Mathematical model for predicting anisotropic effects in plasticity

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Mathematical model for predicting anisotropic effects in plasticity
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Langner, Carl Gottlieb, 1938-
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Thesis--University of Florida.
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Bibliography: leaves 145-151.
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Carl Gottlieb Langner.
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MATHEMATICAL MODEL FOR PREDICTING
ANISOTROPIC EFFECTS IN PLASTICITY








By



CARL GOTTLIEB LANGNER


A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA
1973














DEDICATION



To my wife, Ann, whose constant

help and encouragement made this

work possible.













ACKNOWLEDGMENTS


The author wishes to thank Professor L. E. Malvern for

his help in preparing this dissertation. The author heartily

commends Professor Malvern for his courage and faith that

enabled him to support and defend his student, the author,

even when at times he was not entirely convinced of the

worth of his student's unorthodox ideas.

The author acknowledges discussions with the following

persons which helped to congdal the basic ideas of the radial

element theory: M. A. Eisenberg, I. K. Ebcioglu, E. K. Walsh,

R. E. Reed-Hill, P. R. Paslay, N. Cristescu, Ion Suliciu, C. S.

Ting, and Richard Johnson. The author received technical help

from Guy Demoret, J. D. Macmillan, and Bill Luckhurst. Thanks

are also due the National Science Foundation, which helped

support the author by Grant No. GK-23452, and to the University

of Florida, which provided the majority of computer time used

in this study.


iii













TABLE OF CONTENTS


Acknowledgments . . .

List of Figures . . .


Page

111


Abstract . .. . vii

Chapters
I Introduction. .. . ..... 1
II General Concepts of the Radial Element Theory 17
III A Class of Elemental Stress Vectors for
Predicting Elastic Material Behavior 29
IV A Class of Elemental Stress Vectors for
Predicting Elastic-Plastic Material Behavior. 41
V Stress-Strain Behavior for Axial Loading and
Reversed Axial Loading. . ... 52
VI Stress-Strain Behavior for the Class of
Loadings Involving Fixed Principal Axes .. .68
VII Stress-Strain Behavior for Biaxial Loadings and
for a General Loading History . .. 91
VIII Comparisons between the Theory and Experimental
Results for Commercially Pure Aluminum. ... .108
IX Conclusions and Recommendations for Future Work. 135

Bibliography . . 145


Appendices
A Computer Programs . .
B Tabulated Results . .
C Yield Surfaces in Principal Stress Space. .
D Yield Surfaces in Principal Strain Space. .


S152
. 162
. 170
. 184


Biographical Sketch. . .. ..... 198













LIST OF FIGURES


Figure Page

1 Basic Concepts of the Theory. . 20

2 Spherical Coordinates .... .. .34

3 Distribution of Elemental Stress Vectors
for Axial Loading from an Initially Isotropic
State . . 54

4 Stress-Strain Behavior for Axial Loading,
Small Strain .. ... .. .. 57

5 Stress-Strain Behavior for Axial Loading,
Large Strain. ... '. .... . 58

6 Distribution of Elemental Stress Vectors for
Reversed Axial Loading from an Initially Plastic
State . ..... .. .. 61

7 Axial Loading and Reversed Axial Loadings for X =0 65

8 Axial Loading and Reversed Axial Loadings
for X=1.5. . . .. 66

9 Stress-Strain Behavior for Shear Loadings 74

10 Numerical Errors for Axial Loading; Ordinary
Trapezoidal Integration, X =0 . .. 76

11 Numerical Errors for Axial Loading; Ordinary
Trapezoidal Integration, X =1 .... .... 77

12 Numerical Errors for Axial Loading; Modified
Trapezoidal Integration, =0 . .. .78

13 Numerical Errors for Axial Loading; Modified
Trapezoidal Integration, =1 . 79

14 Surfaces of Constant Offset Strain for an Initially
Isotropic Material. .......... ... 84

15 Surfaces of Constant Offset Strain after an
Axial Loading. . . .85







Figure Page

16 Surfaces of Constant Offset Strain after a Pure
Shear Loading. . . 86

17 Stress-Strain Curves for Biaxial Strain Loadings 98

18 Stress-Strain Curves for Biaxial Stress Loadings 99

19 Loading Paths for Biaxial Strain with
el2 Constant. . . .100

20 Loading Paths for Biaxial Strain with
ell Constant. . .. .101

21 Loading Paths for Biaxial Stress with
s12 Constant . .... 102

22 Loading Paths for Biaxial Stress with
S11 Constant . .... 103

23 Apparatus for Tension-Compression Experiments. 112

24 Specimen for Tension-Compression Experiments 113

25 Comparison of Experiment and Theory for Tension-
Compression-Tension Tests. . ... 115

26 Comparison of Experiment and Theory for
Compression-Tension-Compression Tests. 116

27 Comparison of Experiment and Theory for Axial
Loading and Reversed Axial Loading .. 117

28 Comparison of Experiment and Theory for Axial
Loading and Reversed Axial Loading .. 118

29 Testing Machine for Biaxial Stress Experiments 121

30 Strainometer and Specimen for Biaxial Stress
Experiments. . . ... 122

31 Comparison of Experiment and Theory for Biaxial
Tests, Axial Stress versus Axial Strain. .. .. 127

32 Comparison of Experiment and Theory for Biaxial
Tests, Shear Stress versus Axial Stress. .. .. 128

33 Comparison of Experiment and Theory for Biaxial
Tests, Shear Strain versus Axial Strain. .. .. 129

34 Ratio of Stresses Compared with Slope of Strain
Path for Large Strains .. . .130













Abstract of Dissertation Presented to the Graduate
Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy


MATHEMATICAL MODEL FOR PREDICTING
ANISOTROPIC EFFECTS IN PLASTICITY


By

Carl Gottlieb Langner

August, 1973


Chairman: Lawrence E. Malvern
Major Department: Engineering Science, Mechanics,
and Aerospace Engineering


Existing theories of plasticity do not adequately describe

the anisotropic effects of plasticity. Among the effects not

easily incorporated into existing theories are the Bauschinger

effect and the observed changes in the yield surface during

complex loadings. A new theory of plasticity, the radial

element theory, is introduced in this dissertation, which

predicts stress-strain curves that are similar in shape to

the actual stress-strain curves of many metals. The theory

predicts a Bauschinger effect of the Masing type that compares

closely with experimental data. The theory also agrees fairly

well with experimental data for biaxial loadings of the torsion-

tension type. In addition, yield surfaces are predicted by


vii







this theory that in some ways are similar to those observed

experimentally by others. The most unusual feature of the

theory is the occurrence of a nonsymmetric stress tensor for

loadings that involve rotations of the principal strain axes

relative to the material. The general form of the radial

element theory is not restricted to small deformations or

to any one type of material behavior. Hence, it may find

some use in other fields as well as in plasticity.


viii













CHAPTER I

INTRODUCTION


This work is a contribution to the theory of plasticity,

or more specifically, to the theory of isothermal, time-

independent, stress-strain relations for polycrystalline

metals. A new theory is introduced here for predicting

elastic-plastic stress-strain behavior. The new theory is

designed primarily to predict the three-dimensional

anisotropic effects observed -in metals for reversed loadings

and for loading histories involving combined stresses. These

are precisely the effects that are not predicted very well

by existing theories of plasticity.

We begin our discussion of plasticity with some brief

definitions of the more frequently used terminology. For

detailed discussions of the concepts, terminology, and no-

tations of plasticity theory the reader should consult one

or more of the general expositions [1-5].1 Elasticity is

the property of a solid material once deformed to return

exactly to the shape it had before deformation when all loads



'Numbers in brackets designate reference articles and
books listed in the Bibliography.







are removed. Elasticity implies a one-to-one functional

relationship between stress and strain. Plasticity is the

property of a material whereby permanent or residual strain

develops and is maintained when a certain yield condition

is satisfied or exceeded. Yield condition defines the boundary

between elasticity and plasticity; it is the set of stress

or strain states situated at the outer limits of elastic

behavior.

Stress space and strain space are mathematical vector

spaces having, respectively, the nine components of the stress

or strain tensors as coordinates. In stress or strain space

the yield condition forms a hyper-surface called the yield

surface. Loading point is the point in both stress and

strain space that describes the current state of stress and

strain. Residual strain point is a hypothetical loading

point, corresponding to an elastic unloading from the current

loading point, defined by zero stress. A loading is any

motion of the loading point that involves plasticity. Loading

path is the locus of the loading point as it moves in both

stress and strain space during a loading. Direction of

loading is the tangent to the loading path in either stress

or strain space. Radial loading is a loading such that the

loading path is a straight line passing through the origin

in either stress or strain space. Reversed loading is a

loading, following after an initial loading, in which the







loading point retraces the original loading path back to

and perhaps beyond the origin in either stress or strain

space.

Work-hardening is the phenomenon whereby the yield

stress increases during any approximately radial loading.

More generally, it is the tendency of the yield surface to

expand in directions surrounding the direction of loading

in stress space. Bauschinger effect is the phenomenon whereby

the yield stress, for a hypothetical reversed loading, decreases

or follows the loading point during any approximately radial

loading. More generally, Bauschinger effect is the tendency

of the yield surface to shrink or follow the loading point

in directions roughly opposite the loading direction in stress

space. The concepts of work-hardening and Bauschinger effect

also often include the functional stress-strain relationship

that occurs during a radial loading and a reversed radial

loading, respectively. Two simple hypotheses regarding work-

hardening are isotropic hardening, whereby the yield surface

in stress space expands uniformly while maintaining its

original shape and position, and kinematic hardening, whereby

the yield surface in stress space translates according to a

certain rule while maintaining its original shape and size.

In this work, the material under discussion will always

possess a domain of elasticity in which the stress-strain

relationship is linear and isotropic relative to a current

residual strain. The elastic domain always includes the







current loading point but does not necessarily include the

current residual strain point. Initially the residual strain

is zero and the yield condition is isotropic, so that the

stress-strain behavior for a given loading history is inde-

pendent of the initial direction of loading. Subsequent

yield conditions generally are not isotropic, so that loading

directions are important in determining the stress-strain

relations after initial plastic yielding. In practice, the

initial isotropic state of a polycrystalline metal can be

restored by annealing or by melting and careful solidification.

Throughout this work we assume that the yield condition

is not influenced by mean stress and that the residual or

plastic part of the strain occurs at constant volume. The

first assumption was confirmed experimentally for several

metals by Crossland [6]. From these assumptions it follows

that mean stress and mean strain are elastically related.

Rate effects (creep and relaxation) are ignored; either the

material is inviscid or all loadings occur at the same

constant rate. Thermal effects are ignored; either the material

is insensitive to temperature changes or deformations occur

at the same constant temperature. Since the present subject

is stress-strain relations, we can simply disregard the

equations of motion and the energy relations of a continuum.

There are two main classes or categories of plasticity

theories. The first class we shall call the phenomenological

theories, since the basic ideas of this class of theories




5


are directly concerned with the relationship between stress

and strain. The phenomenological theories were developed in

a long series of papers [7 -28] by many authors. Early

examples of this class of theories are: (1) the theory of

perfect-plasticity which neglects elastic strains, developed

by Saint Venant [8], Levy [9], and von Mises [10]; (2) the

theory of perfect-plasticity which includes elastic strains,

developed by Prandtl [11] and Reuss [15]; and (3) the total

deformation theory of Hencky [12]. These early perfect-

plasticity theories assumed an isotropic yield surface and

no work-hardening, and they required the plastic strain

increment to be proportional to the deviator stress. Because

they ignored work-hardening, these theories do not represent

very accurately the stress-strain behavior of metals. The

Hencky theory related stress to the strain rather than to

increments of strain and it included work-hardening. This

theory has been shown to be unsuitable for describing plastic

behavior for any loadings other than radial and near-radial

loadings [1]. Because of their simplicity, and in spite of

their shortcomings, the above-mentioned theories have been

applied extensively to technological problems [1, 2].

The perfect-plasticity theories were generalized by

Ros and Eichinger [16], Melan [17], Prager [18], Hill [19],

Drucker [20], and others to include work-hardening. The

resulting family of theories is known as plastic potential

theory, because the plastic strain increment is expressed







as a derivative of a potential function with respect to stress.

The simplest and most common form of plastic potential theory

combines the ideas of initial elasticity, a smooth initial

yield surface (usually the Mises condition), isotropic

work-hardening, and a flow rule defined by the requirement

that the direction of the plastic strain increment be the

same as the normal to the yield surface at the loading point.

Further generalizations were made by Koiter [21], Sanders

[22], and Hodge [26], who developed the idea that more than

one yield surface can be operating simultaneously; this idea

permits the formation of corners from an initially smooth

yield surface. The concept of kinematic hardening was

proposed by Ishlinskii [23] and Prager [24, 25], and a

modified form of kinematic hardening was proposed by Ziegler

[27]. Mroz [28] proposed a theory which combines the concepts

of isotropic hardening, kinematic hardening, multiple yield

surfaces, and a field of work-hardening moduli; this theory

can be made to predict work-hardening and a piecewise-linear

Bauschinger effect.

The phenomenological theories possess two advantages

over other plasticity theories. (1) Most of these theories

are simple enough to use in the solution of problems involving

engineering structures. (2) Numerous experimental studies

[29 -32] have demonstrated the validity of the plastic

potential theories for loadings in which the plastic deformation

continues without interruption. See Hill [1, pp. 22-32] for







references to early experimental work. These experimental

studies generally avoided reversed loadings and unloading

followed by reloading in a different direction, situations

in which the phenomenological theories do not provide

accurate predictions.

There are two principal objections to the phenomenological

plasticity theories. (1) The theories lack the unity of a

truly predictive theory. By this is meant that as new ideas

and new experimental data become available the phenomenological

theories must be repeatedly modified and made increasingly

complex with new assumptions in order to account for each

new phenomenon. A more satisfactory theory would not require

this layering of assumption upon assumption, but would predict

all essential phenomena with the simplest and most basic

form of the theory. (2) There does not appear to be any

easy way to include in these theories a realistic Bauschinger

effect or any of the related anisotropic changes in the

yield surface as a function of the loading history that have

been observed experimentally. The first objection is posed

here by the author. The second objection, which concerns

mainly the continuing lack of an adequate hardening hypothesis,

has been stated before in a variety of places; see Zizicas

[3(b)], Naghdi [4, p. 144], Green and Naghdi [5, p. 253],

Iwan [33, p. 612], and Batdorf and Budiansky [34, p. 323].

The following passage from Iwan [33] is pertinent:








One such theory, the incremental theory of
plasticity, describes the three-dimensional
yielding behavior of a material in terms
of a yield surface in stress space, along
with a flow rule and a work-hardening law.
To date, this theory has been applied quite
successfully to the rather large class of
problems which have to do with the monotonic
loading behavior of materials. However, when
the theory is extended to the analysis of a
cyclic and hysteretic behavior, certain
difficulties arise. For the most part, the
difficulties result from the requirement
that the theory be capable of accounting
for a Bauschinger effect. Furthermore,
experimental results indicate that this
Bauschinger effect should have a specific
form; namely, that the stress-strain curves
associated with one-dimensional symmetrical
closed hysteresis loops should be of the
same form as those of stabilized initial
loading curve (or cyclic stress-strain curve
for cyclic hardening or softening systems)
except for an enlargement by a factor of
two. The idea that hysteresis loops should
have this form was first suggested by G.
Masing and is often referred to as Masing's
hypothesis. The recent development of the
concept of kinematic hardening by Prager
has provided a means for introducing a
Bauschinger effect into the incremental
theory of plasticity but, even with this
extension, stress-strain behavior satisfying
Masing's hypothesis has only been obtained
for the special case of a linear work-
hardening law. When the work-hardening
is not linear, it would appear that some
additional assumptions need to be added to
the kinematic hardening hypothesis but as
yet these assumptions have not been set
forth.

The reader who wishes further information on cyclic

loading is referred to two recent papers by Krempl [35]

and Feltner and Landgraf [36]. Masing's hypothesis on the

form of the stabilized hysteresis curve was originally


proposed in the article [37].








Numerous recent papers [32, 38 -53] have reported

experimental investigations of the initial and subsequent

yield surfaces in metals. The metals tested included copper,

brass, mild steel, and various alloys of aluminum. All

papers but one [47] reported biaxial tests performed on

tubular specimens. In [38, 40, 50] the tests involved tension

and internal pressure, and in [42, 48] the tests involved

torsion and internal pressure. The remaining papers reported

combined tension and torsion loadings. Most of the papers

dealt with the question of whether or not a corner is formed

in the yield surface at the loading point. Two methods

have been used to study this question. The first method

consisted of probing with the loading point in various

directions in stress space and thereby detecting a portion

of the yield surface according to some given definitionof

yielding. This was usually done both initially and after a

certain pre-strain. The second method consisted of imposing

on the material a zigzag loading path in stress space and then

measuring and plotting the corresponding plastic strain

increments. According to Drucker [20], if the yield surface

is smooth then the direction of the plastic strain increment

must be normal to the yield surface and independent of the

stress increment. If abrupt changes are observed in the

direction of the plastic strain vector corresponding to the

abrupt changes in the stress vector, then a corner must

exist at the loading point.







The following general conclusions can be drawn from the

experimental papers [32, 38 -53]. The initial yield surface

is closer to the Mises condition than to the Tresca condition.

All subsequent yield surfaces are smooth and convex, and the

direction of the plastic strain increment is always very

close to the yield surface normal at the loading point.

The plastic strain increments appear to depend only very

slightly upon abrupt changes in the directions of the stress

increments. Subsequent yield surfaces exhibit a region of

high curvature surrounding the loading point, but a sharp

corner is not formed. A flattened region generally forms

on the side of the yield surface opposite the loading point

for data plotted in the shear stress-axial stress plane.

The experimentally observed characteristics of the yield

surface depend intimately upon which definition of yield is

used. A proportional limit definition of yield [32, 40, 41,

52, 53] gives yield surfaces which translate away from the

origin in stress space (in the direction of loading) and

which drastically change shape while showing little or no

cross-effect (lateral expansion or contraction) during a

radial loading. A backward-extrapolation definition of

yield [42, 48, 49] leads to yield surfaces which expand

nearly uniformly while translating very little; here work-

hardening and a large cross-effect occur simultaneously.

The sensitivity of the yield surface to the yield definition







led one author [47, 50] to report his results as families

of curves of constant offset strain; this procedure avoids

the subjectivity of the proportional limit definition and

the arbitrariness of any other definition.

The second class of plasticity theory may be called

the physical theories. Examples of the physical theories

are presented in the papers of Batdorf and Budiansky [34],

Besseling [54], Hutchinson [55], Lin and Ito [56, 57], and

Wells and Paslay [58]. Studies of the micro-structure of

metals has led to the following conclusions. (1) Crystallo-

graphic slip is the principal process of plastic deformation

in face-centered-cubic metals at low and intermediate

temperatures [57]. (2) The phenomenon of work-hardening

and the Bauschinger effect are due to locked-in stresses

produced by large numbers of dislocations that accumulate

and become stuck at grain boundaries and around alloying

elements in the material [59, Chapter V]. In various ways

each of the physical theories has incorporated the ideas of

crystalline slip and locked-in stresses. The most notable

achievements of the physical theories are prediction of a

developing corner in the yield surface at the loading point

during plastic deformation [34, 56, 57] and prediction of a

Bauschinger effect of the Masing type [54 58].

None of the physical theories has been entirely successful.

Either the theories are too specialized and complicated to

reliably and economically predict stress-strain behavior for







general loading paths [55, 56, 57], or they require speci-

fication of special functions such as the initial yield function

and the stress-strain function for a monotonic pure tension or

pure shear loading [34, 54, 58]. In the latter case the

physical theories are conceptually not too much different

from the phenomenological theories, most of which also require

specification of various functions, presumably to be obtained

from empirical data.

The theory of Lin and Ito [56, 57] deserves a special

discussion as it is at present the most highly developed

of the physical theories. In this theory the material is

idealized as a cubical array of sixty-four differently

oriented crystals of cubic shape. Each crystal is assumed

to have one slip plane with three equally spaced slip directions.

Heterogeneous slip and stress fields are calculated from single

crystal slip properties subject to conditions of equilibrium

and compatibility. The computed initial yield surface of

this aggregate is very close to the Tresca condition, whereas

the surface in stress space corresponding to a finite offset

strain is close to a Mises condition. This observation

offers one possible explanation why experimentally observed

initial yield surfaces usually appear closer to the Mises

than to the Tresca yield condition. The computed yield surface

after an initial axial loading is found to have a corner at

the loading point, but this corner is "blunt" in the sense

that a corresponding surface of constant offset strain has

a rounded nose instead of a sharp corner.







The theoretical results presented in this dissertation

corroborate the theoretical results of Lin and Ito [56, 57],

as discussed above, including prediction of a Bauschinger

effect of the Masing type. Coincidentally, both the present

theory and the Lin-Ito theory require the specification of

four constants, two elasticities and two plasticity constants,

in order to be completely determined. However, the basic

ideas of the two theories are very different: the Lin-Ito

theory is based on the ideas of slip in a three-dimensional

crystalline aggregate, whereas the present theory does not

make any appeal to the crystalline micro-structure of metals.

The present theory has two advantages over the Lin-Ito

theory, namely, greater accuracy and greater efficiency.

Greater accuracy is achieved since the present theory can be

thought of as representing an infinite rather than a finite

polycrystalline aggregate. Greater efficiency is achieved

since the present theory contains less structural detail,

and this allows a greater quantity of results to be computed

from a given amount of computer time.

For completeness, we mention three other currently

accepted theories of plasticity that do not fit very well

into either of the two categories discussed above. The

first of these is the continuum theory of dislocations

[60, pp. 88-92], which was developed largely by Kr6ner [61].

This theory is concerned with the exact mathematical

description of dislocations in an elastic body, which is







accomplished by replacing the usual Euclidean metric tensor

with a Riemarnnian metric tensor. The second of these theories

is the theory of anisotropic fluids proposed and developed

by Ericksen [62]; here a fluid is assumed to consist of

molecules shaped like dumbbells, This theory has been promoted

as a kind of plasticity theory because in the solution for

Poiseuille (pipe) flow [62(b)] the material at the center

of the flow is found to move as a rigid plug. The third

of these theories is the mathematical theory of plasticity

introduced by Ilyushin [63]. Functional analysis of the type

expounded by Riesz and Nagy [64] is applied to general

loading paths in stress and strain space. Relationships are

assumed to exist between the curvatures of the loading paths

in the two vector spaces. Lensky [65] attempted to connect

this theory with experimental data. The author's brief study

of these three theories has failed to reveal any concrete

predictions of matters vital to plasticity, such as predictions

of stress-strain relations and/or changes in the yield surface

during plastic deformation.

In this chapter we have briefly outlined most of the

existing theories of plasticity, and we have shown that none

of these theories is entirely satisfactory for predicting

all aspects of experimentally observed stress-strain behavior.

This is true even for the restricted situation in which

temperature and time effects can be ignored. We conclude

that there is still need for further theoretical work in







plasticity. It is in the spirit of Zizicas [3(b), p. 448] that

we undertake the development of a new theory.

Instead of waiting for the physical theories
(which take into account the full details of
crystallographic structure) to come all the
way to meet the mathematical theories, it
may prove faster for the mathematical theories
to move and meet the physical theories some-
where halfway between the two. One should
be prepared to examine, or even replace
completely if necessary, the fundamental
assumptions of inr'ceeneltal theory. Certainly
this theory, in its present form, should not
be used as a basis for criticizing or dis-
couraging research work at variance with its
predictions. Such a point-of-view may delay
important developments. The formulation of
a proper set of mathematical stress-strain
relations should be guided primarily by the
aim of improving as much as possible the
description of the physical behavior of
materials. If approximations have to be
made for mathematical convenience, they can
be much better justified after the compli-
cations arising from a generally acceptable
set of stress-strain relations are clearly
demonstrated. A more accurate set always
serves as a reference for comparison with
approximate relations.

This dissertation introduces a new plasticity theory,

hereafter referred to as the radial element theory. This

theory is related to the physical theories mentioned above

in the sense that special importance is attached to the

locked-in stresses. In fact, a new concept is introduced -

a set of elemental stress vectors depending on both the

direction in space and the loading history at a material

point --which exploits the directional aspects of the locked-

in stresses in determining the overall stress at a point in a

material body. The radial element theory is a purely







mathematical theory; its only connection with empirical data

is through specification of four constants: two elasticities

(Young's modulus E and Poisson's ratio v, say), one yield

strain ey, and a work-hardening parameter X.

Among the successes that have been achieved with the

radial element theory are: realistic predictions of work-

hardening and a Bauschinger effect of the Masing type

(Chapter V); relatively simple calculations to obtain

anisotropic yield surfaces for complex loading paths

(Chapter VI); a method of calculating stress-strain behavior

for any arbitrary loading path (Chapter VII); and good

agreement with experimental data on commercially pure aluminum

(Chapter VIII). The experimental data are somewhat limited

in quantity and accuracy, but they include rwo different

loading situations, cyclic tension-compression loading of a

rod and torsion-tension loading applied to a thin-wall tube.

In summary, the most pertinent properties which the radial

element theory aims to predict are: (1) isotropic elasticity

relative to the current residual strain; (2) isotropic

initial yield condition; (3) subsequent yield conditions

which are anisotropic, with the anisotropy determined by

the history of deformations; (4) inviscid plastic flow,

also determined by the deformation history. For tests

conducted at a steady loading rate and at constant temperature,

these properties describe fairly well the mechanical behavior

of most polycrystalline metals.













CHAPTER II

GENERAL CONCEPTS
OF THE RADIAL ELEMENT THEORY


In the theory presented here, the yield surfaces, flow

rules, and hardening laws are not specified explicitly, as

is true of most plasticity theories, but rather they are

determined by the theory once a specific definition of an

elemental stress vector is adopted. The key to the theory

is the relationship that is assumed to exist between the

elemental stress vector and the deformation of a radial

element, which is the material element that reflects the

deformation history in a given direction. Different material

stress-strain behavior results if different relationships

are assumed between the elemental stress vectors and defor-

mations of the radial elements. Stress is defined in terms

of certain integrals of the elemental stress vectors over

an infinitesimal ellipsoid.

Suppose we wish to study the stress-strain behavior

at a point P in a material body. The mathematical theory

for this study can be developed as follows: Consider a

spherical surface embedded in the body in its initially

undeformed condition and centered at the given point P.







The radius R of the sphere may be finite if the body under-

goes only homogeneous deformations. However, if nonhomogeneous

deformations are allowed, then the sphere must be considered

as infinitesimally small R -+ 0. Throughout this dissertation

we assume the material is a "simple material", that is, one

for which the stress tensor at each material point is uniquely

determined by the history of the deformation gradient.

During an arbitrary sequence of deformations the sphere

will deform continuously into a sequence of ellipsoids of

various shapes and orientations. Mathematical description

of the undeformed sphere and of any one of the sequence of

deformed ellipsoids, relative to a fixed coordinate system,

affords the means of defining various finite strain tensors.

For any such strain tensor, the principal directions in the

deformed configuration must coincide with the principal axes

of the ellipsoid, and the principal strain magnitudes must

be related by an invertible function to the principal diameters

of the ellipsoid. The sequence of strain tensors computed

in this manner from a given sequence of deformations comprises

the strain history at the given material point P.

For definiteness and because it is useful, we introduce

here one example of a finite strain tensor: the logarithmic

strain, also sometimes called the natural strain. Let n

(a=1,2,3) be unit vectors directed along the three principal

axes of the deformed ellipsoid; let d be the corresponding

principal diameters; and let "log( )" signify the natural







logarithm function. Then X = d/2R are the principal stretches

and
a = log (X ), a =1,2,3
3 (2 .1)
_= n n log (X)

are the logarithmic principal strains and the logarithmic

strain tensor, respectively. The logarithmic strain has

the following two advantages over other strain measures for

use in connection with the theory of plasticity: (1) mean

and deviator components represent exact measures of volume-

type and distortion-type deformations, respectively, and

(2) experimental results for tension and compression tests

give the same stress-strain curve (up to necking or failure)

when logarithmic strain is used. The latter fact allows

the possibility that an initially isotropic plasticity theory

can agree with experimental data. For the many further

details concerning finite strain measures the reader is

referred to Truesdell and Toupin [66, 241-324] or Malvern

[67, pp. 154-182].

The first fundamental concept of the present theory

is the radial element. Consider a material element which

initially is a differential cone or filament radiating

outward from the center P to a point Q on the surface of

the undeformed sphere. See Figure 1. This radial element,

corresponding to a given fixed direction in the undeformed

state, will elongate (or contract) and shear as the sphere

deforms into a sequence of ellipsoids during a given defor-

mation history.




















elemental
stress
vector


deformation


deformed
ellipsoid


undeformed
sphere


resultant
stress
vector


FIGURE 1. BASIC CONCEPTS OF THE THEORY







The second fundamental concept of the theory is the

elemental stress vector f, which we define to act at each

point Q on the surface of the deformed ellipsoid. This

elemental stress vector is assumed to correspond directly

to the radial element along the line PQ. Let n be a unit

vector in the direction of PQ. Then the set of all elemental

stress vectors f(n) corresponding to the set of all radial

elements forms a vector field over the ellipsoid. Again

refer to Figure 1.

The vector f is assumed to depend exclusively on the

deformation history of the individual radial element upon

which it acts. The overall stress-strain behavior of the

material is governed by this relationship which we assume

to exist between the elemental stress vector and the defor-

mations of the corresponding radial element. For example,

elastic behavior is obtained if this relationship is assumed

to be linear and one-to-one, as proved in Chapter III.

Elastic-plastic behavior is obtained if the relationship

is linear and if the magnitudes of the elemental stress

vectors are limited in the manner specified in Chapter IV.

Suppose we are dealing with a material that exhibits

elastic-plastic behavior, and suppose that the material has

experienced plasticity in certain directions. That is,

suppose we computed the elemental stress vectors elastically

and we truncated those elemental stress vectors f that

exceeded a certain yield criterion. Then the plasticity







will be expressed as a reduction of the stress components

associated with those directions from the values predicted

by elasticity. Anisotropic effects as well as plasticity are

introduced into the theory in this manner. These ideas will

become clearer to the reader as he progresses through

Chapters III and IV.

At this point we must draw a vital distinction between

the elemental stress vector used in the radial element theory

and the ordinary stress vector introduced by Cauchy [66, p. 537].

Cauchy proved that the stress vector t acting on an elemental

surface is related to the surface normal unit vector n by a

homogeneous linear function t(n) =T-n, where T is the stress

tensor. This result follows from the principle of linear

momentum and is known as the Cauchy stress principle. In

the radial element theory f(n) is generalized so that it

no longer satisfies the Cauchy stress principle. This

generalization of the stress vector is necessary in order

to exploit the individual response of material elements

oriented in various directions.

If we wish to preserve intact the classical theory of

continuum mechanics, it becomes necessary to consider the

elemental stress vectors as an entirely separate concept

from the Cauchy stress. The elemental stress vectors can

be viewed as a structural entity of the material similar

to micro-stresses in a dislocation theory or they can be

viewed simply as a mathematical device to be used in







generating stress-strain relations. Regardless of how they

are viewed we shall continue to develop and use the elemental

stress vectors as if they indeed possessed a rigorous mechan-

ical basis.

Ultimately, however, we shall want to use the stress-

strain relations generated by the radial element theory in

the solution of boundary value problems, such as determination

of the bending or twisting of some engineering structure under

a set of loads. For this latter purpose we must ignore the

details of the radial element theory and accept only the

overall stress-strain relations. Finally, we must combine

these stress-strain relations with the established principles

of continuum mechanics [67, Chapter 5], including the Cauchy

stress principle, in order to obtain a meaningful, well-set

mechanical problem.

The following equations, which resemble force and moment

equilibrium equations, are necessary conditions for the con-

struction of a unique stress tensor.

F= f dA=0 M== rxfdA=0 (2.2)
A A

Here r is the radius vector from the center to the ellipsoid

surface and A is the total surface area of the deformed

ellipsoid. Equations (2.2) represent restrictions on the

permissible class of elemental stress vectors; the specific

constitutive assumptions for f reported in Chapters III and

IV do satisfy these restrictions.







The third fundamental concept of the radial element theory

comprises the resultant stress vector t and the couple stress

vector c, defined as follows:

!t(N) = -- f dA, c(N = rx f dA .(2.3)
N AN N AN

Here N is a unit vector normal to the cross-sectional area aN,

and AN is one-half the surface area of the deformed ellipsoid;

both areas aN and AN are determined by N as shown in Figure 1.

Unfortunately, the functions t() and c(N) are nonlinear, in

general, which eliminates them as direct candidates for the

stress tensor and couple stress tensor, respectively.

By comparing equations (2.2) and (2.3), we see that

equations (2.2) are equivalent to

(N = -t (-N) and c (N) = -c-N) (2.4)

Equations (2.4) state that the resultant stress on one side of

a given plane surface is equal and opposite to the resultant

stress acting on the other side of the same surface. If these

equations were not true then the resultant stress would not

be unique, since we would not know which resultant stress

vector, i.e., t(N) or -t(-N to assign to a direction N.

The following procedure has been tried in conjunction

with elastic-plastic behavior for a variety of loadings and

seems to yield a unique stress tensor. Suppose we are given

a set of elemental stress vectors f(n) distributed over a

known deformed ellipsoidal surface. The function f(n) can be

quite general, with discontinuities occurring in the







derivatives and in the function itself, provided the vectors

are real and provided equations (2.2) or (2.4) are satisfied.1

Consider any orthogonal triad of unit vectors N (a =1,2,3),

and compute by means of equation (2.3)1 the three resultant

stress vectors t =t(N ) corresponding to these N In a

Cartesian coordinate system with N as base vectors, the

components of t form a matrix T= [tl ], (i,a=1,2,3).

That is, the components of ti= (tl,t21,t31), t 2= (t12,t22,t32 )

t = (t13,t23,t 3) form the matrix


tll t12 t13
T t21 t22 t23

31 32 t33-

Note that the second subscript identifies the plane on which

the vector acts and the first subscript identifies the component.

Define a positive definite norm in terms of the off-diagonal

elements of T by the expression

2 2+ 2
M = (t12+ t21) (t23+ t32+ (t31+ t13 (2.5)

We can now search for and obtain an orthogonal triad of unit
2
vectors N' by the requirement that the norm M be minimum when
-ay


IThe maximum generality of the function f(n) that satisfies
equations (2.2) and that will permit construction of a stress
tensor has not yet been established.

2The orthogonal triad of vectors IN obtained by minimizing
the norm M probably is unique for any reasonably well-behaved
radial element theory. However, a proof of uniqueness is not
available at this time.







the resultant stress vectors t' =t(N') or the matrix T'=T(N')

are computed relative to N'. In fact, for all specific distri-
-a
butions of f(n) considered in this dissertation, the norm M

can be made zero by a proper choice of N'. Then T' becomes
-a
as diagonal as possible, with t2 =-t', ti3 -t2' t = -t3

and tl' t2, ti3 are the "principal" resultant stresses.

We now define a resultant stress tensor T by the require-

ment that the components of T coincide with the elements of

the matrix T' in the special coordinate system with N' as base
-ar
vectors. That is, we define t! =t! where T'=[t ] is the
la i' -
tensor T referred to the coordinate system defined by N'.
-a
Relative to any fixed Cartesian coordinate system the components

of N' form a rotation matrix R=[N' and the resultant stress

tensor T satisfies the transformation equations

T= R*T'*R T'= RTT-R .(2.6)

The deviator stress tensor S =[s.] is defined by the equations

S= + P1, p = -(tll+ t22+t33)/3 (2.7)

where 1=[ ij] is the unit tensor and -3p is the trace of the

tensor T or the matrix T'. The ordinary stress tensor Z=[cr. .
S1J
can be written as

E = S + i = T + (+ p) 1 (2.8)

where -c is the mean stress to be determined by a separate

relationship between mean stress and mean strain. Equation

(2.7) will prove to be most useful in Chapters V, VI, and VII

of this dissertation.







The above procedure forms the basis of a general computer

program for predicting the stress from a given arbitrary strain

history. This computer program is described in Chapter VII and

is one of the programs listed in Appendix A. The above pro-

cedure is complicated, and it may seem to the reader to be an

arbitrary and unnecessarily elaborate exercise of algebraic

manipulations. The author has considered this matter in detail

and has concluded that the procedure presented is the simplest

and most direct method of constructing a stress tensor from a

general distribution f(n) of elemental stress vectors.

The most controversial feature of the above procedure is

the possibility of predicting a nonsymmetric stress tensor.

I could have defined the deviator stress as the symmetric part

of the deviator of the tensor T, and this would have given the

ordinary symmetric stress tensor. However, my present purpose

is to raise the possibility that the stress tensor may actually

be nonsymmetric for certain loadings, and so I retain the

definitions (2.7) and (2.8). In Chapter VII it is demonstrated

that this procedure does predict a nonsymmetric stress for

plastic deformations that involve rotations of the principal

strain axes.

It may be possible to resolve the couple stress vectors

c(H) into a couple stress tensor by a procedure similar to that

proposed above for the resultant stresses. However, as our

primary interest in this dissertation is with the resultant

stresses, hereafter we shall ignore the couple stresses.







Another definition of stress which possibly could be

more useful for describing material behavior during a finite

strain involves first transferring (via the deformation

function) the elemental stress vectors f from the deformed

ellipsoid to corresponding points on the undeformed sphere.

Then the resultant stress vectors t(N ) are defined as

integrals of the vectors f over the hemispheres of radius R

that are symmetric about a given set of orthogonal unit

vectors N The stress tensor, in this case analogous to

the second Piola-Kirchoff stress tensor, is found from the

vectors t(N ) as before. The two definitions of resultant

stress are not the same, but they can be made equivalent by

appropriate choices of the relationship between the elemental

stress vectors and the deformations of the radial elements.

In the case of stress calculated relative to the undeformed

sphere, we must correlate this stress with one of the strain

tensors associated with the undeformed state.













CHAPTER III

A CLASS OF ELEMENTAL STRESS VECTORS
FOR PREDICTING ELASTIC '.'-FERIAL BEHAVIOR


The general theory presented in Chapter II was valid

for finite strains, and in principle it is possible to con-

tinue this analysis without invoking small strain approximations.

However, by making such approximations the theory is greatly

simplified. Questions of which stress and strain measure

to use are avoided, and computations of yield surfaces are

brought within the scope of a modest computational program.

Therefore, in the remainder of this work we shall restrict

our attention to the case of infinitesimal strains. For

this case, the difference between the deformed ellipsoid and

the undeformed sphere is negligible. The elemental vectors

f(n) and all integration to obtain the resultant stresses

t() may be referred to the surface of the undeformed sphere.

For simplicity we shall always consider this sphere to have

unit radius R=1.

Let a.. be the components of stress as defined by

equation (2.8) and let e.. be the components of the infini-
1J
tesimal strain tensor [ 67, pp. 120-135]. The deviator

components s..,e.. of stress and strain are defined by the

following decomposition:






s.. =c..- o6. ej .= e..- 6ij (i,j =1,2,3)

1 1
0 = 1(l+ 22+ 033) E = -('ll+ 22+ 33)

Here a,e are the mean stress and mean strain and 6.. is the
13
Kronecker delta. These definitions imply

Sll+ s22+ 33 = 0, ell+ e22+e33 =0 .

Throughout the remainder of this work we shall be concerned

mainly with relationships involving the deviator components

s.. and eij, instead of the "actual" or "true" stresses and

strains, since, as usual, we assume plastic behavior to be

confined to these variables. The deviator strain tensor,

with components e. is real and symmetric by definition,

and is assumed known. The deviator stress tensor, with

components s i, is real but not necessarily symmetric, and

is the entity that we seek to compute.

We shall have use of the elasticity constants

E, v- Young's modulus, Poisson's ratio

= E/(1 2v) volume or bulk modulus
} (3.1)
S= E/(l + v) shear modulus(3

and the following special constants depending on a parameter X:

4 -2X 4_X 4 -2 U X
X=AL(4-3 ), 7=A( 3--), C=x+?= =.- (3.2)

Our assumption (page 4) that mean stress and mean strain

are elastically related can be written simply as o=se.



'The shear modulus is customarily written as 2p or 2G;
throughout this dissertation I write it simply as l to save
repeatedly writing the "2."







We require this elastic volume-compressibility relationship

to hold at all times.

A specific constitutive assumption for the dependence

of the elemental stress vector f on the deviator strain

tensor E is now introduced, which predicts linear elastic

behavior when the stress is computed.

f =xE.n+ t(n-E-n) n (3.3)

Here E=[e..] is the deviator strain tensor, n is a general
13
unit vector, and x,7 are the constants defined by equations

(3.2). The motivation for considering f in this form is

that equation (3.3) can be conveniently modified to predict

elastic-plastic behavior, which is the main purpose of this

dissertation. The modified form of equation (3.3) will be

presented in Chapter IV, with further developments presented

in Chapters V, VI, VII.

We can prove that f in the form of equation (3.3) depends

directly on the deformation of a radial element in the direction

of n, which is one of the requirements of the theory expressed

in Chapter II. Since E is the deviator strain tensor, the

inner product E-n represents the deviator strain vector

associated with the unit vector n. A further contraction

of E-n with n gives the magnitude n-E-n of the normal component

of the deviator strain vector, and the vector (n-E-n)n is

the normal component. Hence, equation (3.3) defines f to be

linearly related to the deviator strain vector and the







normal component of the deviator strain vector associated

with the radial element in the direction of the unit vector n.

Further observations can be made by deriving expressions

for the normal and tangential (shear) components of f. The

equations

f = (f n)n= an

f =f-f =x(E.n1-an) (3.4)
-t -nI

where a= n-E.n

show that the normal component of f is proportional to the

normal component of the deviator strain vector in the direc-

tion of n, and that the tangential component of f is

proportional to the maximum shear component of strain associ-

ated with the direction n. From equations (3.2) and (3.4)

it seems reasonable to limit the parameter to the range

0< X< 2, since for this range stretch in a given direction

is accompanied by a tensile normal component f and shear
-n'
in a given pair of directions is accompanied by a tangential

component ft corresponding to the same directions, with f
-t-
oriented so as to support the shear.

To simplify the calculations we now refer the vectors

and tensors to a Cartesian coordinate system that coincides

with the principal strain axes. Thus

e 0 0

E= 0 e2 0 n = (cos 81, cos 82, cos 83)

0 e3







where e.(i= 1,2,3) are the principal deviator strains and

6.(i =1,2,3) are the angles between the vector n and the

coordinate axes. Substituting these results into equations

(3.3) and (3.4) yields

f=[ (xel+ 7a)nI, (xe2 + ?)n2, (xe3 +r a)n3]

f =Can, ft =x(el- )nl,(e2-)n2 (e3-)n3] (3.5)

2 2 2
= n-En= e ln + e2n2 +e n2
1 1 2 2 3 3

Additional expressions for f and f are given by
-n -
3 3
f = ne. cos 28 f =- E 9.e. sin29.
-n 2 1 -t 2 -1 1 1

where 1 = (sin 81, -cot 9 cos 82,-cot l cos 83
1 1 2- s(3.6)
2 = (-cot 92 cos 81, sin 82, -cot 62 cos 93)

A3 = (-cot 83 cos 81, -cot 83 Cos 82, sin 83)

The 8. (i=1,2,3) are unit vectors tangent to the unit
-1

sphere at the point Q determined by the unit vector n. They

each lie in a plane containing one principal strain axis

and the vector n, and they are directed away from the point Q.

Equivalence of the two expressions for f and f can be
-n -t
shown by substitution.

We now prove that the overall force and torque vanish

as required by equations (2.2). Equation (3.5)1 for the

elemental stress vectors is used, which is equivalent to

equation (3.3) expressed in principal strain coordinates.

Introducing spherical coordinates n =cos 8, n2 =sin cos 0,

n3 =sin sin as shown in Figure 2, and imposing the

integration limits 00 <0 21, 0 O8 s, we obtain























































FIGURE 2. SPHERICAL COORDINATES






FI = F d0 d8 sin (fl)

= el d d sin cgos (x + cos28)

+ 7f dO d sin38 cos 9 (e2 cos20 + e3sin2

= el[2 [x (sin28) + (- cos4 8) 1

+ ?[(e2+ e3)0+ (e2- e3) sin 20 ]2 [ sin [83= 0

MI = d0 dG sin B (f3cos 8e2 f2cos 83)

= x(e3- e2) dOJ d9sin38 sin O0cos0

= (e3- e2[ [sin2 2 cos 8 (2 +sin28) =0 .
(e3-e2)[sn 0 [ 3 0

.Proof that the vector equations (2.2), namely F=M=0, are
valid can be completed by repeating the integration for
the other two axes. A simpler proof is obtained by symmetry,
since we can simply permute the indices without affecting
the above results.
We now compute the resultant stress vectors with respect
to the coordinates along the principal strain axes, using
the definition (2.3). We start with the vector t1 = (tllt21,t31)
corresponding to the el-axis, and again we use the relations
nl = os n2 = sin cos 0, n3 =sin sin 0. This time the
integration limits are 0 0 < 27r, 0G'9 /2.

tll = j dO j dEsin E) (fl)
= eL[0x2 [x sin2 ) + (-- cos8) i)T/2

+ ~[(e2+ e3)0+ (e2-e3) sin20]21T [sin o/2

= el(x+) + (e2+e3) = (x+ )el = e







t2 1 Idl dG sin (f2)
21 IT

= dj d9 sin 9 cos 0 (x + sin 9 cos 0)

+ ~ d dB sin28 os (elcos 8 + e3sin 9 sin2f) = 0

t31 d de sinG (f3)

e3 d 2 2 2
e= d dO sin B sin 0 (X +7- sin2 sin 0)

+ 7 dj d9 sin28 sin 0 (elcos 2 + e2sin2 os 2) = 0

The remaining expressions for the resultant stresses can be

obtained simply by a permutation of indices.

S= (t12't22t32) = (0, pe2,0)

3 = (t13t23't33) = (0, 0,Le3)

Following the procedure of Chapter II, we can write

the above results in matrix form.

tll t12 t13 e1 0 0
T = t21 t22 t 23 = 0 e2 0 (3.7)

t31 t32 t33 0 0 e3

The norm M defined by equation (2.5) vanishes for the matrix T

when computed relative to the coordinate system used here.

Hence, by definition, the axes of this coordinate system are

the principal stress axes and by equation (2.7) the quantities

s.=t.. (no summation) are the principal deviator stresses.

These quantities determine a deviator stress tensor S=[sij]

which, because of equation (3.7), is related to the deviator

strain tensor E=[e..] by the classical elasticity equation
21







S = pE (3.8)

Note that the principal axes of stress and strain coincide,

a result that is true in general if and only if the material

behavior is isotropic elasticity. Obviously, the definitions

(3.2)1,2 of the constants x,7 were chosen so that the present

results, equations (3.7) and (3.8), would emerge.

Equation (3.7) is valid only for resultant stresses

computed relative to the principal strain coordinates. It

is important in the present theory to know how the resultant

stress vectors transform during a change of coordinates.

Consider an arbitrary Cartesian coordinate system, related

to the principal strain coordinates through an arbitrary

orthogonal transformation. Equation (3.3) can be written

in component form as
3
f.= x e..n.+7Yn., (i=1,2,3)
j=1 (3.9)
3 3
a F = ejkn nk
j=1 k=l
Introducing spherical coordinates n1 =cos 8, n2 = sin 8 cos ,

n3 = sin 9 sin0, we can rewrite equations (3.9) as

fl = (xell+7C)c +x (cos 6+x(e+ el3sin) sin 8

f2 = (xe22+ ra)sin 9 cos 0+x(e21pcos 8+ e23sin 8 sinf)
(3.10)


a = e cos +e22sin 2 cos 0+e sin 2sin 0

+ e23sin2 sin20+e31sin28 sin+ el2sin 28 cos .







After a lengthy evaluation of integrals having the limits

0Q gs2i, 08
stress vector t = (tllt21t31) in the new coordinate system

as follows:

t = id9 d8 sin (fl) = (x+ ))ell= ell

t21 = d70J dsin8 (f2) = (x+ )e21= e21

t31 = l d sin e (f3) = (x + :)e31= e31

where S = 2L/(3 X). By repeating the integration with

respect to the other two coordinate axes, or simply by

permuting the indices, we find


L2 = (t12,t22,t32) =

-3 = (t13't23,t33) =

The matrix form


T =



shows that

the strain

tensorial.


(e12 pe22, e32)
(e13, 3e23 ,4e33)


tll t12 131 pell 1 e12 e 13

t21 t22 t23 = e21 Ae22 23 (3.11)
t31 t32 t33- e31 ge32 pe33-

the resultant stress T is linearly related to

E, but that in general the relationship is not

The exception occurs when X=l, giving simply


T=LE. Since strain transforms as a tensor, it follows

that in general the resultant stress T does not.

The relationship between the stress and strain tensors

and the resultant stress matrix T =[tij as computed from

equations (2.3) and (3.3) will now be made definite. Let







S=[s..] and E=[e ij be the deviator stress and strain

tensors, related here by equation (3.8). If primed and

unprimed quantities refer to two different Cartesian coor-

dinate systems, then we have the following transformation

equations:
3 3
e..= F F a. a = s. = t/i
13 k=l i=1 ak 3R kR ij
3 3
e'.. = a kia e =s / = t!./ (3.12)
i3 k=l =1 ki j ki ij / 1iij

ij = [p if i= j or =2p/(3 -X) if i /j]

Here A=[a..] is an arbitrary rotation matrix, the elements
13
of which obey the orthogonality conditions [67, p. 27].
3 3 .
Sa a = a =6 = if i
k=l ik jk k=l ki kj ij =0 if ij j
Although the resultant stress matrix T does not transform

as a tensor, we can always construct the deviator stress

tensor S by first finding the principal axes of T, which are

identical to the principal stress axes. Then the elements

of S can be determined from the elements of T by equation

(2.7). The procedure of Chapter II can be rephrased as an

eigenvalue problem. Thus, the principal stress axes N

can be found by solving the equation (if a solution exists)

C (t..+t..-26..t) N = (j,a=1,2,3) (3.13)
i=l 13 ji 1] a t ia
and the eigenvalues t are related to the principal deviator

stresses s by the equations

s = t +t+t3)'3, ( = 1,2,3) .(3.14)







In solving equation (3.13) it must be recognized that, in

general, as for example when plasticity is involved, T is

nonsymmetric and T(N ) is a nonlinear function. The fact

that T(N ) can be nonlinear requires, in general, an iteration

procedure for calculating N involving solution of a sequence

of linear eigenvalue problems of the form (3.13). This topic

will be resumed in Chapter VII.

There may exist definitions of the elemental stress

vector f other than the one proposed by equation (3.3)

which would satisfy the equilibrium requirements F=M=0,

and also predict linear elasticity. For example, one could

imagine the elemental stress vectors for a cubic lattice,

or some other structured lattice, as being zero in all

directions except the principal lattice directions. The vec-

tors f in these lattice directions, for a given nonzero

deformation, would be defined so as to make the resultant

stresses finite. Such theories, which may or may not be

isotropic, would involve complicated expressions for the

elemental vectors and the resultant stresses. In contrast

the present theory, based on equation (3.3), involves only

simple trigonometric functions. Moreover, studies have

shown [60, p. 5] that various microstructural theories of

a given class generally lead to the same macroscopic results.

Accordingly, the present investigation is restricted to

equation (3.3) and to one generalization of equation (3.3)

that involves plasticity.













CHAPTER IV

A CLASS OF ELEMENTAL STRESS VECTORS
FOR PREDICTING ELASTIC-PLASTIC MATERIAL BEHAVIOR


Plasticity is now introduced into the theory by limiting

in a special way the magnitudes of the elemental stress vectors

on the unit sphere. Recall the definition of f adopted in

Chapter III which predicts linear elasticity, namely

f = x E.n+ 7(n-.En)n (3.3)

= A(34 ), 7 = A(3X-_) (3.2)1 2

Here E=[e..] is the deviator strain tensor, n is a general

unit vector, and x,7 are elasticity constants.

For plasticity, the elemental stress vectors are not

uniquely related to the strain, as in equation (3.3), but

are determined by the (time independent) strain history.

The deviator strain history E(m) at a material point can be

defined by a sequence of strain increments dE(m); thus

E(0) = 0 E(m)= E m-l)+dE(m) m=1,2,...,M (4.1)

This discrete representation of the strain history facilitates

computations on a digital computer, as we shall see later.

The limit of a continuously varying strain history can be

approached by letting each increment dE(m) become very






small and at the same time letting the end-point integer M

become very large.

The most reasonable assumption regarding plasticity is

to limit the magnitude of the tangential (shear) component

f of the elemental stress vector f. This is in keeping
-t
with the physical assumption that plasticity is a phenomenon

involving shear stress and unaffected by uniform pressure [68].

Accordingly, equation (3.3) can be generalized to include

plasticity as follows:

f(0) =0 f(m)=f 1i if gt gl, m=1,2,3,...
Y/gt ifgt > Y. 2)
g=f -+x dE n +n (n"dE(n) gt = [g (g-n)2]

where Y is a characteristic yield stress. Equations (4.2)

represent a procedure of repeatedly calculating a vector g

and then the magnitude gt of its tangential component and

then the elemental stress vector f(m) corresponding to a

given deviator strain increment dE(m). The initial condition

f(0)= 0 implies that the material initially is in an isotropic

annealed condition.

Equations (4.2) yield a solution for f(m) in terms of

the deviator strain E(m) only in the case of a radial loading.

Then dE(m) and E(m) are proportional: dE(m) dE; E ( =

(c1+. .+ e) dE. Substituting this result into equations

(4.2) we find for radial loading

f(m)= -g if gt Y-
S/gtif > (4.3)

g=xE(m).n+7nn(n-E(m).n, g =[g.g- (gn)2]







Since g is computed elastically in terms of the (total)
strain E(m) and then truncated to yield f(m equations

(4.3) represent what is called a total-deformation type

theory. Such a theory is valid for radial loadings and

approximately valid for near-radial loadings, but it certainly

would not predict reasonable results for loading paths that

include stress reversals [ 1, p. 47].

The above definition of f(m) could perhaps have been

stated with greater mathematical elegance in terms of time

derivatives and/or the strain invariants. However, the

form presented here is most easily adapted to computer pro-

gramming. Hence, I have adopted equations (4.2), and in the

case of radial loadings equations (4.3), as the basic

definitions of f to be used throughout the remainder of

this work.

Equations (4.2) represent a procedure of computing and

then truncating those elemental stress vectors that exceed

a certain yield condition. The proposed yield condition

tests the magnitude ft of the tangential component of the

elemental stress vector against a characteristic yield stress

Y, which is assumed constant for all elemental vectors and

all time. The present theory is essentially an incremental-

type theory in which each elemental vector obeys the same

elastic-perfectly-plastic law. The proposed yield condition

predicts an initial yield condition of the Tresca type when

expressed in terms of stress or strain, as we shall demon-

strate below.







To find the initial yield condition, we assume residual

stresses and strains are zero initially, so that the material

behaves elastically as discussed in Chapter III. Then we

consider those states of stress or strain for which the

maximum value of ft on the unit sphere is equal to Y, the

characteristic yield stress. Symbolically


sphere ) f(ek)] =Y (4.4)
(sphere) t 13

where e.. represents a point on the initial yield surface

in strain space. In terms of coordinates along the principal

strain axes, we can write the tangential component of f as

t = x[(el-a)nl, (e2-a)n2, (e3-a)n3

2 2 2
f = (ft't)= xD e ne c2 (4.5)
i=li 1
2 2 2
a = elnl + e2n2 +e3n3

Let us first investigate the yield condition for the

special cases of pure shear and axial strain, whereby

e2 =-el, e3 =0, and e2 =e3 =-el/2, respectively. In the
case of pure shear

2 2 2 2 2 2 2 5
a = el(n n2), ft = xel[nf + n2 (n n )2

In the case of axial strain

2 3 2 &
a = el(3n 1), ft xelnl(l n) .

For both cases, the maximum value of ft occurs when

n = (51, F1, 0), as proved below. Let ek, e denote the
deviator yield strains for pure-shear and axial-strain







loadings, respectively, and let sk = ek, S = ey denote

the corresponding deviator yield stresses. That is, at

yield e =e1 =-e2, e3 =0 for a pure-shear loading, and

e =e = -2e2 =-2e3 for an axial-strain loading. Invoking

equation (4.4) and the above expressions for ft, we obtain

the following equivalence relations among the various measures

of yielding:

3 Y 3 -Y 3 X Y
ek= y ey x' sk y x -) (4.6)

We now investigate the yield condition for arbitrary

strain conditions. Introducing spherical coordinates

nl =cos8, n2 =sin cos0, n3 =sin8 sing, we can rewrite

equations (4.4) and (4.5)2 as.


ma [f(8, )/] = ek' where

ft/x = (e cos 9+ e2 sin 9 cos 0+ e3 sin sin -a ) (4.7)
2 2 2 2 2 3
[(el- 2e2cos 0 -2e3sin 0)elcos 8 e2e3sin 2 sin 20 2
S2 2 2 2 2 2 2 2 sn .
+(l-sin 8 cos )e2 cos 0+ (1-sin 9sin 0)e sin2 0

Let us now seek the maximum values of the function ft(9,),

where the range of variables is 0 < 8 0 <-s 21. Note

that ft =0 when =0 and 8 =7 because of the multiplicand

"sin 8." But ft in general is nonzero. Hence max(ft) must

occur in the restricted range 0< 8< i. Necessary conditions

for max(ft) to occur at a given point (8,0) are af /39=

3ft/ 30=0. Thus






8 2 2 2
0 = (ft/x) = (A -B cos C sin 0)sin 28
a 2 2
0 = (ft/x) = (B -C)sin 6 sin 20

where A = (2a-el)el, B = (2a-e2)e2, C = (2a-e3)e3

Suppose 0< 8 <, so that sin 80. Then

(A B os2 -Csin20)cos 8 = 0, (B C) sin20 = 0 .

The first equation is satisfied if 8 =T/2, so that cos 8= 0.

In this case, the second equation becomes

2(B -C) sin20 = (e2 e3)2 sin 4 = 0
2 2 2
since B-C = (2ae2- e2) (2ae3- e) = (e2- e3) cos 20

In general, e2 e3, which implies sin 4 = 0. Thus 0 must

have at least one of the values 0=n7/4 where n=0,1,2,...,8.

Plugging these values into the expression (4.7)3 we find

ft/x = abs (e2 e3) sin 0 cos 0

0 if 0=0, i/2, i7, 31/2, 27r

Sabs (e2 -e3) if 0=T/4, 3T/4, 51/4, 77T/4

The initial yield condition in this case is

ek= m (f/x) = abs (e2-e3)

provided the maximum value of ft occurs at the points 8=7/2,

0 =ni/4 with n=1,3,5,7. Continuing in this manner, or simply

permuting indices, we find the following equation which we

recognize as the Tresca yield condition.

ek = -max (Ie2 e3 e3 el el -e2) (4.8)

This equation states that yield first occurs when the

maximum shear strain (or stress) reaches a critical value.






The directions corresponding to each segment of equation

(4.8) can be specified by unit vectors as follows:

ek = 4le -e : n = \ (1, +1, 0)

ek = e2-e31 : n = 4 (0, +1, +1) (4.9)

ek = e3 -ell : n = (+1, 0, +1)

These unit vectors n indicate the points on the unit

sphere at which yielding first occurs. If the principal

strains have distinct values, then only one line of the

expressions (4.9) can be valid at any one time. Suppose

the first line is valid, which was true for the case of pure

shear analyzed previously, and suppose the deformation is

continued beyond initial yield. Then there will develop

four regions on the unit sphere surrounding the four points

n=1= (+1, +1, 0), inside of which the elemental vectors

are restricted by the plasticity equation ft =Y. Outside

of the four regions ft< Y, and the elasticity equations of

Chapter III continue to govern the elemental stress vectors.

In general, the boundary between the elastic and plastic

regions on the unit sphere can be very complicated. There

is only one situation known to the author in which a closed-

form expression for this elastic-plastic boundary exists.

If the principal strain axes are fixed in the material and

the loading is radial (i.e., the strain magnitudes el,e2,e3

increase proportionally), then the boundary between the elastic

and plastic regions is given by the following equations:






23 3 2 2
ek (ft/)2 = ( e cos ) ( I e cos .)2 (4.10)
i=1 i=1 1
or
2 2 2 2 2
2 (el-2e2cos 0-2e3sin )el cos 8-e2e3sin 8 sin 20 2
= 2 2 2 2 2 2 2 2 sin 6
k +(1-sin 8 cos 0)e2 cos f+(l-sin 9 sin 0) e3 sin 0$

The complexity of the foregoing equations for the elastic-

plastic boundary on the unit sphere even for the restrictive

situation of fixed strain axes and radial loading obviously

limits the number and type of closed-form solutions that may

exist for the stresses when plasticity is involved. Two

closed-form solutions, obtained for pure axial loading and

reversed axial loading, form the subject matter of Chapter V.

Other solutions, which do not necessarily involve fixed axes

or radial loadings, were obtained numerically via the digital

computer and are presented in Chapters VI and VII.

We now prove that if the principal strain axes are

forever fixed in the material, then the principal stress

axes are also fixed and the two sets of axes coincide. The

proof follows from the symmetry of the function f(n) relative

to the principal strain axes. Equations (4.3), which are

valid for radial loadings, can be expressed in principal

strain coordinates as follows:

2 2 2 2 2 2
f=gF(gt/Y), gt=x(e n1 +e2 2+e3 n3 -)

2 2 2
g=x(elnl,e n2,e3) + 770m, a= elnl + en + e3n3 (4.11)

where F(x) = [1 if x 1; 1/x if x >1] .







Let us calculate the elemental stress vectors corresponding

to the four unit vectors n= (a,+b,+c), where a,b,c are
2 2 2
positive constants subject to the restrictions a +b +c =1.

Thus

n(l)= (a, b, c) : f(1)= [x(ela e2b, e3c)+7(a,b, c)]F(gt/Yl
n(2)= (a,-b,) : f(2)= [(ela,-e2b,e3c)+ 7Q(a,-b,c)]F(gt/Y)

n(3)= (a,b,-c) : f (3)=x(ela,e2b,-e 3)+7a(a,b,-c)]F(gt/Y)

n )= (a,-b,-c) : f(4)= [x(ela,-e2b,-e3c)+ 7 (a,-b,-c)]F(gt/Y)
2 2 2 2 2 2 2 !-
where g /Y=(e a +e2 b +e c -a /e

Consider now the sum of these four elemental vectors
4
Z fi) = 4a (xel+ a) F(gt/Y) (1,0,0) (4.13)
i=l(i)
This sum has a nonzero component in the el-direction only.

From equation (2.3) we see that the resultant stress

vector t is computed by integrating the differential vector
-1
dt1 =fdA/i over the hemisphere defined by n1 0, where

n= (nl,n2,n3) is an otherwise arbitrary unit vector. An

equivalent vector t is obtained by integrating the differ-

ential vector

dt = (fl) +f + f + f )dA/IT (4.14)
-1 _(1) -(2)+ (3) (4)

where f are defined above; here each component f dA/7I
-(i) -(i)
is integrated over one quadrant of the given hemisphere.

By comparing equations (4.13) and (4.14) we conclude that

tl has a nonzero component in the el-direction only. Similarly,

we find that t points in the e2-direction and t3 points

in the e3-direction. We conclude that for a radial loading







the resultant stress matrix T=[ti ] is diagonal (t. = Oif ia)

when computed relative to fixed principal strain axes. Recall

that the principal stress axes are identical to the principal

axes of the matrix T.

The same conclusion can be reached similarly for the

case of fixed principal strain axes and arbitrarily varying

strain increments del,de2,de3. Although I omit details in

order to save space, I offer the following outline of this

proof: First assume a given distribution f(0)(n) of elemental

stress vectors which is symmetric in the sense of equations

(4.12). Then introduce an arbitrary strain increment and

compute the new elemental stress vectors f(1)(Q9. Proof

follows by observing that the distribution f(l)(Q) is also

symmetric in the sense of equations (4.12). The general

case in which the principal strain axes rotate relative to

the material is too complicated to allow such simple con-

clusions to be derived.

The following is a discussion of the essential features

of the radial element theory as regards computing elastic-

plastic stress-strain behavior from equation (4.2). This

discussion should help the reader to understand the develop-

ments of the theory presented in the next three chapters.

Consider a loading and unloading of an initially isotropic

material that involves plasticity. Before loading the ele-

mental stress vectors f(n) associated with a given material

particle are all zero; here n represents the set of unit







vectors pointing in all possible directions. During loading

the vectors f(n) vary with the strain according to equations

(4.2), and the stress is computed in terms of these f(n)

by the procedure of Chapter II. When unloading occurs the

stress drops to zero; however, the strain and the set of

elemental stress vectors fo(n) after unloading generally

are not zero. This situation is analogous to a beam which,

after plastic bending, has a residual curvature and locked-in

stresses even when the bending moment is zero. (This beam

analogy applies to many parts of the present theory.)

The importance of the residual stress vectors fO(n)

is that, for any fixed material particle and any moment of

time, this set of vectors contains all the information about

past deformation history sufficient to predict the future

mechanical behavior. Suppose we are given a finite increment

of strain A E along a known strain path. Then knowledge of

f (n) and A E enables us to compute the elemental stress

vectors f(n), and hence the stress, corresponding to this

new increment of strain. Furthermore, from the set of

vectors f (n), we are able to compute the entire yield

surface at each step of a strain history, as will be demon-

strated in Chapter VI.












CHAPTER V

STRESS-STRAIN BEHAVIOR FOR AXIAL LOADING
AND REVERSED AXIAL LOADING


Consider a monotonic loading with fixed principal

strain axes in which the principal deviator strains are

maintained in the constant ratio e2 = e3 =-e1/2. This case

is called axial strain or axial loading, and includes the

common tension and compression tests of a cylindrical

specimen. Referring the deviator strain tensor to the prin-

cipal strain axes, we can define axial loading by the following

expressions:

0e 0 e(0) =0 and e(t) 20
E = 0 -e/2 (5.1)
0 0 -e/2 or e(t) <0 for t>0 .

For this case we can use equations (4.3) to compute the

elemental stress vectors f. We begin by computing

g= (gl 2' 33)
2 2 2 -2
a = n.E.n = e(n -n2/2-n /2)= e(3n -1)




2
gl = (7c + xe)n3 = e n (3Bn1+ a)
2 (5.2)
g2 = (770i xe/2)n2 = e n2 (3T7nl )

g3 = (T7a -xe/2)n3 = e n3(3Tnl- -)






Here a = 2x 7 and p =x + 77. Normal and tangential components
of g are given by

n = Can = e(3nl -1) (nl,n2,n)
3 2 2 2
S= X(E.n-"n- ) = -xe[ (1 nl)n, -nln2, -nln3] (5.3)
( 3 2
g= (= x e nl(l-n n) .

With these results the expression for f becomes


= j if gt = e sin28 (5.4)
S Y/gt if gt/Y > 1 Y y

where 8 is the spherical coordinate defined by n =cos 8.
We recall from equation (4.6) that e =4Y/3x is the yield
strain for axial loading. For convenience we define three
new variables 1, 2', *, as follows:
e
l 2 = + cos =- (5.5)

1 for 0 8< or 2 Then f=g r (5.6)
cos */sin29 for 1 O8< Q 2 or T- 2 <<01- : l

This situation is illustrated in Figure 3. We see that
plasticity develops in two expanding parallel bands on the
unit sphere, centered on the circles 8 = /4 and 8=31~/4,
for the case of axial loading. Resultant stresses are
computed as follows:
tll 2I2 d. '/2 dB sin8(fl)
o o
S2 S d sin (gl) + cos i S2 d sec (g1)
4 2 e (5.7)
cos4 8 sin28). ev .28-sin28
= e[3(- ) +a( 2 )]S +- 26L +ae]8

= p(e +e y -esin ) = s y( -tan + sec )

















mH n ft
4-

0o (degrees)
o0
u 30 60 90 120 150 80



.) ,
bO
a Elastic loading e /e = 1.0
4y

i-d
n

a
r E8 (degrees)


C 30 \60 .90 120 150 /80





Plastic loading el/e = 1.2




FIGURE 3. DISTRIBUTION OF ELEMENTAL STRESS VECTORS FOR
AXIAL LOADING FROM AN INITIALLY ISOTROPIC
STATE







t22 /2 d T d sin 8 (f2)
-IT/2 o

SI dS dB sine (g2) SJ d dB sec 8(g2) (5.8)
o 1 o 2
e 2
2e 49-sin49 2in4in26 26 + ey r3,7(sin ,8 log (Cos 8)]
=-[3??(-32- S4- s[@ 3 -(-)+Clog(cos 6)JS2

= [ ) e + (-t-) sin (+ ( ) (log cos l1- log cos t2)]

The integration limits in equations (5.7) and (5.8) are
given in set theory notation as

S= (-', S1= ((0,81) (82t)), S2 (81' 2)

Here we have used identities such as
2 2
sin =cos2 2 = (l- sin ) sin 21= sin 22 =cos
2 2
sin2 2 =cos i = (1+ sin) cos 2l= -cos 22 = sin .

By symmetry considerations we find

t33 =t22 t23 = t32= t31= t3 = t12 =t21 = 0

The principal stress axes coincide with the principal strain
axes, and so we can use equation (2.7) or equation (3.14) to
solve for the components of the deviator stress tensor.
Letting p=-(tll+ t22+ t33)/3 and L= log cos t-- log cos t2,
we obtain
s 6 1- 4L
p = tan + 2-*see ( ) sin n (- 1
3IT 3 3-


s= -2s= -2s = 2 Sy[ ( tan ) + (1- ) sec
21-X1 4L X
+ T( 3- ) sin + (3 )


S(5.9)






1-X/3
where s = e =Y( 1--- is the yield stress for axial
y y 1-X/21
loading. Equation (5.9) and the following expression for

the deviator strains are the desired results for predicting

stress-strain behavior in the case of axial loading.

el =-22e2 =-2e3 =e see i, 0 < 7/2 (5.10)

Figures 4 and 5 show stress-strain curves calculated

and plotted for various values of the parameter X. We see

that the slope of the stress-strain curves and the stress

relative to yield both increase with increasing values of X.

Hence, we are justified in calling X a work-hardening parameter.

The data of Figure 4 lend support to the suggestion of

Chapter III that X be limited to the range 0
X< 0 we see that the stress reaches a maximum and then falls

with increasing strain. The material is unstable, which is

an unrealistic and undesirable material property. Not obvious

from Figure 4 but true nonetheless is the fact that the slope

dsl/del is greater than the elastic modulus t for X>2 and

for certain values of the strain. When this occurs, the

residual strain becomes negative for a positive loading;

here again the material is unstable [69].

The reader is referred to Tables B-l and B-2 of Appendix B,

where stress and strain are tabulated for five values of the

work-hardening parameter X. These tabular values, based on

the closed-form solution of equations (5.9) and (5.10), will

be compared with the approximate numerical results of Chapter

VI, in order to establish the accuracy of the latter. The




57




" l I I'* 7 i









I : I I I
i i. ; : : I .... : '
: ,i !| t > t [ l I ) : |


,: i ,' ,> '^' : l!' I























,i .I
; i I :i, h i' i i j i *1 l : i : *:: : i |l


.|1 i i !, ; ..:i .. .,, ,: ,! .. ; ii





























.-TI- i -. Deviator Sfriain, e^e ,,+:'.


14 8 10 12


AXIAL LOADING. SMALL STRAIN
2 4 6 8 10 12
FIGURE .' STRSS-STRAN BEHAIOR FO
AXIL LADIG, MAL STAI










:.ll 1:;.




ii..

it






* I .
I-



It I:





1ii -:


htj :


---I -----

I i
---,-C- --









' .* .
I .
.1_... __
, f


- .---

. I'


Z':1717


---' : --| ..
' I I .i.














" ...[_ t .T- _
.-". i 'i
--~--4-
i- ---- -- ---










I -
4I II--
:!....i ... ,si

.; :: 1- i ... .;

... 1 .. ]---
-..




I -
l-ij




i... .i .. ... .-[ .. ..
j '


Lr.

II










ii




* I I
.1]

*1.


I 71'


-- ___ ....Fmm 1 4-
t-P .. I:




Satortess.. ...
f14 T i -- J-i-- T-
-a or 0
:rirrI:::::rflI-I:: :fI1ss s ::;1_1I


,- .. .. .... L.IZ
-.: I- .. I. +
i -.2



..... _.NL-- _















\ .. .. !- i
..--I-.- -. .. i:L
..-..-'


1._1-



i .i :







-it_ --T











It L I '+ .-...-... .. I


FIGURE 5. STRESS-STRAIN BEHAVIOR FOR
AXIAL LOADING, LARGE STRAIN


i







reader is also referred to Program A-i of Appendix A, which

is the computer program used to generate the data of Figures 4

and 5 and Tables B-1 and B-2. The programming language of

Appendix A is BASIC.

The prediction of various stress-strain curves raises

the possibility of fitting the present theoretical results

with experimental uniaxial test data. Perhaps there are

certain metals and certain values of the theoretical parameter

X for which the theory and experiments can be matched reason-

ably well. The reader is referred to Chapter VIII, where

we have attempted to fit the theory with experimental results

on commercially pure polycrystalline aluminum. The present

theoretical results are expressed in terms of the deviator

stresses and strains. In order to make comparisons with

experimental data these quantities must be converted to true

stresses and strains. The following results, easily obtained

and applicable for either uniaxial tension or uniaxial com-

pression, are presented without derivation:

C1 = 3 s /2, 02 = 03 = 0
(5.11)

el = e +sl/20, e2 = 3 = -el/2 + sl/2

We now investigate the case of reversed axial loading.

As before, f(n) has rotational symmetry about the el-axis,

and so depends only on the spherical coordinate angle 8.

Assume a given material body is loaded axially into the

plastic range so that the initial stress-strain behavior is







predicted by equations (5.9) and (5.10) derived above.

Figure 6a shows the distribution of the elemental stress

vectors f as a function of the angle 8 for one such loading.

We observe that plasticity has occurred because the slope

of f(8) is discontinuous. Now superimpose an axial loading

of the opposite sense (i.e., with reversed signs) onto the

given loading, as shown in Figures 6b through 6d. Figure 6b

is approximately the distribution of elementary stress vectors

which gives zero resultant stress. Figures 6c and 6d show

distributions of the elemental stress vectors after yield

has occurred for the reversed loading. Note that Figure 6d

is exactly a mirror image of Figure 6a. If the material

were loaded cyclically between equal tensile and compressive

limits, then the elemental stress vectors would alternate

indefinitely between two mirror image distributions.

In view of the behavior of the function f(8) shown in

Figure 6 and described above, only two additional parameters

are needed to describe the distributions of f during a

reversed loading. We denote the two parameters a1 and *2'

During the initial loading l1 is variable and equal to *

used previously, and 2 = 0. During the reversed loading

i1 is constant and equal to the maximum value of that
occurred in the initial loading, and 12 is variable in the

range 0 2 2 .1' Define a vector field h= (hl,h2,h3) by

hi = (37) cos 28+ a) e cos 8
h2 = (3 cos28-) e sin9 cos 0 (5.12)

h3 = (3B cos28 ) e sin sing




61




.... ._ -8 (fdegreesl .. .

30 60 9120: :150 :180
r-




0 ... ... ... ... .i .-...: \ .: : .--. -

S(a) Initial plastic state e /e 2.3

O






.- (b) Elastic unloading el/e 0.5 : .
i. I .
0




0 : : .. .' '










.. _^,^ ....__i_ i_: _
S(b) Elastic unloading e e .ldi
S: .. ..
e e I 1.2
0

r-4





S- .. .. I .-.- .-. / ...--.. .. .i -






3 0 c: .60: 120 : : 150. :. 180.

S" -. c: .I::-(dc) Reversed plastic state
--H ---- --

Ci H e /e. .. ...







FIGURE 6. DISTRIBUTION OF ELEMENTAL STRESS VECTORS FOR
REVERSED AXIAL LOADING FROM AN INITIALLY
PLASTIC STATE







where 8,0 are the spherical coordinates of Figure 2 and

0B !iT, 080s 21, unless specified otherwise. As before,
a=2x -7 and C =x +7. We can express the elemental stress
vector f in terms of h as follows.
1. Initial elastic loading f=hcos 1, -i T lI 0 (5.13a)
2. Elastic-plastic loading 09s
f=h sec J for eS11 (5.13b)
Lcsc 2 for 8 eS2

3. Elastic unloading < 12 0 < (< 41 constant

f=h Jsee1 for eSll -2heos2 (5.13c)
csc28 for eS2

4. Reversed elastic-plastic loading 0 2 < 1l< I, 1 constant

f=h fsee 1 for eSi -2h see 2 for eS3 (5.13d)
tcsc28 for 8 eS2 lcsc26 for 8 e S
5. Reversed elastic-plastic loading 0 < r1 < 2< 1 -, 1 constant

f=-h see *2 for 8 E:S3 (5.13e)
esec 2 for 9 eSQ
Here Sl'S2,SS3,S are sets of real numbers defined by
I 1 L 2 !L 2
1 4 2 t2 4 2+ 1 4 2' 1 2 =4 2

S = ((Ol'),(,1-2 ),'(f-T_ )), 2 = ((Jt 23 ,(T-_2,- l))

S3 = ((0,1), (C2, -2), ('-Cl, )), S4 = ((Cl,2),I (-C27-T )

Resultant stresses are computed from the equations

t.= (tlit 3 = f dA, (i =1,2,3) (5.15)
li' 3i) f

As before, since the principal strain axes are fixed in

the material, we have the relations







s tii- (tll t22 t33)/3, ( =1,2,3, no sum) (5.16)

Plugging appropriate expressions for f from equations

(5.13) into equations (5.15) and (5.16), and performing
the integration, we find the following closed-form solutions.
Here Ll=log os l -logcos 2 and L2 logoss ~ -logcos 2.
Cases 1 and 2, having to do with initial loadings,were
derived previously and are not repeated here.

3. Elastic unloading 2I 2 <01< < 1 constant

el = -2e2 = -2e3 = ey(ee 1 2 cos c2)

2 2 )2
s = -2s2 = -2s3 = s y-(1- tan a 1)+(l )see 1 (5.17)

2 1 4L1 X
+ )- sin 1+ -( )-2 cos ]
S3- 1 3I3- 2 cos
4. Reversed elastic-plastic loading 0 2 1 Jr< I, *1 constant




s1= -2s2 = -2s3 =s [ (* tan 1)- 2- tan 2) (5.18)

21. 4 *2 2 1-X
+(1- --) sec *1- (2- --) sec 12+ -(--) sin 1

4_1-X- 4 X
(t-) sin 2+ ( ) (Ll- 2L2

5. Reversed elastic-plastic loading 0 < 1 42< 1, 1 constant

el= -2e2 = -2e3 = -e sec 12

2 22 3
s =-2s2 = -2s3 =- (2- tan )+(l- )see 2 (5.19)

2 1-X 2 1
+ W(j) sin 2+]
3-1 3a 3-X







Equations (5.17), (5.18), and (5.19) satisfy "Masing's

hypothesis" [37], whereby the stress-strain curve of reversed

loading is exactly double in size, although rotated 180-

degrees and translated, compared with the stress-strain

curve of the initial loading. Symbolically

1,2. Initial loading: s=s(e), O0e5 e
m
S s'=s -2s(e), sm =S(e)
3,4. Reversed loading: 2( sm=s(e) (5.20)
e'=e 2e, 0 m m
5. Reversed loading: s' =-s(e), e' =-e, eme

For Case 5, where *2 41, the stress-strain curve of reversed

loading is identical in size, but rotated 180-degrees about

the origin, compared with the continuation of the initial

stress-strain curve. This phenomenon, too, is considered

a part of Masing's hypothesis. The present result is to be

expected because Masing's hypothesis is a general property

of mechanical systems composed of elastic-perfectly-plastic

elements [33].

Figures 7 and 8 show computed results of axial and

reversed axial loadings for X=0 and X=1.5, respectively.

The large effect of the parameter X on the stress-strain

behavior can be seen by comparing these graphs. Two related

phenomena appear on both graphs, namely the presence of a

Bauschinger effect and constancy of the elastic range. The

Masing hypothesis, discussed above and embodied in equations

(5.20), is readily discernible from Figure 7 or 8. A special









s: '.


i '








*l



Hi


ili
!ti

ifii
ft-
iiUt


!ii



i





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Elastic limit
for Reversed
loadings









4!.-- .


_-., -


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i : ::.

II


FIGURE 7. AXIAL LOADING AND REVERSED
AXIAL LOADINGS FOR X= 0


1



'
iiiiii
-ce
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t i i i i I I I
'
'
;i::I::::

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fbr revirsed-
load ings'


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~-~-



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i :



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i.
:

-: r


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,,' i :: i I : II 1 ii

!!!ii



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7 7




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tti1








|-H
I


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-w 7

K:! ![:1!

... ...


t t--


FIGURE 8. AXIAL LOADING AND REVERSED
AXIAL LOADINGS FOR X=1.5


Si


I r -






j::i:




I :


:::.: : : ': : f : :: : t I




iiliiii




: : ; r :


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:::: i;:::


A







phenomenon appears in Figure 8 but not in Figure 7, namely

the hysteresis loop of unloading and reloading that involves

plasticity. The reloading curves may be obtained either

from a straightforward extension of the closed-form results

of Chapter V, or by a direct application of the numerical

methods of Chapter VI. Note that the hysteresis loops of

unloading-reloading are present only for certain values of

X and certain values of strain, as defined by the condition

s/s > 2.

One further result relating to uniaxial stress-strain

behavior can be derived from equations (5.9) and (5.10).

Introducing identities such as

sin = cos(26), sec -tan =tan6

log cos 2 = log sin6 = log 6 6 6 /180-...

6 sec =csc(26) = + 62/3 + 76 /15+...

where 6 = 1 = T/4 /2, we can easily show that

lim 1 lim 1 7 4 3 -2X lim X log81 52
el c sy 68-0 Sy 33T3 6 8-0 3 X) (.)

We conclude from equation (5.21) that a limiting stress

corresponding to large values of strain exists if and only

if X=0, in which case

lim s 7T 4
el~oa = 3+3 = 1.4716107....
y












CHAPTER VI

STRESS-STRAIN BEHAVIOR FOR THE CLASS
OF LOADINGS INVOLVING FIXED PRINCIPAL AXES


Consider a class of loadings in which the principal

strain axes are fixed relative to the material, but where

the principal strain magnitudes can vary arbitrarily.

i(0) =0, ie= ei(t) for t> 0
(6.1)
ei = i (l 2 + e3)/3, (i =1,2,3)

This class of loadings is more general than the axial load-

ings studied in Chapter V; axial loadings are included here

as a special case. The following equations govern the

elemental stress vectors and the resultant stresses:

f(0) =0, =g 1 if gt m=1,2,3,... }
Y/gt if gt Y (6.2
f(m-) (,m)(m) [--- 2
g f(m-l)+ dE (mn + n (ndE () gt gg- (g.n

1 27T 1T/2 1T/2 f
t1= O d o d8 fsin8, t= d0 def sinB
L o o -0/2 o
T 7 (6.3)
t3=- d d f sin .
0o

Here f(m-l) and f(m) are the elemental stress vectors before

and after application of a given strain increment dE(m). Again

we refer vectors and tensors to the principal strain coordinates

and we use spherical coordinates 8,0 where convenient.







In Chapter IV we proved that if the principal strain

axes are fixed in the material throughout a deformation,

then the principal stress axes are fixed and coincident with

the strain axes. Since these conditions apply in the present

case, we can compute the principal deviator stresses s.

from the equations

s=tii- (tll+t22+t33)/3, (i=1,2,3, no sum)
S/2 (6.4)
t.= ( t )= d, d f sin9
1 (tli't2i't3i)= J

Equation (6.4)2 follows from equations (6.3) by symmetry

considerations similar to equations (4.12). As noted in

Chapter IV, the boundary between elastic and plastic regions

on the unit sphere can be very complicated for a general

loading. In fact, due to the complicated nature of this

elastic-plastic boundary, there probably does not exist

a closed-form solution for the class of problems discussed

here. Therefore, we shall be forced to derive a numerical

procedure, or algorithm, to be used in conjunction with a

digital computer, for solving equations (6.1) through (6.4).

Consider a finite grid of points covering one octant

of a unit sphere and based on the spherical coordinates

8,0. Thus

8.=iP, j =jP, (i,j =0,,...,N) (6.5)

where N is a grid-spacing integer and P =7/2N. The integral

of a function f(8,0) over a spherical octant can be approximated

by the trapezoidal rule formula as follows:






/2 /2 N N-1
SdZf d8fsin8 = Z a.[ 10 iN + Z f.j]+ e
o o i=0 2 j=1

where fi = f(8e,0) i = (P/2) (cos 8i.- cos 8.i) (6.6)

o = (P/2) (1 cos 81), N = (P/2) cos N-1

If the function f(8,0) is continuous and possesses continuous

first and second derivatives, then the truncation error e

for the trapezoidal integration formula is bounded.

Smax (7T/2) 2 2
abs E< 12N2 (ee + f sin 8)sin 8 (6.7)


Here f and fs are partial derivatives and 0 <8,0 Oi/2.

Equations (6.6) can be derived by summing the product of the

area and the average functional value for each patch of the

finite grid. The inequality (6.7) is derived by expanding

the function f(8,0) in a two-dimensional Taylor's series

[70, p. 187] and substituting the result into equation (6.6).
As an alternative integration formula consider the

following modified trapezoidal rule:

T/2 7/T2 N ,+ N-1
dOj d f sin89= a .[ fiO iN + E f] + e'
o o i=0 2 j=l 1

where f!.=f(8,01)), 8 =01 =iP for i=l,2,...,N-1 (6.8)

98 =0 =P/6, 98=0 = 7/2 -P/6

Here a.(i=0,...,N) are defined in equations (6.6). It can

be shown that the truncation error e' for equation (6.8)

is of order N3, instead of N-2 as in the case of equation

(6.6). That is, the integration formula (6.8) converges







faster than (6.6) as the finite grid is refined. Proof
_3
that e' is order N-3 follows by combining Taylor's series

and the Euler-MacLaurin summation formula [70, p. 154],

both suitably generalized to two dimensions.

Now consider numerical integration of the elemental

stress vectors f over a spherical octant, as required by

equations (6.4). In particular, let us apply equations

(6.6) to the three components fk(k=l,2,3) of f relative

to the principal strain coordinates. Initially during a

loading, the material is elastic and the functions fk(8,J

possess continuous derivatives of all orders. Here the in-

equality (6.7) provides a bound on the truncation error.

After plasticity occurs the functions fk (,) still are

continuous, but the derivatives 8m 80 fk of all orders are

discontinuous along the boundary between elastic and plastic

regions on the unit sphere. Hence, in general, the inequality

(6.7) does not hold true after plasticity has occurred.

Equations (6.6) and (6.8) are the basic numerical

integration formulae to be used in the remainder of this

work, even though the functions to be integrated generally

will have discontinuous derivatives. When plasticity occurs

we simply are forced to establish the numerical accuracy by

comparisons with the exact solutions of Chapter V, instead

of by inequalities such as (6.7). As we shall see, the

numerical error can be made arbitrarily small by using a

sufficiently large grid-spacing integer N.






Let us comment briefly on the choice of integration

formula. I could have chosen a more complicated, higher-

order polynomial formula of the Gauss type, say instead

of the trapezoidal rule. The discontinuous derivatives of

f(8,0) again would have rendered invalid the usual error

estimates. Experience with equations (6.6) and (6.8) has

shown that, except for integration in the elastic range,

there is very little to be gained by using an equation with

truncation error of order N-3 instead of N-2. Therefore, I

doubt whether any significant improvement of accuracy can

be achieved from any higher-order integration formula when

the first derivative of the integrand is discontinuous.

The only direct way of improving accuracy is to refine the

grid. The decisive reason for choosing the trapezoidal rule

formula is that, because of its relative simplicity, for a

given computational effort it allows one to compute the

integrand at a larger number of grid points.

Assume that we are given a sequence of deviator strain

increments de k (m=1,2,3,...) representing a strain history.

The following equations govern the elemental stress vectors

at points (8ij.) of the finite grid:

f(0) = f(m) g 1 if gtY m=1,2,3,...
ijk ijk k Y/g if gt>Y J

S m-l ,) xdek(m) nk+ n(de(m)n+ de (m2 + 3 n3) (6.9
k' ijk Imk nkdek ( 2 2 +de3(m) (2
2 2 2 n 2
gt=g+ g2 +3 (glnl+ g2n2+ g3n3

nl = os i, n2 = sin 8icos 0, n3 =sin i sin j







where i,j =0,1,2,...,N and k=1,2,3 (no summation). The

deviator strains and stresses are computed from the equations

e(O)= 0, e de(m)+ e(m-1), s (t+ t2+ t3)/3
k k k k k k 123
N N-I
t = a!( i0k iNk + D f ), (k=1,2,3)
ki=0 2 j=l (6.10)

where a =N-l(cos Bi- cos 9i+) i= 1,2,...,N-1
-1 -1
S=N-(1- cos 1), = =N cos 9N-

Equation (6.10)2 can represent either the ordinary trape-

zoidal rule or the modified trapezoidal rule, depending on

whether the coordinates i.,0. in equations (6.9) are defined

by equations (6.5) or (6.8)3,4,5. Equations (6.9) and (6.10)

have been incorporated into two computer programs, Programs A-2

and A-3 of Appendix A, representing the ordinary trapezoidal

rule and the modified trapezoidal rule, respectively. These

programs have been used to predict a variety of stress-strain

behavior within the class of loadings defined by fixed prin-

cipal axes.

Figure 9 shows stress-strain curves for shear-type

loadings. The curves were computed for the case of pure

shear, defined by el =-e22 =e(t) and e j=0 if ij ll or 22,

which is included in the class of loadings discussed here.

The results apply equally to the case of simple shear,

defined by e12 = e21=e(t), eij =0 if ij/12 or 21. Simple

shear is equivalent to pure shear plus a rotation, but it

cannot be computed directly from Programs A-2 or A-3, which















I+, L


t: : : : .












11 ii L1
I,


i l. .... ... .. _I I .
Fif. ... .. i' l



SI
"n ut-i t1 i I '

:: : +. '
















II II II II -
I I

J... i -,









1
t 1. l

V L {


S1 ""





;;+ i+: :+:. ;t :::+ ... : ;:+ + :;; :;:+ :++ i+


<


I *





I:

. i- -






42 4




I ii








i: 1i
.i-1-
i r :

-i ^i
. .'

7'T"i:i



! '



I -
i
Ii::;:}::


i
-i











U,
II



II




LU
cLu


' ; '
i,
:" ': i; :


11





*I ,
0) 4,;
I.
LU i:.
LU
LU r-I .


II




N
,r II
,I 1 02
LU

rl
L)
a
LU




*H
C/)


i ii44i I77



, )















..-i + :+
..









Ct "
I ... .
-t-- ? '---

; ? -ir f :t : i.






. ,- ... -
I I'








; I
i i :+; i :
,! 4 +- ....-.. ...J' -


.. ... .. .;:: !

-- ---- *-- -
.. I I


2 O'Z S'I -0" S'


N mJ


cn

P:
C3
n
F4







input only the principal strains. Note that the stress-

strain curves in shear depend on the work-hardening parameter

X in a manner similar to the case of axial loading, but that

the degree of work-hardening is not as great. That is, the

stress compared to yield and the slope of the stress-strain

curve for a given moderate value of strain are less for

shear loadings than for axial loadings.

The question of numerical errors now arises. How

accurate are the numerical results, such as presented in

Figure 9, that are computed using ProgramsA-2 or A-3? To

answer this question we generated a large quantity of numerical

results for the case of axial loading, using Programs A-2

and A-3. These numerical results, some of which are contained

in Tables B-3 and B-4 of Appendix B, then were compared with

highly accurate results computed from the closed-form solutions

of Chapter V.

Figures 10-13 illustrate graphically the numerical

errors for axial loading. The graphs show relative errors

in the computed stress plotted against strain, for two values

of X and for the two integration formulae discussed earlier.

We observe that the numerical error is constant in the elastic

range 0 : el/eki 4/3. However, after plasticity occurs the

variation of numerical error with strain is very complicated,

with many peaks and dips appearing. The peak numerical

errors tend to increase proportionally with the strain;

note that the numerical errors generally are less than 10-2












upper curves:
lower curves:


co
- en
G-a) ml

- 0 n
V 4-J
H 0




-I



'(


0


CM\











I
0






















m
Il
O


--' FIGURE 10

NUMERICAL ERRORS FOR AXIAL LOADING; -,
-.-... ORDINARY TRAPEZOIDAL INTEGRATION, 1= 0




-- ---- --- i *j -* -- **


----_---r-,-----------------------
strain, el/ek : -- -:i*
--- -- -- -- I- - --A


100


i 7 f


r-
O4
*H O

EL

















- .- I


-- ,- .-.


! | I I | I '



N= 20

N= 40


:---:- i-


""


l




77

SI 1.1 I I. I I I. I I I


-upper curves: N=20
2 lower curves: N= 40

U) .. f


-1- *
r M U0-P\\
C)k



r-i 0














.- .-1 k
r-4



























FIGURE 11
. .. NUMERICAL ERRORS FOR AXIAL LOADING;
ao -- ... ":-. ORDINARY TRAPEZOIDAL INTEGRATION, = 1-


1 0 10
H WXi
EO) ,

H 1' 'i"

: U l'*'





C !Of' I U E 1 i; -
II-- -*- UEIALERR O XA ODIG ***
o I- -;' RIAYTAEODLINERTO ,X=1,. -.;.'
._ .- j s r i ,eoe -:^ -: -

-*H- t 0I L -1 lI f i i i
10 10 10




78


I I I I I


I. 1 -. 1 I I I


; upper curves: N=20
lower curves: N= 40


I I'

II

:I i


I
, I ij

* i"' 'fI !


a.I I iI


A


-i )
-(UI (


r-q U
S4-
>t


Ii. -r


T-7r.,1rt *


A


I. J


NUMERICAL ERRORS FOR AXIAL LOADING;
MODIFIED TRAPEZOIDAL INTEGRATION, X= 0 -


-- -4----- -- .-.:,
I::
-v : Zstrain, el/ek
- ~ L ~ h i ._ I ,. ,. I L ~ r I --


S" -


t--
4- --- I--

I I


10 100


;.. .


---


& ~ ~~~ '


----- --


''


I -


i


-c-







I~_ I __~ I


. upper curves: N=20
S lower curves: N= 40i


PmW
41J C)

-H
w e


ECU

I

C--
- (0 i-
4e-1
IO













-
ir

U)~
f>


-a I -ii'


i -. .






FIGURE 13

NUMERICAL ERRORS FOR AXIAL LOADING;
MODIFIED TRAPEZOIDAL INTEGRATION, X= 1




strain, el/ek
-1 t k


I I I i I III


I t I t t...


1 I I 1 -( I 1 I 1


II.







(plotting accuracy) when the strain el/e is less than 10

and when the grid-spacing integer is 20 or larger. The

dips are explained by the fact that for axial loading the

elastic-plastic boundary coincides periodically with the

finite grid. When this occurs the numerical integration

involves only functions with continuous derivatives and hence

the accuracy is greatly improved. For loadings other than

axial loadings the dips in the numerical errors versus strain

curves would not be as pronounced.

The effect of the grid-spacing integer N can be found

by comparing the two sets of curves on each graph: the upper

curve was computed with N=20; the lower curve represents

N=40. When the integer N is doubled, we see that the

numerical error is reduced by a factor of about 3.6, which
-1 85
implies that the error is approximately of order N

This rule applies independent of X and independent of the

integration method. For X=0, the effect of the modified

trapezoidal rule as compared with the ordinary trapezoidal

rule is to reduce the numerical errors by factors of 9 and 3

in the elastic and plastic ranges, respectively. For X=l,

the modified trapezoidal rule reduces the numerical error by a

factor of 12 in the elastic range, and the numerical errors

are approximately unchanged in the plastic range. From the

latter observation we conclude that, in general, there is

no clear advantage in using the modified trapezoidal rule,







or other higher-order integration formulae, instead of the

ordinary trapezoidal rule.

The reader is referred to Table B-5 of Appendix B,

where values of stress and strain are tabulated for the pure

shear type of loading. Here again we see that the numerically

computed stress depends on the parameter X and on the grid-

spacing integer N. In this case, however, we cannot establish

the numerical accuracy by comparisons with exact solutions,

since there are no exact solutions for pure shear loadings.

We recall that classical estimates of the truncation errors

are invalid due to discontinuous derivatives in the functions

fk (,0). Furthermore, classical methods such as Richardson's
extrapolation technique [71, p. 186] are invalid due to the

complicated variations in the numerical errors with N.

Henceforth, in lieu of any rational methods, we assume that

the peak numerical errors versus strain are the same as

shown in Figures 10-13 for any loading that is approximately

radial. The strain measure for an arbitrary loading can be

taken as the accumulated plastic strain, defined by

t 3 3
e = dt [ (eij- ij/p) (eij- s/) (6.11)
o i=1 j=1

In consequence of this assumption we require at least N=20

for any computed data which are to be plotted on a graph and

at least N = 40 for any computed data which are to be included

in a table. The cost penalty of using large values of N

will be discussed later.







We shall have need of the deviatoric plane (or v-plane)

representations of stress and strain, defined as follows:

Consider a three-dimensional vector space with the principal

stresses al, 02, 3 assigned to a set of Cartesian axes.

Any state of stress can be represented by a point a= (0 ,02,03)

in this principal stress space together with a specification

of the principal stress axes. Consider a plane in principal

stress space, called the deviatoric stress plane, defined

as the locus of points that satisfy the equation o1+ 02+ 03 = 0.

The vector s= (sl,s2,s3) composed of the principal deviator

stresses always is contained in the deviatoric stress plane.

Any yield condition that is independent of mean stress can

be represented at most by one closed curve in the deviatoric

stress plane for each orientation of the principal stress axes.

Principal strain space and the deviatoric strain plane

are defined simply by substituting-"strain, e, e" for "stress,

a, s" throughout the preceding paragraph. The following

quantities are associated with the deviatoric planes:

\22 2 1
s = 3/ (s + s + ) deviatoric stress intensity

2 2 23-
e = V3/8 (e1 + e +e ) deviatoric strain intensity
3 23
e = 3/8 [ e (ei- si/t) residual strain intensity
i=l


This statement is true provided the stress tensor is
symmetric. If stress is nonsymmetric, as occurs for certain
loadings discussed in Chapter VII, then the stress couple
must be specified also. The principal stress axes can be
specified by a rotation tensor or by Euler angles.







The quantities s, e, e are proportional to the magnitudes of

the vectors s= (sls2,s3), e= (ele2,e3) eo=e-s

respectively. The factor V3\/8 is included so as to eventually

simplify plotting of the data. With these definitions

s= (4/3,-2/3,-2/3)#.s=1 and s= (1,-1,0)= s = v'/4, etc.

The residual strain eo, as defined here, is a useful concept

only when the principal axes of stress and strain coincide.

One objection which might be raised against the theory

presented in this dissertation is that the predicted initial

yield condition is a Tresca condition, whereas the observed

initial yield condition for most real materials appears to

be better described by a Mises yield condition. See Naghdi,

et al. [41], Mair and Pugh [49], and Phillips, et al. [52,53].

In the deviatoric plane representation the Tresca condition

is an equilateral hexagon and the Mises condition is a circle;

both are symmetric about the origin. We can remove the above-

mentioned objection by considering radial loadings in various

directions and by defining yield as experimentalists must

do in terms of an offset strain or proof strain. Let us

temporarily define yield as the stress intensity s corres-

ponding to some fixed residual strain intensity e along any

radial loading path. Figures 14, 15, 16 show curves of

constant offset strain in the deviatoric plane for several

values of e
0
Figure 14 shows curves of constant offset strain for

an initially isotropic material, defined by f(n) =0 at




















































graph number:
offset strain, e /e :
o y


0 1 2 3 4 5 6
0 .02 .05 .10 .20 .40 1.0


direction of loading: A(2,-1,-1), B(7,-2,-5), C(1,-1, 0)

work-hardening parameter: 0 X=0, A X=1.0, 0 X=1.5

FIGURE 14. SURFACES OF CONSTANT OFFSET STRAIN FOR AN
INITIALLY ISOTROPIC MATERIAL




85














K 7-
i I





r-4 r` Ln o ) cD c





7-1
t- : ? -7-- !- .. .. .:. .. .- ... .. ; ... 1.... _. : i










1 7.








1: 7.
/.. -- ... ... ----- i- -.-





LL- 77. 47 :7N .
i--- ~// ../ // ~.... :- .. .. .... .. L ..... .... ... ... .......




/LlT_.. L -- --I-- -i -- --1:





c-i .

i/i, I HC\-n:D
LiJA .1 U.. -I

i I _______ LI



.L -_ .- .. .. -


I --.: -i t7
I _l




















FIGURE 15. SURFACES OF CONSTANT OFFSET STRAIN
AFTER AN AXIAL LOADING
/ i
t.. /




" --=L .



:2 ___ i.. ._

;_5~_.I ;
t- 7:I7 _..i _!_F .. ; L _L_ ..... Z-L _'_L.JL' ". t ...... ..-.'L .--
... 'q "W -- =- w .- -- : I- ~ -I; : i -!

i :. :i l "
.... : C ...;--.=:-,-i- i---_: ....... i ... r... .~... : : r -, t-.: -' ; = : -; '- -
":. -It.. -- I. j ; I ; ; ; i -

FIGURE 15. SURFACES OF CONSTANT OFFSET STRAIN
AFTER AN AX(IAL LOADING




























































FIGURE 16.


SURFACES OF CONSTANT OFFSET STRAIN
AFTER A PURE SHEAR LOADING







time t=0. The curves are shown as 120-degree segments

rather than as complete closed curves in order to include

data for three values of the work-hardening parameter X on

one graph. If completed,the curves would be symmetric about

the origin in the same sense as the Tresca hexagon. We see

that the "initial yield surface" can assume a variety of

shapes depending on the offset strain used to define yield.

It is apparent that any definition based on an offset strain

in the range .02< e /ey< 1.0 will predict an initial yield

surface that is closer to a Mises circle than to a Tresca

hexagon. This is approximately the range of offset strain

encountered in most experimental determinations of yield.

Figures 15 and 16 show curves of constant offset strain

after the material has experienced plasticity during an

initial axial loading and an initial pure shear loading,

respectively. Here the data are computed for X=0 only.

We see that the "subsequent yield surfaces" are not symmetric

about the origin in the deviatoric stress plane, which implies

the yield conditions are anisotropic. We observe in each

case that the yield surfaces have expanded in the directions

of loading and contracted or flattened in the directions

opposite the loading. This phenomenon may be called the

generalized Bauschinger effect. We also observe that the

theoretical yield surface, defined by zero offset strain,

exhibits a sharp corner at the point of loading, but that

each yield surface defined by a finite offset strain, typical






of an experimental determination, exhibits a "blunt corner"

at the point of loading. The latter observation agrees with

conclusions by the majority of experimentalists who have

investigated the question of whether there are corners in

the yield surface. See Ivey [32], Bertsch and Findley [45],

Mair [51], and Phillips, et al. [52,53].

An interesting and informative presentation of elastic-

plastic stress-strain behavior involves the plotting of

yield surfaces in the deviatoric planes for a sequence of

stress or strain increments. Such a presentation can suggest

rules-of-thumb that might be applicable in a general loading

situation. Program A-4 of Appendix A is a modification of

Program A-2 that computes a sequence of yield surfaces from

a given sequence of strain increments with fixed principal

stress and strain axes. The program first computes a given

state of stress and strain, and then probes in various direc-

tions to find the yield surface defined as the exact

boundary of the elastic domain. Then the procedure is repeated

for the next increment of stress or strain. Program A-4 was

used to generate yield surfaces for several loading paths as

described below.

Appendices C and D present yield surfaces computed for

S=0 and for thirteen different loading paths. Three of the

loading paths are purely radial and ten involve abrupt changes

of direction. The yield surfaces are plotted in the deviatoric

plane in both principal stress and principal strain space,







which enables one to construct the complete stress-strain

histories. The plots were machine-made directly from the

output of the computer program, which at least eliminates

human errors and the tedium of hand-plotting. In some ways

the yield surfaces in principal strain space, contained in

Appendix D, are the most interesting. Here the yield surfaces

gradually develop and pull away from the former yield surfaces

when a new loading direction is taken.

By studying Appendices C and D, we see that most of the

yield surfaces are composed of both straight and curved line

segments. A sharp corner with curved sides tends to form

around the point of loading, whereas the back sides of the

yield surfaces generally retain the character of the initial

yield surface (the Tresca hexagon). The generalized Bauschinger

effect mentioned above is evident throughout Appendices C and

D. For axial loadings the yield surfaces tend to shrink

laterally, but for pure shear loadings the lateral shrinkage,

or cross effect, is small. For all radial loadings the

elastic range in the direction of loading is constant.

Phillips and co-workers investigated experimentally the

effects of temperature and plastic deformation on the shape

of the yield surface for annealed, commercially pure aluminum.

The changes in the yield surfaces observed experimentally by

Phillips, et al. [52,53] and Ivey [32] during plastic

loadings are similar in some respects to the theoretical

results contained in Appendices C and D. Among the







experimental observations are: (1) increased curvature

at the point of loading and flattening on the backside of

the yield surface, (2) translation of the yield surface

away from the origin in the direction of loading in principal

stress space, (3) no cross effect for either shear or axial

loadings, (4) reduction of the elastic range in the direction

of loading for both shear and axial loadings.

While these experimental observations do not entirely

agree with the theoretical results, the differences are not

serious. The author believes that further computations could

be made that would bring the present theory into much better

agreement with the experimental results of Phillips, et al.

Item (1) above is essentially in agreement with the present

theoretical results. Item (2) could be predicted theoretically

by choosing a value of X in the range 1 lX 2. The differences

between item (3) and the theoretical results could perhaps

be traced to the fact that the experimental results are

plotted in the shear-stress versus axial-stress plane whereas

the theoretical results are plotted in the deviatoric planes.

Item (4), dealing with reduction of the elastic range, is

more serious. Prediction of this effect would require a

major modification of the theory, which is not justified at

this time since the simpler present version of the theory

has not yet been fully explored.










CHAPTER VII


STRESS-STRAIN BEHAVIOR FOR BIAXIAL LOADINGS
AND FOR A GENERAL LOADING HISTORY

Suppose we are given an arbitrary sequence of deviator

strain increments comprising a deviator strain history at

a material point.

E() =0, E(m)=E(m-)+ dE(m), m=1,2,3,...

dE (m) de(m)], (i,j=1,2,3) (7.1)

The following equations govern the elemental stress vectors

and the resultant stresses. The notation is the same as

in previous chapters.

f )(n)=, f(m= 1 if gt m=1,2,3,...
-Y/gt if gt > Y (7.2)

g=f-+ xd(mn+nl(n. dE n) = [g(g- (g gn)2 ]
,ff 7T/2 7fT/2 7r/2
l 2 dOl dB fsin t= d/2O dO f sin 9
SffJto 0 lo 7T /2 lo

IT f (7.3)
t3= f dj de fsin 8
3 o o
We assume all vectors and tensors are referred to two coordinate

systems: a given fixed Cartesian coordinate system and an


By a fixed coordinate system we actually mean a set of
coordinate axes fixed relative to the material in the undeformed
state. For use in the deformed state we require the coordinate
axes to be rotated by the same rotation that carries the
principal strain axes from the undeformed state to the deformed
state. This distinction was not necessary in Chapters V and VI
as all vectors were referred to the principal strain coor-
dinates and these axes were fixed in the material.
91







arbitrary Cartesian coordinate system specified by the base

vectors N (c =1,2,3). The spherical coordinates 9,O in
-a
equations (7.3) are defined relative to the arbitrary

coordinate system only. By symmetry considerations we can

rewrite equations (7.3) as follows:


l=I1 + 1 + 13 + 4

t2 = I1 12 13 + 14
t =I -I -I -I
t3= 1 +2 3 4
L3 =I1 + 12 13 14

1 /2 S> (7.4)
where I.= dI dB fsine, (i=1,2,3,4)
1
S = (0,i/2), S3 = (T,37/2)

S2= (,1/2,7), Sq= (3u/2,2)

The reason for considering the decomposition (7.4) is that

it allows us to reduce the integration domain from seven

octants to four octants on the unit sphere. Computer storage

as required for the numerical integration is reduced accord-

ingly. For computational purposes we introduce a finite

grid of points over one hemisphere of the unit sphere, and

we introduce a set of unit vectors defined by these grid

points. If P= n/2N, then

6. =iP, j =j P, (i=0,...,N; j =0,...,4N)
1 J (7.5)
n..= (cos 9., sin .cos .,9 sin .sin .)
-ij i' i

We are now in a position to describe a numerical pro-

cedure for computing the stress history corresponding to a