• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Nonassociated flow rule
 Instability, bifurcation and...
 The classical shear band theor...
 A genuine instability for a nonassociated...
 An unstable bifurcation of an extended...
 More about the instability resulting...
 Conclusion
 Reference
 Biographical sketch
 Copyright














Title: On the instabilities resulting from a nonassociated flow rule
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Title: On the instabilities resulting from a nonassociated flow rule
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Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
    Abstract
        Page viii
        Page ix
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Nonassociated flow rule
        Page 5
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    Instability, bifurcation and uniqueness
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    The classical shear band theory
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    A genuine instability for a nonassociated flow rule with a mohr-coulomb material model
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        Page 90
    An unstable bifurcation of an extended mises model with nonnormality
        Page 91
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        Page 116
        Page 117
    More about the instability resulting from a nonassociated flow rule
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
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        Page 125
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    Conclusion
        Page 127
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    Reference
        Page 133
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        Page 145
        Page 146
    Biographical sketch
        Page 147
        Page 148
        Page 149
    Copyright
        Copyright
Full Text











ON THE INSTABILITIES
RESULTING FROM A NONASSOCIATED FLOW RULE













By

MING LI














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


UNIVERSITY OF FLORIDA


1993




















To My Parents
and Feifei














ACKNOWLEDGMENTS


I would first like to acknowledge Professor Daniel C. Drucker, the chairman of

my supervisory committee, for his tremendous support and encouragement during the

study. The influence on me for the criterion of choosing problems, the way of thinking,

and the philosophy of approaching problems will remain long after this Ph.D. program.

"Try to be an original thinker rather than a follower" is one of the greatest presents he

has given to me. He has provided me the opportunity to read a large number of

classical as well as modem papers in plasticity, instability of materials and structures,

metallurgy, soil mechanics, and many related areas in an absolutely free style. It is

really an honor for me to have such a precious opportunity to enjoy his high character

and integrity, and his feelings for his students.

I am deeply indebted to Professor Lawrence E. Malvem, for the critical reading

of the early draft of this dissertation, for the numerous discussions on a paper I wrote

which is within his familiar fields, though not closely related to this dissertation. The

paper would not have been highly evaluated by the reviewers of Experimental

Mechanics, perhaps not even publishable, without his critical comments.

Professors Martin A. Eisenberg, Peter M. Mataga and Michael C. McVay have

made many contributions to this work. I am grateful for their helpful discussions and








criticism of my work. The delight I found in Professor Mataga's thought-provoking

course on fracture mechanics is also an additional reward.

My thanks should also go to Ann for her kindness and care, which made me feel

so warm in this continent far away from my home when I first came here. I am forever

indebted to my parents, who though not well educated themselves put my education

above everything else. Little would have been done without the full support of Feifei.

The peaceful and enjoyable family she has been maintaining is a necessity for any

successful work in my life.

Finally, the financial support of the Office of Navy Research, Solid Mechanics

Program, Dr. R. S. Barsoum, under Grant Number N00014-87-J-1193 is gratefully

acknowledged.

















TABLE OF CONTENTS


Page


ACKNOWLEDGMENTS ..............................

ABSTRACT .......................................

CHAPTER 1 INTRODUCTION ........................


1.1 Historical Remark ....................
1.2 About This Dissertation ................


CHAPTER 2 NONASSOCIATED FLOW RULE ............

2.1 The Plastic Formulations on the Macroscale ..
2.1.1 The Classical Formulations of
Nonassociated Flow Rule ..........
2.1.2 Nonassociated Flow Rule in Engineering


111
. . . . . iii

......... viii

. . . . . 1

....... ... 1
.......... 2


. . . . . 5

. . . . . 5
.~5


A application .............................. 7
2.2 The Physical Background of Nonnormality ............. 8
2.2.1 The Mechanism of Plastic Deformation of
M etals ................................. 8
2.2.2 Non-Schmid Law and Nonnormality Condition ..... 9
2.2.3 SD Effect and Volume Expansion in Metals ...... 10
2.2.4 Geometry Change on the Microscale, Void
Initiation and Growth ...................... 12
2.2.5 Friction Effect, Microcrack Opening and
Closing ............................... 12
2.2.6 Summ ary .............................. 13

CHAPTER 3 INSTABILITY, BIFURCATION AND UNIQUENESS ....... 18


3.1 Stability .............................
3.1.1 Stability of Path and Stability of
Configuration ....................
3.1.2 Mathematical Requirement for Stability of
Configuration ....................


. ..... 18

....... 18

....... 19








3.2 Bifurcation and Uniqueness ...................... 21
3.3 Stability, Uniqueness and Bifurcation with a
Nonassociated Flow Rule ........................ 24
3.4 Sum m ary ................................... 30

CHAPTER 4 THE CLASSICAL SHEAR BAND THEORY .............. 36

4.1 The Classical Shear Band Theory .................. 37
4.1.1 The Outline of the Classical Shear Band
Theory ............................... 37
4.1.2 Shear Band Localization in Crystals ........... 41
4.1.3 A Consideration of the Classical Shear Band
Approach in Stress Space ................... 43
4.1.4 Vertex Effects ........................... 47
4.2 A Discussion on the Classical Shear Band Theory
with a Nonassociated Flow Rule ................... 48

CHAPTER 5 A GENUINE INSTABILITY FOR A NONASSOCIATED FLOW
RULE WITH A MOHR-COULOMB MATERIAL MODEL .... 54

5.1 A Mohr-Coulomb Model with a Nonassociated
Flow Rule ................................... 54
5.2 A Simple Pattern of Genuine Instability .............. 56
5.3 The Normal and Tangential Driving Forces and
Acceleration ................................. 60
5.4 Discussion on Dynamics ........................ 62
5.5 The Effect of Elastic Response .................... 68
5.6 The Magnitudes of the Unstable Jumps in a Model
Simulating Sand .............................. 70
5.7 Patterns of Parallel and Intersecting Bands ............ 72
5.7.1 Parallel Bands ........................... 73
5.7.2 Intersecting Bands ........................ 74
5.8 Initial Inhomogeneity and Continued Loading .......... 75
5.9 Plane Strain versus Axisymmetry .................. 76
5.10 The Genuine Shear Band Instability and the
Classical Shear Band Instability ................... 79
5.11 Concluding Remarks ........................... 82

CHAPTER 6 AN UNSTABLE BIFURCATION OF AN EXTENDED
MISES MODEL WITH NONNORMALITY ............... 91

6.1 An Extended Mises Model with a Nonassociated
Flow Rule ................................... 92
6.1.1 The Material Model ....................... 92



















CHAPTER 7


6.1.2 A Wedge Path Starting from a Triaxial
Stress State ............................. 93
6.2 An Exploring of Instability of Configuration ........... 95
6.3 An Exploring of Instability of Path ................. 99
6.3.1 A Relative Non-Deforming Plane ............. 99
6.3.2 Energy Considerations ................... 103
6.3.3 Comments on Kinetics ................... 106
6.3.4 The Extent of the Instability of Path .......... 107
6.4 Summary and Concluding Remarks ................ 109

MORE ABOUT THE INSTABILITY RESULTING FROM A
NONASSOCIATED FLOW RULE .................... 118


7.1 On the Mathematical Description of the Initiation
of the Shear Band with Rotating Boundary ...........
7.2 Comments on the Initial Inhomogeneity .............
7.3 Ellipsoidal Type of Localization ..................
7.4 Time and Temperature Effects ...................


118
120
122
123


CHAPTER 8 CONCLUSION ............................


..... 127


8.1 Theoretical Considerations ......................
8.2 Implications for Engineering Practice ...............

REFEREN CES .............................................

BIOGRAPHICAL SKETCH ....................................














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


ON THE INSTABILITIES
RESULTING FROM A NON-ASSOCIATED FLOW RULE

By

Ming Li

May 1993


Chairman: Daniel C. Drucker
Department: Aerospace Engineering, Mechanics & Engineering Science



Instability resulting from a non-associated flow rule is explored. For a Mohr-

Coulomb material model with non-normality, or in the plane strain case for any material

model with non-normality, the model is unstable in any equilibrium configuration as

soon as it enters the plastic range, provided the elastic response along the wedge path

is too small to stabilize the plastic response. This genuine instability of configuration

in the form of a shear band with a rotating boundary is conceptually different from the

classical shear band in many aspects. It is the first counter-example to the commonly

held idea that instability of path always occurs earlier than instability of configuration

in the plastic range. For an elastic-plastic extended Mises model, or any elastic plastic







material model with a smooth yield surface, combined with non-normality, both the

instability of configuration and instability of path are examined. A new type of

instability of path has been discovered that is entirely different from the instability of

path of a Shanley column in the plastic range. The strong tacit, but not always valid,

assumption of continuing equilibrium is responsible for confusion over uniqueness and

bifurcation with a non-associated flow rule. Techniques customary for an associated

flow rule give results for a non-associated flow rule that are of limited value at best.

The possibility is indicated that macroscopic plastic deformation may not develop

continuously, but instead develops in a discontinuous manner reminiscent of breakaway

of dislocation pile-ups in metals on the microscale and frictional response in

geomaterials. However, the jumps may be so small that they are undetectable with the

usual external measuring devices. A reassessment of the reliability and validity of

commercial codes which employ non-associated flow rules is suggested, and alternative

choices are given which are stable in the small in the forward sense.














CHAPTER 1
INTRODUCTION



1.1 Historic Remarks


In the history of the theory of plasticity, stability, uniqueness and bifurcation

have played key roles. Many forward steps in this subject turned out to be landmarks.

Shanley's discussion on the reduced modulus load (Karman load) and the tangent

modulus load for the plastic buckling of a perfect column introduced the concept of

stability of configuration and stability of path (Shanley 1946, 1947). The introduction

of the concept of material stability (Drucker 1950, 1951) laid the foundation for the

subsequent rapid development of the theory of plasticity and its wide range of

engineering applications. The discrepancy between the predictions of physically valid

flow theory and the experimental results on the plastic buckling of simple structures

kept alive the ongoing controversy of whether or not there exist vertices on the current

yield surface (Batdorf 1949). Slip theory (Batdorf and Budiansky 1949), a physically

appealing theory, has such vertices. Hill's general theory on the bifurcation of an

elastic-plastic solid with normality serves as guide in many disciplines of solid

mechanics. Rice's formulation of plastic localization into a shear band significantly

advanced our knowledge of the inelastic deformation of materials. As Bazant put it,








2
"stability stands at the heart of structural and continuum mechanics." (Bazant and

Cedolin 1991, p vii) The question of stability attracted the interest of mechanics people

from generation to generation from Euler to Prandtl and Liapunov, from Karman to

Koiter, Budiansky and Hutchinson. These days, as Needleman and Tvergaard (1992,

p S3) stated, "everybody loves localization problems!"



1.2 About This Dissertation



The questions of stability, uniqueness, and bifurcation with an associated flow

rule are now well understood. However, there is a lack of consensus on the same

questions with a nonassociated flow rule. In most current studies the techniques of the

proof of uniqueness and the approach to bifurcation used for an associated flow rule are

simply carried over to a nonassociated flow rule. More and more publications of this

type on this subject keep appearing.

Also, nonassociated flow rules are extensively used to model the inelastic

behavior of granular materials and other geomaterials in engineering practice. Modern

commercial codes employing a nonassociated flow rule are available and in common

use. Both purely theoretical and practical engineering uncertainties regarding stability,

uniqueness and bifurcation with a nonassociated flow rule call for much more

exploration.

From the micromechanics point of view, nonnormality does have some physical

basis. A non-Schmid effect in crystals, possible pressure-sensitivity in highly dislocated








3
regions in metals and alloys, frictional behavior in granular materials, and significant

microscopic geometry change all lead to nonnormality on the macroscale for inelastic

behavior. The investigation of nonassociated flow rules may provide further

understanding of the inelastic behavior of many materials. It may also provide

suggestions on the suitable mathematical representations of the inelastic deformation for

practical engineering calculations.

This dissertation consists of eight chapters. Following this introductory chapter,

chapter 2 gives the mathematical description of a nonassociated flow rule and the use

of nonassociated flow rules in engineering practice. That chapter is largely devoted to

the discussion of the physical sources of nonnormality. In chapter 3, the concepts of

stability of configuration, stability of path, and the mathematical requirement of stability

in the usual sense are discussed. A critical review of the classical uniqueness proof and

bifurcation approach is presented. A strong tacit but not always valid assumption for

uniqueness and bifurcation with a nonassociated flow rule is pointed out. Chapter 4

briefly outlines the classical shear band theory. Effort is focused on consideration of

the theory in stress space rather than the often employed deformation or "F-space". A

simple explanation is provided for an apparent paradox that has existed in the classical

shear band theory because the direct rigid plastic approach does not coincide with the

limiting case of elastic-plastic approach as the elastic shear modulus tends to infinity.

It is pointed out that results obtained by the application of classical shear band theory

to a nonassociated flow rule are of limited value. In chapter 5, the question of

instability of configuration of the frequently used Mohr-Coulomb model with








4
nonnormality is investigated. A genuine instability with limited total excursion is

exhibited. This is the first counter-example to the common impression that instability

of path in the plastic range always occurs earlier than instability of configuration. A

shear band with rotating boundary is proposed which is conceptually very different from

the classical shear band. Unlike the usual post-bifurcation analysis which expands the

variables of interest in terms of small parameters near the bifurcation point, a

consideration of dynamics of the entire unstable procedure is provided. Chapter 6

examines both instability of configuration and instability of path for an extended Mises

model with nonnormality. An entirely new type of instability of path is exhibited. The

prediction of localization in non-Schmid metal crystals by the theory proposed agrees

favorably with the reported experimental data. A discussion is given for the extension

of the results obtained from this particular model to any material model with a smooth

yield surface. In chapter 7, a tentative mathematical description of the shear band

proposed is presented within the framework of the classical shear band theory. Another

possible pattern of localization with a nonassociated flow rule is suggested. Finally, the

dissertation research reaches its conclusion in chapter 8.














CHAPTER 2
NONASSOCIATED PLASTIC FLOW RULE



2.1 The Plastic Formulations on the Macroscale



2.1.1 The Classical Formulations of Nonassociated Flow Rules

In the now classical formulation of time-independent plastic stress-strain

relations, the current yield surface serves as a plastic potential for the increments (rates)

of plastic strain produced by the increments (rates) of stress that cause plastic

deformation. This is the "associated plastic flow rule". When axes of plastic strain

increment (rate) are superposed on the corresponding axes of stress, the plastic strain

increment (rate) due to any such increment (rate) of stress is a vector in the strain space

normal to the current yield surface at the current yield point. The terminology

"nonassociated plastic flow rule" is employed for a plastic stress-strain relation in which

the current yield surface does not serve as a potential surface, i.e., the plastic potential

surface is not the same as the current yield surface.

The very first reasonably clear formulation of a mathematical theory of plasticity

by St. Venant (1870) and Levy (1870) was actually a nonassociated one. It was the

equivalent of the combination of a potential surface, later known as the Mises criterion

(1928) and the very different Tresca (1870) or maximum shear stress criterion of yield.








6
Nonassociated flow rules of much more general nature were written down by many

others as time went on (Melan 1938) and discussed by Hill in his 1950 treatise on

plasticity. Melan expressed a general plastic formulation as


dePij = -h df (2.1)


where dsij is the plastic strain increment, g is a plastic potential, f is a yield

function and df is given by the component of the stress increment normal to the

current yield surface. The scalar quantity h is a function of current stress state and

the entire history of loading (Hill 1950, p34). Hill, as others before him, then made the

choice of an associated flow rule (the yield function as the plastic potential, i.e., g = f)

for simplicity and convenience.

The concept of stability of material was developed by Drucker in a series of

papers from 1950 on (Drucker 1950, 1951, 1959, 1960, 1964). Stability in this sense

is a very useful classification of material, but not a thermodynamic requirement. It

leads directly to an associated flow rule as well as convexity of the yield surface for all

materials which fall into the "stable" classification. Before the proposal of this concept

of stability of material, the convexity and normality conditions were no more than

convenient assumptions. This concept laid the foundation for subsequent rapid

development of the theory of plasticity and engineering applications to metals and

geomaterials.









2.1.2 Nonassociated Flow Rule in Engineering Application

The stress-strain relations of plasticity were introduced into soil mechanics in the

1950s. The first step was the paper "Soil Mechanics and Plastic Analysis or Limit

Design" (Drucker and Prager 1952) in which the authors applied a perfect plastic three-

dimensional Coulomb criterion to soil mechanics. Later, the concept of work-hardening

was introduced into soil plasticity (Drucker et al. 1957). Inelastic soil mechanics has

since developed as a branch of mechanics (Wroth 1973; Chen and Baladi 1985, p28).

An associated flow rule works well and effectively for many problems of metal

plasticity. However, when the Mohr-Coulomb criterion, or extended Mises criterion

(Drucker and Prager 1952) or any similar simple yield criterion is employed, the

volume change predicted by an associated flow rule is significantly larger than the

experimental result for soils, rocks and other geomaterials.

It seems quite straightforward therefore to employ a nonassociated flow rule in

which the plastic potential and yield function are defined separately to allow a smaller

volume change (Davis 1968). This is illustrated in Figure 2.1. The p axis is the

hydrostatic axis and q is the shear axis. The component of plastic increment in the

negative p direction (with compression as positive in the convention of soil

mechanics) gives the volume expansion. It is obvious that with a proper choice of the

plastic potential a nonassociated flow rule could accommodate a much smaller volume

change than an associated flow rule. Consequently nonassociated flow rules are in

widespread use today to represent the inelastic behavior of clay (Yong and Mohamed

1988), rock (Cristescu 1985, 1989), sand (Cristescu 1991), and geomaterials in general








8
(Frantziskonis et al. 1986; Desai and Faruque 1984; Desai and Hashmi 1989).

Nonassociated flow rules are extensively employed in the commercial codes in present

civil engineering practice.



2.2 The Physical Background of Nonnormality



2.2.1 The Mechanism of Plastic Deformation of Metals

The inelastic or plastic deformation of metals is due to twinning, diffusion flow,

phase transformation, and most important, the motion of dislocations which causes slip

in the crystal (Ashby and Frost 1975). Over the temperature range from near absolute

zero to well above half the melting point, slip dominates plastic deformation (Kocks

1975). Crystal plasticity studies (Taylor 1934, 1938; Lin 1971; Asaro 1983a, 1983b)

usually consider the plastic deformation of crystals with relatively low dislocation

densities or a few lines of dislocation. The crystal deformation then consists of plastic

slip via dislocation motion, elastic response of crystals away from the slip plane, and

rigid rotation of the crystal lattice. Individual dislocation motion is basic. However in

real structural metals and alloys, the average dislocation density may be as high as 1015

lines/m2. The dislocation structure then is in dislocation cells (Mughrabi 1975; Spitzig

1981; Chiem and Duffy 1983; Hasegawa et al. 1986), or, in the terminology of

Kuhlmann-Wilsdorf (1987), of Low Energy Dislocation Structures (LEDS). The

individual dislocation effects or the effects of small groups alone then do not provide

the necessary understanding of the plastic deformation (Drucker 1966); the detailed








9
nature of dislocation interactions is of less importance (Kocks 1985). Instead, the

consideration of the motion of large numbers of dislocation in the average sense

provides the appropriate way to interpret plastic deformation (Drucker 1984, 1987).

A structural metal is a compact aggregate of crystal grains with varying shapes

and orientations. The main difference between a single crystal and a polycrystal is the

presence of grain boundaries. But for highly dislocated metals and alloys the grain

boundary itself provides only a small resistance to the plastic deformation. The

resistance at grain boundaries for grains of the order of 10 to 100 micrometers can not

account for the magnitude for the flow strength of ductile structural alloys (Drucker

1988). The main impediments to dislocation motion, the cause of the work-hardening,

must be elsewhere. Kocks(1985) has proposed extremely highly disordered "lumps",

and Drucker has suggested "lines" and "cords" instead (Drucker 1991). It is these

extremely highly disordered regions which serve as the major block to the motion of

dislocations.



2.2.2 Non-Schmid Law and Nonnormality Condition

The Schmid stress or resolved shear stress for a slip system is the component

of shear stress resolved in the slip plane and in the slip direction. It is treated as the

component of stress that produces forces on dislocations causing them to move. The

Schmid law is an assumption that the resolved shear stress needs to reach a critical

shear stress value to activate the crystal slip in the corresponding slip system. The

normal component of stress on the slip plane does not affect the slip. Rice (1971), by








10
defining a set of finite number of discrete scalar internal-variables, proved that the

normality condition is a direct consequence of the assumption of crystal slip obeying

the Schmid law.

However, some deviation from the Schmid law is inherent in the

micromechanics of dislocation slip itself, although the normal component of stress on

the slip plane may have only a very small effect on the crystalline slip. It is clear that

in processes such as cross-slip of a screw dislocation and climb of an edge dislocation,

stress components other than the Schmid stress affect the process. The cross-slip

mechanism given by Asaro and Rice (1977) is a very good example. One of the main

consequences of non-Schmid stresses entering the yield condition can be the loss of the

normality condition. Within the general framework of Hill and Rice (1972), Qin and

Bassani (1992a, 1992b) pushed forward this point of view in a rigorous mathematical

manner. Nonassociated plastic flow thus may occur in metals and alloys even with low

dislocation densities within the crystals.



2.2.3 SD Effect and Volume Expansion in Metals

The hydrostatic pressure has an almost unnoticeable effect on the yield strength

for most metals and alloys, partly because the stress level caused by external load

usually is much smaller than the atomic force. The normal stress component on the slip

plane caused by the external hydrostatic pressure (the normal stress is not caused only

by external load, e.g., foreign atoms or vacancies can cause high internal normal stress

on the slip plane) then has almost no effect on the plastic slip, though the total normal








11
stress (caused by external load plus internal atomic forces) may have some effect.

Nevertheless, if the external load is high enough to produce a stress level in the crystals

comparable to the stress level caused by the atomic forces, and if the normal component

stress does affect the plastic slip, the hydrostatic pressure then should have some effect

on the yield strength of the crystal. This may be the case for crystals with low

dislocation densities.

Very high strength steels do exhibit somewhat higher yield strength under

uniaxial compression than under tension. This strength differential (SD) effect (Drucker

1973) is equivalent to the strength difference caused by high hydrostatic pressure. It

was reported that the SD effects were 5% to 10% for AISI 4310, 4330 and H-80 steels

(Spitzig et al. 1975, 1976). Like soil, the high strength steels also exhibit a smaller

volume expansion than predicted by a simple yield criterion with its associated flow

rule and manifest pressure dependence (Casey and Jahedmotlagh 1984; Casey and

Sullivan 1985). A nonassociated flow rule may well apply.

A non-Schmid effect may be one of the reasons for the SD effect, but probably,

only a minor reason. It is worth emphasizing over and over again that in real metals

and alloys the dislocation densities are very high. In these metals and alloys there exist

some extremely highly dislocated regions in moderate volume and area fraction, much

more disrupted than the regions appearing as dislocation cell walls. It is these highly

dislocated regions which are thought to dominate the plastic deformation (Drucker

1991). The pressure effect is likely to be much greater in these regions than in the

lightly dislocated regions.









2.2.4 Geometry Change on the Microscale, Void Initiation and Growth

Another source of deviation from normality is geometry change on the

microscale. If numerous micro-defects (e.g. voids) exist in a volume of material, it will

exhibit overall pressure-sensitivity, even though the material itself can be approximated

as pressure insensitive and obeying normality. Figure 2.2 illustrates this situation in an

over-simplified manner (Drucker 1992, plasticity course, University of Florida). A

volume of material consists of material which is itself pressure independent. The

volume of material is subjected to a shear plus a hydrostatic pressure. The

decomposition of these two components clearly demonstrates the equivalence of the

outer hydrostatic pressure applied to the volume to the inner negative hydrostatic

pressure applied on the walls of the voids.

By adopting Gurson's constitutive model for void nucleation and growth in

ductile materials (Gurson 1977), Needleman and Rice (1978) showed that the plastic

increment is not normal to the yield surface for materials involving cavity nucleation

during the plastic flow.



2.2.5 Friction Effect, Microcrack Opening and Closing

Frictional materials and assemblages of frictional materials are not stable in the

sense of Drucker (1954). To a considerable extent, geomaterials are likely to be in this

frictional category. In the simplest frictional system, a block sliding on a rough plane

with or without springs attached on it to represent the elastic effect, the departure from

normality is very large (Drucker 1954; Mandel 1966), Figure 2.3.








13
In granular materials, such as sand and soil, the plastic or inelastic deformation

basically is due to the granular particles slipping by each other and the rotation of the

particles. In such frictional systems it is to be expected that a nonassociated flow rule

should apply.

In the case of rock and concrete, most of the observed plastic or inelastic

response comes from the opening and closing of microcracks which exist in the rock

and concrete in enormous number. As discussed previously, the inelastic behavior of

materials containing a changing number and volume of microdefects during plastic

deformation is pressure sensitive. There is no compelling reason for these types of

material to obey normality.

Certainly, in a strict sense, inelastic behavior of materials is always time-

dependent. Time plays a fundamental role in dislocation motion in metals and alloys,

in frictional motion in sands, and in microcrack closing and opening in rocks and

concretes. However, under usual conditions (strain rate in the range from 10-5 to 102

s-1 and temperature well below half the melting point) most engineering materials can

be modeled very well as time-independent (rate-independent).





2.2.6 Summary

The conclusion is that in the framework of time-independent plasticity, all

materials, including polymers (Richmond and Spitzig 1980), exhibit pressure

dependency to some extent from negligible to appreciable. Normality to a simple yield








14
surface is only an approximation. The frictional response of geomaterials has its analog

in metals under high hydrostatic pressure.

























q


/f (yield surface)



g (potential surface)


Figure 2.1 A nonassociated flow rule
could give much smaller volume change
than an associated flow rule
















(7


i ^~


b- T


-=


- T


=-


Figure 2.2 The equivalence of an outer hydrostatic
pressure to an inner negative hydrostatic pressure for
material which itself is pressure insensitive














N

IF


FRICTION


F
nf A


Figure 2.3 Frictional response deviates
greatly from normality














CHAPTER 3
INSTABILITY, BIFURCATION AND UNIQUENESS




3.1 Stability



3.1.1 Stability of Path and Stability of Configuration

A time-independent system, constrained against free rigid body motion, is stable

in the usual sense when the response of the system to all permissible infinitesimal

perturbation remains infinitesimal. It is unstable when some possible infinitesimal

perturbation would cause a finite response. The terminology of stability of path and

stability of configuration in the common mechanics sense (analogous to the

thermodynamics sense) can be traced back to Karman (1947) in his discussion of

Shanley's remarkable paper (Shanley 1947). The question of stability of path is

investigated by imagining the system to be disturbed infinitesimally in all permissible

ways while it is following a path in load (stress) and in displacement (strain) space to

determine if an alternative path can be followed. If the load is fixed, i.e., the system

is at a point on the path in load (stress) and in displacement (strain) space, the stability

of this equilibrium configuration of the system is investigated by imagining the system

to be subjected to all permissible infinitesimal perturbations to determine if an








19

alternative configuration exists. The response must satisfy compatibility, the equations

of equilibrium or motion, and the stress-strain (load-displacement) relation.

The Shanley column (Shanley 1946, 1947) is an excellent example to illustrate

the difference between the concepts of stability of path and stability of configuration.

At each and every fixed load above the tangent modulus load, but below the reduced

modulus load, a straight column configuration is in stable equilibrium. However, at

each load above the tangent modulus load, when the load increases, the loading path

(uniaxial compression, see Figure 3.1) can bifurcate and an alternative path to the

uniaxial compression path would be followed. At and above the tangent modulus load

the straight configuration is stable but the path of straight configurations is unstable.

This is a stable bifurcation or instability of path in the sense that until a maximum load

is reached each bent configuration is stable and the path followed is stable against

additional bending perturbation. Shanley's illustration is so clear, and the tangential

modulus load rather than reduced modulus load agrees so well with numerous

experimental results on plastic buckling, that it is now often thought that instability of

path always occurs earlier than instability of configuration.



3.1.2 Mathematical Requirement for Stability of Configuration

A loaded system in equilibrium is stable against infinitesimal perturbation, only

if the scalar product of the infinitesimal increment (rate) of load and the corresponding

increment (rate) of displacement is positive for all such possible disturbance in the

neighborhood of the equilibrium configuration. As illustrated by a circular cylinder








20
with a sharp point on one end (Figure 3.2) (Drucker 1960), the cylinder resting on its

flat end on a horizontal plane is a stable configuration,




6P U > 0 or P > 0 (3.1)



while for the unstable configuration, the cylinder resting on its sharp point,





6P 6U < 0 or < 0 (3.2)



in each laterally perturbed position.

The broadening and extension from a system under load to a deformable

material under stress can follow the discussion of Drucker (1950, 1951, 1960, 1964),

and strict and elegant proof of Martin (1975, p 142). A block of nominally

homogeneous material under homogeneous stress normally contains numerous

inhomogeneous microstructures with a very inhomogeneous distribution of stress.

Therefore, physically and mathematically, it is analogous to an ordinary structure under

applied load (Drucker 1965, 1966, 1988), as illustrated in Figure 3.3. A stressed

material is in a stable equilibrium configuration, only if the scalar product of any

possible increment (rate) of stress and corresponding increment (rate) of total strain,

elastic plus plastic, is positive.








21
Stability in the small in the forward sense (Drucker 1951, 1959, 1960, 1964)


6i e~ > 0 (3.3)

is stability in the usual sense. It is a classification of behavior, not a requirement of

thermodynamics.

Suppose that an external agency (Drucker 1950, 1951) quasi-statically applies

small surface traction and body forces which alter the stress state by dai, and produce

small strain increments dej, which are both elastic and plastic. Upon removing the

small forces applied by the external agency, the elastic strain is recovered and thus


6.. iiPi > 0 (3.4)

This is the stability postulate in the sense of Drucker (1951) which corresponds to the

concept of material work hardening. Clearly this stability for a small cycle of stress is

a stronger restriction than the stability in the small in the forward sense. It may be

worth emphasizing here that stability in the small in the forward sense is the usual

concept of stability against infinitesimal perturbation which applies not only to a

structural system under load but also to a continuum under stress. It is not a work

hardening concept.



3.2 Bifurcation and Uniqueness



The customary bifurcation formulation (e.g. Budiansky 1974) is as follows.

Assume first that there exists a fundamental solution u0 that varies smoothly with the








22
load parameter X. Then assume there is another solution u=uo(X)+v(X), in which v is

infinitesimally small. Both solutions satisfy equilibrium, compatibility, corresponding

stress-strain (load-displacement) relations, and traction as well as displacement boundary

conditions. Subtraction of the two solutions to search for a nontrivial solution v gives

a typical bifurcation problem. This usual bifurcation approach actually considers the

stability of a loading path, not the stability of an equilibrium configuration, and

represents Shanley's point of view.

It may be helpful to point out an implicit requirement in the customary

bifurcation approach. The configurations in the vicinity of the current equilibrium

configuration for all permissible loading directions in load or stress (displacement or

strain) space must be in equilibrium. The existence of equilibrium configurations for

some (or even most) loading directions does not guarantee the satisfaction of the

requirement. The requirement of this "continuing equilibrium" is a clear and rigorous

assumption. However, it automatically rules out the possibility that a nonequilibrium

path may be followed after the bifurcation that is unstable to first order.

A time-independent or rate-independent plasticity theory by definition has no

natural time. The rate of stress or strain is only an infinitesimal increment (e.g., Hill

1950, p54; Malvern 1969, p151). Satisfying continuing equilibrium


0 or a = 0 (3.5)
axi axi








23
requires equilibrium be satisfied not only at the inception point of the bifurcation, but

also in the vicinity of the point for all permissible loading directions as well. This is

not always appropriate. A simple rod-spring system (Budiansky 1974, p9) is one of the

simplest cases. Another well known example is the buckling of the thin walled cylinder

under axial load (Karman and Tsien 1941). In both, no neighboring equilibrium state

can be found at all except for unloading. For a material, a Liider's band provides a

good example, in which there is no equilibrium state in the vicinity of the upper yield

point when plastic deformation occurs. If bifurcation calculations assume stability of

each path, the possibility that a nonequilibrium path may be followed is automatically

ruled out. If bifurcation means more generally that at a stage in the loading procedure

there exists more than one solution or mode (Hill 1967b), then there are three categories

that should be clearly distinguished: stable, neutrally stable or unstable bifurcation. The

customary bifurcation calculation presumes stable or neutrally stable bifurcation.

There is an inherent relationship between bifurcation and uniqueness. The

traditional uniqueness proofs ordinarily follow a standard pattern (e.g., Hill 1950;

Drucker 1956; Koiter 1960). Suppose there exist two distinct fields of stress and strain

increment &, , and &2, '2 representing two solutions. Both solutions satisfy the

equilibrium equations and compatibility conditions within the body, traction boundary

condition on S, and displacement boundary condition on Su. The principle of virtual

work then requires the volume integral of (b, 2)'(- -i2) to be zero. If the material

is stable in the small in the forward sense and both solutions follow stable paths, the

integrand is positive if the two solutions are not identical. The proof of uniqueness is








24

then direct. However, if the material is intrinsically unstable, the proof does not

follow.





3.3 Stability, Uniqueness and Bifurcation with
a Nonassociated Flow Rule



Nonassociated flow rules are in wide use today in civil engineering practice to

represent the inelastic response of granular or frictional materials. Commercial codes

which employ a nonassociated flow rule model are available. Nevertheless, there is a

lack of consensus on the uniqueness of the solutions obtained if a nonassociated flow

rule is used, although Drucker (1956) demonstrated much earlier by a simple example

that the combination of Tresca's yield condition with the nonassociated Mises flow rule

would produce nonuniqueness of stress rate. As discussed by Koiter (1960), it seems

likely that the use of associated flow is indeed necessary to ensure uniqueness of

solution of all well-posed boundary value problems. Sandler and Rubin (1987)

demonstrated the nonuniqueness of wave propagation or dynamic problems, under

certain circumstances, when a nonassociated flow rule is used.

Lade and his research group performed a series of very carefully designed

experiments (Lade et al. 1987, 1988; Lade 1988, 1992; Lade and Pradel 1990; Lade and

Yamamuro 1993) to investigate the instability of granular materials, which exhibit

apparent nonassociativity. The specimens were controlled to follow a loading path in

the "wedge" region between the current yield surface and the current potential surface








25

as customarily determined. Both the vertical pressure and the confining pressure were

decreased, but the ratio of the two was increased to produce inelastic deformation.

They reported that the specimens behaved in a fully stable manner, despite the fact the

computed second order plastic work quantity d&iji was strongly negative.

Furthermore, the positive second order elastic work on the path is much too small to

produce a total positive sum of the two, drj ti = Jj (tijP+tj'). They concluded that

"stability in the small in the forward sense" is only a sufficient condition to insure

stability or uniqueness of solution, but is not a necessary condition. In other words, the

material still could behave in a stable manner even when the product of the increment

of stress and the total, elastic plus plastic, increment of strain is negative. A subsequent

paper by Pradel and Lade (1990) put forward the suggestion that the nonassociated flow

rule model was in fact stable because stable paths of loading do exist.

Lack of normality at a smooth point aijo on the current yield surface gives


0 i& < 0 (3.6)

for each loading path direction in a continuous range of directions in the "wedge

region", Figure 3.4. The scalar product of the rate or infinitesimal increment of stress

and the rate or infinitesimal increment of plastic strain is negative.

Similarly, at each point along a straight line stress path in the wedge region


(3.7)


(a i-a.) < 0








26
The scalar product of the total increment of stress and the rate or infinitesimal

increment of plastic strain also is negative.

If the elastic response along the path is small in comparison with the inelastic,

there will be instability both in the large and the small in the forward sense (Drucker

1951, 1960, 1964).


iY eii < 0 (small) (3.8)



( ij-a ) < 0 or o e, > ay if (large) (3.9)

where the total strain rate tj is ei e + ij, the sum of the elastic and the plastic

strain rates.

Then the total work per unit volume done by the initial state of stress along any

such path of deformation


foa 0 dt or fa de l (3.10)
would exceed


fa dei = dE + fo d (3.11)


the energy stored reversibly plus the energy dissipated and stored irreversibly in the unit

volume. Therefore, if any such path exists that can be initiated by a permissible

infinitesimal perturbation and then followed in a kinetically and kinematically consistent

manner, the system would acquire kinetic energy by following it or some other unstable

path from the initial configuration.








27
The above point of view can also be verified via the extended virtual work

(displacement) principle (Truesdell and Toupin 1960, p595; Malvem 1969, p241). Let

Cij j0 uj denote respectively an equilibrium of stress, strain and displacement state

under the body force pbj which satisfy the equation of equilibrium


ao ..o
au- + pb. = 0 (3.12)
ax1


If there exists a stress path consistent with the strain and configuration which permits

the stressed material to run away, i.e., to accelerate and acquire velocity, then the

equation of motion reads

aa..
a + pbj = pVj (3.13)
ax,

which can be rewritten as


80..
+ P( bi ) = 0 (3.14)
axi

Subtraction of equation (3.12) from (3.14) yields


x, y o ) p- p =0 (3.15)

Let us take the compatible set as


Ae = e e
i if U


(3.16)










Au. = Uj uo (3.17)

The principle of virtual work then gives us


f(oyj-o .)Ae ,dV= -fp TVAujdV
V V
= -At- f lpVdV (3.18)

= -AK
where


K = 2 pV2dV (3.19)

is the kinetic energy of the system. The negative fv (aJ-aij)AsjdV will give the

system a positive increment of kinetic energy,i.e.,the system will acquire kinetic energy.

It was then and is now still commonly accepted that stability in the small in the

forward sense is a sufficient condition to ensure uniqueness. For an elastic system

under conservative forces, Hill (1957) pointed out that it is also necessary for

uniqueness and stability by following the argument of the linear theory of vibrations.

He made no comments on elastic-plastic materials. Yet the thought appears to be

spreading that stability in the small in the forward sense is by no means a necessary

condition for uniqueness (Willam and Etse 1990; Bigoni and Hueckel 1990, 1991;

Petryk 1991, 1992; Runesson et al. 1992).

When the stability of a system is examined, all permissible infinitesimal

perturbations must be presumed to be present at all times. For a nonassociated flow

rule most of the paths in stress space from aijo are stable if the specimen can be








29
constrained to follow them. However, the existence of such stable paths is not relevant

for a system free to follow any path. An unstable path will be followed instead by an

unconstrained system if at least one such path is available. The analog given by

Drucker and Li (1992), a block situated at the saddle point of a slightly tilted saddle-

like frictionless surface is helpful though oversimplified as a representation of the

behavior of material. Positive work must be done to move the block in most directions

because the surface on which it rests slopes upward everywhere except for a narrow

angular region on each side of the path of steepest descent. However, under

vanishingly small disturbing forces continually applied in all possible directions, the

block will slide in a generally downward direction in this angular region, picking up

speed as it goes. Attachment of the block through a soft spring to the saddle point does

not change the picture appreciably. The situation is also the same in the presence of

friction with a sufficiently greater tilt of the surface to overcome the frictional resistance

to sliding.

The extension of the uniqueness proof and bifurcation approach from an

associated flow rule to a nonassociated flow rule is neither obvious nor trivial.

Unfortunately, previous uniqueness proofs (e.g., Mroz 1963; Runesson and Mroz 1989)

and bifurcation approaches with a nonassociated flow rule simply carry over the ideas

and methods used in an associated flow rule case but with a potential surface different

from the yield surface. Configurations in the vicinity of the current equilibrium

configurations need not be in equilibrium for all directions of the loading path for a

nonassociated flow rule.








30
The associated flow rule ensures that the material is stable in the small in the

forward sense. The neighboring configurations in all permissible loading directions

then are truly in equilibrium. Continuing equilibrium does hold. In the case of a

nonassociated flow rule, if the elastic response is large enough to overwhelm the plastic

response, i.e., for weakly nonassociated material, the technique for the uniqueness proof

and bifurcation approach does carry over from the associated flow rule to the

nonassociated flow rule. For strongly nonassociated materials the neighboring

configurations for stress path directions in the wedge region will not be in equilibrium.

Then, the extension of the standard uniqueness proof and traditional bifurcation

approach to a nonassociated flow rule will fail. The fundamental reason for the

controversy of uniqueness with a nonassociated flow rule, in the author's opinion,

comes from this seemingly obvious and direct but not appropriate extension, which

leads to many theoretical and practical uncertainties and great confusion for

nonassociated flow rules.





3.4 Summary



Stability in the small in the forward sense is a classification of material behavior,

not a law of thermodynamics. It is stability in the usual sense against infinitesimal

perturbation. The direct simple extension of the uniqueness proof and bifurcation

approach from an associated flow rule to a nonassociated flow rule is based on a strong








31
tacit but not always valid assumption of continuing equilibrium. This is fundamentally

responsible for all controversies regarding the uniqueness and bifurcation with a

nonassociated flow rule.

Nonassociated flow rules are widely used in engineering practice. The

computing uncertainties that arise and the experimental results of Lade and his co-

workers call for further investigation of the instability resulting from a nonassociated

flow rule.

















P


Figure 3.1 For the load below the reduced modulus load
Pr but above the tangent modulus load Pt each and
every straight configuration is stable but the path is
unstable. The bifurcation of path is a stable bifurcation.




































5P 5U>O 5P 8U=0 6P 5U<0


(6Ijij =0


Figure 3.2 The stable, neutrally stable
and unstable configurations of a circular
cylinder with a sharp point on one end
(after Drucker 1960)

















































Figure 3.3 A block of material under
stress is analogous to an ordinary structure
under applied load (after Drucker 1965)

























ij yield
surface


potential
surface


Figure 3.4 A nonassociated flow rule and
the "wedge region"


wedge
region














CHAPTER 4
THE CLASSICAL SHEAR BAND THEORY



The localization of uniform plastic deformation into a planar shear band has been

given an analytical basis by the work of Thomas (1961), Hill (1962), Rice (1973),

Rudnicki and Rice (1975), and Rice (1977) building on the ideas of Hadamard (1903)

about the stress and strain discontinuities in elastic media. Hill (1961) explored the

discontinuity conditions in the elastic-plastic solid in the spirit of Hadamard. Both he

(1961) and Thomas (1961) derived general kinematic and kinetic conditions for a

moving surface of discontinuity. Hill (1962) formulated the condition for a stationary

acceleration wave in an elastic-plastic solid, which is equivalent to the shear band

localization condition. Mandel (1966) expanded on this concept. Rice (1973) pointed

out the possibility that localization into a shear band can be understood as a result of

bifurcation from a uniform deformation mode, and can be predicted from the pre-

localization stress-strain relation. He formulated the conditions of continuing

equilibrium and the "jump" or discontinuity of deformation field across the band.

Rudnicki and Rice (1975) advanced Rice's previous work and clearly formulated the

shear band localization as a bifurcation problem. They applied the condition to an

extended Mises (Drucker and Prager 1952) material with and without vertices in the

subsequent yield surface. Rice's comprehensive and thoughtful survey (Rice 1977)








37
finally settled the framework of the now classical shear band theory. An excellent

review on shear band localization in metals, especially with regard to numerical

investigation, was given by Needleman and Tvergaard (1992).





4.1 The Classical Shear Band Theory



4.1.1 The Outline of the Classical Shear Band Theory

Following the terminology and notation of Rudnicki and Rice (1975) and Rice

(1977), the classical shear band theory is formulated in the following manner. An

initially uniform deformation field is produced by uniform stressing of a homogeneous

material. Conditions are sought for which continued deformation under increasing load

may result in an incipient nonuniform field, with deformation rates in a planar band that

differ from those outside the band. The discontinuity of velocity gradient across the

band boundary is written as




A (V (4.1)








38
where A denotes the difference between the values inside the band and outside the

band, n is the orientation of band and g is an unknown function which varies with

n*x inside the band and is zero outside the band. In terms of the rate-of-deformation

tensor D


1
SDij = Dij-D = (gini+gni) (4.2)

where


1 av. av.
D ( +1 x) (4.3)
U' 2 ax. ax.

Do is the value in the main body of the material outside the band, and D is the value

inside the band. Continuing equilibrium is supposed to be satisfied at the inception of

bifurcation,


a/ 0 (4.4)
axb

As discussed in the previous chapter, this condition of continuing equilibrium means

that equilibrium must be satisfied not only at the inception of bifurcation but at the

immediately subsequent stage as well. The rate of surface traction as well as the

traction itself is continuous across the boundary of the band. Thus equation (4.4)

becomes


niAA6, = 0


(4.5)










The constitutive equation reads as


Sj = LiDkl (4.6)

where Lijkl is a linear material operator. An incrementally nonlinear elastic-plastic solid

later was considered by other authors in the same framework. For the simplest case,

both the material inside and outside the band belong to the same constitutive cone (Rice

1977), i.e., they obey the same stress-strain relation (both continuously deform

plastically), Lijkl = LOijkl and thus


aij = Likl ^Dk (4.7)

Combining equation (4.2), (4.5) and (4.7),


(ni Lijkl n) gk = 0 (4.8)

This is a typical eigenvalue problem and a nontrivial solution for g requires the

vanishing of the determinant of niLijkinl.

Later, Rice and Rudnicki (1980) labelled this case as continuous bifurcation, and

interpreted it as an analog to Shanley column buckling (Shanley 1946, 1947). They

used the term discontinuous bifurcation, if immediately after the bifurcation the material

inside the band and outside the band are not in the same constitutive cone, as for

example when the material inside the band continuously deforms plastically while the

material outside the band unloads elastically. They also showed that continuous

bifurcation corresponds to a lower bifurcation load (Rice and Rudnicki 1980), and noted

that continuous bifurcation is an analog of Shanley's column whereas discontinuous

bifurcation is not. This is true in the strict sense that in a continuous bifurcation all








40
material in the deformation field deforms plastically, just as all material across the

whole section of a Shanley column deforms plastically; while for discontinuous

bifurcation the material outside the band unloads elastically and the material inside the

band continuously loads plastically. However, in a more general sense, Shanley's great

contribution is the introduction of the distinction between the concept of stability of

path and stability of configuration. Then, both the continuous bifurcation and

discontinuous bifurcation are analogous to Shanley's column, because the bifurcation

condition is sought under changing external load. Both are questions of stability of path

rather than stability of configuration.

The continuing equilibrium condition may be formulated in terms of nominal

stress s (first Piola-Kirchhoff stress), as Hill always did, to take into account the

geometry change. However, the traction rate continuity condition, which plays a key

role in the formulation of the classical shear band theory, was shown to be exactly

equivalent in terms of either nominal stress or Cauchy stress (Rudnicki and Rice 1975;

Rice 1977). Also, the Jaumann (co-rotational) stress rate (see, Prager 1961 or Malvern

1969) rather than simple Cauchy stress rate was used in general to take into account the

rigid rotation. The using of the Jaumann rate has an effect of the order of O(h/G), in

which h is the plastic hardening modulus and G is the elastic shear modulus

(Rudnicki and Rice 1975; Rice 1977).

Following the establishing of this framework there were numerous publications

on many aspects of shear band localization for a variety of materials, e.g., porous media

(Loret and Prevost 1991, Loret and Harireche 1991). The number is still growing (e.g.








41

Lee 1989; Ottosen and Runesson 1991; Aubry and Modaressi 1992). The theory was

extended to hypoelastic material (Kolymbas and Rombach 1989) and hypoplastic

material (Wu and Sikora 1991) and constitutive incremental nonlinearity was also

considered (Kolymbas 1981; Desrues and Chambon 1989).

A great number of applications to granular materials (Vardoulakis 1989;

Schaeffer 1990; Chandler 1990; Molenkamp 1991a, 1991b; Bardet and Proubet 1992)

has appeared. There is an interest not only in the critical localization modulus, but also

the inclination angle of the shear band that is initiated, and the prediction of the

thickness of the band as a function of the diameter of the granular particles

(Vardoulakis 1980, 1988; Bardet 1990; Jenkins 1990).



4.1.2 Shear Band Localization in Crystals

Localization into a band in crystals is of particular interest. Plastic deformation

in crystals is inherently nonuniform and highly localized, especially when the crystals

are subjected to large strains in the work-hardening stage. Asaro (1983a) classified the

three most often observed nonuniform deformations in crystals as "deformation bands",

"kink bands" and "shear bands". The experimental results obtained by Price and Kelly

(1964) are interesting and important. To determine whether the material inside the band

was "softer" than the remainder of the crystals, the specimen was loaded until "coarse

slip bands" appeared, and then the specimen was unloaded. After the surface of the

crystal was polished, the specimen was reloaded. It was found that when the shear

stress reached the previous level for the formation of coarse slip bands, the coarse slip








42
bands appeared again but never at the same place as the previous band. This suggested

that the material inside the band actually was work-hardening during the formation of

the band rather than softening. The experimental results of Chang and Asaro (1981)

gave the same indication. It is believed that the coarse slip bands were caused by the

thermally activated breakaway of dislocation barriers formed during work-hardening.

By adopting the classical shear band formulation, Asaro and Rice (1977) made

important progress in the understanding of this form of localization in crystals. One of

the most important results of their investigation concerns the non-Schmid effects on

localization and the estimation of the Strength Differential (SD) effects. They obtained

the approximate relation, hcr 0.07 SD2 G, where her is the critical plastic hardening

modulus and G is the elastic shear modulus. Localization in crystals, on the other

hand, is generally observed to occur in the range of hc/G = 5x103 5x10-4 (Chang

and Asaro 1981). Such values correspond to a SD of 0.27 to 0.085 according to their

approximate relation. However, reported SD values are extremely small for ordinary

mild steel and only reach the order of 0.10 to 0.07 in high-strength martensitic steels

(Spitzig et al. 1975, 1976). The conclusion is therefore obvious, the hydrostatic

pressure-sensitivity is unlikely to have a significant effect on the onset of classical shear

bands in crystals. It may be noted that in this approach and in the later approach of

Qin and Bassani (1992), the nonassociativity in crystals is attributed only to non-

Schmid effects.









4.1.3 A Consideration of the Classical Shear Band Approach in Stress Space

Equation (4.2) can be rewritten as D = AD + Do. Therefore, basically, the

classical shear band approach is to superpose a plane shear band pattern (a particular

type of perturbation) on the uniform deformation pattern to seek the condition under

which this superposed pattern can grow during the loading procedure. This is a

question of stability of path. It is helpful, at the outset, to exhibit one of the basic

assumptions in the analysis, equations (4.1) and (4.2). The spatial coordinate system

is chosen as shown in Figure 4.1 with x, in direction normal to the band. Equation

(4.1) becomes (Rudnicki and Rice 1975),



av, (4.9)
A (x) = g(xl)8j1



Consequently, equation (4.2) becomes


a Di = = (g ji+g8 il (4.10)

It is clearer to write out each component


AD11 = g,

^D12 = AD21 = 2
2 (4.11)
AD13 = AD1 = 1g3
AD22 = AD33 = AD23 = AD32 = 0








44

The deformation pattern has no relative deformation in the x2 x3 plane, i.e., a relative

instantaneously non-deforming plane exists in the deformation field which divides the

material inside the band from the material outside the band. There can be relative pure

shear in the x, x3 plane and in the x, x2 plane; also there can be relative

extension in the x, direction (Figure 4.1). Therefore the shear band type localization

is a plane strain picture superposed on a uniform deformation field. It is not surprising

that the shear band, as interpreted by Rice (1977), appears more easily under conditions

of plane strain than axial symmetry.

The consideration of the classical shear band approach in stress space, which

differs from the usual consideration in deformation space or "F-space" (Rice 1977), is

informative. Equation (4.7) clearly suggests that it is the difference of the stress rate

Abr which gives the difference in the deformation pattern AD provided the material

operator L possesses an inverse. As Rice (1977) emphasized, the localization

condition is not that the constitutive equation (4.6) possesses no inverse.

The continuity of traction rate across the band n A&i = 0 only requires the

continuity of three components of stress rate, dIr,, 612, 13 (the coordinate system is

chosen as in Figure 4.1). The other three components of the stress rate, 622, 633, 023, are

not necessarily continuous across the band, and actually, at least one of them should be

discontinuous. It is the discontinuous stress rate components which permit the

discontinuity in the deformation field corresponding to the shear band pattern (Figure

4.2).








45
An apparent paradox, which was pointed out by Lippmann (1976), according

to Rice and Rudnicki (1980), arose in the shear band approach of Rudnicki and Rice

(1975) and Rice (1977). The direct rigid plastic approach does not coincide with the

limiting case of elastic-plastic approach as the elastic shear modulus G tends to

infinity. Let us reexamine this paradox. As Rice (1977) described, the response of

the rigid plastic material can be generally represented as,


h D. = P, (Qn& M) (4.12)

where h is the plastic hardening modulus, D is the rate-of-deformation tensor. P is the

outer normal unit "vector" of the potential surface which gives the "direction" of plastic

deformation, and Q is the outer normal unit "vector" of the yield surface. Both P

and Q are second order dimensionless symmetric tensors. If the material inside the

band and outside the band belong to the same constitutive cone immediately after the

bifurcation, the material inside the band and outside the band possess the same h, P

and Q. From equation (4.12),


h ADi = Piu (Q Ab kl) (4.13)

The assumed shear band pattern, equation (4.2), requires P to have the form of (Rice

1977, Rice and Rudnicki 1980),


P = (ni1y+nj i) (4.14)








46

where g is some vector having the form of Xg. As we did for equation (4.2), if the

spatial coordinate system is chosen with the band normal in the direction of x, (Figure

4.1), the components of P will be


P11 = R
P 12= P21 2 1
2 (4.15)
P13 = P31 = 13
P = P =P =P =0
P22 =33 = P23 = P32 =



This is the condition in which there exists an instantaneously non-deforming plane,

x2 x3 plane, in the material (Rice 1977, Rice and Rudnicki 1980). Rice thought of

this as extremely restrictive. It is, in fact, as may be seen by examining this picture in

stress space. The stress state reaches a point on the yield surface corresponding to a

plane strain state before the bifurcation occurs. Therefore the direct rigid plastic

approach deals with a problem of instability of path when the stress point in stress

space has already reached a plane strain state, rather than with a general stress state as

in the elastic-plastic approach. The explanation of the paradox is simply that for the

rigid plastic material the stress state can move along the current yield surface without

any restriction and without any further plastic deformation. An elastic response, no

matter how small, provides a barrier to moving purely along the current yield surface.

In the rigid plastic case, the increment of strain (rate) does not determine the increment

of stress (rate) in an unique manner. As shown in Figure 4.3, both stress increments

correspond to the same increment of strain. Therefore the direct rigid plastic approach








47

only appears to consider an instability of path at a general stress state. It actually

considers only a plane strain state to which the stress state moves instantaneously.



4.1.4 Vertex Effects

As an aside, plastic flow localization has similarities to the plastic buckling

problem. A possible vertex effect enters the discussion. Within the framework of rate-

independent plasticity, it was observed that even when geometric imperfections were

taken into account the physically valid flow (or incremental) theory did not predict a

plastic buckling load in accord with experimental results when isotropic hardening was

assumed. Instead, predictions by the generally abandoned nonlinear-elastic-like

deformation (or total) theory agreed reasonably well with the experimental results

without the assumption of geometric imperfection. Batdorf (1949) showed that good

agreement with flow theory could be obtained by introducing a vertex corerr) on the

current yield surface. This served as a "driving force" to explore the existence of

vertices on the current yield surface. Batdorf and Budiansky (1949, 1954) proposed a

theory based on slip, a physically appealing concept, which leads to vertices at the

current loading point on subsequent yield surfaces. Hill's (Hill, 1967a) more rigorous

general study, the work of Hutchinson (1970), and the J2 corer theory of Christofferson

and Hutchinson (1979) convinced many of the existence of vertices on the yield

surfaces for metals, although most careful experimental investigations showed well-

rounded noses rather than covers. This aspect of behavior remains quite controversial.

For frictional materials or geomaterials in general, theory based on simple slip, similar








48

to that of the simple Schmid type in metals, also indicates vertices (Rudnicki and Rice

1975). Storen and Rice (1975) investigated the localized necking in thin sheets under

biaxial stretching using a flow theory with vertices, because the simple flow theory with

a smooth yield surface predicts that a perfect sheet can not exhibit localized necking

under the bi-stretching condition. Mear and Hutchinson (1985) examined vertex effects

on flow localization. Just as in plastic buckling and localized necking, the vertex has

a very strong destabilizing effect leading to earlier shear band localization, because at

the vertex point an abrupt change in the loading direction gives a much lower

bifurcation load than for a smooth yield surface. Rice (1977) has given an extensive

discussion of vertex effects on shear band localization.





4.2 A Discussion of the Classical Shear Band Theory
with a Nonassociated Flow Rule



Let us now return to the nonassociated flow rule. For an associated flow rule,

the bifurcation analyses in general (Hill 1958, 1959, 1967b, 1978) and the shear band

localization in particular (Hill 1962, Rudnicki and Rice 1975, Rice 1977) is now well

understood. The corresponding theory for a nonassociated flow rule, however, is much

less developed although it has been discussed by Rudnicki and Rice (1975) and Rice

(1977) to some extent, and attempts were made by some other investigators (Needleman

1979; Raniecki and Bruhns 1981). It was recognized that nonnormality produces a

strong tendency toward localization. One of the very interesting aspects of the results








49

obtained by Rudnicki and Rice (1975) is that localization can never occur within the

regime of work-hardening for elastic-plastic materials with associated flow rules. For

the elastic-plastic material (extended Mises) with nonnormality, however, it was shown

that localization can occur in the hardening regime (though close to zero hardening),

if the stress state falls into a narrow range which depends on the extent of

nonassociativity. Needleman(1979) has also shown in the plane strain case that large

enough nonnormality may make it possible for the localization to occur regardless of

the hardening.

In his consideration of an acceleration wave and the dynamic growth of a

disturbance, Rice coined a very descriptive terminology, "flutter" instability (Rice

1977; Loret and Harireche 1991; Loret 1992) to describe the possibility of the flutter

growth of a disturbance when normality does not apply. But it should be noted that the

possibility of an propagation of the acceleration wave depends both on the material

constitutive parameters and the direction of its propagation (Hill 1962). This

propagation direction in geometric space is connected with the loading direction of the

material in stress space. Any type of planar wave in homogenous materials can only

propagate if the material response for the propagation direction is stable. Restoring

force is needed to bring the material particles back to their original positions after the

wave front passes. For unstable materials, the smallest perturbation causes the material

particles to depart significantly from their original positions.

Rice(1977) showed that, if the material model is rigid plastic with normality,

localization can only occur with zero or negative hardening; while for rigid plastic








50

material with smallest degree of nonnormality, localization could initiate at any stage

of hardening for any general stress state. However, the slightest elastic response will

bring the critical hardening modulus back close to zero, a result obtained by Rudnicki

and Rice (1975). Again, the basic reason for this peculiarity of elastic response having

such a drastic effect on their prediction of localization is that, for a rigid plastic material

the stress state can move freely without any further plastic deformation along the

current yield surface from a general stress state to a plane strain state.

Most recent publications on aspects of bifurcation, uniqueness and localization

concentrate on the nonassociated flow rule. There is a great lack of consensus (Laborde

1987; Chau and Rudnicki 1990; Bigoni and Hueckel 1990, 1991; Runesson et al. 1991;

Chau 1992; Bigoni and Zaccaria 1992a 1992b; Loret 1992; Neilsen and Schreyer 1993).

As emphasized in the previous chapter, continuing equilibrium need not be satisfied

under all circumstances. Consequently, interpretations for a nonassociated flow rule

with the assumption of continuing equilibrium are of limited value, or even entirely

inappropriate.
























0
C~11


outside the band


-12
n
X3


inside the band


(712 (13


Figure 4.1 The spatial coordinate system
is chosen with the x, direction normal to
the band, x2 and. x3 parallel to the band























inside the band
G/


outside the band

A& = & 0


Figure 4.2 The bifurcation of stress path
of a uniform deformation. The difference
between the two stress rates permits the
occurrence of a shear band

















































Figure 4.3 Both the stress increments 6,
and 62 correspond to the same increment of
plastic strain














CHAPTER 5
A GENUINE INSTABILITY FOR A NONASSOCIATED FLOW RULE
WITH A MOHR-COULOMB MATERIAL MODEL



The first material model to be examined is the Mohr-Coulomb with a

nonassociated flow rule. The elastic response is assumed to be small in comparison

with the plastic response. There are two primary reasons for this choice. One comes

from the world of practice. A nonassociated Mohr-Coulomb material model is most

often employed in civil engineering practice to represent the inelastic behavior of sand

with and without cohesion. The other is a purely theoretical matter of convenience.

This simple material model suffices to elucidate the instability, a genuine instability in

the form of a shear band with a rotating boundary, that is conceptually very different

from the classical shear band with a fixed direction. This chapter largely follows the

work of Drucker and Li (1992, 1993) with some minor changes and many more details.





5.1 A Mohr-Coulomb Model with a Nonassociated Flow Rule



The isotropic cohesionless Mohr-Coulomb model is represented at each stage by

a hexagonal pyramid in a three dimensional principal stress space with its apex at the








55
origin. When oc is the intermediate principal stress, the governing yield plane

becomes the line, f(yield surface), in the two dimensional principal stress space aA, OB

of Fig. 5.1. The extension of this intersection of the current yield surface with a plane

through the current stress state and perpendicular to the third principal stress axis passes

through the point (0,0,ac) in the three dimensional principal stress space and appears

at the origin of the two dimensional stress space. The normal to the current yield

surface in the three dimensional principal stress space has a zero component in the

direction of the third principal stress axis cc and consequently appears in full in the

two dimensional principal stress space CA,acB A plastic loading path from stress point

1 to stress point 2 pushes the yield surface out and rotates it counterclockwise, as

illustrated in the inset of Fig.5.1, increasing its angle 4 with the 450 line. For

simplicity, the current potential surface, to which the plastic strain rate or increment is

normal, also is taken to be a plane perpendicular to the paper. It appears as the line in

Fig. 5.1 at a counterclockwise angle P to the 450 line.

In accord with the plastic stress-strain relation


P = ag ( 6 ) (5.1)
eGo a kl

where f is the yield function and g is the potential function. There is no plastic

deformation in the direction of ,c. A value of P = 0 indicates plastic

incompressibility of the material; P > 0 indicates volume expansion as the material

deforms plastically. If P = -, the potential surface and the yield surface coincide, the

model obeys an associated flow rule. For ease of description and analysis here, the

potential surface is assumed to remain plane and constant in orientation. A rotating or








56
a curved potential surface would add some complexity but would not have a significant

effect as long as its normal continued to have a zero or very small component out of

the plane of the paper.

From Fig.5.1, when the elastic strain rates are neglected, the components of the

strain rate in the principal stress directions are:


= eP= 0Pl cos(450 + 3 ) (5.2)
,B = e = -IPl sin(450 + 1)

where I p I is the magnitude of the total plastic strain rate "vector". The principal

stress rates are


oA = -161 sin(450+()-a) (53)
6B = -lo1 cos(450+--a)


where | is the magnitude of the total stress rate "vector".

The outward pointing component of r, normal to the current yield surface, is

| 6sin a Therefore, (5.1) may be rewritten in the physically appealing form

= P l sin (5.4)
h

where h is a plastic modulus.



5.2 A Simple Pattern of Genuine Instability



Consider a volume of material subjected to the stress state (oY, o ,,H) with aC

as the intermediate principal stress. The material has entered the plastic range and the








57

stress state (ac, cr ,a) corresponds to a point on the current Mohr-Coulomb yield

surface in the principal stress space. Suppose there is some way to hold the material

under the stress state and then release the constraint. Now consider whether the system,

the volume of material under the given stress, is stable in the usual sense against all

possible infinitesimal perturbations. A uniform deformation of the material and a stable

behavior of the system is consistent with the fixed stress boundary condition and the

constitutive relations. The question is one of stability of this configuration.

As discussed in chapter 4, a nonassociated flow rule leads to the fact that the

material is unstable in the small in the forward sense. To demonstrate instability, all

one needs to do is to ensure that at least one unstable path exists for which the kinetics

and kinematics are consistent. The following argument is simply to construct one such

solution.

At fixed external loading (oa, oa ,Ha), a simple kinetically and kinematically

consistent pattern of instability exists in the form of a shear band with rotating plane

boundaries. Within this band, shown shaded in Fig. 5.2, the model material would be

free to follow and so would follow some unstable stress path in the wedge region. This

shear band pattern of instability is only one possibility, but it suffices to show that the

model is unstable at each fixed state of stress (oav,I,,H) in the plastic range. For

negligible mass-acceleration terms, as the acceleration grows from its initial value of

zero, the material outside the deforming band remains at the fixed stress state and does

not deform. Consequently, the boundary plane dividing the deforming band from the

non-deforming upper block must be inextensional. Its initial orientation, 00 in Fig.5.2,








58

and subsequent orientations 0 = 00 + xV are determined by the direction of zero

extensional strain rate tT in the band, as the principal stresses Ar, aB in the band

rotate and change in magnitude from their initial values av, rH The intermediate

stress ac keeps the same direction as the initial or but changes in magnitude from that

initial value. The requirement of zero extensional strain rate in the C direction is met

for negligible elastic strain rates because the normal to the current potential surface

appears in full in the two dimensional stress space. Its component out of the plane is

zero, (5.1).

A free body diagram of the upper portion of the specimen at each stage of the

unstable motion is shown symbolically in Fig.5.3. The traction components drawn in

the upper block, N," and ,NT" are simply the static equivalent of the fixed external

tractions CV,,H applied to the upper portion of the specimen. The intermediate

principal stress r, does not enter the kinetics. The initial state of static equilibrium

changes to an accelerating pattern of deformation in which the (equivalent) normal and

shear traction components on one side of the dividing plane are not the same as on the

other.

A consistent kinetics cannot be obtained at constant external load with a fixed

orientation of the band. The inextensional planes must rotate to provide the needed

inward or negative acceleration component aN required by ON" > oN .

In this simple model, the direction of zero extensional plastic strain in the band

remains at a fixed angle 00 to the directions of the principal stresses there. Therefore,

the directions of oA and CB must rotate by the same amount W as the plane








59

boundaries of the band, provided the elastic strain increments are negligible and the

potential surface maintains a constant orientation. Moderate elastic strain increments

would cause only a small change in the orientation of the plane boundary and in this

rotation.

At the inception of the instability and all during the rotation:


1 1
e8= (A+eB) B (A-B) cos20
2 2
1 1 (5.5)
eN = -(8,+e) + B A- BCOS280
2 2
?yrN = (A-eB)sin260

and


1 1
S= (GAA+aB) + -(O A -B)COS200
2 2 (5.6)
1
T, = (A-(A B)sin2O0


The condition s; = 0 (5.2), and the first equation of (5.5) give the initial angle

between the boundary plane and the horizontal (and the continuing angle between the

boundary plane and the direction of CB).


cos20o = -tan3 or sin( 200-- ) = tan3
2

For an incompressible material, 0 = 0 and 00 = 450 For a small to moderate

volume expansion angle, P tanp does not differ much from sin1 and 200 7/2 is

about equal to P .









Equations (5.2) to (5.7) give


( 6N 6Nu )N + (NT NT" ) YNT =
loP sina ( + (5.8)
sinm[ (a + P)]
h



a negative value only if the stress path lies in the wedge region, P > a This is
equivalent to instability in the large in the forward sense.





5.3 The Normal and Tangential Driving Forces and Accelerations



The surrogate tractions for the upper block oNu TNT and the tractions inside

the band ,N, TNT are shown pictorially in the Mohr's circle plot of Fig.5.4. The initial

point M is at (twice) the initial orientation, 00, of the plane of zero strain. As the

band boundary rotates by y the point u (Nu", TNT), which represents the fixed state

of stress oV,CH in the upper block, moves counterclockwise by 2w around the circle.


aN (= -(a+ ) + -(a v-aH)cos20
2 2 (5.9)
S= -j(oav-H)sin26

When aA, aB follows a straight line wedge path at an angle a to the initial yield

surface through cv, aH as shown in Fig.5.1, the stress rates &A I B, I I in Fig.5.1

and (5.3) may be replaced by the stress increments Ao A' B | AC I Subsequent

yield surfaces have increasing values of but a increases by the same amount so








61

that a a remains constant. Because (5.6) is linear in stress, the point b

representing the tractions inside the band, ca and %T, moves along a straight line path

from M toward some point P on the circle. The traction increments AxNT and Acs

are negative and given by:


A - [ cos(4-a) + cos260 sin(4-a) ]
r2 (5.10)
IAol
AT, IA- sin20o sin(O-a)


The angle 68 between MP and the horizontal is given by


SA NT (5.11)
tan, -
AoN



It is easy to observe from Fig.5.4 that 200 82 = 900. Employing the equations

(5.3), (5.6) and (5.7) gives


tan6 = tan(-ca) 1-tan21 (5.12)
1-tan(4-a)tanp

and


tan, = tanp1 (5.13)
1 -tan2i3

One can show from equation (5.12) and equation (5.13) that 6, > 62 is valid only under

the condition of ( P > a ; the stresses CA B must follow a path in the wedge

region. The locus of the traction oN TNT will follow a straight line secant of the

locus circle of the surrogate traction of the upper block aN,, ,TT" .








62

The main component of the driving force is the excess of rNTu over ,T or

the difference between the vertical coordinates of u and of b The excess of Cou

over oN, the difference between the horizontal coordinates produces an inward or

negative component of acceleration a .





5.4 Discussion of Dynamics



Although great point has been made of the need to be sure that both the kinetics

and kinematics of the problem are satisfied appropriately, no real dynamic analysis has

been given. For simplification the incompressible material is examined here. The

treatment can be extended to compressible material without difficulties in principle.

The incompressibility of the material gives tN =0. The geometric relations are:



VN =- BN (5.14)
VT = BNT



where B is the instantaneous or current thickness of the band. The rate of change of

B, B = -2L0 (Fig.5.2), in which L is the instantaneous or current length of the band.

The equation of motion of the upper block in the N direction (for unit thickness in the

third direction) is











oN O = maN +hmVN
dVN
= m(-OV + ) + mhV (5.15)
dt
= m6OV

and in the NT direction


T TT = mat + maT
dVT
= m(6dV +_) + nVr (5.16)
= mV, + th V


where m is the instantaneous or current mass of the upper block. Equation (5.15)

indicates 6 = 0 at the inception of the instability. The process of running away starts

from equilibrium with an arbitrarily infinitesimal perturbation of V.

From the second equation of equations (5.14)


(5.17) VT = 'NT + B~'N,
= By 2LO,


The rate of change of mass ih = 1/2 pL26 Therefore equation (5.16) can be rewritten

as


r r m = m(B 2L8 7) + -pL2 OBj'N
2 (5.18)
= NBj + L(2-2pL)6T1
= mByi, + L(2m--pLB)O69 .
2








64

Because m is much greater than pLB the mass of the band material, the

effect of the change of mass on the equation of motion in the NT direction can be

neglected. Substitution of the second equation of equations (5.14) into equation (5.15)

gives,


o -o = m OV = -mB"T. (5.19)

The rate of VT is composed of two parts, equation (5.17). The first part is due to the

stress change and the second part is the contribution of the narrowing of the band. The

rate of change VT is positive and thus the second term can not override the first term

for a consistent kinematics. Actually, the second term may be negligible to a first

approximation. Suppose B~NT is three times large as 2LOj or much larger still;

then TT" TNT = 4mL6,NT Let us assume B/2L is the order of one tenth, then

oN N," is of the order of one twentieth of TNTu NT. Therefore, the shear traction

component dominates the dynamics of the upper block and crN remains almost equal

to uNU during the process.

Equation (5.15) also indicates that aN is always a little bit smaller than cr" and

thus the point (aN rNT) travels a little faster than the corresponding point (aNu Nu")

with almost the same value of aN Fig.5.4. The difference between TNT" and NT

provides the major portion of the driving force. When this driving force vanishes, as

it does at point P (Fig.5.4), the system overshoots a little and then comes to rest.

The estimation of the overshoot is also of interest. The rate of energy input to

the system by the external force ,a and oH at time t is LB'NTVT and the rate

of energy dissipated by the band material is LB'CNT'NT where L and B are the








65

instantaneous band length and thickness (Fig.5.3). Thus the kinetic energy developed

in the system from the starting time t = 0 to the time at which the driving force

vanishes, t = to, is

to AYNT
( NT-
fLB( NT-tmix dt= LB(T -T ) dy, (5.20)
0 0

where aYNT is the total shear strain increment from the starting point of instability to

the point of vanishing driving force. The energy dissipated in the overshooting process

is


ti 6YNT
f LBt, dt= f LB-r dy (5.21)
to 0

where 8yNT is the shear strain increment during the overshoot. Equating the two

energies


AYNT 8YNT
f LB(t-u-T dy = f LBr dy (5.22)
0 0

or


LB(tu-tr ) Ay, = LB, 6Sy (5.23)

where the overline denotes the average over the corresponding strain increment.

Therefore










U
-LBNT NT ) NTN mA"Y (5.24)
LB NT T NT

Because the difference of xrN zNT is in the order of a tenth of ,NT (Fig.5.4), the

overshoot is of the order of a tenth of the total strain jumps.

The mass of the system to which the surrogate free body of Fig.5.3 applies has

been left unspecified, as has the thickness or mass of the shear band. Certainly it is an

oversimplification to assume that only the specimen need be considered, because the

dynamic loads to which it is subjected depend strongly upon the fluid and solid loading

devices as well as the specimen itself. Furthermore, the mass of the upper block

increases continually as the boundary planes rotate and the mass of the deforming band

decreases. If the free body diagram of Fig.5.3 is interpreted as showing the forces

acting on the upper block, the time rate of change of momentum of the block would

include a term for the time rate of increase in mass. If the free body diagram instead

represents half of the band plus the upper block, then the total mass does not change

but the equivalent term is introduced by the transfer of moving mass in the band to

stationary mass in the block and the other changes in velocity in the band. Also, elastic

and plastic wave propagation would accompany any unstable jump in stress and

displacement.

There are many uncertainties of detail. For example, the implicit assumption is

made of uniformity of the state of stress and strain along the length of the band.

Therefore, the traction boundary conditions are not met where the band reaches the

loaded lateral surface of the specimen. This would be of little consequence for a very








67

thin shear band. However, unless the band has appreciable thickness, the instability

would be severely limited by the geometry of the rotation.

Nevertheless, the essence of dynamics and geometry has been included in this

analysis of a model of homogeneous material obeying a nonassociated flow rule. It is

clear that permissible infinitesimal disturbances will cause the stresses, strains, and

displacements of each equilibrium configuration in the plastic range to move or "jump"

along a continuous unstable path to another configuration a finite "distance" away.

The plastic stress-strain relations by definition, whether associated or

nonassociated, are time-independent. This idealized response provides great flexibility

because it has no time scale. Consequently, much of the real complexity can be

ignored. A reasonable picture of the instability can be obtained on the assumption that

the upper block is rigid and its mass large enough compared with the mass of the band

to take the driving force equal to the acceleration multiplied by a fixed mass.

The unstable motion of this nonassociated flow rule model will come to a halt

just a little after point P is reached, or earlier should the thickness of the band go to

zero sooner. The stress at each point in the band then reverts to CTV,CI,H, the initially

imposed state of stress, along an elastic (unloading) path of increasing principal stress

magnitudes lying inside (below) the current yield surface. The stress state once again

becomes homogeneous throughout the entire model. However, the material of the

model now is far from homogeneous. No deformation, therefore no hardening, has

occurred outside the band. Within the band, different portions of the material have

experienced different plastic strains and therefore different amounts of hardening.








68

The unstable jumps will be still smaller when the band is very thin, because the

thickness B will go to zero before the full rotation


max = 61 6 (5.25)

can be reached.





5.5 The Effect of Elastic Response



The effect of including a moderate elastic response in the model of the material

along with the plastic does call for some additional comment. The details of the

instability certainly become more complex. With the usual assumption of isotropy, the

principal elastic strains in the band would be in the direction of, and given in magnitude

by, the principal stresses. The elastic strain rates are given by the stress rates as


.e oA (&B+6C)
e v
E E
B. B C A) (5.26)
E E
.e 6C OA +&B)
v
E E

Therefore, for the extensional strain rate in the direction of ac to be zero, ac would

have to change along with GA, cB. Also, 4T would have elastic strain rate terms due

both to the rate of change and to the rotation of the principal stresses.









Writing ij = ij e + ptP the condition tT = 0 gives 00


-(1+v)(1-2v)cos(4-a) + sinp-sina
cos20 =
h(1+v)sin((-a) cosp-sina (5.27)
E
= tan + ( () )
E

which converts back to the equation (5.7) if h/E is negligible. Consequently the

angular orientation of the plane of zero total extensional rate in the band would differ

only a little from that found on the basis of zero plastic strain rate. If the angle a is

not too small, the plastic strain rate would be much larger than the elastic strain rate;

the orientation already computed is close enough. In general, the change of this

orientation is a smooth and moderate function of the elastic to plastic strain rate ratio.

So also are the net driving tractions in the directions normal and tangential to the

boundary plane.

Also, the elastic response is taken into account,


( 6N aN" )eN + ('NT NT" ) NT =
a- sinasin[)-(a+p)] + (1+v)[(1-2v)cos2Q(-a) + sin2( -a)] + O( (h)2
h E E
(5.28)

The first term is the term in the equation (5.8) which is the plastic part, while the

second term corresponds to the elastic part and involves only the elastic properties,

Young's modulus E, Possion's ratio v, and the stress path which is described by its

direction D a and by its magnitude r Obviously, the elastic part is of the order
of O(h/E) of the plastic part.








70

Consequently, the onset and the character of the instability of the model would

not alter significantly when an elastic response is added to the plastic, provided the

plastic modulus h is very much smaller than Young's modulus E.



5.6 The Magnitudes of the Unstable Jumps
in a Model Simulating Sand



The unstable jumps predicted to occur in the stress, strain, and displacement of

a Mohr-Coulomb model depend strongly on its material properties and the wedge path

chosen. For convenience and interest, the 1987 paper by Lade and coworkers served

as a guide for the following choices in the equations and Fig.5.1: cy = 110 kPa (approx

1.1 kg/cm2); aH = 30 kPa; a=50 ; = tan'(av /GH) 450 = 300 ; expansion angle

0 = 200; wedge angle ( P = 100 ; plastic modulus h = 0.13 MPa; and Young's

modulus E = 40 MPa (approx 400 kg/cm2).

As already described, it is not necessary to solve the dynamic problem in order

to estimate the jumps closely enough for the purpose here. A simple geometric

approach based on Fig.5.4 is sufficient. Substitution of the values of 4, a, and p into

(5.7) and (5.11), gives 8, = 27.60 and 62= 20o 7n/2 = 21.30 .

The maximum possible rotation, yma of the boundary planes of the band

(5.25) is a moderate 6.30, half the central angle on the Mohr's circle subtended by the

arc or the chord from M to P. If the initial band thickness is not large enough to

accommodate this rotation the current thickness goes to zero before this maximum value

of W is reached and the unstable motion ceases earlier.








71
The jumps in the tractions of the material inside the band, from point M to

point P along the chord, are given by the length of the chord (av H)sin(81-62) = 8.8

kPa and its slope angle 81: AON = 8.8 cos68 = 7.8 kPa ; Ar, = 8.8 sin8, = 4.1

kPa.

The jump in the stress state from (5.10) is | AT | = 14.7 kPa. Then, from (5.3)

with stress increments instead of stress rates, AGA = 13.8 kPa and Ao, = 5.0 kPa .

Thus AaA/aV is about 12.5%, and AcB/GH is about 16%.

The magnitude of the corresponding total strain jump from (5.4) is

approximately 0.01 when increments replace rates and the small effect of the change

in a along the path is ignored. From a similar modification of (5.2) that neglects the

changing direction of P in the physical space: AAP = +1 Asp I cos(450 + p) = 0.004 ;

AsBP = Ag Psin(450 + P) = 0.009 .

In one sense, the stress jump in the band is represented by the 8.8 kPa length

of the chord MP in Fig.5.4 and is quite large. However it is only the much smaller

difference between traction points u and b that can be detected by an observer

external to the band. The difference in the normal traction is very small. The

maximum difference between the shear tractions is the maximum distance in the radial

direction between the chord MP and the arc MP divided by the cosine of the angle 61

between that radius and the vertical axis:


(a V-O)[l-cos(61-62)] (5.29)
2 cos6,








72

The result is 0.27 kPa, which is only 0.3% of MP or about 0.25% of the initial

vertical pressure oy.

It is also of interest to give a rough estimation of the jumps in stress and strain

for high strength steels. Let us only consider cases of uniaxial tension or compression.

The experimental data of Spitzig et al. (1975, 1976) serves here as a guide for choices

of corresponding numbers. 4 = 50 and P = 1/15 4 = 0.330. The plastic hardening

modulus h is about 1/100 G, or 3x10' psi (2.07 GPa); the yield strength ao of those

high strength steels is also about 3x105 psi (2.07 GPa);. Again, if the unstable path is

chosen to be in the middle of the wedge, a & 1/2 4. The magnitudes of the hidden

jumps in stress and strain, and the magnitude of traction jump observable from the

outside are approximately


Ao, AoJ 1
sinl = 4.35%
Yo oY 2

AeA AeB Y sin2Y cos =- 1.9 10-3 (5.30)
h 2 2
AT sin24 = 0.0476% AT = 142 psi (0.98MPa)
rY 4





5.7 Patterns of Parallel and Intersecting Bands



The behavior of the single band exhibits the basic character of the genuine

instability of configuration under fixed loads ov, o,, aH. However, the instability is

incipient everywhere in the homogeneous specimen of the Mohr-Coulomb model








73

material that has been assumed here. Many bands would develop sequentially or

simultaneously, not just one.



5.7.1 Parallel Bands

The schematic picture on the left of Fig.5.5, shows a set of such parallel bands,

separated by undeformed regions. Each point of the material initially in each band has

been hardened to some extent. The greater the rotation y at the point, the greater the

hardening. The W ,a or most hardened region, shown by full lines of shading, tapers

off in the dashed zone to unhardened material in the rest of the specimen.

If the bands are thought of as occurring sequentially, as each is formed the stress

state in the material behind the moving boundary reverts to the original stress state, cy,

ao and oH, along some elastic path. A subsequent band then can initiate in any

undeformed and therefore unhardened region. The plane boundaries of each

undeformed region are at the needed initial orientation for a band to form. With surface

traction boundary conditions all around, the entire specimen finally would be covered

by parallel inhomogeneous bands of varying initial thickness and final hardness, Fig.5.5.

The final picture would be similar if the parallel bands initiate (almost) simultaneously

instead of sequentially. The "almost" indicates that a single thin band at any instant of

time minimizes the problem that the traction boundary conditions are not properly met

in the vicinity of the intersection of a band with the surface of the specimen.

If the boundary condition is imposed of no relative motion of points at the top

and bottom of the specimen, the bands shown there would be constrained from forming.








74

Only a portion of the material would participate in the unstable motion. With half of

the material in the specimen involved in the deformation, the total computed jump of

displacement in the vertical direction would be less than 0.5 mm in a specimen 200 mm

high.





5.7.2 Intersecting Bands

The formation of a single band forces a continuing unsymmetric change in the

geometry of the specimen at fixed load. Parallel bands can form without impediment,

but a shear band that attempts to cut across an existing one would have to traverse

already hardened material in the region of intersection and so would require a higher

load. However, there is no more reason for a band to initiate at +00 than -00 A

symmetric pattern produced by two symmetric shear bands that may be thought of as

initiating at their intersection is shown in Fig.5.6.

The material in each of the shear bands follows the equivalent of the single shear

band boundary plane rotation and wedge path of changing stress, strain, and

displacement. It is the motion of the top block (vertically down) relative to each side

block (horizontal) that is the equivalent of the motion of the upper block in the single

band picture of Fig.5.3. The material in the diamond shaped region of intersection also

follows a different wedge path in principal stress space but the directions of the

principal stresses there remain vertical and horizontal as the region deforms. All needed

continuity of surface tractions and displacements across the straight line boundaries








75

between this region and the shear bands can be shown to be satisfied. No plastic

deformation occurs in the remaining material.

This picture of "intersecting" shear bands that do not shear through each other,

but instead are offset on each side, can be generalized to any number of bands as shown

in the inset of Fig.5.6. However, the need to initiate the entire set simultaneously from

each of the intersection sites makes it a rather unlikely picture. Initiation from a single

small region, not necessarily at the mid-width as in the symmetric sketch of Fig.5.6,

seems more likely. Should that occur, the hardened bands generated would not allow

additional shear bands to cut across them at the fixed load.





5.8 Initial Inhomogeneitv and Continued Loading



The picture presented of a simple rotating shear band instability does not carry

over directly to a model that includes the initial inhomogeneity likely in any real

material or to the continued loading of an initially homogeneous model which hardens

inhomogeneously from the f, of an unhardened region to the f2 of a fully hardened

region, Fig.5.1. Suppose the loads applied to either model to be increased. A region

below yield will respond elastically while the yield surfaces of less hard regions are

successively pushed outward as the local stress state reaches each one. The incipient

instability at each point of the material that has reached yield certainly would be

constrained by the surrounding material's elastic response. If the path of loading is








76

made stable by the constraint of the material below yield or by some set of added

constraints, all the yield surfaces might be pushed outward to f2. The specimen then

would be homogenous and able to respond in the unstable shear band manner.

This indicates no more than that the configuration will be unstable when this

higher load is reached or earlier, not that a shear band clear across the specimen at that

load is the appropriate computed unstable response of the inhomogeneous model. It

seems far more likely instead that a succession of local or regional instabilities of very

limited excursion would set in as the load is increased.







5.9 Plane Strain versus Axisvmmetry



The shear band localization is a plane strain pattern superposed on the uniform

deformation (possibly zero) that takes place outside of the band. Mathematically, it is

easier to pose the problem under a total plane strain condition and easier to see the

character of the bifurcation and post-bifurcation for a particular material model (Hill

and Hutchinson 1975, Hutchinson and Tvergaard 1981, Valkonen et al. 1987, Novak

and Lauerova 1991). Numerically, it is easier to define the boundary condition and

easier to approach because it is a 2-D rather than a 3-D problem (Tvergaard et al.

1981). Experimentally, it is easier to perform and observe without losing the qualitative

features of this type of instability (Anand and Spitzig 1980). Physically, the








77

localization occurs much more easily for plane strain than for axisymmetry (Rudnicki

and Rice 1975, Rice 1977, Peric et al. 1992).

There are a large number of publications dealing with bifurcation under plane

strain conditions. They consider not only the shear band mode but also a mode of

diffused type (geometric type of instability), to seek the lowest bifurcation load. Within

the scope of the work of Hill and Hutchinson (1975), which basically dealt with a

incompressible incrementally linear material, Needleman (1979) followed the same

approach for a nonassociative incompressible material. His work was extended by Chau

and Rudnicki (1990) to a compressible material with nonnormality. Needleman (1989)

also investigated the dynamic development of a shear band initiated from an internal

inhomogeneity, and the work of Bardet (1991) generalized the work of Hill and

Hutchinson (1975) to a compressible material.

The significance of a Mohr-Coulomb model is that the intermediate principal

stress does not enter the yield condition. The material model of a Mohr-Coulomb yield

surface combined with a flat potential surface with unvarying normal direction is

actually exactly a plane strain situation. As described in section 5.1, there is no plastic

strain increment in the direction of intermediate principal stress. The elastic strain

increment in this direction also is made zero by choosing the stress path as oc =

v(aA+GB). The only nonzero strain rate components are the shear along the band

direction and extension perpendicular to the band direction. For a general plastic

material model (e.g, extended Mises), the stress state of a plane strain condition must

be a point in the principal stress space at which there is no plastic strain increment in







78

one of the principal stress directions. It is then locally (or in the small) similar to the

picture for a Mohr-Coulomb model.

A triaxial test is most often performed to determine the stress-strain relation of

geomaterials. A typical test procedure of this type usually is to load the specimen

hydrostatically to a stress state (on, oH, oH), then keep two of the stress components

fixed while increasing the other component to oy For the Mohr-Coulomb yield

condition, the stress point is just at a corer of the current yield surface. However, as

soon as the stress point deviates sightly from the triaxial test stress state, the stress point

is on one or the other flat yield surface (Fig. 5.7). Hereafter throughout the entire

dissertation, the term "triaxial stress state" means the "triaxial test stress state".

For the nonassociated Mohr-Coulomb material model employed in the present

chapter, the character of the genuine instability demonstrated remains the same under

the triaxial stress state as under an arbitrary stress state. Two distinctions between the

two conditions may be noted. Under a triaxial stress state, vertex effects help the

material to move from the triaxial stress state to a plane strain stress state. However,

the amount of this motion is infinitely small. Another distinction is subtle. If the

elastic response is taken into account, to maintain the major and minor principal stresses

unchanged while an unstable path is followed starting from a triaxial stress state, ac

must remain the intermediate principal stress. ac changes in accord with oa = v(oA

+ OB). Substitution of the expressions given in equation (5.3) yields the requirement

of tan (--a) < 1-2v. For the usual models of common geomaterials this condition

is automatically satisfied as soon as the materials enter the plastic range.








79

The triaxial stress condition is the most stable condition against localization into

a shear band for a material model with a smooth yield surface. It is necessary to point

out a basic error in a recent attempt to investigate localization into a shear band of a

circular cylinder under axisymmetric tension and compression (Chau 1992 section 4).

The assumption in this paper of a jump in the velocity gradient across the band is not

valid for a shear band, because the formulated jump condition was axisymmetric. It

corresponds to a conical deformation discontinuity rather than a shear band type. The

shear band type jump is always nonaxisymmetric and at or close to a plane strain

condition. Although such a cone-like nonuniformity may initiate, the boundary

condition prevents its development.







5.10 The Genuine Shear Band Instability and the
Classical Shear Band Instability



The instability demonstrated is termed "genuine", because it is a real instability,

an instability of configuration which occurs at each and every point in stress space as

soon as the material enters the plastic range, provided the elastic response along a

wedge path is too small to stabilize the plastic. The instability in the classical shear

band approach is an instability of path, or an instability in the constitutive description

(Rudnicki and Rice 1975). The bifurcation is actually stable or neutrally stable (if it

occurs on the rising portion of the stress-strain curve or at the peak).








80

The genuine instability occurs much earlier than the conventional shear band

bifurcation during the loading procedure. This is the first counter-example, to the

author's knowledge, to the commonly held idea that instability of path always

corresponds to a lower load than instability of configuration in the plastic range

(Shanley 1946, 1947).

For the genuine instability demonstrated, the material in the main body outside

the band stops deforming after the initiation of the band, neither plastically loading nor

elastically unloading. The boundary rotates as the instability develops. For the classical

shear band with a fixed band direction, however, the material outside the band may

continuously load plastically (continuous bifurcation) or may elastically unload

(discontinuous bifurcation) after the bifurcation point.

For the genuine instability, the transition from rigid plastic to elastic-plastic is

smooth and the effect of elastic response is mild. This is in sharp contrast to the drastic

effect of the elastic response in the classical shear band theory.

Unlike the classical shear band theory which insists on continuing equilibrium,

in the shear band with rotating boundary the traction is continuous but the traction rate

is discontinuous at the inception of the instability. Consequently, the initial acceleration

is zero but not its rate of change. In the initial stages, the net driving force increases.

In the later stages, however, this net driving force decreases and eventually becomes

zero (Fig.5.4). Therefore, the process is self-limiting and the total excursion is

moderate. The stress and strain of the material inside the band undergo a significant

but bounded "jump". The amount of the jump is highly dependent on the wedge angle








81

and goes to zero as the wedge angle goes to zero. This type of instability is an intrinsic

feature of a nonassociated flow rule when the elastic response does not produce stability

in the small in the forward sense. It is in sharp contrast to the initially stable

conventional shear band bifurcation in the hardening range which leads to a continually

accelerating instability, connoting complete failure, at higher loads.

When this jump motion of a nonassociated flow rule model comes to a halt, the

stress at each point in the band reverts to the initially imposed state of stress along an

elastic (unloading) path of increasing principal stress magnitudes lying inside (below)

the current yield surface. The stress state once again becomes homogeneous throughout

the entire model. However, the material of the model now is far from homogenous.

Even within the band, different portions of the material have experienced different

plastic strains and therefore different degrees of hardening. Instability of configuration

is much more drastic than instability of path. In either case, however, it is necessary

to examine the behavior of the system beyond its initial unstable response in order to

assess the importance of the instability (Drucker and Li 1992). A bifurcation study is

really not complete without some attention to stability in the dynamical sense

(Budiansky, 1974) beyond the bifurcation point. As pointed out in the previous section,

the cone-like nonuniformity (Chau 1992) may initiate but the instability is restricted

only to its bifurcation point. In other words, it is not an instability at all.

The usual postbuckling analysis expands the variables of interest in terms of

small parameters near the bifurcation point (e.g., Sewell 1965, 1972; Budiansky 1974;

Hutchinson 1974). The procedure can not go very far from the bifurcation point. The








82

analysis in this chapter, however, considers the dynamics of the entire process of

instability.





5.11 Concluding Remarks



This predicted instability of configuration may or may not represent the actual

behavior of a real material specimen. It is a purely mathematical consequence of the

selection of a simple nonassociated flow rule material model. Where reality lies

involves the entirely different and more important question of how to choose a model

that appropriately describes the real behavior of any particular material such as sand.

Nevertheless, the model selected does shed light on the general character of the

behavior of any model obeying a nonassociated flow rule. It also suggests the

possibility that if metals and alloys, as well as geomaterials, actually do obey a

nonassociated flow rule, plastic deformation would not develop smoothly in each region

when the applied load follows a continuous path. Plastic strain would instead progress

by small unstable "jumps". With a wedge angle of no more than a few degrees, if such

jumps occurred, they would not be easily observable in most experiments.

A genuine instability of configuration and related nonuniqueness is an intrinsic

phenomenon of the nonnormality, at least for a Mohr-Coulomb model, if the elastic

response is too small to stabilize the plastic. It leads to the likelihood that the

computational results obtained by using a code employing a nonassociated flow rule








83

model may lose the physical behavior of material, and instead, reflect the errors

accumulated in the computer. It is worthwhile to reassess the reliability of those codes

employing nonassociated flow rule in engineering practice.

The resulting lack of uniqueness is difficult if not impossible to handle in a

routine manner. If it represents the essence of the physical reality that is under study

it must, of course, be included correctly. If not, there is a clear advantage to using

associated flow models that are stable in the small in the forward sense. Choices

include unconventional formulations of the type proposed by Drucker and Seereeram

(1987) that are unstable in a small cycle, or a modem version of the fully stable

conventional work-hardening model with moving yield surfaces and "cap" proposed

much earlier by Drucker, Gibson, and Henkel (1957). The more complicated the

loading paths the more elaborate the conventional or unconventional model needed to

match the data in fine detail. A model that is stable in the small in the forward sense

does exhibit stability of configuration as well as of path for small perturbations. Such

stability of a model of material has much to commend it, quite apart from the adequacy

of its representation of physical reality.





















A e
or
pP


Sf (yield surface)


S(potential surface)
g (potential surface)


(GH,Ov)




or A C




/
/ GA


f2 fl


I /

(OGH.O(H)




GH GB


Figure 5.1 Mohr-Coulomb yield surface
and "wedge region" in two dimensional
principal stress'space





































C7-r n

B A








\/
/

















Figure 5.2 Rotation of boundary planes of
the band and the principal stress directions











































ON TNT


Figure 5.3 Equivalent of a free body
diagram of part of the band and the upper
portion of the triaxial test specimen


































N0,2r- 7c/2


Figure 5.4 The normal and shearing components of
traction on planes parallel to the instantaneous midplane
of the band. b is for the band; u, on the Mohr's circle,
represents the surface tractions o cGH applied to the
specimen.




















fully hardened region


unhardened region less hardened region


Figure 5.5 Patterns of parallel bands:
(left)some regions not hardened, (right) all
regions hardened to some extent


















































Figure 5.6 The stress and deformation
fields of a set of intersecting bands


















initial yield
surface








subsequent
yield surface







aA





















Figure 5.7 The initial and current yield
surfaces of a Mohr-Coulomb model. A
triaxial test stress state is just at a corer of
the current yield surface














CHAPTER 6
AN UNSTABLE BIFURCATION OF AN EXTENDED MISES MODEL
WITH NONNORMALITY



An extended Mises yield condition (Drucker and Prager 1952) is a typical model

with a smooth yield surface. In principle, the results obtained from this material model

can be extended to any material model with a smooth yield surface, for example, the

Lade-Duncan model (Lade and Duncan 1973). It was used by Drucker (1973) and

modified by many following him (Spitzig et al. 1975, 1976; Richmond and Spitzig

1980; Casey and Jahedmotlagh 1984; Casey and Sullivan 1985) to interpret the SD

effects in high strength steels. It has intensively been employed to investigate

instability and localization (e.g., Rudnicki and Rice 1975).

The most often reported plastic localization in metals in the hardening regime

is in the form of a planar shear band. It may be the only possible pattern compatible

with the boundary condition of a specimen. The bands must be moderately thin to

satisfy the surface traction condition at the ends of the band closely enough. (Another

possibility will be discussed in chapter 7.)

It is assumed in this chapter that the pattern of plastic localization in the

hardening regime is in the form of a shear band. A basic characteristic of a shear band

is that there exists a "relative" or "absolute" non-deforming plane in the deformation




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