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THESIS

LIBRARY

Michigan State
University

 

nr’

 

ABSTRACT

NONLINEAR, QUASI—STATIC BEHAVIOR OF SOME
PHOTOELASTIC AND MECHANICAL
MODEL MATERIALS

by Ernst W. Kiesling

A study was made of the effects of time upon the
birefringent and mechanical properties of four plastics:
Polyester Resin CR-39 (Cast Optics Co., Inc.); Polyester
Resin PS—l and Epoxy Resin PS—2 (Photolastic, Inc.); and
Polyester Resin Palatal P6—K (Badische Anilin und Soda—
Fabrik9 Germany)° The isothermal, quasi—static behavior
under constant, uniaxial stress conditions was examined
exhaustively—~for approximately five decades of logarithmic
timeV3 and for a range of stress sufficiently broad to cover
the linear and much of the nonlinear range (where non—
linearities appeared)o

The stresses limiting the linear range of behavior-—
called linear limit stresses—wwere determined approximately
for both one percent and five percent deviations from
linearityo The variation with time after loading of these
linear limit stresses was also examined°

Data is presented graphically for all materials.
Certain characteristic functions defined by viscoelastic

and photoviscoelastic theories are given for Polyester

 

 

Ernst W. Kiesling

Resin CR-39 in both the linear and nonlinear regions. Both
these theories are discussed in some detail.

Several wave lengths of radiation, covering the
visible range, were employed in the investigations of bire-
fringence. Two phenomena—-dispersion of birefringence and
the possible variation with wave length of linear limit
stresses-—could thus be studied.

Tapered models were used to obtain birefringence and
mechanical data essential to this study. The moire method,
used in conjunction with tapered models, provided a con-
venient means of obtaining continuous strain-stress data

from a single model and loading.

NONLINEAR, QUASI—STATIC BEHAVIOR OF SOME
PHOTOELASTIC AND MECHANICAL

MODEL MATERIALS

BY

Ernst w. Kiesling

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and Materials Science

1966

I
a
flu!

ACKNOWLEDGMENTS

The advice and encouragement given by the members of
the Guidance Committee——Drs. W. A. Bradley (Chairman),

G. L. Cloud, J. L. Lubkin, and C. P. Wells——are gratefully
acknowledged. Dr. J. T. Pindera gave valuable guidance
during the early stages of this research effort.

Funds and laboratory facilities for the experimental
program were provided by the Department of Metallurgy,
Mechanics and Materials Science, and by the Division of
Engineering Research. The first year of advanced graduate
study was made possible by a National Science Foundation

Science Faculty Award.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . ii

LIST OF ILLUSTRATIONS. . . . . . . . . . . . . . . . . Vi

LIST OF SYMBOLS O O O O O O O O O O O O O O O O O O O O Xi

Chapter

I.

II.

III.

INTRODUCTION 0 O O O O O O O O O O O O O O O O 1

Objective and Scope

Problem of Model Similarity
Primary Objectives of Thesis
Collateral Objectives
Organization of Thesis

CALIBRATION TESTS. O O O O O O O Q 0 O O O O O 12

Part A. Description of Material Properties. 12

Introduction

Calibration of Mechanical Model Materials
The Photoelastic Effect

Calibration of Photoelastic Model Materials
Complete Description of Material Properties

Part B. Calibration Procedure . . . . . . . 35

Possible Calibration Models

Model Used in This Study

Model Preparation and Clamping

Test Procedure—-Photoelastic Calibration
Test Procedure-~Mechanical Calibration
Materials Tested

BIREFRINGENT AND MECHANICAL PROPERTIES

OF (ZR-39 o o o o o o o o o o o o o o o o o o 52

Birefringence vs. Stress, Time Constant
Birefringence vs. Time, Stress Constant
Strain vs. Stress, Time Constant

Strain vs. Time, Stress Constant

Linear Limit Stress vs. Time

iii

CONTENTS-~Continued

 

Chapter Page

Birefringence vs. Strain, Time Constant
Material Coefficients in Linear Range

IV. VISCOELASTIC STRESS ANALYSIS . . . . . . . . . 73

Introduction

Aspects of the Problem

Forms of Representation of Linear Visco-
elastic Behavior

Forms of Representation of Linear Photo—
viscoelastic Behavior

Nonlinear Viscoelasticity

Nonlinear Photoviscoelasticity

Limitations of Results Presented

V. VARIATION OF OPTICAL PROPERTIES WITH WAVE
LENGTH O O O O O O O O O O O O O O O O O O O l l 5

Investigation Performed

Dispersion of Birefringence

Variation of Linear Limit Stress with Wave
Length

VI. DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS
FOR FURTHER RESEARCH . . . . . . . . . . . . 128

Achievement of Objective
Experimental Techniques

Material Property Descriptions
Properties of Particular Materials
Recommendations for Further Research

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . 145
Appendix

I. ANALYSIS OF STRESS DISTRIBUTION IN TAPERED
MODEL. 0 O O O O O O O O O O O O O O O O O O 151

II. TECHNIQUES OF MOIRE STRAIN ANALYSIS. . . . . . 161

Difficulties Presented by a "Conventional"
Method

Suitability of Moire Method

Production of Line Arrays—~General Methods

Grid Application Method Employed

iv

CONTENTS—~Continued

 

Appendix Page
III. DATA FOR OTHER MATERIALS TESTED. . . . . . . . 176
Polyester Resin PS-l

Epoxy Resin PS-2
Polyester Resin Palatal P6-K

Figure

2.1

LIST OF ILLUSTRATIONS

Representative plot of 5 = 8(6), t = const
(tl,t

constant load. . . . . . . . . . . . . .

2,t3, . . .) for uniaxial stress,

Linear limit stress and relaxation modulus
functions of time for uniaxial stress,

constant load. . . . . . . . . . . . . .

Representative plot of 8 = 8(t), o = const
(01,02,03, . . .) for uniaxial stress,

constant load. . . . . . . . . . . . . .

Isochromatic fringe order n vs. stress for

two wave lengths, X1

Variation of linear limit stress with time,

based on birefringence . . . . . . . . .

Fringe constant f and stress—optic coefficient

Co as functions of time for given wave

length 0 O O O O O O O O O O O O O O O 0

Order of interference n vs. time for various

stress levels, uniaxial stress . . . . .

Order of interference n vs. strain for t =

CODSt and. A : Al 0 o o o o o o o o o o o o 0
Order of interference n vs. time for 8 = const
and )\ = X1 0 o o o o o o o o o o o o o o o 0

Equipment used to scribe scale on models .

Sketches illustrating (a) the effect of bend—

ing of the clamp jaws, and (b) improvement

by modification. 0 O O O O O O O O O O 0

vi

and X2. 0 O O O O O O O

Page

19

20

21

29

30

31

31

33

33

38

41

 

Figure

2.12

3.2

3.7a

ILLUSTRATIONS--C0ntinued

 

Page

Polarisc0pe used in photoelastic studies:
(a) light source, (b) condensing lens,
(d) dispersing screen, (P) polarizer,
(Q1) quarter wave plate, (M) model,
(02) quarter wave plate, (A) analyzer,

(F) filters, and (C) camera. . . . . . . . . 43

Apparatus used in mechanical model tests:
(a) light source, (b) collimating lens,
(M) model, (D) "dummy" model or analyzer,

(d) condenser lens, and (C) camera . . . . . 48
Optical system used in mechanical model tests. 48

Illustration of method of attaching moire

analyzer to model. . . . . . . . . . . . . . SO

Isochromatic fringe order n vs. distance
along centerline, t = tl,t2, . . . ,tn
for CR-390 o o o o o o o o o o o o o o o o o 53

Isochromatic fringe order n vs. stress, t =

tl,t2, o o o ,tn for CR_39 o o o o o o o o o 55

Isochromatic fringe order n vs. time, 0 =
01,62, 0 o o ,Gn for CR—39 o o o o o o o o o 57

Moire fringe order m vs. distance along
centerline, t = tl,t2, . . . ,tn for CR-39 . 59

Percentage elongation vs. stress, t =

tl’t2’ o o o ,tn for CR—39 o o o o o o o o o 61

Percentage elongation vs. time, 0 =
01,02, 0 o 0 ,0n for CR—39 o o o o o o o o o 63

Linear limit stresses vs. time for bire-
fringence and strain, time range: 0-240
hours, for CR—39 o o o o o o o o o o o o o o 64

vii

ILLUSTRATIONS—~Continued

 

Figure Page

3.7b Linear limit stresses vs. time for bire-
fringence and strain, time range: 0-10
hours, for CR—Bg o o o o o o o o o o o o o o 65

3.8 Isochromatic fringe order n vs. percentage
elongation, t = tl,t2, . . . ,tn for CR-39 . 69
3.9 The functions CG(t), fO(t), and E(t)
for CR—39. . . . . . . . . . . . . . . . . . 72
4.1 Functions 80(t) and D(t) for CR—39 . . . . . . 96

4.2 Plot of (n - no) vs. time for three stress
levels for CR—39 . . . . . . . . . . . . . . 107

4.3 Kernels Dl(t), D2(t,t) and D3(t,t,t)
for CR-390 o o o o o o o o o o o o o o o o o 109

4.4 Isochromatic fringe order n vs. time, experi-
mental and calculated values, 0 = 2,600
pSi, for CR—39 o o o o o o o o o o o o o o o 110

4.5 Isochromatic fringe order n vs. time, experi—
mental and calculated values, 0 =
01,62, 0 o 0 ,0n for CR—39 o o o o o o o o o 1].].

5.1 Isochromatic fringe order n vs. stress, 1 =
X1,XZ, o o o ,Xn for CR"39 o o o o o o o o o 120

5.2 Normalized retardation curves for CR—39. . . . 122

5.3 Normalized dispersion of birefringence
for CR-390 o o o o o o o o o o o o o o o o o 124

5.4 Isochromatic fringe order n vs. stress, 1
X1,12, . . . ’Xn’ showing linear limit

stresses for CR-39 . . . . . . . . . . . . . 126

Al.l Sketch of wedge with concentrated load at the

apex O O O O O O O O O O O O O O O O O O O O 152

Figure

A1.2

A1.3

A104

A3.1

A302

A3.3

A3.4

A3.5

A3.6

A3.7

A3.8

A3.9

A3.1O

ILLUSTRATIONS—-Continued

Sketch showing shape and approximate

dimension of tapered model used. . . . . .

Plot showing variation of the stress con-
centration factor k along the centerline

of the photoelastic model. . . . . . . . .

Photographs of (a) isochromatic fringe

pattern and (b) moire fringe pattern . . .

Isochromatic fringe order n vs. stress, t =

tl,t2, o o o ,tn for PS—lo o o o o o o o o

Isochromatic fringe order n vs. time, o =

01,02, 0 o 0 ,6n for PS-l. o o o o o o o 0

Percentage elongation vs. stress, t =

tl’tZ’ o o o ,tn for PS—lo o o o o o o o 0

Percentage elongation vs. time, 0 =

01,02, 0 o o ,On for PS—l. o o o o o o o o

Isochromatic fringe order n vs. percentage

elongation, t = tl’t2’ . . . ,tn for PS—l.

Isochromatic fringe order n vs. stress, t =

tl,t2, o o o ,tn for PS—2o o o o o o o o o

Isochromatic fringe order n vs. time, o =

01,02, 0 o 0 ,0n for PS_20 o o o o o 0 o 0

Percentage elongation vs. stress, t =

tl,t2, o o . ,tn for PS-2. o o o o o o o 0

Percentage elongation vs. time, o =

61,02, 0 o 0 ,6n for PS_20 o o o o o o o o

Isochromatic fringe order n vs. percentage

elongation, t = t t ,tn for PS—2.

2, O O 0

ix

1’

Page

156

158

160

177

178

179

180

181

183

184

185

186

187

Figure

A3.11

A3.12

A3.13

A3.14

A3015

ILLUSTRATIONS——Continugg

Isochromatic fringe order n vs. stress, t =

tl’tz, c o o ,tn for P6‘Ko o o o o o o o n

Isochromatic fringe order n vs. time, o =
01,02, 0 o o ,On for P6_Ko a o o o o o n 0

Percentage elongation vs. stress, t =
tl’t2, o o I ,tn for p6—K. o o I o a c o 0
Percentage elongation vs. time, o =

01,02, 0 a 0 ,6n for p6-Ko I o o o o o o I

Isochromatic fringe order n vs. percentage

elongation, t = tl,t2, . . . ,tn for P6—K.

Page

189

190

191

192

193

LE

LIST OF SYMBOLS

uniaxial stress

normal stresses along axes of
coordinates

principal stresses

stress tensor or typical component
of stress tensor

0 + o

11 * G22 33

5 the stress deviator

aka

0.

lj ijokk’

uniaxial strain

principal strains

8 + 8

11 22 + 5

33

strain tensor or typical component
of strain tensor

6

the strain deviator

mud

Sij ‘ijgkk’

isochromatic fringe order or order
of isochromatic or order of inter—
ference

indices of refraction along
coordinate directions

moire fringe order
time

linear limit stress

xi

n‘ozz)1.o' n<czz)5.o

s‘“22’1.o’ s‘ozz)5.o

D(t)
E(t)
J(t)

E(t)

E(t)

a(t)

SYMBOLS--Continued

 

linear limit stresses for bire—
fringence at 1.0% and 5.0% devi—
ations from linearity, respectively
linear limit stresses for strain at
1.0% and 5.0% deviations from
linearity, respectively

abbreviation for nanometer, one

6

_ -3 .
nanometer = 10 mm or 10 microns

a material "fringe constant"

thickness of model
wave length of radiation

stress—optic and strain—optic
coefficients, respectively

linear differential operators with
respect to time

shear modulus

bulk modulus

extension compliance

relaxation modulus

creep compliance in shear

bulk compliance

Poisson's ratio

difference in secondary principal
stresses for unit sustained bire-
fringence

birefringence for a unit sustained

difference in secondary principal
stresses

xii

 

SYMBOLS——Continued

(nl - n2), the difference in
secondary principal values of index
of refraction

angle between secondary principal

optical directions and coordinate
axes

angle between secondary principal
stress directions and coordinate

axes

angle between secondary principal
strain directions and coordinate

axes

kernels of integrals, for represent—
ing viscoelastic behavior

dispersion of birefringence

n1, relative retardation

R(l) . .
, normalized retardation

R(XO)

%%, normalized dispersion of bire—

fringence

stress concentration factor

xiii

CHAPTER I

INTRODUCTION

Objective and Scope

 

The demands for more efficient use of materials in
today's technology, particularly in areas of air and space
travel, have elevated the importance of accurate stress
analysis to unprecedented heights. These demands have
fostered, or forced, rapid developments in the methods of
stress analysis, both analytical and experimental. Though
the theory of elasticity offers mathematically exact solu-
tions to some problems, difficulties multiply sharply when
boundaries or loads are complicated. Comparable or even
greater difficulties may arise in viscoelasticity and
plasticity. Development of the vast capability in data
processing has broadened the range of problems which can be
satisfactorily solved analytically but has not enabled the
analytical approach to encompass all problems.

Some problems which defy analytical solution yield
to experimental methods. One such method is photoelas-
ticity, which has been used successfully in a number of
cases. Although the photoelastic effect has been known
since 1816, when Sir David Brewster published his account,
it became a useful tool to the engineer much more recently.

1

2

The Treatise on Photoelasticity, first published in 1931 by

 

E. G. Coker and L. N. G. Filon, Opened the eyes of many to
the potential of the method and established a firm founda-
tion for its further development. The invention of
"Polaroid" provided a relatively inexpensive means of
producing large beams of polarized light. In recent years,
and continuing at present, the development of new materials
(plastics) possessing desirable optical characteristics has
expanded the application of the method to a wider variety
of problems, including those in the realm of viscoelas—
ticity and plasticity.

The moire method has emerged as another important
tool to the experimental stress analyst, despite its late
introduction—~less than twenty years ago. Present inten-
sive research on the method promises to make it more useful
in the near future.

Upgrading in techniques and equipment has made its
impact. The addition of courses in experimental stress
analysis in engineering curricula, coupled with more par—
ticipation in professional organizations devoted to the
advancement of experimental stress analysis, should further
enhance the capability and broaden the applications of both
the methods mentioned.

The distinct advantage of these experimental methods
over most others, that of yielding quantitative representa-

tions of stress or strain over an entire field, not only

 

3
excites the imagination of this writer but also partially
explains his optimism regarding their future.

Despite progress in all areas of experimental stress
analysis, many difficulties and sources of error in their
application remain. Among them is model similarity, par—
ticularly as it pertains to properties of model materials.
This study aims at elucidating some aspects of this problem
and giving data and recommendations to help resolve the
problem.

The mechanical and optical properties of several
materials commonly used in photoelastic analyses were
studied and are herein reported. Any impact that the
results of this study may have, therefore, will be most
pronounced for the analyst solving problems in the elastic
domain. The method of determining the material properties,
however, is felt to be unique and applicable to a much
broader class of materials, including low—modulus ones SUCh
as might be useful to model studies of viscoelasticity.

One chapter of the thesis is devoted to discussing the
experimental approach to solving viscoelastic stress
analysis problems.

The moire method is employed in studying mechanical

properties. Existing literature on the theory and inter-

Pretation of moire patterns is extensive. Relatively

little has been reported, however, regarding the PrOduCtlon

0f Suitable grids and the techniques of their application.

 

4
A discussion of various methods tried in the course of this
study, therefore, is included.

This study investigates the phenomenological
behavior only. The optical and mechanical properties are
investigated without consideration of causes or effects at
flm molecular level. Furthermore, only the relatively
long—time-—i.e., quasi—static—-behavior at constant

temperature is considered.

Problem of Model Similarity

 

Certain laws of similarity must be observed in
structural—model analysis, model photoelasticity, photo-
viscoelasticity, photoelastic coatings, or in any model
analysis. That complete physical similarity requires
geometric similarity between model and prototype presents
no real problem in photoelasticity, since plastics used for
models can be cast and/or machined to practically any
desired shape. In the usual application of photoelastic
Coatings, a layer of plastic is bonded to the surface of
the Prototype so that geometric similarity is automatic.

But complete physical similarity imposes other
requirements upon the model material. The theory of elas—
ticity generally assumes a Hookean material--i.e., one for
WhiCh stress—strain relations are linear, single—valued,
and independent of time. Despite such a sweeping idealiza—

tion, Valuable solutions for a broad class of problems can

be obtained by this theory. Apparently SUCh an

5
idealization is satisfactory for a number of materials and
conditions.

With such an assumption and those of homogeneity and
isotropy the stress distribution in a restricted class of
elasticity problems is found to be independent of the
elastic constants of the material. This is true for "plane
strain" or "generalized plane stress" problems in which the
resultant force over each boundary vanishes separately and
the boundary conditions involve only stresses. Many
practical problems, such as those in plates with singly-
connected boundaries, fall into this class. The independ-
ence of stress distribution from elastic constants "opens
the door" to model studies, such as photoelasticity.

The elastic constants of the material cannot be
eliminated from the equations formulating three-dimensional
or general two—dimensional elasticity problems. The stress
distribution is found to depend upon one elastic constant,
such as Poisson's ratio. But in investigating several
problems of this more general type Clutterbuck [5]1 found
that the differences in stress distributions due to the
effects of Poisson's ratio are quite small—~50 small that
the effects can seemingly be regarded as secondary ones.
Photoelasticity thus provides a potential for obtaining

approximate solutions to a wide variety of problems.

 

V,—

1Numbers in brackets refer to the List of References
at the end of the thesis.

6

If model materials were available for which the
constitutive equations were linear, single-valued, and
independent of time, then the problem of physical simi-
larity would be quite trivial. But the plastics usually
used for structural or photoelastic models are viscoelastic
or photoviscoelastic so that stress-strain relations or
stress-birefringence relations are not independent of time.
These relations may be rather complicated functional rela—
tions, such as s = 8(o,t) or n = n(8,t), where n denotes
the fringe order of the birefringence.

Fortunately, some model materials are momentarily
linear for a broad range of stress—~i.e., at any given time
after a constant load is applied the stress—strain and
birefringence—stress relations are linear. This linearity,
at a given time, makes such materials more useful for
models and fulfills another important requirement of com-
plete physical similarity. But this “broad range of
stress," for which momentarily linear behavior is mani—

fested, is limited and is time dependent in most plastics.

 

The range and the time dependence vary greatly with mate—
rials, but in the case of almost all plastics which exhibit
momentarily linear behavior the stress which limits the
linear range is considerably below the ultimate or fracture
strength at a given time after loading. This makes an
analysis based on linearity vulnerable to serious error:

if the initial stress is too high and/or if a given stress

 

7
is sustained too long before desired information is
collected, gross anomalies can result. The latter of these
possibilities becomes particularly relevant when a visco-
elastic prototype is being studied by means of models,

since extended time periods are sometimes necessary.

Primary Objectives of Thesis

 

Determination of linear limit stresses.—-Determining

 

the value of stress which limits the momentarily linear
range of behavior for a number of plastics subjected to
constant axial load is one of the primary objectives of
this study. This value of stress, called the "linear limit

stress" and denoted by o is, of course, a function of

72’
time. That such a limit exists and that it is time-
dependent is well-known, but few published results can be
found regarding it, none showing explicit values for it at
a given time after loading or for its variation with time.
The linearity and its limiting values pertain to

both birefringence—stress and strain—stress relations. In
an earlier publication by the author [50] the opinion was
expressed that the linear limit stress for birefringence,
as a function of time, might be accepted also for strain.
At least one published result [18], in which time effects
were considered negligible, contradicted this opinion. In
this study both mechanical and optical properties are con-

sidered and the linear limit stresses for the two compared.

 

 

8
A discussion of the eXperimental data obtained, a
description of the models and testing procedures, and a
list of the materials chosen for the study are presented in
Chapter II. This is followed by the presentation of
principal results in Chapter III.

Laws of viscoelasticity, experimental results.-—The

 

mathematical theory of viscoelasticity has provided a
fruitful field of investigation in recent years. High
temperature applications of conventional structural mate-
rials, the increasing use of plastics in industry (e.g.,
tubing in refineries), increased interest in soil and rock
mechanics, and other factors have all kindled interest in
the subject. Linear viscoelasticity may now be considered
well-established, with many elegant theoretical develop—
ments and with considerable experimental evidence to
support its applicability to a large group of problems.
Nonlinear viscoelasticity, on the other hand, has not yet
achieved this status.

Because of the viscoelastic nature of plastics used
in structural and photoelastic models and because of the
increased importance of the experimental approach to visco—
elastic stress analysis, some understanding of the theory
and applications of viscoelasticity is important, if not
imperative, to successful work in this area of experimental
stress analysis. A brief review of some of the theories is

presented in Chapter IV, along with plotted results

9
correlating the experimental data gathered in this study

with some of the theories.

Collateral Objectives

 

Variation of 03$ with wave length of radiation.--

Though a number of attempts have been made to explain the

 

cause and the nature of birefringence (or double refraction
or retardation) and its relation to stress, strain, wave
length of radiation, and other variables, none has been
completely successful. The order n of the isochromatic
fringes in a uniaxial stress state is often assumed to be
inversely proportional to the predominant wave length of
radiation. Several investigators have observed and
reported deviations from such a relationship. Pindera and
Cloud [49] and Cloud [4] very recently reported their find—
ings on the dependence of the order of isochromatic and
consequently of the photoelastic coefficients upon wave
length of radiation. This dependence is known as disper-
sion of birefringence.

Since the order of isochromatic depends upon the
wave length of radiation, even when the birefringence—
stress relationship is linear for a certain range of
stress, it seems reasonable to ask whether the linear limit

stress for birefringence 0 might also vary with wave

0, ,0.
length. To pursue this question, approximately ten differ-
ent wave lengths ranging from 0.410 microns to 0.800

microns were used at various times throughout the tests.

 

10

The results indicating the dependence of o on wave length

22
are given in Chapter V.

Dispersion of birefringence.-—Data obtained from the

 

investigations above readily yield information on the dis-
persion of birefringence. Some of the materials tested are
the same as those tested by Cloud [4], and hence these
results are merely a supplement to or a continuation of his
work. However, this study includes other materials, as
well as a different (but narrower) range of wave lengths.

Chapter V also reports on these findings.

Organization of Thesis

 

In the ensuing, Chapter II discusses the plots
desired for data presentation, the models and testing
procedures employed to obtain the data, and a list of mate-
rials tested. This is followed by the presentation of
principal results in Chapter III.

The theory of viscoelasticity, both linear and non-
linear, is discussed in Chapter IV. Some plots of "charac-
teristic functions" describing viscoelastic behavior in the
linear range are given, together with mathematical expres—
sions intended to describe both linear and nonlinear
behavior.

Chapter V discusses the collateral objectives just
mentioned: dispersion of birefringence and variation of

linear limit stress with wave length.

11

This work closes with a summary, conclusions, and
some recommendations for future research in Chapter VI.
Appendix I gives the theoretical and experimental analysis
of the stress distribution in the tapered model used
throughout this work. Appendix II reviews briefly the
prior developments of the moire method, as well as the
efforts of this study.

The results presented graphically in Chapters III
through V are for Polyester Resin CR-39 supplied by Cast

Optics Co., Inc. Similar results for other materials

tested are presented in Appendix III.

 

 

CHAPTER II

CALIBRATION TESTS

Part A. Description of Material Properties

 

Introduction

 

The materials tested in this study are inherently
viscoelastic or photoviscoelasticaai.e., the response to
stress, whether strain or birefringence, is a function of
time. The viscoelastic properties of such materials may be
defined in various ways; for an incompressible, isotropic,
linear viscoelastic material, Alfrey [1] listed seven
distinct methods. He then showed that they are all mathe-
matically equivalent. For a real material some measure of
compressibility, such as Poisson's ratio, must also be
known. At least as many ways are available for specifying
the photoviscoelastic properties.

The choice of specification depends upon the
intended use of the information. For visualization of
material response the graphical representation seems most
useful. Such representation suffices for experimental
stress analysis, whereas analytical work usually favors a
representation in mathematical form~~e.g., a constitutive
equation.

12

l3

Suitable mathematical forms are discussed in
Chapter IV. Part A of the present chapter describes
various graphical forms of representation and how one form
is deduced from another. Part B describes the testing
procedure followed in this investigation.

All calibration models employed in this study were
formed from plates of uniform thickness. The stress state
applied was basically one—dimensional, constant throughout
the thickness of the model. Although the results obtained
from such one-dimensional tests cannot be guaranteed to be
general, particularly in the nonlinear range, they give
valuable insight into the general behavior. In the linear
range onewdimensional tests furnish results which are
sufficient for the complete characterization of the matem
rial, whether it be elastic or viscoelastic.’1 These
results can be used to predict with confidence the behavior

of the material under more general states of stress.

D

In addition to yielding data of value to the str

.55

f

analyst, onewdimensional tests yield valuable data for the

k4

comparison of different materials and compositions. A1.
of these features, coupled with the relative ease of
conducting such tests, seem to be sufficient to justify
their use. The present studies are confined to onem

dimensional loading, partly because the additional

 

l o n u o - u —

This is further discussed in the next section and
again in Chapter IV where the uniaxial stress state is
resolved into deviatoric and dilatational parts.

l4
complication of time-dependence has been investigated
exhaustively.

The question of how long constant load should be
maintained in a creep test also has several aspects.
Photoelastic stress analysis usually requires knowledge of
material properties for no more than a few hours. This may
be much too short a time interval for a structural model
test of a viscoelastic material.

According to Leaderman [32] the behavior of
polymers, as described by the creep compliance, changes
significantly over six to ten decades of logarithmic time,
depending upon the type of polymer. Supposing the first
measurement possible at 0.001 hours (3.6 sec) after load-
ing, six decades of logarithmic time would dictate the
duration of test to be 1,000 hours and ten decades over
1,140 years! Hence, to cover the entire range of time
other tests must be devised. These may be dynamic tests,
wherein shortmterm effects are determined, or tests at
different temperatures, providing information sufficient
for the application of the timewtemperature correspondence
principle.1

In this study the time range covered by the
isothermal creep tests is from a few seconds to ten days
(240 hr)~-i.e., approximately five decades of logarithmic

time.

 

1See Eirich [16], vol. II, pp. 67ff.

15

In the majority of solid mechanics problems which
have been successfully analyzed by model techniques
(analogs) the prototype material may reasonably be assumed
linear (elastic or viscoelastic), homogeneous, and
isotropic. If data from a model test is to be interpreted
conveniently and accurately, the model material must
exhibit linearity, homogeneity, and isotropy. The last two
are quite easy to attain-—at least to an acceptable degree
of approximation-~in most potential model materials. If
the response to stress of a given material is linear at any
instant of time after loading and if this linearity extends
over a sufficiently broad range of stress to allow accurate
measurement of the response, then the material may be
useful for models.

Results of a model test can be no more accurate than
the investigatorVS knowledge of the model material’s
behavior. Such knowledge can be gained only through
carefully conducted calibration tests.

Calibration of Mechanical
Model Materials

 

 

Two aspects of the calibration of model materials
deserve careful attention. First, the range of stress for
which the constitutive (stressmstrain) equation is linear
is limited and time-dependent. Second, the material
properties may significantly change with time. Then the

ordinary material constants, such as the modulus of

 

16
elasticity E and Poisson's ratio u, may have to be
redefined or even abandoned. In such cases a suitable
viscoelastic description of the material behavior might be
enlisted.
When time effects are significant, the equations
describing material behavior become functional ones which

include time-~for example:

0')
ll

s(d,t) (2.1)

or

Q
II

o(s,t). (2.2)

The problem of defining such relations1 is not, in
general, simple. If, however, the strain is proportional
to stress at a given time after loading—~i.e., the behavior
is "momentarily linear”——then the well—developed theories
of linear viscoelasticity can be employed to determine
specific forms of relations (2.1) and (2.2).

A graphical approach is employed in this study to
portray both the mechanical and optical (birefringent)
behavior of the materials tested. This behavior is deter—
mined as the response to a onemdimensional stress state,
suddenly applied and constantly maintained. The usefulness
of the data given is not limited, however, to one—

dimensional applications. As Theocaris [64,65] has

 

1Some possible forms of such relations are given in
Chapter IV.

17
explained, "a simultaneous measurement of the mechanical
and optical quantities of a linear viscoelastic material in
a simple tension test supplemented with one initial value
of another characteristic function . . . suffices for the
complete characterization of the material." For the other
characteristic function mentioned, Theocaris used a
"lateral contraction ratio," defined as the ratio of
lateral strain (negative) to axial strain in a creep test
with constant axial stress. This ratio might be thought of
as a "time—dependent Poisson's ratio"; indeed, it is so
called by Daniel [10].

As explained in Chapter IV, this ratio might
accurately be assumed constant for the materials and tests
herein reported. Its value can be easily determined for
each material tested. With this value and the one-
dimensional mechanical and optical properties determined by
the creep tests of this study, other functions, such as the
bulk and shear compliances, relaxation moduli, and optical
coefficients, may be evaluated [65]. All of these
functions and coefficients are defined in Chapter IV.

The strain response to a constant stress is
expressed in a form suitable for graphical representation

by:

s = 8(0), t = const (tl,t2,t3, . . .). (2.3)

 

18
If the stress—strain relation is linear at any given time-—

say ti——after loading, then a relation of the form

E(ti) = D(ti)0 (2.4)

may be written, where D(ti) is the momentary value of the
"extension compliance." But D(t) is related1 to a "relaxa-

tion modulus" E(t) by:

D(t) = E—lmn. (2.5)

Thus Equation (2.4) may be written:

E(ti) =m. (2.6)

l

The striking resemblance of Equation (2.6) to the
familiar Hooke‘s law for one-dimensional stress should be
noted. It seems that the entire discussion and development
thus far presented might be recast within the framework of
a "modified, time—dependent" Hooke's law° Although this
point is not pursued in depth, it is further discussed in
Chapter VI.

Both the extent of the linear range and momentary
values of E(t) can conveniently be obtained from a plot of
the relations expressed by Equation (2.3). Figure 2.1

shows schematically such a plot for a uniaxial stress state.

 

1The reader is again referred to Chapter IV for
further definitions and discussion.

 

 

19

m

 

 

 

Fig. 2.l—-Representative plot of s = 8(0),
t = const (tl,t2,t3, . . .) for uniaxial

stress, constant load.

If a sufficient number of curves are shown in a plot,
such as Figure 2.1, then a curve of E(t) as a function of
time can be drawn (see Figure 2.2). Equation (2.6) may

then be replaced by:

E(t) = my, (2.7)

and the response to stress can be found at any specific
time within the range of the calibration test.

Since the stress-strain curve gradually deviates
from linearity, the exact limit of linearity cannot be
defined, much less determined. However, the limit can be

defined and quite accurately determined for an arbitrarily

 

 

20
chosen deviation from linearity. This deviation is
illustrated as a percentage of strain (1.0%) in Figure 2.1.
The stress at this deviation is called the linear limit

) where the subscript on

stress and is denoted by 8( l 0’

the left indicates that the linear limit stress is based on

can

strain and the subscript on the right denotes the

percentage deviation from linearity.

 

zz . E

         

)

5(022 1.0

 

 

l
I
I
|
I
I
I
l

 

(‘f‘-———-__-___

t t t Time

1 2 n

Fig. 2.2~-Linear limit stress and relaxation
modulus as functions of time for uniaxial
stress, constant load.

Again, if a sufficient number of curves are drawn in
a plot, such as Figure 2.1, then a curve of linear limit
stress (for strain) as a function time can be drawn, as in
Figure 2.2.

A plot of the strain as a function of time for a

given (constant) stress aids visualization of the nature of

 

21
mechanical creep. Figure 2.3 represents such a plot. It
can be deduced from Figure 2.1, or it may be required for
the construction of Figure 2.1, depending upon how the data

is obtained. This is discussed later in the chapter.

 

 

 

Q

 

.._-_—-t--———.-.-—d

 

 

t t t Time t

[.1
N
(.10
5

Fig. 2.3—-Representative plot of s = E(t),
o = const (01,02,03, . . .) for uniaxial

stress, constant load.

The Photoelastic Effect

 

Some materials possess the property of resolving
incident light rays into two orthogonal components and
transmitting them on orthogonal planes (see [4], p. 16).
This phenomenon is known as "double refraction" or "bire—
fringence." The optical properties on the two planes, in

general, will be different so that the two components will

22
be transmitted with different velocities. When they emerge
from a plate of the material, there is a difference in
phase——i.e., a relative retardation——between the two compo-
nent waves that is proportional to the thickness of the
plate and to the difference in velocities of the two
components.

In many synthetic resins this double refraction
depends upon the nature and intensity of stress. For
normal incidence on flat plates of such materials, sub-
jected to plane stress, the transmission of light obeys two
laws which form the basis of photoelastic stress determina—
tion [13]:

1. The planes of transmission coincide with the
principal—stress axes.l Only on the principal
planes of stress is light transmitted.

2. The velocity of transmission in each principal plane
depends upon the intensities of the principal
stresses in these two planes.

If the principal-stress intensities differ, so will the
velocities of transmission in the principal planes, result-
ing in a phase difference in the two components propor—
tional to the difference between the principal stresses.
Any method that can be employed to determine this phase

difference can be used to measure the difference between

 

1This is generally true for certain high-modulus
materials but may not be true in some low-modulus ones.
See Chapter IV for further discussion.

 

23
the principal stresses. A device called a "polariscope"
serves such a purpose. The function of each of its compo—
nents, possible variations in arrangement of components,
and interpretations of the patterns it produces are
explained in many books on photoelasticity, such as [6,13,
18,28].

Basically two types of photoelastic patterns
(fringes) are produced. Both of these are sufficient, and
generally necessary, for the complete determination of
plane stresses (or the corresponding strains) in an elastic
body. One type of fringe, called the "isoclinic," is the
locus of points having constant stress direction. The
other, called the "isochromatic," is the locus of points
having constant difference between principal stresses. In
uniaxial stress fields, such as are usually employed in
calibration tests, only the isochromatics are needed.

Daylight, or white light, is made up of a number of
constituent vibrations possessing different wave lengths
which can be distinguished from one another through the
sense of color. The wave lengths of visible light run from
about 390 nm (nm is abbreviation for nanometer; one nano—
meter 2 10"6 mm or 10",3 microns) to 770 nm. When light
enters the polariscope, the double refraction of the model
produces a phase difference which depends upon the wave
length. Hence if white light is used, each constituent

wave length is affected differently at a given point in the

 

24
model. As a result, every color in the visible spectrum
appears between integral orders of isochromatics, where an
integral order is represented by a particular color and
fractional orders by bands of other colors.

If monochromatic light is used——i.e., light
consisting of one wave length or color-—the isochromatic
fringes appear simply as dark or light bands. Here inte-
gral orders are represented by either the dark or the light
bands, depending upon the setup of the polariscope, while
fractional orders are represented by various levels of
intensity between the light and dark. Since the phase
difference produced by the relative retardation of the
model depends upon wave length, so also does the order of
isochromatic at a given point. As previously stated, it
also depends upon the stresses in the model and upon its
thickness.

Monochromatic light is generally preferred for
photoelastic analysis. Though it can be produced in a
number of ways, the method most commonly employed is that
of "filtering" a chosen wave length from the continuous
emission of a light source. Filtering consists of blocking
out all but a certain wave length, or rather, a limited
range of wave lengths. Information on types of filters,
their specification, and instructions for their use may be
found in a number of sources, a very concise one being

available from Eastman Kodak [29].

 

25

Usually only one wave length is used in a photo—
elastic investigation; the choice is governed by available
light sources and filters, sensitivity and transmission
characteristics of photoelastic material, and other factors
such as personal preference.

In some plastics the birefringence, or the order of
isochromatic, is found to be directly proportional to the
difference in principal stresses for a range of stress.

This may be expressed in one standard form as [18]:

d
1'1 =7z-E—C-j-(O'l - C72), (2.8)

where (typically)

 

fly: f = a "fringe constant" for the material,
1b
in x order ’
.d = thickness of the photoelastic model (in),
n = order of interference, or fringe order,
01,02 = principal stresses;

or alternatively as:

Cod
n =-—T— (01 - 02), (2.9)
where
CC = stress-optic coefficient for the material,
l = wave length of radiation.

A similar equation can be written in terms of strain as:

Cad
n = _T_ (81 - 82), (2.10)

26

where
C8 = strain—optic coefficient for the material,
81,82 = principal strains.

The linearity between birefringence and stress (or
strain) manifested by some plastics makes them useful as
model materials for photoelastic analysis. The evaluation
of the difference between the principal stresses at some
point in the prototype depends upon determining the order
of interference n [Equation (2.8)] at the correSponding
point in the model of known thickness d and on knowing the
fringe constant f for the model material. Knowledge of the
difference between principal stresses at various points in
the model can then be combined with information gained from
the isoclinics to "separate" the stresses and to determine
the stress field. Several techniques, such as shear
difference or oblique incidence, have been employed to
achieve this [18].

Calibration of Photoelastic
Model Materials

 

 

Evaluation of the fringe "constant" f of Equation
(2.8) for a given wave length of radiation, or of CO in
Equation (2.9), or of C8 in Equation (2.10) is a calibra—
tion of the stress-optic or strain—optic properties of the
material. These values are sometimes assumed to be inde—
pendent of time, temperature, and other variables. Indeed,

some manufacturers and suppliers go so far as to provide

27

values of the constants without stipulating the conditions
under which tests are to be performed. Indiscriminate use
of such constants can lead to gross errors. Only if
careful calibration tests are carried out under conditions
of temperature, time, and humidity very similar to those of
the model test can a high degree of accuracy be expected.

As implied above, creep phenomena are manifested in
the birefringent properties of most model materials, as
well as in the mechanical properties. The results of this
study indicate that birefringence is, in fact, more closely
related to strains than to stresses. The effect of time

upon f or C in general, cannot be neglected. If constant

0"
load is maintained, then Equation (2.8) should perhaps be

written:

_ d
n(ti) - m (01 - C52), (2.11)

where the subscript on t again indicates that reliable
values can be stated only at specific times after a con-
stant load is applied. The sequel to Equation (2.1) is

thus a functional relation, such as:
n = n(o,t). (2.12)

The complications of trying to express mathemati-
cally such a relation can again be avoided by expressing

the order of isochromatic n as a function of stress at

 

 

28

given times after loading—-i.e.:

n = n(o), t = const (tl’t t3, . . .). (2.13)

2’

For many photoelastic model materials the n—o
relation, at a given time, is linear. The extent of the
linear range is again limited and time—dependent.

If the stress state is uniaxial, then the quantity
(Cl — 02) becomes simply o. In many photoelastic analyses
a one-dimensional calibration test is conducted to deter-
mine momentary values of the photoelastic coefficients.
These momentary values are then used in an equation, such
as Equation (2.11), to determine the more general states of
stress in photoelastic models of the same materials.

For materials exhibiting‘significant time effects,
such applications should seemingly be made within the
framework of a photoviscoelastic theory instead of a
"momentary-elastic" one. However, in an overwhelming
number of cases, close agreement with theoretical results
has been obtained using the momentary—elastic "theory."
This not only attests to the validity of such a theory for
the optical (birefringent) behavior but also implies that a
similar theory might be valid for the mechanical behavior--
e.g., a "time-dependent Hooke's law." Such a theory could
be expected to apply only to the behavior of relatively

"high—modulus" or "stiff" materials.

29
Both the extent of the linear range and the fringe
constant f for a given wave length of radiation can be
deduced from a plot of the relation expressed by Equation

(2.13). Such a plot is shown in Figure 2.4.

 

 

t
n
n ‘
t3
t2 t
1
t
n
x = x2 t3
V t2
t1
\n
x = x1
(5

Fig. 2.4-—Isochromatic fringe order n vs.
stress for two wave lengths, X1 and l2.

If enough such curves for each wave length are drawn
and enough wave lengths are considered, then sufficient

data can be deduced to determine:

the linear limit stress (1.0% deviation

1. )

n(01£ 1.0’
from linearity) based on birefringence for a given

wave length, and its variation with time.

)

2. The variation of with wave length for a

n(Gz£ 1.0

given time.

 

30

3. The fringe constant f as a function of time for a
known thickness and given wave length.

4. The stress-Optic coefficient Co as a function of
time for a given wave length.

5. The variation of CO with wave length for a given
time-—i.e., information on dispersion of bire-
fringence.

6. The relation n = n(t), G = const (01,02, . . .).

All six of the above relations can be plotted;
however, most photoelastic analyses employ only one wave
length of radiation for both the calibration of the mate-
rial and the model study. Dispersion of birefringence then
becomes irrelevant. Typical sample curves of 1, of 3 and 4,
and of 6 may prove very useful and are illustrated in

Figures 2.5, 2.6, and 2.7, respectively.

 

 

Time t

Fig. 2.5——Variation of linear limit stress
with time, based on birefringence.

 

31

 

 

 

 

t3 Time t

Fig. 2.6-—Fringe constant f and stress~optic coeffi—
cient Co as functions of time for given wave length.

 

 

Time

Fig. 2.7.--Order of interference n vs. time
for various stress levels, uniaxial stress.

32
Curves showing mechanical properties can be combined
with those of photoelastic ones for other useful relations.
If common times are used in determining the relations [see

Equations (2.3) and (2.13)]

8 = 8(0), t = const (tl,t2,t3, . . .), see Figure 2.1,

n = n(0), t = const (tl,t2,t3, . . .), see Figure 2.4;

then they may be combined to yield

n = n(8), t = const (tl,t t .) (2.14)

2’ 3, o I

for a given wave length x. If the times are not common in
the above relations, Equation (2.14) may still be deduced

from

8 = 8(t), 0 = const (01,02,03, . . .), see Figure 2.3,

n = n(t), 0 = const (01,02,03, . . .), see Figure 2.7,

if common stress values are chosen in plotting Figures 2.3
and 2.7. Figure 2.8 shows a representative plot of
Equation (2.14).

One further extension of the data can be executed.

From Figure 2.8 the relation

n = n(t), 8 = const (61,82,83, . . .) (2.15)

33
may be obtained. A plot of Equation (2.15) might appear as

in Figure 2.9.

 

 

Fig. 2.8——Order of interference n vs.

 

 

strain for t = const and l = ll.
n
8
n
83
V82
8
l
X — X1
Time

Fig. 2.9-—Order of interference n vs.
time for 8 = const and l = ll.

 

34

Complete Description of
Material Properties

 

 

All of the material properties necessary for
mechanical or photoelastic model studies may be deduced

from plots of the following relations:

8 = 8(0), t

const (tl,t2,t3, . . .)

8 = 8(t),

Q
II

const (01,02,03, . . .)

 

n = n(0), t = const (tl,t2,t3, . . .)

(2.16)
n = n(t), 0 = const (01,02,03, . . .)
n = n(8), t = const (tl,t2,t3, . . .)
n = n(t), 8 = const (81,62,83, . . .).

The relations involving n—-i.e., those expressing
the birefringent properties of photoelastic materials—~are
assumed to be determined for one wave length of radiation.
As previously stated, this requires only that the same wave
length be used in the model study as in the calibration.
Furthermore, relations (2.16) take no account of the effect
of temperature upon the material properties; this effect is
very pronounced. Constant (and identical) temperature is,‘
of course, desirable for calibration of the material and

the model study, either mechanical or photoelastic. If

35

constant temperature cannot be maintained, then it is
essential that the temperature cycle be the same for both.

The times expressed in relations (2.16) are measured
from the "instant" a constant load is applied. Since the
load cannot be instantaneously applied, some error is
introduced by the loading procedure. The smallest meaning—
ful times in relations (2.16), as well as earliest times at
which readings should be taken in the model test, depend
upon the manner and speed of the loading. This point is

further discussed in Part B of this chapter.

Part B. Calibration Procedure

 

Possible Calibration Models

 

Theoretically, any model for which there is a known
analytical solution for the stress distribution can be used
for calibration. In practice, limitations of experimental
technique, available equipment, and the accuracy required
may govern the selection. Possible models include pris-
matic tension bars or "stepped" tension bars with prismatic
sections, circular disks in diametral compression, beams in
pure bending, and concentrated loads on "semi-infinite"
plates.

Each of these has distinct advantages as well as
disadvantages. Since none of the models mentioned were
used successfully for obtaining the data essential to this

study, only their disadvantages will be stressed.

 

36

For a given load, only one level of stress prevails
in a prismatic tension bar. Hence only one value of strain
or one value of the isochromatic fringe order can be
obtained for a given load at a given time. Determination
of the properties for a range of stress and time requires
either a number of models or repeated or step loading on a
single model. When a number of models are used, they must
be machined and handled identically. If repeated loading
is used, the specimen must be allowed to recover completely
between loads and the repeatability of response must be
checked. For complete recovery annealing is sometimes
required. If step loadings are employed, creep effects are
more difficult to account for, especially if response is
nonlinear. Should "crazing" of the material occur, as it
did at high stresses in some of the materials tested, then
neither repeated nor step loadings would seem satisfactory.

The stresses in a circular disk in diametral
compression are biaxial [68] and somewhat difficult to
relate to strain or isochromatic order at any given point
on the disk. If this difficulty is avoided by considering
only one point in the disk, such as the center, then
repeated or step loads are necessary.

Beams in pure bending, though not presenting the
same problems as the prismatic bar or the disk, are some-
what more difficult to load than the other models

mentioned. In addition, if large deflections occur, as

 

37
they often do in plastics, the elementary beam theory must
be abandoned in favor of a more complex theory. Then
problems similar to the ones mentioned for the disk appear.

Although the stress distribution for a concentrated
load on a semi—infinite plate is quite easily related to
strain or order of isochromatic [13], the loading is some-
what difficult, especially for thin plates. The "semi-
infinite" condition also imposes restrictions on the
smallest permissible size of plate.

Another possible model, which better meets the
requirements for this study, is a tapered model or "wedge."
Coker and Filon [6] suggested the use of a wedge, loaded
symmetrically in compression, for use as a compensator.

The frozen stress method was applied to a modified wedge
model by Frocht [18] to produce a permanent compensator.'
The wedge was further modified and used by Pindera [47] in
studying rheological photoelastic properties of various
materials. The first known attempt to use the moire method
on a tapered model for studying rheological mechanical
properties is made in this study. It also includes a
slightly modified technique in machining to ensure the

desired symmetry.

Model Used in This Study

 

The obvious advantage of the wedge over the models
mentioned above lies in the fact that a range of stress is

produced by a single load, this stress being distributed

38
radially and varying continuously. Hence a single model
produces data for a chosen range of stress at any given
time after a load is applied. That the stress distribution
in a wedge is purely "radial" was shown by Frocht [l8] and
is given in Appendix I of this thesis, along with the
procedure used for determining the stress at any point on
the centerline of the modified wedge used in this study. A
sketch showing the approximate shape of the model is also
presented in Appendix I.

For the photoelastic calibration test a scale was
scribed along the centerline (previously scribed) of a
tapered model and the smallest division of the scale chosen
as either 0.10 inch or 0.20 cm. A vernier height gage
moving on a surface plate, shown in Figure 2.10, allowed a
high degree of accuracy in the scribed scale. The width of
the model at specified
points along the center—
line was then measured
with either a micrometer
caliper or a measuring
microscope. The radial
(or actual) stress at
these specified points

was computed from the

 

nominal stresses and the

Fig. 2.lO—-Equipment used stress concentration
to scribe scale on models.

39
factors given in Appendix I. Plotting of stress vs.
distance along the centerline followed.

The model was loaded in tension by means of dead
weight on the lever of a stationary load frame within the
field of a circular polariscope. Photographs were taken of
the model showing isochromatic fringes, as well as the
scale on the model, at chosen times after application of
the load. From these photographs data was obtained for a
plot of isochromatic fringe order n vs. distance along the

centerline. Then the relation between stress and fringe

 

order could be obtained for every time at which a photo—

graph was taken-—i.e., the relation

n = n(0), t = const (tl,t t3, . . .)

2,

was obtained.

With the aid of the moire method, data for
mechanical properties was obtained from the same type of
tapered model. Since the stress in the wedge portion of
the model is radial, it follows that the strain is also
radial. A grid of density 500 lines per inch was applied
(see Appendix II) along the centerline, with the grid lines
perpendicular to the centerline of the model. A matching
grid on a "dummy" model placed in front of the active one
produced moire fringes whose spacing depends upon relative
displacement between the two grids. Following the same

procedure (from scratching the scale to photographing

 

4O
fringes) as explained above for the photoelastic
calibration, a plot of the moire fringe order m vs.
distance along the centerline was obtained. As explained
in Appendix II, the slope of such a curve at any point
yields the strain at that point. After these slopes were
obtained, at chosen points corresponding to chosen values

of stress, the relation

8 = 8(6), t = CODSt (tl,t2,t3, o o 0)

was obtained.
Other details of the test procedure are outlined in

the following sections.

Model Preparation and Clamping

The tapered models were cut to coarse dimensions
with a band saw and finished with a high-speed routing tool
integrated into a Ratiobar Pantograph. Symmetry could not
be achieved with a single mounting of the model on the
table of the pantograph. The problem was circumvented by
drilling, on the centerline of the model near each end, a
small hole which fit tightly over a stationary pin on the
machining table. After one side of the model was machined,
it was turned over and aligned by means of the pins for
machining of the other side.

When brittle materials were used, the edges of the

model were hand polished with three grades (1,2,0) of

41
metallurgical emery polishing paper followed by two grades
(1.0 and 0.3 micron) of powdered alumina abrasive to
partially eliminate stress concentrations arising from
minute machining notches or scratches.

The jaws of the mounting clamps were padded with
thin sheets of nearly pure aluminum to prevent surface
damage to the model. As one further means of reducing
stress concentration, the effects of bending of the clamp
jaws were minimized. This was shown to be necessary by
fracture at the clamps of several specimens. Both the

problem and the solution are illustrated in Figure 2.11.

Clamping

P1 1
I I 1 I

l I I : :

‘ecimen-y } {Specimen—3

///f/[//////I :JJ ' , V////////]//1

 

‘—

_——_‘
..-—q
—-_
--
.-
-

 

 

    

_}Washer

1 T

Approximate
stress
’ distributions, . . .

(a) (b)

 

 

 

 

 

Fig. 2.11-—Sketches illustrating (a) the
effect of bending of the clamp jaws, and
(b) improvement by modification.

 

42

Test Procedure—~Photoelastic
EElibration

 

 

After the model was completely prepared and
measured, it was mounted in the upper clamp of the load
frame. The lever, on a knife—edge fulcrum, was then
balanced with the load hanger in place. Hence no correc—
tion for the weight of the lower clamp was required. After
the model was secured in the lower clamp, dead—weight load
was applied to the hanger.

Usually the load was applied within about two
seconds, in a single step. When relatively large loads
demanded a two-step loading process, five to six seconds
were consumed. Time measurement began when the full load
was reached. Because of the loading time, measurements
within the first few seconds seemed meaningless. Hence
first measurements were usually taken after four to six
seconds in case of single-step loads and ten to twelve
seconds for two—step loads.

Duration of the constant load was intended to be 240
hours in all tests; however, in several tests fracture of
the model occurred in less time. When it occurred in less
than about 100 hours, the test was repeated; otherwise the
results were used. Though creep tests ideally maintain
constant stress on the specimen, rather than constant load,
the difference is small for high-modulus materials where

changes in cross—section are small.

 

43

Recovery data was collected when the model carried
the load for the full 240 hours. Though not essential to
this study, such data might be useful for checking super-
position theories of viscoelastic materials, to name only
one possibility.

The load frame was within the field of a standard
type of circular polariscope consisting of a light source,
condensing lens, dispersing screen, polarizer, quarter wave
plate, model, quarter wave plate, analyzer, filters, and
camera. Figure 2.12 shows the polariscope. The light
source, as well as the camera and filter wheel, was
repositioned for the photograph. Space limitations pre—
vented taking a composite photograph with all elements in

normal position.

 

Fig. 2.12—~Polariscope used in photoelastic studies:
(a) light source, (b) condensing lens, (d) dispers—
ing screen, (P) polarizer, (Q1) quarter wave plate,
(M) model, (02) quarter wave plate, (A) analyzer,
(F) filters, and (C) camera.

44

The function of each of these elements has been
explained by a number of authors. Frocht's [18] presenta—
tion was concise and appealing; Jessop and Harris [28] also
included a thorough explanation of possible errors arising
from the improper use of some arrangements.

A "mixed setup" of the circular polariscope was
employed for producing the isochromatic fringes pertinent
to this study. The quarter wave plates were kept crossed

and 45° from the plane of polarization. Generally a "light

 

field," produced by crossing the planes of the polarizer
and analyzer, was used because it offered the advantage of
making visible the boundaries of the model.

As stated, the circular polariscope causes only the
isochromatic fringes to appear, each fringe being the locus
of points having a constant difference between principal
stresses. Since the model is symmetrical about its center—
line, the isochromatic fringe pattern is symmetrical.
Furthermore, in the wedge portion of the model the iso-
chromatic fringes are radial, as a consequence of the
radial stress distribution there. Isoclinics, representing
constant stress direction, are not required.

Because a range of wave lengths was used with the
same polariscope, the performance of the quarter wave
plates was checked. Their performance must be questioned,
because a plate can give a relative retardation of exactly

one-quarter wave length for only the one particular wave

45
length for which it is designed. Mindlin [40] showed,
however, that if crossed plates are not quarter wave plates

for the wave length used, but are nevertheless identical,

 

then the isochromatic fringe pattern is unaffected. The
only effect is the incomplete removal of the isoclinics.
Jessop and Harris [28] showed further that if the plates
are identical and each produces a relative phase retarda—
tion of one-fourth wave length plus some error, then the

maximum error introduced into the fringe pattern will be

 

the algebraic difference of their individual errors when

 

they are crossed but will be the algebraic gum when the
plates are used parallel. Since they were crossed in this
investigation, the accuracy of their over—all performance
depended only upon their being identical.

The check for possible error consisted simply of
comparing the location of isochromatic fringes produced by
two "setups" of the polariscope for a wave length other
than the one for which the quarter wave plates were
intended. One setup was the mixed circular one used
throughout. The other consisted of putting the quarter
wave plates' axes in the plane of vibration of the
polarizer, 45° from the model axis;l hence these plates

produced no effect upon fringe formation.

 

l O I O O I 0
Both lSOCllnlCS and isochromatics were produced in

the second setup. To eliminate the objectionable effect of
having a zero-order isoclinic along the centerline of the
model, the plane of polarization was shifted 45° from the
centerline. The zero-order, 45° isoclinic does not appear
on the centerline.

46

This comparison revealed no difference in the
isochromatic fringe location for any wave length used in
the investigation; the quarter wave plates apparently
qualify as "identical."

Nine filters, all manufactured by Zeiss Jena, were
used to isolate wave lengths of 410, 436, 500, 546, 578,
600, 650, 750, and 800 nm, emitted by a 200 watt high
pressure mercury lamp by Osram. Within this range of wave
lengths the mercury lamp emits high intensity "peaks" at
405-408, 436, 546, and 577-79 nm. However, the highest
intensity is produced at 365—66 with a rather broad band.
Coupled with the relatively broad transmission band of a
Zeiss Jena 405 nm interference filter, this led to diffi-
culty in attempting to isolate the 405 nm wave length,
which seemed a natural one to use. Fortunately, a 410 nm
filter transmitted very little of the 365 nm emission band
but enough of the 405 nm band so that the 410 nm wave
length could be isolated.

At frequent intervals throughout the tests, pictures
were taken of the isochromatic fringes with at least three
wave lengths, usually 410, 578, and 800 nm. Analyses of
the effects of time—~i.e., the creep characteristics—~were
deduced from these. At one or two selected times during
the test, pictures were taken using all nine wave lengths.
These provided information on the possible variation of

linear limit stress with wave length, as well as data on

47
the dispersion of birefringence. Taking a series of nine
pictures required approximately six minutes; only after the
load had been on the model for a long period of time-—say
100 hours-—was such a series taken. The effect of creep
during the six-minute interval could then be neglected.

All pictures were taken with a Nikon F camera
(35 mm) fitted with a Coligon Zoom lens, f = 95 mm to f =
205 mm. The long focal—length lens allowed sufficient
separation of the camera from the model so that the maximum
deviation from normal incidence was less than 3°.

Kodak High Speed Infrared film was used throughout
the photoelastic studies. Exposures varied from one second
at f-l6 for wave lengths of high intensity (peak outputs of
the mercury light source) to sixty seconds at f—6.3 for
wave lengths filtered from the "background" or continuous
radiation of the source.

Film negatives were projected and magnified (15X)
for observation. Possible distortion by camera or projec—
tor lenses or of the film was not important, since the
reference scale was scribed on the model. Slight distor—
tion by the projector lens was detectible.

Effit Procedure——Mechanical
Calibration

 

 

The load frame used and the loading procedure for
the mechanical model tests were identical to those

described for the photoelastic ones. Figure 2.13 shows

 

48
the essential parts of the apparatus. The light source and
camera were repositioned to permit a composite photograph.
The optical system is illustrated in Figure 2.14.
To approximate a point light source a small hole (0.039—in
diameter) was drilled through an opaque diaphragm and

introduced into the optical system.

       

‘ ‘- Ar nit;
Fig. 2.13——Apparatus used in mechanical model tests:
(a) light source, (b) collimating lens, (M) model,
(D) "dummy" model or analyzer, (d) condenser lens,
and (C) camera.

Collimating Condensing

Light lens;7 f lens 77
F

p |_____*_]__ _____. __

Model—4//[[Analyzer -Z{:amera

Fig. 2.14—-Optical system used
in mechanical model tests.

 

 

 

 

 

 

Diaphragm

49

A 200 watt high pressure mercury lamp, a double
convex condenser lens, and a blue filter were contained in
a Leitz Model 250 lamp housing. The collimator and con-
denser consisted of a matched pair of plano—convex lenses
of 13—inch diameter and 39-inch focal length. A Linhof
Technika camera with a Schneider-Kreuznach Xenar 1:4.5/300
lens was used to photograph the moire fringe pattern.

Kodak Panatomic-X (4 in x 5 in) sheet film was used.

Since model and analyzer were placed in a parallel-
1ight field, spacing between them was not critical,
although fringe quality improved with smaller spacing. The
dummy model, made of the same material as the active one,
was attached to the active model near one end by means of a
small C-clamp. A thin shim between the models, spanning a
small length on each side of the clamp, provided the neces—
sary complete separation between the active and dummy
models outside the clamped region. Location of the shim
and clamp are illustrated in Figure 2.15.

Since the load lever was balanced after the analyzer
was attached but before the model was secured in the lower
loading clamp, no corrections for the weight of analyzer or
clamp were necessary.

Loading procedure and duration, as well as the
timing of pictures, were the same as for the photoelastic

models described earlier. A check for symmetry of loading

50
was accomplished with the aid of a small circular

polariscope held by hand.

Load on model———> [1
Shim
if
—"'.1‘—-'
Small clamping force-f

Active model————*- -+———-Dummy model

 

 

 

 

Fig. 2.15——Illustration of method of
attaching moire analyzer to model.

All tests, both mechanical and optical, were
conducted at 68°F : 1°F. With the humidistat existing in
the laboratory precise control of the relative humidity

could not be maintained. It was observed on a recording

hygrometer to vary between 55% and 70%.

51

Materials Tested

 

Because the primary aims of this study are to
investigate certain possible sources of error and to
suggest ways of improving the accuracy of photoelastic
stress analysis, those materials commonly used in photo-
elasticity were selected for testing. Polyester Resin
CR-39 was the first choice, because of its extensive use
in photoelasticity; it is probably the most commonly used
of all birefringent materials in this country. This
material was supplied by Cast Optics Co., Inc.

Also selected for examination were two materials
commonly used for photoelastic coatings but believed to be
potentially useful also for models. These were supplied by
Photolastic, Inc.: Epoxy Resin PS-2 and Polyester Resin
PS-l. These, as well as CR—39, are available in sheets of
various sizes and thicknesses.

Polyester Resin Palatal P6—K, manufactured by
Badische Anilin und Soda—Fabrik (Germany), was also tested.
The material in liquid form was donated by the manufac—
turer. Plates were cast in 1/4-inch and 3/8-inch thick-
nesses and then cured.

In summary, the materials tested were:

Polyester Resin CR-39.
Epoxy Resin PS-2.
Polyester Resin PS-l.

Polyester Resin Palatal P6-K.

CHAPTER III

BIREFRINGENT AND MECHANICAL PROPERTIES OF CR-39

Birefringence vs. Stress,
Time Constant

 

 

Details of the testing procedure and of the
graphical presentation of results were given in the pre—
ceding chapter. Discussion in this chapter is therefore
limited mostly to the results obtained.

The birefringence data presented in this chapter is
based on the shortest wave length used in the study—-
namely, 410 nm. Since the isochromatic fringe order is
inversely related to wave length, the highest possible
number of fringes was thus obtained.

Figure 3.1 shows the fringe order n as a function of
length along the centerline of the specimen. The extensive
use of the 410 nm wave length was not anticipated when the
test was conducted. Consequently, two pictures with
another wave length were taken before the first one with
410 nm at one minute after the load was applied. The
accuracy and fineness of the scribed scale (smallest divi—
sion 0.20 cm) allowed accurate location of both integral
and half—order fringes. The data furnished very smooth
curves with no observable irregularities.

52

53

¢_

.mmumo mom up. . . . .mn.an u n .mcfiaumncmu macaw
QUGMDmHol.m> C umpuo mmcfium UHDmEOHLUOmHIIH.m .mam

AEoVJ

.N. O. m w ¢

JN

 

 

d d J _ u u d d d d

.5 mud n u .ecozfix .8 -mo

 

 

 

..oo ..o ..o /” ‘
..oo .9 co ///”
.0» .oo ... //

.0» .oo .1.
.8 .oo .13
.oo ..oo.
.oogon_
.00 gets

do

u ‘ upJo obupd

 

 

9!

w.

ON

54

A measuring microscope permitted corresponding
accuracy in measuring the width of the specimen at a number
of locations. The thickness was measured at several points
with a micrometer caliper. The nominal stress was computed
from the area and the load of 320.85 pounds (64.17 magni—
fied 5.00 times by the load lever). By using the values of
the "stress concentration factor" described in Appendix I,
the actual stress was computed at each of the points where
the area was known. A plot of stress vs. length along the
centerline (not shown) yielded no detectible scatter.l

The fringe order for each of the times represented
in Figure 3.1 was then read at about fifteen values of
stress, ranging from 800 psi to 2,900 psi. One additional
set of points corresponding to fringe order and stress in
the shank is also available. Data was thereby obtained for
a plot of fringe order as a function of stress for each
time represented. This plot is shown in Figure 3.2.

The nonlinearity of the stress—optic behavior is
clearly evident in this plot. The linear limit stress at

)

one percent deviation from linearity, denoted by n(0££ l 0’
is marked by an "x" on each curve. For the assumption of
linearity to be accurate within one percent, it appears

necessary that the fringe order remain below seven fringes

 

1The term "scatter" usually implies the testing of

more than one specimen and some sort of data averaging for

each point plotted. Tapered models were used in this study
to avoid such scatter. Herein, scatter denotes observable

irregularities or lack of smoothness.

55

ommlmu “om c#o o o o «mynfl

#

n p .mmmhwm

 

 

.m> c Hmouo mmcfiuw UflymeounuomHiim.m .mflm
, 2...: b
oo~.n 0.8.... comm ooofi 8.6.. com. com oo.¢
I 3.01.5203} .mmumo
A:
r
1 Aqnzjbflh l-o
r. °._Aqubvc llx
/ I
I //
r //l ///(I .oo..o no
a / I, . ..oo .2 ..o
1 / :00 .008.
r /, /.on.oo .1.
. I .8 .oo ..3
. /I .oo ..oo.
r /I .oo :3.
. r b P b P P > _ b IIL [ h p [ b p b p p P L r L t F

 

 

2L

 

 

u ‘upso obugu

0.

ON

Va

 

56
within the first hour after loading. This number depends,
of course, upon the wave length of radiation and the
thickness of the model.

It should also be observed that within the first
hour after loading a much higher fringe value (or stress)
is permissible if a deviation from linearity of, say, five
percent is allowed. The linear limit stress at five
percent deviation from linearity is marked by an "O" in
)

Figure 3.2. The difference between n(OM) and

1.0 n(GM 5.0

is seen to decrease with increasing time.

The error in assuming the linearity at small
stresses to extend to higher stresses may be considerable.
At 3,000 psi, for example, this error is approximately 7%
at one minute, 11% at one hour, 28% at 25 hours, and 56% at
240 hours.

Birefringence vs. Time,
Stress Constant

 

 

At the eight times represented in Figure 3.2 the
fringe order at any stress within the prescribed range may
be obtained, thus providing data for plotting fringe order
as a function of time with O = constant. Figure 3.3 shows
such a plot. Nonlinearity at a given time may again be
observed as increasing distance between consecutive curves
as the stress level increases.

Any irregularities (or "scatter") of raw data would

have shown in Figure 3.1 and have been "smoothed” by

57

QQMIINHU HOW no» 0 o o oNbaHb
.m> c umpuo mmcflum vameOHSUOmHIIm.m .oflm

332: 06:.

n o .mep

 

 

 

 

 

 

 

 

 

 

 

 

oom ow. ow. om ov o
F P b . . o
A u _ u A _ J 4 d 4 d a u - u ‘ q s u q
I .5361. .5532 .mm-mo 1
ll.\
:0 00¢ lb ]|I|I|II\
11 e
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T
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.uaOON_-b \\\
... ..uOO¢.-b K m ”m:
. .. U
1 2.89.6 w
0
2.002 .6 W
O
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..aooo~..b .
1 N. u
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3.39.9.6
1
1 Jim—
i|l\\.. 283.6
.2. 33.6 . 4
r 110m
L . . _ _ L P . . _ . p r p _ _ . p . L

 

 

 

 

58
careful plotting to a large scale (21 in x 33 in). The
last two figures were deduced from Figure 3.1; thus no
irregularities were apparent in these plots.

Strain vs. Stress,
Time Constant

 

 

With the aid of the moire method of strain analysis,
evaluation of mechanical properties followed a procedure
very similar to that used in evaluating birefringence
properties. In Figure 3.4 the moire fringe order is shown
as a function of distance along the centerline of the
specimen for eight values of time after the load was
applied. The grids were apparently distorted in the course
of applying them to the specimen and analyzer. This gave
rise to an initial moire pattern represented by the lowest
curve shown.

The first picture of the moire pattern was taken
approximately four seconds after the load was applied.

With a fringe order of approximately twenty, locating both
integral and half-order fringes provided approximately
forty points for plotting a curve. This was increased to
about seventy-eight points at 150 hours, after which the
specimen fractured. The scatter of data points on this
plot was again insignificant. Though a more dense moire
grid would increase the number of points available for
plotting, these additional points would not aid materially

in defining the shape of the curve.

59

.mmumu mom a». . . . .mn.ap u p .mcaaumpcmo
macaw mUGMMmHo .m> E Hmouo mosauw muaoZII¢.m .mHm

 

     

 

 

 

 

A..;oc.. 4
o _ u n ¢ 0 m s m
b p p p . p . P 0 _ O
a . q a 4 a a q 4 d a . _ q . 4
TI IlIIII-I‘ull‘liuil‘ll‘ll‘lu‘I-II' L
1' . I \‘\‘x I]
V\ a. II v
I l
vl Itl m
H 65):: 08 ....N.
v mml mo 1
T 1
1 im—
/////Ioo .co .0050 H
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/////l. .oo.o.go 1
/ :00 OO: _ 11 VN
/|l.. ...OO 00 no. 1
.oo :00 lrmN
L
1. . .oo ‘00— L
.oogon_ ..mn
r L
.l .1
h _ _ p p _ b b L p L . .r a b . 0”

m ‘Joplo obugu ”you

60

Stress was plotted as a function of distance along
the centerline of the specimen.

The strain at a given stress and time is represented
by the slope of one of the curves at a particular point.
Slopes were measured by a graphical technique, employing a
set of mirrors mounted on a 90° machining block. One
mirror was placed tangent to the line at the chosen point,
the other perpendicular to it, thus yielding an image of
the line tangent at the point and an image continuing the
line at the point.

This method of measuring slope is, of course,
susceptible to significant error. However, readings could
be repeated to within about four percent. The possible
increase in accuracy resulting from a sophisticated numeri-
cal evaluation of the slopes did not seem to be justified.

The slope evaluations yield data for plotting strain
(or percentage elongation) as a function of stress for each
of the times represented in Figure 3.4. Such a plot is
shown in Figure 3.5 with the data points indicated. Some
lack of smoothness is seen to exist. Because of the ini—
tial moire pattern, the data points actually represent the
difference of two slope measurements, increasing the
possible error. Despite this the scatter does not seem
excessive, and the results are believed quite reliable.

Nonlinearity of the mechanical behavior at high

stresses is again rather obvious. The departure from

61

OOOn

T

- T

 

OONN

oovw . ooa

mnsmo

.mmnmu now up. . . . .Nu.Hu u p
.wmmuym .m> coaummsoam manpcmuummllm.m .OHm

com.

:3. b

009

 

CON—

o.oaquvw II o

33.3w I u x

rO

....MO

:66

.10..

 

..N

uouobuma obmuooud

62
linearity (vertical deviation from extended straight line)
seems more rapid than for birefringence, as indicated by

comparison with Figure 3.2. The stress at one percent

deviation from linearity 8( ) is again marked by an

CM. 1.0

"x" on each curve and 8(OM)5 0 by an "0." Linear limit

stresses will be further discussed in a subsequent section.

Strain vs. Time,
Stress Constant

 

 

From Figure 3.5 data can be deduced for a plot of
strain (or percentage elongation) vs. time for a fixed
stress level. Figure 3.6 shows such a plot for twelve
levels of stress. Deviations from linearity again appear
as increasing distances between consecutive curves at
higher levels of stress.

Since the plot of Figure 3.5 "smooths" the measured
strain data, the data points of Figure 3.6 furnish regular,

well—defined curves.

Linear Limit Stress vs. Time

 

After the optical and mechanical linear limit
stresses are determined from Figures 3.2 and 3.5,
respectively, it is possible to show them as functions of
time. This is done in Figures 3.7a and 3.7b; different
time scales are necessary to show clearly their variation
throughout the time interval for which they were

determined.

63

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66

The linear limit stress at one percent deviation
from linearity is seen to be considerably lower for bire-
fringence than for strain. Though little information is
available for comparison, such a wide difference was not
expected. Coolidge [7] reported that for a given rate of
loading the stress-strain curve for CR-39 is a straight
line up to about 3,000 psi, whereas linearity of the
stress-fringe relation extends to "somewhere between 2,500
and 3,000 psi." The resin he tested was manufactured by
the Pittsburgh Plate Glass Company. The method of testing
employed in his investigation would not seem to permit
adequate allowance for creep effects. Frocht [18] found
that in Bakelite (BT—6l-893) the "proportional limit" for
birefringence is over twenty percent higher than for
strain. Creep of this Bakelite was apparently considered
negligible. Approximate values of n(oaa)l.0 for CR-39
manufactured in Great Britain may be deduced from data
presented by Pindera [47]. The values seem to be consider-
ably higher than those of the American—made CR—39 tested in
this study. However, the ultimate strength is also
considerably higher for the British-made resin.

It must be concluded that the linear range of
behavior, as well as other properties of CR—39, varies
considerably with different manufacturers. Properties may
vary from one lot to the next with the same manufacturer.

Age and storage conditions may also affect some or all of

67

the properties. Pindera's CR-39 is known to have been
stored for five years, whereas the material used in this
study was tested within six months after purchase. The
qualitative results of this study seem consistent with
earlier findings; the quantitative differences emphasize
the need for careful calibration.

Comparison of Figures 3.2 and 3.5 seems to indicate
a more gradual departure from linearity in the
birefringence—stress relation than in the strain-stress
relation, especially at times greater than one hour. The
linear limit stresses for birefringence and strain at five
percent deviation from linearity are more nearly equal than
those at one percent, except at very short times. This can
be seen in Figures 3.7a and 3.7b. The crossing of the two
curves at five percent deviation is puzzling and may be due
to accumulated experimental error. Two quantities are
being compared which are difficult to determine in the
first place and which are of roughly the same magnitude.
Irregularities are therefore not surprising. The crossing
of the curves may actually portray the time behavior of the
material. This does not seem entirely implausible but may
be worthy of further detailed study.

Birefringence vs. Strain,
Time Constant

 

 

From the plots previously shown, the relation of

birefringence vs. strain at a given time after loading may

68
be deduced by the method outlined in Chapter II. Such a
plot is shown in Figure 3.8. The largest strain
encountered at one minute after loading was slightly
greater than one percent. The relations are, of course,
linear up to the fringe values or strains corresponding to
the optical linear limit stress. Then there is a "breaking
over" toward higher strain, indicating that at a given time
after loading the strain increases faster than fringe value
with increasing stress. Thus birefringence deviates more
gradually than strain from linearity. No reference to such
behavior could be found in the literature for CR-39, either
to confirm or contradict these results.

The inflection of the curves at still higher values
of strain (or birefringence) may warrant additional
confirmation. It is not believed attributable to experi-
mental error. An error analysis would be somewhat diffi—
cult to perform and would be of questionable accuracy. A
statistical analysis of the scatter of the data points on
the birefringence—stress and strain-stress plots might
provide a meaningful estimate of the consistent error
committed. Because of the regularity of the birefringence—
stress plot, such analysis should seemingly be applied only
to the strain—stress plot. But errors in measuring the
slope of the moire fringe order vs. length curves are
mostly reSponsible for the scatter of the strain—stress

plot. The repeatability of the slope measurements (within

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70

about 4%) offers some estimate of the possible errors in
strain values. Consistent errors of i 4% could cause the
inflection, but errors of this magnitude are not believed
present. Identical techniques were employed in testing the
other materials (see Appendix III); no inflection appeared
in birefringence-strain plots. Thus if consistent error is
responsible for the inflection of the plot for CR-39, such
error is apparently unique to this one test.

Further reduction of data, such as that necessary
for a plot of fringe value vs. time for constant strain,
does not seem advisable. Data for such a relation can be
obtained directly from a relaxation test, and the plot of
birefringence vs. strain for t = constant (as shown in
Figure 3.8) can be obtained in a single step. Relaxation
tests are thus recommended to determine whether the inflec—
tion indicated in Figure 3.8 is a true representation of
this material"s behavior.

At a given strain Figure 3.8 shows a decrease in
birefringence with increasing time. This is expected for
CRm39 by comparison with results presented by Clark [3]
but, as he points out, is not true of all materials.

Material Coefficients
in Linear Range

 

 

The photoelastic and mechanical material
coefficients, such as E(t), fo(t)’ and Co(t)’ can be

deduced from the curves presented. As indicated, these

71
are functions of time, rather than constants as is
sometimes assumed. It should further be emphasized that
they are defined for the linear range of behavior.

From the slopes of the linear portions of the curves
in Figures 3.2 and 3.5, fO yielding CO and E(t) are
deduced. These functions are plotted in Figure 3.9. Their
strong time-dependence during the first two or three hours,
the time during which most static tests are performed,
should be noted. The possible error which may result from
assuming these functions to be constants, independent of

time, is evident.

72

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CHAPTER IV

VISCOELASTIC STRESS ANALYSIS

Introduction

 

From the discussion and the graphical presentation
of results in the two preceding chapters, the change with
time of the mechanical and optical properties of the mate-
rials is quite apparent. This viscoelastic behavior is
manifested as creep under constant stress or relaxation
under constant strain. In most discussions, strain and
stress are considered the time—dependent functions in creep
and relaxation tests, respectively. The simplest way to
describe the viscoelastic behavior is then to define the
extension modulus D(t) or one of the shear or bulk compli—
ances or moduli (to be defined) over the whole range of
time and temperature. However, other time—dependent func-
tions such as birefringence may be considered, with an
analogous photoviscoelastic description given in terms of
the stress-optic or strain-optic coefficients in creep or
relaxation. In the following discussion of viscoelastic-
ity, strain is usually considered the basic time—dependent
function in the discussion of the creep tests performed in
this study. Birefringence data was also obtained, and the
isochromatic fringe order is considered the basic

73

74
time-dependent function in a similar discussion of
photoviscoelasticity.

In this chapter some elements of the existing
theories of linear viscoelasticity and photoviscoelasticity
are discussed. Graphs of some "characteristic functions"
defined by these theories are presented. Nonlinear
theories are then discussed and applied to the optical
creep test of CR—39.

Though viscoelasticity and photoviscoelasticity have
much in common, discussions of them are separated in the
following. The similarities and differences are pointed

out where they do not seem obvious.

Aspects of the Problem

 

The problem of viscoelastic stress analysis has
three aspects [ll]: determination of material properties,
interrelations among the various descriptions of visco»
elastic behavior, and the methods of stress analysis. The
first of these, determination of material properties, is
the main concern in this study. The other two aspects
warrant consideration even in a study of this type, since
they determine the form of representation of material prop—
erties most suitable for a particular problem, as well as
define the range of time, stress, temperature, etc. for
which material properties must be determined. The first
two aspects might indeed be considered as one [27]—-namely,

that of specifying the constitutive equations of the

75
material which are necessary for the analytical solution of
boundary-value problems.

Most of the theoretical development of viscoelastic—
ity,and surely most of the problems which have been solved,
assume linear viscoelastic behavior--i.e., the time-
dependent functions are approximately linearly proportional
to the applied constant stresses or strains.l Many photo-
elastic and model materials, including those tested in this
study, are found to be linearly photoviscoelastic or visco-
elastic, provided the applied quantities do not exceed
certain limiting values. The theory of linear viscoelas-
ticity can then be applied to solution of boundary-value
problems involving such materials.

Although there does not exist as yet a systematic or
unified body of theoretical results on linear viscoelastic-
ity comparable to that available in classical elasticity
theory, it may be considered well-established. Work such
as that of Gurtin and Sternberg [27] established a firm
mathematical basis for the theory and gave a clear formula-

tion of the problem.

 

1Since the linear limit stress (in a creep test)
decreases with time, the designation "linearly viscoelas-
tic" is more restrictive than "momentarily linear"--i.e.,
the former implies specification of a time interval
throughout which behavior is linear. Within this time
interval the behavior will be momentarily linear for a
wider range of stress than that for which it is linearly
viscoelastic. Reference to Figure 2.1, page 19, might
clarify this assertion.

 

76

The isothermal stress analysis problem for an
isotropic medium is formulated as follows: Given a body,
occupying a closed region of space in its undeformed state,
subjected to prescribed surface tractions Ti(xk’t) over
part ST of the boundary and to prescribed surface displace-
ments Ui(xk,t) over the remaining part Su of the boundary,
determine the stress and strain distributions throughout
the body. If the body is loaded or displaced at t = 0,

then initial and boundary conditions take the form:

0.. = 8.. = u. = 0 for t 55 0 (4.1)
ij ij i
oij(xk,t)nj(xk) = Ti(xk,t) on ST for t :> 0 (4.2)
ui(xk,t) = Ui(xk,t) on Su for t >1 0, (4.3)
where

ui = displacement component in the direction of the ith
coordinate (i = 1,2,3),

nj = unit outward normal vector on the boundary,

xk = kth coordinate of point.

The equations of motion and the (small) strain—displacement

equations are:

...... x, = (4.4)
ngj + l P <§ 2

011. d)u.
=é l+—"‘l . (4.5)
13 (jx. (5xi
where
Xi = body force component in the direction of the 1th

coordinate,
f) = density of medium.
Finally, the constitutive equations of the viscoelastic

material may be stated in general as:

P(sij) = Q(eij) (4.6a)
P'(Oii) = Q'(8ii), (4.6b)
where
_ l .
sij — Oij --§ 6iijk’ the stress dev1ator,
_ l . .
eij — Sij --§ 6ijgkk’ the strain deViator,
P,P', 0,0' = linear operators with respect to time (to

be further explained).

The solution of the stress analysis problem consists
of solving the system of Equations (4.2)-(4.6). Time vari-
ations, if any, of the traction and displacement boundaries
must be prescribed. This can cause severe complications;
in most solved problems such variations are not considered.
In quasi-static problems the inertia terms in the equation
of motion are negligible; the right-hand side of Equation
(4.4) then becomes zero. Most problems so far solved are

of this type.

78

In the methods of stress analysis mentioned at the
beginning of this section—~i.e., in arriving at a solution
of Equations (4.2)-(4.6)--severa1 possible approaches have
emerged with, of course, a number of variations of each.

In one such approach the governing equations are solved
directly, which is sometimes possible by integrating sepa-
rately with reSpect to the time and space variables. Lee
[33] gavra an example of this in an excellent survey
article. A more common approach applies the Laplace trans-
form to the system of equations, thus removing the time-
dependence, so that the viscoelastic problem becomes an
elastic one in the transformed variables. The so—called
"correspondence principle" is thereby invoked. This method
applies directly only to problems in which the portions ST
and Su do not vary during the loading process [34]. A dis-
cussion of this approach, along with reference to a number
of solved problems, may again be found in the survey
article by Lee [33].

In addition to the mathematical complexities often
encountered in the two methods mentioned above, they
require the expression of material behavior in mathematical
form-~i.e., the viscoelastic operators of Equations (4.6a)
and (4.6b) must be known. Such expressions can easily be
too complicated to handle in a mathematical solution or can
be too simple to adequately describe the behavior of the

material, or both. This is particularly true in dynamics

79
problems where very short-time response is required.
Physically meaningful solutions, nevertheless, have been
obtained either by fitting mathematical functions to
experimental data on material properties or by using such
data directly in numerical solutions with the aid of a
computer [34]. This latter method removes the limitation
to the type of boundary—value problem which can be trans»
formed into an associated elastic one.

Another approach is the completely experimental one.
The importance of such an approach becomes apparent when
the complexity of the viscoelastic stress analysis problem
is considered. As in elasticity, solutions of only those
problems with simple boundary conditions and geometry seem
feasible analytically. When the correspondence principle
is used, the viscoelastic solution indeed depends upon the
availability of the solution to the associated elastic
problem. It appears then that the problem of viscoelastic
stress analysis is generally more complex than, or at best
just as complex as, the elastic one.

One experimental approach for viscoelastic stress
analysis is photoviscoelasticity-—an extension of photo-
elasticity for elastic analyses--a logical extension indeed
in View of the fact that most photoelastic model materials
are inherently viscoelastic. Some aspects of this experi-

mental approach are given in the following sections.

80

Stress, strain, and birefringence in a viscoelastic
material are three uniquely interrelated second-order
tensors. Mindlin [39] derived a stress-strain-optic law in
the form of second-order linear differential operators for
an idealized four-element model, assuming the material to
be incompressible. Mindlin's work and that of others who
followed are reviewed by Daniel [11] and extended to forms
suitable for stress analysis purposes. Daniel gave the
stress-optic law in both differential and integral form.
The experimental approach necessary for calibration of the
material and solution of the stress analysis problem was
outlined and then applied to a dynamics problem.

The following sections give a rudimentary account of
some methods of representation of the viscoelastic and
photoviscoelastic properties of materials. Only a few of
the many possible representations appear, and these are
specialized to the needs of this study—~namely, to the iso-
thermal, quasi~static case. Reference is made to a number
of sources where more general developments are given.

Forms of Representation of

Linear Viscoelastic
Behavior

 

 

 

Among the most comprehensive works in linear visco-
elasticity are those by Leaderman [31], Alfrey [1], Gross
[25], Staverman and Schwarzl [62], Ferry [1?], and the
three-volume one edited by Eirich [16]. Reference to many

recent publications may also be found in Lee's paper [33].

81
Equations (4.6) gave the constitutive equations for
a viscoelastic material in terms of deviatoric and dilata—
tional parts of stress and strain. The corresponding

equations for an isotropic elastic material are:

5.. = 2Ge.. (4.7a)
lJ 1]
0.. = 3K8.., (4.7b)
ii ii
where
G = shear modulus,

K = bulk modulus.

Equations (4.6) may conveniently be written:

P(sij) = 20(eij) (4.8a)
P'(oii) = BQ'(sii), (4.8b)
where P, P', Q, and Q' are again linear operators with

respect to time, identical to those of Equations (4.6)

except for the constant coefficients. Hereafter any refer-

ence to these operators assumes those of Equations (4.8).
In differential—operator form these operators are

expressed as:

m r n r
P = E p & , Q = Z q &
r tr 1: C] tr
r=0 9 r=0

 

 

, (4.9a,b)

82

 

 

m' n'
r r
P' = E p} ‘3 r’ 0' = E q; (3 r’ (4.9c,d)
r=0 (3t r=0 6t

where the coefficients pr, q pg, and q; are constants of

r’
the material. The number of such constants-~i.e., the
values of m, n, m', and n'--necessary to describe a mate—
rial depends upon the viscoelastic behavior of the material
and, of course, upon the accuracy desired. Frequently, the
number required is high enough to make the mathematical
treatment of the equations formidable.

The constants in the differential operators can be
related to an idealized mechanical system composed of
springs and dashpots. These mechanical systems then serve
as pictorial representations of the material behavior. For
such purposes the constants are usually specialized to
correspond to the elements of simple mechanical systems,
such as the Kelvin model (spring and dashpot in parallel),
Maxwell model (spring and dashpot in series), or various
combinations of these. The idea of the representation by
mechanical models can be extended to an arbitrarily large
number, indeed to an infinite number of elements. The
operators are then characterized not by a finite number of
constants but by continuous functions called distribution
functions [62]-—"distribution of retardation times" or
"retardation spectrum" for the continuous Kelvin model used

to describe creep, and "distribution of relaxation times"

83
or "relaxation spectrum" for the continuous Maxwell model
used to describe relaxation.

The spectra give a picture of the springs and dash—
pots of the model which in turn can be thought of as
representing molecular processes. Hence the spectra give a
close representation of the molecular processes occurring
in viscoelasticity, a fact which is important to the
chemist or physicist relating chemical constitution of a
material to its properties. Perhaps this is the best
justification for their use to describe material behavior.

A more general representation is given by visco-
elastic operators in integral form, which result from
considering Boltzmann's superposition principle [1] for
continuous stress or strain histories. Such a super-
position principle is valid for (or equivalently describes)
linear viscoelastic behavior. In integral-operator form,
the deviatoric strain response to any time-varying stress

sififit) is given by:
'.)

t
.. l dsij(T)
Slj(t) :; 'Z J(t-T) T dT, (4.10)

...CD

where J(t), called the "creep compliance" in shear, repre-
sents the strain response to a unit sustained shear stress.

A similar expression for pure dilatation is:

84

1 t ddii(T)
8ii(.t) = '3' E(t-T) —-a-:E--—— CIT, (4.11)

where E(t) is the "bulk compliance."

When the material has no stress or strain history--
i.e., it is initially "dead"-—then the lower limits of the
integrals are replaced by zero.

Corresponding equations may be written in terms of
"relaxation moduli" for the stress response to time—varying
strain. Such moduli, as well as the compliances defined
above, usually characterize the longutime behavior of
materials. Though they could be used to describe dynamic
behavior as well, provided their time variation for suffi-
ciently short times were known, they are usually abandoned
in dynamic studies in favor of complex moduli and
compliances. These may be obtained from the response to
sinusoidal loading of variable frequency. Since only creep
tests were performed in the present study, only the creep
compliances are of value here.

It should be mentioned that all of the moduli and
compliances of an isotropic, iscoelastic medium are
related and can, in principle, be found when only one of
these so~called [62] "characteristic functions" is known
for all times (or frequencies). Gross [25] presented many
transformation formulas for this purpose. Not only are

some of the transformations difficult to carry out, but

85
usually no one characteristic function is known for a
sufficiently wide range of time or frequency to make all
the transformations practicable. Some functions might then
be thought of as complementing others for different time
intervals rather than being equivalent.

The Equations (4.10) and (4.11) for deviatoric and
dilatational strain response to time-varying stress may
further be specialized for the case of uniaxial tension.
An "extension compliance" D(t) can be deduced from the
creep and bulk compliances, since a deviatoric and hydro»
static stress can be superposed to give uniaxial tension.
In integralwoperator form, the axial strain response to

timenvarying uniaxial stress is:

t
s(t) =f D(t-T) Egéll d’L'. (4.12)
0

Note that the lower limit of zero has been employed,
assuming no previous stress history.

[If the loading consists of the sudden application of
stress 00 at time zero which is then maintained constant,

the strain response found from Equation (4.12) is:

E(t) = GOD(t). (4.13)

The extension compliance will be independent of 0 within

0

the range of stress for which the material is linearly

viscoelastic.

86
For relaxation-type tests the stress response to

time-varying axial strain is:

t

- ds(1)

0(t) -;/h E(t-T) —7fi7—— d1, (4.14)
0

where E(t) is the relaxation modulus or extension modulus.
For a suddenly applied (at time zero) strain which is then

maintained constant the stress response is given by:

C(t) = 80E(t). (4.15)

The extension compliance and modulus are formally

t
‘jf E(t-T)D(T)dT = t. (4.16)

For moderate rates of creep an approximate relation

is [11]:

E(t) ’5: W0 (4017)

To Show how the creep and bulk compliances can be
deduced from the extension compliance (and other easily
accessible information) for the special case of sudden
application of constant load at time zero, it is necessary

simply to insert the deviatoric and dilatational parts of

87
the stress and strain tensors for uniaxial tension into the

appropriate operators (4.10) and (4.11). The tensors are:

60 O 0 811 0 0
0 O 0 O 0 s

33

Now from Equation (4.10) the (deviatoric) shear creep

compliance is (see also List of Symbols):

 

 

J(t) - 2eij(t) _ 2ell(t) _ 2811(t) - 822(t) — 833(t).
‘ - " W " _____._._ _ 3
5ij 511 G0
(4.19)
and the bulk compliance from Equation (4.11) is:
3s..(t) 3[s (t) + s (t) + s (t)]
E(t) : “ii.— : ll 22 33 o (4020)
O'.. 0
ii 0

Thus if one of the transverse strains is measured along
with the axial strain, then the other transverse strain can
be computed (assuming isotropy); or equivalently, Poisson‘s
ratio1 n(t) may be obtained. When Poisson‘s ratio is
employed, the above equations become:

2811(t)[1 + u(t)]

J(t) = (4.21)
O'
O

 

 

l o O C O o
"POisson's ratio" is generalized beyond ltS normal

meaning and is here defined simply as the negative ratio of
the lateral to the axial strain. For other definitions see
page 73 of Stuart [63].

88
and

3811(t)[l - 2u(t)]
E(t) = O . (4.22)
O

 

For many polymers Poisson's ratio is found to be
nearly constant with time. Unplasticized high polymers at
room temperature are in their glassy state [64]. It is
known [62] that Poisson's ratio may quite accurately be
assumed constant over a wide range of time for such
materials. Daniel [10] found it to vary little over four
decades of time in a lowmmodulus material with true axial
strains as high as fifteen percent. Equations (4.21) and
(4.22) are even further simplified if the time-dependence
of PoissonVS ratio is neglected. A simple tension test,
yielding the extension compliance, along with the known or
measured value of Poisson“s ratio, then suffices to com-
pletely characterize the linear viscoelastic, mechanical
behavior of an isotropic material.

Before proceeding to the forms of representation of
photoviscoelastic properties, one further point regarding
the mechanical behavior should be noted. In the most
general case of loading, the principal axes of stress do
not coincide with the principal axes of strain in a visco~
elastic medium. This could be expected from a comparison
with plasticity theory, where such coincidence breaks down
unless the directions of principal stresses are constant.

Although few examples of this have been observed in

89
plastics, Frocht and Thomson [23] found such a lack of
coincidence in cellulose nitrate when the loading was such
that the directions of principal stress rotated.

Both stress and strain enter only one set of
equations in the formulation of a viscoelasticity problem
[Equations (4.l)—(4.6)]-—namely, the constitutive
equations. There the stresses and strains are divided into
deviatoric and dilatational parts. Hence it would seem, in
principle, possible to solve analytically the general
viscoelaSoic problem wherein principal axes of stress and
strain do not coincide. The method used to solve the equa-
tions, however carefully chosen, would likely lead to a
lengthy and involved analysis. Experimental methods, one
of which is briefly discussed in the following section,
offer an alternative approach. Even here, a number of
complications arise.

On the other hand, Mindlin [39] showed that if the
stress can be represented as a product of two functions,
one of which depends only on position coordinates (xk) and

the other only on time, i.e.,

oij(xk,.) = Cij(xk)f(t); (4.23)

then the principal axes of stress and of strain coincide.
This condition excludes, for example, moving loads, which

would cause principal stress directions to change, and

 

90
initial stresses, unless these stresses are of the same
type as those represented in Equation (4.23)o
Forms of Representation of

Linear PhotoviECOelastic
BehaviOr

 

 

 

As stated previously, stress, strain, and bire»
fringence are three uniquely interrelated second-order
tensors. Daniel's representation [11] in integral form of
the stress~optic law for viscoelastic materials seems most
practical for stress analysis purposes. Though his work
draws upon that of Mindlin [39] and others, his formulation
seems to be the best suited for inclusion in this study. A
slightly different notation is used in the equations pre-
sented below, reflecting that of Theocaris [64].

If the direction of light propagation is considered

the zwdirection of a cartesian system of axes, then the

“x
I’D
1.-
Q!
(‘1'
A

Wire retardation (birefringence) observed in the usual
applications of tw0wdimensional photoelasticity is the
maximum difference of birefringence in two mutually perpenm

dicu lair di'

6
m

fictions in the xmy plane. These directions are

Gal.

ed the secondary principal optical directions, and the

I 3
...

corresponding values of the index of refraction are called
the secondary principal values. Similar terminology is
used for stresses and strains.

Stresswbirefringence relations may now be written

91

t
. _ d
OXX ..., oyy —f BUR-1:) a? (nxx — nyy) d1. (4.24)
0
where
O 3 o = normal stresses along axes of coordinates,
xx yy
n , n = indices of refraction along coordinate
XX YY

directions,

Bo(t) = time—varying difference in secondary
principal stresses for unit sustained
birefringence;

or as:

t
d
Oxx - ny =‘jf Bo(t‘1)'3? [nO(T) cos 2$n(T)] d1,
0 (4.25)

where

V
U

nO(t) (n - n difference in secondary principal
values of index of refraction
(relative birefringence),

@n(t) 2 angle between secondary principal
optical directions and coordinate
axes.

The secondary principal stress difference (01 — O2)

may be related directly to the relative birefringence by:

6

92

t 2
Cl - 02 = j[. BG(t—T)-%? [nG(T) sin 2¢n(T)] d1
0

l
t 2 .2
+ ~/F BO(t—T)-%? [nO(T) cos 2@n(T) dT] . (4.26)
O

The principal stress directions are available from:

t
BG(t-T)-%? [nG(T) sin 2®n(1)] d1

 

5%

tan 2®G(t) = t a

BO(t—T)-§? [nO(T) cos 2¢n(T)] dT
(4.27)

where @O(t) = angle between secondary principal stress

directions and coordinate axes.
Finally, the shear stress is given by:
t
OXY =-%‘jf BO(t-T) g? [nO sin 2¢n(T)] dT. (4.28)
0

Thus the photoviscoelastic determination of the
secondary stress difference at a point in a viscoelastic
body, as a function of time, requires a continuous record

of birefringence and isoclinic data in addition to

93
knowledge of the stress—optic coefficient B0(t). This is
in contrast to photoelasticity where such a stress differ—
ence is determined by the momentary values of fringe order
and Cc(t).

To separate the principal stresses the shear
difference or oblique incidence methods commonly used in
photoelasticity can be applied [21]. Some modifications
accounting for the difference of @n and $0 may be
necessary. This accomplished, the strain can be determined
using the theory already presented. An alternative proce-
dure for determining strain is to measure it directly,
possibly by employing the moire method. Such an approach
would require no prior knowledge of mechanical properties.

Evaluation of Bo(t), the time-varying difference in
principal stresses for unit sustained birefringence, in
principle, could be determined directly from an optical
relaxation test. But such a test, wherein birefringence is
maintained constant, is very difficult to conduct.
However, if a(t) represents the time-varying birefringence
for a unit sustained difference in secondary principal
stresses (creep test), then d(t) and 80(t) are related as

follows:

t
[ or.(t-1:)BG(T)d1: = t (4.29a)
O

or

94

t
_jr a(T)BO(t-T)dT = t. (4.29b)
O

For moderate rates of creep,

[30m Egg, (4.30)
which permits the determination of Bo(t) by means of a
relatively easy-to—perform creep test such as the one
performed in this study. For a body with symmetrical shape
and loading ¢n(t) can be made zero and Equation (4.26)

becomes:

t
d
01 - 02 =-]F BG(t-T)-a? [nd(r)] dr. (4.31)
O

For a suddenly attained birefringence nO which is main—

tained constant from time zero (optical relaxation test)

Equation (4.31) becomes:1

01 — 02 = Bo(t)n0. (4.32)

The analagous equation for an optical creep test is:

n(t) = a(t)(dl - 02). (4.33)

 

1Comparison with Equation (2.8) reveals that
2fq(t)

E(t) = —-a———.

95
In the present study a uniaxial stress state was

employed and a(t) was determined from
n(t) = a(t)O, (4.34)

which, by use of Equation (4.30), determined Bd(t).

Plots of Bo(t) and D(t)l as functions of time are
shown in Figure 4.1. Since these characteristic functions
are defined for linear optical and mechanical behavior,
they can describe the behavior only for a limited range of
stress and time as determined in the previous chapter.

Some similarities and differences in the treatment
of linear viscoelasticity and linear photoviscoelasticity
may be noted. It has been pointed out that, generally,
mechanical coincidence in a viscoelastic body does not
prevai1--i.e., the principal—stress and principal-strain
axes do not coincide. This is in contrast to elasticity
where they do coincide, at least for isotropic media.
Furthermore, in photoelasticity the principal—optical axes
coincide with the principal-stress axes so that the meaning
of isoclinics is clear. But the question now arises
whether the breakdown in mechanical coincidence is accom-
panied by a breakdown in optical coincidence; or equiva—
lently, what is the meaning of the isoclinic parameters in
the presence of general viscoelastic or plastic behavior.

Do the isoclinic parameters give the directions of the

 

Y—

1Refer to Equation (4.13) for definition of D(t).

96

 

 

 

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97
secondary principal stresses, the secondary principal
strains, or neither? The theory above assumes neither.
Equations for the strain—optic laws similar to the
stress—optic ones above would require the introduction of
another angle $8, the angle between the secondary
principal-strain axes and the coordinate axes. The above

question, rephrased, becomes: is Tn = m or is Qn = T

G, 8,

or is $0 f @n # $8? In the most general case it must be
assumed that the last condition prevails. Theocaris [64]
pointed out, however, that the "glassy" region of visco-
elastic behavior, where deformation is predominantly
elastic, and the "rubbery" region, where the creep rate is
quite low, correspond to quasi-elastic conditions wherein
the axes of secondary principal stress, strain, and bire-
fringence coincide. He observed marked deviations from
coincidence in the "transition" range, where creep rate is
high, in a highly (60%) plasticized epoxy polymer.

Frocht and Thomson [22] found similar deviations in
an American cellulose nitrate but found that after several
hours the secondary principal optical directions coincided
with secondary principal stress directions--i.e., @n = $0.
But in a German cellulose nitrate Frocht and Cheng [19,20]
noted no such time delay. Although mechanical coincidence
broke down when principal stresses underwent rotation,

optical coincidence (@n = $0) continued to exist. _Such

.results are quite remarkable and far—reaching in regard to

98
the applications of photoviscoelasticity or photo-
plasticity. It appears that time spent in material

selection might be profitably invested.

Nonlinear Viscoelasticity

 

The theories of linear viscoelasticity and photo—
viscoelasticity, which were briefly introduced in preceding
sections, describe the mechanical and optical behavior for
only limited ranges of stress and time. Even within these
ranges some anomalies are probably present but cannot
easily be detected. Thus the descriptions "linear" and
"nonlinear" are somewhat arbitrary. The nonlinear region
might be defined as that region where deviations from
linearity can definitely be detected or cannot be over—
looked. That such a region exists for some of the mate—
rials tested in this study becomes quite evident from
inspection of the plots of Chapter III or Appendix III.
Though considerable effort has gone into formulating
theories of nonlinear viscoelasticity, the subject is not
yet well-developed nor is there agreement on the most
satisfactory approach. This is to be expected in view of
the relatively recent beginning of any concentrated
research efforts, the many types and causes of deviations
from linearity, and the mathematical complexity of the
subject relative to its linear counterpart.

One factor limiting development is the lack of

accurate experimental data. As late as 1965 Theocaris [64]

99
wrote, concerning nonlinear behavior of polymers in their
glassy state where strains are relatively small: "Experi—
mental evidence is rather sparse and the only significant
contributions which exist concern the nonlinear behavior in
the rubbery zone." This study hopefully provides some
evidence of the type mentioned.

Most attempts to describe nonlinear behavior fall
into one of the three categories: (1) empirical power laws
(see Marin and co-workers [38]), (2) postulated behavior in
terms of models (with nonlinear springs and modified dash-
pots), and (3) integral representation in the form of a
modified superposition principle. All three of these have
their roots in corresponding representations of linear
behavior. Of the three, the last one seems most natural
and most successful; further discussion will be limited to
it.

Green, Rivlin, and Spencer [24] established the
basis for the nonlinear theory in a series of highly mathe—
matical articles. Quoting Theocaris again, ". . . it seems
that only simple cases may be satisfactorily considered by
this theory." The theory has been successfully applied by
a number of investigators to such simple cases. Ward and
Onat [69] presented a clear and concise summary of the
theory as it applies to the behavior under simple stress
conditions. Some elements of their presentation are

reflected in the one below. Lockett [35] considered a more

100

general case and showed how the material functions
(properties) for a general triaxial loading can be deter-
mined experimentally for a material whose response can be
adequately expressed by a constitutive equation involving
integrals of the first, second, and third orders. The
constitutive equation is expressed in matrix form; the
number of tests shown necessary for completely determining
the material functions appears to be quite impracticable.

Onaran and Findley [46] specialized Lockett's
results to simple stress states and presented examples to
show the effectiveness of a multiple integral functional
relationship for describing the response of polyvinyl
chloride. The same approach is used at the end of this
chapter.

The assumption that the elongation of a simple
tension specimen at time t depends on all the previous

values of the rate of loading can be expressed mathemati-

cally by:
t
E(t) =6? 3531431] , (4.35)
T=-w
where
E(t) = unit strain of specimen at time t,
0(1) = time-dependent uniaxial stress,
F = a functional.

Should the functional F be linear and continuous,

this response can be represented by Equation (4.12).

101
It has been shown (see [69] for references) that if
F is continuous and nonlinear it may be represented to any

desired degree of accuracy by:

t dO(Tl)
E(t) = Dl(t-Tl) ? dTl
l

t t dG(Tl) do(r2)
+ D2(t-Tl,t—T2) T7175?— d’Eld’L'Z + o o o

t t dO(Tl)
+ o o o Dn(t-Tl, o o o ,t"Tn) W- o o o

do(1n)
—a-,-E—— dTl o o o dTn. (4.36)

n

Spencer and Rivlin [61] showed that for an initially
isotropic material the number of such integrals required
does not exceed five. For most applications thus far
attempted two or three have proven sufficient for accept—
able accuracy.

If the material has no load history prior to time
t = O, the lower limit of the integrals can be replaced by
zero. Furthermore, if the stress-—say oi--is suddenly
applied at time zero and subsequently maintained constant

(creep test), then Equation (4.36) becomes:

102

2
5(t,oi) _ Dl(t)oi + D2(t,t)oi + . . .

+ Dn(t, . . . ,t)o§. (4.37)

The first term is linear in stress; subsequent terms
might be regarded as corrections to the linear one to
account for nonlinear behavior.

A number of investigators, including Onaran and
Findley [46], have found three terms of Equation (4.37)
sufficient to adequately describe the behavior of several
polymers for a reasonably wide range of stress and time.

Then:

_ 2 3
8(t,Ui) — Dl(t)o’i + D2(t,t)0'i + D3(t,t,t)di. (4.38)

Hence loading programs having three independent
values of Oi provide three equations for determining the
functions Dl(t), D2(t,t), and D3(t,t,t). The function
Dl(t) is determined completely, while the other two are
determined only where their arguments take on equal values.
Though Dl(t) is determined completely, it should be noted
[35] that it depends on each of the constants Oi and there-
fore will not be identical with the function obtained in
the linear theory from a single value of oi.

If experimental values of E(tl’oi) for the three

chosen values of Oi are used in Equation (4.38), then

103
values for D1, D2, and D3 are obtained only for t1, the
time represented by the data. When such values are
computed at enough different times, the form of the func-
tions Dl(t), D2(t,t), and D3(t,t,t) can be determined.

Alternatively, if analytical curves can be fitted to
the data for the three chosen values of oi, then analytical
expressions can be used in Equation (4.38) to yield the
functions Dl(t), etc. directly. Though this approach is
somewhat empirical, it extends the purely empirical
approach by providing the functions D1(t), 02(t,t), and
D3(t,t,t), which are the kernels of the more general
"superposition" integrals in Equation (4.36).

The problem in this latter approach is to find the
analytical expression which satisfactorily fits the data
for the three chosen levels of stress. The accuracy of
such a fit depends upon the range of stress and time
employed, as well as upon the form of the expression and
upon the behavior of the material itself-—e.g., upon the
"smoothness" of the transition from the linear to the
nonlinear region.

Onaran and Findley [46] have found an expression of

the form

5(t) = so + gth, (4.39)

where

104

8

O a curve-fit constant,

g,h = constants (to be explained),
to yield satisfactory results for several polymers. After
rearranging Equation (4.39) and taking the logarithm of

both sides, it becomes:
log (8 - 80) = log g + h log t. (4.40)

Hence on a full logarithmic plot of (8 - 80) vs. t, h is
the slope of the line at time t and g is an intercept. The
constants 80 may be chosen to make the curve a straight
line of best fit, thereby making h a constant. When these
constants are determined for three independent stress
levels oi, Equation (4.39) can be used with Equation (4.38)
to determine the functions (kernels) Dl(t), D2(t,t), and

D3(t,t,t).

Nonlinear Photoviscoelasticity

 

Virtually all work on nonlinear behavior reported in
the literature deals with the mechanical properties of the
materials tested. Similarity of the strain—stress and
birefringence—stress plots of Chapter III seems to imply
that any analytical description of the mechanical behavior
would parallel that of the optical behavior——i.e., bire—
fringence is very closely related to strain throughout the
linear and into the nonlinear range. Observing this and
considering birefringence data of this study more accurate

than the strain data, it was decided to apply the scheme

105
outlined above to determine similar kernels for an
integral-type representation of the photoviscoelastic
properties of the material tested.
Since axial tension tests were used in this study,
questions regarding coincidence of optical and mechanical
axes do not enter. In all of the equations of the previous

section n(t) might simply be substituted for E(t) where:

n(t) = nl(t) - n2(t), (4.41)

so that n(t) is the difference of secondary principal
indices of refraction--i.e., the relative birefringence or
simply the fringe value.

Equation (4.38) then implies:
n(t o ) = D (t)o + D (t t)o2 + D (t t t)o3 (4 42)
’ i 1 i 2 ’ i 3 ’ ’ i' °

Equation (4.39) suggests:

n(t) = no + gth; (4.43)
and Equation (4.40) leads to:
log (n - no) = log g + h log t. (4.44)

The largest stress plotted in the optical test of
CR-39 was 2,858 psi. The stress levels 600 psi, 1,600 psi,
and 2,600 psi were chosen for the three independent values

of stress for use in Equation (4.42). The time range

106
available was one minute to 240 hours (14,400 min). The
range one minute to 104 minutes was chosen for representa-
tion because of its convenience on the logarithmic time
scale.

The method of determining nO in Equation (4.44) to
cause a "best fit" straight line is not apparent in the
literature surveyed. The following scheme was therefore
devised. Using data from the curves of birefringence vs.
stress at t = constant (Chapter III), a plot of n vs. log
was made1 for the three stress levels mentioned above.
From this plot values of n were read at one minute, 102
minutes, and 104 minutes at each of the three stress
levels. With these three values of n, at a given stress,
nO was then computed to force a straight line on the log
(n - no) vs. log t plot through the three points. Figure
4.2 shows this plot. Then 9 was evaluated as the ordinate
at one minute and h as the slope of the line for each
stress2 in accordance with Equation (4.44). This yielded
the desired expressions for the left side of Equation
(4.42) which may then be written in matrix form, using the

values found, as:

 

1The plot of n vs. t at o = constant in Chapter III
does not yield sufficient accuracy at 102 minutes.

2In all publications reviewed the slopes were found
to be the same for all values of stress. The variation of
h in the present case is evident in the exponents on t in
Equation (4.45).

107

 

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108

'— —H r— "r—‘H

 

 

 

 

 

 

2.10 + .28t'1291 600 6002 6003 01
5.86 + .48t'1895 = 1,600 1,6002 1,6003 02
9.77 + .76t’2386 2,600 2,6002 2,5003 D3 .
d "' d _"‘(4.45)

When the first matrix on the right is inverted, the
desired expressions for the kernels are found.

The kernels are shown as functions of time in
Figure 4.3. It should be noted that D2(t,t) is negative
throughout.

Greater insight into the contribution of each kernel
to the total birefringence is afforded by Figure 4.4, where
the three terms on the right side of Equation (4.42) are
shown separately, along with their sum and the experimen-
tally determined curve of birefringence vs. time at 2,600
psi. The nonlinearity is seen to be quite pronounced; and
the deviation between the linear viscoelastic contribution,
given by the Dl(t) kernel, and the actual behavior is seen
to increase with increasing time. The calculated curve
with all three kernels gives excellent agreement with the
experimental results. Recall, however, that the stress of
2,600 psi was used in determining the kernels.

The accuracy of the calculated values of bire—
fringence for a wide range of stress can be judged from
Figure 4.5, where the curve of n vs. t (o = const) from
Chapter III is shown, along with calculated values of n.

Agreement is seen to be quite good (throughout the time

 

109

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l12

range) for the stress range of 600 psi to 2,600 psi but
poor for the stress of 2,858 psi. This is not surprising,
since the stresses of 600, 1,600, and 2,600 psi were used
to determine the kernels. Furthermore, the nonlinearity
becomes very pronounced at stresses above 2,600 psi. This
may be observed either in Figure 4.5 or in Figure 3.2 of
Chapter III.

Closer agreement between experimental and calculated
results for the higher stress level of 2,858 psi, of
course, is possible by using this value of stress as one of
the three for calculating the kernels. However, this would
probably cause greater discrepancies at the lower levels of
stress.

Modifying Equation (4.42) to yield closer agreement
with experimental results is also possible. Onaran and

Findley suggested the following form:

2 3
n(t,oi) = Dl(t)oi + D2(t,t)(o§) + D3(t,t,t)(oi) ,

(4.46)
where
. = —
Oi Oi all when Oi2> Ott’
t =
Ci 0 when Oi'< GEL.

As previously pointed out, the kernel Dl(t) represents the
linear contribution to the response, and the other two
represent the deviations from linearity. Since this devi-
ation is more pronounced at higher levels of stress, the

above modification can be expected to improve considerably

 

113
the representation at these higher levels without loss of
accuracy at lower levels, where behavior is essentially
linear.

Since the linear limit stress 02$ changes signifi-
cantly with time, some question arises concerning the
proper value to use in Equation (4.46). It seems that the
lowest value, corresponding to the longest time to be

considered in the representation, should be used.

This modified theory was not used for the CR-39

 

reported here, because the former representation seemed
adequate for the stress range of 600 psi to 2,600 psi. For
the time range of 240 hours, 2,600 psi is dangerously near
the fracture strength of the material. Although it carried
a stress of 3,085 psi for the full 240 hours in the optical
test, it fractured in less than 167 hours at a stress of
2,858 in the mechanical test. The latter specimen was
probably damaged while applying the moire grid, though such
damage was not apparent. In earlier optical tests the
material fractured in less than 47 hours at 3,700 psi and
in less than 2 hours at 4,170 psi.

Limitations of Results
Presented

 

 

All results given above are based on constant
uniaxial stress sustained for 240 hours. Hence no informa-
tion on the behavior of the material in recovery or upon

unloading is given, even though such data was collected.

114
Furthermore, this study makes no attempt to determine
whether the stress—optic law for more general stress states
differs materially from the one for uniaxial stress. On
the basis of some published results, significant differ-
ences are not expected even in the nonlinear range. For
example, in cellulose nitrate Frocht and Thomson [23] found
that for equal values of principal stress difference
essentially the same birefringence was obtained under a
biaxial state of stress as under a uniaxial state, although
strains of nearly twenty percent were encountered. As
pointed out previously, a thorough investigation of
behavior under combined stress states extending into the

nonlinear region requires an exhaustive testing program.

CHAPTER V
VARIATION OF OPTICAL PROPERTIES WITH WAVE LENGTH

Investigatioanerformed

All results presented in previous chapters involving
the relative retardation or birefringent properties of
CR—39 were based upon one wave length of radiation-~namely,
410 nm. The photoelastic coefficients, such as CO and f0,
in the linear range for this one wave length were deter-
mined as functions of time. The extent of the linear
) ) —-was likewise deter—

range—-i.e., or

1.0 n(OM, 5.0

mined as a function of time for one wave length.

n(ozz

Most optical material properties depend to some
degree upon wave length of radiation. An extreme example
is the high transmission of some wave lengths and complete
blocking out of others, a phenomenon useful in the produc-
tion of filters and in x-ray techniques, to name only two
familiar cases. The birefringent properties are likewise
known to depend upon wave length. This dependence is known
as dispersion of birefringence. Since a number of wave
lengths, ranging from 410 nm to 800 nm, were employed in
this investigation, data on this phenomenon was obtained.
Further discussion and some data on dispersion of bire-
fringence are given in the following section.

115

116

In view of the dependence of birefringence on wave
length, it seems logical to question whether the linear
limit stress at a given deviation from linearity-~say
n(dzz)l.0-'might also depend upon wave length. This was,
in fact, the primary purpose of using several wave lengths.
Although the results of this investigation are not conclu-
sive in answering the question, some useful results were

obtained and are presented in the last section of this

chapter.

Dispersion of Birefringence

The time-dependence of the photoelastic coefficients
in Equations (2.7)-(2.9) was previously discussed. Disper-
sion of birefringence implies that the coefficient CO is
not independent of wave length, nor is the coefficient fO
directly proportional to wave length. It is equivalent to
say that the fringe order n is not inversely proportional
to wave length.

Dispersion of birefringence may be regarded as a
source of error, a nuisance, or an asset, depending upon
its recognition and utilization. If several wave lengths
are employed in a photoelastic investigation and the dis—
persion of birefringence ignored, significant errors may
result even though a relatively narrow range of wave
lengths is employed. Such errors, of course, can be
avoided by careful calibration tests employing the same

wave lengths used in the investigation, resulting then in

117
some inconvenience. For at least one material, dispersion
of birefringence was successfully used to measure the
degree of plastic deformation in photoplasticity investiga—
tions [42]. In this case dispersion might well be
considered an asset.

Several types of measurements and descriptions of
dispersion of birefringence have been employed. Coker and
Filon [6] reported the results of their many investigations
on glass and referenced the work of several other investi-
gators. Though their methods of measurement varied consid-
erably, results obtained were sufficiently consistent that
a description of dispersion of birefringence was possible.
For many glasses they found that the variation with wave
length of the stress-optic coefficient CO was adequately

described by:

 

C = , (5.1)

where CO and 10 are constants for a given glass. The
authors pointed out that "the formula is not to be regarded
as of universal application" and that even some glasses
must be considered outside the range of applicability of
the formula.

Monch [41] expressed dispersion of birefringence as

a percentage by the equation:

118

(n1)Na - (n1)

Hg
D = . 100, (5.2)
(n1)Na

 

where (n1)Na and (n1)Hg are the products of fringe order
and wave length for yellow sodium (1 = 589.3 nm) and blue
mercury (l = 435.8 nm) radiation, respectively. In later
work Monch and Loreck [42] substituted red light (A =

655 nm) for the yellow sodium above, resulting in greater
values of D and therefore greater accuracy. Although the
above description with either value served the desired
purpose-—namely, representing the degree of plastic defor—
mation in celluloid——the arbitrariness of the choice of
wave lengths employed should be noted. No insight is given
into the fundamental nature of dispersion of birefringence
at any wave lengths other than those used in the descrip—
tion, covering a rather narrow range.

The shortcomings of previous descriptions as general
measures of dispersion of birefringence were discussed by
Pindera and Cloud [49]. A new description, called
"Normalized Dispersion of Birefringence," was developed
which removed most of the shortcomings of those previously
presented. Perhaps most significant, it allows dispersion
of birefringence to be determined as a continuous function
of wave length. The procedure outlined below follows

closely that given by the authors cited. Some

119
modifications, reflecting mostly personal preference, are
introduced in the following.

In the creep test performed on CR-39 a sequence of
pictures at various wave lengths was taken, beginning at
100 hours after the constant load was applied. Since the
entire sequence was obtained in about six minutes, creep
effects were negligible. From these pictures data was
obtained for the plot of fringe order vs. length along the
tapered specimen. Since the stress was known as a contin-
uous function of the length, fringe order could then be
plotted as a function of stress at each of the wave
lengths. This plot is shown in Figure 5.1 for eight wave
lengths.1

Then at o = 1,200 psi (within the linear range of
behavior) the fringe order n was read for each wave length.
These values and the "relative retardation" R (R = ml) were

tabulated. In the linear range these values are, of

course, linear functions of stress. The ratios

_ R(l)
r -rn'57’ (5'3)
where 10 = 546 nm is an arbitrarily chosen "reference" wave

length, were then computed and tabulated.

 

1Pictures at 500 nm and 700 nm exhibited irregular
intensity fringes and therefore were not used.

120

.mmlmu mom ax. . . . .mxnax n K .mmwnpm
.m> c umpuo mosanm UHDDEOMSUOmHIIH.m .mfim
:2: b

 

 

OO .vd ooo.~ Dow;

00 N._ 00m 00¢

- fi

1 a q

 

.536 no ”.09 1 .mm . mo

4\\\,///, x
/ , / .5. com ...
/ , .52: L
/ ////| gE.ommuA .
/ / .5: 00m u A

.6: Fun "A
/ ...... mg "4
.E: wnv "A

.6: 0;. «A

d d d d d

 

 

obugxg

Q

9

u ‘10910

t.

m.

121
The ratios of Equation (5.3) were plotted as a
function of wave length. Pindera and Cloud called this a
"Normalized Retardation Curve." The rate of change of
normalized retardation with respect to wave length was then
defined as the "Normalized Dispersion of Birefringence" and

denoted by D Thus

x.

_ dr
DX _ 8")?" (5.4)

Both r and D as defined by Equation (5.3) and

1’
Equation (5.4), respectively, are independent of stress in
the linear range but may well depend upon stress in the
nonlinear range. This might be expected on the basis of
the previously mentioned results for celluloid. In fact,
the dependence of r on stress could seemingly be an
indication of nonlinearity.

To investigate this possibility for CR—39,
normalized retardation curves at one level of stress in the
linear range (1,200 psi) and several levels in the non—
linear range were plotted. Figure 5.2 shows a reduction of
such a plot in which only two levels of stress in the non—
linear range are included. Some variation with stress is
evident.

The curve for the stress in the linear range is
Similar in many respects to the one given by Cloud [4] for

CR—39. Cloud found no systematic variations with uniaxial

Stress: variations were random. He analyzed his results

122

.mmlmU MOM mm>HSU COHpMUMMQOH UONHHMEMOZIIN.m

confirm.

 

 

 

 

 

 

$0.0

123
statistically and showed a range of r values at each wave
length employed.

Results of this study are subject to significant
error because of the sensitivity of r to wave length and
the lack of precise determination of the wave lengths used.
Manufacturer's specifications of filters employed were
checked using a hand spectroscope (Zeiss) and the values
were accepted.

The slope of the normalized retardation curve in the
linear range of stress was evaluated at several points.
From these slopes the normalized dispersion of birefring—
ence curve, Figure 5.3, was plotted. Again some similari-
ties to the corresponding curve obtained by Cloud for CR-39
are evident, as well as some striking differences. The
dispersion found by Cloud was negative throughout the range
of wave lengths employed at 66 hours after the load was
applied. Although the increased time of 100 hours used in
this study could account in part for the differences, it
could not be solely responsible for them. Further inves-
tigations seem necessary to reconcile the differences.

In an attempt to correlate dispersion of birefring-
ence phenomena to the degree of nonlinearity of birefring-
ence, a plot of normalized retardation vs. stress was
constructed for each wave length. Readings were taken at
several values of stress in the nonlinear range. However,

since no conclusions could be drawn from the plot, it is

124

.mmlmu mom DUCDWCHMMDHHQ mo coamumdmflo pmNHHMEMOZIIm.m

cowfib

 

M mmlmu

.3

O
O
H

II

00m

 

 

 

 

 

 

 

N+

125
not shown. Figure 5.2 provides a glimpse of the difficulty
encountered. The curves for 2,000 psi and for 2,800 psi
are hardly distinguishable from each other, despite wide
differences in the degree of nonlinearity.

It must be concluded that the normalized retardation
and hence the normalized dispersion of birefringence are
not effective measures of the nonlinearity of CR-39, even
though they cannot be assumed independent of stress.

The magnitude of the error committed in assuming
that photoelastic coefficients are independent of wave
length (in the linear range) can be estimated from Figure
5.2. The possible error in the visible spectrum (390 nm to
770 nm) for CR-39 appears to be about seven percent.

Variation of Linear Limit
Stress with Wave Length

 

 

Figure 5.1 readily yields data on the dependence of
the linear limit stress on wave lengths. This is presented

)

) and

in Figure 5.4, where 1.0 n(ozg

are given for

n(ou 5.0

each wave length.

Considerable variation of the linear limit stress

values may be observed. Values of ) are within

mm” 1.0

about ten percent of a "mean" value of 1,355 psi; values of
n(dzz)5.0 are Within seven percent of 1,720 p51. These
variations seem quite random, however, perhaps indicating

only the magnitude of experimental error rather than any

126

 

.mmtmu Mom mummmuwm uHEHH HDQCHH

c
mcflsogm . A.

.m> s umouo mmcflum UHmeOHnuolelw.m

00 ¢.N 000.N 000..

W . 4 . . A . . . . .

.536 "u “.09 1 .3 (mo

/ / / .5: one ...A
/ // .8: com n K
////.E: @VD "K

o o o nNKnHJA

:2: 6
com;

q q u

d 4

 

Seen»

.6: ppm "A

.E: wnv u A

.52.. O_¢ "K

n A .mwmhum

.mflm

fi

, . ...... com...

A

com

 

 

‘0

(w
u ‘ JOpJO Ghana

9

¢.

w.

m.

127
dependence on wave length. Such dependence cannot be
detected in the results of this study.

Causes and consequences of the irregularities of
linear limit stress values are further discussed in the
next chapter. It might here be mentioned that some irregu-
larities must be expected. Limiting-value types of
material properties (such as proportional limit, endurance
limit, or ultimate strength) are not easily determined.
Usually a large number of specimens is tested and the data
is subjected to a statistical analysis. Precise values of

such properties cannot be obtained from a single specimen.

CHAPTER VI

DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS

FOR FURTHER RESEARCH

Achievement of Objective

 

The primary objective of this study——to investigate
the effect of time upon the birefringent and mechanical
properties of several plastics--was achieved. The iso-
thermal, quasi-static behavior under constant, uniaxial
stress conditions was examined quite exhaustively--for
approximately five decades of logarithmic time, and for a
range of stress sufficiently broad to cover not only the
linear region but also much of the nonlinear region for
those materials exhibiting nonlinear behavior.

Data is presented graphically for all materials.
Certain characteristic functions defined by viscoelastic
and photoviscoelastic theories are determined for CR—39 in
both the linear and the nonlinear regions. Such charac—
teristic functions can be deduced from data presented in
Appendix III for the other materials tested.

The linear limit stresses--i.e., the stresses
limiting the momentarily linear range of behavior—-were
determined as functions of time for the response to stress
of both strain and birefringence. The linear limit

128

129
stresses at both one percent and five percent deviation
from linearity were determined. For CR—39 these stresses
are plotted as functions of time in Chapter III. They are
shown on birefringence-stress and strain—stress plots in
Appendix III for the other materials. Further discussion
of specific properties of each material appears later in
this chapter.

A number of wave lengths of radiation were employed
in the birefringence investigations, providing data for the
study of two phenomena: dispersion of birefringence and
variation of linear limit stresses with wave length. Such

data was analyzed only for CR—39.

Experimental Techniques

 

Tapered models were successfully employed to obtain
optical and mechanical data essential to this study. Few
references can be found to the use of such models by other
investigators. Their use in conjunction with the moire
method for collecting mechanical strain data is believed to
be without precedent.

Tapered models offer the advantage over most
commonly used calibration models of allowing measurement of
the response to stress over a broad range of stress. In
principle, the response can be measured as a continuous
function of stress within the existing range. This avoids
the necessity of testing a large number of models or of

using repeated or step loadings (which would be necessary

130
for prismatic models) to obtain data such as that required
by this study. Because stresses near the ultimate strength
were reached, neither repeated nor step loadings would have
been appropriate. The use of tapered models thus made
feasible a very extensive investigation into the effects of
time on the materials tested. More general use of such
models for studies of material behavior is recommended.

Although the models and techniques employed yielded
results acceptably accurate for this study, some improve-
ment in models and techniques is possible. The equipment
available for this study would have allowed longer models.
Somewhat less than a lO—inch—long gage portion of both the
photoelastic and the mechanical specimens was actually
useful because of the interference produced by the load in
the stress distribution near the ends. The fields of both
the polariscope and the optical system employed in observ—
ing moire fringes were thirteen inches in diameter. The
use of longer models would permit a slightly more accurate
determination of stresses and would reduce the significance
of errors in determining fringe locations. However, this
improvement would be very small in the present case.

A small improvement might also result from using
higher—density line arrays in the moire analysis. A suffi—
Ci‘entnumber of points was available for plotting very
SI“Both curves of fringe order vs. length. But higher—

dearlsity arrays would yield not only a larger number but

131
also narrower fringes, permitting more accurate fringe
location determinations. The same advantage could be
gained by using an optical sensing device in conjunction
with an x—y recorder, as described in Appendix II.

Improvement in technique is needed chiefly in slope
determinations of the moire fringe order vs. length curves.
Better results might be obtained by use of an analytical
(numerical) approach wherein smooth curves are fitted to
the experimental data points and the slopes of these curves
are evaluated numerically at desired points to yield the
desired strains. These strains could then be related to
stresses at the points where strains are known and stress-
strain curves could be plotted. Birefringence-stress
curves could be plotted in much the same way. From these
curves linear limit stresses for any chosen deviation from
linearity could be evaluated.

It seems that the most reliable check of the
over-all accuracy of such an ambitious numerical program
would be a comparison with results obtained by the proce—
dures followed in this study. The proposed numerical
approach might conceivably reduce the irregularity of data
point locations on some plots, such as those for strain vs.
stress or linear limit stress vs. time. The effects of
such irregularity were minimized by the graphical tech—
niques employed in this study. It was felt that the

increase in accuracy, if any, which might have resulted

132
from using a numerical approach rather than a graphical
one, could not justify the expenditure of time required by
both the computer and the programmer to pursue such an
approach.

The results of the study of dispersion of bire-
fringence in CR-39 are not conclusive, even though they
compare quite favorably with some results previously pub—
lished by other authors. The main source of error is
believed to be in the determination of the wave lengths-—a
very crucial determination for such studies. A hand—held
spectroscope was used to check the manufacturer's specifi-
cations of the filters used. Although no deviations from
the Specifications were detected, the sensitivity of the
spectroscope readings (about i 2 nm) was not considered
adequate for the purpose at hand.

Some variation with stress of the normalized
retardation curve was detected. The variation was not con-
sistent enough, however, to indicate a definite relation-
ship between normalized retardation and stress, or between
normalized retardation and degree of nonlinearity in the
birefringence response to stress.

No dependence upon wave length of the linear limit
stresses could be detected. There is probably no good
reason to expect such dependence, at least within the
limited range of wave lengths employed. However, this part

of the study yielded valuable information regarding the

133
magnitude of possible error in all previous determinations
)

of linear limit stresses. Apparently the n( values

022 1.0
found for CR-39 might be in error by as much as ten
percent. But this does not mean that a serious error would
result from assuming linearity to extend to a stress level
that is ten percent higher than the actual value of
n(ozz)1.0° It simply means that the deviation from
linearity of the response (birefringence or strain) might
be, say, two percent rather than one percent.

Hence the errors which may be present in the
reported values of the linear limit stresses do not
seriously limit the usefulness of the results of this
study. No serious errors will result from assuming the
behavior of the materials to be linear up to the linear
limit stresses reported, even if these values are not pre-
cisely determined. Serious errors can result, however,
from assuming that a material such as CR—39 exhibits linear

behavior up to the ultimate strength. Hopefully this point

has been made eminently clear by this study.

Material Property Descriptions

 

One fundamental aspect of the forms of material
property descriptions and of their interpretation deserves
further discussion. Suitable viscoelastic and photovisco-
elastic forms of representation were discussed in Chapter
IV. In Chapter II it was implied that the characteristic

functions describing the mechanical properties might be

134
thought of as time-dependent elastic coefficients and might
be used in a modified, time-dependent Hooke's law. The
strain response to one-dimensional constant stress was

given as:

E(t) = m. (6.1)

The quantity E(t) was called a relaxation modulus, but the
resemblance to a time—dependent Young's modulus was pointed
out.

Similarly, the commonly used birefringence-stress

relations were stated as:

n = 42%— (C51 - 02) (6.2)
O
and
Cod
It was shown that the coefficients f0 and CO
actually vary with time. Again in this case it was implied

that "momentary" values of these coefficients, as deter-
mined by one—dimensional tests, can be used in the more
general stress situations represented by Equations (6.2)
and (6.3). Thus a "momentary—photoelastic" condition of
the material is implied, although the materials tested are
admittedly photoviscoelastic.

The accuracy of many previously conducted photo—

elastic investigations, using materials such as those

 

135

tested in this study, indicates that the assumption of
momentary "photoelasticity" is valid. This furthermore
suggests that the assumption of momentary elasticity is
valid in the case of mechanical properties. In this study
no serious effort was directed toward establishing a firm
theoretical basis for such assumptions, nor was an
intensive search made of the literature for possible
establishment by other investigators. Some ideas or first
impressions are believed to be worth stating and are
therefore given below.

In the discussion of viscoelasticity (Chapter IV)
the relationship between the relaxation modulus E(t) and

the extension compliance D(t) was given as:

t
J/~ E(t—T)D(T)dT = t. (6.4)
0

But for "moderate rates of cree " an a roximate relation
PP

is:
E(t) ; DTET’ (6.5)

Then the one-dimensional strain response to stress is given
by Equation (6.1). But this equation assumes a modified
form of "Hooke's law"; the relaxation modulus E(t) is con-
sidered simply to be a time-dependent Young's modulus.

Hence the accuracy of the approximation involved in

136
Equation (6.5) is believed to be closely (if not directly)
related to the momentary-elastic assumption.
Some insight into the problem might be gained by
approaching it from another point of view. Differentiating
with respect to time the one-dimensional Hooke's law

equation

Q
ll

Es, (6.6)

we obtain:

Es + Es, (6.7)

Q
II

where the dots denote time derivations and E is Young's

modulus. Now if the behavior is truly elastic, E must be

zero, since E is a constant. Then Equation (6.7) becomes:

0

6: E8. (6.8)

For a momentary-elastic material the sequel to

Equation (6.8) must be:

6:. D(t)§:. (6.9)
But by starting with a time—dependent Hooke's law

0 = E(t)8, (6.10)

we obtain by differentiation:

6: E(t)s + D(t)é. (6.11)

For Equation (6.11) to be identical to Equation (6.9) we

137

must have:

E(t) E 0. (6.12)

Thus if the time-dependent Hooke's law assumption is to

yield satisfactory results, the condition
E(t) 3'0 (6.13)

must prevail. This condition is implied by the "moderate
rates of creep" condition leading to Equation (6.5).
Whether or not the two are more rigorously and fundamen—
tally related, both are believed to be applicable to the
behavior of the materials tested in this study.

Properties of Particular
Materials

 

 

Polyester Resin CR-39.—-The properties of this

 

material were discussed from many viewpoints in Chapters
III-V. It was selected for testing and for extensive
discussion because it is probably the most widely used
photoelastic material in this country. It has many desir-
able properties: It is clear and very smooth-surfaced,
thus possessing excellent transmission properties and
yielding very distinct, regular fringes. Furthermore, it
is in an intermediate range of optical sensitivity and
exhibits linear behavior throughout a reasonably wide
range. It is inexpensive and readily available in sheets
of various sizes and thicknesses. It will undoubtedly

continue to be widely used.

138

However, this material must be carefully calibrated,
not only for the effects of time upon the photoelastic
coefficients and the mechanical characteristic functions
but also for the extent of the linear range of behavior.
This is particularly important because CR-39 is available
from a number of suppliers and its properties are known to
vary with the manufacturer.

The results of this study show that the linear limit
stresses vary greatly with time after loading. Further»
more, within a few minutes after load is applied the linear
limit stresses are much lower than the ultimate strength.
This makes an analysis based on the assumption of linearity
up to near-failure vulnerable to serious error, since large
deviations from linearity are clearly possible.

Polyester Resin PS-l.=~Although this material is

 

intended primarily for photoelastic coatings, its proper~
ties make it very useful for models also. The results
obtained from the tests on this material are presented
graphically in Figures A3.1-A3.5 of Appendix III.

The material is seen to exhibit a distinct linear
range of behavior, both mechanical and optical. The
change of linear limit stress with time is small. The

values of n(d ) are lower than the corresponding values

RE 1.0

for strain 5(Gflz)l.0°

reported for CR—39.

This is the same result herein

139

Deviations from linearity are relatively "slow":
that is to say, errors would be small in assuming linearity
to extend well beyond the linear limit stresses reported.
Furthermore, the birefringence-strain plot (Figure A3.5) is
very nearly linear up to the highest strains encountered,
about two percent. This is a very important property in
regard to the material's potential as a photoelastic
coating.

Figures A3.2 and A3.4 show that time effects are
extremely small after the load has been applied for three
or four hours——i.e., the creep rate is very low.

The material is very nearly clear; some roughness of
the surface detracts from its otherwise good transmission
characteristics. It is very sensitive optically and is
easily machined. It was found to yield (not fracture) at a
stress above 9,000 psi. Therefore, it may be a useful
material for photoplasticity studies. Its lack of brittle—
ness reduces the hazards normally associated with stress
concentrations.

Over-all, PS-l is considered to be an excellent
photoelastic model and coating material.

Epoxy Resin PS-2.——The behavior of PS—2 is similar

 

in many respects to that of CR—39. It exhibits a much
lower creep rate, however, and a smaller variation with
thma of linear limit stresses. Curves showing the data

Obteiined for this material appear in Figures A3.6-A3.10.

140
No deviation from linearity in either the optical or
mechanical response to stress was detected within the first
few minutes after load was applied. This is contrary to
earlier findings of the optical behavior as reported in
[50]. A different lot number of the material was used in
this study than in the earlier investigation.

The linear limit stresses at one percent deviation
from linearity for optical behavior are slightly lower than
the corresponding ones for mechanical behavior.

The creep rate of PS-2 is somewhere between those of
CR—39 and PS—l. Figure A3.10 shows that the birefringence-
strain relation is linear for a wide range of strains
(above 1%) and deviates very little from linearity for
strains up to nearly two percent. Like PS-l, it is very
useful for coatings. Birefringence is seen to decrease
with time at a given strain.

The material is ductile: it fails in shear at a
uniaxial tensile stress above 12,000 psi. It has a deep
amber color'which presents some problems concerning fringe
definition (low contrast). It has very smooth surfaces and
is very sensitive optically (comparable to PS—l). Like
PS—l, it is available in sheets of various sizes and
thicknesses.

Polyester Resin Palatal P6-K.--Figures A3.11-A3.15

 

show the optical and mechanical properties of P6-K. No

deviation from linearity was detected in the optical

141
behavior within the ranges of stress and time employed.
Some nonlinearity was observed in the mechanical behavior,
but the deviations from linearity are seen (Figure A3.13)
to be small and to occur only after constant stress was
maintained for several hours.

The Specimens used for the two investigations
(optical and mechanical) were from different plates which
were cured separately. Although the plates were taken from
the same "lot" and approximately the same curing cycle was
used, the differences in the behavior (mechanical and
Optical) might be due to variations in the handling of the
specimens.

The creep rate is very low. Birefringence is seen
(Figure A3.15) to decrease rapidly with time at a given
strain.

The material was donated by the manufacturer in
liquid form. It was mixed in the following proportions by
weight: 100 parts resin, 1.50 parts catalyst, and 0.05
parts activator. It was cast between glass plates in
sheets of nominal l/4—inch and 3/8—inch thicknesses. A
curing temperature of 90°C was maintained for 40 hours.
The material was tested after it was stored, at room
temperature, for over a year.

Though relatively insensitive optically, the broad
range of linearity in the birefringence response to stress

makes it a very useful photoelastic model material.

142

Fracture occurred in the photoelastic specimen after
200 hours at a stress of 5,950 psi. The test for mechani-
cal properties was terminated after 100 hours because of an
air conditioner failure.

Palatal P6-K exhibits good optical transmission
characteristics with only a slight amber tint. It is quite
brittle and is susceptible to chipping during the machining
process.

Recommendations for
Further Research

 

 

Several suggestions for continuing or extending the
work of this study seem in order. These are given below:

1. Constant strain, relaxation-type tests, both optical
and mechanical, should be conducted to check some
results deduced from the creep tests of this study.
Such tests on CR—39 would be particularly enlighten—
ing in View of the possible inflection in the
birefringence—strain plot (Figure 3.8). A direct
check could be made on the moduli, compliances, and
photoelastic coefficients reported in this study.

2. Only quasi-static loadings were employed in this
investigation. Yet accurate calibration tests are
equally or even more important to dynamic studies
using models. Some experimental studies of wave
propagation, for example, have been reported with

wholly inadequate calibrations, or even a complete

4.

5.

143
absence of them. Thus, the short-time response to
periodic and/or impact loadings should be accurately
determined. Constant strain rate loadings might
also be employed. Such studies would constitute a
valuable extension to the results here reported, if
the same materials were used.
Photoviscoelasticity promises to become a powerful
tool in viscoelastic stress analysis. An intensive
search for suitable model materials must be con—
ducted if the method is to achieve prominence.
These materials must, of course, be carefully exam-
ined (calibrated). Photoviscoelastic theories and
techniques must be scrutinized, tested, and further
developed.
The momentary-elastic assumption, discussed at
various points throughout this writing, seems justi—
fied for high—modulus materials exhibiting a low
creep rate. But the question remains open of when
(for what creep rate, say) the momentary—elastic
theory must be abandoned in favor of a more general
viscoelastic theory. Theoretical and/or experi—
mental studies to help answer this question would be
very valuable.
Linear limit stresses and certain other characteris-
tic functions describing material behavior were

determined in this study for the response to

144
one-dimensional stress. The ability of these
quantities or functions to predict behavior under

more general stress states should be checked.

6. Little experimental evidence has been provided for

the study of nonlinear viscoelasticity under com-
bined stress states. A number of theories and
assumptions, therefore, lack experimental verifica-
tion. This is particularly true for nonlinearities
associated with small strains. Nonlinearities of
this type are obviously exhibited by some of the
materials (e.g., CR—39) tested in this study.
Further investigations, employing combined stress
states extending into the nonlinear range, might
provide valuable data for further theoretical and
experimental studies.

The moire method of strain (displacement) analysis
has been the subject of very intensive study during
the past few years. Further development is believed
needed, particularly in the techniques of grid
production and application. The possible use of
diffraction gratings, especially, should be further

investigated and developed.

10.

11.

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Clark, A. B. J., "Static and Dynamic Calibration of a
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Cloud, G. L., "Infrared Photoelasticity: Principles,
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Diruy, M., "L'Analyse des Contraintes par la Methods
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Dolan, T. J., and W. M. Murray, "Photoelasticity,"
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Douglas, R. A., C. Akkoc, and C. E. Pugh, "Strain-
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Durelli, A. J., and V. J. Parks, "Moire Fringes as
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Eirich, F. R., Rheology, Theory and Applications,
vols. I, II, and III, Academic Press, New York, 1960.

 

Ferry, J. D., Viscoelastic Properties of Polymers,
John Wiley and Sons, New York, 1961.

 

Frocht, M. M., Epotoelasticity, vols. I and II, John
Wiley and Sons, Inc., New York, 1941.

 

Frocht, M. M., and Y. F. Cheng, "An Experimental Study
of the Laws of Double Refraction in the Plastic State
in Cellulose Nitrate-~Foundations for Three—dimensional
Photoplasticity," Photoelasticity, ed. M. M. Frocht,
Pergamon Press, New York, 1963, pp. 195—216.

 

Frocht, M. M., and Y. F. Cheng, "On the Meaning of
Isoclinic Parameters in the Plastic State in Cellulose
Nitrate," Jour. Applied Mech., 22 (1962), 1-6.

Frocht, M. M., and R. Guernsey, Jr., "Further Work on
the General Three—dimensional Photoelastic Problem,"
Jour. Applied Mech., 22, 2 (1955), 183-89.

Frocht, M. M., and R. A. Thomson, "Further Work on
Plane ElastOplastic Stress Distributions," Photo-
elasticity, ed. M. M. Frocht, Pergamon Press, New

Frocht, M. M., and R. A. Thomson, "Studies in Photo-
plasticity," Proc. Third U.S. Natl. Cong. Applied
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Green, A. E., and R. S. Rivlin, "The Mechanics of
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Mech. and Anal., Part I, l (1957), 1—21; Part II, with
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60), 387-404. '" "

Gross, B., Mathematical Structure of the Theories of
.Xiscoelasticity, Hermann, Paris, 1953. FSee also:
Jour. Applied Phys., 18, 212 (1947); 19, 257 (1948);
and 22, 1035 (1951). “' “'

 

 

Guild, J., The Interference Systems of Crossed Dif-
fraction Gratings, CIarendon Press, Oxford, 1956.

 

 

Gurtin, M. E., and E. Sternberg, "On the Linear Theory
of Viscoelasticity," Archive for Rational Mech. and
Anal., 11, 4 (1961), 291—356.

Jessop, H. T., and F. C. Harris, Photoelasticity
Principles and Methods, Dover PubliCations, New York,
I949. '

 

 

"Kodak Wratten Filters," Scientific and Technical Data
Book B—3, Eastman Kodak Co., Rochester, N.Y., 1965.

Kolsky, H., "Experimental Studies of the Mechanical
Behavior of Linear Viscoelastic Solids," Technical

Report No. AM—18, Dept. of Defense, ARPA, Materials
Research Program, April, 1965.

Leaderman, H., Elastic and CreepProperties of Fila-

mentous Materials, Textile FoundatiEn, Washington,
1943.

 

 

Leaderman, H., "Viscoelastic Phenomena in Amorphous
High Polymeric Systems," Rheology, vol. II, ed. F. R.
Eirich, Academic Press, New York, 1958.

 

Lee, E. H., "Viscoelastic Stress Analysis," Proc.
First Symposium on Naval Structural Mech., Pergamon
Press, New York, 1960, pp. 456-82.

Lee, E. H., and T. G. Rogers, "Solution of Visco—
elastic Stress Analysis Problems Using Measured Creep
or Relaxation Functions," Jour. Applied Mech., 30
(1963), 127-33. '__

Lockett, F. J., "Creep and Stress—relaxation Experi-
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'3 (1965), 59-75.

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Low, I. A. B., "The Moire Method Using Diffraction
Gratings," presented at 1966 S.E.S.A. spring meeting,
Detroit, Michigan, May 4-6, 1966.

Low, I. A. B., and J. W. Bray, "Strain Analysis Using
Moire Fringes," Engineering, 213 (1962), 566.

Marin, J., and J. E. Griffith, "Creep Relaxation of
Plexiglas IIA for Simple Stresses," Jour. Eng. Mech.
Div., Proc. A.S.C.E., 82, EM—3 (1956), Paper 1029,
1-20. ‘-

Mindlin, R. D., "A Mathematical Theory of Photovisco-
elasticity," Jour. Applied Phys., 22 (1949), 206-216.

Mindlin, R. D., "Analysis of Doubly Refracting Mate-
rials with Circularly and Elliptically Polarized
Light," Jour. Optical Soc. Am., 31 (1937), 288.

Ménch, E., "Die Dispersion der Doppelbrechung bei
Zelluloid als Plastizitatsmas in der Spannungsoptik,"
Zeitschrift fur angewandte Physik, g, 8 (1954), 371-75.

M6nch, E., and R. Loreck, "A Study of the Accuracy and
Limits of Application of Plane Photoplastic Experi-
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Press, New York, 19* , pp. 169-84.

Morse, 5., A. J. Durelli, and C. Sciammarella,
"Geometry of Moire Fringes in Strain Analysis," Jour.
Eng. Mech. Div., Proc. A.S.C.E., EM—4 (1960), pp. 105-
126.

Nielson, L. E., Mechanical Properties of Pol mers,
Reinhold Publishing Corp., Chapman and Hall, L .,
London, 1962.

 

O'Haven, C. P., and J. F. Harding, "Studies of Plastic
Flow Problems by Photo-grid Methods," Proc. Soc.
Exptl. Stress Anal., 2, 2 (1944), 59-70.

Onaran, K., and W. N. Findley, "Combined Stress-creep
Experiments on a Nonlinear Viscoelastic Material to
Determine the Kernel Functions for a Multiple Integral
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(1965) , 299-327. "

Pindera, J. T., "Einige rheolgische Probleme bei
spannungsoptischen Untersuchungen," Internationales
spannunQSOptischen Symposium, Berlin, April, 1961,
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Pindera, J. T., "Remarks on Properties of Photovisco-
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Pindera, J. T., and G. L. Cloud, "On the Description
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Pindera, J. T., and E. W. Kiesling, "On the Linear
Range of Behavior of Photoelastic and Model Mate-
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West Berlin, March, 1966, VDI-Berichte No. 102, 1966,
pp. 89-940

Post, D., "The Moire Grid-analyzer Method of Strain
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Ross, B. E., C. A. Sciammarella, and D. Sturgeon,
"Basic Optical Law in the Interpretation of Moire
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Exptl. Mech., 5, 6 (1965), 161—66.

Sciammarella, C. A., "Basic Optical Law in the Inter—
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Sciammarella, C. A., "Techniques of Fringe Interpola-
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Sciammarella, C. A., and F. Chiang, "The Moire Method
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APPENDIX I

ANALYSIS OF STRESS DISTRIBUTION IN TAPERED MODEL

As stated in Chapter II, any model for which there
is a known analytical solution for the stress distribution
can be used for calibration. Such a solution for a wedge
loaded with a concentrated force at the apex is given
below. Combining this with experimental evidence obtained
for a material with known properties, the value of radial
stress along the centerline of the tapered model used
throughout this study is obtained.

Timoshenko and Goodier [68] obtained the solution by
assuming stresses which satisfy the boundary conditions.
An unknown constant is evaluated from equilibrium con-

siderations. Frocht [18] used the Airy stress function

m z Cre sin 8 (Al.l)

to obtain the stresses

0']: .. 2c 233—9, (111.2)
Ge :3 O, (A103)

151

152
where 9 is measured from the centerline and r from the apex
as shown in Figure Al.l. The boundary conditions are
satisfied by such stresses, and it may easily be verified
that the stress function satisfies the necessary compati-

bility equation.

 

Fig. Al.l—-Sketch of wedge with
concentrated load at the apex.

The constant C is evaluated by considering the

equilibrium of a portion of the wedge bounded by the

153
straight sides and the surface r = constant. The constant

C is thus found to be:

 

_ P
C - d(2a + sin 2&7’ (A1°5)
where d is the thickness of the model.
Combining Equations (A1.2) and (Al.5) yields:
0 = 2P cos 0 (Al.6)

 

r rd(2a + sin 2d)’

The equation relating order of isochromatic and

stress was given in Chapter II as:

 

Cod
n = T (01 - 02). (A1.7)
In the case of the wedge,
01 = Or and 02 = 0,
so
C
_ o 2P cos 9
n _ T—' r(2a + sin 2a) ' (Al'8)

Equation (A1.8) shows that on the centerline of the model,
when 9 = 0, the isochromatic fringe of a given order is
circular. For the model used in this study a = 7.12°.
Since cos 7.l2° = 0.9922, the fringes may be regarded as

circular for the entire wedge portion of the model used.

154

It is interesting that the solution presented above,
through Equation (A1.8), does not restrict the value of the
angle a to being small. Hence the solution applies as well
to a semi—infinite plate, where a =-g.

For the modified wedge used in this study (soon to
be described), the small part of the wedge is transformed
to a section of uniform width. Hence the apex of the wedge
is not on a boundary. The radius r, therefore, is not
easily measured. However, the width of the model can be
measured with accuracy and speed. This can be used to

determine the nominal stress which can in turn be related

to the radial stress or as follows:

 

 

 

 

Onom =-———E——— from Figure A1.1,
d(AlA2)

B = r cos 9, AlB = B tan a = A1A2/2,
AlA2 = 2'_§ tan a = 2r cos 9 tan a,

o = P (A1 9)

nom 2rd cos 6 tan a' °

Let
r nom

where k is a stress concentration factor. From Equations

(Al.6), (Al.9), and (Al.10),

155

 

 

2 cos 9 (k) 1

2a + sin 2a = 2 cos 9 tan d’

from which

4 cos2 9 tan a
2d + sin 2a

 

k = . (Al.ll)

Along the centerline of the model (wedge), where the fringe

 

order for birefringence or moire patterns is read, 6 = 0
and
_ 4 tan a
(k)8=0 _ 2a + sin 2a (Al°12)

For models used, a = 7.13°. The value of k along the

centerline becomes:

(k)e=0 = 1.011. (Al.13)

Thus

r nom

The above analysis is valid only in that portion of the
wedge—shaped portion of the specimen where no disturbance
in the stress distribution exists. The expression for or
shows that or —+- m as r —a’ 0; hence it is necessary to
modify the wedge shape in such a way that the smallest
cross—section is capable of carrying a load sufficient to
cause a meaningful stress in the largest portion of the
wedge. This is accomplished by transforming the smaller

portion of the wedge into a prismatic "shank," as shown in

156
Figure Al.2. This shank is long enough to provide a
uniform axial stress over a length sufficient to accurately
measure order of isochromatic, longitudinal strain, and
transverse strain. On the end of the prismatic portion
opposite the wedge, smooth transition to a corner is made,
thus providing a point of zero stress where the zero-order
isochromatic is found. The small holes near each end were
used only to align and hold the model for machining, as

discussed in Chapter II.

Transition region

 

 
 

 

 

 

 

Sh7nk {
t— % s44rWedge ::
. aJ -..---
:_f [Sr—J:
0. 40" beginning
of fillet
L 1 1 1
V I T
2" 2.1" 11"

Fig. Al.2—-Sketch showing shape and approximate
dimension of tapered model used.

Since the region of transition from the wedge to the
shank does not lend itself to mathematical analysis for the
stress distribution, an experimental method is used. The
stress concentration factor k, defined by Equation (Al.10),
is determined continuously from the wedge portion, where
its value is 1.011, through the transition region and into

the shank portion, where its value is 1.00. This

157
experimental determination is easily made with a material
known to have a momentarily linear stress—birefringence
relation for stresses below those encountered in the shank
during the determination. Polyester Resin Palatal P6-K in
the cured state was known from previous tests1 to possess
such properties and was therefore selected.

Because the stress-birefringence relation is linear,
each integral-order isochromatic fringe represents a given
difference in principal stresses. But since the distribu-
tion is radial in the wedge, this given difference is
simply an increment of the radial stress or. On a picture
of the isochromatic fringe pattern, taken with a constant
load on the model, the location of each integral fringe was
determined on the centerline. In the wedge portion, where
k is known to be 1.011, the radial stress at each fringe
location is computed from the nominal stress. Now, when
the increment of or represented by each integral isochroma—
tic fringe is known, the value of the apparent or can be
determined at each point on the centerline where a fringe
crosses, through the transition region and into the shank.
At each of these points the nominal stress is computed.

Now the stress concentration factor k is determined at the

 

1Since both stress distribution and isochromatics
are radial in the wedge portion of the model (Fig. Al.2),
the momentary linearity of the stress-birefringence rela-
tion can be checked up to the maximum stress in the wedge
portion. This stress should not be exceeded in the shank
during the test for determination of the stress concentra-
tion factor.

158
discrete points where fringes cross the centerline. The
known value of k in the shank affords a check.

The value of k may now be plotted against the
distance along the centerline. The plot used throughout
the photoelastic calibration studies is shown in Figure
Al.3. The beginning of the fillet, located by means of a
micrometer caliper, served as a reference point. Two
scales are shown on the abscissa because in scribing the
scale on some models metric units were employed; on others

the scale was in inches.

 

 

 

 

 

 

1.024
r nom
1001‘ . ' . o
in from beginning
of fillet
1.00 ~\
\
\
99. \\ cm from beginning
' \ of fillet
\
\
.98- \
\
.97-

 

Fig. Al.3-—Plot showing variation of the
stress concentration factor k along the
centerline of the photoelastic model.

Additional points on the plot may be obtained by
considering half—order fringes or by changing the load on

the model.

159

The external load at the large end of the taper
disrupts the radial distribution of stress as given by
Equation (Al.6) near the end. The width there is approxi-
mately 2.5 inches. The isochromatic fringes are very broad
in the largest portion of the taper, due to the small
stress gradient, making fringe location difficult. Hence
no readings were taken in the portion 3—4 inches from the
large end. The lowest useful stress was then approximately
twenty percent of that in the shank.

A similar calibration could be performed for the
strain distribution. It is more complicated because an
integral moire fringe does not represent a particular
strain but only a certain displacement relative to another
fringe. Furthermore, the effect on the moire pattern is
not as pronounced as on the isochromatic pattern for the
following reason. The isochromatic order is related to the

difference in principal stresses (0 — 02) or, in this

1
case, (or — 09). Hence any small transverse stress 09
affects the isochromatic order directly. The strain in the
radial direction would, on the other hand, be affected by a
transverse stress component only through the Poisson
effect. Since Poisson's ratio cannot exceed the value of
0.5 (in an elastic material), the effect of such a trans-

verse stress component on axial strain is less pronounced

than upon the isochromatic fringe pattern.

160

Poisson's ratio is less than 0.4 for the materials
tested in this study. Irregularities of the moire fringe
pattern in the transition region were not apparent. On the
mechanical models the stress concentration factor was
therefore assumed 1.00 in the shank and up to the point of
crossing in Figure A1.3, and 1.011 for the remainder of the
model.

Typical isochromatic and moire fringe patterns are

shown in Figure Al.4.

F'" «A 5

:5

   

 
       
   

mum

l

:2
All!
m

, I

(a) (b)

7!

mun» )

Fig. Al.4——Photographs of (a) isochromatic
fringe pattern and (b) moire fringe pattern.

APPENDIX II

TECHNIQUES OF MOIRE STRAIN ANALYSIS

Difficulties Presented by
a "Conventional" Method

 

In order to plot strain as a function of stress, at
a given time after loading, including not only the linear
but also a portion of the nonlinear region, at least eight
points seem necessary (see Figure 3.5). Though the general
shape of the stress—strain curve can be established with
only a few points, the accurate evaluation of the linear
limit stress value at a specified deviation from linearity
of strain requires eight or more.

These considerations led to the attempt at measuring
the average strain over a 4—inch gage length by means of a
mechanical extensometer on eight tension specimens of
uniform width. The extensometer consists of an invar steel
bar attached at one of its ends to the specimen, with the
bar resting lightly in a guide cemented to the specimen
near the other end. The displacement of a fine line
scratched on the bar, 4 inches from the attached end, rela—
tive to a similar line on the guide, was measured with a
telemicrosc0pe supplied by Gaertner Scientific Corporation.

161

162

This method promised three distinct advantages:

(1) high sensitivity (better than 1 x 10"4 in/in); (2) no
adverse effects, such as stiffening or indentation of the
specimen; and (3) repeated use of a single extensometer.

Inherent in this method are some disadvantages.

Only one specimen at a time can be tested without special
load frames, designed to accommodate multiple specimens,
and provision for readily and accurately moving the tele-
microscope to obtain successive readings on all specimens
is also necessary. A load frame holding nine specimens was
used by Marin and Griffith [38], with electrical resistance
gages measuring strain. The expense of building such a
load frame, with the additional telemicroscope carriage,
was beyond the means of the present study.

The series of eight or more separate tests require
at least eighty days for obtaining the creep data desired
in this study; nevertheless, it was attempted. Eight
specimens of Polyester Resin PS—l, with a prismatic portion
exceeding 5.2 inches in length and a width of 0.40 inch,
were cut from a plate approximately 0.130 inch thick. From
the data obtained by separate tests at different stress
levels curves of strain vs. time were drawn for each stress
level. Six tests at successively higher stress levels,
extending well into the nonlinear region, were followed by
two at stress levels by then known to be near the linear

limit stress. Although it was difficult to repeat readings

163
closely, little scatter was evident in the data. From this
plot data was deduced for strain vs. stress at selected
times.

In this strain—stress plot glaring discrepancies
appeared. The eight points available for plotting each
curve were scattered excessively for purposes of accurately
determining the linear limit stress. The points corre—
sponding to the last two tests appeared four to five per—
cent above the curve established by the other six points.

Several causes, acting alone or in combination,
might be responsible for such erratic behavior. As men—
tioned in Chapter II, when more than one specimen is used
in a calibration test, they must all be machined and
handled identically. For convenience and consistency all
eight specimens were machined at once. But then how are
they to be handled identically when they must be stored for
various periods of time awaiting testing? Since the time-
edge effect was known to be significant on some plastics,
the machined surfaces were coated with silicone grease and
the specimens were stored in tightly closed polyethylene
bags. Perhaps this precaution was not necessary-—or per—
haps it was necessary but not effective. Possibly it even
caused a degradation of the data.

Lack of homogeneity could also cause the discrep—
ancies. Frocht and Thomson [23] found variations of up to

seven percent in birefringence, in eight different models

164
cut from a single sheet of celluloid. Some variation in
mechanical properties seems just as likely.
Reloading for shorter time periods produced roughly
the same magnitudes of strain response but a different rate
of creep, particularly at the highest stresses encountered.

No data of value to this study was obtained.

Suitability of Moire Method

 

Because of the time required by further attempts to
use the method discussed above, it was abandoned in favor
of one employing the moire method on a tapered specimen
with a shape identical to that used in the optical test.
Its use on such a model yields strain data for any (every)
value of stress within a certain range, this range quite
naturally being about the same as for the test of optical
properties on the same material. Hence one model yields
all the stress—strain data required by this study.

The theory and methods of interpretation of moire
patterns are well—established [15,43,51,55,56]. Theocaris
[66] recently reported on past efforts and presented an
extensive bibliography on the moire method.

Low sensitivity has generally limited the usefulness
of the method to the measurement of the relatively large
strains. With two initially identical line arrays of
density 500 lines/inch each fringe, formed by superposition
of the two arrays after one is deformed, represents a rela-

tive displacement of 0.002 inch. By counting half fringes

165
a relative displacement of 0.001 inch can be detected.
Since strain in a given direction is given by the displace—
ment gradient in that direction, fringes must be closely
spaced in regions where the gradient is small or changes
rapidly if serious errors are to be avoided. Fortunately,
no such rapid changes were present in the tapered specimen
used for this study.

Accuracy of the moire method can be improved by many
techniques [9]: (l) Higher—density (i.e., smaller—pitch)
arrays produce more fringes with consequent increase in
precision. (2) Variable~pitch arrays allow compensation-
type determinations of fractional fringe orders, thus
increasing accuracy. This is essentially the same as using
an initial mismatch of pitches, producing an initial pat—
tern, and subsequently yielding a larger number of fringes.
(3) Photocells can be used to measure light intensity at a
point. The intensity, in turn, can be related [53] to the
maximum and minimum intensities corresponding to integral
and half—order fringes, thereby yielding a continuous rela—
tionship between light intensities and displacements, and
hence strains. Sciammarella and associates [52] obtained
better results by this method using a 300 line/inch grid,

6 inch/inch, than by

yielding minimum strains of 20 x 10—
the conventional method using 1,000 lines/inch in a pre-
vious test under similar conditions. (4) Various methods

of fringe interpolation have been successfully employed

166
[54,58]. Sciammarella and Lurowist [57] obtained partial
fringes by observing the normal moire pattern in a field of
polarized light, thus taking advantage of the diffraction
characteristics of line arrays. Low [36] achieved the same
result using diffraction gratings instead of ordinary line
arrays. These gratings promise several advantages other
than the one mentioned.

None of these refinements seemed necessary in this
study. The investment in time and auxiliary equipment
required to utilize them seemed unwarranted. As mentioned
above, no rapid changes in the displacement gradient
occurred. It was seen in Chapter III that a 500 line/inch
array yielded an adequate number of points for plotting a
smooth curve of fringe order vs. length along the specimen.
Loss of accuracy resulted mostly from evaluating the slope
of this line rather than in constructing it, and this
evaluation might be improved by means of a suitable data—'
smoothing technique if necessary.

The choice of 500 lines/inch as the density for
model and analyzer was somewhat arbitrary. The pitch most
appropriate for a given application can be specified only
within some range. It should be small enough to yield
sufficient response for the smallest strains anticipated
but should not be so small as to produce so many fringes in
regions of highest strain that they can no longer be

clearly distinguished. The difficulty and expense inherent

167
in producing fine arrays will usually dictate that the

largest permissible pitch be used.

Production of Line Arrays--
General Methods

 

 

Methods of producing line arrays vary widely,
depending upon such factors as the density of the array,
type of model material used, equipment available, and the
number of arrays required. Reference arrays in various
densities are available on a stable transparent backing,
such as glass, with cost depending upon the density and
over—all size. Duplicates of these can be made by contact
printing. Suitable arrays can also be made by photograph-
ing larger, inexpensive, coarse screens.

Line arrays on the model are much more difficult to
produce. Of course, they must be accurate in line width
and spacing and must follow the model perfectly during
deformation yet offer no significant resistance to deforma—
tion. Model arrays are usually produced in one of five
ways: (1) The array is scribed or indented into the
surface. (2) An array is photographed or contact printed
on stripping film, and a very thin film carrying the emul-
sion is cemented to the specimen. (3) The array is
printed, by contact, in a layer of light—sensitive emulsion
coating the specimen. (4) Commercially available gages
with arrays etched in thin metal may be cemented to the

specimen. When the cement has cured, a metal backing is

168
stripped off, leaving only a series of thin metal filaments
attached to the specimen. (5) A diffraction grating is
"printed" by contact in a layer of liquid resin, coating
the specimen.

The problem of choosing the method most suitable for
a particular application has many aspects, some of which
were mentioned at the beginning of this section. Applica—
tion of an array on plastics presents problems not present
for metals: chemicals used for etching may attack the
plastics, scribing may introduce stress concentrations or
fail to yield sufficient contrast for observable fringes,
and the heating required for the cure of some coatings is
excessive for plastics.

The first method mentioned above——namely, scribing
or indenting the array into the surface—-can hardly be
considered for plastics. It has, however, been used quite
successfully on other materials. Using the mechanism from
a rotary microtome, Douglas, Akkoc, and Pugh [14] obtained
consistently successful gratings of line densities to
12,700 lines/inch on flat metal specimens. Bell [2]
obtained rulings in the form of cylindrical threads with
densities as high as 30,720 lines/inch using a specially
modified lathe.

The use of stripping film, as mentioned secondly
above, offers the advantage of using directly an array on

film, thus eliminating the additional step of contact

169
printing on the emulsion-covered specimen as required by
(3). Since stripping film was used extensively in this
study, its use will be further discussed in the following
section.

Before a suitable cement for bonding the stripping
film to the specimen was found, the third method was tried.
In this method the specimen is coated with a light—
sensitive emulsion which is, in turn, exposed by contact
printing to the desired array on glass or film. Because
most of the available emulsions are relatively insensitive
and because intimate contact is required, a vacuum printing
frame and carbon—arc lamp are usually used in the opera-
tion. These were made available for use in this study by
two local firms——one a lithographer, the other a photo—
engraver.

Although this method is probably the most widely
used for applying arrays to models, success was not
achieved in the course of this study. The greatest diffi—
culty was the lack of an inexpensive array on glass or film
which transmitted the light exposing the emulsion with uni-
form intensity. Glass plates with the desired 500 line/inch
array scratched into the surface were known to be available.
Their cost was prohibitive ($16 per square inch from one
manufacturer) for the size required in this study (approxi-

mately lO-inch length). Film reproductions of these arrays,

170

which were later purchased, were not known to be available
at the time the method was attempted.

An array of 500 lines/inch was, however, obtained on
film by photographing a coarse, inexpensive screen of
65 lines/inch. The lack of uniform light transmission by
the screen caused a corresponding nonuniformity in the
film, which was further magnified in the array printed on
the specimen. The exposure of the emulsion coating the
specimen ranged from too low, so that the emulsion was
removed in developing, to too high, so that the lines could
not be distinguished. The images on film, however, were
sufficiently uniform in intensity for the production of
high quality and high contrast moire fringes when two such
films were placed in contact. Since these images could
successfully be produced on stripping film, the method
employing such film directly was again pursued and finally
used successfully. Before discussing the technique used
in applying the stripping film, the last two methods men—
tioned above——i.e., use of commercially available, bondable
gages and diffraction gratings——will be briefly discussed.

Bondable gages promise the advantages of easy
application, lack of stiffening since the grid is adver-
tised as consisting of a series of unconnected "dots," and
extremely high contrast since the array is formed in metal
which is transferred to the specimen. Their chief dis-

advantages would seem to be relatively high cost and

171
limited size (e.g., at present the largest available
500 line/inch grid is 3.6 in x 3.6 in at $55). Application
in this study, where an area of 0.40 inch by 10 inches was
needed, required cutting the largest available grid into
strips and joining three strips end to end.

In view of the promised advantages, one of the
bondable gages described above was purchased and bonded, in
strips, to two specimens, one of which was to serve as the
analyzer to measure the strain response of the other.

After testing it was discovered that instead of a series of
square dots the grid actually consisted of a series of
crossed lines, thus aborting some of its greatest potential
advantages——those of offering no resistance to deformation
and of following arbitrarily large strains. Though the
slight stiffening was not believed significant, no further
use was made of the bondable gages, since cross-lined metal
screens of the same density were found to be available from
another manufacturer in an ll—inch by ll—inch size for
approximately the same cost as quoted above for the bond—
able gage of one—ninth the area. One of the metal screens
was purchased to make contact prints on film of the cross—
lined grid, but it was not used directly as a gage.

The diffraction—grating method used by Low [36]
offers at least three distinct advantages: ease of appli—
cation, high density, and inherent fringe-interpolation

potential. The grating is available as a series of

172
"grooves" in hardened gelatin on a glass backing plate.
The specimen to which the grating is to be applied is
coated with a liquid resin into which the grooves are
pressed and held until the resin "sets" or cures. The
master grating is then removed and can be used again.
Gratings with 5,000 rulings per inch have been successfully
employed; 2,500 rulings per inch are more common and
generally adequate.

With the proper optical system the fundamental moire
fringes are reduced to fine sharp lines, thus facilitating
accurate location. But in addition, subsidiary fringes can
be produced which are fractional—order fringes. These are
useful for interpolating displacements within the funda—
mental period, an important factor for very small strains
or high—strain gradients.

Although the increased accuracy of such a method was
not required in the present study, the low cost and appar-
ent ease of grid production might have dictated its use.
Unfortunately, information about the method came too late
for inclusion in this study.

Grid Application
Method Employed

 

 

Of the five previously mentioned methods of applying
arrays to the specimen and analyzer, three were tried but
only one proved successful-—namely, the one employing

stripping film. In this method the desired array is

173
imprinted in the emulsion of the film, either by photog-
raphy or by contact printing. A coarse screen of 65
lines/inch was photographically reduced to 500 lines/inch
on stripping film in the first attempts of this study.
Later the commercial availability of a 500 line/inch film
reproduction was recognized, and one was purchased. It was
used to make contact prints on stripping film. These
prints were more uniform but gave slightly less contrast
than the photographs.

The stripping layer or membrane carrying the emul-
sion is quite stable, considering its thickness (between
0.0004 and 0.0005 in). This stability is necessary for the
usual transfer process of stripping (wet or dry) and
cementing to a new base. While its stiffness is not sig—
nificant in stiffening the specimens to which it was
applied, it is also inadequate for allowing one inexperi-
enced in its handling to transfer, after stripping, a piece
0.40 inch by 10 inches to a specimen without extensive
distortion! If, however, a cement could be found giving
adequate adhesion to allow the backing of the stripping
film to be peeled off after cementing, then the problem of
distortion would be drastically reduced. The stability of
the membrane creates further demands upon the cement, since
no creep of the cement is allowed if the bonded membrane is

to follow the specimen upon straining.

174

Eastman 910 cement was found adequate for the task.
Its "instant" curing adds another advantage. A liquid
cement of the same material as the specimen has been used
[12]. Besides the lack of availability in liquid form of
some materials, the excessive curing time of most materials
handicaps their use as a cement.

The procedure used in cementing the thin strip to
the specimen was developed without the advantages of prior
experience or any instructions. The specimen to which the
membrane was to be bonded was thoroughly "degreased" and
cleaned with powdered abrasive. The strip of film was
cleaned with a cloth moistened with a very mild acetic acid
solution. The film, both membrane and backing, was then
positioned and one side taped. The exposed part of the
specimen surface, as well as the edges and the film back—
ing, was then coated with a releasing agent to prevent the
"squeezed out" cement from adhering to the specimen. Care
was necessary to avoid getting the releasing agent under
the film.

The film was then raised with the taped edge acting
as a hinge. The catalyst was applied to the film and a
layer of the cement was applied to the specimen near the
taped edge of the film. The specimen, film side down, was
then placed on a surface plate, and uniform pressure, via
weight on a straight bar, was applied from the back (top)

of the specimen. With high pressure applied for a few

175
seconds the cement cured. Excess cement was then wiped
off, along with the releasing agent. The backing was
stripped from the membrane making the specimen, with line
array attached, ready for testing. Other details of the

testing procedure are given in Chapter II, Part B.

APPENDIX III

DATA FOR OTHER MATERIALS TESTED

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