A Principles of Hyperplasticity Part 1 pdf

Principles of Hyperplasticity

G.

T. Houlsby and A.

M. Puzrin

Principles of
Hyperplasticity
An Approach to Plasticity Theory
Based on Thermodynamic Principles

123
G.

T. Houlsby, MA, DSc, FREng, FICE
Department of Engineering Science
Parks Road
Oxford
OX1 3PJ
UK
A.

M. Puzrin, DSc
ETH Zurich
Institute of Geotechnical Engineering
CH 8093
Zurich
Switzerland
British Library Cataloguing in Publication Data

Houlsby, G.T.
Principles of hyperplasticity: an approach to plasticity
theory based on thermodynamic principles
1. Plasticity 2. Thermodynamics
I. Title II. Puzrin, A.M.
531.3'85
ISBN-13: 9781846282393
ISBN-10: 184628239X

Library of Congress Control Number: 2006936877

ISBN 978-1-84628-239-3 e-ISBN 1-84628-240-3 Printed on acid-free paper

© Springer-Verlag London Limited 2006

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Preface
This book is about the interplay between plasticity theory and thermodynamics.
Both of these theories deal with materials in which dissipation occurs, and yet
there are surprisingly few points of contact between the two classical theories.
The purpose of this book is to bridge this gap by formulating plasticity theory
entirely within the context of thermodynamics.
The book is aimed at researchers in the field of constitutive modelling, and
those who have to implement some of the sophisticated theoretical models in
use in modern practice. Whilst this book is not restricted to any particular range
of applications in engineering, much of the motivation for the book comes from
the special problems posed by geotechnical materials, so it should be of particu-
lar relevance to those working in geomechanics.
Structure of this book
After some introductory material in Chapter 1, a presentation of classical plas-

ticity theory is given in Chapter 2. This chapter does not contain new material,
but sets out the background and the terminology for the later chapters. Simi-
larly, in Chapter 3, we present basic thermodynamic concepts, taking this as far
as the thermodynamics of continua.
In Chapter 4, we set out the hyperplastic formulation, and this chapter forms
the core of the book. Much emphasis is placed on the fact that this approach al-
lows us to define plasticity models by specifying just two scalar functions. In
Chapter 5, we describe some simple applications, and examine different forms of
energy and dissipation functions to enable the reader to become familiar with the
mathematical forms that these functions take for different cases.
In Chapter 6, we discuss some of the more advanced approaches to plasticity
theories, as a preparation for Chapters 7 and 8. In Chapter 7, we extend the ap-
proach to the use of multiple internal variables. We find that multiple independ-
ent dissipation mechanisms are related to multiple yield surfaces. Then we use
this approach to develop significantly more complex models employing multiple
vi Preface
yield surfaces (comparable to the nested yield surface models frequently em-
ployed in geotechnical engineering).
In Chapter 8, we further extend the concept to an infinite number of internal
variables. When the finite number of internal variables is replaced by a continu-
ous field of variables, the resulting models allow smooth transitions between
elastic and plastic behaviour. This is an important development conceptually,
and introduces the need for functionals (as opposed to functions). However, the
use of some mathematical techniques which may be unfamiliar to some readers
is amply repaid by the benefits that follow. We term the use of a continuous field
of internal variables “continuous hyperplasticity.”
Chapters 9 and 10 are devoted to examples from geomechanics, addressing
issues such as effective stress modelling, the treatment of the small-strain
nonlinearity of soils, critical state soil mechanics, friction, dilation, and non-
associated flow.

Most of the book is concerned with rate-independent materials, but in Chap-
ter 11 we briefly examine ways these ideas are extended to materials with some
rate dependence. Ziegler (1983) devotes much attention to rate-dependent mate-
rials, but Chapter 11 is focused on elastic-viscoplastic modelling of materials,
concentrating on materials that are dominated by plastic behaviour, yet include
some rate effects.
The strength of the hyperplastic method is that the entire constitutive re-
sponse of a material is expressed through two scalar functions (or functionals).
In Chapter 12, we explore how this method can be extended to include other
features of material behaviour such as thermal effects and conduction phenom-
ena. The chapter focuses particularly on the behaviour of porous media.
In Chapter 13, we re-express hyperplasticity theory in the terminology of con-
vex analysis, which allows to be expressed many of the concepts encountered
earlier more rigorously. The only reason we do not employ this approach earlier
is to make the book as accessible as possible to a wide readership, by avoiding
where possible unfamiliar mathematical methods. Appendix D provides an in-
troduction to this subject. In particular, we find that convex analysis proves
convenient for expressing models that include constraints, and it also provides a
more rigorous link between dissipation and yield.
Chapter 14 contains a number of disconnected applications of the approach to
mechanics that we have described elsewhere in this book. The breadth of applica-
tions illustrates that the hyperplasticity approach is a powerful unifying tech-
nique for studying many problems of constitutive modelling in mechanics.
In Chapter 15, we summarise the hyperplasticity approach in a very general
form and point the way to future developments.
In developing the ideas that are presented in this book, we have found that our
understanding of the subject has at each stage been enabled by the identification
of appropriate mathematical techniques to describe the phenomena of interest.
Therefore, we include appendices on some of the mathematical techniques that
we employ in this book; in particular, we describe the Frechet derivatives used in

Preface vii
the analysis of functionals, we discuss the Legendre transform, and we provide an
introduction to convex analysis.
There are occasions in the book where we find it convenient, for didactic pur-
poses, to repeat material. We prefer to do this rather than require the reader to
have to refer too much to material found in different chapters.
In a book such as this, it is inevitable that a great deal of specialised terminol-
ogy must be used. In some disciplines (notably medicine), it is customary to give
technical terms long, obscure names of classical derivation. Engineers have an
even more disconcerting habit of using commonplace words, which have every-
day meanings, to express different and precise technical terms. Examples in this
book are “stress”, “strain”, “elastic”, “plastic”, “yield”, “normal”, “function” and
many other words. Where a specialised meaning is implied, we shall generally
draw attention to this by showing a phrase in italics when it is first used and
defined.
Acknowledgments
We wish to express our gratitude to many people who have influenced this book.
It was the late Professor Peter Wroth who guided the first author toward the
rigorous study of the interplay between plasticity theory and thermodynamics,
and who was first his supervisor and later a much respected colleague. The roots
of the ideas expressed here lie in the work of the late Professor Hans Ziegler,
who was a considerate correspondent with the first author. Many of the ideas lay
dormant for several years until a very successful collaboration with Professor
Ian Collins, who brought to bear a number of mathematical techniques (notably
the Legendre Transform) which are central to the developments described here.
Professor Michael Sewell provided, in a brief conversation, the key to the
treatment of part of the formulation: that of treating
F
and F as separate vari-
ables. Professor Martin Brokate is thanked for very fruitful discussions at the

Horton conference in 2002.


Guy Houlsby (Oxford University)
Alexander Puzrin (ETH Zurich)

Contents
1 Introduction 1
1.1 Plasticity and Thermodynamics 1
1.1.1 Purpose of this Book 1
1.1.2 Advantages of Our Approach 2
1.1.3 Generality 2
1.1.4 Ziegler’s Orthogonality Condition 3
1.1.5 Constitutive Models 4
1.2 Context of this Book 4
1.3 Notation 5
1.4 Some Basic Continuum Mechanics 6
1.4.1 Small Deformations and Small Strains 6
1.4.2 Sign Convention 8
1.5 Equations of Continuum Mechanics 8
1.5.1 Equilibrium 9
1.5.2 Compatibility 9
1.5.3 Initial and Boundary Conditions 9
1.5.4 Work Conjugacy 10
1.5.5 Numbers of Variables and Equations 11
2 Classical Elasticity and Plasticity 13
2.1 Elasticity 13
2.2 Basic Concepts of Plasticity Theory 16
2.3 Incremental Stiffness in Plasticity Models 19
2.3.1 Perfect Plasticity 20

2.3.2 Hardening Plasticity 22
2.3.3 Isotropic Hardening 25
2.3.4 Kinematic Hardening 27
2.3.5 Discussion of Hardening Laws 28
2.4 Frictional Plasticity 28
x Contents
2.5 Restrictions on Plasticity Theories 30
2.5.1 Drucker's Stability Postulate 31
2.5.2 Il'iushin's Postulate of Plasticity 32
3 Thermodynamics 35
3.1 Classical Thermodynamics 35
3.1.1 Introduction 35
3.1.2 The First Law 36
3.1.3 The Second Law 38
3.2 Thermodynamics of Fluids 40
3.2.1 Energy Functions 42
3.2.2 An Example of an Internal Energy Function 43
3.2.3 Perfect Gases 44
3.3 Thermomechanics of Continua 47
3.3.1 Terminology 47
3.3.2 Thermoelasticity 48
3.3.3 Internal Variables and Dissipation 49
4 The Hyperplastic Formalism 53
4.1 Introduction 53
4.2 Internal Variables and Generalised Stress 53
4.3 Dissipation and Dissipative Generalised Stress 54
4.3.1 The Laws of Thermodynamics 54
4.3.2 Dissipation Function 55
4.3.3 Dissipative Generalised Stress 56
4.4 Yield Surface 56

4.4.1 Definition 56
4.4.2 The Flow Rule 57
4.4.3 Convexity 58
4.4.4 Uniqueness of the Yield Function 58
4.5 Transformations from Internal Variable to Generalised Stress 59
4.6 A Complete Formulation 59
4.7 Incremental Response 62
4.8 Isothermal and Adiabatic Conditions 66
4.9 Plastic Strains 67
4.10 Yield Surface in Stress Space 68
4.11 Conversions Between Potentials 69
4.11.1 Entropy and Temperature 69
4.11.2 Stress and Strain 70
4.11.3 Internal Variable and Generalised Stress 70
4.11.4 Dissipation Function to Yield Function 70
4.11.5 Yield Function to Dissipation Function 71
Contents xi
4.12 Constraints 71
4.12.1 Constraints on Strains 72
4.12.2 Constraints on Plastic Strain Rates 73
4.13 Advantages of Hyperplasticity 74
4.14 Summary 74
5 Elastic and Plastic Models in Hyperplasticity 77
5.1 Elasticity and Thermoelasticity 77
5.1.1 One-dimensional Elasticity 77
5.1.2 Isotropic Elasticity 78
5.1.3 Incompressible Elasticity 78
5.1.4 Isotropic Thermoelasticity 79
5.1.5 Hierarchy of Isotropic Elastic Models 80
5.2 Perfect Elastoplasticity 81

5.2.1 One-dimensional Elastoplasticity 81
5.2.2 Von Mises Elastoplasticity 83
5.2.3 Rigid-plastic Models 84
5.3 Frictional Plasticity and Non-associated Flow 84
5.3.1 A Two-dimensional Model 85
5.3.2 Dilation 86
5.3.3 The Drucker-Prager Model with Non-associated Flow 87
5.4 Strain Hardening 88
5.4.1 Theory of Strain-hardening Hyperplasticity 88
5.4.2 Isotropic Hardening 91
5.4.3 Kinematic Hardening 96
5.4.4 Mixed Hardening 101
5.5 Hierarchy of Plastic Models 102
6 Advanced Plasticity Theories 105
6.1 Developments of Classical Plasticity Theory 105
6.2 Bounding Surface Plasticity 105
6.3 Nested Surface Plasticity 107
6.4 Multiple Surface Plasticity 110
6.5 Remarks on the Intersection of Yield Surfaces 112
6.5.1 The Non-intersection Condition 112
6.5.2 Example of Intersecting Surfaces 112
6.5.3 What Occurs when the Surfaces Intersect? 115
6.6 Alternative Approaches to Material Non-linearity 117
6.7 Comparison of Advanced Plasticity Models 118
7 Multisurface Hyperplasticity 119
7.1 Motivation 119
7.2 Multiple Internal Variables 120
xii Contents
7.3 Kinematic Hardening with Multiple Yield Surfaces 121
7.3.1 Potential Functions 121

7.3.2 Link to Conventional Plasticity 121
7.3.3 Incremental Response 123
7.4 One-dimensional Example (the Iwan Model) 125
7.5 Multidimensional Example (von Mises Yield Surfaces) 128
7.6 Summary 131
8 Continuous Hyperplasticity 133
8.1 Generalised Thermodynamics and Rational Mechanics 133
8.2 Internal Functions 134
8.3 Energy and Dissipation Functionals 134
8.3.1 Energy Functional 134
8.3.2 Generalised Stress Function 135
8.3.3 Dissipation Functional 136
8.3.4 Dissipative Generalised Stress Function 136
8.4 Legendre Transformations of the Functionals 137
8.4.1 Legendre Transformations of the Energy Functional 137
8.4.2 Legendre Transformation of the Dissipation
Functional 138

8.5 Incremental Response 138
8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 142
8.6.1 Potential Functionals 142
8.6.2 Link to Conventional Plasticity 143
8.6.3 Incremental Response 145
8.7 Example: One-dimensional Continuous Hyperplastic Model 146
8.8 Calibration of Continuous Kinematic Hardening Models 148
8.9 Example: Calibration of the Weighting Function 148
8.9.1 Formulation of the One-dimensional Model 148
8.9.2 Analogy with the Extended Iwan’s Model 149
8.9.3 Model Calibration Using the Initial Loading Curve 150
8.9.4 Unloading Behaviour 151

8.10 Example: Calibration of the Plastic Modulus Function 151
8.10.1 Formulation of the Multidimensional
von Mises Model 151

8.10.2 Model Calibration Using the Initial Loading Curve 154
8.10.3 Analogy with an Advanced Plasticity Model 155
8.11 Hierarchy of Multisurface and Continuous Models 155
9 Applications in Geomechanics: Elasticity and Small Strains 159
9.1 Special Features of Mechanical Behaviour of Soils 159
9.2 Sign Convention and Triaxial Variables 159
9.3 Effective Stresses 160
Contents xiii
9.4 Dependence of Stiffness on Pressure 162
9.4.1 Linear and Non-linear Isotropic Hyperelasticity 163
9.4.2 Proposed Hyperelastic Potential 167
9.4.3 Elastic-plastic Coupling in Clays 172
9.4.4 Effects of Elasticity on Plastic Behaviour 175
9.5 Small Strain Plasticity, Non-linearity, and Anisotropy 176
9.5.1 Continuous Hyperplastic Form of a Small
Strain Model 177

9.5.2 Derivation of the Model from Potential Functions 178
9.5.3 Behaviour of the Model During Initial Proportional
Loading 180

9.5.4 Behaviour of the Model During Proportional Cyclic
Loading 184

9.5.5 Concluding Remarks 186
10 Applications in Geomechanics: Plasticity and Friction 187

10.1 Critical State Models 187
10.1.1 Hyperplastic Formulation of Modified Cam-Clay 187
10.1.2 Non-uniqueness of the Energy Functions 190
10.2 Towards Unified Soil Models 191
10.2.1 Small Strain Non-linearity: Hyperbolic
Stress-strain Law 191

10.2.2 Modified Forms of the Energy Functionals 193
10.2.3 Combining Small-strain and Critical State Behaviour 195
10.2.4 Examples 198
10.2.5 Continuous Hyperplastic Modified Cam-Clay 203
10.3 Frictional Behaviour and Non-associated Flow 204
10.3.1 The Dissipation to Yield Surface Transformation 205
10.3.2 The Yield Surface to Dissipation Transformation 207
10.3.3 Tensorial Form 209
10.4 Further Applications of Hyperplasticity in Geomechanics 209
11 Rate Effects 211
11.1 Theoretical Background 211
11.1.1 Preliminaries 211
11.1.2 The Force Potential and the Flow Potential 213
11.1.3 Incremental Response 215
11.2 Examples 216
11.2.1 One-dimensional Model with Additive Viscous Term 216
11.2.2 A Non-linear Viscosity Model 219
11.2.3 Rate Process Theory 221
11.2.4 A Continuum Model 223
xiv Contents
11.3 Models with Multiple Internal Variables 224
11.3.1 Multiple Internal Variables 225
11.3.2 Incremental Response 225

11.3.3 Example 226
11.4 Continuous Models with Internal Functions 228
11.4.1 Energy Potential Functional 228
11.4.2 Force Potential Functional 229
11.4.3 Legendre Transformation of the Force
Potential Functional 230

11.4.4 Incremental Response 230
11.4.5 Example 231
11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay 233
11.5.1 Formulation 233
11.5.2 Incremental Response 234
11.5.3 Comparison with Experimental Results 235
11.5.4 Extension of the Model to Three Dimensions 238
11.6 Advantages of the Rate-dependent Formulation 239
12 Behaviour of Porous Continua 241
12.1 Introduction 241
12.2 Thermomechanical Framework 242
12.2.1 Density Definitions, Velocities, and Balance Laws 243
12.2.2 Tractions, Stresses, Work, and Energy 245
12.2.3 The First Law 246
12.2.4 Equations of Motion 248
12.2.5 The Second Law 248
12.2.6 Combining the First and Second Laws 249
12.2.7 The Internal Energy Function 251
12.2.8 The Dissipation Function and Force Potential 251
12.2.9 Constitutive Equations 252
12.2.10 Discussion 254
12.3 The Complete Formulation 255
12.3.1 Modifications to Account for Tortuosity 255

12.4 Legendre-Fenchel Transforms 256
12.5 Small Strain Formulation 257
12.6 Example 258
12.7 Conclusions 261
13 Convex Analysis and Hyperplasticity 263
13.1 Introduction 263
13.2 Hyperplasticity Re-expressed in Convex Analytical Terms 264
13.3 Examples from Elasticity 265
13.4 The Yield Surface Revisited 268
13.5 Examples from Plasticity 270
Contents xv
14 Further Topics in Hyperplasticity 273
14.1 Introduction 273
14.2 Damage Mechanics 274
14.3 Elementary Structural Analysis 277
14.3.1 Pin-jointed Structures 277
14.3.2 More General Structures 279
14.3.3 Assemblies of Rigid Elements 281
14.4 Bending of Prismatic Beams 284
14.5 Large Deformation Rubber Elasticity 286
14.6 Fibre-reinforced Material 288
14.7 Analysis of Axial and Lateral Pile Capacity 290
14.7.1 Rigid Pile under Vertical Loading 290
14.7.2 Flexible Pile under Vertical Loading 294
14.7.3 Rigid Pile under Lateral Loading 297
14.7.4 Flexible Pile under Lateral Loading 298
15 Concluding Remarks 301
15.1 Summary of the Complete Formalism 301
15.2 Legendre-Fenchel Transforms 303
15.3 Some Future Directions 303

15.4 Concluding Remarks 304
Appendix A Functions, Functionals and their Derivatives 305
A.1 Functions and Functionals 305
A.2 Some Special Functions 306
A.3 Derivatives and Differentials 307
A.4 Selected Results 309
A.4.1 Frechet Derivatives of Integrals 309
A.4.2 Frechet Derivatives of Integrals Containing
Differential Terms 310

Appendix B Tensors 311
B.1 Tensor Definitions and Identities 311
B.2 Mixed Invariants 313
B.2.1 Differentials of Invariants of Tensors 313
Appendix C Legendre Transformations 315
C.1 Introduction 315
C.2 Geometrical Representation in (n

+

1)-dimensional Space 315
C.3 Geometrical Representation in n-dimensional Space 317
C.4 Homogeneous Functions 318
C.5 Partial Legendre Transformations 319
C.6 The Singular Transformation 320
xvi Contents
C.7 Legendre Transformations of Functionals 321
C.7.1 Integral Functional of a Single Function 321
C.7.2 Integral Functional of Multiple Functions 322
C.7.3 The Singular Transformation 323

Appendix D Convex Analysis 325
D.1 Introduction 325
D.2 Some Terminology of Sets 325
D.3 Convex Sets and Functions 327
D.4 Subdifferentials and Subgradients 328
D.5 Functions Defined for Convex Sets 329
D.6 Legendre-Fenchel Transformation 331
D.7 The Support Function 332
D.8 Further Results in Convex Analysis 334
D.9 Summary of Results for Plasticity Theory 334
D.10 Some Special Functions 336
References 339
Index 345






Notation
Variables
e
a semidiameter of the linear elastic region
L
a semidiameter of the small strain region
i
a acceleration
ij
a
stress or strain (see Table 4.4)

A pile area
ij
b
stress or strain (see Table 4.4)
c (1) constraint function
(2) specific heat
(3) concentration of fibres
(4) undrained shear strength
p
c
the specific heat at constant pressure
v
c the specific heat at constant volume
1
c ,
2
c ,
3
c compliances
ijkl
c
compliance matrix
d dissipation function or dissipation functional
e
d dissipation function corresponding to the energy func-
tion e
t
d total dissipation
ijkl
d stiffness matrix

e (1) specific energy function: any of u, f, h or g
(2) voids ratio
E elastic modulus
xviii Notation
f
(1) specific Helmholtz free energy
(2) yield surface
g
(1) specific Gibbs free energy
(2) plastic potential
s
g
shear modulus constant
G elastic shear modulus
0
G small strain shear modulus
h enthalpy

K
ˆ
h plastic modulus function
H plastic modulus
H the Heaviside step function
I
moment of inertia
C
I indicator function
123
,,III invariants of a second order tensor
J

coupling modulus
23
,
J
J invariants of the deviator of a second order tensor
k
shear strength

Kk
function defining the size of a yield surface in the field
i
k
i-th bar axial stiffness
m
k
permeability coefficient
K
k
thermal conductivity coefficient
K
elastic bulk modulus
i
l
i-th bar length
L
pile length
ijkl
L
linear operator in hypoplasticity
m

parameter of elliptical surface
i
m
mass flux
M
(1) slope of critical state line in
c
,
p
q plot
(2) moment
n
parameter of elliptical surface
i
n
outward normal to the boundary
N the number of variables
c
N vertical bearing capacity factor
C
N
normal cone
Notation xix
h
N
horizontal bearing capacity factor
ij
N
linear operator in hypoplasticity
p

pressure, mean (effective) compressive stress
a
p
atmospheric pressure
r
p
reference pressure
x
p
preconsolidation pressure in terms of mean stress
L
p
transform of pore pressure to Lagrangian coordinate
system
i
P
nodal external forces
ij
P
deformation gradient
q
deviator stress invariant
i
q
heat flux

Q
heat supply
r
non-linear viscous model parameter

R
(1) the gas constant
(2) parameter of the logarithmic function
(3) radius of the curvature (of a prismatic beam)
(4) pile radius
s
specific entropy
S
(1) the modified signum function
(2) entropy
(3) surface
ij
S
the modified signum tensor function
t
time
i
t
traction
i
T
i-th bar tension
u
(1) specific internal energy
(2) horizontal pile displacement
i
u
displacement
w
u

pore water pressure
U
internal energy
v
specific volume
i
v
velocity
V
volume
xx Notation
w
(1) force potential
(2) water content
(3) vertical pile displacement

W
rate of work input
p
W
plastic work
x
temperature or entropy (see Table 4.4)
i
x
spatial coordinate
L
x
normalized limiting strain
r

x
normalized strain at the stress reversal
y
yield function
y
canonical yield surface
e
y
yield surface corresponding to the energy function e

DFK
ˆˆ
ˆ
,, 0
g
ij ij
y
field of yield functions
z
(1) temperature or entropy (see Table 4.4)
(2) flow potential

D (1) coefficient of thermal expansion
(2) parameter of the logarithmic function
(3) damage parameter
D
v
volumetric kinematic internal variable
D
s

deviatoric (shear) kinematic internal variable
f
D
damage model material constant
D
ij
internal kinematic variable

DK
ˆ
ij
internal variable function
E
dilational constant
c
E
ij
tensor defining direction of loading in deviatoric stress
space
F
r
damage model material constant
F
ij
,
F
ij
generalised stress

FK

ˆ
ij
dissipative generalised stress function

FK
ˆ
ij
generalised stress function
J
shear strain
J
C
gauge function
Notation xxi

*K
distribution function
G
(1) distance between the stress and image points
(2) the Dirac impulse function
G
ij
Kronecker delta (unit tensor):
G 1
ij
if
ij
,
G 0
ij

otherwise
'
ij
Green-Lagrange strain
H
s
deviatoric (shear) strain
H
v
volumetric strain
H
ij
strain tensor

H
e
ij
elastic strain tensor

H
p
ij
plastic strain tensor
M
c
effective angle of internal friction
K
(1) internal coordinate (typically a dimensionless meas-
ure of the size of the yield surfaces)

(2) stress ratio
qp

K
i
entropy flux
N
slope of swelling line in consolidation plot
O
slope of compression line in consolidation plot
O
,
/
scalar multipliers
O
i
principal stretches

O
n
an n-th non-negative multiplier

OK
ˆ
a non-negative multiplier function
P
(1) frictional constant
(2) viscosity
Q
Poisson’s ratio

S
ij
Piola-Kirchhoff stress tensor
8 the domain of
K
, usually
>
@
0;1

T
temperature
U
density
U
ij
back stress

UK
ˆ
ij
back stress function
V normal stress
xxii Notation
V
C
support function
V
y
yield stress

V
ij
stress
V
ij
effective stress
W
shear stress
Z
ij
rigid body rotations
[
hardening parameter
\ angle of dilation
Subscripts, superscripts and diacritics
e
elastic
p
plastic
w
pore fluid (water)
s
soil skeleton

n n-th in a finite series of similar variables
o
a
initial value of
a


a
rate of
a
with time,
{w w

aat


a
material derivative,
{
,

ii
aaav

ˆ
a
any function of the internal variable
K
,

{K
ˆˆ
aa

c
ij

a
deviatoric part of second order tensor
ij
a
,
c
{G
ij ij ij kk
aa a

,i
a
derivative with respect to spatial coordinate
i
x
,
{w w
,ii
a
y
x

w
x
a
subdifferential of
a

Discontinuous functions
In examples of one-dimensional models, we make much use of the absolute

value of a variable:
xx if t 0x , xx if  0x . On differentiation, we ob-
tain

ww Sxx x, where

S x is a generalized signum function such that

S1x if ! 0x ,

S1x if  0x , and

S x takes an undefined value such
Notation xxiii
that

d d1S 1x if 0x . The function

S x is closely related to the conven-
tional signum function, also often written

sgn x or

sg x . The difference is
that the signum function is usually defined such that

s
g
n0x for 0x . We
need to introduce the definition in which


sgn x can take any of a range of
values at
0x so that it properly defines the value of the subdifferential of x
(see Appendix D). Strictly in convex analysis notation, we define the subdiffer-
ential

wSxx, and

S x is a set-valued function.
In analyses of continuum models, the equivalent of
x is
ij ij
xx . On differ-
entiation, we obtain


kl kl
ij
ij
kl kl
dxx
x
dx
xx
. This bears a close analogy to

S x .
It is undefined if
0

ij
x
, but otherwise gives (in a generalised sense) the direc-
tion of
ij
x
. For convenience [and by analogy with

S x ], we define the function

S
ij
ij ij
kl kl
x
x
xx
. Note also that

SS 1
ij ij ij ij
xx for z 0
ij
x , and we require
that

dd0S S 1
ij ij ij ij
xx for
0

ij
x
.
We use the notation
for Macaulay brackets, such that xx if t 0x
and
0x
if  0x . Macaulay brackets can also be defined in terms of the abso-
lute value (or vice versa)


2
xx
x
.
Chapter 1
Introduction
1.1 Plasticity and Thermodynamics
1.1.1 Purpose of this Book
This book is concerned with the structure of theories used for the constitutive
modelling of rate-independent materials, and their application to solving engi-
neering problems. Countless theories have been proposed for such materials,
based on concepts such as linear and non-linear elasticity, and the many vari-
ants of plasticity theory. Whilst some theoretical models are based on clearly
articulated fundamental principles, others are little better than a collection of
(sometimes inconsistent) arbitrary equations. We take as our starting point an
assumption that the former approach is required.
The purpose here is not to put forward particular theories for specific mate-
rials, although examples of particular theories will be given, nor is it to attempt
an all-embracing theory with extravagant claims of generality. Instead, a frame-

work will be described within which a rather broad class of theoretical models
may be defined. Individual models are specified by the choice (subject to cer-
tain constraints) of certain functions. The constitutive behaviour, i.

e. the entire
incremental stress-strain response, then follows automatically from the proce-
dures described here. The advantages of this approach are outlined below.
The approach used here is based on what is often termed generalised thermo-
dynamics. It places strong emphasis on the use of internal variables to describe
the past history of the material (and alternatively is called thermodynamics with
internal variables). The First and Second Laws of Thermodynamics are enforced
directly in this formulation, which is described in detail in Chapter 4, so that any
model defined within this framework will automatically obey these Laws. It is only
necessary to draw attention to this fact because many constitutive models that
have been published define behaviour that may violate one or the other of these
Laws. Models that violate thermodynamics cannot be used with any confidence to
2 1 Introduction
describe material behaviour (except perhaps under some rather particular and
well-defined conditions).
1.1.2 Advantages of Our Approach
The motivation here is to describe a rigorous and consistent framework, within
which models can be developed, to describe a wide range of engineering materi-
als. In particular, the framework can accommodate models for geotechnical
materials (e.

g. soils, powders, concrete, rocks) that exhibit what is called non-
associated plastic flow (see Sections 2.2 and following). The principal advantage
of embedding such models in a thermodynamic framework is the confidence it
gives that they cannot produce thermodynamically unreasonable results.
A second advantage is that the framework makes considerable use of potential

functions. The use of potentials is closely related to variational and extremum
principles. Although such concepts are not pursued in this book, this offers the
possibility of deriving, in the future, theorems about constitutive models. The
Lower and Upper Bound Theorems of plasticity theory were, for instance, de-
rived for plasticity models that conform to certain strict criteria, and the coinci-
dence of the yield surface and plastic potential plays an important part in their
development. Within the framework described here, it may be possible to prove
more general theorems, or at the very least establish the range of applicability of
existing theorems.
Thirdly, the approach used here may allow a number of competing models
to be cast within a single framework, and so allow them to be more readily
compared.
The implementation of sophisticated models of material behaviour makes in-
creasing use of computation. The commonest, and most general, method used is
finite element analysis, and this requires (usually) the constitutive model to be
cast as an incremental stress-strain relationship. In this book, much emphasis is
therefore placed on the necessity that a model should be expressed in this way,
even when the original functions describing the material may be in quite a differ-
ent form.
1.1.3 Generality
No claim is made here that the framework described is entirely general. For ex-
ample, there are some constitutive models, which could not be described using
the approach we adopt here, that are developed within the approach commonly
termed rational thermodynamics. In that approach the behaviour is expressed as
general functionals of the history of deformation, rather than (as used here)
functions of internal parameters that somehow encapsulate this history. In prin-
ciple, an infinite number of internal parameters would be needed to describe
1.1 Plasticity and Thermodynamics 3
some of the models that employ functionals. In practice, however, provided that
sufficient ingenuity is employed in selecting them, a fairly small number of inter-

nal parameters can be used to construct close approximations to the more gen-
eral models. In Chapter 8, we explore models where the number of internal pa-
rameters does in effect become infinite, and the method described there offers
some of the advantages of rational thermodynamics.
1.1.4 Ziegler’s Orthogonality Condition
An important limitation is that the generalised thermodynamics approach used
here uses Ziegler’s orthogonality principle, which is explored in more detail in
Chapter 4, and then plays a major role in the remainder of this book. The prin-
ciple forms the basis of the important book by Ziegler (1977). The name derives
from Ziegler’s assumption that the dissipation function acts as a potential, so
that the “dissipative generalised stress” is orthogonal to level surfaces of the
dissipation. The principle can be viewed in a variety of ways, but the most useful
is to view it as a stronger statement than the Second Law of Thermodynamics
[see Chapter 15 of Ziegler, (1983)]. The Second Law requires that energy be dis-
sipated; the orthogonality principal requires the dissipation to be maximal.
Some authors accept the principle as “true” (in the same sense that the Second
Law of Thermodynamics is almost universally regarded as “true”), and these
authors would presumably accept the formulation derived here as having
a rather general status. Others regard Ziegler’s principle as unproven (some
reject it as overrestrictive), so that materials obeying it are merely a subset of
a wider class of possible models. We do not intend to enter this debate, and are
content therefore with the idea of Ziegler’s principle as a classifying hypothesis
and with the models described here as a subset of all those possible. Neverthe-
less, this subset is certainly large, and can encompass a very wide range of rate-
independent materials within a single framework.
It should also be stated that, as far as the authors are aware, it has not yet
been possible to devise an experimental test of the veracity of the orthogonality
principle (indeed it is far from clear whether such a test could be devised), and
so no “proof” exists that the principle is false. This is in contrast, for instance, to
Drucker’s postulate, where there is clear proof (both in terms of conceptual

models and experimental data) that the postulate is not always true, but must
simply be regarded as a classifying postulate; see Section 2.5. The status of the
principle is illustrated in Table 1.1.
4 1 Introduction
Table 1.1. Status of Ziegler’s orthogonality principle
Second Law of
Thermodynamics
x (Almost) universally accepted as true.
x Strong theoretical grounds for acceptance.
x No experimental counterexamples.
Ziegler’s orthogo-
nality principle
x Accepted by some as true, but by others regarded as a classi-
fying principle.
x Possible to devise conceptual counterexamples, but there are
no experimental data (known to the authors) that clearly vio-
late the principle.
Drucker’s stability
postulate
x Known to be a classification only.
x Many counterexamples in the form of conceptual models and
experimental data.
1.1.5 Constitutive Models
In the application of thermodynamic principles to constitutive models for sol-
ids, two approaches are possible. Firstly, the models may be developed within
a framework that embodies the Laws of Thermodynamics. Secondly, models
may be derived using arbitrary procedures, and thermodynamic principles ap-
plied to these models post hoc. In this book, the first approach is adopted. The
principal advantage of this method is that it allows a consistent and well-defined
approach to constitutive modelling to be adopted.

The constitutive modelling of the complex behaviour of solids inevitably
leads to a significant amount of mathematics, and some of the techniques we
introduce may be unfamiliar to some readers. We deliberately try to keep the
mathematics as simple as possible, and introduce only those concepts which we
believe provide powerful tools to aid the analysis. We provide appendices intro-
ducing a number of mathematical techniques that are important to the theme of
this book, specifically on tensors, on the algebra and calculus of functionals, on
the Legendre transform, and on convex analysis. Where possible, we introduce
new concepts by building up from relatively simple examples.
1.2 Context of this Book
The work described here has its roots principally in the work of Ziegler (1977,
1983), as developed for plasticity theory by the authors and published in a series
of papers: see Houlsby (1981, 1982, 1992, 1996, 2000, 2002); Collins and Houlsby
(1997); Houlsby and Puzrin (1999, 2000, 2002); Puzrin and Houlsby (2001a, b, c;
2003); and Puzrin et

al. (2001).
However, the approach described here also has much in common with the
work of others, particularly from the French school of plasticity: see, for instance,