 5©\* 0*013 gf/ym. By assuming that the

fiber is bonded to its neiglUsors and the matrix in the same effective

way, and further assuming that the average distance between
12

neighboring fibers is given by 100 ym, we can estimate that the

force differential between two crossings can be at most 1.3 gf.

Let us now assume that the minimum breaking load for the

single fiber is 5.2 gf. According to the above calculation, it

takes 4 crossings to fully develop that force. Since this takes

place at both ends, it requires a minimum length of 800 pm or

0.8 mm for a fiber to fully develop a load of 5.2 gf between D and

E as shown in Fig. 10(b) . It is reasonable to expect that there

are a good portion of fibers longer than 0.8 mm. Every cycle

of loading will then subject those fibers longer than 0.8 mm to

break with a probability given by the fiber-breaking load distribution

similar to that given in Fig. 8(b). On a statistical basis, a

measure of the load carrying capacity of the specimen can be

constructed from the area under the curve given by the load distribution of Fig. 10(b). In Fig. 10(c), we see how an application of

the fiber-breaking criterion can reduce the load-carrying capacity

of the same fiber after a break is assumed to take place mid-point

between D and E. This mechanism produces a loss of modulus by

keeping the strain constant, the load-carrying capacity decreasing,

and the "effective" fiber length distribution altered in favor of

shorter fibers. Fig. 10(d) shows how the fiber load distribution

changes during unloading with the implication that the damage per

cycle as caused by microscopic fiber-breaking is irreversible.

Since the same calculation is carried out for a microscopic state

with a time- or cycle-dependent fiber length distribution, the

damage per cycle must necessarily vary with each cycle. This nonlinear damage-to-cycle result is consistent with the experimental

data on currency paper as shown in Fig. 4.

6 . 2 Failure Mechanism Type "B"

If it turns out that the fiber-to-fiber bond distribution, as

a material property, changes with time or number of cycles more

rapidly than the fiber-breaking load distribution, then we must

consider a mechanism where the bond strength is controlling rather

than the fiber-breaking load as shown previously. Since bond

strength is usually analyzed over several crossings, it does not

make sense to emphasize a minimum value and its probability as was

done for the fiber-breaking load. Our new mechanism, therefore,

will be formulated for a critical bond strength to be given by the

mean of a given distribution. In addition, the failure mechanism

is considered as a process of bond transfer through a typical matrix

containing a void which increases its diameter with time. In Fig.

11(a), we view the fiber as being loaded until the critical bond

strength is reached. In Fig. 11(b) and 11(c), we describe how

local shear yielding can take place as a consequence of the increase

in void diameter. The mechanism is such that there is no loss in

bond-transfer capacity, but there is an increase in fiber motion

following the loss of the matrix stiffness. A loss of modulus

is again a natural consequence of this failure mechanism because

as the load capacity is maintained constant, the strain can increase

and the "equivalent" void diameter distribution is altered in

favor of larger holes. Assuming that Unloading has no influence
13

on the micro- geometry of the specimen, we see that the damage

under this mechanism is again irreversible. Using a similar

argument as given for the Type "A" mechanism, it is seen that

the damage per cycle cannot be a constant and therefore the damage

is a nonlinear function of the number of cycles

.
14

7. DISCUSSION

In each of the two mechanisms of microscopic failures described

above, there is a built-in algorithm for the feature of a rapid

loss of modulus during the first few thousand cycles and a gradual

leveling of the modulus during subsequent cycling. For different

grades of paper, it is quite conceivable that either mechanism or

both may be controlling during separate stages of degradation.

For currency paper, evidence from scanning electron photomicrographs of flexed specimens shows that the problem is much more

complicated than postulated in the two mechanisms. For instance,

all of the processes causing micro-structure degradation are

probably occurring simultaneously, albeit at substantially different

rates. Mathematical modelling with limited experimental data

should never be pushed beyond the stage where there is no forseeable

hope of verifying critical assumptions. At this stage, it is not

unreasonable to conclude that a novel approach to the problem of

modelling the degradation of paper has been found. It remains

to be seen whether this approach is more profitable and physically

more meaningful than those due to Corte [17], Page [18,19], Dodson

[20] and others.
15

8 . ACKNOWLEDGMENT

We wish to thank C. T. J. Dodson (University of Lancaster,

England) , R. Kirsch and H. J. Oser (both of the National Bureau

of Standards) , for helpful discussions and encouragement in the

course of this investigation.
16

9. BIBLIOGRAPHY

1. Clark, J. d'A., Tech. Assoc. Papers , 25 , 285-290 (1943).

2. Jayne, B. A., Tappi , 42 , 461-467 (1959).

3. Corte, H. , "The Porous Structure of Paper," in Funda. Papermaking Fibres , Trans. Symp. Caipb., Sept. 1957 (F. Bolam, e d J , pp. 301-331. Tech. Sect. , British Paper and Board Maker's

Assoc. (Inc.), London (1958).

4. Le Cacheux, P., Papeterie , 75 , 659-675 (1953).

5. Kallmes, 0., and Corte, H., Tappi , 43 , (9); 737-752 (1960).

6. Van den Akker, J. A., "Some Theoretical Considerations on the

Mechanical Properties of Fibrous Structures," in Trans. Symp . Oxford, Sept. 1961 , (F. Bolam, ed.), pp. 205-247. Tech.

Sect., British Paper and Board Maker's Assoc. (Inc.), London

(1962)

.

7. Forgacs, 0. L. , Pulp Paper Mag. Canada , Convention Issue:

T-89 to T-118 (1963)

.

8. Graminski, E. L. , "The Effect of Flexing on the Mechanical

Properties of Paper," to appear in Proc. Symp. Cambridge , Sept. 1973 (Tech. Sect., British Paper and Board Maker^s

Assoc. , London)

.

9. Andersson, 0., and Sjoberg, L. , Svensk Papperstidning, 56 , (1960); 615-624 (1953).

10. Moore, G. A., Praktische Metallographie , 9_ (2), 76-97 (1971).

11. Graminski, E. L. , and Toth, E. E. , "Evaluation of Currency

and Stamp Paper," NBS Report 10564, April 1, 1971.

12. Wilson, W. K. , Forshee, B. W. , and Carter, T. J., Evaluation

of Currency and Stairp Papers, NBS Report 8316, March 16, 19 64.

13. Weibull, W. , A Statistical Theory of the Strength of Materials , Published in English as Handbook No. 151 by the Royal

Swedish Institute for Engineering Research, Stockholm (19 39).

14. Johnson, N. L. , and Kotz, S., Continuous Univariate Distributions , Vol. 1, Chapt. 20, pp. 250-271, Boston, Houghton-Mifflin (1970).

15. Duncker, B. , and Nordman, L. , Paperi ja Puu , 10 , 539-552 (1965).

16. Schniewind, A. P., Nemeth, L. J., and Brink, D. L. , Tappi , 4 7

(4), 244-248 (1964).
17

17. Corte, H., "Statistical Geometry of Random Fibre Networks/'

in Proc. Intl. Conf. Struct., Solid Mech. & Eng. Design in

Civil Eng. Mat Is , Southampton, (M. Te ' em, ed. ) , pp. 341-347

,

London: Wiley (1971).

18. Page, D. H., Tydeman, P. A., and Hunt, M. , "The Behavior of

Fibre-Fibre Bonds in Sheet under Dynamic Conditions," in

Trans. Symp. Oxford Sept. 1961 (F. Bolam, ed.), pp. 249-263.

Tech. Sect. , British Paper and Board Maker's Assoc. (Inc.),

London (1962)

.

19. Page, D. H. , Tappi , 52 (4), 674-681 (1969).

20. Dodson, C. T. J., "The Failure of Bonds During the Fracture

of a Random Fibrous Network," in Proc. Intl. Conf. Struct. , Solid Mech. , & Eng. Design in Civil Eng. Matls . , Southampton,

1969 (M. Te'em, ed.), pp. 357-366, London: Wiley (19 71)

.

21. Graminski, E. L. , and Toth, E. E., "Evaluation of Currency

and Stamp Paper," NBS Report 10090, June 1969).

USCOMM-NBS-DC

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(a)

RANGE OF

HORIZ. MOTION

OF ROLLERS

4 -—

y{

23.5 cm (±)

0 5cm

+

TYPICAL POINT "E"

EXTREME FIBER

STRAIN RANGE

M N —DiSTAIICE

\

/

TIME (sec)

4%

il STRAIN

0.02 sec

(b)

TIME (sec)

Figure 2: Time and Motion Plot for Rollers and Typical Point "E"

(A) Horizontal Motion of Rollers vs. Time; (b) Extreme Fiber Strain vs. Time.
TENSILE STRAIN (%)

Figure 3: Loaa-Elongation Curves for Currency Paper in Cross

Direction Before and After 80,000 Flexes.
900

T—

r

T—

r

CM

E

E

800

700

\

\ (M

E

E

600

<n 500

D

-J

o 400

o

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o

S>—^

5,000

9 DYNAMIC MODULUS

O QUASI-STATIC MODULUS

(LOADING ONLY)

3O 200

100

XX ^ O -O - 1,000

J L J L i

20,000 40,000 60»000 80,000

NUMBER OF FLEXES

Figure 4: Change of Young's Modulus of Currency Paper in Cross

Direction Due to Flexing over 3.18 mm. Diameter Rollers.
3.0

2.0

1.0

FIXED Xo= 1.0

I

I

I

I

I

I

II

/

I

\

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I \m=5.5

'

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1.0

(X-Xu)

2,0 3.0

Figure 5: Weibull Probability-Density Function with Fixed

and Varying m.
1.0

U ^ \

/ \

0.6 h/ '

0.6

T

FIXED m= 1.5

I,

1/ 0.4 i,I

L

0.2

\ ^vXo=I.O

Xo=0.8N ^\

1.0 2.0 3.0

(X-Xy)

Figure 6: Weibull Probability-Density Function with Fixed m and

Varying x^.
(/) ocUJ

CD

(i. U.

o

q:

LU

CD

2.0 3.0 4.0

FIBER LENGTH (mm)

<

Io

hij. o

UJ

o

oc

2.0 3.0 4.0

FIBER LENGTH (mm)

Figure 7: Frequency Distribution of Fiber Lengths for Currency

Paper Before and After 80,000 Flexes.
q

CVJ

II

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u.

IIoX 3

IE '

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20,000 40,000 60,000

NUMBER OF FLEXES

80,000

Figure 9: Air Permeability vs. Number of Flexes for Currency

Paper in Cross Direction.
MEAN BONO STRENGTH FIBER TENSILE

LOAO OlSTRIBUTION

A r =±iB

/-^ MINIMUM BREAKING LOAO

^

0 ^

C C'

*

C C' ^

(0)

(b)

(c)

Figure 10: Failure Mechanism Type "A" Due to Fiber-Breaking.
VOID OlAMr

VOID OIAM.

BEFORE YIELDING

CRITICAL BOND STRENGTH

FIBER TENSILE

y~ LOAD DISTRIBUTION

I

I iC

CRITICAL BOND

STRENGTH

r

Al^

r

I

I

I !c

B

B

(a)

(b)

VOID DIAM,

AFTER YIELDING

CRITICAL BOND

STRENGTH

A/£: J—

B

(c)

Figure 11: Failure Mechanism Type "B" Due to Loss of Matrix

Stiffness

.

I NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 76-1062

2. Ciov't Acces.sion No.

3. Recipient'.s Acces.sion No.

4. TITLE AND SUBTITLE

A Micro-Degradation Model for Paper

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S)

J. I, Fong, R. G. Rehm, E. L. Graminski

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Projcct/Task/Work Unit No. 2050141 and 3110443

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

Bureau of Engraving and Printing

U.S. Department of the Treasury

Washington, D. C. 20234

jll5. SUPPLEMENTARY NOTES

13. Type of Report & Period June 1, 1975

February 29. 1976

14. Sponsoring Agency Code

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

Motivated by scanning electron micrographs showing micro-structural degradation

of several grades of paper under repetitive loadings (Graminski, 1973), we present

a crude statistical approach in defining two micro-variables , namely, an "effective"

fiber length, and an "equivalent" void diameter. The approach is based on the

mathematical properties of a 3-parameter statistical distribution originally due

to Weibull (1939). Two micro-strength parameters, namely, the fiber breaking

load, and the fiber- to- fiber bond strength, are also given a Weibull statistical

analysis. Two distinct mechanisms for explaining the loss of modulus due to

repetitive flexing are presented for constructing a self-consistent microscopic

degradation model of paper.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons) Durability; fatigue; fiber length; flexing; low cycle

fatigue; mechanical properties; micros tructure; modelling; paper; pore size;

statistical analysis; Weibull distribution.

18. AVAILABILITY |^ Unlimited

Ix For Official Distribution. Do Not Release to NTIS

1

^ Order From Sup, of Doc, U.S. Government Printing Office Washington, D.C, 20402. SD Cat. No. C13

1

^ Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

21. NO. OF PAGES

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

22. Price

USCOMM-DC 29042-P74