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 thoughtprovoking
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 N0001487J1193 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 NonSchmid 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 MOHRCOULOMB MATERIAL MODEL .... 54
5.1 A MohrCoulomb 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 NonDeforming 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 NONASSOCIATED FLOW RULE
By
Ming Li
May 1993
Chairman: Daniel C. Drucker
Department: Aerospace Engineering, Mechanics & Engineering Science
Instability resulting from a nonassociated flow rule is explored. For a Mohr
Coulomb material model with nonnormality, or in the plane strain case for any material
model with nonnormality, 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 counterexample to the commonly
held idea that instability of path always occurs earlier than instability of configuration
in the plastic range. For an elasticplastic extended Mises model, or any elastic plastic
material model with a smooth yield surface, combined with nonnormality, 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 nonassociated flow rule. Techniques customary for an associated
flow rule give results for a nonassociated 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 pileups 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 nonassociated 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
elasticplastic 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 nonSchmid effect in crystals, possible pressuresensitivity 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 "Fspace". 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 elasticplastic 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 MohrCoulomb model with
4
nonnormality is investigated. A genuine instability with limited total excursion is
exhibited. This is the first counterexample 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 postbifurcation 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 nonSchmid 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 timeindependent plastic stressstrain
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 stressstrain 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 stressstrain 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 workhardening
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 MohrCoulomb 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
KuhlmannWilsdorf (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 workhardening,
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 NonSchmid 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 internalvariables, 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 crossslip of a screw dislocation and climb of an edge dislocation,
stress components other than the Schmid stress affect the process. The crossslip
mechanism given by Asaro and Rice (1977) is a very good example. One of the main
consequences of nonSchmid 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 H80 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 nonSchmid 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 microdefects (e.g. voids) exist in a volume of material, it will
exhibit overall pressuresensitivity, even though the material itself can be approximated
as pressure insensitive and obeying normality. Figure 2.2 illustrates this situation in an
oversimplified 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 105 to 102
s1 and temperature well below half the melting point) most engineering materials can
be modeled very well as timeindependent (rateindependent).
2.2.6 Summary
The conclusion is that in the framework of timeindependent 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 timeindependent 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 stressstrain (loaddisplacement) 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) quasistatically 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
stressstrain (loaddisplacement) 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 timeindependent or rateindependent 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 rodspring 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 wellposed 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 ia.) < 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)
( ija ) < 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(oyjo .)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 (aJaij)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 elasticplastic 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 elasticplastic 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 elasticplastic 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 stressstrain 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 rateofdeformation
tensor D
1
SDij = DijD = (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 elasticplastic 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 stressstrain 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 PiolaKirchhoff 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 (corotational) 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 workhardening 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 workhardening 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 workhardening.
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 nonSchmid 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 5x104 (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 highstrength martensitic steels
(Spitzig et al. 1975, 1976). The conclusion is therefore obvious, the hydrostatic
pressuresensitivity 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 nondeforming 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 "Fspace" (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 elasticplastic 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 rateofdeformation 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 nondeforming 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 elasticplastic 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 nonlinearelasticlike
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 bistretching 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 workhardening for elasticplastic materials with associated flow rules. For
the elasticplastic 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 MOHRCOULOMB MATERIAL MODEL
The first material model to be examined is the MohrCoulomb 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 MohrCoulomb 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 MohrCoulomb Model with a Nonassociated Flow Rule
The isotropic cohesionless MohrCoulomb 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 stressstrain 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 MohrCoulomb 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 massacceleration 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
nondeforming 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 (AB) cos20
2 2
1 1 (5.5)
eN = (8,+e) + B A BCOS280
2 2
?yrN = (AeB)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 vaH)cos20
2 2 (5.9)
S= j(oavH)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(4a) + cos260 sin(4a) ]
r2 (5.10)
IAol
AT, IA sin20o sin(Oa)
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) 1tan21 (5.12)
1tan(4a)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(22pL)6T1
= mByi, + L(2mpLB)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( NTtmix 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 LBr dy (5.21)
to 0
where 8yNT is the shear strain increment during the overshoot. Equating the two
energies
AYNT 8YNT
f LB(tuT dy = f LBr dy (5.22)
0 0
or
LB(tutr ) 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 stressstrain relations by definition, whether associated or
nonassociated, are timeindependent. 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)(12v)cos(4a) + sinpsina
cos20 =
h(1+v)sin((a) cospsina (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)[(12v)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 MohrCoulomb 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(8162) = 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 VO)[lcos(6162)] (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 = 1.9 103 (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 MohrCoulomb 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 midwidth 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
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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 postbifurcation 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 2D rather than a 3D 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
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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 MohrCoulomb model is that the intermediate principal
stress does not enter the yield condition. The material model of a MohrCoulomb 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 MohrCoulomb model.
A triaxial test is most often performed to determine the stressstrain 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 MohrCoulomb 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 MohrCoulomb 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) < 12v. For the usual models of common geomaterials this condition
is automatically satisfied as soon as the materials enter the plastic range.
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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 conelike 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 stressstrain 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 counterexample, 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 elasticplastic 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 selflimiting 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
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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 conelike 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
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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 MohrCoulomb 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 workhardening 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 MohrCoulomb yield surface
and "wedge region" in two dimensional
principal stress'space
C7r 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 MohrCoulomb 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
LadeDuncan 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" nondeforming plane in the deformation
