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

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 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. 9 8 7 6 5 4 3 2 1 Springer Science+Business Media springer.com

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 particular relevance to those working in geomechanics.

Structure of this book After some introductory material in Chapter 1, a presentation of classical plasticity 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. Similarly, 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 allows 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 approach to the use of multiple internal variables. We find that multiple independent 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 employed 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 continuous 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 nonassociated flow. Most of the book is concerned with rate-independent materials, but in Chapter 11 we briefly examine ways these ideas are extended to materials with some rate dependence. Ziegler (1983) devotes much attention to rate-dependent materials, 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 response 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 phenomena. The chapter focuses particularly on the behaviour of porous media. In Chapter 13, we re-express hyperplasticity theory in the terminology of convex 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 introduction 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 applications illustrates that the hyperplasticity approach is a powerful unifying technique 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 purposes, 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 terminology 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 everyday 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 variables. 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

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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

4.12

4.13 4.14

xi

Constraints .......................................................................................... 71 4.12.1 Constraints on Strains........................................................... 72 4.12.2 Constraints on Plastic Strain Rates...................................... 73 Advantages of Hyperplasticity .......................................................... 74 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

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Contents

7.3

7.4 7.5 7.6

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 One-dimensional Example (the Iwan Model)................................ 125 Multidimensional Example (von Mises Yield Surfaces) ............... 128 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

9.4

9.5

xiii

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 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

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Contents

11.3

11.4

11.5

11.6

Models with Multiple Internal Variables........................................ 224 11.3.1 Multiple Internal Variables................................................. 225 11.3.2 Incremental Response ......................................................... 225 11.3.3 Example ................................................................................ 226 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 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 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 A.1 A.2 A.3 A.4

Functions, Functionals and their Derivatives ............................... 305 Functions and Functionals .............................................................. 305 Some Special Functions.................................................................... 306 Derivatives and Differentials ........................................................... 307 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 C.1 C.2 C.3 C.4 C.5 C.6

Legendre Transformations .............................................................. 315 Introduction ...................................................................................... 315 Geometrical Representation in (n + 1)-dimensional Space .......... 315 Geometrical Representation in n-dimensional Space ................... 317 Homogeneous Functions ................................................................. 318 Partial Legendre Transformations .................................................. 319 The Singular Transformation.......................................................... 320

xvi

Contents

C.7

Appendix D D.1 D.2 D.3 D.4 D.5 D.6 D.7 D.8 D.9 D.10

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 Convex Analysis.................................................................................. 325 Introduction ...................................................................................... 325 Some Terminology of Sets ............................................................... 325 Convex Sets and Functions .............................................................. 327 Subdifferentials and Subgradients .................................................. 328 Functions Defined for Convex Sets................................................. 329 Legendre-Fenchel Transformation ................................................. 331 The Support Function ...................................................................... 332 Further Results in Convex Analysis ................................................ 334 Summary of Results for Plasticity Theory...................................... 334 Some Special Functions.................................................................... 336

References................................................................................................................ 339 Index......................................................................................................................... 345

Notation

Variables ae

semidiameter of the linear elastic region

aL

semidiameter of the small strain region

ai

acceleration

aij

stress or strain (see Table 4.4)

A bij

pile area stress or strain (see Table 4.4)

c

cp

(1) constraint function (2) specific heat (3) concentration of fibres (4) undrained shear strength the specific heat at constant pressure

cv

the specific heat at constant volume

c1 , c2 , c3

compliances

cijkl

compliance matrix

d

dissipation function or dissipation functional

d

e

dissipation function corresponding to the energy function e

dt dijkl

total dissipation stiffness matrix

e

(1) specific energy function: any of u, f, h or g (2) voids ratio elastic modulus

E

xviii

Notation

gs

(1) specific Helmholtz free energy (2) yield surface (1) specific Gibbs free energy (2) plastic potential shear modulus constant

G

elastic shear modulus

G0

small strain shear modulus

h hˆ  K

enthalpy plastic modulus function

H H I IC

plastic modulus the Heaviside step function moment of inertia indicator function

I1 , I 2 , I3

invariants of a second order tensor

J

coupling modulus

J 2 , J3

invariants of the deviator of a second order tensor

k

shear strength

k  K

function defining the size of a yield surface in the field

ki

i-th bar axial stiffness

km

permeability coefficient

kK

thermal conductivity coefficient

K li

elastic bulk modulus i-th bar length

L Lijkl

pile length linear operator in hypoplasticity

m

parameter of elliptical surface mass flux

f g

mi

ni

(1) slope of critical state line in pc, q plot (2) moment parameter of elliptical surface outward normal to the boundary

N Nc

the number of variables vertical bearing capacity factor

NC

normal cone

M n

Notation

xix

Nh

horizontal bearing capacity factor

N ij

linear operator in hypoplasticity

p

pressure, mean (effective) compressive stress

pa

atmospheric pressure

pr

reference pressure

px

preconsolidation pressure in terms of mean stress

pL Pi

transform of pore pressure to Lagrangian coordinate system nodal external forces

Pij

deformation gradient

q

deviator stress invariant

qi

heat flux

Q

heat supply

r R

non-linear viscous model parameter (1) the gas constant (2) parameter of the logarithmic function (3) radius of the curvature (of a prismatic beam) (4) pile radius specific entropy (1) the modified signum function (2) entropy (3) surface the modified signum tensor function

s S

Sij

ti

time traction

Ti

i-th bar tension

u

(1) specific internal energy (2) horizontal pile displacement displacement

t

ui

uw U v vi

pore water pressure internal energy specific volume velocity

V

volume

xx

Notation

(1) force potential (2) water content (3) vertical pile displacement

w

W W x

rate of work input p

xi

plastic work temperature or entropy (see Table 4.4) spatial coordinate

xL

normalized limiting strain

xr

normalized strain at the stress reversal

y

yield function

y

canonical yield surface

ye

yield surface corresponding to the energy function e



ˆ ij , Fˆ ij , K yˆ g D



0

field of yield functions

z

(1) temperature or entropy (see Table 4.4) (2) flow potential

D

Dv

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

Ds

deviatoric (shear) kinematic internal variable

Df

damage model material constant

Dij

internal kinematic variable

ˆ ij  K D

internal variable function

E

dilational constant

Eijc

tensor defining direction of loading in deviatoric stress

Fr

space damage model material constant

Fij , Fij

generalised stress

Fˆ ij  K

dissipative generalised stress function

Fˆ ij  K

generalised stress function

J

shear strain

JC

gauge function

Notation

xxi

*  K

distribution function

G

Gij

(1) distance between the stress and image points (2) the Dirac impulse function Kronecker delta (unit tensor): Gij 1 if i j , Gij

'ij

otherwise Green-Lagrange strain

Hs

deviatoric (shear) strain

Hv

volumetric strain

Hij

strain tensor

e Hij

elastic strain tensor

 p Hij

plastic strain tensor

Mc

effective angle of internal friction

K

(1) internal coordinate (typically a dimensionless measure of the size of the yield surfaces) (2) stress ratio q p

Ki

entropy flux

N O O,/

slope of swelling line in consolidation plot slope of compression line in consolidation plot scalar multipliers principal stretches

Oi n O Oˆ  K

P

an n-th non-negative multiplier a non-negative multiplier function

Sij

(1) frictional constant (2) viscosity Poisson’s ratio Piola-Kirchhoff stress tensor

8

the domain of K , usually >0;1@

T U

temperature density

Uij

back stress

Uˆ ij  K

back stress function

V

normal stress

Q

0

xxii

Notation

VC

support function

Vy

yield stress

Vij

stress

Vij

effective stress

W

Zij

shear stress rigid body rotations

[

hardening parameter

\

angle of dilation

Subscripts, superscripts and diacritics e p

elastic plastic

w s

pore fluid (water) soil skeleton

n

n-th in a finite series of similar variables

ao

initial value of a

a

rate of a with time, a { wa wt

a

material derivative, a { a  a,i vi

aˆ

any function of the internal variable K , aˆ { aˆ  K

aijc

deviatoric

part

of

second

order

tensor

aij ,

aijc { aij  Gij akk

a,i

derivative with respect to spatial coordinate

xi ,

a,i { wy wxi wxa

subdifferential of a

Discontinuous functions In examples of one-dimensional models, we make much use of the absolute value of a variable: x x if x t 0 , x  x if x  0 . On differentiation, we obtain w x wx S  x , where S  x is a generalized signum function such that S  x 1 if x ! 0 , S  x 1 if x  0 , and S  x takes an undefined value such

Notation

xxiii

that 1 d S  x d 1 if x 0 . The function S  x is closely related to the conventional signum function, also often written sgn  x or sg  x . The difference is that the signum function is usually defined such that sgn  x 0 for x 0 . We need to introduce the definition in which sgn  x can take any of a range of values at x 0 so that it properly defines the value of the subdifferential of x (see Appendix D). Strictly in convex analysis notation, we define the subdifferential w x S  x , and S  x is a set-valued function. In analyses of continuum models, the equivalent of x is xij xij . On differentiation, we obtain

d



It is undefined if xij

xkl xkl



xij

dxij

xkl xkl

. This bears a close analogy to S  x .

0 , but otherwise gives (in a generalised sense) the direc-

tion of xij . For convenience [and by analogy with S  x ], we define the function



Sij xij

xij xkl xkl

 

. Note also that Sij xij Sij xij

 

that 0 d Sij xij Sij xij d 1 for xij We use the notation and x

1 for xij z 0 , and we require

0.

for Macaulay brackets, such that x

x if x t 0

0 if x  0 . Macaulay brackets can also be defined in terms of the abso-

lute value (or vice versa) x

x x 2

.

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 engineering problems. Countless theories have been proposed for such materials, based on concepts such as linear and non-linear elasticity, and the many variants 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 materials, although examples of particular theories will be given, nor is it to attempt an all-embracing theory with extravagant claims of generality. Instead, a framework will be described within which a rather broad class of theoretical models may be defined. Individual models are specified by the choice (subject to certain constraints) of certain functions. The constitutive behaviour, i. e. the entire incremental stress-strain response, then follows automatically from the procedures described here. The advantages of this approach are outlined below. The approach used here is based on what is often termed generalised thermodynamics. 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 materials. In particular, the framework can accommodate models for geotechnical materials (e. g. soils, powders, concrete, rocks) that exhibit what is called nonassociated 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, derived for plasticity models that conform to certain strict criteria, and the coincidence 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 increasing 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 different form.

1.1.3 Generality No claim is made here that the framework described is entirely general. For example, 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 principle, 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 internal parameters can be used to construct close approximations to the more general models. In Chapter 8, we explore models where the number of internal parameters 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 principle 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 dissipated; 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. Nevertheless, this subset is certainly large, and can encompass a very wide range of rateindependent 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 orthogonality principle

x Accepted by some as true, but by others regarded as a classifying principle. x Possible to devise conceptual counterexamples, but there are no experimental data (known to the authors) that clearly violate 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 solids, 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 applied 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 introducing 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,

1.3 Notation

5

Lemaitre and Chaboche (1990), Maugin (1992, 1999), Coussy (1995). In this context, what we call a hyperplastic material has much in common with what the French authors define as a “standard material”, although we believe we have adopted a rather stricter formalism for the necessary definitions. There are numerous other works in which the thermodynamics of continua are addressed. Many attempt great generality [e. g. the works by Eringen (1962), Truesdell (1977), and Holzapfel (2000)], describing the mechanics in terms of appropriate tensor notation for large strain analysis, and often developing a somewhat axiomatic approach in which thermodynamics is introduced at a very general level. The disadvantages of this approach are twofold. Firstly, these works are impenetrable to all but the highly specialised reader. Secondly, we have found that, despite their generality, their treatment of plasticity theory is often superficial. Plasticity theory does not sit easily within formulations oriented principally toward the treatment of materials in which the dissipation is viscous (rate dependent). Often the authors of more general texts revert to “conventional” analysis of plasticity problems and make little connection between plasticity and thermodynamics. Our approach is different in that (a) thermodynamics and hyperplasticity are intimately connected and (b) we make our ideas more accessible by adopting just small strain analysis, which makes much of the presentation more straightforward.

1.3 Notation This book makes much use of vectors and tensors, and for these we use the subscript notation. Thus Vij is a shorthand for a second-order tensor which could

ª V11 V12 V13 º « » be written out in full in matrix form as «V21 V22 V23 » . We adopt the summa«¬V31 V32 V33 »¼ tion convention over a repeated index, so that, for instance, Vii { V11  V22  V33 . Although the subscript notation can become rather cumbersome at times, and many authors prefer the boldface notation in which the above tensor would simply be written as V , it has the advantage that there is absolute clarity about products, particularly contracted products. If the boldface notation is used, one has to pay meticulous attention to the order of multiplication, and also elaborate notation for different types of product is necessary. Note that in any equation, dimensional consistency requires that the subscripts must be balanced. Thus aij bij or aij bik ckj are meaningful statements, whilst aij

bik is not. A problem arises when we wish to indicate the arguments of

a function within an equation. Suppose, for instance, that f kl is a function of

6

1 Introduction

a tensor xij . Then we could write ykl



f kl xij . The subscripts within the list of

arguments of the function are there merely to indicate a tensorial argument and play no other role in the equation. We shall usually just use ‘ij ’ for the subscripts in such cases. The unit tensor (Kronecker delta) is given by Gij , where Gij 1 for i j and

Gij

0 for 1. The deviator of a tensor is indicated by a prime notation, thus

Vijc { Vij  13 Gij Vkk . Other details of the terminology we shall use for tensors are

given in Appendix B. We denote the time differential by the dot notation; thus y { wy wt and

 y { w 2 y wt 2 . Spatial differentiation is denoted by a comma notation, so that if xi are the coordinate directions, then y,i { wy wxi . In Chapter 12, we use the notation y for the material derivarive y y  y,i vi , where vi xi is the velocity of a material point.

1.4 Some Basic Continuum Mechanics 1.4.1 Small Deformations and Small Strains In this book, we shall restrict our attention (with the exception of Chapter 12) to problems of small strain. It has become usual in continuum mechanics papers and textbooks to present ideas in the very general mathematical forms that can be used for large strain problems. However, it is our view that this adds a level of complexity which can make the underlying ideas less clear in some cases. Whilst the ideas that are described in this book can certainly be extended to large strain (see Chapter 12), one should not underestimate the difficulties involved; some areas would need considerable attention to matters of detail. It is worth, however, reflecting at this stage on precisely what we mean by the small strain assumption. We consider a process of deformation in which the initial position t 0 of a material point in a body is given by the Cartesian coordinates Xi . At some later time t, the position of the same material point, measured in the same coordinate system, is xi . Furthermore, we define the displacement of the point as ui xi  Xi . The deformation gradient tensor is defined as Aij wxi wX j , and the displacement gradient tensor is wui wX j

Aij  Gij . We define a small displace-

ment gradient process as one in which wui wX j  1 , where aij defines an appropriate norm of the tensor aij . Possible choices of the norm are max aij , i, j

1.4 Some Basic Continuum Mechanics

¦ aij

, or

7

aij aij . Use of any of these definitions ensures that the condition

i, j

wui wX j  1 requires that each individual component of wui wX j is itself small. For any small deformation process, it follows that Aij | Gij , so that

wui wX j

wui wxk wxk wX j

wui wu Akj | i Gkj wxk wxk

wui . It follows therefore that for such wx j

processes, it is not necessary to distinguish between differentiation with respect to the initial coordinates Xi (a Lagrangian formulation) and the current coordinates xi (an Eulerian formulation). For convenience (except in Chapter 12), we shall use an Eulerian formulation. Now we turn our attention to definitions of strains. Many definitions are in general possible, but the strains most commonly defined for large displacement 1 § wui wu j wuk wuk ·   ¨ ¸ and processes are the Green-Lagrange tensor Eij 2 ¨© wX j wXi wXi wX j ¸¹ 1 § wui wu j wuk wuk ·   ¨ ¸ . Both of these reduce to the Euler-Almansi tensor eij 2 ¨© wx j wxi wxi wx j ¸¹ zero for processes that involve pure translation and rotation. For small displacement gradient processes as defined above, the quadratic terms become of second order, and both of these tensors become approximately 1 § wui wu j · 1 § wui wu j ·   ¨ ¸| ¨ ¸ equal to the Cauchy small strain tensor Hij 2 ¨© wx j wxi ¸¹ 2 ¨© wX j wXi ¸¹ and Hij  1 . Clearly, small displacement gradient processes involve small strains, but the reverse is not necessarily true. A distinction is necessary between small strain problems and small displacement problems. It is possible to have a process in which the displacements are large, but the strains themselves are small everywhere. Trivial examples involve rigid body rotations (zero strain), but others involve small non-zero strains but large displacements. For instance, a thin metal strip can be bent such that the displacements are certainly of the same order as the dimensions, and yet the strains can still be very small throughout the strip. Small strain means that, for every element of the continuum, the linear strain (fractional change of length) and shear strain (change of angle) are “small”. However, how small is small? The answer lies in the precision required of the theory, but for most practical purposes, 1% is certainly small, whilst 10% is approaching the magnitude at which the approximations in the theory may be important. In the following, we shall strictly adopt definitions that are applicable only to small deformations, which is a stricter criterion than small strains. However, the constitutive responses we shall describe are applicable to the wider class of

8

1 Introduction

problems involving small strains, and need be restricted only to small strains, and not necessarily to small deformations. Small deformation, according to the definition above, means that the displacements of any part of the continuum are small compared to the dimensions, and therefore need not be considered in any way commensurate with them. A result is that an apparently self-contradictory assumption is made, so that although the continuum undergoes displacements, no changes of geometry need be considered. The issue of how small is small again arises, and again the answer depends on the problem. It is important to realize, however, that the magnitude of displacements that might be regarded large, as far as the serviceability of a structure are concerned, would be quite unrelated to the magnitude of the displacements at which the mathematics of small displacement theory would no longer be valid. For instance, the serviceability of a building may be considered compromised if deflections are greater than about 1 300 of its linear dimension (due to aesthetic concerns about damages to finishes, etc.), yet by any standard, displacements of only 1 300 of the dimensions are “small” in the mathematical sense. For the small strain analysis in this book, we shall employ the Cauchy stress Vij (force per unit area) and the linear or Cauchy strain Hij , as defined above.

1.4.2 Sign Convention Throughout the entire book, except for Chapters 9 and 10, we adopt the usual convention of tension positive from continuum mechanics. Unfortunately, in soil mechanics and geotechnical engineering, it is usual to adopt a compression positive convention. Only in Chapters 9 and 10, which deal with applications in geomechanics, we reverse the sign convention and use compression positive, to make these chapters more accessible to geotechnical engineers.

1.5 Equations of Continuum Mechanics In the solution to any problem in continuum mechanics, four issues must be addressed: x x x x

Equilibrium Compatibility Constitutive relationships Initial and boundary conditions.

The bulk of this book is concerned with the third of these topics for a particular class of materials, but it is worthwhile devoting some attention to the other three, so that the context of the constitutive relationships is properly understood.

1.5 Equations of Continuum Mechanics

9

1.5.1 Equilibrium The stresses in a continuum cannot simply be specified arbitrarily, but must satisfy certain equilibrium relationships. These are well-known and are derived by examining the equilibrium of a small element in a varying stress field. Firstly, the moment equilibrium of the element establishes the symmetry of the stress tensor: V ji

Vij

(1.1)

Secondly, the direct equilibrium equations give wVij wx j

or more compactly Vij , j  Fi

 Fi

Uüi

(1.2)

Uüi , where Fi is the body force per unit volume

(usually equal to Ug i , where g i is the gravitational acceleration vector), U is the density, and ui the displacement. In large displacement theory, the term üi would be augmented to üi  ui, j u j , which is the convective derivative of the velocity and represents the total time rate of change of the velocity of the element currently at the coordinate position under consideration. In static (or quasi-static) problems, the right-hand side of Equation (1.2) is zero.

1.5.2 Compatibility Similarly, the strains cannot be arbitrarily specified, but have to satisfy certain compatibility relationships. These arise because (in three dimensions) six independent strain components are derived, by definition, from the gradients of only three displacements: Hij

or more compactly Hij

1 2

1 § wui 2 ¨¨ wx © j



wu j · ¸ wxi ¸¹

(1.3)

ui, j  u j,i . This definition of the strains is a com-

plete statement of the compatibility requirements, although some authors prefer to eliminate the displacements and derive relationships between higher order derivatives of the strains.

1.5.3 Initial and Boundary Conditions Initial values of the stresses and/or the displacements [and hence through Equation (1.3) the strains] need to be specified. The stresses must, of course, obey the equilibrium Equations (1.1) and (1.2). It is common (but not essential) to treat

10

1 Introduction

the displacements and strains as zero at the initial conditions. Later in this book, we shall use internal variables (see Chapter 4 and following), and these will also need to be specified as part of the initial conditions, because they define the initial state of the material. The boundary conditions fall into two categories. On some sections of the boundary, the displacements will be specified. This is straightforward. On other sections, there will be boundary conditions on the tractions. These are often loosely referred to as “stress” boundary conditions, but we prefer to make a careful distinction between stresses (which are second-order tensors) and tractions on a free surface, which are defined as forces per unit area on the surface. The traction ti is therefore a vector. It is the tractions that can be defined at a boundary, and there is then an equilibrium relationship between these tractions and the stresses in an element immediately within the boundary: Vij ni

tj

(1.4)

where ni is the unit outward normal to the boundary. There is also the possibility of mixed boundary conditions, either in which some displacement components and some traction components are specified, or in which some relationship between displacement and traction is defined. The most common example of the first type is the case where the shear traction on a surface is defined, together with the normal displacement component. This method would be used to describe, for instance, a contact with a smooth rigid body. An example of the second type would be a case where contact with an elastic support was to be simulated by expressing some relationship between the increment in displacement and the increment of traction. There are also occasions where boundary conditions can be expressed in differential form. For instance, in analyzing a problem of steady flow past some obstruction, it may be convenient to express the boundary conditions far upstream (and/or far downstream) by noting that the differentials of certain quantities must vanish asymptotically in the direction of flow. Since we are concerned here principally with constitutive behaviour, we do not pursue the issue of boundary conditions further, but note that these need to be specified with some care if the equations to be solved are to form a well-posed mathematical problem. Further discussion is outside the scope of this book.

1.5.4 Work Conjugacy A very important concept in continuum mechanics is that of work conjugacy of stresses and strains. A proper set of definitions satisfies the condition that the stresses and strains are work conjugate in the sense that the rate of work input to the material, per unit volume, is given by the product of the stresses with the strain increments:

1.5 Equations of Continuum Mechanics

W

Vij H ij

11

(1.5)

It is straightforward to show that the conventional small-strain definitions of the Cauchy stress and the linear strain satisfy this condition.

1.5.5 Numbers of Variables and Equations Within a three-dimensional continuum, a solution is required for 15 variables (6 stresses components, 6 strain components and 3 displacements). The equilibrium Equation (1.2) provides three equations that involve only the stresses, and the strain definitions (1.3) provide six equations that involve the strains and displacements. For a solution to the problem, we are missing six equations, and this is where the constitutive relations come in. They provide six relationships between the stresses (or more usually their increments) and the strains (or strain increments), thus satisfying the need for six more equations, and also providing a link between stresses and strains. The constitutive relations are quite different in character from the equilibrium and compatibility relationships. Firstly, they are algebraic rather than differential equations. Secondly, and more importantly, the equilibrium and compatibility relationships are the same for all materials and (subject to the usual provisos about the applicability of continuum mechanics to a given problem) are universally “true”. Each of the constitutive equations is different for different materials – this statement is a tautology, because it is these very differences in the constitutive relations that define the differences in the characteristics of the materials. Furthermore, the constitutive relations are simply approximations to the behaviour of real materials; none of which will behave exactly according to the idealisations employed. Thus constitutive relations are never “true” for a real material; they can only provide solutions that approximate what happens in reality to a certain degree of precision. Of course, in certain branches of mechanics (notably the elasticity of metals within a limited stress range), the precision of the constitutive model is so good that, for all practical purposes, it can be treated as a “true” model. In this book, however, we treat much more complex materials, and for these a certain lack of precision is inevitable. Whoever uses the models should understand this and must try to assess its implications for the problem in hand.

Chapter 2

Classical Elasticity and Plasticity

2.1 Elasticity Fung (1965) provides elegant definitions for the different forms of elasticity theory, and we follow his terminology here. A material is said to be elastic if the stress can be expressed as a single-valued function of the strain:

Vij



where fij Hij



(2.1)

fij Hij

is a second-order tensor-valued function of the strains. If



fij Hij is linear in Hij , then it can be written Vij



fij Hij

dijkl Hkl , where dijkl

is a fourth-order tensor constant usually called the stiffness matrix: this is the common case of linear elasticity. The incremental stress-strain relationship for an elastic material is written

V ij

 H

wfij Hij wHkl

kl

(2.2)

Alternatively, if the stress-strain relationship is originally expressed in incremental form, the material is described as hypoelastic:

V ij



where fijkl Hij , Vij







fijkl Hij , Vij H kl

(2.3)

is a fourth-order tensor valued function of the strains or

of the stresses, or even (rarely) of both. Note that we distinguish between the fourth-order tensor function fijkl and the second-order tensor function



fij . If fijkl Hij , Vij fijkl Hij , Vij dijkl .





is a constant, then the material is linear elastic and

14

2 Classical Elasticity and Plasticity

Alternatively, if the stresses can be derived from a strain energy potential, then the material is said to be hyperelastic:

Vij



in which f Hij



wf Hij

(2.4)

wHij

is a scalar valued function of the strains. (Again note that we

distinguish the scalar function f from the two tensor functions fij and fijkl ). It follows that

V ij

 H

w 2 f Hij

wHij wHkl

kl

(2.5)



If f Hij is a quadratic function of the strains, then the material is linear elastic, and



w 2 f Hij

wHij wHkl

dijkl .

Example 2.1 Linear Isotropic Elasticity The following strain energy function can be used to express linear isotropic elasticity in hyperelastic form: Hii H jj Hijc Hijc (2.6) f 3K  2G 6 2 where K is the isothermal bulk modulus and G is the shear modulus. Differentiating the above1 gives the elastic form, wf Vij fij Hij K Hkk Gij  2GHcij (2.7) wHij



Differentiating once more leads to the hyperelastic incremental form,

V ij

w2 f H kl wHij wHkl

dijkl H kl

K H kk Gij  2GH ijc

(2.8)

from which one can derive dijkl

1

2G · § ¨K  ¸ Gij Gkl  2GGik G jl 3 ¹ ©

See table B.1 in Appendix B for some differentials of tensor functions.

(2.9)

2.1 Elasticity

15

Comparison of Equations (2.1) and (2.4) reveals that every hyperelastic matewf Hij . Conversely, however, an elastic material is also elastic, with fij Hij wHij rial is hyperelastic only if

  fij  Hij is an integrable function of the strains.

Similarly, comparison of Equations (2.2) and (2.3) reveals that all elastic mawfij Hij . Again, the converse is terials are also hypoelastic, with fijkl Hij , Vij wHkl not true, and a hypoelastic material is elastic only if fijkl Hij , Vij can be ex-











pressed as an integrable function of the strains only. Thus hypoelasticity is the most general form, followed by elasticity and hyperelasticity. This hierarchy is illustrated in Figure 2.1. What therefore are the advantages of the more restrictive forms? The first advantage is the compactness of the formulation. A hyperelastic material requires solely the definition of a scalar function f Hij for its complete specification. An elastic material re-



quires the definition of a second-order tensor function, and a hypoelastic material requires a fourth-order tensor function. The second advantage relates to the Laws of Thermodynamics. It is quite possible to specify an elastic or hypoelastic material so that, for a closed cycle of stress (or of strain), the material either creates or destroys energy in each cycle. This is clearly contrary to the First Law of Thermodynamics. Furthermore, for a hypoelastic case, it is possible to specify a material for which a closed cycle of stress does not necessarily result in a closed cycle of strain, thus contradicting the notion of elasticity in its sense that it implies that no irrecoverable strains occur over a cycle of stress. Thus there are sound reasons why hyperelasticity should always be the preferred form of elasticity theory. In this book, we build on this concept to express models for plastic materials, i. e. those that do display irrecoverable behaviour in a way that is consistent with thermodynamics.

Hyperelastic Elastic Hypoelastic Figure 2.1. Classes of elasticity theory

16

2 Classical Elasticity and Plasticity

2.2 Basic Concepts of Plasticity Theory Before moving on to the new formulation of plasticity theory that is the main subject of this book, it is useful to present the conventional formulation of plasticity theory. This serves as a basis for comparison for the approach presented in later chapters. The first and fundamental assumption of plasticity theory is that the strains can be decomposed into additive elastic and plastic components: Hij

e  p Hij  Hij

(2.10)

e The elastic strains Hij can be specified by any of the means used for elasticity, as discussed in Section 2.1 . For the reasons described there, the hyperelastic approach would of course be preferred. The changes of elastic strain are thus solely related to changes in stress. The plastic strains are of a different nature, and are defined in quite a different way. The rules used to define the plastic strains have become well established over many years, but they are based on empirical observation, first of metals and later of many other materials. The concept of a yield surface is introduced. This is a surface in stress space, defined through a yield function by f Vij , ! 0 . For





the time being, we shall just consider this as a function of the stresses, although later we shall consider cases where it is also a function of other variables. Changes in plastic strain can occur only if the stress point lies on the yield surface, i. e. f Vij 0 . If the stress point falls within the yield surface (which is





conventionally defined as the region where f Vij  0 ), then no plastic strain increments occur, and the response is incrementally elastic. We refer to this region as “within” the yield surface, even for the quite common cases where the surface is not closed in stress space. Stress states outside the yield surface, i. e. for which f Vij ! 0 , are not attainable.



Example 2.2 The von Mises Yield Surface The von Mises yield surface is specified by the function f Vij Vijc Vijc  2k2 0 , where k is the yield stress in simple shear. In prin-



cipal stress space, the surface is a cylinder, centred on the space diagonal, 3  V1  V2 2   V2  V3 2   V3  V1 2  k2 0 . In plane stress  V2 0 , this 2 reduces to the ellipse V12  V1V3  V32  3k2 0 . The strength in pure tension  V2 V3 0 is V1 3k .

2.2 Basic Concepts of Plasticity Theory

17

It is important to note that the form of the yield function is not uniquely defined. The same surface as defined in Example 2.2 could equally well be defined, Vcij Vcij  1 0 or for instance, by any of the functions, f Vcij Vcij  2k 0 , f 2k2 Vcij Vcij  1 0 . All define the same surface, but the functions take different f 2k values at points that do not lie on f 0 . Even the dimensionality of the functions is different. The original expression in Example 2.2 had the dimension of stress squared, the first of the alternatives given above has the dimension of stress, and the other two are dimensionless. At this stage, we are unable to define a “preferred” yield function, but later, in Chapter 13, we shall see that convex analysis does allow us to select a preferred yield function (which we call the canonical yield function). This happens to be in the last of the forms given above. When yield occurs at a particular point on the yield surface, it is found empirically for many materials (at least to a first approximation) that the ratios between the plastic strain components are fixed, irrespective of the stress increments. One way of defining this mathematically is to define a flow rule, in which the ratio of strain increments is related directly to the stress state. This method is sometimes used for models that involve just two dimensions of stress, but for more complex cases, it becomes very cumbersome, and the ratios between the strain increments are defined instead by a plastic potential, from which the flow rule can be derived.

V1 V3

V1

V1

 V3

3k

3k , V3

k

Figure 2.2. The von Mises yield surface in plane stress

0

18

2 Classical Elasticity and Plasticity

The plastic potential, like the yield surface, is a function of the stresses and other variables, and is usually written g Vij , ! 0 . The direction of the plastic





strain increment is then defined as normal to the plastic potential:

 p H ij

O



wg Vij

(2.11)

wVij

where O is a plastic multiplier which is as yet undetermined. In metal plasticity, it is usual for the yield function f Vij , !



plastic potential g Vij , !







0 and the

0 to be identical functions. This choice is based

both on a wealth of empirical evidence and also on sound theoretical grounds (see Section 2.5 ). This important case is called “associated flow” or “normality” (in the sense that the plastic strain increments are normal to the yield surface, and not in the sense that “normal” means that this represents the usual behaviour of materials). There are many advantages of theories that adopt associated flow, sufficient that many practitioners go to elaborate means to avoid using non-associated flow theories. However, for some materials, notably soils, the empirical evidence for non-associated flow is so overwhelming that it is essential to address this more complex case. It is usual (although not essential) to define the plastic potential so that g 0 at the particular stress point at which the strain increment is required. This means that (except for associated flow) it is necessary to introduce some additional dummy variables, say x, into the plastic potential, defined so that g Vij , x 0 at the particular stress point on the yield surface.





Note that we follow here the common notation in plasticity theory and use f for the yield function and g for the plastic potential. Later in this book, we shall need to use f and g for the Helmholtz and Gibbs free energies, as is common practice in thermodynamics. We shall attempt to make the meaning of the variables clear whenever there is any danger of ambiguity. The yield surface defines the possibility of plastic strain increments, and the plastic potential then defines the ratios between the plastic strain increments; but what remains to be defined is the magnitude of the plastic strains. It is in this area that the greatest variety of ways of specifying plasticity models is found, and it is not possible to present an approach that encompasses all the different forms in the literature. Apart from perfectly plastic materials (see Section 2.3 below), the magnitude of plastic strain is defined by establishing a link between the plastic strains and the expansion (or movement or contraction) of the yield surface. Probably the most common approach is to define the yield surface so that it is a function of the stresses and some hardening parameters [, which are usually scalars, but could be tensors. Thus the yield surface is written f Vij , [ 0 . The hardening pa-





rameters are, in turn, defined in terms of the plastic strains. The most common

2.3 Incremental Stiffness in Plasticity Models

19

form is that the hardening parameters are simply functions of the plastic strains  p ȟ ȟ İij . In this case, the hardening parameters can be eliminated from the





p formulation and the yield function simply expressed in the form f Vij , Hij



0.

This form of hardening is called strain hardening. An alternative is that, although the hardening parameters cannot be expressed as functions of the plastic strains, evolution equations can be defined for the hardening parameters in terms of the plastic strain rates. The evolution equations may also involve other variables, so that a rather general form is



 p  p [ [ Vij , Hij , [, H ij



(2.12)

(We hesitate to say that the above is the most general possible form because the imagination of plasticity theorists in devising ever more complex approaches seems almost boundless.) One important special case occurs when [ is identified p with plastic work, in which case we can write [ W  p Vij H ij . The term work hardening is strictly applicable only to this case, although it is frequently applied much more loosely to any hardening process. To recapitulate, a strain hardening plasticity model can be completely defined by the following assumptions: x Decomposition of the strain into elastic and plastic components; x Definition of the elastic strains, using, for instance, a hyperelastic law that requires specification of a single scalar function;  p 0 ; x Definition of a yield surface f ıij , İij







 p x Definition of a plastic potential g ıij , İij , x



0.

Thus the model requires three scalar functions to be defined for its complete specification. Unfortunately, many plasticity theories do not adhere to this simple pattern, and some adopt a series of ad hoc rules and assumptions. Not only can this be confusing, but it also makes comparisons between competing theories difficult. At worst, it can lead to theories that are not internally self-consistent.

2.3 Incremental Stiffness in Plasticity Models For any constitutive model, one of the most important operations is derivation of the incremental stress-strain relationship. For plasticity theories, the method adopted is different for the special case of perfect plasticity. So we shall treat this case first, before going on to the case of hardening plasticity.

20

2 Classical Elasticity and Plasticity

2.3.1 Perfect Plasticity In perfect plasticity, the yield surface remains fixed in stress space, so there is no hardening (or softening). Thus the yield surface is only a function of the stress f Vij 0 . The plastic potential will have the form g Vij , x 0 , where x are the







dummy variables introduced to satisfy the (not strictly necessary) condition that g 0 at each stress point on the yield surface. The incremental stress-strain relationship is obtained by combining the equations below. The strain decomposition is written in incremental form: H ij

e  p H ij  H ij

(2.13)

The elastic strain rates are defined by a stiffness matrix:

e dijkl H kl

V ij

(2.14)

Note that the above form can always be derived, irrespective of whether the elastic strains are specified through hyperelasticity, elasticity, or hypoelasticity. The yield surface is written in differential form, noting that during any process in which the plastic strains are non-zero, not only is f 0 , but also f 0 . This incremental form of the yield surface is usually referred to as the consistency condition:

wf V ij wVij

f

0

(2.15)

Finally, we need the plastic strain rate ratio, obtained from the plastic potential:

 p H ij

O

wg wVij

(2.16)

Substituting (2.13) and (2.16) in (2.14) gives V ij

e dijkl H kl



 p dijkl H kl  H kl



§ wg · dijkl ¨ H kl  O ¸ wVkl ¹ ©

(2.17)

Combining Equations (2.15) and (2.17), we obtain

f

wf V ij wVij

§ wf wg · dijkl ¨ H kl  O ¸ 0 wVij wVkl ¹ ©

(2.18)

which leads to the solution for the plastic multiplier O:

O

wf dijkl H kl wVij wf wg d pqrs wV pq wVrs

(2.19)

2.3 Incremental Stiffness in Plasticity Models

21

Equation (2.19) is then back-substituted in (2.17) to give

V ij

§ wf dmnab H ab ¨ wV dijkl H kl  ¨ mn wg ¨ wf ¨ wV d pqrs wV rs © pq

· ¸ wg ¸d ¸ ijkl wVkl ¸ ¹

(2.20)

With some interchanging of the dummy subscripts, this can be rewritten V ij

 ep dijkl H kl

(2.21)

 ep where dijkl is the elastic-plastic stiffness matrix defined as  ep dijkl dijkl 

wg wf dmnkl wVab wVmn wf wg d pqrs wV pq wVrs

dijab

(2.22)

Thus, given the specification of the elastic behaviour, the yield surface, and the plastic potential, the incremental constitutive behaviour can be obtained by applying a purely automatic process to obtain the incremental stress-strain relationship. This is important because no further ad hoc assumptions are necessary. Note also that the derivation involves solely matrix manipulation and differentiation. Both processes can be readily carried out using symbolic manipulation packages.

Example 2.3 Von Mises Plasticity with Isotropic Elasticity Using the yield surface as defined in Example 2.2, which is also the plastic potential as the von Mises model uses associated flow, then f g Vijc Vijc  2k2 , so that wf wVij wg wVij 2Vijc . 2G · § From Example 2.1, dijkl ¨ K  ¸ Gij Gkl  2GGik Gil . Noting that for this 3 ¹ © wg wf case, dijkl dklij 4GVijc , substitution in Equation (2.22) gives wVkl wVkl 2GVijc Vckl 2G ·  ep § dijkl ¨ K  ¸ Gij Gkl  2GGik G jl  c c V pq Vij 3 ¹ ©

(2.23)

which can be further simplified to 2G · G  ep § dijkl ¨ K  ¸ Gij Gkl  2GGik G jl  2 Vijc Vckl 3 ¹ © k

(2.24)

22

2 Classical Elasticity and Plasticity

It is important to note that this stiffness matrix is singular, so that it cannot be inverted to give a compliance matrix. This is a feature common to all perfect plasticity models.

2.3.2 Hardening Plasticity Incremental Response for Strain Hardening For strain-hardening plasticity, Equations (2.13), (2.14), and (2.16) take the same form, but because the yield surface is now also a function of the plastic strains, the consistency condition takes the form f

wf wf  p V ij  H  p ij wVij wH

0

(2.25)

ij

Substituting the flow rule (2.16), we obtain f

wf wf wg V ij  O  p wVij wH wVij

0

(2.26)

ij

It is more convenient in this case to obtain the solution for O in terms of the stress increment rather than the strain increment (as was used for perfect plasticity): wf V ij wVij O  wf wg  p wHij wVij

(2.27)

wf wg is often termed the hardening modulus, and we  p wHij wVij 1 wf  p 1 wg wf  . The hardening modulus is not can write O V ij or H ij Vkl h wVij wVkl h wVij a defined parameter of the model, but is derived at a given stress point in terms of the yield function and plastic potential. It is identically zero in a perfect plasticity model. The most convenient way to proceed is to use the elastic compliance matrix,

The quantity h 

e H ij cijkl V kl

(2.28)

2.3 Incremental Stiffness in Plasticity Models

23

so that

 cijkl V kl  H ijp

H ij

which leads immediately to H ij matrix is

(ep ) cijkl

cijkl 

cijkl V kl 

1 wg wf V kl h wV ij wV kl

(2.29)

ep V , where the elastic-plastic compliance cijkl kl

1 wg wf h wV ij wV kl

wg wf wV ij wV kl cijkl  wf wg  p wV wH mn mn

(2.30)

If the stiffness matrix is required, it can be obtained either by numerical inversion of the compliance matrix, or by solving for the plastic multiplier, as before, in terms of the strains. Starting with the consistency condition,

f

wf wf V ij   p H ij p wVij wHij § wf wg d ijkl ¨¨ H kl  O wVij wV kl ©

(2.31)

· wf wg ¸¸  O  p wHij wVij ¹

0

the solution for O becomes

O

wf d ijkl H kl wV ij

(2.32)

§ wf wf ¸· wg ¨ d pqrs  ¨ wH rsp ¸¹ wV rs © wV pq

The solution then proceeds exactly as for the perfectly plastic case, except that this time d ep takes the form ijkl

ep d ijkl

wg wf d mnkl wV ab wV mn d ijkl  § wf wf ·¸ wg ¨ d pqrs  ¨ wV pq wH rsp ¸¹ wV rs © d ijab

(2.33)

Note that if wf wH rsp 0 then Equation (2.33) reduces to the result for perfect plasticity, (2.22). However, the equation for compliance, Equation (2.30), becomes singular in the perfectly plastic case.

24

2 Classical Elasticity and Plasticity

Incremental Response for Work Hardening Because of its historical importance, we set out here the particular case of work hardening, but this section can be omitted and the reader can proceed to Section 2.3.3. For work-hardening plasticity, Equations (2.13), (2.14), and (2.16) p again apply, but now we write the yield surface in the form f V ,W  0 ,



where W  p

ij



 V ij H ijp .

The consistency condition takes the form,

wf wf V ij  W  p  p wV ij wW

f

0

(2.34)

Substituting the definition of the plastic work and the flow rule (2.16), we obtain

f

wf wf wg V ij  O V ij  p wV ij wV ij wW

0

(2.35)

so that

O

( p)

and recalling that H ij is h



wf



wf V ij wV ij

(2.36)

wf

wg V  p ij wV wW ij

1 wg wf V kl , the hardening modulus in this case h wV ij wV kl

wg

V ij . wV ij wW  p

The analysis proceeds exactly as for the strain hardening case, except that this time

(ep ) cijkl

cijkl 

1 wg wf h wV ij wV kl

cijkl 

wg wf wV ij wV kl wf

wg V  p mn wV wW mn

(2.37)

2.3 Incremental Stiffness in Plasticity Models

25

The alternative analysis for the stiffness matrix starts with the consistency condition,

f

wf wf V ij  W  p  p wV ij wW § wf wg d ijkl ¨¨ H kl  O wV ij wV kl ©

(2.38)

· wf wg ¸¸  O V ij wV ij wW  p ¹

0

and the solution for O becomes

O

wf d ijkl H kl wV ij

(2.39)

§ wf · wg wf ¨ d pqrs  V rs ¸ ¨ wV pq ¸ wV rs wW  p © ¹

ep takes the form Again, the solution proceeds as before, and this time d ijkl

ep

d ijkl

wg wf d mnkl wV ab wV mn d ijkl  § wf · wg wf ¨ d pqrs  V rs ¸ ¨ wV pq ¸ wV rs wW  p © ¹ d ijab

(2.40)

2.3.3 Isotropic Hardening If, the yield surface expands (or contracts) but does not translate as plastic straining occurs, then this is said to be isotropic hardening (or softening). This is illustrated in Figure 2.3a for a simple one-dimensional material that hardens linearly and isotropically with plastic strain. The material yields at A at a stress c, and during plastic deformation AB, it hardens and the stress increases to c1. It is then unloaded, and reverse yielding occurs at C when the stress is c1. Further hardening occurs on CD, so that, when the material is reloaded, the expansion of the yield surface is such that the yield at E occurs above the original curve AB, and further hardening occurs along EF. Figure 2.3b illustrates isotropic hardening in two dimensions. The surface expands but does not change shape.

26

2 Classical Elasticity and Plasticity

Example 2.4 Von Mises Yield Surface with Isotropic Hardening Consider a modification of the von Mises yield surface, as specified by the 2 p  p Vc Vc  2 k İ p 0 , where k İij , the yield function f ıij , İ ij ij ij ij





^  `



stress in simple shear, is now a function of the plastic strain. A typical form

  k 0  h H cij p H cij p , which would result (for a straight strain path) in linear hardening for h ! 0 , or softening for h  0 . would be k

Such a model is, however, rather too simplistic in that if the material is first strained plastically in one direction (and hardens) and then strained plastically in the opposite direction, the plastic strain will reduce, and the material will therefore soften. More realistically, hardening occurs on any further plastic straining, irrespective of the direction. This more realistic model can be achieved by defining a hardening parameter [ such that

[

  H cij p H cij p , and defining the strength in terms of this hardening pa-

rameter, k k 0  h[ . This modification has no effect on the predicted response of straight initial loading paths, but models typical isotropic hardening processes more realistically for more complex strain paths.

Figure 2.3. Isotropic hardening: (a) stress-strain curve in one dimension; (b) change of size of yield surface in two dimensions

2.3 Incremental Stiffness in Plasticity Models

27

2.3.4 Kinematic Hardening On the other hand, if the yield surface translates, but does not change size, as plastic strain occurs, then this is said to be kinematic hardening. Figure 2.4a, which should be contrasted with Figure 2.3a, shows the response of a simple kinematic hardening material that hardens linearly with plastic strain. The loading curve OAB is identical to that of isotropic hardening, but on unloading from B, yield occurs at Cc , such that the size of the elastic region is 2c (the stress at Cc therefore is c1  2c ). Hardening occurs on reverse loading CcDc , but on reloading, yield occurs at Ec , which falls on the original line AB, and the hardening once again occurs along EcFc . Figure 2.4b (which should be contrasted with Figure 2.3b) shows the translation of a yield surface for a kinematic hardening model in two dimensions.

Figure 2.4. Kinematic hardening (a) stress-strain behaviour in one dimension; (b) translation of yield surface in two dimensions

Example 2.5 Von Mises Yield Surface with Kinematic Hardening Consider a modification of the von Mises yield surface, specified by the func-



tion f Vij , H ij  p

 Vc  Uc  Vc  Uc  2k ij

ij

ij

ij

2

0 , where Uijc





p Uijc Hij  , called

the back stress, is a function of the plastic strain. A simple form would be Uijc

p hH' ij  , which would result (for a straight strain path) in linear harden-

 p ing. This hardening relationship could also be written U cij hH ijc , so that the translation of the yield surface is in the same direction as the direction of

28

2 Classical Elasticity and Plasticity

the plastic strain. This type of hardening is often referred to as Prager’s trans ij P Vcij  Ucij , where P is a lation rule. Alternatively, one could write Uc scalar multiplier, so that the translation of the yield surface is in the same direction as Vcij  Ucij . This is often known as Ziegler’s translation rule. For









the special case of the von Mises type of yield surface with associated flow, the two translation rules are identical: U cij

hH cij

hO

wg wVcij





2hO Vcij  Ucij .

2.3.5 Discussion of Hardening Laws It is impossible to distinguish between kinematic and isotropic hardening if one considers only the plastic behaviour of materials during initial monotonic loading, as illustrated by curves OAB in Figures 2.3a and 2.4a. It is during unloading and subsequent reloading that the differences between the theories are exhibited. During isotropic hardening, the elastic region changes its size but not shape, whereas during pure kinematic hardening, it translates but does not change size (Figure 2.3b and 2.4b). Some materials exhibit one type of behaviour whilst others exhibit the other. For instance, the very successful “critical state” family of models for the behaviour of soft clays uses isotropic hardening. In these models, hardening is linked to volumetric strain rather than shear strain, and this concept proves vitally important in modelling geotechnical materials. On the other hand, an important extension of kinematic hardening (pursued in detail in Chapter 7) allows the modelling of “Masing” type of hysteretic behaviour, and this proves realistic for many cyclic loading applications. Of course, mixed forms of hardening, involving both expansion and translation of the yield surface are also employed, for instance, in modelling the behaviour of soils under complex loading paths. Further discussion of different types of hardening in some more recent developments of plasticity theory is given in Chapter 6. Finally, there is the possibility that the yield surface changes shape as well as its size or location. Some advanced theories for soils employ yield surfaces which change shape; see, for example, Whittle (1993).

2.4 Frictional Plasticity We shall return later in Chapter 10 to the more realistic modelling of frictional materials, but it is useful at this stage to introduce some of the simple concepts of fricional plasticity, as this is the most important application in which nonassociated flow occurs. We consider a simple conceptual model in which a material is subjected to a normal stress V and a shear stress W , with corresponding strains H and J . If the material had a cohesive strength c (independent of the normal stress) and exhibited associated flow, we could write the yield surface

2.4 Frictional Plasticity

29

and plastic potential as f g W  c 0 . It is straightforward to show that (since wg wV 0 ) this model involves plastic flow at constant volume. Note that in the following discussion of friction, we maintain our usual convention of tensile positive. Those readers more familiar with the compressive positive terminology used in frictional problems in soil mechanics will need to take special care. The yield surface for a perfectly plastic frictional can be defined as W P*V 0

f

(2.41)

where P is the apparent coefficient of internal friction. We note that for this frictional material, the stress V will always be negative (i. e. compressive). Within the yield surface, the stress-strain behaviour is elastic and could be written ı K

İ

Ȗ

IJ G

(2.42)

During plastic flow, the stress-strain behaviour is governed by the flow rule: p İ 

Ȝ

wg wı

p Ȗ 

Ȝ

wg wIJ

(2.43)

where g is the plastic potential, which we write in the form,

g

W  EV  x

0

(2.44)

where E is a coefficient which we discuss below, and x is a constant chosen to ensure that the plastic potential surface always passes through the current stress state on the yield surface. Clearly, if E = P*, then the yield surface and plastic potential are identical and the flow rule (2.43) becomes associated. In this case, it follows from (2.43) that p p İ  ȜE Ȝµ* and Ȗ  ȜS  W [See the notation section for the definition of the signum function S(x)]. The rate of plastic work is then determined as p p p W  VH   WJ  OP * V  OWS  W O  P * V  W , and by substituting the expression for the yield surface, we obtain W  p 0 . Thus a “frictional” material with associated flow is not frictional at all. It dissipates no work plastically! Now consider plastic strains in more detail. It is simple to show from the flow p p rule that İ  Ȗ  ȕS IJ . Since O is a positive multiplier, it also follows that  p Ȗ  p ȕS Ȗ  p or W and J  p have the same sign, so that we can write İ



p p H  E J  . For a positive value of E, the volumetric plastic strain is always

positive and the material dilates. The apparent “frictional” strength in an associated material is entirely due to this dilation. More realistically, E  P * and

30

2 Classical Elasticity and Plasticity

Figure 2.5. Frictional yield surface

p W  O  EV  W OV  P * E , which is positive because V  0 , so that plastic work is dissipated in this case. The case E ! P * would generally be disallowed on “thermodynamic” grounds as this would involve negative plastic work. It is common to identify plastic work with thermodynamic dissipation. We shall see later in this book that this is in general an oversimplification, but in this instance, the identification of plastic work with dissipation is correct. p For E 0 , H  0 , and the material deforms at constant volume. The yield surface and flow vectors for the general case are shown in Figure 2.5.

2.5 Restrictions on Plasticity Theories In the discussion of frictional materials, we have just encountered the fact that certain restrictions may be placed on parameters on “thermodynamic” grounds. Two important restrictions on plasticity theories have been applied by many

2.5 Restrictions on Plasticity Theories

31

users in the past, and these are discussed below. Both bear a superficial similarity to thermodynamic laws, and both lead to normality relationships, but neither embodies any thermodynamic principles.

2.5.1 Drucker's Stability Postulate Drucker (1951) proposed a “stability postulate” for plastically deforming materials. Although not a thermodynamic statement, it bears a passing resemblance to the Second Law of Thermodynamics and is therefore referred to as a “quasithermodynamic” postulate for classifying materials. It can be stated in a variety of equivalent ways, but represents the idea that, if a material is in a given state of stress and some “external agency” applies additional stresses, then “The work done by the external agency on the displacements it produces must be positive or zero” (Drucker, 1959). If the external agency applies a stress increment GVij that causes additional strains GHij , then the postulate is that GVij GHij t 0 . The product GVij GHij is often called the second order work. In the one-dimensional case shown in Figure 2.6a, the postulate states that the area ABC must be positive. Strain-softening behaviour is thus excluded. In the one-dimensional case, a strain-softening material is mechanically unstable under stress control, and this is linked to the identification of the postulate as a “stability postulate”. Unfortunately, this has led to the interpretation that a material which does not obey the postulate will exhibit mechanically unstable behaviour. The obvious corollary is that a material which is mechanically stable must therefore obey the postulate. The identification of the postulate with mechanical stability for the multidimensional case is, however, erroneous. The conclusion that mechanically stable materials must obey Drucker’s postulate is therefore equally erroneous.

Figure 2.6. One-dimensional illustrations of (a,b) Drucker’s postulate and (c) Il’iushin’s postulate

32

2 Classical Elasticity and Plasticity

If the external agency first applies then removes the stress increment GVij , p such that the additional strain remaining after this stress cycle is GHij , then it p also follows from the postulate that GVij GHij t 0 . Thus in the one-dimensional case shown in Figure 2.6b, the area ABD must be zero or positive. We do not elaborate the proof here, but it can be shown that Drucker’s postulate leads to the requirement that the flow is associated for a conventional plasticity model (i. e., that the yield surface and plastic potential are identical). Furthermore, it follows that the yield surface for a multidimensional model must be convex (or at least non-concave) in stress space. Strictly these results apply to uncoupled materials, in which the elastic properties do not depend on plastic strains. For coupled materials in which the elastic properties are changed by plastic straining, the results are modified, but for realistic levels of coupling the effects are rather minor. Many advantages follow from the use of associated flow. For instance, for perfectly plastic materials with associated flow, it is possible to prove that (a) a unique collapse load exists for any problem of proportionate loading and (b) this collapse load can be bracketed by the Lower Bound Theorem and Upper Bound Theorem. In numerical analysis, associated flow guarantees that the material stiffness matrix is symmetrical, which has important benefits for the efficiency and stability of numerical algorithms. Furthermore, for many materials (notably metals), associated flow is an excellent approximation to the observed behaviour. For all these reasons, it is understandable therefore why many practitioners are reluctant to adopt models that depart from associated flow. Frictional materials (soils), however, undoubtedly exhibit behaviour which can only be described with any accuracy with non-associated flow. If a purely frictional material with a constant angle of friction were to exhibit associated flow, then it would dissipate no energy, which is clearly at variance with common sense. With some reluctance, therefore, we must seek more general theories, which can accommodate non-associated behaviour.

2.5.2 Il'iushin's Postulate of Plasticity The “postulate of plasticity” proposed by Il’iushin (1961) is similar to Drucker’s postulate, but significantly it uses a cycle of strain rather than a cycle of stress. It is simply stated as follows. Consider a cycle of strain which, to avoid complications from thermal strains, takes place at constant temperature. It is assumed that the material is in equilibrium throughout, and that the strain (for a sufficiently small region under consideration) is homogeneous. The material is said to be plastic if, during the cycle, the total work done is positive, and is said to be elastic if the work done is zero. The postulate excludes the possibility that the work done might be negative. This is illustrated for the one-dimensional case in Figure 2.6c, where the postulate states that the area ABE must be non-negative.

2.5 Restrictions on Plasticity Theories

33

The postulate has certain advantages over Drucker’s statement because it uses a strain cycle. Drucker’s statement depends on consideration of a cycle of stress, which is not attainable in certain cases such as strain softening. On the other hand, almost all materials can always be subjected to a strain cycle. The exceptions are rather unusual materials which exhibit “locking” behaviour (in the one-dimensional case this involves a response in which an increase in stress results in a decrease in strain). It may be in any case that such materials are no more than conceptual oddities, and we have never encountered them. A more significant limitation is Il’iushin’s assumption that the strain is homogeneous. It may well be that for some cases (e. g. strain-softening behaviour), homogeneous strain is not possible, and bifurcation must occur. Il’iushin’s postulate seems even more like a thermodynamic statement (and specifically a restatement of the Second Law) than Drucker’s postulate, but again it is not. A cycle of strain is not a true cycle in the thermodynamic sense because the material is not necessarily returned to identically the same state at the end of the cycle. The specific recognition that a cycle of strain would result in a change of stress is an acknowledgment that the state of the material changes. In later chapters, it will be seen that one interpretation of this is that a cycle of strain may involve changes in the internal variables. Il’iushin’s postulate is therefore no more than a classifying postulate. Even though it holds intuitive appeal – that a deformation cycle should involve positive or zero work – it is possible to find materials (both real and conceptual) that violate the postulate. Such materials would release energy during a cycle of strain, and in so doing would change their state. Il’iushin showed that his postulate also leads to the requirement that, in a conventionally expressed plasticity theory, the plastic strain increment vector should be normal to the yield surface; in other words, the yield surface and plastic potential are identical. Since many materials, notably soils, violate this condition, we must conclude on experimental grounds that Il’iushin’s postulate is overrestrictive, and in later chapters, we seek a broader, less restrictive framework.

Chapter 3

Thermodynamics

3.1 Classical Thermodynamics 3.1.1 Introduction In the following, we establish the thermodynamic terminology we shall use, keeping as close as possible to conventional usage in the thermodynamics of fluids. We shall not attempt here a comprehensive introduction to thermodynamics. Numerous texts deal with this thoroughly. We shall assume therefore a certain familiarity with thermodynamic principles and provide simply a reminder of some important points. It is important to note that our objective is not to provide a rigorously established generalization of thermodynamics as a field theory. For reasons discussed in Chapter 4, this is an area which is fraught with difficulties. Instead, our more limited objective in later chapters is to set out a formalism for plasticity theory that is consistent with accepted thermodynamic principles. We shall therefore define materials which are a subset of all those that could be described within a rigorously defined thermodynamic approach. It is for the reader to decide whether this subset is sufficiently wide that it describes materials of practical importance. First we establish some basic definitions. A thermodynamic closed system (for brevity just system) is a body of material separated from its surroundings by certain walls. The state of the system is characterized by a certain number of state variables. A complete definition of the system will also require knowledge of certain constants (for instance, the mass of material within the system), but these will be of less direct concern to us here. For instance, if the system were to consist of a certain mass of a “perfect gas,” then the chosen state variables could be the volume of the gas and the temperature T. A proper choice of state variables is such that they are both necessary and sufficient to describe the current state of the system at the level of accuracy that a particular application demands.

36

3 Thermodynamics

Any quantity that can be uniquely determined as a function of the state variables is called a property. In the example above, for instance, the pressure p is a property of the system because pv RT for a perfect gas, where v is the specific volume (volume per unit mass) and R the gas constant. Such a relation is called an equation of state. The existence of such relationships means that the roles of state variables and properties can be interchanged. For instance, pressure and temperature could be considered state variables, and the specific volume would be determined as a property. Such an interchange of variables will be an important theme in Chapter 4 and afterwards in this book. The concepts of state variables and properties are rigorously defined only for systems that are in thermodynamic equilibrium. If this restriction were to be enforced strictly, however, classical thermodynamics could be applied only to processes that are infinitesimally slow. In practice, the concepts of classical thermodynamics can be applied successfully to rapidly changing systems, and we allow this possibility here. There is of course an important discipline of nonequilibrium thermodynamics, but there are a number of different approaches to studying it, and a discussion is beyond the scope of this book. In classical thermodynamics, the “Zeroth”, First, Second and Third “Laws” are defined. As for any “Laws” of nature, they are empirically based and are therefore unprovable: they could be falsified by counterexample but cannot be proven. However, they fit into a sufficiently logical framework that they are almost universally accepted as “true” by the scientific community. The Zeroth Law states that two bodies that are each in thermal equilibrium with a third body are also in thermal equilibrium with each other. It provides a rigorous basis for the definition of the temperature T as a property of a body that is internally in thermal equilibrium. (Throughout this book, we use T for the thermodynamic, or absolute, temperature, which is always positive.) We shall return to the First and Second Laws below. They establish the existence of two further properties of a body in thermodynamic equilibrium: the internal energy and the entropy. The Third Law then requires that entropy is zero at zero temperature. Whilst important in other contexts, it does not enter our discussions here.

3.1.2 The First Law We consider a closed system isolated from its surroundings by certain walls. A process involves interaction between the system and its surroundings and can involve transfer of two types of energy: heat flow Q into the system from the surroundings and mechanical power input W , also from the surroundings. The First Law is usually stated in the following form: for a system in thermodynamic equilibrium, there is a property of the system, called internal energy U, such that

Q  W U

(3.1)

3.1 Classical Thermodynamics

37

Note very importantly, however, that Q and W are not each separately integrable with time to give properties Q and W, and in fact no such properties exist. The above equation is a somewhat simplified form of the First Law, in that it does not expressly account for energy input from, for instance, a gravitational field or from the flow of electrical current. Also it ignores the fact that some of the input power may, in general, cause an increase in kinetic and potential energies, which are conventionally separated from internal energy. However, Equation (3.1) expresses the essential principle of conservation of energy: the sum of all the sources of power input to a body is equal to the rate of increase of the energy of the body. Furthermore, the most important sources of power we must consider are mechanical power and heat supply. There is, however, one term that is sometimes mistakenly included in Equation (3.1) but which we explicitly exclude, and that is a source of heat from within the body itself. In a number of texts, it will be found that an additional term of this sort is added, but the introduction of such a term, allowing energy to be magically conjured up inside the body, is nothing more nor less than a complete denial of the validity of the First Law! Why then do some authors resort to this approach? The explanation is that they find it necessary when they attempt to define the properties of a system in terms of an inadequate set of state parameters. An example offers the simplest explanation. Suppose that the body in question consists of a radioactive metal, surrounded by conducting walls. It will be observed that a significant amount of heat leaves the body, whilst the body itself appears to suffer no change in state. It is tempting to attribute this observation to a “heat source” somewhere within the body. However, a proper description of the state of the body requires the amounts of the different isotopes present to be known, and each of these will have a different internal energy. As one isotope decays to another, the internal energy of the body decreases, and it is this decrease that is reflected in the outward flow of heat. If the internal composition in terms of isotopes is (mistakenly) ignored, then it becomes necessary to include the mysterious internal heat source in Equation (3.1). It is often convenient to consider a system that is unchanged by a certain process. This simply means that all state variables (and therefore all properties) of the system are the same after completion of the process as they were at the beginning. They may of course have taken some different values at some point during the process. Consider the process shown in Figure 3.1a, in which an unchanged system receives an amount of work W X from the surroundings and also receives some amount of heat Q . Since the system is unchanged, U 0 , and a trivial application of the first Law shows that Q  X , so that the system must reject to the surroundings exactly as much energy as it received in work. Many simple devices operate in the way shown in Figure 3.1a. For instance, a frictional brake (once it has reached a steady temperature) converts work input to heat output. The pure conversion of work to heat is called dissipation.

38

3 Thermodynamics

W

 Q

X

Q

X

X

 W

X

Figure 3.1. Possible and impossible processes

3.1.3 The Second Law In the following, it will be necessary to consider a number of processes in which systems interact with their surroundings. The surroundings themselves will have certain properties, and in particular the temperature of the surroundings will be important. A part of the surroundings which is sufficiently large that its properties can be regarded as unchanged by any interactions with the system is said to be a reservoir. The Second Law is considerably more subtle than the First and can be expressed in a number of equivalent ways. It applies certain restrictions to the processes that can occur. For instance, one of the basic consequences of the Second Law is that work can be dissipated in the form of heat, but that heat cannot be changed into work without some side effects occurring too. Thus the process in Figure 3.1a (in which X is a positive quantity) is physically possible, whilst that in Figure 3.1b is not, even though it too obeys the First Law. We shall return to this example shortly. There are a number of ways by which the Law can be expressed, but the most useful is in the form of the Clausius-Duhem inequality. To express the First Law, we had to introduce the concept of a property called the internal energy U. The Second Law is also best expressed by making the hypothesis that there is a further property, called entropy S. Most people find entropy a more abstract concept than internal energy, and perhaps the most useful approach is initially to treat it simply as a mathematical abstraction without seeking a physical meaning. The Clausius-Duhem inequality states that, for a system that exchanges heat with n reservoirs at temperatures Ti , the change of entropy is such that n  Q S t ¦ i T i 1 i

(3.2)

where Q i is the rate of heat input from reservoir i. It can readily be seen that inequality (3.2) does not permit the process in Figure 3.1b. For the unchanged system, S 0 , so that inequality (3.2) requires that Q d 0 for a system exchanging heat with only a single reservoir (since the temperature must be positive). A consequence of the Second Law is that heat cannot spontaneously flow from a colder place to a hotter one. Consider the process shown in Figure 3.2 in which an unchanged system exchanges heat with two reservoirs at different temperatures. The First Law clearly requires that Q1  Q 2 0 . We shall assume

3.1 Classical Thermodynamics

Q1

39

Q 2

Figure 3.2. An unchanged system exchanging heat with two reservoirs

that Q1 is positive and Q 2 negative. Since S 0 for the unchanged system, the Second Law requires 0t

Q1 Q 2  T1 T2

(3.3)

Q1 Q1 , then using the fact that T1 , T2 and Q1 are  T1 T2 all positive gives T1 t T2 , so that in a process of pure heat transfer, heat can only flow from a hotter place to a colder one. Now consider the process shown in Figure 3.3, in which an unchanged system (a heat engine) receives heat Q1 from one reservoir at T1 and rejects a fraction of this (so that Q 2 is negative and Q 2  Q1 ) to another reservoir at T2  T1 . The process produces a power output W , which (since U 0 for the unchanged system) is determined by the First Law as W Q1  Q 2 ! 0 . Now we consider the maximum possible thermodynamic efficiency of the system, which is the ratio of useful output work to the input heat flow:

Rearranging this as 0 t

W Q1

Q1  Q 2 Q1

Q 1 2 Q1

(3.4)

Since S 0 for the unchanged system, from the Second Law, 0t

Q1 Q 2  T1 T2

(3.5)

which can be rearranged as Q 2 T2 d Q1 T1

Q1

(3.6)

Q 2

W Figure 3.3. A heat engine exchanging heat with two reservoirs

40

3 Thermodynamics

(using the fact that both Q1 and T2 are positive). Substituting this in the expression for the efficiency, we obtain: T W (3.7) d1 2  Q1 T1 so that the maximum possible efficiency of such a process is 1  T2 T1 , and it can be shown that this can be attained only by an ideal system in which there is no dissipation (in which case, the equality holds in (3.2)). We can see that 100% efficiency is approached only as T2 o 0 . So to achieve maximum efficiency from a heat engine, one needs to have available a reservoir at near absolute zero temperature! Countless other examples of the implications of the Second Law can be found in thermodynamics textbooks, but these examples serve our purpose here to demonstrate the role of entropy and the important qualitative difference that the Second Law introduces between mechanical work and heat supply. An important concept in thermodynamics is that of reversibility. A process is reversible if all the directions of the heat and work flows into or out of the system can be reversed simultaneously and the resulting process still obeys the Second Law. It is straightforward to show that reversibility is possible only when equality rather than inequality holds for the Clausius-Duhem relationship for the process. So for all reversible processes, S

n

Q

¦ Tii

(3.8)

i 1

In practice, almost all the materials we shall encounter in this book do not exhibit purely reversible behaviour, in that they are dissipative. We shall address this in much more detail below, but first we explore the behaviour of simple reversible materials to gain some familiarity with thermodynamic functions.

3.2 Thermodynamics of Fluids The classical thermodynamics of fluids uses the intensive quantities, pressure p and temperature T, which are properties that do not depend on the amount of fluid in the system. There are also extensive quantities, which are quantities that (for given values of the intensive quantities) take values directly proportional to the mass of the system. The volume is the most obvious example, but internal energy and entropy are also extensive. It is convenient to normalize all extensive quantities to obtain specific values of them, i. e. values per unit mass. We have already introduced the specific volume v (volume per unit mass). The specific internal energy is u and the specific entropy is s. By convention, lowercase letters are used for specific quantities.

3.2 Thermodynamics of Fluids

41

Consider again a simple material that can undergo reversible processes. In practice, this means that the process must be sufficiently slow that the sample can always be considered in a state of equilibrium, and that the sample does not dissipate energy. (There will be much further discussion of dissipation in the remainder of this book). A perfect gas, which is discussed further below, would be an example of such a material. We write the First and Second Laws for a volume element of the material in the form, qi ,i  p

v v

u v

(3.9)

s § qi · t¨ ¸ v © T ¹ ,i

(3.10)

where we note that if qi ist the heat flux per unit area, the rate of heat supply per unit volume is qi,i , and the work input per unit volume is  pv v . The terms in u and s are divided by v to convert them to a per-unit-volume basis. Now consider the reversal of the process. We make two observations. Firstly the sign of s and qi in the first part of inequality (3.10) will change, so that we can deduce that the equality must hold: s v

§ qi ¨ © T

· ¸ ¹,i

(3.11)

Secondly, we expand the right-hand side of this equation as § qi · ¨ T ¸ © ¹,i

qi ,i T



qi T,i T2

(3.12)

and make the empirical observation that the direction of heat flow is always in the direction opposite to the temperature gradient, so that the sign of T,i will change at the same time as qi , and the second term on the right-hand side of (3.12) does not change sign as the process is reversed. The only way (3.12) can be true for both forward and reverse processes is therefore, in the limit as the temperature gradients become sufficiently small, that this term becomes negligible. It follows that for a reversible process, s v

qi ,i

(3.13)

T

Now we can eliminate the divergence of the heat flux between Equations (3.9) and (3.13) to obtain for our “reversible” material,

u Ts  pv

(3.14)

Since the directions of temperature gradient and heat flux are opposed, the term qi T,i T is always positive and is called the thermal dissipation.

42

3 Thermodynamics

3.2.1 Energy Functions Bearing in mind that the First Law states that u is a property of a material, it can be expressed as a function of an appropriate choice of state variables. Equation (3.14) suggests that just such a choice would be v and s, and we choose these as our independent variables, writing u u  v , s . It follows that wu wu v  s wv ws Comparing with Equation (3.14), we obtain u

(3.15)

wu · § wu · § (3.16) ¨ p  ¸ v  ¨ T  ¸ s 0 wv ¹ © ws ¹ © Now if v and s are an appropriate choice of state variables, they can be varied independently, from which we immediately deduce the classical results, wu (3.17) p  wv wu (3.18) T ws This demonstrates the vitally important role of the internal energy u; it serves as a potential from which (with knowledge of the independent quantities v and s) one can determine both the dependent variables p and T. Thus from a single function u u  v, s , we deduce (by the thermodynamic principles described above) two relationships p p  v , s and T T  v , s . This illustrates two advantages of the thermodynamic approach that are central to the theme of the remainder of this book. The thermodynamic approach is mathematically concise and efficient. In this case, we specify only one function and from that deduce all other aspects of the constitutive behaviour (in fact two further functions). On the other hand, if we had arbitrarily specified p p  v , s and T T  v, s as our starting point, it would be quite possible (indeed likely) that we would choose functions that were not consistent with derivation from an energy potential, and so these functions would violate thermodynamic principles. Three other energy functions may be defined; the specific Helmholtz free energy f, specific enthalpy h, and specific Gibbs free energy g. The choice of which of the four energies are used depends on which variables are used as independent state parameters. The choices are between pressure and specific volume and between temperature and entropy. The four energies are related through a series of Legendre transforms, shown in Table 3.1. The Legendre transform plays an extremely important role in this book and is discussed extensively in Appendix C. Readers not familiar with this transformation should study Appendix C Sections 3.1–3.5 before proceeding further. Callen (1960) also provides a useful discussion of the Legendre transform in this context.

3.2 Thermodynamics of Fluids

43

Each of the different forms of energy function are most convenient for different types of problem. For instance, for isothermal (constant temperature) problems, the Helmholtz or Gibbs free energies are most convenient because they employ temperature as a state variable. In contrast, internal energy or enthalpy are more convenient for isentropic (constant entropy) problems. The latter are important because entropy changes are solely related to heat transfer for a reversible process. An adiabatic process is one in which no heat transfer occurs (the system is thermally insulated). Reversible adiabatic processes are isentropic. The starting point is internal energy, and when expressed as a function of the extensive parameters v and s, the intensive parameters p and T are obtained by the differentials shown in the first column of Table 3.1. The Helmholtz free energy is a Legendre transform of the internal energy, in which the roles of s and T are interchanged. The enthalpy and the Gibbs free energy are obtained by further Legendre transformations. The well-known Maxwell’s relations arise from further partial differentiation of the expressions in the last row of Table 3.1. Table 3.1. Energy definitions for classical thermodynamics of fluids

Internal energy

u u  v, s

wu wv wu ws

Helmholtz free energy

f f

f  v, T u  sT

Enthalpy

h h  p, s h u  pv

wf wv wf  wT

p 

p 

v

T

s

T

wh wp wh ws

Gibbs free energy

g  p, T g h  sT f  pv wg v wp wg s  wT

3.2.2 An Example of an Internal Energy Function To understand the role of internal energy further, we briefly explore the implications of specific terms in the internal energy function. Consider an internal energy that is a simple second-order function of specific volume and entropy:

u u0  u1v  u2 s  u3v 2  u4 vs  u5 s 2

(3.19)

The pressure and temperature are obtained by applying Equations (3.17) and (3.18) to give wu u1  2u3v  u4 s wv wu u2  u4 v  2u5 s ws

p 

(3.20)

T

(3.21)

44

3 Thermodynamics

Further differentiation then gives the incremental relationships which can be expressed conveniently in matrix form: ª dp º ª 2u3 u4 º ªdv º (3.22) »« » « » « ¬ dT ¼ ¬ u4 2u5 ¼ ¬ ds ¼ Now we consider the roles of the different terms in the internal energy expression. The constant term u0 disappears on differentiation and plays no part in the relationships between the properties p, v, T and s. It simply determines the reference point for internal energy. The linear terms u1v and u2 s disappear on the second differentiation, and play no part in the incremental relationships between p, v, T and s. They simply determine reference points for these properties. The quadratic term u3v 2 determines the incremental relationship between pressure and specific volume, that is to say the stiffness of the material. Specifically, the isentropic (adiabatic) bulk modulus is 2u3v . Any higher order terms in v would relate to more complex stiffness relationships. The quadratic term u5 s 2 determines the incremental relationship between temperature and entropy. Entropy changes are in turn related to heat flow to or from the material, so this term defines the specific heat of the material. In particular, the specific heat at constant volume is 1  2u5 T . Any higher order terms in s would define more complex specific heat relationships. The cross term u4 vs introduces a coupling between volume and temperature (and symmetrically between pressure and entropy). Therefore it defines the thermal expansion of the material. We shall not pursue this simple model further here, but simply use it to illustrate the fact that each term in the internal energy expression is related to a particular aspect of the physical behaviour of the material. Once a familiarity is established with the aspects of behaviour that are related to particular terms in the energy expression, then the process can be reversed. If it is wished to specify a material with certain chosen properties, it is possible to deduce the expected form of the energy expression. Sometimes, on differentiation of the energy, it will be found that unexpected phenomena will be predicted. For instance, we observe that any coupling between volume and temperature must be accompanied by a similar coupling between pressure and entropy. Not only are such relationships required theoretically, but they are also confirmed by experiments.

3.2.3 Perfect Gases Since one of the simplest of all materials to be described by thermodynamic functions is a perfect gas, it is useful to show how this is described by the thermodynamic functions described above. This section can, however, be omitted at a first reading, and the reader can proceed directly to Section 3.3.

3.2 Thermodynamics of Fluids

45

Firstly, perfect gases are non-dissipative, so the behaviour of the gas is specified entirely by knowledge of any one of the four energy functions u, f, g or h. To define the behaviour of a gas, it is necessary to specify some reference values for the units. The most convenient way to do this is to take a reference temperature To and a reference specific volume vo . The properties of the gas are defined by two material constants, which we will take as R (the gas constant) and cv (which we shall demonstrate is the specific heat at constant volume). Both constants have the dimension of specific entropy. It is convenient to define the following derived quantities: a reference pressure po RTo vo , a quantity that we shall show is the specific heat at constant pressure c p R  cv , and the ratio of the specific heats J c p cv . We consider here the simplest form of perfect gas, which has constant values of specific heats. The internal energy of a perfect gas is §v · u  v , s cv To ¨ o ¸ © v ¹

R cv

§ s · exp ¨ ¸ © cv ¹

from which, using the standard results, we immediately derive R c c § s · wu RTo § vo · v v p  exp ¨ ¸ ¨ ¸ wv vo © v ¹ © cv ¹ J § s · §v · po ¨ o ¸ exp ¨ ¸ © v ¹ © cv ¹

T

wu ws

§v · To ¨ o ¸ © v ¹

§v · To ¨ o ¸ © v ¹

J1

R cv

§ s · exp ¨ ¸ © cv ¹

(3.23)

(3.24)

(3.25)

§ s · exp ¨ ¸ © cv ¹

Dividing the two above equations to eliminate the entropy, we obtain p po vo , which can be rearranged to give the famous equation of state for T To v a perfect gas pv RT . The equations can also be manipulated to eliminate in turn each of the other variables p, v, and T to give alternative equations of state, which we express as J 1J § p · § T · § s · (3.26) ¨ ¸ ¨ ¸ exp ¨ ¸ © po ¹ © To ¹ © cv ¹  J1 § s · T§ v · (3.27) exp ¨ ¸ ¨ ¸ To © vo ¹ © cv ¹ p§ v · ¨ ¸ po © vo ¹

J

§ s · exp ¨ ¸ © cv ¹

(3.28)

46

3 Thermodynamics

Carrying out the appropriate Legendre transforms, it is straightforward to derive the following alternative energy expressions: h  p, s u  pv f  v , T u  Ts g  p, T h  Ts

ªp § s J pvo « o exp ¨ J 1 «¬ p © cv

ª § T «1  log ¨ «¬ © To

1J

·º ¸» ¹ »¼

(3.29)

· § v ·º ¸   J  1 log ¨ ¸ » cv T ¹ © vo ¹ »¼

ª § T « J  J log ¨ © To ¬«

· § p ¸   J  1 log ¨ ¹ © po

·º ¸ » cv T ¹ »¼

(3.30)

(3.31)

To demonstrate that cv and c p are specific heats, it is most straightforward first to recast the expressions for the internal energy and enthalpy in terms of temperature rather than entropy. Simple manipulation shows that both are linear functions of temperature: u uT  v, T cv T , and h hT  p, T c p T . For an infinitesimal process of pure heating at constant volume, the First Law gives dq  pdv dq du where dq is the heat supplied, so that the specific heat at dq wuT cv . Similarly, for an infinitesimal procconstant volume is simply dT wT v ess at constant pressure, the First Law gives dq  pdv du dh  pdv  vdp dq whT cp . dh  pdv , so that dq dh and the specific heat in this case is dT wT p To allow comparison with some materials that we shall encounter later, it is worth exploring some of the other properties of a perfect gas. Using the logarithmic strain H log  v vo , the (bulk) coefficient of thermal expansion is wH 1 wv D . The equation of state leads immediately to D 1 T . Similarly, wT p v wT p dp dp the isothermal bulk stiffness can be derived as K T  v p , and the dH T dv T dp dp isentropic (adiabatic) bulk stiffness as K s  v Jp . Therefore, dH s dv s perfect gases have the property that their stiffness is proportional to the pressure. Note for completeness that the above equations contravene the Third Law because they imply that as T o 0 , s o f , rather than s o 0 . This problem cannot be circumvented simply by shifting the reference point for entropy (e. g. by replacing each occurrence of exp  s cv by exp   s  so cv ). We conclude therefore that the above relations are applicable only within a certain range of temperatures for which the ideal gas idealisation is reasonable, and that at a sufficiently low temperature, modification would be required to make the relationships consistent with the Third Law.

3.3 Thermomechanics of Continua

47

3.3 Thermomechanics of Continua 3.3.1 Terminology In the following, we use the terminology of classical thermodynamics as far as possible, but some minor changes are convenient. We now highlight the areas where our notation departs from the terminology used in the thermodynamics of fluids, as described above. In applying thermodynamics to solids undergoing small strains, it is necessary to replace the role of the pressure by the stress tensor Vij and the specific volume by the small strain tensor Hij . Using the conventional tensile positive convention, p  13 Vkk and v vo 1  Hkk , where vo is the initial specific volume. In a direct mapping of the conventional notation to that for a solid, p would therefore be replaced by Vij and v by vo 13 Gij  Hij . Thus pv would be replaced by



Vij vo 13 Gij

 Hij





vo 13 Vkk

  Vij Hij



pvo  vo Vij Hij .



rather u  Hij , s )

It is clearly more straightforward to express u u Hij , s

than

§ §1 · · u u u ¨ vo ¨ Gij  Hij ¸ , s ¸ , and it follows (using that 3 © ¹ ¹ © Vij Uo wu wHij , where Uo 1 vo is the initial density. This then suggests the





transforms h u  vo Vij Hij and g h  sT u  vo Vij Hij  sT f  vo Vij Hij . The values of h and g are therefore slightly different from those in the classical definition because  pv z vo Vij Hij . The value of f u  sT is not, however, changed from that used in the classical approach. The differences are mentioned here purely for clarification, and have no material affect on the subsequent derivations, because these derivations all involve differentials of the energy functions. The other three energies may be expressed in the form f f Hij , T ,













h h Vij , s , and g Vij , T , and the following relationships are then readily

obtained: Vij

wf wHij

Uo

wu wHij

Uo

Uo

wh wVij

Uo

and Hij

wg wVij

In small strain analysis, the factor Uo simply appears as a multiplier throughout the analysis. If the specific extensive quantities are all converted to a perunit-volume basis, rather than per-unit-mass, then this factor disappears, considerably simplifying the notation. Note that u, f, h, g, and s now have slightly different meanings from those in the classical formulation, but their roles are

48

3 Thermodynamics

Table 3.2. Energy definitions for use in small strain continuum mechanics

Internal energy



u u Hij , s

Vij T

wu wHij wu ws



Helmholtz free energy



f

f Hij , T

f

u  sT

Vij s



Enthalpy



Gibbs free energy





h h Vij , s h u  Vij Hij

wf wHij wf  wT

Hij



T

wh ws



g Vij , T g h  sT f  Vij Hij wg Hij  wVij wg s  wT

wh wVij

analogous. For clarity, the definitions equivalent to those in Table 3.1 for classical thermodynamics of fluids are given in Table 3.2 for small strain continuum mechanics.

3.3.2 Thermoelasticity Before moving on to the more complex behaviour of dissipative materials, it is worth describing non-dissipative (thermoelastic) materials. Fung (1965) provides an excellent classification of the different forms of elastic model, which has been covered in Section 2.1. Within this classification, the formulation used here requires all materials to be hyperelastic because this is the only means by which the material is guaranteed to obey the First Law of Thermodynamics. This section can be omitted on first reading, and the reader can proceed to Section 3.3.3. The behaviour of a hyperelastic material is defined entirely by a Helmholtz free energy function f f Hij , T . The stresses are given by Vij wf wHij (see





Table 3.2), and the incremental response is obtained by further differentiation: w2 f w2 f  H kl  T wHij wHkl wHij wT

V ij

(3.32)

in which the term w 2 f wHij wHkl is the isothermal stiffness matrix. Alternatively, one can start from the Gibbs free energy (often in this context referred to as the (negative) complementary energy) g g Vij , T . This is the Legendre transformation of f in which the roles of the stress and strain are interchanged (see Table 3.2). It follows that Hij wg wVij and



H ij



w2 g w2 g  V kl  T wVij wVkl wVij wT



(3.33)

3.3 Thermomechanics of Continua

49

w 2 g w 2 g is the isothermal compliance matrix and the matrix wVij wVkl wVij wT of coefficients of thermal expansion. Note that g and f cannot be separately and arbitrarily defined because they are functionally related by the Legendre transformation (Appendix C). Once one is specified, the other can be found, although for certain choices of the functions, it may not be possible to express one or the other in terms of conventional mathematical expressions. Further transformations of internal energy or enthalpy are useful if adiabatic conditions are to be considered. The complete set of the energy functions for isotropic thermoelasticity is presented in Section 7.1. Hill (1981) explores many of the relationships between the different moduli which can be derived from the above approach; in particular, he examines the relationships between isothermal and adiabatic moduli.

The term

3.3.3 Internal Variables and Dissipation Most materials which will be of concern to us in this book do not in general undergo reversible processes, but show irreversible or dissipative behaviour. A feature of this type of behaviour is that the response of the material depends not only on the current values of the state variables that we have already introduced, but also on the history of how the material arrived at that state. There are broadly two approaches to dealing with this problem: x “Rational mechanics” in which the response is made not simply a function of the current state but a functional of the whole history of state. Thermodynamic principles are then applied to ensure that the equations for the evolution of the dependent variables are consistent with the Laws of Thermodynamics. x “Generalised thermodynamics” in which the history of the state is encapsulated within certain internal variables. A full description of the state then requires both the variables we have already introduced and the internal variables. This approach is also called “thermodynamics with internal variables” (TIV). This is the approach that we shall adopt here.

For convenience, we shall introduce an internal variable Dij which is tensorial in form and kinematic, that is to say strain-like in nature. Although it is not strictly necessary, the internal variable will often be found to play the same role  p as the plastic strain Hij . It may be convenient to think of it in these terms, but we use the Dij notation to emphasize the generality. The internal energy now





takes the form u u Hij , Dij , s , and so we write u

wu wu wu H ij  D ij  s wHij wDij ws

(3.34)

50

3 Thermodynamics

Now consider the Second Law, which we rewrite in the form, §q s  ¨ i ©T

· ¸ ¹,i

dt t0 T

(3.35)

Note by comparison with inequality (3.10) that the factor v has disappeared because we are now using specific quantities per unit volume, applicable only for small strains. We call dt the total dissipation, which must always be nonnegative. Noting the First Law in the form u Vij H ij  qi,i , we can combine the two Laws to eliminate the divergence of heat flux and obtain u Vij H ij  Ts 

qi T,i T

 dt

(3.36)

Equating this with (3.34) and grouping terms, we obtain § qi T,i wu · wu · wu § D ij   dt ¨ Vij  ¸ H ij  ¨ T  ¸ s  ¨ ¸ wH w wD T s © ¹ ij ¹ ij ©

0

(3.37)

The quantity  qi T,i T is readily identified as the dissipation due to heat flow and is often termed thermal dissipation. The remaining part of the dissipation d dt    qi T,i T is termed mechanical dissipation. Whilst the Second Law requires only that the total dissipation be non-negative, it is common to assume that the processes of mechanical dissipation and heat flow are independent, so that the thermal and mechanical dissipations are each required to be nonnegative. Thus we would write § wu · wu · wu § D ij  d 0 ¨ Vij  ¸ H ij  ¨ T  ¸ s  ¨ ¸ wH w s wD © ¹ ij ij © ¹

(3.38)

with the requirement that d t 0 . We shall return to the matter of total and mechanical dissipation in Chapter 12. At this point, different treatments of the thermodynamics of irreversible behaviour of materials tend to diverge. A common point of agreement is, however, to consider that the relationships Vij wu wHij and T wu ws derived for nondissipative material continue to apply, so that we can write 

wu D ij wDij

dt0

(3.39)

At the very minimum, one can then specify certain evolution equations for D ij , and ensure by applying checks to them that they are always consistent with the inequality in Equation (3.39). We find, however, that this approach is unsatisfactory for two reasons. Firstly, it is mathematically inconvenient in that a set of tensorial evolution equations have to be specified. Secondly, it is essential that all possible conditions be checked to ensure compliance with the inequality, and

3.3 Thermomechanics of Continua

51

this process may not be straightforward. In the next chapter, we describe an approach in which we avoid both of these problems, at the expense of imposing slightly more stringent conditions than are made necessary by (3.39). In the above development, we avoided some important issues relating to the rigour of the definitions of properties. These are defined only when the material is in a state of thermodynamic equilibrium. This requires that all processes occur on a timescale that is long with respect to any relaxation time relevant to the phenomena that are occurring. In practice, the application of “equilibrium thermodynamics”, even to a rapidly evolving process, is very successful. In this book, we shall be concerned mainly with rate-independent materials, and when these exhibit irreversible behaviour, they effectively achieve a state of “frozen” inequilibrium in the thermodynamic sense, in which relaxation times are infinitely long. Therefore we have to make the bold assumption that quantities strictly defined for thermodynamic equilibrium conditions also are applicable to these states of “frozen” inequilibrium.

Chapter 4

The Hyperplastic Formalism

4.1 Introduction In the previous chapter, we used thermodynamics to derive certain restrictions on the form of constitutive models. The approach used there was entirely conventional within the context of thermodynamics with internal variables. We arrive simply at an inequality, Equation (3.39), which succinctly expresses the Second Law in the form that the dissipation should be positive. This inequality is insufficient, however, to determine the evolution equations for internal variables. In this chapter, we introduce an additional assumption that allows us to satisfy the dissipation inequality but which also allows us to derive the evolution equations for internal variables. The assumption we make is essentially identical to the “orthogonality condition” introduced by Ziegler (1977), although we develop the analysis in a different way. For completeness in the following, we repeat some of the analysis from the previous chapter, although with a different emphasis.

4.2 Internal Variables and Generalised Stress In Section 3.3.3, we introduced the concept that the thermodynamic state of a material depends on internal kinematic variables as well as on strain and temperature. For convenience, we shall consider a single kinematic internal variable of tensorial form Dij . This is chosen because, in the following, it will be found that the internal variable is often conveniently identified with the plastic strain. Generalization either to other forms of internal variable (in particular to a scalar) or to multiple internal variables is straightforward. The internal energy now takes the form u u Hij , Dij , s , and the transformed energies take the corresponding forms f













f Hij , Dij , T , h h Vij , Dij , s , and

54

4 The Hyperplastic Formalism





g Vij , Dij , T . Corresponding to the kinematic variable, we define a “generalised stress” Fij wu wDij . Then it follows from the properties of the Legendre transformations between the energies that wf wg wh . Fij    wDij wDij wDij g

4.3 Dissipation and Dissipative Generalised Stress The principal motivation for the introduction of internal variables is to enable the study of dissipative materials. The development used here differs from that used by Ziegler (1983) and by Collins and Houlsby (1997), but is entirely consistent with this earlier work. The reason for the new approach is that we consider that it leads more straightforwardly to the potential formulation and is more accessible to those unfamiliar with thermodynamics.

4.3.1 The Laws of Thermodynamics In generalised thermodynamics, a central hypothesis is that the state of a material is entirely determined by the values of a certain set of independent variables: the kinematic variables (strain and internal variables) and the temperature. Properties are functions of state. The First Law of Thermodynamics states that there is a property (the internal energy) u such that

W  Q u

(4.1)

where W Vij H ij is the mechanical work input and Q qk,k is the heat supply to an element of volume (recall that we use here the comma notation to indicate a spatial differential). The Second Law of Thermodynamics can be expressed in various ways, but the most convenient here is to state that there is a property (the entropy) s such that §q · s t  ¨ k ¸ © T ¹, k

(4.2)

where  qk T is the entropy flux. Equation (4.2) can be restated as Ts  qk ,k 

qk T,k T

t0

(4.3)

We recall that the first two terms Ts  qk,k d are the mechanical dissipation. The third term  qk T,k T is the thermal dissipation. The thermal dissipation is

4.3 Dissipation and Dissipative Generalised Stress

55

always non-negative because heat flux is always in the direction of the negative thermal gradient. The third term becomes small by comparison with the first two for slow processes, so it is argued that the mechanical dissipation must itself be non-negative. Requiring both Ts  qk,k d t 0 and   qk T,k T t 0 is a slightly more stringent condition than the Second Law (4.3) but is widely accepted. In the following, we shall require that d t 0 . (Total dissipation is treated in Chapter 12.)

4.3.2 Dissipation Function From (4.1) and the definition of d, it follows that

u Vij H ij  qk,k

Vij H ij  Ts  d

(4.4)

The internal energy is a function of the state. In writing this function, it is convenient to choose entropy rather than temperature as the independent variable, so that we write (as above) u u Hij , Dij , s , and further that



u



wu wu wu H ij  D ij  s wHij wDij ws

(4.5)

Noting that the increments of the variables are independent of the state and comparing (4.4) and (4.5), Vij wu wHij and T wu ws . Furthermore, noting the definition of the generalised stress Fij wu wDij , then it follows that

d Fij D ij

(4.6)

The dissipation function must of course depend functionally on some set of variables, and we assume that the dissipation function is a function of the thermodynamic state of the material and also of the rate of change of state. It is found in the following that it is sufficient to consider just those mechanisms where dissipation depends only on the rate of change of the internal variable, D ij . The dissipation function is written variously as

   



d du Hij , Dij , s, D ij t 0 ½ ° f  d d Hij , Dij , T, Dij t 0 °° ¾ d d h Vij , Dij , s, D ij t 0 ° ° d d g Vij , Dij , T, D ij t 0 ° ¿

(4.7)

according to which form of the energy is specified. In principle, it would be possible to define the dissipation function in terms of a set of variables different from the energy function, but only in rather particular circumstances might this be useful.

56

4 The Hyperplastic Formalism

4.3.3 Dissipative Generalised Stress In each case, we define the “dissipative generalised stress” as Fij

wd e , where e wD ij

stands for any of u, f, h, or g. For a rate-independent material, the dissipation must be a homogeneous first-order function in the rates D ij because (for fixed ratios between the rates) the magnitude of dissipated energy must be directly proportional to the magnitude of deformation. For a homogeneous first-order function, Euler’s theorem gives wd e D ij wD ij

Fij D ij

d

(4.8)

Comparing Equation (4.6) with (4.8), we immediately obtain

 Fij  Fij D ij

(4.9)

0

 Fij  Fij

0 since the D ij ’s are apparently arbitrary. But we must note that Fij may be a function of D ij , so that

At first sight, this appears to imply that





we can draw from (4.9) only the much weaker conclusion that Fij  Fij is always orthogonal to D ij . Ziegler (1983) argues, however, that the stronger



statement Fij  Fij



0 can be made, and we follow this approach here. As

discussed in Chapter 1, we do not regard the debate whether Ziegler’s hypothesis is proven or not as important here. Instead, we examine the subset of idealised material models for which Fij  Fij 0 . We conclude that this subset is





sufficiently wide to provide realistic descriptions of many materials, including some that involve frictional dissipation, that is to say dissipation which depends on the applied pressure and (in the terminology of plasticity theory) nonassociated flow. The fundamental constitutive hypothesis (equivalent to Ziegler’s orthogonality condition) therefore is that Fij Fij . Although, throughout the following, we shall assume that this is the case, Fij and Fij need be kept as separate variables for formal purposes.

4.4 Yield Surface 4.4.1 Definition The roles of the rate of the internal variable D ij and the dissipative generalised stress Fij can now be interchanged by a further Legendre transform (Appendix C). As discussed in Collins and Houlsby (1997), this transformation

4.4 Yield Surface

57

is a degenerate special case of the Legendre transformation because the dissipation is homogeneous and first-order in the rates. The degenerate transform of a first-order function of D ij is a function of the conjugate variables Fij . However, it has the remarkable property that the value of the function is always identically zero. Thus the transform of the dissipation results in an equation that must be satisfied by generalised stresses. This relationship is none other than the yield function, expressed in terms of generalised stresses. The existence of the yield surface therefore arises as a consequence of the rateindependence of the material behaviour. Rate independence requires that the dissipation be first-order in the rates, and this in turn implies the existence of the yield surface. This contrasts sharply with the conventional approach to plasticity theory, in which the existence of a yield surface is the starting point for the theory. There is a further important distinction. In hyperplasticity, the yield surface is expressed as a function of the generalised stresses, not the true stresses. Later, we shall explore the relationship between the two. We express the yield surface by first writing

w Fij D ij  d 0

(4.10)

Since w 0 , we find that it can be determined only to within an arbitrary multiplicative constant. Therefore, it is useful to decompose w as w Oy 0 where O is an arbitrary non-negative multiplier. In later sections, we shall find it useful to explore the two solutions O 0 and y 0 . We can identify y as the yield surface, which is written in one of the following forms:

y y y y

 y f  Hij , Dij , T, Fij y h  Vij , Dij , s, Fij y g  Vij , Dij , T, Fij yu Hij , Dij , s, Fij

0½ ° 0 °° ¾ 0° ° 0° ¿

(4.11)

4.4.2 The Flow Rule The differential of the transformed function Oy 0 gives the flow rule: D ij

O

wy e wFij

(4.12)

As discussed below, D ij can be identified with conventionally defined plastic strain rates. Collins and Houlsby (1997) discuss the fact that Equation (4.12) links these strain rates to the differential of the yield surface with respect to generalised stresses Fij . Contrast this with conventional plasticity theory in Chapter 2, in which they would be related to the differential of the plastic

58

4 The Hyperplastic Formalism

potential (the yield surface for the case of associated flow) with respect to stresses Vij . Collins and Houlsby also discuss the fact that a yield surface in stress space can be derived by elimination of generalised stresses from Equations (4.11). They also demonstrate that non-associated flow (in the sense of conventional plasticity theory) can be derived within this framework and is intimately linked to stressdependence of the dissipation function. This issue is addressed in Section 4.10.

4.4.3 Convexity wy e t 0 (because wFij O t 0 ). This has a straightforward geometric interpretation and is simply the condition that the surface y e 0 contains the origin in generalised stress space and satisfies certain convexity conditions. It does not require, however, that the yield surface should be strictly convex either in generalised stress space or in stress space.

Since d Fij D ij t 0 , it follows that the condition on y e is Fij

4.4.4 Uniqueness of the Yield Function There are also relationships for each of the passive variables xij of the form: wd e wx ij

O

wy e wx ij

(4.13)

where x stands for any of Hij , Vij , Dij , s , or T. These relationships demonstrate that there is a close relationship between the functional forms of d e and y e . Note that, because of the nature of the singular transformation, the functional e form of y is not uniquely determined. In particular, the dimension of y e is not determined. However, the product Oy e must have the dimension (stress) u (strain rate) . If, for instance, O is chosen to have the dimension of strain rate (i. e. the same dimension as D ij ), then it follows that y e must be a homogeneous first-order function in stress. Note, however, that quantities with the dimension of stress might include the stresses Vij , generalised stresses Fij , and material properties with the dimension of stress. An alternative is that O could be chosen with the dimensions of (stress) u (strain rate) , in which case the yield function must be dimensionless. We place here no particular requirement on the form of the yield function. In Chapter 13, in which we express hyperplasticity in a convex analytical framework, we will find that it is possible to select a preferred form for the yield function, and we shall call this the canonical yield function.

4.6 A Complete Formulation

59

4.5 Transformations from Internal Variable to Generalised Stress For each of the functions e (u, f, h or g), a further transformation is possible, changing the independent variable from Dij to Fij in the form e e  Fij Dij . Correspondingly, the relevant passive variable in d e or y e is changed from Dij we we we to Fij . After the transformation, note the results Dij and , wFij wxij wxij where xij is any of the passive variables Hij , Vij , s or T. This last result gives alternative forms for the differentiation to obtain the appropriate complementary variables.

4.6 A Complete Formulation Adopting the approach described above, the constitutive behaviour is entirely defined by the specification of two potentials. The first is an energy potential, and the second either a dissipation function or the yield surface. There are a total of 16 different possibilities, however, for the choice of the potentials, representing all permutations of the following possibilities: x choice of u, f, h or g or for the energy function x dissipation function d e or yield surface y e x transformation between Dij and Fij for the energy function

The possibilities are illustrated in Table 4.1. In principle any of the 16 formulations could be used to provide a complete specification of the constitutive behaviour of a material. In each case, two potentials are specified. Technically, it would be possible to specify the energy potential from one of the 16 boxes and the dissipation or yield function from another, but presumably such a mixed form would be adopted only in rather special circumstances. The choice of formulation will depend on the application in hand. For instance, the four forms of the energy potential in classical thermodynamics are adopted in different cases (e. g. isothermal problems, adiabatic problems, etc.). On differentiating the energy function and dissipation or yield functions with respect to the appropriate variables, the relationships in Table 4.2 are obtained. Once the chosen two scalar functions have been specified, the entire constitutive behaviour can be derived from the differentials in the appropriate box in Table 4.2, together with the condition Fij Fij .

60

4 The Hyperplastic Formalism

Table 4.1. The 16 possible formulations

u or u

Energy function Dissipation function

Dij

de t 0 Fij



y

e

Dij

0 Fij



Dissipation function

de t 0



ye

Dij

0 Fij

h or h

h Vij , Dij , s h









f Hij , Fij , T





 Hij , Fij , T, D ij f  Hij , Dij , T y f  Hij , Dij , T, Fij f  Hij , Fij , T y f  Hij , Fij , T, Fij

d

f

g or g

g Vij , Dij , T wg wh Hij   wVij wVij



g Vij , Fij , T

 Vij , Fij , s, D ij h  Vij , Dij , s y h  Vij , Dij , s , Fij h  Vij , Fij , s y h  Vij , Fij , s , Fij h



f Hij , Dij , T d f Hij , Dij , T, D ij



 Vij , Dij , s, D ij

h Vij , Fij , s

d Yield surface



 Hij , Fij , s, D ij u  Hij , Dij , s y u  Hij , Dij , s, Fij u  Hij , Fij , s y u  Hij , Fij , s , Fij d

Fij



u

Energy function

Dij



u Hij , Fij , s d

Yield surface



u Hij , Dij , s d u Hij , Dij , s , D ij

f or f











d g Vij , Fij , T, D ij





g Vij , Dij , T y g Vij , Dij , T, Fij







g Vij , Fij , T y g Vij , Fij , T, Fij





4.6 A Complete Formulation

61

Table 4.2. Results from differentiation of energy and dissipation functions

u or u

Energy function Dissipation function d e t 0

Dij

f or f wf wu Vij Vij wHij wHij wf wu T s  ws wT wf wu Fij  Fij  wDij wDij

wd u wD ij wu Vij wHij wu T ws wu Dij wFij Fij

Fij

Fij Yield surface

ye

Dij

0

wu wHij wu T ws wu Fij  wDij Vij

wy u wFij wu Vij wHij wu T ws wu Dij wFij D ij

Fij

wdu wD ij

D ij

O

O

Fij

wd f wD ij

wf wHij wf s  wT wf Dij wFij Vij

wd f Fij wD ij wf Vij wHij wf s  wT wf Fij  wDij D ij

O

wy f wFij

wf wHij wf s  wT wf Dij wFij Vij

wy u wFij D ij

wy f O wFij

g or g

h or h

wh Hij  wVij wh T ws wh Fij  wDij Fij

wd h wD ij

Hij



T

wh ws

Dij

wh wVij

wh wFij

wd h Fij wD ij wh Hij  wVij wh T ws wh Fij  wDij D ij

O

Hij



T

wh ws

wy h wFij

wh wVij

Dij

wh wFij

D ij

wy h O wFij

wg wVij wg s  wT wg Fij  wDij Hij



wd g wD ij wg Hij  wVij wg s  wT wg Dij wFij Fij

Fij

wd g wD ij

wg wVij wg s  wT wg Fij  wDij

Hij



wy g wFij wg Hij  wVij wg s  wT wg Dij wFij D ij

O

D ij

O

wy g wFij

62

4 The Hyperplastic Formalism

4.7 Incremental Response In the numerical analysis of problems involving non-linear materials, the incremental form of the constitutive relationship is usually required. This, for instance, often forms a central part of a finite element analysis. Therefore, one of the most important criteria that needs to be applied to the formulation of any model is that the incremental form of the constitutive relationship should be derived solely by applying standard procedures, without the need to introduce either ad hoc procedures or additional assumptions. Within classical plasticity theory, more or less standardized procedures are adopted to derive incremental response [see for example Zienciewicz (1977)], although the mathematical treatment of the hardening behaviour tends to vary considerably. Differentiation of the energy expressions in Table 4.2 leads straightforwardly to the results in Table 4.3 where the (symmetrical) matrix >ucc@ is defined as

>ucc@

ª w 2u « « wHij wHkl « 2 « w u « wD wH « ij kl « w 2u « «¬ wswHkl

w 2u wHij wD kl w 2u wDij wD kl w 2u wswD kl

w 2u º » wHij ws » » w 2u » wDij ws » » 2 w u » » ws 2 ¼»

(4.14)

and the matrices >ucc@ , > f cc@ , ª¬ f cc¼º , >hcc@ , ª¬hcc¼º , > g cc@ and > g cc@ are similarly defined with appropriate permutation of the energy functions and independent variables. These incremental relationships are true for both dissipation and yield function formulations. However, in general, the explicit stress-strain response can be obtained only for those formulations based on the yield functions and only of the

Table 4.3. Incremental results obtained from energy expressions

u or u

f or f

­ V ij ½ ­ H kl ½ ­ V ij ½ ­ H kl ½ °  ° ° ° ° °°  °  cc cc u F D ® ij ¾ > @ ® kl ¾ ®Fij ¾ > f @ ®D kl ¾ °  ° ° s ° ° ° °  ° ¯ ¿ ¯ s ¿ ¯ T ¿ ¯ T ¿ ­ V ij ½ ­ H kl ½ ­ H kl ½ ­ V ij ½ ° ° ° ° ° ° ° ° ®D ij ¾ >ucc@ ®Fkl ¾ ®D ij ¾ ª¬ f ccº¼ ®F kl ¾ ° T ° °  ° ° ° ° ° ¯ ¿ ¯ s ¿ ¯ s ¿ ¯T¿

g or g

h or h ­ H ij ½ ­ V kl ½ °  ° ° ° cc h F ® ij ¾ > @ ®D kl ¾ °  ° ° ° ¯ s ¿ ¯ T ¿ ­H ij ½ ­V kl ½ ° ° ° ° ® D ij ¾ ª¬hccº¼ ® F kl ¾ °  ° ° ° ¯ s ¿ ¯ T ¿

­ H ij ½ °  ° ®Fij ¾ ° ° ¯ s ¿

­ V kl ½ ° ° cc > g @ ®D kl ¾ ° T ° ¯ ¿

­H ij ½ ° ° ® D ij ¾ ° ° ¯ s ¿

­V kl ½ > g cc@ ®° F kl ¾° ° T ° ¯ ¿

4.7 Incremental Response

63

Table 4.4. Substitution of variables for different formulations

e aij bij

u Vij Hij

f Vij Hij

h Hij Vij

g Hij Vij

x z

T s

s T

T s

s T

y e type. For each of these forms the incremental relationships can be written (noting that F ij F ij ) in the following form:

­ aij ½ ° ° ®F ij ¾ ° ° ¯ x ¿

ª w 2e « « wbij wbkl « 2 « w e « wD wb « ij kl « w 2e « «¬ wzwbkl

w 2e wbij wD kl w 2e wDij wD kl w 2e wzwD kl

w 2e º » wbij wz » » ­ bkl ½ w 2e » ° ° ®D kl ¾ wDij wz » ° ° » z ¯ ¿ w 2e » » wz 2 »¼

(4.15)

where substitutions for e, aij , bij , x, and z are to be taken from the appropriate column of Table 4.4. Equation (4.15) is used together with the flow rule: D ij

O

wy e wFij

(4.16)

The multiplier O is obtained by substituting the above equations in the consistency condition, which is obtained by differentiating the yield function: y e

wy e  wy e wy e wy e D ij  F ij bij  z  wbij wDij wz wFij

0

Together with the orthogonality condition in its incremental form F ij can be used to derive O 

Aijeb

ez

A b  z e ij B Be

(4.17)

F ij , this

(4.18)

where for convenience, we define the notation, Aijeb

wy e wy e w 2e  wbij wFkl wD kl wbij

(4.19)

Aez

wy e wy e w 2e  wz wFkl wD kl wz

(4.20)

64

4 The Hyperplastic Formalism

§ wy e wy e w 2e · wy e  ¨ ¸ ¨ wDij wFkl wD kl wDij ¸ wFij © ¹ This leads to the following incremental stress-strain relationships: Be

ª w 2e w 2e eb Cmnkl  « b b b w w w wD ij mn « ij kl ­ aij ½ « 2 2 ° ° « w e  w e C eb ° x ° « wzwbkl wzwDmn mnkl °F ° « ® ij ¾ « w 2e w 2e eb ° D ° « Cmnkl  ij ° ° « wDij wbkl wDij wDmn ° O ° « ¯ ¿ eb Cijkl « «  Aijeb Be «¬ Finally, this can be simplified to ­ aij ½ ° ° ° x ° °F ° ® ij ¾ ° D ° ° ij ° ° O ° ¯ ¿

(4.21)

w 2e w 2e ez º Cmn  » wbij wz wbij wDmn » » 2 2 w e w e ez  Cmn »» 2 wzwD wz mn » ­°bkl ½° (4.22) 2 2 » ®° z ¾° w e w e ez ¯ ¿ Cmn  » wDij wz wDij wDmn » » Cijez » » ez e A B »¼

ebb ª Dijkl « « Debz kl « eb « DijklD « « C eb ijkl « «  Aeb Be ¬ kl

Dijezb

º » Dezz » » ­b ½ ° ° DijezD » ® kl ¾ » ¯° z ¿° Cijez » »  Aez Be »¼

(4.23)

where w 2e w 2e eb  Cmnkl wEij wbkl wEij wDmn

(4.24)

ezb Dkl

w 2e w 2e eb  Cmnkl wzwbkl wzwDmn

(4.25)

DijeEz

w 2e w 2e ez  Cmn wEij wz wEij wDmn

(4.26)

eEb Dijkl

Dez

w 2e wz 2

eb Cmnkl ez Cmn

and E stands for either D or b.



w 2e ez Cmn wzwDmn

(4.27)

Aeb wy e  kle B wFmn

(4.28)

Aez wy e  e B wFmn

(4.29)

4.7 Incremental Response

65

The first two rows of the matrix in Equation (4.23) describe the incremental relationships among the stresses, strains, temperature, and entropy. The third and fourth rows are the evolution equations for the generalised stress and the internal variable. The final row allows evaluation of the plastic multiplier O for the increment. The forms of the relationships, after the appropriate substitution of variables, are given in Table 4.5. The above solution applies only when plastic deformation occurs, i. e. D ij z 0 , and O ! 0 . If the above solution results in O  0 , then it implies that elastic unloading has occurred. In this case, the consistency equation no longer applies but is simply replaced by the condition O 0 . For this case, it is straightforward to show that the above relations are replaced by

­ aij ½ ° ° ° x ° ° ° ®F ij ¾ ° D ° ° ij ° ° O ° ¯ ¿

ª w 2e « « wbij wbkl « 2 « w e « wzwb kl « « w 2e « « wDij wbkl « 0 « «¬ 0

w 2e º » wbij wz » » w 2e » wz 2 »» °­bkl °½ ® ¾ w 2e » °¯ z ¿° » wDij wz » 0 »» 0 »¼

(4.30)

Table 4.5. Summary of incremental form of constitutive relations

H kl

s

­ V ij ½ °  ° ° T ° °F ° ® ij ¾ ° D ° ° ij ° ° O ° ¯ ¿

­ V ij ½ ° ° ° s ° ° ° T ®F ij ¾ °  ° ° Dij ° ° O ° ¯ ¿

uHH Dijkl

ª « « DusH « kl uDH « Dijkl « « C uH « ijkl « AuH «  kl «¬ Bu ª D f HH « ijkl « f TH « Dkl « f DH « Dijkl « fH « Cijkl « « Af H «  klf ¬ B

V kl

DijuHs

º » Dus » » DijuDs » ­H kl ½ »® ¾ s Cijus » ¯ ¿ » Aus » »  Bu »¼ f HT Dij º » fT » D » f DT » Dij » ­H kl ½ ®  ¾ fT » T Cij » ¯ ¿ » AfT »  f » B ¼

­ H ij ½ °  ° ° T ° °F ° ® ij ¾ ° D ° ° ij ° ° O ° ¯ ¿

­ H ij ½ ° ° ° s ° °F ° ® ij ¾ °  ° ° Dij ° ° O ° ¯ ¿

hVV ª Dijkl

« « DhsV « kl hDV « Dijkl « « C hV « ijkl « AhV «  kl ¬« Bh

DijhVs º » hs » D » DijhDs » ­V kl ½ »® ¾ s Cijhs » ¯ ¿ » Ahs »  h » B ¼»

ª D g VV « ijkl « g TV « Dkl « g DV « Dijkl « gV « Cijkl « « Ag V «  klg ¬ B

º » gT » D » g DT »  Dij » ®­Vkl ¾½  gT » T Cij » ¯ ¿ » Ag T »  g /» B ¼

g VT

Dij

66

4 The Hyperplastic Formalism

The choice of formulation is determined by the application in hand, and to a certain extent by personal preferences. The u and h formulations are particularly convenient for problems where changes in entropy are determined (e. g. adiabatic problems), whilst the f and g formulations are appropriate for those with prescribed temperature (e. g. isothermal problems). The u and f formulations correspond to strain-space based plasticity models and are particularly applicable when the strains are specified. Conversely the h and g formulations correspond to the more commonly used stress-space plasticity approaches and are particularly convenient for problems with prescribed stresses. However, by appropriate numerical manipulation, it is possible to use any of the formulations for any application. For instance, the g formulation leads directly to the compliance matrix. This can be straightforwardly inverted to give the stiffness matrix.

4.8 Isothermal and Adiabatic Conditions Isothermal conditions can be imposed straightforwardly by the condition T 0 . They are most conveniently examined using either the Helmholtz free energy or Gibbs free energy forms of the equations. Thus the isothermal elastic-plastic f HH g VV stiffness matrix is Dijkl and the isothermal compliance matrix is Dijkl (both from Table 4.5). For elastic conditions, these reduce to

w2 f w2 g and , wHij wHkl wVij wVkl

respectively. Adiabatic conditions are slightly more complex. In reversible thermodynamics, the adiabatic condition (no heat flow across boundaries) is associated with isentropic conditions, but in the presence of dissipation, the adiabatic condition becomes Ts  d Ts  Fij D ij 0 . Adiabatic conditions are most conveniently expressed using the internal energy or the enthalpy forms of the equations. Multiplying the fourth line of the appropriate matrix equations is Table 4.5 by Fij and substituting the adiabatic condition Ts Fij D ij gives Fij D ij

uH  Fij Cijkl Hkl  Fij Cijus s Ts

(4.31)

Fij D ij

hV Fij Cijkl V kl  Fij Cijhs s Ts

(4.32)

or

4.9 Plastic Strains

67

which can simply be rearranged to solve for s in terms of either the stress or strain increment. Substituting in the first line of the appropriate matrix equation in Table 4.5 gives the adiabatic stiffness or compliance behaviour as V ij

ª uH « DuHH  DuHs FmnCmnkl ijkl ij « T  F pqC us pq «¬

H ij

ª hV « DhVV  DhVs FmnCmnkl ijkl ij « T  F pqC hs pq «¬





º » H » kl »¼

(4.33)



º » V » kl »¼

(4.34)

or



Similar substitutions for the entropy increment are necessary in the second to fifth lines of the equations to solve for the other incremental quantities. Note that for the elastic case, adiabatic and isentropic conditions are idenw 2u and tical, and the stiffness and compliance matrices are simply wHij wHkl w 2h , respectively. wVij wVkl

4.9 Plastic Strains So far, no particular interpretation has been placed on the internal variable Dij . By a suitable choice of Dij , Collins and Houlsby (1997) showed that it is normally possible to write the Gibbs free energy so that the only term that involves both Vij and Dij is linear in Dij : g







g1 Vij  g 2 Dij  g 3 Vij Dij

(4.35)

Furthermore, if g 3 is also linear in the stresses, then Collins and Houlsby (1997) showed that no elastic-plastic coupling occurs. In this case, it is always possible (again by suitable choice of Dij ) to choose g 3 Vij Dij . For this case, it follows that wg Hij  1  Dij (4.36) wVij Fij

Vij 

wg 2 wDij

(4.37)

The interpretation of the above is that Dij plays exactly the same role as the p conventionally defined plastic strain Hij . It is convenient to define elastic strain

68

4 The Hyperplastic Formalism

e Hij

Hij  Dij

wg1 wVij . Furthermore, the generalised stress simply differs

from the stress by the term wg 2 wDij , and it is convenient to introduce the e e “back stress” defined as Uij Vij  Fij wg 2 wDij . Note that Hij Hij Vij and





Uij

Uij Dij . For this case, the development of the incremental response equations can be considerably simplified by noting that the differential w 2 g wVij wD kl w 2 g wDij wVkl Gij Gkl . By using the back stress and the elastic strain, further Legendre transformations are possible that can lead to certain simpler forms of the other energy functions, but this topic is not further pursued here.

4.10 Yield Surface in Stress Space Consider the case where a material is specified by choosing the Gibbs free energy g g Vij , Dij and the yield function y g y g Vij , Dij , Fij 0 . Note that









because the yield function is the Legendre transform of the dissipation function, either can be used to specify the material. Noting that Fij Fij wg wDij, we can express the generalised stress as a function of the true stress and internal variable Fij





Fij Vij , Dij . Substituting

this in the expression for the yield surface, we obtain





y g Vij , Dij , Fij Vij , Dij



where y * Vij , Dij







y * Vij , Dij



0

(4.38)

is the yield function in true stress space. Differentiating

(4.38), we obtain wy g wy g wy g dVij  dDij  dFij wVij wDij wFij · wy g wy g wy g § w2 g w2 g dVij  dDij  dVkl  dD kl ¸ ¨ ¨ ¸ wVij wDij wFij © wDij wVkl wDij wD kl ¹ wy * wy * dVij  dDij wVij wDij

(4.39)

Now for an uncoupled material in which g 3 Vij Dij in Equation (4.35), w g wDij wVkl Gik G jl . Equating terms in dVij from (4.39) then gives 2

wy g wy g  wVij wFij

wy * wVij

(4.40)

4.11 Conversions Between Potentials

69

We observe that the plastic strain increments are in the direction wy g wFij . They will be “associated” in the conventional sense, i. e. normal to the yield surface in true stress space if they are in the direction wy * wVij . Clearly, this is only the case if wy g wVij 0 , that is, if the yield function is independent of the stresses (or, exceptionally, if wy g wVij is always parallel to wy g wFij ). From (4.13), we observe that wy g wVij 0 only if wd g wVij 0 , so that associated flow only occurs if the dissipation is independent of the true stress. Conversely, if the dissipation depends on the stress, it is an inevitable consequence of our approach that flow should be non-associated in the conventional sense. Frictional materials involve dissipation which depends on the stresses, and so we conclude that frictional materials will always involve non-associated flow. This observation is entirely consistent with experimental observations on granular materials.

4.11 Conversions Between Potentials In the formulation described here, much emphasis has been placed on the concept that, once two scalar functions are known, then the entire constitutive behaviour of the material is determined. Emphasis has also been placed on the fact that there are many possible combinations of functions that can be used, and that these are interrelated through a series of Legendre transformations. Different functions may be required for different applications. For instance, a hypothesis about the constitutive behaviour of a material might best be expressed as an assumption about the form of the dissipation function, whereas the incremental response is most conveniently derived from the yield function. The ability to transfer between the various functions is therefore vitally important.

4.11.1 Entropy and Temperature The simplest transformations are those between u and h, in terms of entropy, and f and g, in terms of temperature. Take the example of the u to f transformation. The equation T T Hij , Dij , s wu ws has to be solved for





wT w 2u z 0 . Once s ws ws 2 is known in terms of T, it is a trivial matter to substitute s for T throughout the equation f u  sT . The inversion is particularly simple for certain common forms of the energy function (e. g. quadratic in s), but for more complex forms, the inversion may need considerable ingenuity, or may not even be expressible in conventional mathematical functions. All other transformations between entropy and temperature are possible, subject to analogous conditions. s





s Hij , Dij , T . This can be achieved, provided only that

70

4 The Hyperplastic Formalism

4.11.2 Stress and Strain Transformations between u and f, in terms of strain, and h and g, in terms of stress, are similar to those involving temperature to entropy changes, except that this time the variables to be eliminated are tensorial in form. Taking the u to h transformation as the example, the equations Vij Vij Hij , Dij , s wu wHij have









to be solved for Hij Hij Vij , Dij , s . This involves in general the solution of n equations in n independent variables, where n is the number of independent Hij ’s (usually six). This can be achieved, in principle, provided that the determinant of the Hessian matrix

wVij wHkl

w 2u z 0 . Once Vij is known in wHij wHkl

terms of Hij , it is again trivial to substitute Vij for Hij throughout the equation h u  Vij Hij . The inversion is again simple for certain common forms of the energy function (e. g. quadratic in Hij ), but can become extremely intractable for more complex forms. All other transformations between stress and strain are possible, subject to analogous conditions.

4.11.3 Internal Variable and Generalised Stress Transformations between u, f, h, and g, in terms of the internal variable, and u , f , h , and g , in terms of the generalised stress, can be achieved under conditions that are analogous to those applying to the stress and strain transformations discussed in the preceding section.

4.11.4 Dissipation Function to Yield Function The transformations from the dissipation to the yield function differ from those discussed above because they involve the special case of the transform of a homogeneous first-order function. The equations Fij wd e wD ij are homogeneous of degree zero in D ij . Therefore, it is possible to divide all these equations by any one of the D ij ’s, resulting in n equations in n  1 variables, where n is the number of D ij ’s (generally six). If it is possible to form the resolvant by eliminating n  1 variables, leaving one equation which does not contain the D ij ’s, then this equation (when expressed in the appropriate form) is the yield surface y e

0 . The condition for the existence of the resolvant is that

the Hessian matrix

w 2d e wD ij wD kl

0 . Note the sharp contrast with the cases

4.12 Constraints

71

previously discussed in which the condition for solving n equations in n variables was that the determinant of the Hessian should be non-zero. In this case, we are seeking the condition that n equations in n  1 variables are consistent, and that condition requires that the determinant of the Hessian should be zero. In Section 10.3.1, we give an example of a transformation from a dissipation function to a yield function for a non-trivial case.

4.11.5 Yield Function to Dissipation Function The transformation of the yield function to the dissipation function is also nonstandard because it involves a singular transformation. The rate equations D ij O wy e wFij are first divided by O to give n equations in n variables D ij O . These equations can be solved for Fij in terms of D ij O , if

w2 ye z 0 . Now, wFij wFkl

the value of O must be found, and this is achieved by substituting the solution for Fij in the yield condition y e 0 to give an equation in D ij O which is solved for O in terms of the D ij ’s. This result is then used to convert the solutions for Fij , in terms of D ij O , to solutions that are simply in terms of D ij . Finally, this result is substituted in the expression d e Fij D ij . It will be found that the resulting expression is (as required) homogeneous of degree one in the D ij ’s. Even if the determinant of the Hessian

w2 ye wFij wFkl

is equal to zero, it may

nevertheless be possible to resolve the n  1 equations (the yield condition together with the n equations for the D ij ’s) to eliminate O, solve for the Fij ’s in terms of the D ij ’s, and determine the dissipation function. In Section 10.3.2, we give an example of a transformation from a yield function to a dissipation function for a non-trivial case.

4.12 Constraints The development of some models is most efficiently achieved by introducing constraints. Typically these might be constraints on either the strains (e. g. incompressible behaviour) or on the rates of the internal variables (e. g. dilational constraints for granular materials). A full treatment of constraints would not be appropriate here, but some simple cases are of sufficient importance that some discussion is necessary.

72

4 The Hyperplastic Formalism

4.12.1 Constraints on Strains If there is a constraint on strains, e. g. the incompressibility condition Hkk 0 , then the most convenient starting point is from consideration of u or f. We consider the case where f is specified. Writing the constraint as c Hij 0 , we introduce the effect of the constraint by using the standard method of Lagrangian multipliers. Instead of using f, we define a new function f c f  /c , which by virtue of the condition c 0 is numerically equal to f. Now we define the stresses as



wf c wHij

Vij

wf wc / wHij wHij

(4.41)

To obtain g, we perform the Legendre transformation on f c : gc

f c  Vij Hij

(4.42)

and the properties of the transform lead directly to Hij wg c wVij . However, we can note that g c f c  Vij Hij f  /c  Vij Hij f  Vij Hij g , and therefore Hij wg wVij , which is of course the same as the normal result in the absence of a constraint. If instead g had been specified with no constraint on the stresses, we would as usual write Hij wg wVij . If these equations are not independent, we find that there must be a relationship between the strains, and correspondingly that we cannot solve for the equivalent stress. We express the relationship between the strains as the constraint c Hij 0 . Define g c g and then use the Legendre



transform,

f c g c  Vij Hij leading to Vij

(4.43)



wf c wHij . Note that because of the constraint c Hij



possible to establish the functional form of f c Hij



0 , it is not

uniquely because any

multiple of c Hij can be added to f c without affecting Equation (4.43). We can express this by using f c f  /c , where / is an arbitrary constant, as the definition of f. Thus we obtain Vij

wf wc / wHij wHij

(4.44)

Finally note the asymmetry that a constraint on the strains corresponds to an indeterminacy in the stress. This indeterminacy is of course closely related to the Lagrangian multiplier /. An example of the use of a constraint of this sort is given in Section 5.1.3.

4.12 Constraints

73

4.12.2 Constraints on Plastic Strain Rates The other case where it is particularly useful to introduce constraints is in the definition of plastic behaviour through the use of a dissipation function. For example, Houlsby (1992) uses a constraint to introduce the effect of dilation into a plasticity model. The case of a constraint on plastic strain rates c D ij 0 is of most interest. Clearly, c must be a homogeneous equation in the rates, and for consistency with the dissipation function, we shall choose to write it as a homogeneous first-order function. In this case, following a procedure similar to that described above, we define a modified dissipation function d ce d e  /c . The definition of the generalised stress becomes



Fij

wd ce wD ij

wd e wc /  wDij wD ij

(4.45)

The yield function is obtained by the singular transformation Oy ce Fij D ij  d ce 0 , and it follows from the properties of the transformation that D ij

O wy ce wFij . Note that because they serve exactly the same role and

both are equal to zero, it follows that we need make no distinction between y ce and y e . If instead y e had been specified, we would as usual write D ij O wy e wFij . If the equations for the D ij ’s are not independent, there must be a relationship between the rates, and correspondingly there will be a component of the Fij ’s that cannot be resolved. The relationship between the rates can be expressed as a constraint c D ij 0 . Define y ce y e and use the Legendre transform,



d ce

Fij D ij  Oy ce

(4.46)

It follows from the properties of the transform that Fij wd ce wD ij . Note that (in a way similar to the case described in the preceding section), it is not possible to establish d ce D ij uniquely because any multiple of the constraint equation



can be added to d ce without affecting Equation (4.46). Again, we can express this by using d ce d e  /c , where / is an arbitrary constant, as the definition of d e . Thus we obtain Fij

wd e wc / wD ij wD ij

An example of this type of constraint is given in Section 10.3.1.

(4.47)

74

4 The Hyperplastic Formalism

4.13 Advantages of Hyperplasticity The motivation for this work comes principally from the development of constitutive models for geotechnical materials, which usually exhibit frictional (i. e. pressure-dependent) behaviour and non-associated flow, although the formulation described here could also find a wider application. Many theoretical models for soils, concrete and rocks have been proposed, involving a huge variety of methods, assumptions and procedures. The purpose here has been to provide a coherent framework within which models could be developed without the need for additional ad hoc assumptions and procedures. Whilst making no claims of total generality, it is our belief that this framework is sufficiently general that realistic models of geotechnical materials can be developed within it. A central theme of hyperplasticity is that, once two scalar potentials have been specified, the entire constitutive response can be derived. This approach can be implemented readily in a computer program capable of predicting the entire stress-strain response of a material subject to a specified sequence of stress or strain increments. The material model is specified solely by the expressions either for g and y g or alternatively for f and y f , which are most convenient for isothermal conditions. All differentials necessary to determine the incremental response can be evaluated either by numerical differentiation or analytically, if a symbolic manipulation package is used.

4.14 Summary It is convenient at this point to restate the complete formalism developed here in a succinct form, to highlight precisely which assumptions are necessary to develop the formalism. Assume that the local state of the material is completely defined by knowledge of (a) the strain Hij (measured from a suitable reference configuration), (b) the entropy s, and (c) certain internal variables Dij . The constitutive behaviour of the material will be then completely defined by specifying two thermodynamic potential functions of state. The first potential is the specific internal energy, a function of state u u Hij , s, Dij , which is a property satisfying the First Law of Thermodynamics:





u Vij H ij  qk,k

(4.48)

where Vij is the stress that is work-conjugate to the strain rate and qk is the heat flux vector.

4.14 Summary

75

The second potential is the specific mechanical dissipation, a function of state and the rate of change of internal variable d d Hij , s , Dij , D ij , satisfying the Second Law of Thermodynamics in the form,



Ts  qk,k



dt0

(4.49)

where T is the non-negative thermodynamic temperature. (Total dissipation, including the thermal dissipation term  qk T,k T , will be treated in Chapter 12). Adding Equations (4.48) and (4.49), we obtain u  d Vij H ij  Ts





(4.50)



is differentiable and d d Hij , s , Dij , D ij a homogeneous first-order function of D ij , we write Assuming that u u Hij , s, Dij

u



is

wu wu wu H ij  D ij  s wHij wDij ws

(4.51)

wd D ij wD ij

(4.52)

d

which on substitution in (4.50) yields: § wu · wd · § wu · § wu  Vij ¸ H ij  ¨  T ¸ s  ¨  ¨ ¸ D ij ¨ wHij ¸ ¨ © ws ¹ © wDij wD ij ¸¹ © ¹

0

(4.53)

Assuming Ziegler’s orthogonality condition, wu wd  wDij wD ij

0

(4.54)

and that the processes of straining and change of entropy are mutually independent, we obtain from Equation (4.53) Vij T

wu wHij

(4.55)

wu ws

(4.56)

Equations (4.54)(4.56) are sufficient to establish the constitutive behaviour. For convenience of the derivation of incremental response, we define generalised wu wd and dissipative generalised stress Fij and use stress Fij  wDij wD ij Equation (4.54) in the form Fij

Fij .

Chapter 5

Elastic and Plastic Models in Hyperplasticity

5.1 Elasticity and Thermoelasticity 5.1.1 One-dimensional Elasticity A one-dimensional elastic model may be specified by the Helmholtz free energy (in this case also equal to the internal energy) f u EH2 2 . In the context of elasticity theory, this is usually known as the strain energy. Differentiation then gives V df dH EH . Alternatively, the model may be specified by the Gibbs free energy, which in this case is equal to the enthalpy g h V2 2E . In the context of elasticity, the quantity  g is often known as the complementary energy. Again differentiation gives H  dg dV V E . The model and its behaviour are illustrated in Figure 5.1. When the spring has been loaded so that the state is at point A, the significance of f and –g is that

Figure 5.1. One-dimensional elasticity

78

5 Elastic and Plastic Models in Hyperplasticity

they represent the areas shown in the figure. For the special case that the energy functions are quadratic functions, the areas f and –g are equal.

5.1.2 Isotropic Elasticity The extension of the one-dimensional case to a continuum is straightforward and well known, and has already been cited in Chapter 2. For isotropic linear elasticity (without thermal effects), the Helmholtz free energy, which is equal to the internal energy, is

f

K Hii H jj  GHijc Hijc 2

u

(5.1)

where K is the bulk modulus and G the shear modulus. From this, it immediately follows by differentiation that Vij K Hkk Gij  2GHijc , which can be decomposed into volumetric and deviatoric parts as Vkk 3K Hkk and Vijc 2GHijc . Alternatively, the behaviour can be specified by the Gibbs free energy, which in this case is equal to the enthalpy: 1 1 (5.2) Vii V jj  Vijc Vijc 18K 4G 1 1 From this, it follows by differentiation that Hij Vkk Gij  Vijc , which again 9K 2G can be decomposed into volumetric and deviatoric parts as H kk Vkk 3K and Hijc Vijc 2G . g

h 

5.1.3 Incompressible Elasticity Incompressible isotropic linear elasticity (without thermal effects) is a useful example of the introduction of a constraint. It is most conveniently first approached by considering the limit K o f in the Gibbs free energy formulation: 

g

1 Vijc Vijc 4G

(5.3)

From this, it immediately follows that Hij Vijc 2G . These equations for strains are not independent, and taking the trace of the strain simply gives Hkk 0 as required for incompressible behaviour. The lack of independence is reflected in the fact that there is no constitutive relationship that determines the trace of stress, so that Vkk is undetermined and simply appears as a reaction. When the Legendre transform is taken to obtain the Helmholtz free energy, we get f

GHijc Hijc

(5.4)

5.1 Elasticity and Thermoelasticity

79

But the fact that the strains are not independent has to be introduced by specifying the constraint c Hkk 0 . The stresses are then obtained by differentiation of the augmented free energy f c f  /c , where / is an undetermined multiplier. Thus the stresses are wf c Vij 2GHijc  /Gij (5.5) wHij

Taking the trace, we obtain Vkk 3/ , so that we identify the undetermined multiplier associated with the incompressibility constraint as the mean stress Vkk 3 , which again appears as a reaction and is undetermined by the constitutive relationship. Thus either the f or g formulation gives the same behaviour. Note, however, that it is not necessary to express the constraint explicitly in one, whilst it is in the other.

5.1.4 Isotropic Thermoelasticity Linear isotropic small strain thermoelasticity can be expressed by using any of the following energy expressions: Hii H jj Hijc Hijc D3K T0 T s2 (5.6) u 3KX  2G  sHkk  0  sT0 6 2 c c 2 f

3K h 

g

Hii H jj 6

 2G

Hijc Hijc 2

 D3K  T  T0 Hkk 

c  T  T0 2 T0

2

(5.7)

T s2 1 Vii V jj 1 Vijc Vijc DT0   sVkk  0  sT0 3KX 6 2G 2 cX cX 2

(5.8)

2 1 Vii V jj 1 Vijc Vijc cX  T  T0    D  T  T0 Vkk  3K 6 2G 2 T0 2

(5.9)

where K is now identified as the isothermal bulk modulus, G is the shear modulus (which is the same for isothermal and adiabatic conditions), D is the coefficient of linear thermal expansion, c is the heat capacity per unit volume at constant strain, T0 the initial temperature and X 1  9K D2 T0 c . The value





of X is typically very close to unity, and represents the magnitude of the difference between adiabatic and isothermal behaviour. Each of the above forms is quadratic, and on differentiation results in a linear response. The terms in the stresses or strains result in the elastic response. Those in temperature or entropy result in the heat capacity of the material. The coupling terms (e. g. between stress and temperature) result in the thermal expansion. For example, differentiation of (5.9) gives wg 1 1 Hij  Vkk Gij  Vijc  D  T  T0 Gij (5.10) wVij 9 K 2G

80

5 Elastic and Plastic Models in Hyperplasticity

The role of D as the coefficient of thermal expansion is obvious from (5.10), which simply indicates an additive strain term that is proportional to the change in temperature. The role of c is less obvious. However, differentiating (5.6) with respect to entropy, we obtain T

wu ws



D3K T0 T Hkk  0 s  T0 c c

(5.11)

and D3K T0 T T  H kk  0 s c c

(5.12)

However, for a process of pure heating at constant volume, H kk 0 and qk ,k . It is qk ,k Ts , so that for such a process at T T0 we can write T  c clear that c is the heat capacity per unit volume at constant strain (analogous to cv for a gas). From any of (5.6)–(5.9), it is possible to derive the incremental response: ªH kk º « » « H ijc » « s » ¬ ¼

0 3D º ª V kk º ª1 3K « » « 0 1 2G 0 »» « V ijc » « «¬ 3D 0 cX T0 »¼ «¬ T »¼

(5.13)

which can of course be manipulated into a variety of other forms.

5.1.5 Hierarchy of Isotropic Elastic Models An advantage of the hyperelastic approach (here extended to include thermal effects) is that models can easily be classified and placed in a hierarchy on the basis of energy functions. Simpler models can be identified as special cases of more complex models. For example, Table 5.1 shows the relationships among general thermoelasticity, elasticity without thermal effects, and the special case

Table 5.1. Hierarchy of isotropic elastic models

Model

Gibbs free energy

Thermoelasticity

2 1 Vii V jj 1 Vijc Vijc cX  T  T0    D  T  T0 Vkk  3K 6 2G 2 2 T0 1 Vii V jj 1 Vijc Vijc   3K 6 2G 2 1 Vijc Vijc  2G 2

Elasticity Incompressible elasticity

g g g

5.2 Perfect Elastoplasticity

81

of incompressible elasticity. The differences are clear in the expressions for Gibbs free energy. With a little practice, it becomes possible to identify the properties of the model rapidly from the form of the energy expressions. As familiarity with the expressions is gained, it is then possible to reverse this process and deduce (or at the very least guess) suitable forms of the expressions to produce particular features in a model.

5.2 Perfect Elastoplasticity 5.2.1 One-dimensional Elastoplasticity Now we turn to models with dissipation. The first example is a one-dimensional elastic plastic system in which a spring and sliding element (with a limiting stress k) are placed in series as shown in Figure 5.2. The mechanical behaviour is expected to be as shown in Figure 5.3a.

Figure 5.2. One-dimensional perfectly plastic model

Figure 5.3. Cyclic stress-strain behaviour of the perfectly plastic model: (a) in true stress space, (b) in generalised stress space

82

5 Elastic and Plastic Models in Hyperplasticity

The easiest starting point is to consider the Helmholtz free energy, which now becomes E  H  D 2 2

f

(5.14)

so that V wf wH E  H  D . Carrying out the Legendre transform, we can derive g

E  H  D 2  E  H  D H 2

f  VH 

E  H  D 2  E  H  D D 2



V2  VD 2E

(5.15)

Differentiation of this expression gives, as expected, H wg wV  V E  D , so that the strain consists of two additive parts: (a) the elastic strain V E , which is just proportional to the stress and (b) the plastic strain D . We can observe that it is the term VD in the Gibbs free energy expression which determines that the role of D is that of conventional plastic strain. We can also observe that differentiation of (5.15) gives F wg wD V . If the f formulation is used, F wf wD  E  H  D , which leads to the same result F V . Although the generalised stress is equal to the true stress, we keep it as a separate variable for formal purposes. Now we turn to the dissipative part of the model. The dissipation will simply be given by k D

d

(5.16)

where we have to take the absolute magnitude of D because the dissipation is positive irrespective of the sign of D . Differentiation gives F

wd wD

k S  D

(5.17)

where S  D is the generalised signum function, as discussed in the Notation section of this book. It is the differential (or more properly the subdifferential) of D with respect to D . Consider first the case when D z 0 . Then, it follows from (5.17) that (a) the sign of F is the same as that of D and (b) F k . When D 0 , the definition of S  D requires that  k d F d k . It is clear therefore that the yield surface is defined by the equation F  k 0 . The LegendreFenchel transformation of the dissipation function is therefore written w

Oy

O F  k 0

(5.18)

5.2 Perfect Elastoplasticity

83

It follows that ww wF

D

O

wy wF

O SF

(5.19)

where by virtue of (5.18) either O 0 (which corresponds to elastic behaviour) or F  k 0 (plastic behaviour). Note the important result that the yield surface is derived from the dissipation, and is not introduced by a separate assumption.

5.2.2 Von Mises Elastoplasticity The von Mises plasticity model can be described by defining Dij as plastic strain (together with g 2 Dij 0 so that there is no back-stress, and therefore Vij Fij ). Using the g formulation with isotropic elasticity gives therefore,

 g



1 1 Vii V jj  Vijc Vijc  Vij Dij 18K 4G

(5.20)

We can then use either the dissipation function, d

together with the constraint D kk y

k 2 D ij D ij

(5.21)

0 , or the yield function,

Fijc Fijc  k 2

(5.22)

0

It is straightforward to show that the yield function can be derived from the dissipation function and vice versa. For instance, differentiation of (5.21) gives Fijc



where Sij D ij

wd wD ij



k 2 Sij D ij

(5.23)

is the generalised tensorial signum function as defined in the

 

Notation section of this book. If D ij z 0 , we can use Sij D ij Sij D ij rive Fijc Fijc

1 to de-

2k 2 from (5.23), hence giving the Von Mises yield surface

y Vijc Vijc  2k 2 0 (since Gcij Fijc ). The value of the strength in simple shear is k and in uniaxial tension or compression is k 3 . The flow rule is given by D ij

O

wy wFij



2O Sij Fijc



2O Sij Vijc

which can be recognized as the associated flow rule.

(5.24)

84

5 Elastic and Plastic Models in Hyperplasticity

5.2.3 Rigid-plastic Models Rigid-plastic materials are special cases in which energy is dissipated only, and not stored. They can be described without use of internal variables because there is no elastic strain. The total strain is simply equal to the plastic strain. To describe such materials without internal variables, it would be necessary to develop a theory in which the dissipation could depend on the strain rate, e. g. d d Hij , H ij . Rather than develop such a theory, we use instead the approach





already adopted, but simply introduce a set of constraints cij Hij  Dij 0 . Using the f formulation, we start from f 0 and, because of the constraints, write f c f  /ij cij and Vij Fij

wf c wHij 

wf c wDij

/ij

(5.25)

/ij

(5.26)

Then, as before, we use the dissipation function d k 2 D ij D ij together with the constraint c D kk

0 , so that Fijc



k 2 Sij D ij . Combining this with

Equations (5.25), (5.26) and the constraint equation gives Vijc



k 2 Sij H ij ,

from which both the yield surface and the flow rule follow. Note that the mean stress Vkk / kk Fkk is undetermined by the constitutive law because of the c D kk 0 constraint, and appears purely as a reaction. The same result could also have been obtained by specifying the yield surface as y Fijc Fijc  k 2 0 and deriving the expression for D ij and hence (from the constraint) H ij . If the g formulation is used, one simply starts from the expression g Vij Dij , from which it immediately follows that Hij wg wVij Dij and Fij wg wDij Vij . The rest of the formulation follows as above. Note that if the two starting functions are g and y, then no additional constraints need to be introduced. This is an example of a more general observation that the g and y formulation often offers the simplest route for deriving constitutive behaviour.

5.3 Frictional Plasticity and Non-associated Flow In the previous section, we demonstrated that hyperplasticity can reproduce simple elastic perfectly plastic models where the size of the yield surface in the deviatoric plane does not depend on the stress state. This class of models describes a wide range of materials – from metals to saturated clays. However, most granular materials exhibit a property called friction, characterized by dependence of

5.3 Frictional Plasticity and Non-associated Flow

85

shear strength on normal stress. As will be demonstrated below, the hyperplastic formulation can easily reproduce this behaviour with an interesting and important restriction – the flow rule for these models cannot be associated.

5.3.1 A Two-dimensional Model We shall start by introducing a simple frictional model defined in a twodimensional stress space  V, W , where V and W are the normal and shear stresses. The corresponding normal and shear strains are H and J, respectively. Consider the model specified by 

g

V 2 W2   VD H  WD J 2K 2G

(5.28)

PV D J

d

(5.27)

complemented by the constraint D H  E D J

c

(5.29)

0

This results in the modified dissipation function

 PV  /E D J

d c d  /c

 /D H

(5.30)

with / as a Lagrangian multiplier. The standard procedures applied to d c produce:

FV FW

wd c wD H wdc wD J

/ (5.31)



  PV  /E S D J

which, by eliminating /, can be combined to produce the yield surface in the generalised stress space (see Figure 5.4): y

F W  PV  EFV

0

(5.32)

with the flow rule wy wFV

D H

O

D J

wy O wFW

OE (5.33) OS  F W

wg wg W in (5.32) and (5.33) V and F W FW  wD J wD H gives a yield surface identical to that derived from the conventional plasticity

Substitution of FV

FV



86

5 Elastic and Plastic Models in Hyperplasticity

Figure 5.4. Yield surface and plastic potential for a frictional model

model (2.41) in true stress space (with P* = P + E) and the flow rule (2.43) with the plastic potential (2.44), as shown in true stress space in Figure 2.5.

5.3.2 Dilation As seen from Figure 5.4, within the framework of the above model, when E > 0, plastic shearing leads to positive increments of normal plastic strains which cause an increase in volume. This is a well-known phenomenon in the behaviour of dense granular materials, called dilation. Experimental data, however, show that the associated flow rule would grossly overestimate the amount of dilation. An associated flow rule is obtained in this model only if P = 0. The ability of this hyperplastic model to accommodate non-associated flow by allowing E > 0 is important for realistic modelling of the behaviour of frictional materials. Furthermore, in this model, the purely empirical observation of P* = P + E is explained for dense materials that exhibit dilation. As demonstrated in Chapter 4, the Second Law of Thermodynamics demands that mechanical dissipation be non-negative. Applied to Equation (5.28), this requirement yields P t 0. In the case P = 0, then P* = E, the dissipation is zero, and the flow is “associated” in the conventional sense. Clearly such a model of frictional behaviour is unrealistic because of the implication of zero dissipation.

5.3 Frictional Plasticity and Non-associated Flow

87

There are no restrictions on the sign of E. Clearly, when E = 0, we observe incompressible behaviour, which is achieved in the critical state. Choosing E < 0 allows for contraction, exhibited by some loose granular materials, at least for a limited range of strains.

5.3.3 The Drucker-Prager Model with Non-associated Flow Extension of the two-dimensional hyperplastic model into the general six-dimensional stress-strain space is formulated through the following potential functions: Vii V jj Vijc Vijc (5.34) g    Vij Dij 18 K 4G

G d  M kk 3

2 D cij D cij 3

(5.35)

complemented by the constraint c

D ii  B

2 D ijc D ijc 3

0

(5.36)

This results in the modified dissipation function, dc 

M Gkk  3/B 2 D ijc D ijc  /D cii 3 3

(5.37)

with / as the Lagrangian multiplier. The standard procedures applied to d c produce Fij



D ijc M Vkk  3/B 2  /Gij 3 3 D mn c D mn c

(5.38)

From which we derive Fii Fijc



3/

M Vkk  BFkk 3

(5.39) D ijc 2 3 D mn c D mn c

(5.40)

which can be rearranged to eliminate the rates of the plastic strains and give the yield surface in generalised stress space, expressed as y

3 § M Vkk  BF kk · Fijc Fijc  ¨ ¸ 0 2 3 © ¹

(5.41)

which is closely analogous to the simple two-parameter model derived in Section 5.3.1.

88

5 Elastic and Plastic Models in Hyperplasticity

5.4 Strain Hardening 5.4.1 Theory of Strain-hardening Hyperplasticity The general hyperplastic framework was presented in Chapter 4. Here we revisit briefly some of its highlights, relevant for generation of strain-hardening hyperplastic models.

Potential Functions The constitutive behaviour of a dissipative material can be completely defined by two potential functions. The first function is either the Gibbs free energy g ıij , Įij , ș or the Helmholtz free energy f İij , Įij , ș . We shall omit the









dependence on temperature in the following because we are not considering thermal effects. If g is specified, then the strain is obtained by Hij wg Vij , Dij wVij and a “generalised stress” from Fij Fij

 wg  Vij , Dij

wDij . For brevity in the following, we shall often omit the list

of arguments where a function appears within a differential, so that we would write, for instance, simply Fij wg wDij . By a suitable choice of Dij , it is possible to write the Gibbs free energy so that the only term which involves both Vij and Dij is linear in Dij : g







g1 Vij  g 2 Dij  g 3 Vij Dij

(5.42)

Furthermore, if g 3 is also linear in the stresses, then it is always possible (again by a suitable choice of Dij ) to choose simply g 3 Vij Vij :

 g  Vij , Dij g1  Vij  g 2  Dij  Vij Dij

(5.43)

The second function required to define the constitutive behaviour is (assuming that g is the specified energy function) either a dissipation function d d g Vij , Dij , D ij t 0 or a yield function y y g Vij , Dij , Fij 0 . For a rate-









independent material, dissipation is a homogeneous function of order one of the plastic strain rate tensor. In this case, dissipation and yield functions are related by a degenerate Legendre transformation. The dissipative generalised stress is defined as Fij wd g Vij , Dij , D ij wD ij . Then from a property of the transforma-



tion, it follows that D ij

Owy

g

 Vij , Dij , Fij

wFij , where O is an arbitrary non-

negative multiplier. The formulation is completed by the orthogonality assumption of Ziegler (1977), which is equivalent to the assumption Fij Fij .

5.4 Strain Hardening

89

Link to Conventional Plasticity The strain hardening hyperplastic formulation, although based entirely on specification of two potential functions, intrinsically contains all the components of conventional plasticity theory. As mentioned above, the yield function, although in generalised stress space, is obtained as a degenerate Legendre transform of the dissipation function. When the Gibbs free energy can be written in the form of (5.43), no elastic-plastic coupling occurs, and it follows that Hij





wg Vij , Dij



wVij



 D

wg1 Vij wVij

(5.44)

ij

The interpretation of the above is that Dij plays exactly the same role as the  p (elastic strain is defined as conventionally defined plastic strain Hij  e H  D wg wV ). The generalised stress in this case also acquires Hij 1 ij ij ij



a clear physical meaning: Fij





wg Vij , Dij wDij Vij  wg 2 Dij wDij . The generalised stress simply differs from the stress by the term Uij Dij wg 2 Dij wDij , known as the “back stress”. In conventional kinematic





hardening plasticity, the “back stress” would normally be associated with the stress coordinates of the centre of the yield surface. The flow rule is given by the expression D ij

O



wy g Vij , Dij , Fij



(5.45)

wFij

where O is an arbitrary non-negative multiplier. When the yield (and dissipation) function does not depend on stress explicitly, this flow rule is associated in both true and generalised stress spaces. Finally, the hardening rule is also specified by the two potential functions. Isotropic hardening is defined by dependency of the dissipation d d g Vij , Dij , D ij t 0 or yield function y g Vij , Dij , Fij 0 on the internal









variable Dij . Kinematic hardening is defined by dependency of the “back stress” Uij Dij wg 2 Dij wDij on internal variable Dij , as defined by the second term





of the Gibbs free energy function. The translation rule for the yield surface can be recognized in the expression



U ij Dij

V ij  F ij

 D

w 2 g 2 Dij

wDij wD kl

kl

(5.46)

90

5 Elastic and Plastic Models in Hyperplasticity

Dissipation and Plastic Work It is important to note the difference between the dissipation d Fij D ij , which we require to be non-negative on thermodynamic grounds, and the rate of plas p tic work W p Vij H . Consider, for instance, uncoupled plasticity for which ij



we can write g Vij , Dij







g1 Vij  g 2 Dij  Vij Dij . In this case, as discussed  p above, the internal variable plays the role of the plastic strain Dij { Hij . The

difference between the rate of plastic work and dissipation is therefore  p W p  d Vij H ij  Fij D ij Uij D ij . Using (5.46), we see that dissipation and plastic w2 g2 D kl 0 , which can only be true for general wDij wD kl w2 g2 0 . Thus the condition that the increments of the plastic strain if wDij wD kl dissipation is equal to the plastic work reduces to a simple condition on the form of the energy function.

work are identical only if

Incremental Stress-strain Response In a way similar to conventional plasticity, two possibilities exist in the description of the incremental response of a strain-hardening hyperplastic material. Consider a non-frictional material, either its state is within the yield surface ( y g Dij , Fij  0), in which case no dissipation occurs, and O = 0. If the material







point lies on the yield surface ( y g Dij , Fij



0 ), then plastic deformation can

occur, provided that O t 0 . In the latter case, the incremental response is obtained by invoking the consistency condition of the yield surface:



y g Dij , Fij



wy g wy g D ij  F ij wDij wFij

0

(5.47)

which is solved for the multiplier O:

O

wy g V ij wFij wy g w 2 g 2 wy g wy g wy g  wFij wDij wD kl wFkl wDij wFij

(5.48)

Differentiation of Equation (5.44) and substitution of (5.45) in the result and in (5.46) give the incremental stress-strain response: H ij



wy g wg1 V kl  O wVij wVkl wFij

(5.49)

5.4 Strain Hardening

91

and the update equations for internal variable and generalised stress: D ij F ij



V ij  U ij Dij

O

wy g wFij

(5.50)

V ij  O

w 2 g 2 wy g wDij wD kl wFkl

(5.51)

The multiplier O defined from Equation (5.48) is derived from the consistency condition y g Dij , Fij 0 . Therefore, it applies only when y g Dij , Fij 0 and













O ! 0 . For all other cases (i. e. either when y g Dij , Fij  0 or when (5.48) results in a negative O value), then O 0 .

5.4.2 Isotropic Hardening One-dimensional Example Consider a model with constitutive behaviour completely defined by the following specific Gibbs free energy potential function g  V, D : g  V, D 

1 2 V  VD 2E

(5.52)

From (5.43), we can derive wg V D wV E wg F  V wD

(5.53)

H 

(5.54)

The second function required is a dissipation function: d  D, D k  D D t 0

(5.55)

where k  D ! 0 . The dissipative generalised stress F is obtained as F wd  D, D wD k  D S  D , so that the yield function is most conveniently expressed as y F2   k  D

2

0

(5.56)

which defines the linear elastic range ª¬ k  D ; k  D º¼ for the generalised stress F . Differentiation of the yield surface gives D O

wy wF

2OF

(5.57)

92

5 Elastic and Plastic Models in Hyperplasticity

The incremental response of the hyperplastic model is found from Equations (5.48)–(5.51): V (5.58) H  D E D 2Ok  D S  D (5.59) F V

(5.60)

where we can obtain the solution,

O

The notation

­ V 2 , when F2  k  D 0 °° 2 k D w k wD  ® ° 2 0, when F2  k  D  0 °¯ is used for Macaulay brackets (i. e.

(5.61)

x

x, x ! 0 ;

0, x d 0 ). Two cases now follow. Either O d 0 , in which case no dissipation occurs (“elastic” behaviour), D 0 : x

V F

EH

(5.62)

or O ! 0 , in which case dissipation occurs (“plastic” behaviour), D z 0 , and we can derive E wk wD (5.63) V H E  wk wD which for linear hardening, k  D k  H D , yields V

EH S  D H E  H S  D

(5.64)

The constitutive behaviour described by these incremental equations is shown in Figure 5.5 for a first loading with positive (OAB) or negative (OCD)

Figure 5.5. Behaviour of model with linear isotropic hardening: (a) in true stress space; (b) in generalised stress space

5.4 Strain Hardening

93

Figure 5.6. Behaviour of unmodified and modified models with linear isotropic hardening on reversal of plastic strain direction

plastic strain, followed by an unloading. From Figure 5.5a, it is clearly seen that the initial linear elastic range > k; k @ of the model undergoes expansion. For plastic loading, we observe strain hardening with the tangent modulus E1 EH  E  H . In generalised stress space, where the effect of the elastic strain is eliminated, the behaviour appears as rigid-plastic with linear strain hardening/softening, as in Figure 5.5b. A further modification is required if this isotropic hardening model is to provide realistic modelling on subsequent cycles of unloading and reloading. The expression k  D k  H D would imply softening if (after first having positive plastic straining) D were later to be reduced. The hardening should be a function of the accumulated plastic strain. This can be achieved by introducing a second kinematic variable E, which is defined through the constraint equation c E  D 0 . The hardening is then expressed in the form k  E k  HE . For a first loading, this modification makes no difference to the model, but it gives realistic behaviour on unloading, as illustrated in Figure 5.6.

Multidimensional Example (the von Mises Yield Surface) The following model is an example of isotropic hardening hyperplasticity in sixdimensional stress space. The constitutive behaviour of the model is again defined by two potential functions. In this case, these are supplemented by the plastic incompressibility condition, D kk 0 , which is introduced as a constraint (see Chapter 4). The first function is the specific Gibbs free energy:



g Vij , Dij





1 1 Vll Vkk  Vijc Vijc  Vij Dij 18K 4G

(5.65)

94

5 Elastic and Plastic Models in Hyperplasticity

where a prime notation is used to denote the deviator of a tensor. It follows that Hij





wg Vij , Dij



1 1 Vkk Gij  Vijc  Dijc 9K 2G

wVij Fij





wg Vij , Dij



wDij

(5.66)

(5.67)

Vij

From this, we immediately obtain 1 Vijc  Dijc 2G 1 Hkk Vkk 3K

(5.68)

Hijc

(5.69)

The second function is a dissipation function,



d g Dijc , D ijc

k  Dijc

2D ijc D ijc t 0

(5.70)



where k Dijc ! 0 is the strength in simple shear. The plastic incompressibility condition is introduced as a side constraint, c D kk 0 , by employing a Lagrange multiplier / and considering the augmented dissipation function,



d cg Dijc , D ij

k  Dijc

2D ijc D ijc  /D kk t 0

(5.71)

The deviatoric and hydrostatic parts of the dissipative generalised stress tensor are obtained by differentiating the augmented dissipation function: Fij



wd c g Dijc , D ij





k Dijc

wD ij

2D ijc 2D ijc D ijc

 /Gij

(5.72)

From this, it follows that Fijc



k Dijc F kk

2D ijc 2D ijc D ijc 3/

(5.73) (5.74)

As expected, the mean dissipative generalised stress is undetermined by the constitutive equation. The plastic strain rates are eliminated from Equation (5.73) generating the von Mises yield function: yg

Fijc Fijc  2k 2

0

(5.75)

2OFijc

(5.76)

which in turn leads to the flow rule: D ij

O

wy g wFij

5.4 Strain Hardening

95

The incremental stress-strain response during plastic flow is easily derived using (5.48)–(5.51): 1 V ijc  D ijc 2G 1 H kk V kk 3K D ijc 2OFijc

H ijc

F ij

(5.77) (5.78) (5.79)

V ij

(5.80)

where we can show that

­ Fijc V ijc ° ° O ® 2k Dijc Fijc wk Dijc ° °¯ 0,



 wDij

, when Fijc Fijc  2k 2

0

(5.81)

when Fijc Fijc  2k2  0

This response is completely consistent with the stress-strain behaviour of the following isotropic hardening plasticity model. The linear elastic region is bounded by a von Mises yield surface, which can be expressed in true stress

 

2

0 (Figure 5.7). Within the yield surface, the stressspace as Vijc Vijc  2 k Dij strain behaviour is governed by Hooke’s law: 1 İ kk ıkk 3K (5.82) 1 İijc ıijc 2G

V1

V ' F '

V3

dD '

V2

Figure 5.7. Schematic layout of the isotropic hardening von Mises yield surface in the S plane

96

5 Elastic and Plastic Models in Hyperplasticity

During plastic flow, these expressions give the elastic component of the total strain. An associated flow rule is implied by the model, together with the plastic incompressibility condition, so that D ijc 2OVijc and D kk 0 . In linear harden-



ing, when k Dijc

k  H Dijc Dijc , the stress-strain response of the model to one-

dimensional loading is similar to that presented in Figure 5.5. To obtain realistic unloading behaviour, it is again necessary to introduce a variable representing the accumulated plastic strain, e. g. through the constraint c E  D ijc D ijc 0 and then defining k  E k  HE . The response will then be as for the modified model in Figure 5.6.

5.4.3 Kinematic Hardening One-dimensional Example Consider a model with constitutive behaviour completely defined by the following specific Gibbs free energy potential function g  V, D : g  V, D 

1 2 H 2 V  D  VD 2E 2

(5.83)

The second function required is a dissipation function (which in this case is not a function of D): d  D k D t 0

(5.84)

where k > 0. The dissipative generalised stress F is obtained as F wd  D wD kS  D , so that the yield function is most conveniently expressed as F2  k 2 0

y

(5.85)

which defines the linear elastic range > k; k @ for the generalised stress F. Differentiation of the yield surface gives D O

wy wF

2OF

(5.86)

The incremental response of the hyperplastic model is found from Equations (5.48)–(5.51): H

V  D E

(5.87)

D 2Ok S  D

(5.88)

F V  U V  2OHk S  D

(5.89)

5.4 Strain Hardening

97

where we can derive

­ S  D V , when F2  k2 0 ° O ® 2kH ° 0, when F2  k 2  0 ¯

(5.90)

Two cases again follow. Either O d 0 , in which case no dissipation occurs (“elastic” behaviour): V F

EH

(5.91)

or O ! 0 , in which case dissipation occurs (“plastic” behaviour), so that F 0 (from the consistency condition) and EH (5.92) H EH The constitutive behaviour described by these incremental equations is shown in Figure 5.8. From this figure, it is clearly seen that the linear elastic range > k; k @ of the model does not undergo any expansion, just being translated along the stress axis following the current stress state. In generalised stress space, where the effects of “back stress” and elastic strain are eliminated, the behaviour appears as rigid-perfectly plastic (Figure 5.8b). This behaviour is consistent with the stress-strain behaviour of the St-Venant model with linear kinematic hardening. The original St-Venant model (Figure 5.2) is a spring with elastic coefficient E in series with a sliding element with slip stress k. This model simulates one-dimensional linear elastic-perfectly plastic stress-strain behaviour. The linear hardening is introduced by incorporating another spring with elastic coefficient H (Figure 5.9) in parallel with the slid(e) ing element. An elongation of the E spring gives elastic strain H , whereas an elongation of the H spring gives plastic strain D, their sum gives the total strain H. During initial loading OA, before the stress reaches the value of the slip stress k, the behaviour is linear elastic and is governed by the elongation of the E V

Figure 5.8. Cyclic stress-strain behaviour of the St-Venant model with linear kinematic hardening: (a) in true stress space; (b) in generalised stress space

98

5 Elastic and Plastic Models in Hyperplasticity

Figure 5.9. Schematic layout of the St-Venant model with linear kinematic hardening

spring, i. e. the total strain is equal to the elastic strain. After the stress reaches the value of slip stress k, the sliding element slips and the H spring is extended or compressed. The corresponding behaviour is elastoplastic with a linear hardening characterized by the tangent modulus E1 , AB in Figure 5.8: V Vk dV EH (5.93)  , Ÿ E1 E H dH E  H The kinematic nature of this kind of hardening becomes clear when a stress reversal takes place. In this case, the stress in the sliding element drops below the slip value of k and the sliding element is locked. The behaviour will be linear elastic, BC; the stress V gradually decreases, so that at a certain stage, the stress stored in the H spring due to the previous loading leads to development of negative stress in the sliding element. When this stress reaches the value of –k, the sliding element slips in the direction opposite to that during loading and the extension of the H spring again changes, CD. The behaviour is elastoplastic with a linear hardening characterized by the tangent modulus E1 from Equation (5.93). Note that Martin and Nappi (1990) developed a version of kinematic hardening plasticity similar to that described here, but based their approach on the Helmholtz free energy rather than the Gibbs free energy. We find the Gibbs free energy approach more attractive, because the link to conventional stress-based plasticity theories is more transparent. e H H  D

Multidimensional Example (the von Mises Yield Surface) The following model is an example of kinematic hardening hyperplasticity in six-dimensional stress space. The constitutive behaviour of the model is again defined by two potential functions. In this case, they are supplemented by the plastic incompressibility condition D kk 0, which is introduced as a constraint (see Chapter 4). The first function is the specific Gibbs free energy:



g Vij , Dij





1 1 h Vll Vkk  Vijc Vijc  Dijc Dijc  Vij Dij 18K 4G 2

(5.94)

5.4 Strain Hardening

99

where the prime notation is used to denote the deviator of a tensor. It follows that Hij







wg Vij , Dij

1 1 Vkk Gij  Vijc  Dijc 9K 2G

wVij Fij



wg Vij , Dij





hDijc  Vij

wDij

(5.95)

(5.96)

From this, one can derive 1 Vijc  Dijc 2G 1 Hkk Vkk 3K

(5.97)

Hijc

(5.98)

The second function is a dissipation function



d g D ij

k 2D ijc D ijc t 0

(5.99)

where k ! 0 is the strength in simple shear. The plastic incompressibility condition is introduced as a side constraint by employing the Lagrange multiplier / and considering the augmented dissipation function:



d cg D ij

k 2D ijc D ijc  /D kk t 0

(5.100)

The deviatoric and hydrostatic parts of the dissipative generalised stress tensor are obtained by differentiating the augmented dissipation function: Fij



wd cg D ij wD ij Fijc

2D ijc

k

k

2Dckl Dckl 2D ijc

(5.101)

(5.102)

2Dckl Dckl

Fkk

 /Gij

3/

(5.103)

As expected, the mean dissipative generalised stress is undetermined by the constitutive equation. The plastic strain rates are eliminated from Equation (5.102), generating the von Mises yield function: yg

Fcij Fcij  2k 2

0

(5.104)

2OFcij

(5.105)

which in turn leads to the flow rule:

Dij

O

wy g wFij

100

5 Elastic and Plastic Models in Hyperplasticity

The incremental stress-strain response during plastic flow is easily derived using (5.48)–(5.51): 1 V ijc  D ijc 2G

H ijc

H kk D ij F ij

(5.106)

1 V kk 3K 2OFijc

V ij  U ij

(5.107) (5.108)

V ij  2hOFijc

(5.109)

where we can derive

­ Fijc V ijc , when Fijc Fijc  2k 2 0 °° 2 O ® 4k h ° 0, when Fijc Fijc  2k 2  0 °¯

(5.110)

Again there are two cases. In both cases the volumetric behaviour is purely elastic. Either O d 0 (elastic behaviour), in which case, 1 V ijc 2G F ij V ij

(5.111)

H ijc

(5.112)

or O ! 0 , in which case H ijc

h  2G V ijc 2Gh F ij 0

(5.113) (5.114)

This response is completely consistent with the stress-strain behaviour of the following kinematic hardening plasticity model. The linear elastic region is bounded by a von Mises yield surface, which can be expressed in true stress space as Vijc  Uijc Vijc  Uijc  2k 2 0 (Figure 5.10). Within the yield surface,







the stress-strain behaviour is governed by Hooke’s law: 1 İ kk ıkk 3K (5.115) 1 İijc ıckk 2G During plastic flow, these expressions give the elastic component of the total strain. An associated flow rule is implied by the model, together with the plastic incompressibility condition, so that D cij



2O Vijc  Uijc



and D kk

0

(5.116)

5.4 Strain Hardening

101

dD'

V1

F'

V'

U'

V3

V2

Figure 5.10. The kinematic hardening von Mises yield surface in the S plane

The model exhibits the Prager translation rule in the form U ijc Ziegler translation rule in the form U ijc



hD ijc , or the



2Oh Vijc  Uijc , which in the case of the

von Mises yield surface and associative flow rule are identical. The stress-strain response of the model to one-dimensional cyclic loading is similar to that presented in Figure 5.8.

5.4.4 Mixed Hardening Mixed hardening is obtained in hyperplastic formulation by combining isotropic hardening (dependence of the dissipation function on Dij ) with kinematic hardening (special form of the Gibbs function).

One-dimensional Example A one-dimensional model with mixed hardening is completely defined by the following specific Gibbs free energy potential function g  V, D : g  V, D 

1 2 H 2 V  D  VD 2E 2

(5.117)

The second function is a dissipation function (in this case a function of D): d  D, D k  D D t 0 where k  D ! 0 .

(5.118)

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5 Elastic and Plastic Models in Hyperplasticity

Multidimensional Example (the von Mises Yield Surface) The following model is an example of mixed hardening hyperplasticity in sixdimensional stress space. The constitutive behaviour of the model is again defined by two potential functions. In this case, they are supplemented by the plastic incompressibility condition D kk 0 which is introduced as a constraint. The first function is the specific Gibbs free energy:





g Vij , Dij



1 18 K

Vll Vkk 

1

h Vijc Vijc  Dijc Dijc  Vij Dij 4G 2

(5.119)

The second function is a dissipation function



d g Dijc , D ijc

k  Dijc

2D ijc D ijc t 0

(5.120)



where k Dijc ! 0 is the strength in simple shear.

5.5 Hierarchy of Plastic Models The main features of hyperplasticity are the compactness of the formulation and a clear link to conventional plasticity. As an illustration of these features, the hierarchy of plasticity models within the thermomechanical framework is given in Table 5.2. A Gibbs free energy formulation for isothermal conditions is taken as an example. Let us start with classification with respect to elastic-plastic coupling. A material is decoupled when the energy function can be presented as a sum of the following three terms: g1 depends on stress only, g 2 depends on internal variable, and a mixed term is the product of stress and internal variable tensors. Table 5.2. Hierarchy of plastic models

Feature

Type Coupled

Elasticity Decoupled Non-associated Flow rule Associated Perfectly plastic Isotropic Hardening Kinematic Mixed

Gibbs free energy



g Vij , Dij



  g1  Vij  Vij Dij  g 2  Dij g1  Vij  Vij Dij  g 2  Dij g1  Vij  Vij Dij g1  Vij  Vij Dij g1  Vij  Vij Dij  g 2  Dij g1  Vij  Vij Dij  g 2  Dij g1 Vij  Vij Dij  g 2 Dij

Dissipation

 d  Vij , Dij , D ij d g  Vij , Dij , D ij d g  Dij , D ij d g  Vij , D ij d g  Vij , Dij , D ij d g  Vij , D ij d g  Vij , Dij , D ij d g Vij , Dij , D ij g

5.5 Hierarchy of Plastic Models

103

Considering further only decoupled materials, let us investigate the flow rule classification. Associated flow requires the dissipation function to be independent of stress. In other words, when the dissipation function does depend on stress, as for frictional materials, normality in stress space is impossible. Finally, consider a classification for hardening. In perfectly plastic materials, the third term in the Gibbs free energy function vanishes, whereas the dissipation function should be independent of the internal variable. Strain hardening again omits the third term of the Gibbs free energy, but requires the dissipation function to depend on the internal variable. Kinematic hardening, on the contrary, leaves the dissipation function independent of the internal variable but requires all three terms in the Gibbs free energy function. Finally, mixed kinematic and isotropic hardening requires both the three terms in the Gibbs free energy function and the dependency of the dissipation function on the internal variable. As is seen, this simple classification covers the most important cases of conventional plasticity. In fact, generalisation of this framework allows derivation of even broader classes of modern plasticity models using the thermomechanical approach. First, introduction of multiple kinematic internal variables leads to the thermomechanical formulation of kinematic hardening plasticity with multiple yield surfaces. In the next step of generalization continuous internal functions instead of a discrete set of internal variables are introduced. A direct application of this generalisation is the thermomechanical formulation of kinematic hardening plasticity with an infinite number of yield surfaces. These applications will be considered in the following chapters.

Chapter 6

Advanced Plasticity Theories

6.1 Developments of Classical Plasticity Theory Chapter 2 describes the approach used for basic plasticity theories, and Chapter 5 allows these theories to be reformulated within a thermodynamically consistent framework. However, countless developments of classical theories will be found in the literature. The developments are principally to allow more realistic modelling of behaviour in unload-reload cycles. It is typically found that the model of material behaviour in which there is a single sudden transition from elastic to plastic behaviour as a single yield surface is encountered represents too dramatic a change in response. The transition from elastic to plastic behaviour, especially for certain materials such as soils, can be more progressive. Also, unload-reload loops tend to exhibit a certain amount of hysteresis in many materials, whereas the simple approach described above gives no hysteresis. Three of the most important methods used to introduce a more progressive development of plastic strain are discussed below: (a) bounding surface plasticity, (b) nested surface plasticity, and (c) multiple surface plasticity. All of these techniques involve introduction of plastic strains within the yield surface  p 0 described above. f ıij , İij





6.2 Bounding Surface Plasticity In bounding surface plasticity (see e. g. Dafalias and Herrmann, 1982), the yield surface f 0 plays a rather different role from the classical formulation. It is renamed the bounding surface because strictly no plastic strain is allowed inside a yield surface. To emphasize the change in role (and for consistency with the presentation of multiple surface plasticity below), we shall write the bounding

106

6 Advanced Plasticity Theories



 p surface as F Vij , Hij



0 . The domain enclosed by the bounding surface is not

elastic; though for some stress trajectories within this surface, an elastic response can be obtained. The stress point P Vij always lies within or on the







bounding surface. For each P Vij , a unique “image” point R VijR

on the

bounding surface, F 0 , is defined, (see Figure 6.1), according to a specific mapping rule which is a part of the constitutive relations. One possibility is the simple “radial” rule: assuming that the origin O always lies within a convex bounding surface; R VijR is defined as the intersection of F 0 and the





(Figure 6.1). Analytically, this

ı

(6.1)

straight line connecting the origin with P Vij can be expressed by ıijR



with E obtained from F ȕıij , İ(ijp)





( p)

ȕ ıij , İij

ij

0.



As a result of this mapping rule, it follows that through each point P Vij , a loading surface homothetic to the bounding surface with respect to the origin is indirectly defined by a particular value of E. It is assumed that if the incremental stress vector at the point P Vij is directed inside this surface, the be-



haviour is elastic and governed by Equation (2.7). Mathematically, the unloading condition can be expressed by noting that the gradient of the loading surface

F=0 d

(p) ij

R P O

ij

Figure 6.1. Radial mapping rule for defining an image point in bounding surface plasticity

6.3 Nested Surface Plasticity

107



at the point P Vij

is simply equal to the gradient of the bounding surface wF V ij  0 . F 0 at the “image” point R VijR . Unloading will correspond to wVijR





If the incremental stress vector at each point P Vij

is directed outward from

the loading surface, the behaviour is elastoplastic. For simplicity, we shall assume that associated flow applies, so that the surface F 0 also serves as the plastic potential. The plastic components of the incremental strain vector are defined by the equations,

 p H ij

1 wF wF V R kl h wVijR wVkl

(6.2)

where h is the hardening modulus, defined as a function of the distance from the bounding surface. Thus h h  G , where G PR is the distance between the stress point P and its image point R on F 0 in Figure 6.1. To preserve the original plasticity relations when the bounding surface is reached, we require wF wF that h  0 H  , and so that reduced plastic strains are given within  p wH wVij ij

the bounding surface h  G ! h  0 . In practice, the form of h  G should be determined by calibration against experimental unloading-reloading curves. A variety of forms have been suggested in the literature. The advantage of the bounding surface approach is that, apart from the definition of the function h  G , it involves little extra complexity by comparison with the plasticity theory from which it is derived. Bounding surface plasticity can be successful in describing hysteresis for large unload-reload loops, typically those which involve complete load reversals. For small unload-reload loops, however, it predicts a quite unrealistic “ratcheting” effect, as plastic strain occurs on every reload. It is also incapable of accounting for the effects of immediate past history on the stiffness of the response. These problems can be circumvented, to a certain extent, by the introduction of multiple yield surfaces, as described below.

6.3 Nested Surface Plasticity Plasticity models employing nested yield surfaces have become common, particularly in modelling the behaviour of soils. Although some models employ many yield surfaces, it is sufficient here to illustrate the main principles using just two surfaces. Such a model was described, for instance, by Mroz et al. (1979) for soils.

108

6 Advanced Plasticity Theories

Figure 6.2. Inner and outer yield surfaces for two-surface model



 p Assume that within the outermost yield surface F Vij , Hij



0, there exists

a smaller inner yield surface (Figure 6.2), which may translate and expand or contract and is defined by  p f Vij  Uij , Hij 0 (6.3)





where the “back stress” Uij defines the position of the yield surface and can be interpreted as the stress coordinates of its “centre”. We shall assume in the following that the associated flow rule applies, although this is not essential. We shall also assume, although again this is not essential, that the inner surface has the same shape as the outer one. The translation rule for the inner yield surface (defining changes in Uij ) and its dependence on the plastic strain should be specified. The rule is constructed so that, as the inner surface moves, it is constrained to remain entirely within the outer surface: hence the terminology “nested surface” model. Furthermore the two surfaces are allowed to touch only at the current stress point when it reaches the outer surface. It is assumed that if the stress state is inside the inner yield surface, or if it is on the yield surface and the plastic multiplier O would be negative, the behaviour is elastic and governed by Equations (2.7). If the stress state is on the inner yield surface and the plastic multiplier O is positive, the behaviour is elastoplastic, and the plastic components of the incremental strain vector are defined from the equations: wf 1 wf wf  p H ij O V kl (6.4) wVij h wVij wVkl

6.3 Nested Surface Plasticity

109

which satisfy the associated flow rule for surface f. The value of h, the plastic hardening modulus, is discussed further below. The domain enclosed by outermost yield surface F ıij , İ(ijp) 0 in this for-





mulation is, as for the bounding surface model, not elastic and for stress trajectories within this surface, plastic flow occurs once f 0 and O ! 0 . In the case of kinematic hardening, the consistency condition:



f ıij  ȡij , İ(ijp)



0

(6.5)

is not sufficient to define the hardening modulus h because it contains the additional unknown variables Uij . This problem is usually dealt with in the following manner. First it is again necessary to define an image point on the outer surface, corresponding to any stress point on the inner one. This is done, see Figure 6.2, in a way different from the bounding surface model. The image point R is defined simply as the point on the outer surface at which the direction of the normal is wf wF D the same as the that of the normal to the inner surface, i. e. , where R wVij wVij D is a positive scalar. Next it is usually assumed that the direction of movement of the “centre” of the inner surface is defined by a rule that ensures that the surfaces remain properly nested. It can be shown that this can be achieved by specifying that the relative motion of the point P with respect to its conjugate point R has to occur in the direction of PR: V ij  V ijR



P Vij  VijR



(6.6)

where P is a positive scalar. The value of P is determined by the form of the hardening rule. Either a fixed hardening modulus for the inner surface can be chosen, in which case there is a sudden change of stiffness when the outer surface is reached, or (as for the bounding surface model) the hardening modulus for the inner surface can be adjusted as a function of the distance from the outer surface, so that a smooth transition is achieved as follows. Assuming that the large-scale surface F ıij , İ(ijp) 0 is a classical strain





hardening plasticity yield surface with the associated flow rule, its hardening modulus H is determined from the consistency condition F ıij , İ(ijp) 0 as



H



wF ( p) wHij

wF wVij



(6.7)

The hardening modulus h for a stress point on the inner yield surface depends on the relative configuration of these two surfaces. It becomes equal to H

110

6 Advanced Plasticity Theories

when these two surfaces come in contact with each other. The modulus can, for instance, be specified by an expression of the form, J

hG

§ G · H  ¨ ¸  h0  H © G0 ¹

(6.8)

where as before G PR is the distance from the current stress point to the image point. The form ensures that h  0 H and h  G0 h0 . This formulation is very close in concept to bounding surface plasticity. This form of plasticity, however, avoids the problem inherent in the bounding surface models of ratcheting for small cycles of unloading and reloading. The more sophisticated rules for defining the image point and the translation of the inner yield surface provide a more realistic way for describing hysteretic behaviour and the effect of past loading history. This process can be taken a step further by introducing a set of nesting surfaces within the domain between f 0 and F 0 . These surfaces can translate and expand or contract due to plastic straining. They are capable of encoding in a more subtle way the details of the past stress history. The relative motion between each adjacent surface is defined by rules similar to (6.6) to ensure that they remain nested. The hardening modulus can be defined by an interpolation formula such that it is h0 on the innermost yield surface, and equal to H on the outermost surface. For instance, it can be assumed if the stress point is in contact with n surfaces out of a total of N: J

§ N n · h H ¨ ¸  h0  H © N 1 ¹

(6.9)

The configuration of the nesting surfaces and therefore the subsequent stiffness for a particular stress path depends on the past history of loading. The material response for any loading history may be studied by following the evolution of the configurations of the nested surfaces. This model possesses a multi-level memory structure because, for cyclically varying stress, only a certain number of surfaces undergo translation; the other surfaces may change only isotropically. This approach can be extended to an infinite number of surfaces, although for practical computations, a finite number is necessary.

6.4 Multiple Surface Plasticity Although it has certain advantages, the translation rule for multiple yield surfaces that requires that the surfaces remain nested is not strictly necessary (see Section 6.5). The principal advantage of the nested approach is that this allows the determination of a single plastic strain component, with its magnitude established by one of the above procedures for the hardening modulus.

6.4 Multiple Surface Plasticity

111

An alternative approach is to introduce multiple yield surfaces, but to treat each as independent, giving rise to a separate plastic strain component. The total strain is the sum of the elastic strain and the plastic strain components: N

Hij

e  p  n Hij  ¦ Hij

(6.10)

n 1



p n n Each yield surface is specified in the form f  ıij , İij 



0 , where for

simplicity the yield surface depends only on the plastic strains associated with that surface, and is not coupled to other yield surfaces by dependence on their plastic strains. In principle, non-associated flow can be specified readily by defining plastic potentials distinct from the yield surfaces so that n  p  n n wg . H ij O wVij Combining the elasticity relationship and the flow rule, one obtains: ı ij

N n · § n wg ¸ dijkl ¨ İ kl  O ¨ wıij ¸ n 1 © ¹

¦

(6.11)

and the plastic multipliers are eliminated by the consistency conditions for each of the yield surfaces: n n wf  wf   p n V ij  H  p n ij wVij wHij

n n n wf  wf   n wg  V ij  O  p n wVij wVij wHij

0

(6.12)

The analysis proceeds exactly as for the single surface model with the elasticplastic matrix determined as ­ n n wg  wf  ° d dmnkl ijab N ° wVab wVmn °  ep dijkl dijkl  ® n § n wf  · wg n 1 ° wf ° ¨¨ wV d pqrs   p n ¸¸ wV wHrs °¯ © pq ¹ rs

¦

½ ° °° ¾ ° ° ¿°

(6.13)

Strictly, the summations in the above equations are only for the “active” yield n surfaces, for which f n 0 ; on the other surfaces, simply, O 0 . It can be seen that the multiple surface method is simpler in concept than the nested surface method. It does not involve the plethora of ad hoc rules about translation and hardening of the inner surface, most of which are introduced simply to guarantee “nesting” rather than to reproduce any well-defined feature of material behaviour. The multiple surface models do, however, have all the advantages of nested surface models in modelling hysteresis and stress history effects. An example of this type of model was given by Houlsby (1999). This approach, too, can be extended to an infinite number of surfaces.

112

6 Advanced Plasticity Theories

6.5 Remarks on the Intersection of Yield Surfaces 6.5.1 The Non-intersection Condition The use of kinematic hardening plasticity with multiple yield surfaces has a history of more than 30 years. It has proved a very convenient framework for modelling the pre-failure behaviour of soils and other materials, allowing a realistic treatment of issues such as non-linearity at small strain and the effects of recent stress history. The growing interest in modelling small strain behaviour of soils has recently resulted in the development of many so-called “bubble” models, such as those described by Stallebrass and Taylor (1997), Kavvadas and Amorosi (1998), Rouainia and Muir Wood (1998), Gajo and Muir Wood (1999), Houlsby (1999), Puzrin and Burland (2000), and Puzrin and Kirshenboim (1999). In all the above models, except Houlsby (1999), the “translation rules” are specified to avoid intersection of the yield surfaces, and it is commonly believed that this non-intersection condition must be met, but some publications express a contrary view. This subject is discussed here because the non-intersection condition leads to unnecessary complications in kinematic hardening hyperplasticity with multiple yield surfaces. As discussed above, the simple translation rules used by Ziegler or Prager correspond to simple forms of Gibbs free energy. If non-intersection is to be imposed, much more complex energy expressions are required, in which terms involving cross-coupling between different plastic strain components must appear. Puzrin and Houlsby (2001a) argue that the condition is not necessary, but is required only when an incrementally bilinear constitutive law is to be derived. Sometimes it is claimed that, even if not strictly necessary, the non-intersection condition should be accepted on pragmatic grounds. Incremental bilinearity (and hence non-intersection) certainly offers some advantage in computation. The main one is that, if an updated stiffness approach is taken in finite element analysis, the incremental stress-strain relationship is known (for plastic loading), reducing the need for iteration. At the opposite extreme, if an incrementally non-linear approach (e. g. as in hypoplastic theories) is used, the incremental stress-strain relationship cannot be determined without prior knowledge of the path during the increment. If intersection of yield surfaces is allowed, an intermediate case occurs: the response is incrementally multilinear (see the discussion below). In practice, this does not prove to be a significant disadvantage, since for most relatively smooth stress paths, the incremental plastic response can be determined in advance for each increment.

6.5.2 Example of Intersecting Surfaces To demonstrate that the non-intersection condition is not strictly necessary, we describe here a model with two yield surfaces that are allowed to intersect. It will

6.5 Remarks on the Intersection of Yield Surfaces

113

V1 F(1) U(1)

(2)

F

V' U(2)

V2

Figure 6.3. Two-dimensional kinematic hardening plasticity model

be seen that this model poses no theoretical difficulties. Consider the plasticity model with two kinematic hardening yield surfaces in a two-dimensional stress space, as shown in Figure 6.3. The yield surfaces are 1

2 f

V

where 2 U

^V1 , V2 `T

T

 V  U  V  U   k  V  U  V  U   k

1 f

2

is

the

1

T

2

2

stress

0

1

(6.14) 2

0

2

1 U

vector;

T

^ȡ ,ȡ ` 1 1

1 2

and

 2  2 T are the coordinates of the centres of the yield surfaces, and ȡ1 , ȡ 2

^

`

k1 and k2 are their radii. Plastic yielding and hardening are calculated using an associated flow rule:

1 H p  2 H p

1 1 wf Ȝ wV 2 2 wf Ȝ wV

 2Ȝ   V  U

1 1 2Ȝ  V  U 2

2

(6.15)

114

where

 2 Hp

6 Advanced Plasticity Theories

1 1 2 O and O are non-negative multipliers; H p

T

^İ , İ ` 1 p1

1 p2

and

^İ , İ ` are the plastic strain vectors associated with each of the sur2 p1

2 p2

1  2 faces, so that the total plastic strain vector is given by H p H p  H p . Plastic hardening is calculated using Prager’s translation rule (which in this case is identical to Ziegler’s): 1 U 

1 E1H p

(6.16)

2 U 

 2 E2 H p

(6.17)

and

Finally, the elastic component of this model is defined by V EHe , where T H e ^He1 , He 2 ` is the elastic strain vector, so that the total strain vector is given by H He  H p . The model defined by Equations (6.14)–(6.17) is a particular case of the multi-surface model (7.38)–(7.39) which will be derived in Section 7.5 within the hyperplastic framework. Prager’s and Ziegler’s translation rules are known to violate the nonintersection condition. Consider as an example the case presented in Figure 6.4. During loading, the stress state P was reached, where the two surfaces touch each other (if they do not touch at only one point, they intersect and the proof is completed). Next a stress reversal took place and the stress state moved inside 1 the yield surface f  such that the current stress state V was reached, which is on this yield surface but not on the outer yield surface. The next stress increment dV is such that plastic response of the yield surface f 1 will occur, and 1 V  U1 1 , as will cause a strain increment dH p directed along the vector F prescribed by the associated flow rule (6.15). Then, according to Prager’s translation rule (6.17), the instantaneous displacement dU1 of the centre Q of the yield surface f 1 will also be directed along the vector F1 V  U1 . Therefore, if the current stress state V is located so that the angle D between the vectors F and U1 is acute, the instantaneous displacement vector dU1 will have a component directed along the ray QP. In this case, when the stress increment dV takes place, the point P on the yield surface f 1 moves into the exterior of the yield surface f  2 , and the surfaces intersect.

6.5 Remarks on the Intersection of Yield Surfaces

115

V2

dV V

F(1) U(1)

D

P

Q

O

V1

Figure 6.4. Example of a violation of the non-intersection condition

6.5.3 What Occurs when the Surfaces Intersect? There are no significant detrimental effects when yield surface intersect, provided that the plastic loading and consistency conditions are applied separately to each yield surface [see, for example, de Borst (1986)]. In this case, the constitutive relationship simply becomes multilinear instead of bilinear. Consider, for example, the kinematic hardening model with two yield surfaces described by Equations (6.14)–(6.17). The incremental stress-strain response of this model is derived by applying consistency conditions f 1 0 and 2 f  0 separately to each surface as appropriate. For this case, the following incremental relationships can be obtained: V 1 1 2 2 (6.18) H  2 Ȝ  V  U  2 Ȝ  V  U E where





1 Ȝ



­ 1 T V ° V U 1 ° , when f  0 ® 2E k 2 1 1 ° 1 °0, when f   0 ¯







(6.19)

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6 Advanced Plasticity Theories

V2 Zone 1

Zone 2

Zone 4

O

Zone 3

V1

Figure 6.5. Intersecting yield surfaces

2 Ȝ

and

­  2 T V ° V U 2 ° , when f  0 ® 2E k 2 2 2 ° 2 ° 0, when f   0 ¯





x, x ! 0; x

are Macaulay brackets (i. e. x

(6.20)

0, x d 0 ).

Assuming that the surfaces intersect at the current stress state in Figure 6.5, four different types of behaviour are encountered, depending on which of the four possible zones the incremental stress vector is directed into.

Zone 1:

­ Ȝ1 ! 0 ° Ÿ H ® 2 °¯Ȝ ! 0



1 V  U V  E E1k12

T



V

2

V

E2k22

Zone 2:

V  E



1 V  U

E1k12

1

(6.21) T

 V  U ­ Ȝ1 ! 0 ° Ÿ H ® 2 °¯Ȝ d 0

 V  U

T



V

 V  U 2

 V  U 1

(6.22)

6.6 Alternative Approaches to Material Non-linearity

Zone 3:

­ Ȝ1 d 0 ° Ÿ H ® 2 °¯Ȝ ! 0

Zone 4:

­ Ȝ1 d 0 ° Ÿ H ® 2  ¯°Ȝ d 0

T 2 V  U V V 2  V  U E E2k22









V E

117

(6.23)

(6.24)

Equations (6.21)–(6.24) represent an example of an incrementally multilinear constitutive relationship, as opposed to a bilinear one obtained when the nonintersection condition is satisfied. During loading, zones 2 and 3 would be encountered only in rather rare circumstances which would involve rather contorted stress paths. Many other recent developments in generalised plasticity, hypo-, and hyperplasticity are based on the use of incrementally non-linear and multilinear constitutive relationships. The main conclusion is that the non-intersection condition is necessary only when a bilinear constitutive law has to be derived. Intersection of yield surfaces, when treated properly, leads to multilinear constitutive relationships, which are consistent with recent developments in plasticity theory. Note that we make no case here that every model that allows intersection of yield surfaces may be theoretically consistent. It would be quite possible to formulate such a model so that it was either theoretically unacceptable or produced unjustifiable results. The case we present is simply that intersection of yield surfaces is allowable and on occasions may offer advantages.

6.6 Alternative Approaches to Material Non-linearity Plasticity theory is not the only method that has been used to model the irreversible and non-linear behaviour of rate-independent materials. For completeness, two further alternatives should be mentioned. Endochronic theory [Valanis (1975); Bazant (1978)] enjoyed some popularity at one time, but has now largely fallen into disuse. Initially it was an attempt to model irreversibility within a thermodynamic context and without recourse to yield surfaces. It concentrated instead on the use of an “intrinsic time”, which was typically identified with some measure of plastic strain. Incremental relations relating stresses, strains, and intrinsic time increment were proposed. Unfortunately, the main purpose of endochronic theory – to avoid yield surfaces – was the cause of its downfall. Real materials that exhibit rate-independent, irreversible behaviour also exhibit the phenomenon of a yield surface. Thus it became necessary to modify endochronic theory to include yield surfaces artificially. The theories became increasingly contrived, and are now rarely used. Hypoplasticity is closely related to endochronic theory, although it does not employ an intrinsic time. Instead, rate equations are proposed specifying the

118

6 Advanced Plasticity Theories

stresses in terms of the strain rates. These equations make much use of tensor analysis to identify the most general forms of first-order (but not necessarily linear) expressions for stress rate in terms of strain rate. For example, Kolymbas (1977) assumes a direct incrementally non-linear stress-strain relationship: V ij

Lijkl H kl  N ij H kl H kl

(6.25)

where Lijkl and N ij are linear operators. The early theories did not use yield surfaces, but (for the same reasons as encountered by endochronic theory) more recent theories have become increasingly complex to introduce the phenomenon of yield surfaces. The theories are still popular in some quarters, but in our view are unlikely to find long-term favour.

6.7 Comparison of Advanced Plasticity Models As seen from the above examples of different plasticity formulations, their common feature is the existence of an outer or bounding surface F 0 (in soils often defined by the degree of material consolidation). In classical plasticity strain hardening models, this surface is assumed to be a yield surface, containing an entirely elastic domain. To incorporate plastic flow within this surface, bounding surface models, nested surface models, and multiple surface models have been developed. In bounding surface models, the F 0 surface is treated as a bounding surface, and a loading surface passing through the current stress state is defined using a specific mapping rule. This mapping rule also defines the distance in the stress space between the stress state and the bounding surface, and the postulated hardening rule depends on this distance. The disadvantage of these models is the unrealistic “ratcheting” behaviour for small unload-reload cycles. In nesting surface models, the stress history of cyclic loading may be followed, and the ratcheting problem avoided by the more sophisticated rules for the evolution of the surfaces. Unfortunately, a number of ad hoc assumptions have to be introduced specifying the motion of the surfaces. True multiple surface models (without the nesting requirement) avoid these assumptions, and are simpler in concept than nested surface models. They can accommodate nonassociated flow more easily. They too have a disadvantage. Since each surface acts independently, each must be checked for yield, whilst for nested surface models, it is known that the surfaces are engaged in order from the innermost to the outermost. All multiple surface models can in principle be extended to an infinite number of surfaces. There is no definitive choice between the more sophisticated plasticity models. In the following, however, we shall develop hyperplasticity versions of multiple surface models. It will be seen that these then lead naturally to a further extension into models with an infinite number of surfaces.

Chapter 7

Multisurface Hyperplasticity

7.1 Motivation The purpose of this chapter is to present a more general framework for hyperplastic modelling of the kinematic hardening of plastic materials. In Section 5.4.3, a simple example of a single kinematically hardening yield surface was presented. The elastoplastic stress-strain behaviour of this simple model was bilinear. The stiffness is controlled by elastic moduli within the yield surface and by the hardening moduli at the surface. The limitations of this simple model become clear when its behaviour is compared to that of some real materials, in particular soils. In soils, the true linear elastic region is often negligibly small, and plastic yielding starts almost immediately with straining. The behaviour therefore appears to be highly non-linear even within the large-scale yield surface. It also appears that soil has a memory of stress-reversal history within the large-scale yield surface. A simple single surface kinematic hardening model cannot simulate these features. In an attempt to solve this problem, Iwan (1967) and Mroz (1967) introduced the concept of multiple yield (or loading) surfaces, as discussed in Chapter 6. In multiple yield surface kinematic hardening models, the size of the true linear elastic region can be reduced, even to the limiting case in which it vanishes completely. The stress-strain behaviour becomes piecewise linear and can follow more closely the true non-linearity of the material. Importantly, the model has a discrete memory of stress reversals, reflected in the relative configuration of the yield surfaces. Generalization of the multiple surface concept to an infinite number of yield surfaces produces models with a continuous field of yield surfaces. These models allow the simulation of the true non-linear stress-strain behaviour and a continuous material memory, and will be the subject of Chapter 8.

120

7 Multisurface Hyperplasticity

7.2 Multiple Internal Variables For simplicity, in Chapter 4, we considered materials which could be characterised by a single kinematic internal variable Dij , which was in the form of a second-order tensor. The kinematic internal variable can often be conveniently identified with the plastic strain. The significance of the single internal variable is that a single yield surface is derived, on which there is an abrupt change from elastic to elastic-plastic behaviour. The generalisation of the results to some other cases is straightforward; for instance, a scalar internal variable can be obtained simply by dropping the subscripts from the variables Dij , Fij , and Fij in Chapter 4. The generalisation to some more complex cases is marginally more complex. For instance, N second-order tensor internal variables would mean that the function for the Gibbs free energy g Vij , Dij , T in Chapter 4 is simply generalwg 1 N ised to g Vij , Dij ,!, Dij , T . The corresponding differential Fij  is wDij wg n replaced by Fij where n 1! N . The corresponding forms and re n wDij sults for other energy functions and differentials follow a similar pattern. When Legendre transformations are made between different functions, the number of 2 N possible transformations becomes enormous (for instance, there are 2 possible forms of the energy function). However, it is likely that only a small fraction of the possible forms would be of practical application, and so no systematic presentation of the forms with multiple internal variables is given here. If any of the N internal variables are scalars rather than tensors, then all that is necessary is to drop the subscripts from the appropriate variables. The main reason for the introduction of multiple internal variables is to allow the definition of models with multiple yield surfaces. These can be used for a variety of purposes, e. g.:









x modelling separately compression and shear effects, as may be appropriate for some granular materials (i. e. “cone and cap” models); x modelling anisotropy by using multiple kinematically hardening yield surfaces; x modelling the memory of stress reversals; and x approximation of a smooth transition from elastic to plastic behaviour.

The last of these purposes is perhaps the most important. The use of internal variables (within the thermodynamic framework) is an extremely powerful method for describing the past history of an elastic-plastic material, but suffers from the disadvantage that it inevitably leads to abrupt changes between elastic and elastic-plastic behaviour. Although using multiple internal variables allows these changes to be divided into a number of smaller steps, a completely smooth

7.3 Kinematic Hardening with Multiple Yield Surfaces

121

transition can be achieved only by introducing an infinite number of internal variables. Such an idea leads to the concept of an internal function rather than internal variables. The generalisation of the results given in Chapter 4 to internal functions is rather more complex than the generalisations to multiple variables discussed above and will be the subject of Chapter 8.

7.3 Kinematic Hardening with Multiple Yield Surfaces 7.3.1 Potential Functions The model with a single yield surface presented in Section 5.4.3 can be extended to multiple yield surfaces by modifying the two potential functions that define the constitutive behaviour. The specific Gibbs free energy becomes a function of n the stress and a finite number of internal variables Dij ,  n 1,!, N , where N is the total number of the yield surfaces. We choose the Gibbs free energy in the following form:



1 N g Vij , Dij ,!, Dij



N

N

n 1

n 1



n n n g1 Vij  Vij ¦ Dij  ¦ g 2 Dij



(7.1)

The dissipation function also becomes a function of the finite number of intern nal variables and their rates D ij ,  n 1,!, N :



1  N 1 N d g Vij , Dij ,!, Dij , D ij ,!, D ij N

¦d

g n

n 1



1



 N n Vij , Dij ,!, Dij , D ij t 0

(7.2)

where we assume that the dissipation can be decomposed into additive terms involving each individual plastic strain tensor. For a rate-independent material, the dissipation is a homogeneous first-order function of the plastic strain rate tensor. For associated plasticity, the dissipation function is independent of the stress. We shall consider only such cases in the remainder of this chapter, and so we drop the dependence on Vij in Equation (7.2).

7.3.2 Link to Conventional Plasticity In the conventional formulation of multiple surface kinematic hardening plasticity, calculation of incremental stress-strain response requires the equations to be defined explicitly for all the yield surfaces. Then, for each yield surface, the following rules are specified:

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7 Multisurface Hyperplasticity

x the flow rule (or more usually the plastic potential function), x the strain hardening rule, x the translation rule.

All these equations and the resulting incremental stress-strain response can be derived from the potentials (7.1) and (7.2) using Legendre transformations and their properties for active and passive variables (see Appendix C). The yield function is related to the dissipation function (7.2) by the Legendre n transform, where the rate of each internal variable D ij is interchanged with the n corresponding dissipative generalised stress Fij , n 1,!, N :

n F ij



1  N 1 N wd g Dij ,!, Dij , D ij ,!, D ij





g n 1  N n wd  Dij ,!, Dij , D ij

n wD ij

n wD ij



(7.3)

This transformation is a degenerate special case of the Legendre transformation because the dissipation is homogeneous and first order in the rates. Therefore, for each n 1,!, N , n n n g n g n (7.4) O y  Fij D ij  d  0 where y g n



g n 1  N n y  Dij ,!, Dij , Fij



n 0 is the nth yield function and O is

an arbitrary non-negative multiplier. As seen from Equations (7.4), all N yield functions are contained in the equation of the dissipation function (7.2) in a compact form. The Gibbs free energy function (7.1) allows the definition of the strain tensor: 1  2 N wg Vij , Dij , Dij ,!, Dij N wg1 Vij n (7.5) Hij    ¦ Dij wVij wVij n 1





 N

where we see that the sum of the internal variables

¦ Dij n

plays exactly the

n 1

p same role as the conventionally defined plastic strain Hij and each individual n internal variable Dij can be interpreted as a component of plastic strain associ-

ated with the plastic flow on the nth yield surface. It is also convenient to define e the elastic strain Hij wg1 wVij . The flow rule for the nth yield surface is obtained from the properties of the Legendre transformation (7.4): g n 1  N n wy  Dij ,!, Dij , Fij n  n  (7.6) D ij O , n 1,!, N n wFij





7.3 Kinematic Hardening with Multiple Yield Surfaces

123

We restricted the dissipation function to exhibit no explicit dependence on the true stresses, so it follows that the normality represented by Equations (7.6) in generalised stress space also holds in true stress space. n The dependence of the dissipation function on internal variables Dij is transferred to each yield function by the Legendre transformation (7.4) with n Dij a passive variable. Therefore, the strain hardening rule is obtained automatically through the functional dependence of the yield function on plastic n strains Dij . We shall limit our analysis here to materials with a dissipation n function (and hence yield functions) independent of internal variables Dij , i. e. to materials undergoing pure kinematic hardening. The generalised stress is defined by differentiating the Gibbs free energy function with respect to the internal variable:

n F ij





1  2 N wg Vij , Dij , Dij ,!, Dij n wD



Vij 

 ,

n n wg 2 Dij

ij

n wDij

n 1,!, N

(7.7)

It is convenient at this stage to introduce the “back stress” associated with each internal variable that is simply defined as the difference between the true and generalised stress. By applying Ziegler’s orthogonality principle (in the form n F(ijn) F(ijn) ), the “back stress” Uij can be expressed as



n n Uij Dij

n Vij  Fij

 ,

n n wg 2 Dij n wDij

n 1,}, N

(7.8)

which, after differentiation for the nth yield surface, gives



n n Uij Dij

n V ij  F ij



n n w 2 g 2 Dij n D , n 1,!, N  n n kl wDij wD kl

(7.9)

n As in conventional plasticity, Uij defines the coordinates of the centre of the nth

yield surface in true stress space. Equation (7.9) can therefore be interpreted as the translation rule for the nth yield surface.

7.3.3 Incremental Response In a similar way to the description of the incremental response of a hyperplastic material with a single yield surface, two possibilities exist for each of the N yield

124

7 Multisurface Hyperplasticity



n g n surfaces. If the material state is within the nth yield surface §¨ y  Fij  0 ·¸ , © ¹ n then no dissipation occurs and O 0 . If the material point lies on the yield n surface §¨ y g  n Fij 0 ·¸ , then plastic deformation can occur provided that © ¹ n O t 0 . In the latter case, the incremental response is obtained by invoking the consistency condition for the yield surface:



g n wy   n F n ij wFij

g n y 

0

(7.10)

n Substitution of (7.6) and (7.9) in (7.10) leads to the solution for the multiplier O : g n wy  V  n ij wFij

n O

wy

g n

n

wFij

2

n

(7.11)

g n

w g2 wy n n n wDij wD kl wFkl

Differentiation of Equation (7.5) and substitution of (7.6) in both the result and in (7.9) gives the incremental stress-strain response, H ij



 V

wg1 Vij

wVij wVkl

N

kl



¦ On

n 1

g n wy  n wFij

(7.12)

and the update equations for the internal variables and generalised stress,

n F ij

n D ij

g n n wy O n wFij



n V ij  O 

n n V ij  Uij Dij

(7.13)

n g n w 2 g2 wy  n n n wDij wDkl wFkl



n n n where O is defined from Equation (7.11) when y  Fij n Otherwise O



(7.14)

n 0 and O ! 0 .

n n 0 (when y n Fij  0 or when (7.11) gives a negative O

value). Description of the constitutive behaviour during any transient loading ren n quires the values of Fij and Dij ,  n 1,! N to be known throughout, but they are updated using Equations (7.13) and (7.14). A summary of the comparison between single and multiple surface models is given in Table 7.1.

7.4 One-dimensional Example (the Iwan Model)

125

Table 7.1. Examples of comparisons between single and multiple internal variable formulations

Single internal variable Variables

Vij , Hij T, s Dij , Fij , Fij

Typical energy function

g Vij , Dij , T

Typical dissipation function

d g Vij , Dij , T, D ij

Typical yield function

y g Vij , Dij , T, Fij

Typical derivatives

Hij

Incremental response

Equations (4.22)–(4.29)



Multiple internal variables

Vij , Hij T, s n n n Dij , Fij , Fij





1 N g Vij , Dij ,!, Dij , T













1 N 1 N d g Vij , Dij ,!, Dij , T, D ij ,!, D ij 0

wg wVij wg Fij  wDij wg s  wT



1 N n g n y  Vij , Dij ,!, Dij , T, Fij





0

wg wVij wg n Fij  n wDij wg s  wT



Hij



Equations (7.11)–(7.14)

7.4 One-dimensional Example (the Iwan Model) We first illustrate multiple surface models using a one-dimensional example. This type of model was described (although using slightly different terminology) by Iwan (1967). The constitutive behaviour of the model is defined by two potential functions: g  V, D1 !D N 

N 1 2 1 N V  ¦ Hn Dn2  V ¦ Dn 2E 2n 1 n 1

(7.15)

N

d  D 1 !D N

¦ kn D n

(7.16)

n 1

For convenience (and without any loss of generality), we shall choose the numbering of the internal variables such that kn ! kn1 for all n 2! N . The dissipative generalised stress Fn is obtained from (7.3): Fn wd wD n kn S  D n , so that the nth yield function is given by

yn

Fn  kn

0

(7.17)

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7 Multisurface Hyperplasticity

The incremental Equations (7.11)–(7.14) reduce to H

N V n  2 O kn S  D n E n 1

¦

D n F n

V  U n

(7.18)

n 2O  kn S  D n

(7.19)

n V  2O  H n kn S  D n

(7.20)

­ S  D n V , when Fn kn ° ° where On ® 2kn Hn . ° 0, when Fn  kn °¯ n For each yield function, one of two cases takes place. If O d 0 , no dissipation related to the nth yield function occurs, so that D n 0 and F n V . Altern natively O ! 0 , in which case dissipation occurs (“plastic” behaviour), so that

F n

0 , and for monotonic loading, § 1 N* 1 · H ¨  ¸ V ¨E H ¸ n 1 n¹ ©

¦

(7.21)

where N* is the largest n for which Fn  kn 0 , i. e. the number of the largest active yield surface. The cyclic stress-strain behaviour of the Iwan model during initial loading, unloading, and subsequent reloading, governed by the same incremental relations (7.18)–(7.20), is presented in Figure 7.1. This stress-strain behaviour is identical to that described by Iwan (1967), who described a model built from one spring with elastic coefficient E and a series of sliding elements with slip stresses kn , each in parallel with a spring with corresponding elastic coefficient H n (Figure 7.2). An elongation of the E spring gives e elastic strain H , whereas an elongation of each of the H springs contributes n

the plastic strain Dn to the total plastic strain; the sum of elastic and all plastic strains gives the total strain H . During initial loading, before the stress reaches the value of the first slip stress k1 , the behaviour is linear elastic and is governed by the elongation of the E spring, i. e. the total strain is equal to the elastic strain. After the stress reaches the value of the slip stress k1 , the first sliding element slips and the H1 spring becomes active. The corresponding behaviour is elastoplastic with a linear hardening characterized by the tangent modulus E1 EH1  E  H1 . After the stress reaches the value of slip stress kN * , the N*th sliding element slips and the H N * spring becomes active. The corresponding behaviour is elastoplastic with linear

7.4 One-dimensional Example (the Iwan Model)

V UN U2

D1

H(e)

kN

E2

k2

U1

D2

127

DN EN E

E1

2k1

k1

2k2 E

2kN H

E1

E1

E2

E EN

Figure 7.2. Cyclic stress-strain behaviour of the Iwan model

hardening characterized by the tangent modulus EN * , which can be determined 1 1 N* 1 from the relationship . The strain is calculated as: ¦ EN * E n 1 Hn e H H 

N*

¦ Dn

n 1

V N * V  kn  E n 1 Hn

¦

(7.22)

When stress reversal takes place, the stress in each sliding element drops below the slip value of kn and the sliding element becomes locked. The behaviour will be linear elastic; the stress V gradually decreases, so that at a certain stage, the stress in the H1 spring due to the previous loading leads to development of negative stress in the first sliding element. When this stress reaches the value of

H1

H2

HN

E V

k1

k2

kN

D1

D2

DN H

Figure 7.1. Schematic layout of the Iwan model

H(e)

128

7 Multisurface Hyperplasticity

k1 , the sliding element slips in a direction opposite that during loading and the H1 spring becomes active again. The behaviour is elastoplastic with linear hardening characterized by the tangent modulus E1 . A further decrease in stress would bring the stress in the N*th sliding element to kN * . This would cause it to slip with shortening of the H N * spring, so that the stress-strain behaviour is characterized by the tangent modulus EN * as defined above. If unloading to higher stresses occurs, then sliding elements which had not previously been activated may now become active.

7.5 Multidimensional Example (von Mises Yield Surfaces) A simple example of a hyperplastic model with multiple yield surfaces in sixdimensional stress space is an extension of the model in Section 5.4.3. The model is again defined by two potential functions, supplemented by the plastic n) incompressibility condition D(kk 0:



1 N g Vij , Dij !Dij





1 1 Vll Vkk  Vijc Vijc 18 K 4G

N 1 N n n n n h Dijc Dijc  Vij Dij  2n 1 n 1

¦



1 N d g D ij !D ij



(7.23)

¦

N

¦ kn

n n 2D ijc D ijc t 0

(7.24)

n 1

n where k is the parameter defining the size of the nth yield surface. It follows that

Hij





1 N wg Vij , Dij !Dij



wVij Hijc

N 1 1 n Vkk Gij  Vcij  Dcij 9K 2G n 1

¦

N 1 n Vijc  Dijc 2G n 1 1 Hkk Vkk 3K

¦

(7.25)

(7.26) (7.27)

The plastic incompressibility condition is included by employing a Lagrange multiplier / and considering the augmented dissipation function:



1 N dc g D ij ,!, D ij



N

¦ k n

n 1

n n 2D ijc D ijc  /

N

¦ D kk t 0

n 1

n

(7.28)

7.5 Multidimensional Example (von Mises Yield Surfaces)

129

The deviatoric and isotropic parts of the dissipative generalised stress tensor are obtained using (7.3): wd cg n wD ij

n

Fij

n Fijc

n k

n k

n F kk

n 2D ijc n n 2D ijc D ijc

 /Gij

n 2D ijc

(7.29)

(7.30)

n n 2D ijc D ijc

(7.31)

3/

The plastic strain rates are eliminated from Equation (7.30), generating the nth von Mises yield function, g n y 



n n n Fcij Fijc  2 k

n Differentiation of (7.32) yields wy g n wFijc

2

0

(7.32)

n 2Fijc , so that using (7.11)–

(7.14), we obtain H ijc

n F ij

N 1 n V ijc  D ij 2G n 1 1 H kk V kk 3K n n n D ij 2O  Fijc

n V ij  U ij

¦

(7.33) (7.34) (7.35)

n n n V ij  2h O  Fijc

(7.36)

­ n ° 2 Fijc V cij cij n Fcij n  2 k n F , when 0 ° 2 ° n where O ® 4 kn h n ° 2 ° n n n 0, when Fijc Fijc  2 k  0 ° ¯ Again, there are two cases for each yield surface. In both cases, the volumetric n behaviour is purely elastic. If O d 0 , no dissipation related to the nth yield n n  n  function occurs, so that D ij 0 and F ij Vij . Alternatively, O ! 0 , in n which case dissipation occurs (“plastic” behaviour), so that F ij 0 , and for







monotonic loading,

130

7 Multisurface Hyperplasticity

*

ª 1 N 1 º «  » V ijc «¬ 2G n 1 h n »¼

¦

H ijc



n n n where N* is the largest n such that Fijc Fijc  2 k

(7.37) 2

0.

The above hyperplastic model is equivalent to the kinematic hardening multiple von Mises yield surfaces model presented in Figure 7.3 (for simplicity, the decomposition of the stress Vij into additive terms of back stress U(ijn) and generalised stress F(ijn) is shown for only one of the yield surfaces). A model of this type is described by Houlsby (1999). The set of N von Mises yield surfaces in true stress space is given by







n n n Vijc  Uijc Vijc  Uijc  2 k

2

0

(7.38)

The elastic component of strain is calculated according to Hooke’s law. An associated flow rule is assumed together with the plastic incompressibility condition, so that

n D ijc



n n O Vijc  Uijc

n D kk



(7.39) (7.40)

0

dD'1 dD'2

V1

V'

dD'3

F'1 U'1

V3

V2

Figure 7.3. Schematic layout of the kinematic hardening multiple von Mises yield surfaces in the S plane

7.6 Summary

131

The translation rule resulting from the formulation can be expressed either in n n n the form of the Prager (1949) translation rule: U ijc h D ijc , or in the form of  n 2O(n)h(n) Vcn  Uc n the Ziegler (1959) translation rule: Uijc , which in this ij ij





case are identical. The Mroz (1967) translation rule would also give the same result for proportional loading, but not for other cases. The stress-strain response of the model to one-dimensional cyclic loading is similar to that presented in Figure 7.1.

7.6 Summary In this chapter, we have generalised results previously obtained in Chapter 4 for plastic materials with a single tensorial internal variable to the case of multiple internal variables. The motivation is to allow the development of more sophisticated models and, in particular, to incorporate plastic strains within a largescale yield surface and the material memory for modelling cyclic and transient behaviour. The case of an infinite number of internal variables (i. e. an internal variable function) is considered in Chapter 8. It involves replacing energy and dissipation functions by equivalent functionals, resulting in a continuous hyperplastic formulation.

Chapter 8

Continuous Hyperplasticity

8.1 Generalised Thermodynamics and Rational Mechanics As mentioned in Chapter 3, the theoretical approaches to the mechanics of inelastic materials can be divided into two main classes, which are often termed generalised thermodynamics and rational mechanics. The generalised thermodynamics approach (which is used here) makes much use of internal variables to describe the history of loading, and the current response is expressed in terms of functions of the stress and/or strain state and the internal variables. The rational mechanics approach [see, for example, Truesdell (1977)] instead expresses the response in terms of functionals of the history of the material (usually through the history of strain and temperature). Both approaches have advantages and drawbacks. Rational mechanics achieves great generality, but at the expense that it has so far proved difficult to express simple material models for inelastic materials within this framework. Generalised thermodynamics has been a very successful framework for simple models, but has the disadvantage that the use of internal variables sometimes oversimplifies the response. In particular, it is difficult to express smooth transitions of behaviour using internal variables. In this chapter, we address this shortcoming of generalised thermodynamics by extending the concept of the internal variable to that of an internal function. The response is then expressed in terms of functionals, and so offers some of the advantages achieved by rational mechanics. It is suggested that this approach may provide some link between the two frameworks of generalised thermodynamics and rational mechanics, although we have not pursued that route. One reason that the rational mechanics approach has not found favour in some quarters is that it requires the specification of a tensor-valued functional (the stress as a functional of the strain history). This is clearly a challenging task. An advantage of the approach adopted here is that the functionals which

134

8 Continuous Hyperplasticity

have to be determined are scalar-valued, and therefore are expected to have a simpler functional form.

8.2 Internal Functions In general, the internal function will be expressed in terms of a variable K which ˆ ij  K . we will term the internal coordinate; so we write the internal function as D In many cases, K will not have any obvious physical interpretation, but in some cases it may. For instance, in a model with N internal variables, these are provided with a set of indices i 1!N . Generalising this idea to a continuous field of internal variables, the internal coordinate K simply replaces the index i, and the domain Y of K replaces the set of integers 1!N . Often it will be convenient simply to take Y as the set of real numbers from 0 to 1. In a particular model, each plastic strain component may be associated with a sliding element with a particular slip stress ranging from zero to some maximum value, say k. In that case, the slip stress for each sliding element could be taken as Kk, and K has a simple physical interpretation. This is pursued as an example in Section 8.7. ˆ ) to distinguish any variable which is We use the “hat” notation (e. g. D a function of the internal coordinate from a previously used variable with the same name.

8.3 Energy and Dissipation Functionals 8.3.1 Energy Functional Chapter 4 presents a general formulation in which a number of alternative forms of energy functions were used. Here, we shall pursue only one of these alternatives. Other forms can be obtained by analogous developments. We take the example of the Gibbs free energy. The free energy function will now become ˆ ij  K , Tº . Note that we use square brackets [ ] a free energy functional g ª¬Vij , D ¼

to distinguish a functional from a function. In loose terms, a functional may be defined as a “function of a function”. We shall assume for the present that the functional can be written in the particular form: ˆ ij , Tº g ª¬Vij , D ¼

³ gˆ  Vij , Dˆ ij  K , T, K *  K dK

(8.1)

8

where Y is the domain of K. Other more general forms of functional are possible, but the form in Equation (8.1) proves of practical use and importance. It is convenient to introduce for generality the (non-negative) weighting function *  K

8.3 Energy and Dissipation Functionals

135

within the integral in Equation (8.1), as this adds a useful element of flexibility to a later aspect of the formulation. Alternatively, *  K can simply be taken as unity, and its role simply absorbed within the function gˆ . In some cases, it may be more convenient to consider the free energy as the sum of a function and a functional:









ˆ ij , T º g1 Vij , T  gˆ2 Vij , Dˆ ij  K , T, K *  K dK g ª¬Vij , D ³ ¼

(8.2)

8

For simplicity, however, we shall first describe just the functional form, Equation (8.1), here. In any case, g1 in Equation (8.2) can be included with gˆ 2 within the integral simply by dividing by the constant ³ *  K dK . 8

8.3.2 Generalised Stress Function Chapter 4 uses a generalised stress, which is work-conjugate to the internal kiwg . Corresponding, therefore, to the nematic variable and is defined by Fij  wDij kinematic internal function is a generalised stress function Fˆ  K . For the single ij

internal variable, it is easy to show (using definitions in Chapter 4) that

g

Hij V ij  Fij D ij  sT

(8.3)

For multiple internal variables, this simply becomes N

g

Hij V ij 

¦ F(ijn)D (ijn)  sT

(8.4)

n 1

Generalising for the case of a functional, we obtain the Frechet time derivative of the Gibbs free energy (see Appendix B), which yields the result, g

ˆ ij  K *  K dK  sT Hij V ij  Fˆ ij  K D

³

(8.5)

8

where we have introduced the definition of the “generalised stress function”, Fˆ ij



wgˆ ˆ ij wD

(8.6)

Equation (8.5) can be seen as a generalisation of (8.4) when the finite number of internal variables becomes infinite, and it is treated as a continuous function.

136

8 Continuous Hyperplasticity

8.3.3 Dissipation Functional From the First Law of Thermodynamics, the definition of the mechanical dissipation d (Chapter 4), it follows that

g

Vij H ij  sT  d

(8.7)

Comparing with Equation (8.5), we can see that d

³ Fˆ ij  K Dˆ ij  K *  K dK

(8.8)

8

In Chapter 4, the dissipation is defined as a function of state and rate of change of internal variable. Similary, we can write a dissipation functional in the form: ˆ ij , T, D ˆ ij º d g ªVij , D ¬ ¼

g ³ dˆ  Vij , Dˆ ij  K , T, Dˆ ij  K , K *  K dK t 0

(8.9)

8

which can be compared with Equation (4.7). We use the superscript g simply to indicate that the arguments of the function are those used for the Gibbs free energy: numerically d d g .

8.3.4 Dissipative Generalised Stress Function By analogy with the definition Fij

wd g , we define: wD ij wdˆ g ˆ wD

Fˆ ij

(8.10)

ij

ˆ ij for rate-independent materials, it follows Since dˆ g will be first order in D from Euler’s theorem that



ˆ ij  K , T, D ˆ ij  K , K dˆ g Vij , D



ˆ ij  K Fˆ ij  K D

(8.11)

³ Fˆ ij  K Dˆ ij  K *  K dK

(8.12)

and so, ˆ ij , T, D ˆ ij º d g ªVij , D ¬ ¼

8

Equations (8.8) and (8.12) can be compared with Equations (4.6) and (4.8). It follows from (8.8) and (8.12) that

³  Fˆ ij  K  Fˆ ij  K Dˆ ij  K *  K dK

0

(8.13)

8

which compares with Equation (4.9). At first glance, Equation (8.13) would suggest Fˆ ij  K  Fˆ ij  K 0 , but strictly it implies the much weaker condition that





8.4 Legendre Transformations of the Functionals

137

 Fˆ ij  K  Fˆ ij  K

ˆ ij are orthogonal functions. We shall adopt the strong and D condition Fˆ ij  K Fˆ ij  K here as a generalisation of Ziegler’s orthogonality. This condition is entirely consistent with the Second Law of Thermodynamics but is a more restrictive statement. In a similar way to Chapter 4, we shall adopt it here simply as a constitutive hypothesis: it defines a class of materials that satisfy thermodynamics. Wider classes of materials that satisfy thermodynamics but violate our constitutive hypothesis could exist. However, the class defined by this hypothesis proves very wide, encompassing realistic descriptions of many materials. Furthermore, these descriptions are very compact in that only two scalar functionals need to be defined. We have adopted the Fˆ and Fˆ notation to indicate the fact that these quantities are always equal in our formulation (by hypothesis), but are separately defined [Equations (8.6) and (8.10)].

8.4 Legendre Transformations of the Functionals 8.4.1 Legendre Transformations of the Energy Functional In the original approach, a variety of Legendre transformations between energy functions were used, e. g. f Hij , Dij , T g Vij , Dij , T  Vij Hij . Transformations







that involve variables (as opposed to functions of internal coordinates) are simiˆ ij , T º ˆ ij , T º  Vij Hij . Those involving lar to the original form, e. g. f ª¬Hij , D g ª¬Vij , D ¼ ¼ the internal function and the generalised stress function are slightly more complex. Thus instead of the original g Vij , Dij , T g Vij , Fij , T  Fij Dij we have





ˆ ij  K , T, K gˆ Vij , D





gˆ  Vij , Fˆ ij  K , T, K  Fˆ ij  K Dˆ ij  K

(8.14)

together with ˆ ij D



wgˆ wFˆ

(8.15)

ij

(See Appendix C for details of the Legendre transformation methods for functionals.) We can also define g ª¬Vij , Fˆ ij , T º¼

³ gˆ  Vij , Fˆ ij  K , T, K *  K dK

8

(8.16)

ˆ ij , T º  Fˆ ij  K Dˆ ij  K *  K dK g ª¬Vij , D ¼

³

8

138

8 Continuous Hyperplasticity

8.4.2 Legendre Transformation of the Dissipation Functional The only relevant transformation is the singular transformation from the dissipation functional to the yield functional. The original transformation was, for instance, of the form Oy g Vij , Dij , T, Fij Fij D ij  d g Vij , Dij , T, D ij 0 together g

with the result D ij

O









wy . This now becomes (see Appendix C.7.3): wFij



ˆ ij  K , T, Fˆ ij  K , K Oˆ  K yˆ g Vij , D





ˆ  K  dˆ g V , D ˆ  K , K Fˆ ij  K D ij ij ˆ ij  K , T, D ij



(8.17) 0

together with the result that ˆ ij D

g

wyˆ Oˆ  K wFˆ ij

(8.18)

which is the analogy of the normality condition in conventional “associated” plasticity. Note, however, that in the conventional approach, the plastic strain rate (the internal variable rate) is given by the differential of the yield function with respect to the stress; here it is given by the differential with respect to the generalised stress. This allows the current formulation to encompass nonassociated flow. It is possible to define a yield functional by the integration ˆ ij , T, Fˆ ij º ˆ ij  K , T, Fˆ ij K *  K dK 0 , but it is unclear Oy g ª¬Vij , D Oˆ  K yˆ g Vij , D ¼

³





8

that this would serve any useful purpose.

8.5 Incremental Response Chapter 4 demonstrates how, given knowledge of the energy function and the yield function, it is possible to derive the entire incremental response for an elastic-plastic material within the adopted formalism. This is of particular importance because non-linear material models are frequently implemented in finite element codes for which an incremental response is required. The derivation of the incremental response begins with differentiation of the energy function, giving the results summarised in the sixth row of Table 8.1. Further differentiation gives the rates of the variables. This is set out for the single internal variable in general form as Equation (4.15), which, for the particular case of the Gibbs free energy, takes the following form, easily obtained by double differentiation of the Gibbs free energy:

8.5 Incremental Response

­ H ij ½ ° ° ®F ij ¾ ° ° ¯ s ¿

ª w2 g « « wVij wVkl « 2 « w g « wD wV « ij kl « w2 g « «¬ wTwVkl

w2 g wVij wDkl

w2 g º » wVij wT » » ­ V kl ½ w2 g » ° ° ®D kl ¾ wDij wT » °  ° »¯ T ¿ 2 » w g » wT2 »¼

w2 g wDij wDkl w2 g wTwD kl

139

(8.19)

In the new formulation, this becomes §

w 2 gˆ

w 2 gˆ

w 2 gˆ

·

³ ¨¨ wVij wVkl V kl  wVij wDˆ kl Dˆ kl  K  wVij wT T ¸¸ *  K dK

H ij

8©

Fˆ ij  K

w 2 gˆ w 2 gˆ  w 2 gˆ ˆ kl  K  V kl  D T ˆ ij wVkl ˆ ij wD ˆ kl ˆ ij  K wT wD wD wD

§ w 2 gˆ w 2 gˆ  w 2 gˆ  · ˆ  V  D K  T ¸ *  K dK ¨  ³ ¨ wTwVkl kl wTwDˆ kl kl wT2 ¸¹ 8©

s

(8.20)

¹

(8.21)

(8.22)

Table 8.1. Examples of comparisons between different formulations

Single internal variable Variables

Vij , Hij T, s Dij , Fij , Fij



Typical energy g Vij , Dij , T function(al)



d g Vij , Dij , T, D ij

Typical yield function

y g Vij , Dij , T, Fij

Typical derivatives

Hij

Incremental response

³ 



Typical dissipation function(al)

Internal function

Vij , Hij T, s ˆ ij  K , Fˆ ij  K , Fˆ ij  K D ˆ ij  K , T, K *  K dK g gˆ Vij , D 8



dg



g ³ dˆ  Vij , Dˆ ij  K , T, Dˆ ij  K , K *  K dK

8





wg wVij wg Fij  wDij wg s  wT 

Equations (4.22) to (4.29)

0



ˆ ij  K , T, Fˆ ij  K , K yˆ g Vij , D

Hij



wg wVij

Fˆ ij  K 

s 

wg wT



wgˆ



0

³ wVij *  K dK

8

wgˆ ˆ ij  K wD wgˆ  *  K dK wT

³

8

Equations. (8.33) to (8.41)

140

8 Continuous Hyperplasticity

Equations (8.20)–(8.22) are used together with the flow rule, Equation (8.18), to derive w 2 gˆ

§

w 2 gˆ

wyˆ g

w 2 gˆ

·

³ ¨¨ wVij wVkl V kl  wVij wDˆ kl Oˆ  K wFˆ kl  wVij wT T ¸¸ *  K dK

H ij

8©

wyˆ g w 2 gˆ w 2 gˆ ˆ w 2 gˆ  V kl  O  K  T ˆ ij wVkl ˆ ij wD ˆ kl ˆ ij wT wD wD wFˆ kl wD

(8.24)

§ w 2 gˆ wyˆ g w 2 gˆ  · w 2 gˆ ˆ  V  O K  T ¸ *  K dK ¨  kl ³ ¨ wTwVkl ˆ kl wTwD wFˆ kl wT2 ¹¸ 8©

(8.25)

Fˆ ij  K

s

(8.23)

¹

The multiplier function Oˆ is obtained by substituting the above equations in the consistency condition, which is obtained by differentiation of the yield function. Equation (4.17) results in the condition, y g

wy g wy g wy g  wy g V ij  D ij  T F ij wVij wDij wT wFij

0

(8.26)

and for the functional approach, this now becomes

yˆ g  K

wyˆ g wyˆ g  wyˆ g  wyˆ g  ˆ ij  K  V ij  D T Fˆ ij  K 0 ˆ ij wVij wD wT wFˆ ij

(8.27)

which leads immediately (on substitution of Equations (8.18) and (8.24) to wyˆ g wyˆ g ˆ wyˆ g wyˆ g  V ij  O  K  T ˆ ij wVij wD wFˆ ij wT wyˆ g § w 2 gˆ wyˆ g w 2 gˆ ˆ w 2 gˆ  · V kl  O  K  T¸ 0 ¨ ˆ ij wVkl ˆ ij wD ˆ kl ˆ ij wT ¸ wFˆ ij ¨© wD wD wFˆ kl wD ¹

(8.28)

From this, we obtain ˆ g V  K ˆ g T  K A A ij Oˆ  K  V ij  T Bˆ g  K Bˆ g  K

(8.29)

where

ˆ g V  K A ij ˆ g T  K A

wyˆ g wyˆ g w 2 gˆ  ˆ kl wVij wVij wFˆ kl wD

(8.30)

wyˆ g wyˆ g w 2 gˆ  ˆ kl wT wT wFˆ kl wD

(8.31)

§ wyˆ g wyˆ g w 2 gˆ Bˆ g  K ¨  ¨ wD ˆ ˆ ˆ ˆ © ij wFkl wD kl wDij

· wyˆ g ¸ ¸ wFˆ ij ¹

(8.32)

8.5 Incremental Response

141

Note that Equations (8.29)–(8.32) are analogous to Equations (4.18)(4.21). Finally, Equation (8.29) is substituted in Equations (8.23)–(8.25) to obtain the complete incremental relationships, which can be expressed In a similar way to Equation (4.23): g VV g VT ª º Dijkl Dij « » « » g TV gT Dkl D « » ˆ ˆ « » ­V kl ½ g DV g DT Dˆ ijkl  K Dˆ ij  K « »®  ¾ « »¯ T ¿ g V g T Cˆijkl  K Cˆij  K « » « » ˆ g V  K Bˆ g  K  A ˆ g T  K Bˆ g  K » « A ¬ ij ¼

(8.33)

Dijkl

§ w 2 gˆ w 2 gˆ gV ·  ¨ ³ ¨ wVij wVkl wVij wDˆ mn Cˆmnkl ¸¸*  K dK ¹ 8©

(8.34)

Dijg VT

³ ¨¨ wVij wT  wVij wDˆ mn Cˆmn  K ¸¸*  K dK

(8.35)

§ w 2 gˆ · w 2 gˆ ˆ g V  Cmnkl  K ¸*  K dK ¨¨ ¸ ˆ wTwVkl wTwDmn ¹ 8©

(8.36)

§ w 2 gˆ · w 2 gˆ ˆ g T Cmn  K ¸*  K dK ¨¨ 2  ¸ ˆ mn wT wTwD ¹ 8©

(8.37)

w 2 gˆ w 2 gˆ gV  Cˆ  K ˆ kl wVij wD ˆ mn mnkl wVij wD

(8.38)

w 2 gˆ w 2 gˆ ˆ g V  C  K ˆ kl wTwD ˆ mn mnkl wTwD

(8.39)

­ H ij ½ ° ° ° s ° ° ˆ ° ®Fij  K ¾ ° ˆ ° ° Dij  K ° ° ˆ ° ¯ O  K ¿ where g VV

g TV Dkl

D gT

§ w 2 gˆ

w 2 gˆ

gT

8©

· ¹

³

³

ˆ g DV Dˆ ijkl  K

ˆ g DT Dˆ kl  K

ˆ g V  K wyˆ g A gV ˆ Cmnkl  K  kl Bˆ g  K wFˆ mn

(8.40)

ˆ g T  K wyˆ e A gT Cˆmn  K  ˆ g B  K wFˆ mn

(8.41)

Equations (8.33)–(8.41) are analogous to Equations (4.23)–(4.29).

142

8 Continuous Hyperplasticity

Thus we can see that the entire constitutive response of the material (expressed through the incremental stress-strain relationships and the evolution equations for internal variables) can be derived from the original two thermodynamic functionals. In Chapter 4 we discuss a number of cases in which constraints are imposed (for example, on the rates of the internal variables). Constraints may also be necessary within this new formulation, but have not been addressed here. The purpose here has been to set out the basic theory of a new approach to plasticity theory with an infinite number of yield surfaces. The following chapters will pursue examples in detail. It is useful, however, to set out a simple example to demonstrate how the formalism can be used. In section 8.6 we develop the general equations for a kinematic hardening plasticity model, and in section 8.7 describe a particular model for the one-dimensional case.

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces The advantage of the multiple surface models is clearly that they are able to fit the non-linear behaviour of certain materials more accurately across a wide range of strain amplitudes. This is important, for instance, in modelling geotechnical materials (Houlsby, 1999). The disadvantage is that a large number of material parameters (associated with each yield surface) are necessary. In this section, we take the modelling of non-linearity to its logical conclusion by introducing an infinite number of yield surfaces. Paradoxically, this reduces the number of material parameters required to specify the models, although at the expense that certain functions also have to be chosen.

8.6.1 Potential Functionals The hyperplastic formulation for multiple yield surfaces (Section 7.3) can be further extended to describe a continuous field of kinematic hardening yield surfaces of the type originally suggested by Mroz and Norris (1982). The general formulation of continuous hyperplastic models is given in Sections 8.2–8.5. Below we show how the continuous hyperplastic models are capable of reproducing decoupled associated kinematic hardening plasticity with a continuous field of yield surfaces. For this case, the specific Gibbs free energy is a functional ˆ ij  K : (rather than function) of the stress and an internal variable function D







ˆ ij º g1 Vij  Vij D g ª¬Vij , D ³ ˆ ij  K *  K dK  ³ gˆ2 Dˆ ij  K , K *  K dK (8.42) ¼ 8

8

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces

143

where Y is the domain of K. The function *  K is a weighting function, such that *  K dK is the fraction of the total number of the yield surfaces having a dimensionless size parameter between K and K  dK . The dissipation functional, which is a functional of internal variable function ˆ ij  K , is also required: and its rate D ˆ ij , D ˆ ij º d g ªVij , D ¬ ¼

g ³ dˆ  Vij , Dˆ ij  K , Dˆ ij  K , K *  K dK t 0

(8.43)

8

Furthermore, in the following, only dissipation functionals with no dependence on stress are considered. This automatically leads to models in which the flow rule (in the conventional sense in plasticity theory) is associated.

8.6.2 Link to Conventional Plasticity





ˆ ij , D ˆ ij , K within The field of yield functions is related to the function dˆ g Vij , D the dissipation functional (8.43) by the Legendre transform (see Appendix C), ˆ ij  K is interchanged with the where the rate of internal variable function D dissipative generalised stress function Fˆ ij  K . Noting that here we are considering only cases where the dissipation does not depend on the stress, the dissipative generalised stress function Fˆ ij  K is defined by



ˆ ij  K , D ˆ ij  K , K wdˆ g D ˆ  K wD

Fˆ ij  K



(8.44)

ij

The transformation from the dissipation to the yield function is a degenerate special case of the Legendre transformation due to the fact that the dissipation is homogeneous and first order in the rates. Therefore, this transformation results in the following identity:



ˆ ij , Fˆ ij , K Oˆ  K yˆ g D



ˆ ij  K , Fˆ ij  K , K where yˆ g D







ˆ ij  K  dˆ g Dˆ ij , Dˆ ij , K Fˆ ij  K D



0

(8.45)

0 is the field of yield functions and Oˆ  K is a non-

negative multiplier. As seen from Equation (8.45), a complete field of yield functions is contained in the equation of the dissipation functional (8.43) in a compact form. The Gibbs free energy functional (8.42) allows the definition of the strain tensor: Hij



ˆ ij º wg ª¬Vij , D ¼ wVij



wg1 ˆ ij  K *  K dK  D wVij

³

8

(8.46)

144

Here

8 Continuous Hyperplasticity

³ Dˆ ij  K *  K dK plays exactly the same role as the conventionally defined

8

 p plastic strain Hij . It is convenient also to define the elastic strain  e wg wV . Hij 1 ij The flow rule for the field of yield surfaces is obtained from the properties of the degenerate special case of the Legendre transformation (8.45) relating yield and dissipation functions (see Appendix C): g

wyˆ ˆ ij  K Oˆ  K D wFˆ ij

(8.47)

We restricted the dissipation function to exhibit no explicit dependence on the true stresses, so again it follows that the normality presented by Equation (8.47) in the generalised stress space also holds in the true stress space. The dependence of the dissipation functional on the internal variable funcˆ ij  K is transferred to the field of yield functions by the Legendre transtion D ˆ ij  K plays the role of a passive variable. Therefore, formation (8.45) where D the strain hardening rule is obtained automatically through the functional dependence of the yield function on the internal variable (or plastic strain) funcˆ ij  K . tion D The generalised stress function is defined by Frechet differentiation of the Gibbs free energy functional (8.42) with respect to the internal variable function, resulting in: wgˆ Fˆ ij  K Vij  2 ˆ ij wD

(8.48)

Again it is convenient to introduce at this stage the “back stress” function Uˆ ij  K associated with the internal variable function and defined as the difference between the true stress and generalised stress function. By applying Ziegler’s orthogonality principle in the form Fˆ ij  K Fˆ ij  K , the back stress function can be expressed as wgˆ 2 ˆ ij wD

(8.49)

w 2 gˆ 2  ˆ  K D ˆ ij wD ˆ kl kl wD

(8.50)

Uˆ ij  K Vij  Fˆ ij  K

which, after differentiation, yields

Uˆ ij  K V ij  Fˆ ij  K

Equation (8.50) is interpreted as the translation rule for the field of yield surfaces when the dissipation function (and hence also the yield function) exhibits no explicit dependence on the true stresses.

8.6 Kinematic Hardening with Infinitely Many Yield Surfaces

145

8.6.3 Incremental Response Two possibilities exist for each value of K. If the material state is within the yield ˆ ij  K , Fˆ ij  K , K  0 , no dissipation occurs and Oˆ  K 0 . If the surface, yˆ g D









ˆ ij  K , Fˆ ij  K , K 0 , then plastic material point lies on the yield surface, yˆ g D deformation can occur provided that Oˆ  K t 0 . In the latter case, the incremental response is obtained by invoking the consistency condition of the field of yield surfaces:



ˆ ij  K , Fˆ ij  K , K yˆ g D



wyˆ g  wyˆ g  ˆ ij  K  D Fˆ ij  K 0 ˆ ij wD wFˆ ij

(8.51)

Substitution of (8.47) and (8.50) in (8.51) leads to the solution for the multiplier Oˆ  K function:

Oˆ  K

wyˆ g V ij wFˆ ij wyˆ g w 2 gˆ 2 wyˆ g wyˆ g wyˆ g  ˆ ij wD ˆ kl wFˆ kl wDˆ ij wFˆ ij wFˆ ij wD

(8.52)

Differentiation of Equation (8.46) and substitution of (8.47) in both the result and in (8.50) gives the incremental stress-strain response,

H ij



wyˆ g wg1 V kl  Oˆ  K *  K dK wVij wVkl wFˆ ij

³

(8.53)

8

and the update equations for the internal variable and generalised stress functions: wyˆ g ˆ ij  K Oˆ  K D wFˆ ij w 2 gˆ 2 wyˆ g Fˆ ij  K V ij  Uˆ ij  K V ij  Oˆ  K ˆ ij wD ˆ kl wFˆ kl wD Oˆ  K multiplier is defined from Equation

The ˆ ij  K , Fˆ ij  K , K yˆ g D

ˆg

y

 

(8.54)

(8.55)

(8.52) when ˆ Otherwise, O  K 0 [when 0 , and ˆ ij  K , Fˆ ij  K , K  0 or when (8.52) gives a negative value of Oˆ  K ]. D



Oˆ  K ! 0 .

Description of the constitutive behaviour during any loading requires a proˆ ij  K , K8 , and this is achieved by cedure for keeping track of Fˆ ij  K and D using Equations (8.54) and (8.55).

146

8 Continuous Hyperplasticity

8.7 Example: One-dimensional Continuous Hyperplastic Model We now give a simple example of a continuous hyperplasticity model to introduce the techniques used to manipulate the functionals to derive a constitutive response. Consider the two functionals, g



1

1

³

³

3

ˆ2 E 1  K D V2 ˆ dK  V D dK 2E 2  a  1 2 0

(8.56)

0

1

³  kK Dˆ dK

dg

(8.57)

0

so that 3

gˆ



ˆ2 E 1  K D V2 ˆ  VD 2E 2  a  1 2

(8.58)

 kK Dˆ

(8.59)

dˆ g It follows that H 

V ˆ dK  D E ³

(8.60)

0

3

wgˆ Fˆ  ˆ wD Fˆ

1

wg wV

V

wdˆ ˆ wD

E 1  K

2  a  1

ˆ D

 kK S  Dˆ

(8.61) (8.62)

ˆ from the above gives the yield function, Eliminating D 2 yˆ Fˆ 2   kK

(8.63)

so that the “flow rule” is wyˆ ˆ Oˆ D wFˆ

ˆˆ 2OF

(8.64)

Derivation of the stress-strain law involves noting that, on first yield from an initially unstressed state, combination of (8.61) and (8.62) gives (together with Fˆ F ), for the elements that are yielding, 3

E 1  K ˆ  ˆ V kKS D D 2  a  1



(8.65)

8.7 Example: One-dimensional Continuous Hyperplastic Model

147

or ˆ D

 V  kKS Dˆ E21aK1 3

(8.65)

Substitution in (8.60) gives V  E

H

K*

2  a  1

³  V  kKS  Dˆ E 1  K 3 dK

(8.66)

0

where K * specifies the size of the largest yield surface which has become active ˆ 0 . For simplicity, now consider monotonic loading for which and V  kK * S D H ! 0 , so that S Dˆ 1 .

 

Differentiation of the strain expression gives H

V  E

K*

2  a  1

2  a  1

³ V E 1  K 3 dK   V  kK * E 1  K * 3 K *

(8.67)

0

But the last term is zero (since V  kK* 0 ). It follows that dH dV

H V

1  E

K*

2  a  1

³ E 1  K 3 dK

(8.68)

0

and differentiating again with respect to time: x

2  a  1

§ dH · ¨ ¸ © dV ¹

3

E 1  K *

K *

2  a  1 V 3 E 1  K * k

(8.69)

so that d 2H dV

x

 dH dV

2  a  1

V

Ec 1   V k

2

3

(8.70)

This may be readily integrated, noting that appropriate initial conditions can be obtained from Equations (8.66) and (8.68), to give the hyperbolic stress-strain curve: 2

H

V  a  1 V  E E k  V

(8.71)

The asymptotic strength is V k , the initial stiffness E, and the secant stiffness at V k 2 is E a . It can be shown that on stress reversal the model gives pure kinematic hardening behaviour, and the stress-strain curve obeys the “Masing” rules.

148

8 Continuous Hyperplasticity

8.8 Calibration of Continuous Kinematic Hardening Models In the above example, we showed how the choice of a particular pair of functionals leads to a material model with a specific stress-strain curve. In certain circumstances, it may be of value to reverse this process. We may wish to calibrate the model by specifying the shape of the stress-strain curve and from this, deduce the form of the functionals. In defining a model using the above approach, there is considerable freedom in the way the functionals can be expressed. For instance, in generalizing the Iwan model with a finite number of yield functions to a model with an infinite number, there is a choice of x the way the hardening stiffnesses (plastic moduli) hˆ  K are distributed, x the way the strength parameters kˆ  K are distributed, x the use of the weighting function *  K .

In this chapter, we explore two alternative ways of calibrating models. First, in Section 8.9, we show how the weighting function *  K can be calibrated, using a one-dimensional model example. Next, in Section 8.10, calibration of the plastic moduli function is demonstrated in a multidimensional example.

8.9 Example: Calibration of the Weighting Function 8.9.1 Formulation of the One-dimensional Model In this one-dimensional example, we consider that hˆ  K H is constant, and kˆ  K kK . The function *  K is left undetermined at first, and it will be seen that the form of this function will be determined from the shape of the stress-strain curve. The model constitutive behaviour is defined by two potential functionals: ˆ@  g > V, D

1

1

³

³  Dˆ  K

H V2 ˆ  K *  K dK  V D 2E 2 0

ˆ º d g ª¬D ¼

2

*  K dK

(8.72)

0

1

³ kK Dˆ  K *  K dK

(8.73)

0

The dissipative generalised stress function Fˆ  K is obtained from (8.44): ˆ kKS D ˆ  K , so that the field of yield functions is given by Fˆ  K wdˆ g wD





yˆ g  K

Fˆ  K  kK 0

(8.74)

8.9 Example: Calibration of the Weighting Function

149

The back stress is defined from (8.49): Uˆ  K V  Fˆ  K

wgˆ2 ˆ wD

(8.75)

ˆ  K HD



ˆ  K Differentiation of (8.74) yields wyˆ g wFˆ S D



ˆ 0 , so that and wyˆ g wD

the incremental Equations (8.52)–(8.55) reduce to: 1

H

V ˆ  K *  K dK  D E ³

(8.76)

0



ˆ  K Oˆ  K S D ˆ  K D



(8.77)



ˆ  K Fˆ  K V  Uˆ  K V  Oˆ  K H S D



where

Oˆ  K



­ S D ˆ  K V ° , ° H ® ° 0, °¯



(8.78)

when Fˆ  K  kK 0 when Fˆ  K  kK  0

For each yield function, one out of two cases takes place. If Oˆ  K d 0 , no disˆ  K 0 and Fˆ  K V . sipation related to the Kth yield function occurs, so that D Alternatively, Oˆ  K ! 0 , in which case dissipation occurs (“plastic” behaviour), so that Fˆ  K 0 and for monotonic loading, substitution of (8.78) in (8.77) and (8.76) gives ª 1 1 K* º H «  ³ * dK» V «E H 0 » ¬ ¼ where K * is the largest K for which Fˆ  K  kK 0 .

(8.79)

8.9.2 Analogy with the Extended Iwan’s Model This incremental behaviour is identical to the constitutive behaviour of the continuous Iwan (1967) model, defined as an extension of the Iwan model described in Section 7.4. In this extension, the number of slip elements with slip stresses k  K kK is continuous and described by the distribution function *  K , so that *  K dK is the fraction of the total number of slip elements having slip stress between kK and k  K  dK . This model simulates one-dimensional elastic-non-linear plastic stress-strain behaviour. Elongation of the E spring gives e elastic strain H , and elongation of the distribution of the H  K springs conˆ  K to the total plastic strain; their sum gives the tributes the plastic strain D total strain H. It is assumed here that the elastic coefficients of all H  K springs are the same and equal to H.

150

8 Continuous Hyperplasticity

To complete the analogy between the hyperplastic formulation (8.79) of incremental response and the behaviour of the continuous Iwan (1967) model during monotonic loading, described by Equations (8.80), it is necessary to interpret the parameter K * . This parameter characterizes the size of the largest active yield surface. When the current generalised stress state first reaches the yield surface associated with K * , this surface is not yet involved in plastic flow, ˆ  K * 0 . Then, from so that the plastic strain associated with this surface D Equation (8.75), Fˆ  K * V , and from Equation (8.74) for the yield surface Fˆ  K * kK * . Therefore, in this simple case, the parameter K * can be interpreted as dimensionless current stress V k , which makes the formulation (8.79) identical to that of the continuous Iwan model during initial loading.

8.9.3 Model Calibration Using the Initial Loading Curve The stress-strain behaviour of the model is similar to that in Figure 7.1; the difference is that the stress-strain curves now become smooth. The behaviour is elastic-non-linear plastic with the tangent modulus ET  V in initial monotonic loading given by

1

1 1  E H

ET  V

H

Vk

*  K dK , which can be derived from

³ 0

V 1  E H

Differentiating the expression

Vk

³

 V  kK *  K dK

(8.80)

0

dH dV

1 1  E H

Vk

³

*  K dK with respect to V

0

gives (using standard results for the differential of a definite integral in which the limits are themselves variable): d 2H dV

2

1 §V· *¨ ¸ kH © k ¹

(8.81)

The weighting function *  K is therefore uniquely related to the second derivative of the initial “backbone” curve H  V . For example, the simple hyperbolic k V is generated by the distribution function stress-strain curve H E kV *  V k d 2H 2k 2 H 2 . , so that *  K 2 3 kH E 1  K 3 dV E k  V

8.10 Example: Calibration of the Plastic Modulus Function

151

8.9.4 Unloading Behaviour When a stress reversal takes place at V Vr , the unloading stress-strain curve is obtained as H

V 1  E H

 V V r

2k

 V  kK *  K dK 

³ 0

1 H

Vr k

³

 Vr  kK *  K dK

(8.82)

 Vr V 2k

So the tangent stiffness on unloading is given by  V V 2k 1 1 1  ³ *  K dK . ETU  V E H 0 §V V· Noting that ETU  V ET ¨ r ¸ , it is straightforward to demonstrate that © 2 ¹ the unloading curve can be derived from the loading curve by the Masing rules, that is to say, it is obtained by taking the loading curve, reversing it, doubling its dimensions in both the stress and strain directions, and finally shifting its origin to the point at which unloading commences. It can also be shown that for more complex loading-unloading-loading sequences, this model, which employs pure kinematic hardening, obeys the more general Masing rules. r

8.10 Example: Calibration of the Plastic Modulus Function 8.10.1 Formulation of the Multidimensional von Mises Model As in the previous section, the model of Section 7.5, with N von Mises yield surfaces can be extended to become a model with a continuous field of yield surfaces. The model is defined by two potential functionals, again supplemented by the plastic incompressibility condition: ˆ ij º g ª¬Vij , D ¼



Vll Vkk Vijc Vijc  18K 4G 8

 Vij

8

³ Dˆ ij  K *  K dK  ³ 0

ˆ ijc º d g ª¬D ¼

0

8

³ kˆ  K

hˆ  K 2

(8.83) ˆ ijc  K D ˆ ijc  K *  K dK D

ˆ ijc  K D ˆ ijc  K *  K dK t 0 2D

(8.84)

0

where  0;8 is the domain of K . The function *  K is again a weighting function and hˆ  K and kˆ  K are functions related to the plastic modulus and size of the

152

8 Continuous Hyperplasticity

yield surfaces, respectively. The possible range of choices of these functions demonstrates the flexibility of the hyperplasticity framework. Two out of the three functions can be chosen arbitrarily, and the third is defined from the monotonic initial stress-strain curve. In Section 8.9, to make the hyperplastic formulation consistent with that of the continuous Iwan (1967) model, hˆ  K was chosen to be constant hˆ  K H , kˆ K was chosen to be linear kˆ  K kK , and *  K was











left to fit the stress-strain curve. In the present section, a different possibility is demonstrated by assuming *  K 1 and kˆ  K kmin   kmax  kmin K , where kmin and kmax are the sizes of the smallest and the largest yield surfaces in the field. In this case, the function hˆ  K is left to fit the stress-strain curve. Using these assumptions, the potential functions (8.83) and (8.84) can be rewritten ˆ ij º  g ª¬Vij , D ¼

Vll Vkk Vijc Vijc  18K 4G 1

1

ˆ ij  K dK   Vij D

³ 0

ˆ c º d g ªD ¬ ij ¼

³

hˆ  K

0

2

1

³  kmin   kmax  kmin K

(8.85) ˆ ijc  K D ˆ ijc  K dK D

ˆ ijc  K Dˆ ijc  K dK t 0 2D

(8.86)

0

The plastic incompressibility condition is introduced as a side constraint by ˆ  K and considering the augmented dissipaemploying a Lagrange multiplier / tion functional: ˆ ij º dc g ª¬D ¼

1

³^ kmin   kmax  kmin K

`

ˆ  K D ˆ ijc  K Dˆ ijc  K  / ˆ kk  K dK t 0 2D

0

(8.87) Denoting further kˆ  K kmin   kmax  kmin K , the dissipative generalised stress is obtained using (8.44):

Fˆ ij  K

wdˆcg ˆ ij wD

kˆ  K

ˆ ijc  K 2D ˆ ijc  K D ˆ ijc  K 2D

 /  K Gij

(8.88)

so that

Fˆ ijc  K kˆ  K

ˆ ijc  K 2D ˆ ijc  K D ˆ ijc  K 2D

ˆ  K Fˆ kk  K 3/

(8.89) (8.90)

8.10 Example: Calibration of the Plastic Modulus Function

153

Elimination of the rate of the internal variable function from Equations (8.89) generates the field of the Von Mises yield functions: 2 yˆ g  K Fˆ ijc  K Fˆ ijc  K  2  k  K 0

(8.91)

The “back stress” tensor defining the position of the centre of the yield surface in stress space is calculated using (8.49): Uˆ ij  K Vij  Fˆ ij  K

Differentiation of (8.91) yields wyˆ g wFij using (8.52)–(8.55), we obtain:

wgˆ 2 ˆ ij wD

ˆ ijc  K hˆ  K D

(8.92)

ˆ ij 2Fˆ ijc  K and wyˆ g wD

0 , so that

1

H ijc

1 V ijc  2³ Oˆ  K Fˆ ijc  K dK 2G

(8.93)

1 V kk 3K

(8.94)

0

H kk

ˆ ij  K 2Oˆ  K Fˆ ijc  K D

(8.95)

Fˆ ij  K V ij  Uˆ ij  K V ij  2Oˆ  K hˆ  K Fˆ ijc  K

(8.96)

­ Fˆ ijc  K V ijc 2 ° , when Fˆ ijc  K Fˆ ijc  K  2  k  K 0 2 ° ˆ where Oˆ  K ® 4k  K h  K ° 2 0, when Fˆ ijc  K Fˆ ijc  K  2  k  K  0 °¯ Again there are two cases for each yield surface. In both cases, the volumetric behaviour is purely elastic. If Oˆ  K d 0 , no dissipation related to the Kth yield ˆ ij  K 0 and Fˆ ij  K V . Alternatively, Oˆ  K ! 0 , in function occurs, so that D which case dissipation occurs (“plastic” behaviour), so that Fˆ  K 0 and for ij

monotonic proportional loading, H ijc

§ 1 K* 1 · ¨ ³ dK ¸ V ijc ¨ 2G ¸ hˆ K 0  © ¹

where K * is the largest K such that Fˆ ijc  K Fˆ ijc  K  2  k  K

(8.97) 2

0.

154

8 Continuous Hyperplasticity

8.10.2 Model Calibration Using the Initial Loading Curve To define the function hˆ  K , it is necessary to interpret the parameter K * . During initial loading, when the current stress state reaches the active yield surface yˆ  K * , this surface is not yet involved in plastic flow, so that the plastic strain ˆ ijc  K * 0 . Then, from Equation (8.92), associated with this surface D



Fˆ ijc  K * Vijc , and from Equation (8.91) of the yield surface yˆ K* , K*

J 2V  kmin

(8.98)

kmax  kmin

where J 2V Vijc Vijc 2 is the second invariant of the deviatoric stress tensor. Consider a proportional monotonic loading described by Vijc Eijc V where c Eij is a tensor defining the direction of loading in deviatoric stress space chosen so that J 2E Eijc Eijc 2 1 and V is a scalar stress multiplier. Then Equation (8.97) can be rewritten as ª 1 K* 1 º H «  ³ dK» V « 2G 0 hˆ  K » ¬ ¼

(8.99)

V  kmin . Integration of Equation (8.99) produces kmax  kmin the normalized stress-strain curve H f  V , which can be fitted to the experimental data. The fitting is performed by proper choice of the function hˆ  K .

where Eijc H Hijc and K*

Differentiation of Equation (8.99) yields the following relationship between the hˆ  K function and the second derivative of the experimental normalized stressstrain curve: d2H dV

2

 kmax

1 § V  kmin ·  kmin h ¨ ¸ © kmax  kmin ¹

For example, the hyperbolic stress-strain curve H

(8.100)

k V is related to the E kV

function: § V  kmin · h¨ ¸ © kmax  kmin ¹

By setting kmax

§ d 2H · 1 ¨ ¸ kmax  kmin ¨© dV2 ¸¹

k and kmin

1

3

E k  V 1 kmax  kmin 2k 2

0 , this leads to hˆ  K

E 1  K 3 . 2

8.11 Hierarchy of Multisurface and Continuous Models

155

8.10.3 Analogy with an Advanced Plasticity Model The incremental behaviour of this hyperplastic model is identical to the constitutive behaviour of the model with a continuous field of yield surfaces, defined as an extension of the model with multiple von Mises yield surfaces described in Section 7.5. In this extension, the field of von Mises yield surfaces is described by Equation (8.91), with kˆ  K kmin   kmax  kmin K where kmin and kmax are the sizes of the smallest and the largest yield surfaces in the field. This model simulates six-dimensional elastic-non-linear plastic stress-strain behaviour. The elastic component of strain is calculated according to Hooke’s law. An associated flow rule is implied, together with the plastic incompressibility condition, so that



ˆ ijc  K 2Oˆ  K Vijc  Uˆijc  K D



ˆ kk  K 0 D

(8.101)

The formulation implies either the Prager (1949) translation rule in the form, ˆ ijc  K , or alternatively, the Ziegler (1959) translation rule in the Uˆ ijc  K hˆ  K D form Uˆ c  K 2Oˆ  K hˆ  K Vc  Uˆ c  K , which as noted earlier, are identical in ij



ij

ij



this case. The Mroz (1967) translation rule would again give the same result for proportional loading. The stress-strain response of the model to proportional cyclic loading is similar to that presented in Figure 7.1. The difference is that the stress-strain curves are smooth.

8.11 Hierarchy of Multisurface and Continuous Models In Chapters 7 and 8, we have systematically presented a compact hyperplastic framework for kinematic hardening of plastic materials. In Chapter 7, we generalize a model with a single yield surface to the case of multiple surfaces. In Chapter 8, the case with an infinite number of yield surfaces is considered. The description of this last case is given within the framework of “continuous hyperplasticity.” The models developed can reproduce smooth transitions from elastic to plastic behaviour, including cases where the truly elastic region vanishes altogether. The stress-reversal history is memorized within the internal variable function. The continuous hyperplasticity approach allows materials to be modelled in which the past strain history is in effect represented by an infinite number of internal variables but compactly represented as an internal function. This approach, although rooted entirely within the methods of “generalised thermodynamics”, offers some of the advantages that are more often associated with the “rational mechanics” approach. However, the approach developed here allows constitutive

156

8 Continuous Hyperplasticity

models to be derived entirely from two potential functions: once these are specified, only standard manipulation is required to derive the entire response. At each stage of generalization, an incremental stress-strain response is derived, and the link with conventional plasticity is demonstrated. Examples of one- and multi-dimensional hyperplastic models are presented together with their conventional plasticity interpretations. Because they are formulated using hyperplasticity, these kinematic hardening models are guaranteed to obey the First and Second Laws of Thermodynamics. They can also be set within a simple hierarchical framework, as shown in Table 8.2, for one-dimensional kinematic hardening models, and in Table 8.3 for multi-dimensional kinematic hardening plasticity models. All the above models are based on the concept of a number of elements, each consisting of a slider and a spring, arranged in series as in Figure 7.2. The plastic strains in each of these elements are therefore additive. An alternative way of generalising from a single plastic strain to many plastic strain components involves arranging the elements in parallel, as shown in Figure 8.1. Such models are most naturally expressed in terms of the Helmholtz free energy rather than the Gibbs free energy. The defining functions are given in Table 8.4 for the single surface, multisurface, and continuous cases. We use E for the internal variable in the parallel model rather than D, as it plays a slightly different physical role. Likewise, we use different variables for the spring stiffnesses and slider strengths, as these too play slightly different roles. Einav (2004) pursues the analysis of parallel models.

G1

c1 E1

G2

c2

E2 GN

V cN

EN

F H

Figure 8.1. Parallel models for multiple surface plasticity

8.11 Hierarchy of Multisurface and Continuous Models

157

Table 8.2. Hierarchy of one-dimensional kinematic plasticity models (series case)

Gibbs free energy g

Dissipation function d

2

V 2E V2 H D 2 g  V, D    VD 2E 2 g  V, D1 !D N g V 

Elasticity Single surface plasticity Multiple surface plasticity

N V2 N H n 2 ¦ D n  V ¦ Dn 2E n 1 2 n 1 ˆ@ g > V, D

N

n 1

ˆ º d ªD ¬ ¼

1 V Hˆ 2 ˆ dK  V D ˆ dK   D 2E 2 2

1

³

³

0

d  D 1 !D N

¦ kn D n



Continuous plasticity

d  D k D

1

³ kˆ Dˆ dK 0

0

Table 8.3. Hierarchy of multidimensional kinematic plasticity models (series case)

Gibbs free energy g

g Vij









Dissipation function d

Vijc Vijc

Vii V jj

 18K 4G Vii V jj Vijc Vijc  g Vij , Dij  Single surface 18K 4G d D ij k 2D ijc D ijc plasticity h  Dijc Dijc  Vij Dij 2 1   N  Vii V jj  Vijc Vijc g Vij , Dij !Dij 18K 4G 1 N d D ij !D ij N n h n n   Multiple surface ¦ Dc Dc N 2 ij ij plasticity n n n n 1 k 2D ijc D ijc ¦ N n 1 n  Vij ¦ Dij Elasticity











n 1

Continuous plasticity

ˆ ij º  g ª¬Vij , D ¼

Vii V jj

Vijc Vijc

 18K 4G 1 ˆ 1 h ˆ ijc D ˆ cij dK  Vij D ³ D ³ ˆ ijdK 2 0

0

ˆ c º d ªD ¬ ij ¼ 1

³ kˆ 0

ˆ ijc D ˆ ijc dK 2D

158

8 Continuous Hyperplasticity

Table 8.4. Hierarchy of one-dimensional kinematic plasticity models (parallel case)

Helmholtz free energy f Elasticity Single surface plasticity Multiple surface plasticity

Continuous plasticity

FH 2 2 F H2 G  H  E f  H, E  2 2 f  H, E1 !EN f H

F H2 N Gn 2  ¦  H  En 2 n 1 2 f ªH, Eˆ º ¬ ¼ 2

Dissipation function d

2

1

2 FH Gˆ  H  Eˆ dK 2 2

³  0



 c E d  E 1 !E N

d E

N

¦ cn E n

n 1

 d ªEˆ º ¬« ¼»

1



³ cˆ Eˆ dK 0

Chapter 9

Applications in Geomechanics: Elasticity and Small Strains

9.1 Special Features of Mechanical Behaviour of Soils The motivation for many of the results presented in this book comes principally from applications to geotechnical materials. In this chapter, we show how some special features of the mechanical behaviour of soils (such as the concept of effective stresses; frictional behaviour; non-associated flow; dependence of stiffness on pressure, density and loading history; stress induced anisotropy; critical state; and small strain non-linearity) can be expressed within the hyperplastic constitutive framework developed in earlier chapters.

9.2 Sign Convention and Triaxial Variables As mentioned in Chapter 1, it is common in soil mechanics and geotechnical engineering to adopt a compressive positive convention for stresses, so for this chapter and Chapter 10 only, we reverse our convention for stresses and strains and adopt compression as positive. We shall give several examples in this chapter and the next which are presented in terms of the “ p, q ” variables common in soil mechanics. These are particularly convenient for the analysis of the triaxial test, which is a compression test on a cylindrical specimen in which the axial stress is V1 and the radial stress is V3 . The circumferential stress is assumed (in a uniform specimen) to be equal to the radial stress. The so-called “triaxial” variables are the mean stress p and deviator stress q defined as p

1  V1  2V3 3 q V1  V3

(9.1) (9.2)

160

9 Effective Stresses

Corresponding to them are the volumetric and shear strains defined by

Hv Hs

H1  2H3

(9.3)

2  H1  H3 3

(9.4)

For more general stress states, the stress and strain variables are p q

1 Vkk 3 3 Vijc Vijc 2

Hv

Hkk

Hs

2 Hijc Hijc 3

(9.5)

(9.6)

It can easily be shown that for the triaxial test, the variables defined in Equations (9.1) and (9.2) satisfy the work conjugacy condition W pH v  qH s .

9.3 Effective Stresses An essential feature of modelling geotechnical materials is that the role of the pore fluid pressure is usually expressed through the concept of effective stress, as introduced by Terzaghi (1943). The purpose of this section is to demonstrate how the developments in earlier chapters can be used in terms of effective stresses for saturated granular materials. In earlier chapters, the analysis has been carried out entirely in terms of total stresses, in which case the input power (per unit volume) is Vij H ij , and we identified the mechanical dissipation as d T s  q . Houlsby (1979) showed that k,k

the power input to a saturated granular material with incompressible grains and pore fluid is Vij H ij  wk u,ckw , where Vij is the effective stress (we use Vij here rather than the more usual Vijc to avoid confusion with the deviator of the stress tensor), wk is the artificial seepage velocity (or Darcy velocity) of the pore water, and u,ckw is the gradient of the excess pore water pressure (water pressure less elevation head, ucw uw  Uw gz where z is the downward vertical coordinate). The effective stress is defined (noting our compressive positive convention here) as Vij Vij  uw Gij where uw is the total pore water pressure. The First Law can then be written u Vij H ij  w k u,ckw  qk ,k

(9.7)

9.3 Applications in Geomechanics: Elasticity and Small Strains

161

and the Second Law is written in the form derived from the Clausius-Duhem inequality: Ts  qk,k 

qk T,k T

t0

(9.8)

The mechanical dissipation can now be split into the dissipation related to the deformation of the soil skeleton and the part that results from the flow of pore water. It is clear that we can identify the term wk u,ckw as the input power that is dissipated in the flow of pore water, so we can write d

Ts  qk,k

d  wk u,ckw

(9.9)

where d is the dissipation associated with deformation of the soil skeleton. The Second Law becomes d  wk u,ckw 

qk T,k T

t0

(9.10)

So we see the close analogy between the energy dissipated by the heat flux qk and the water flux wk . Although the Second Law requires only that the sum of the three terms in inequality (9.10) be non-negative, we again assume that each individual term is non-negative and therefore that d t 0 . From Equation (9.9), we obtain wk u,ckw  qk ,k Ts  d , so that the First Law becomes u Vij H ij  Ts  d

(9.11)

and the analysis proceeds exactly as in Section 4.1, but with Vij replacing Vij and d replacing d throughout. Thus we see that the development in Chapter 4, which was in terms of total stresses (in the absence of pore pressure), is equally applicable in terms of the conventionally defined effective stresses for a saturated granular material under the usual assumption of incompressible grains and pore fluid. Houlsby (1997) developed expressions for the power input to an unsaturated granular material (i. e. one composed of grains, liquid, and gas), subject to some simplifying assumptions. An analysis similar to that presented above can be applied to the unsaturated case. Appropriate definitions for effective stress can be made so that the development in earlier chapters is applicable, but additional terms appear in the equations, related to the degree of saturation and the difference between pore water and pore air pressures. The application to unsaturated materials is not, however, straightforward, and further work is required in this area. A more detailed study of the processes of soil deformation and flow in porous media is presented in Chapter 12.

162

9 Dependence of Stiffness on Pressure

In the remainder of this chapter and in Chapter 10, we work entirely in terms of effective stresses, and for brevity we use Vij and d instead of Vij and d .

9.4 Dependence of Stiffness on Pressure In recent years, a considerable amount of experimental research has been carried out to investigate the mechanical behaviour of soils undergoing very small strains, for which the response is usually assumed to be reversible. The interpretation of the data shows that the initial soil stiffness (or small strain stiffness) is a non-linear function of the stress (specifically the mean effective stress). The stiffness is also affected by other variables, such as the voids ratio, and/or the preconsolidation pressure, which we address in Section 9.4.3. The small strain tangent stiffness depends on the stress level, and typically the elastic moduli vary as power functions of the mean stress. Simple elastic or hypoelastic models of this non-linearity can result in behaviour that violates the laws of thermodynamics. To ensure that an elasticity model is thermodynamically acceptable, we use the hyperelastic approach here to derive models which allow for variation of elastic moduli as power functions of mean stress. Analysis of many geotechnical problems depends on a realistic representation of the non-linear dependence of the initial stiffness on stress, and we first explore how this has often been achieved in the past. The usual approach is to adopt “hypoelastic” formulations (Fung, 1965) in elastic-plastic models, in which varying tangent moduli are defined. For instance, it is common to adopt the following procedure to calculate elastic moduli for the modified Cam-clay model. The bulk modulus K is usually defined through the pressure-dependent pc 1  e expression K , and the shear modulus G is then obtained by assumN ing a constant Poisson ratio v. Such a model leads to a non-conservative “elastic” response (Zytynski et al., 1978). Instead we adopt the “hyperelastic” approach, which naturally leads to a conservative elastic response, guaranteed to obey the first law of thermodynamics. It is worth remarking here that it is well-recognised that the soil stiffness is also significantly dependent on the strain amplitude. This raises more difficult problems of hysteresis and energy loss and is addressed later in Sections 9.5 and 9.6. Several models have been developed to reproduce the reversible behaviour of soils, and these are reviewed by Houlsby et al. (2005), who also review briefly some typical experimental observations of the small strain stiffness of soils and their semi-empirical interpretations. We present here a hyperelastic, isotropic energy potential capable of accounting for the non-linear dependence of elastic stiffness on stress. We first develop this for triaxial conditions and later for more general stresses.

9.4 Applications in Geomechanics: Elasticity and Small Strains

163

9.4.1 Linear and Non-linear Isotropic Hyperelasticity Experimental evidence suggests that the small strain bulk and shear stiffnesses of soils can be well represented as power functions of the mean effective stress, and we write them in the following forms:

§ p· K =k¨ ¸ ¨p ¸ pr © r¹

n

§ p· G = gs ¨ ¸ ¨p ¸ pr © r¹

(9.12) n

(9.13)

where k and g s are dimensionless constants; pr is a reference pressure, often conveniently taken as atmospheric pressure; and n is an exponent 0 d n d 1 . The possibility that the exponent n could be different for the bulk and shear moduli might be considered, but is not pursued further here, as this leads to considerable additional complexity in the mathematical development. In the following, we will focus our attention on deriving stress-dependent stiffness from potentials that are expressed as functions of invariants either of the strain or of the stress tensor, so that the material behaviour described will be fundamentally isotropic, although it will be seen below that “stress-induced” anisotropy is predicted under certain conditions. Expressed in terms of the triaxial variables, the Helmholtz free energy f (also equal to the elastic strain energy) is written as a function of the strains, i. e. f =  Hv , Hs . It then follows that p=

wf wHv

q=

wf wHs

(9.14)

and further that the tangent bulk and shear moduli are defined by K=

wp w 2 f = wHv wHv2

3G =

wq w 2 f = wHs wH2s

(9.15)

Furthermore, it can be shown that off-diagonal terms may in general appear in the incremental stiffness matrix, ª dp º ª K «dq » = « J ¬ ¼ ¬

J º ª d Hv º 3G »¼ «¬ dHs »¼

(9.16)

164

9 Dependence of Stiffness on Pressure

where J=

wp wq w2 f = = wHs wHv wHv wHs

(9.17)

When J is non-zero, the material behaves incrementally in an anisotropic manner, even though f is an isotropic function of the strains. This is a case of stress-induced anisotropy. Although elastic behaviour can be derived by differentiation, as in Equations (9.15) and (9.17), this has certain disadvantages. The resulting expressions for the moduli are in terms of the strains, which can be inconvenient, because moduli expressed as functions of stress are usually of more practical use. Therefore, it is useful to use instead the Gibbs free energy function g (which in the context of elasticity theory is also the negative complementary energy):

g = f   pHv  qHs When g is expressed as a function of the stresses g be derived as

(9.18)

g  p, q , the strains may

Hv = 

wg wp

(9.19)

Hs = 

wg wq

(9.20)

c3 º ª dp º c2 »¼ ¬«dq ¼»

(9.21)

= 2

wHv w 2g = 2 wp wp

(9.22)

= 2

wHs w 2g = 2 wq wq

(9.23)

wHv wHs w 2g = = wq wp wpwq

(9.24)

and the terms in the compliance matrix, ªdHv º ª c1 « »=« ¬ dH s ¼ ¬ c 3

are c1 = c2 = c3 =

3G 3KG  J K 3KG  J J

3KG  J

2

=

The Helmholtz and Gibbs free energy expressions for linear elasticity (which corresponds to n 0 in (9.6) and (9.7)) are each quadratic in form: 3g §k · f = p r ¨ Hv2 + s Hs2 ¸ 2 2 © ¹

g=

1 pr

§ 1 2 1 2· q ¸ ¨ p + 6gs ¹ © 2k

(9.25) (9.26)

9.4 Applications in Geomechanics: Elasticity and Small Strains

165

with k and g s dimensionless constants. From the above, it is straightforward to derive p = k p rHv , q = 3g s p rHs , K = k p r , G = g s p r , and J = 0 . The expressions that give non-linear elasticity (i. e. K v pn ) under purely isotropic stress conditions (i. e. without q and Hs terms) can also be established unambiguously. For n z 1 , the expressions for f and g must be f=

pr k 2  n

g=

 k 1  n Hv  2n 1n p 2n

p1rnk 1  n  2  n

(9.27)

(9.28)

From either of them, one can derive 1n

§ p· k 1  n Hv = ¨ ¸ ¨p ¸ © r¹

(9.29)

and n

§ p· K n 1n = k¨ ¸ = k  k 1  n Hv ¨ ¸ pr © pr ¹

(9.30)

For n = 1 , the above expressions become singular. A difficulty also arises when n = 1 that, if the volumetric strain is taken as zero at p 0 , then it is infinite for all finite stresses. This problem can be avoided by shifting the reference point for zero volumetric strain from the origin ( p 0 ) to p pr . This is achieved by changing (9.27) and (9.28) to f= * where Hv

Hv 

 2n 1n

pa

 k 1  n H*v k 2  n

(9.31)

1 and k 1  n

g=

p 2n p1rnk 1  n  2  n



p k 1  n

(9.32)

This modifies (9.29) to 1n

§ p· 1  k 1  n Hv* = ¨ ¸ ¨p ¸ © r¹

(9.33)

and (9.30) to n

§ p· K = k¨ ¸ = k k 1  n Hv* ¨p ¸ pr © r¹



n 1n



(9.34)

166

9 Dependence of Stiffness on Pressure

but note that this does not effect the expression for stiffness in terms of pressure. The asymptotic expressions for n = 1 are f=

pr k

(9.35)

exp  kHv

p§ § p · · g =  ¨ ln ¨ ¸  1 ¸ k ¨© ¨© pr ¸¹ ¸¹

(9.36)

From either of them, one can derive: p = exp  kHv pr

(9.37)

K = kp

(9.38)

and

Equations (9.25) and (9.26) apply for n 0 for any triaxial stress states, whilst (9.31) and (9.32) [or (9.35) and (9.36) for n 1 ] apply when n z 0 but only on the isotropic axis. It is our purpose in the following to obtain more general expressions which apply both to any triaxial stress states and for n z 0 , and reduce to each of the above equations in the appropriate special cases. This generalisation can be done in a variety of ways. We first consider (for simplicity) the case where the reference point for volumetric strain is at p 0 . Three possible ways of generalising Equations (9.27) and (9.28) are as follows: (a) f is of the form,



2 2 f = Hm v A Hv + BH s



(9.39)

where A, B and m are constants. It can be shown that no simple form of g exists for this case. (b) f is of the form,



f = A Hv2 + BH2s

m



(9.40)

which results in g of the form



g =  C p 2 + Dq 2

m  2m 1



(9.41)

(c) g is of the form,



g =  p m Ap 2 + Bq 2



(9.42)

It can be shown that no simple form of f exists in this case. Einav and Puzrin (2004a) pursue this form of energy function and show [following Houlsby (1985)] how it also has the disadvantage that it introduces a limiting stress

9.4 Applications in Geomechanics: Elasticity and Small Strains

167

ratio, implying that some stress states are unattainable. Both the inability to derive the f function and the inaccessibility of certain stress states are significant drawbacks of this form of function. All the forms described above exhibit a constant Poisson's ratio under isotropic stress conditions. At least at present, available experimental data are insufficient to distinguish definitively between the above three approaches, so the selection is based on the simplicity of the mathematics. The approach that proves most versatile is (b), so this is adopted in the following.

9.4.2 Proposed Hyperelastic Potential Triaxial Formulation Following approach (b) above, the generalisation of Equations (9.25) and (9.31) that we seek for the function f must consist of a quadratic function of H*v and Hs , raised to an appropriate power. Inspection of the forms of Equations (9.25) and (9.31), followed by some calculation to determine some appropriate constant factors, shows that the required general expression is 2  2n  22n pr  2n 1n § *2 3g s Hs · f ¨¨ Hv  ¸  k 1  n k2  n k 1  n ¸¹ (9.43) © n n 2  1    pr * kHvo 1  n k2  n





3g s H2s . Note that H*v is used instead of Hv to move the origin k 1  n for volumetric strain to p pr , for consistency with the n 1 case. From the above, the stresses and moduli may be obtained by differentiation as in Equations (9.14)–(9.17). The resulting expressions for the moduli are in terms of the strains. It can be shown (after some manipulation) that the Gibbs free energy expression which is the Legendre transform of the expression in Equation (9.43) is

where H*2 vo

H*2 v 

g



§ 2 k 1  n 2 · q ¸ ¨p + 1n 3g s pr k 1  n  2  n © ¹ 1

po2n

p  1n  pr k 1  n  2  n k 1  n

where p o2 = p 2 +

k 1  n q 2 3g s

.

 2n

2



p k 1  n

(9.44)

168

9 Dependence of Stiffness on Pressure

Although Equations (9.43) and (9.44) may appear complex, it can be seen that they have the basic structure of (9.40) and (9.41) (allowing for the shift of origin for strain). The particular forms of the functions are chosen so that, after differentiation, the moduli reduce to simple expressions. It follows from the above that Hv =

· p 1 § ¨  1¸ k 1  n ¨ p1n p n ¸ o © r ¹ q Hs = 1n p r 3g s p no

(9.45)

(9.46)

and c1 =

c2 =

§ np2 · ¨1  2 ¸ k 1  n p1rn pon ¨© po ¸¹ 1

§ nk 1  n q 2 · ¨1  ¸ 3g p1rn pno ¨© 3g po2 ¸¹ npq c3 =  3g s p1rn p no  2 1

(9.47)

(9.48)

(9.49)

Some of the above expressions (9.46)–(9.49), are valid for n = 1 as well as n z 1 , but this is not the case for (9.43)–(9.45). Noting that for n 1 , po p , the kq2 · 1 § asymptotic values of the compliances for n 1 are c1 ¨1  ¸, kp ¨© 3g s p2 ¸¹ q 1 , and c3  . The asymptotic expressions for n = 1 replacing c2 3g s p 3g s p2 (9.43)–(9.45) are § p 3g kH2 · f = r exp ¨ kHv  s s ¸ ¨ k 2 ¸¹ © g=

p § § p · · q2 ¨ ln ¨ ¸  1 ¸  k ¨© ¨© p r ¸¹ ¸¹ 6 g s p

q2 1 § p· Hv = ln ¨ ¸  k ¨© p r ¸¹ 6 g s p 2

(9.50)

(9.51)

(9.52)

Finally we can note that the stiffnesses are given by K c2 D , 3G c1D , and 2n c3 D , where D 3kg s pr2  po pr . The implications of the above choice of free energy (and hence complementary energy) are as follows:

J

9.4 Applications in Geomechanics: Elasticity and Small Strains

169

On the isotropic axis simple modulus expressions may be obtained, and J = 0 . On this axis, it is also possible to define Poisson’s ratio Q =  3k  2 g s  6k + 2 g s [alternatively expressed in the form g s k = 3 1  2Q 2 1+ Q ]. For more general stress points, the expressions for moduli are more complex, but the most important feature is that the moduli are still power functions of the mean stress (although they also depend on the stress ratio). The fact that J z 0 for general stress states implies stress-induced anisotropic elastic behaviour. The shapes of shear strain and volumetric strain contours are given directly by Equations (9.45), (9.46), and (9.51). Within the range of stress ratios of interest, the volumetric strain contours are similar (but not identical to) parabolae symmetrical at about the p -axis and convex toward the origin. Shear strain contours are curves, convex upward in the region of the  p, q plot accessible for reasonable soil properties. For n = 1 , the shear strain contours become straight lines radiating from the origin. Some undesirable features of the model proposed by Houlsby (1985) and extended by Borja et al. (1997), in particular the crossing of volumetric strain contours, are absent. Figure 9.1 shows contours of shear and volumetric strains for n 0.5 . Note that the volumetric strain contours correspond to undrained stress paths for elastic behaviour. For n z 0 , the approximately parabolic undrained stress paths indicate that (other than on the isotropic axis) the response of the soil is incrementally anisotropic. This stress-induced anisotropy arises as a natural consequence of the hyperelastic formulation, and corresponds well to observations of soil behaviour. 800

q

600

400

200

0 0

200

400

600

800

p'

Figure 9.1. Example of volumetric (solid lines) and shear (dashed lines) strain contours for n = 0.5

170

9 Dependence of Stiffness on Pressure

Figure 9.2. Shear and volumetric strain contours presented by Shaw and Brown (1988), based on experimental data on crushed limestone

Many studies have presented evidence that the stiffness of sands can be expressed as a power function of stress level, but a special feature of the hyperelastic approach adopted here is that it predicts the related effect of the curvature of strain contours. Figure 9.2 is reproduced from Shaw and Brown (1988), who follow the approach of Pappin and Brown (1980) in plotting “resilient” shear and volumetric strain contours derived from an extensive series of cyclic tests on granular material. Importantly, they show that the volumetric strain contours are approximately parabolic and curved approximately as in Figure 9.1. The shear strain contours (which Shaw and Brown assume to be straight) are also very similar to those in Figure 9.1. Comparable data for clays were presented by Borja et al. (1997).

General Stress Formulation The results described above can be generalised to other than triaxial stress states, if the free energy f is written as a function of the strains H ij and the complementary energy g as a function of the stresses Vij . In this case, the normal expressions for the stresses and strains, as derivatives of the free energies, are used. The stiffness matrix is d ijkl =

and the compliance matrix is

w Vij w H kl

=

wf w H ijw H kl

(9.53)

9.4 Applications in Geomechanics: Elasticity and Small Strains

c ijkl =

w Hij w V kl

=

wg w Vijw V kl

171

(9.54)

The Helmholtz free energy (for n z 1 ) is written as before: pa *  2n 1n k 1  n Hvo k 2  n



f



(9.55)

§ ·§ 1 1 · 2 g s Hijc Hijc . H jj  ¨¨ Hii  ¸¨ ¸ ¸¨ k 1  n ¹© k 1  n ¹¸ k 1  n © The Gibbs free energy for n z 1 is as before:

2 where now Hvo

po2n Vkk  1n  pr k 1  n  2  n 3k 1  n

g

(9.56)

c Vmn c VmmVnn k 1  n Vmn . + 9 2g s For n = 1 , the Gibbs free energy is

where po2 =

V § § V jj · · 3Vijc Vijc g =  ii ¨ ln ¨ ¸  1¸  3k ¨© ¨© 3 p r ¸¹ ¸¹ 4 g s V kk

(9.57)

The tangent compliance matrix can be written: n

cijkl

§ pr · 1 ¨ ¸ pr k 1  n © po ¹

­° n § Vmm Gij k 1  n Vijc + ® 2 ¨¨ 2g s ¯° po © 9 

n ­ª

1 § pr · ¨ ¸ pr © po ¹

Gij Gkl 9

+

· § Vmm Gkl k 1  n Vckl · + ¸ ¸¸ ¨ 9 2g s ¹ ¹©

k 1  n § 1 ·½ ¨ Gik G jl  Gkl Gij ¸ ¾ 2g s © 3 ¹¿

c · Gij Gkl nVmm ° § 1 nVcmn Vmn  Vcij Gkl  Gij Vckl ¸ ® «¨¨  2 g s po2 ¸¹ 9 18 g s po2 °¯ «¬© k 



(9.58)



1 § 1 · nk 1  n c c º ½° Vij Vkl » ¾ ¨ Gik G jl  Gkl Gij ¸  2g s © 3 ¹ 4 g s2 po2 »¼ ¿°

The latter form is also applicable for n 1 . It can also be shown that the stiffness matrix can be expressed as n

dijkl

§p · pr ¨ o ¸ © pr ¹

§ Vij Vkl 1 § ·· ¨¨ nk 2  k 1  n Gij Gkl  2 g s ¨ Gik G jl  Gkl Gij ¸ ¸¸ (9.59) 3 © ¹¹ po ©

172

9 Dependence of Stiffness on Pressure

It can easily be shown that tests performed in triaxial systems equipped with vertically fitted bender elements allow measurement of the stiffness component d1212 2G , where the vertical and radial directions are 1 and 2, respectively. Under triaxial stress conditions, the above expressions reduce to those obtained earlier. The form of the stiffness matrix in Equation (9.59) has two important consequences. Firstly and obviously, the terms depend on the stresses (not just the mean stress). This means that the stiffness can be determined only by reference to the complete stress system. Secondly and less obviously, it can be shown that (other than on the isotropic axis) the incremental stiffness cannot be expressed just in terms of isotropic stiffnesses, that is to say, Equation (9.59) does not simply imply stress-dependent values of the parameters K and G. The response can be represented only by anisotropic elasticity. This is an example of “stressinduced” anisotropy: it has nothing to do with the fundamental structure of the material, which is isotropic, but is induced by the stress field. Of course, real soils may also exhibit structural anisotropy. Extension of the concepts discussed here to structurally anisotropic materials is not addressed here. The compliances or stiffnesses expressed in Equations (9.58) and (9.59) can be used directly, for instance, in a finite element program for general stress states, ensuring fully conservative elastic behaviour when the moduli are functions of pressure. The expressions require just three dimensionless constants k, g s , and n.

9.4.3 Elastic-plastic Coupling in Clays Incorporation of the density and/or loading history dependency of elastic stiffness into the hyperplastic framework implies even more subtlety in the elastic response. Changes in density are normally expressed through the volumetric plastic strain, and the loading history is reflected in the overconsolidation ratio (OCR) and preconsolidation pressure, which can again be expressed through the volumetric plastic strain. In this case, the specific Helmholtz free energy f Hij , Dij and the specific Gibbs free energy g Vij , Dij depend on the kine-









matic internal variable tensor Dij , most often associated with plastic strain, so p that in the following we adopt Dij { Hij (see Chapter 4). The incremental elastic stress-stain response is in this case affected by the plastic strain increment as well. This is called “elastic-plastic coupling.” In this case, great care is required to make a careful distinction between the “plastic” and “irreversible” components of strain. Collins and Houlsby (1997) discuss elastic-plastic coupling within the hyperplastic framework and address its consequences for the overall behaviour of soils. An important consequence of a plastic-strain-dependent complementary energy relates to the decomposition of the elastic strain tensor, as shown by

9.4 Applications in Geomechanics: Elasticity and Small Strains

173

Hueckel (1977). This can be demonstrated in the hyperplastic formulation as follows. If g



g Vij , Dij



(9.60)

the strain increment is determined by differentiation as H ij



w2 g w2 g V kl  D kl wVij wVkl wVij wDkl

(9.61)

Alternatively, if the Helmholtz free energy form is used, we derive w2 f w2 f H kl  D kl wHij wHkl wHij wD kl

V ij

(9.62)

The effect of plastic strain increments on the elastic component of soil behaviour is avoided only if the corresponding energy potentials can be decomposed in the following way:





f Hij , Dij







In

g Vij , Dij

where w2 f wHij wD kl



f 2 Dij { g 2 Dij .





(9.63)



(9.64)



g1 Vij  Vij Dij  g 2 Dij

this

case

only,

w2 g wVij wD kl

Gij Gkl

and

w2 f , so that the total strain can be decomposed e e wHij wHij e into elastic and plastic components Hij Hij  Dij and the incremental elastic 

w2 f wHij wHkl



f1 Hij  Dij  f 2 Dij



stress-strain response is given by V ij

w2 f e H e  e kl  wHij wHkl

(9.65)

 H ij



w2 g V kl wVij wVkl

(9.66)

e

Any other form of the energy potentials leads to coupling between elastic and plastic components of model behaviour. The proposed Gibbs free energy function (9.44) in Section 9.4.2 does not include any dependency of the reversible behaviour on the internal variables that characterise the hardening of soil. The elastic stiffness of clays is influenced by the preconsolidation pressure; see e. g. Houlsby and Wroth (1991). Typically, the

174

9 Dependence of Stiffness on Pressure

stiffness can be expressed as a power function of the preconsolidation pressure as well as of the mean effective stress. The following modification of the Gibbs free energy is proposed to extend it to clay soils: r § pr · § po2n Vkk · ¸ g =  ¨ ¸ ¨ 1n  © pc ¹ ¨© pr k 1  n  2  n 3k 1  n ¸¹

(9.67)

Equation (9.61) shows that the elastic strain increment can be decomposed r into a first term defining the reversible strain H ij and a second term called





c the coupled strain H ij . The latter accounts for the change of stiffness as plas-

tic straining occurs, and the former expresses the stiffness of the material at fixed plastic strain. It can be observed that bender element testing on elasticr plastic coupled soils provides data on the reversible part of the strain H ij



rather than on the elastic strain. On clays, each measurement is performed without inducing any variation in the preconsolidation pressure pc (i. e. D kl 0 ); thus each individual test gives data only on the first compliance term § w2 g · ¨ ¸ in the decomposition Equation (9.61). ¨ wVij wVkl ¸ © ¹ We now compare predictions obtained by (9.67) and the observed behaviour of a reconstituted clay. The experimental data are from Rampello et al. (1997). 200000

K 

Go (kPa)

160000

120000

80000

40000

100

200

300

400

500

p (kPa) Figure 9.3. Comparison between moduli from Equation (9.86) and data from Rampello et al. (1997)

9.4 Applications in Geomechanics: Elasticity and Small Strains

175

A series of tests was carried out in computer-controlled triaxial cells, equipped with vertically fitted bender elements. The samples were loaded and unloaded along radial stress paths, defined by values of the stress ratio K q p of 0.0, 0.5, and 0.7. During the compression stages, bender element measurements were performed for different values of the effective stress and overconsolidation ratio. Figure 9.3 shows the values of G against the mean pressure for K 0.5 , using circles for the experimental data, dotted lines for the empirical expression proposed by Rampello et al. (1997), and continuous lines for the prediction derived from (9.67) and using the parameters k 692 , g s 415 , n 0.56 , r 0.29 , and the reference pressure pr 100 kPa. The values of the preconsolidation pressure for K ! 0 have been derived assuming that the shape of the yield surface is that of the modified Cam-clay ellipse with M 0.8 . The comparison between Equation (9.67) and the data is very satisfactory. Note that k has been estimated assuming that, under isotropic stress conditions, Poissoncs ratio Q is equal to 0.25.

9.4.4 Effects of Elasticity on Plastic Behaviour Following from the previous section, to reflect soil properties realistically, the specific Helmholtz free energy f Hij , Dij and the specific Gibbs free energy



g Vij , Dij







should be functions of the kinematic internal variable (most often

plastic strain) tensor Dij . However, this dependency may have further important implications, if the energy potential is assumed to represent a potential also for the generalised stress tensor Fij



wf wDij



wg wDij

(9.68)

leading to the incremental relations in the form of F ij



w2 f w2 f D kl  H kl wDij wD kl wDij wHkl

(9.69)

F ij



w2 g w2 g D kl  V kl wDij wD kl wDij wVkl

(9.70)

or

w2 f w2 g and , which have wHij wD kl wVij wD kl been demonstrated in the previous section to cause effects of plastic strains on elastic behaviour. However, here they cause an inverse effect: plasticity is affected by the elastic component of behaviour. When this inverse coupling is

Clearly, we obtain the same coupling terms

176

9 Small Strain Plasticity, Non-linearity, and Anisotropy

ignored, the First Law of Thermodynamics will be violated. Only when energy potentials can be expressed as (9.63) or (9.64) is this coupling not present, but w2 f w2 g even in this case, the coupling terms and are not equal to wHij wD kl wVij wD kl zero! Differentiation of Equations (9.69) and (9.70) gives in this case w2 g wVij wD kl w2 f wHij wD kl



Gij Gkl

w2 f wHij wHkl



(9.71)

w2 f e e wHij wHij

(9.72)

so that expressions (9.69) and (9.70) are reduced to F ij



w 2 f2 w2 f e D kl  H kl e e   wDij wD kl wH wH

(9.73)

w2 g 2 D kl  V ij wDij wD kl

(9.74)

ij

F ij



kl

respectively, which can both be rewritten as a “translation rule”:

F ij

V ij  U ij

(9.75)

w2 g2 w 2 f2 D kl D kl is the “back” stress. wDij wD kl wDij wD kl The above result is closely linked to another important effect that the elastic strains may have on the plastic behaviour of coupled materials. Collins (2002) demonstrated that for coupled materials, derivation of an elastic-plastic response from basic thermomechanical principles may lead to non-associated flow. Two types of coupling that affect the flow rule were distinguished: “modulus coupling” and “dissipative coupling”. The former is attributed to elastic moduli dependency on plastic strains; the latter is related to the dependency of the dissipation rate on stresses. It was demonstrated that for a decoupled dissipation rate (i. e. independent of stresses), the modulus coupling inevitably produces a non-associated flow rule. Therefore, even when the elastic-plastic coupling has been accounted for correctly by using Equations (9.62) and (9.69) or (9.70), using associated plasticity to derive plastic strains may lead to violation of thermomechanical principles.

where U ij

9.5 Small Strain Plasticity, Non-linearity, and Anisotropy Puzrin and Burland (1998) presented a model for the small and intermediate strain behaviour of overconsolidated clays. The purpose of the model was to

9.5 Applications in Geomechanics: Elasticity and Small Strains

177

reproduce the characteristic S-shaped curve in a plot of normalised secant shear stiffness G Go against shear strain log  J and to generalise this variation of stiffness to any stress path. The model was based on the concepts of plasticity theory. It employed an inner yield surface which encloses a region of stress space in which the response is purely elastic, and an outer surface representing the outer boundary of non-linear behaviour. Once the outer surface is reached, further monotonic loading is at constant stiffness. The plastic strains which occur if the stress point lies between the inner and outer surfaces were calculated using an empirically determined relationship, logarithmic in form (Puzrin and Burland, 1996), that fits the behaviour of clays well. The model was rigorously calibrated against a number of tests on overconsolidated clay. The model does not incorporate any features related to gross plastic strains or to failure, and these would need to be added for a comprehensive description of soil behaviour. In this section, we reformulate the model within the continuous hyperplastic method presented in Chapter 8 [see also Puzrin et al. (2001)]. Because the Puzrin and Burland (1998) model reproduces the behaviour of overconsolidated clays well, further comparisons between the model and test data are not given here. Instead we simply demonstrate that the same material behaviour can be expressed in hyperplasticity. For proportional loading paths from the initial stress point, the continuous hyperplastic model produces behaviour identical to that of the earlier model. For other paths, including those with multiple stress reversals, the continuous hyperplastic model behaviour is identical to that of a classical plasticity model with an infinite number of kinematic hardening elliptical yield surfaces, each one with the associated flow rule and Ziegler’s translation rule.

9.5.1 Continuous Hyperplastic Form of a Small Strain Model The Puzrin and Burland (1998) model was expressed using triaxial stress and T strain parameters. For brevity, we shall adopt a vector notation V ^ pc q` , with the corresponding strains denoted by H

^Hv

T

Hs ` . The initial stresses are

T

V 0 ^ p0c q0 ` . Consider a model with constitutive behaviour defined by the following two potential functionals. For simplicity, the weighting function *  K has been taken as unity, and K is a dimensionless measure of the size of the yield surfaces. The specific Gibbs free energy functional is given by 1

ˆ@ g > V, D



1  V  V0 T D1  V  V0   V  V0 T Dˆ  K dK 2

³ 0

1

1 ˆ  K T B1D ˆ  K dK  hˆ  K D 2

³ 0

(9.76)

178

9 Small Strain Plasticity, Non-linearity, and Anisotropy

The dissipation functional is given by ˆ º d ªD ¬ ¼

1

³  ae   aL  ae K

T Įˆ  K B 1 Įˆ  K dK t 0

(9.77)

0

­dH ½ where ^dH` ® v ¾ is the internal variable vector function and D is an elastic ¯ dH s ¿ J º ªK * stiffness matrix defined by D « » [using the anisotropic elastic form ¬ J 3G * ¼ described by Graham and Houlsby (1983)]. The matrix B is defined by § 1 m · , where n2 K * 3G * and m J 3G * . This particular form of > B@ ¨¨ 2 ¸¸ m n  © ¹ B simplifies the relationship between elastic and plastic behaviour. The parameter ae defines the size of the inner yield surface (within which behaviour is purely elastic), and aL the size of an outer yield surface (defining the outer limit of the variable stiffness region). On further monotonic straining, once the outer surface is reached, the tangent stiffness is constant. Finally, hˆ  K is a plastic modulus function which defines the shape of the stress-strain curve in the non-linear region (see Section 8.10). The specific form of this function will be addressed later.

9.5.2 Derivation of the Model from Potential Functions The dissipative generalised stress vector function Fˆ  K is defined by differentiating the dissipation functional (9.77): ­°Fˆ p °½ ˆ  K a  K B1 D Fˆ  K ® ¾ (9.78) ˆ ˆ  K T B1 D ˆ  K ¯° Fq ¿° D where a  K ae  K aL  ae . We can observe that the internal variable rates can be eliminated from Equation (9.78), leading to the following continuous ˆ  K : field of elliptical yield functions that are independent of D T 2 yˆ g  Fˆ  K , K Fˆ  K BFˆ  K  a  K

0, K >0;1@

(9.79)

where a  K ae  K aL  ae is the semidiameter of each elliptical yield surface in the p direction. In the true stress space, this field of elliptical yield surfaces is given by

 ı  ı0  ȡˆ  K T B  ı  ı0  ȡˆ  K  a  K 2

0, a  K  >ae ; aL @

(9.80)

where Uˆ  K is the “back stress” vector function and the quantity  V 0  Uˆ  K defines the stress coordinates of the centre of each elliptical yield surface.

9.5 Applications in Geomechanics: Elasticity and Small Strains

179

As seen, the yield functions constituting the field do not depend explicitly on ˆ  K ; therefore, the yield surfaces do not undergo any size the internal function D or shape changes or rotation during plastic loading. Hardening of the model is purely due to kinematic translation. The two yield surfaces bounding this field, with semidiameters ae and aL , correspond to the boundaries of the “linear elastic region” (LER) and “small strain region” (SSR), respectively, as defined by Puzrin and Burland (1998). The “back stress” function is obtained by differentiating the specific Gibbs free energy functional (9.76): ­°Uˆ p ½° ˆ  K Uˆ  K ® ¾ V  V0  Fˆ  K hˆ  K B1 D ˆ ¯°Uq ¿°

(9.81)

The flow rule for the field of yield surfaces is obtained by differentiating the yield functions (9.79): ˆ  K 2Oˆ  K BFˆ  K 2Oˆ  K B  V  V 0  Uˆ  K (9.82) D where Oˆ  K is a non-negative multiplier defined from the consistency condition. As seen from Equations (9.82), the associated flow rule holds separately for each yield surface, in both the generalised and the true stress space. An expression for the strain vector and its decomposition into elastic and plastic components is obtained from Equation (9.76): e p H H  H

1

ˆ  K dK D1  V  V 0  D

³

(9.83)

0

The translation rule for the field of the yield surfaces is given by Equations (9.81) and (9.82): ˆ  K 2hˆ  K Oˆ  K  V  V  Uˆ  K (9.84) Uˆ  K hˆ  K B 1 D 0

This rule, known as the Ziegler translation rule [Ziegler (1959)], states that during plastic loading, the incremental displacement of the centre of each yield surface occurs along the radius vector connecting this centre with the current stress state. The incremental stress-strain response of the model is given by 1

H D1V  2 Oˆ  K B  V  V 0  Uˆ  K dK

³

(9.85)

0

where

are Macaulay brackets and Oˆ  K is defined from Oˆ  K

 V  V0  Uˆ  K BV 2hˆ  K a2  K

(9.86)

180

9 Small Strain Plasticity, Non-linearity, and Anisotropy

The plastic strains are updated using Equations (9.82) and (9.86): ˆ  K 2 Oˆ  K B  V  V  Uˆ  K D 0

(9.87)

Then the “back stresses” are updated using Equation (9.81).

9.5.3 Behaviour of the Model During Initial Proportional Loading Elasticity with Cross-coupling During initial loading emanating from the initial stress state V 0 and before the stress state reaches a yield surface with semidiameter a  K , the plastic strain ˆ  K associated with this yield surface is equal to zero. Then, from component D Equation (9.81), it follows that the back stress Uˆ  K 0 , so that the field of yield surfaces is initially centred around the initial stress state V 0 :

 V  V 0 T B  V  V 0  a  K 2

0, K >0;1@

(9.88)

The initial proportional loading emanating from the initial stress state V 0 is described by the following expression: aa (9.89) V  V  V0 aa where aa

 V  V 0 T B  V  V 0

(9.90)

ˆg

When aa  ae , it follows from (9.79) that y  V  V 0  Uˆ  K , K  0 K >0;1@ so that from (9.86), Oˆ  K 0 and the stress-strain behaviour is elastic, with crosscoupling described by the first term in Equation (9.85): H D1 V

(9.91)

This form of constitutive behaviour was proposed by Graham and Houlsby (1983) and used by Puzrin and Burland (1998) to describe the stress-strain behaviour within the LER.

Logarithmic Normalized Stress-strain Curve The functional form of hˆ  K can now be defined following the development of Section 8.10. When aa  >ae ; aL @ , it follows from (9.79) that ­ yˆ g  V  V 0  Uˆ  K , K 0 K >0; Ka @ ° ® g °¯ yˆ  V  V 0  Uˆ  K , K  0 K > Ka ;1@

(9.92)

9.5 Applications in Geomechanics: Elasticity and Small Strains

181

aa  ae , so that aa a  Ka . aL  ae Then, substituting Equation (9.89) in (9.86) and the result in (9.84) and solving differential Equation (9.84) for Uˆ  K , K>0; Ka @ , we obtain

where Ka

§ a  K · Uˆ  K ¨ 1  ¸  V  V0 aa ¹ ©

(9.93)

Substitution of (9.81) and (9.93) in expression (9.82) yields ­ Ka D § a  K · °½ 1 ° H ®1  ¨1  ¸ dK ¾ D  V  V 0 *ˆ aa ¹ ° °¯ 0 3G h  K © ¿

³

(9.94)

Defining V L as the stress point at which the stress path reaches the SSR boundary, we then define a normalised stress y such that  V  V0 y  V L  V0 . It also follows that y aa aL . We define also a normalised strain x such that H x D 1  V L  V 0

(9.95)

The normalisation is chosen so that x y within the linear elastic region. At the boundary of the LER, y x xe ae aL . Equation (9.94) may be rewritten in normalised form as Ka

x

y

D

§

³ 3G*hˆ  K ¨© y  0

a  K · ¸ dK aL ¹

(9.96)

aa  ae y  xe It should be noted that Ka is itself a function of y. aL  ae 1  xe Double differentiation of (9.96) with respect to y yields, after some manipulation, D § d2x · § y  xe · 1 hˆ  K hˆ ¨ ¨ 2¸ ¸ * © 1  xe ¹ 1  xe 3G ¨© dy ¸¹ 2



1

D § d y · § dy · 1 ¨ ¸ 1  xe 3G* ¨© dx 2 ¸¹ ¨© dx ¸¹

1

(9.97)

3

Equation (9.97) establishes the relationship between the shape of the normalised stress-strain  x, y curve and the function hˆ  K . Any plasticity model requires a specific form of the hardening law to be chosen, usually on the grounds of fitting experimental data. Puzrin and Burland (1996 and 1998) found that the following logarithmic function fitted experimental data extremely well. It is expressed as a normalized stress-strain curve describing both

182

9 Small Strain Plasticity, Non-linearity, and Anisotropy

Figure 9.4. Normalized stress-strain curve

deviatoric and volumetric behaviour during any initial proportional loading (Figure 8.2). Note, however, that no particular importance is attached to the specific mathematical form adopted here, and other expressions may provide a comparable quality of fit to experiments. The normalized equation is

^

y xe   x  xe 1  D  ln 1  x  xe

R

`

(9.98)

where § 1  xe 1  c · 1  xL  xe ln 1  xL  xe  R ¨ ¸  x L  1 © xL  xe xL ¹ xL  1 D R  xL  xe  ln 1  xL  xe

(9.99) (9.100)

xL is the normalized limiting strain at the SSR boundary and b is the ratio between the tangent stiffness at the SSR and LER boundaries. Double differentiating (9.98) and substituting it in (9.97), we obtain the following expression for the stiffness function: hˆ  K

^

R



R1

`

3 X 1  D  ln X  DR  X  1  ln X 1 D 1  xe 3G* DRX  ln X R2  1  X ln X   R  1  X  1

where X  K is defined by the relationship K

 X  1

1  D  ln X .

(9.101)

R

1  xe The small strain constitutive model is completely defined by Equations (9.76), (9.77), and (9.101). It can be shown that substitution of Equation (9.101) in incremental Equations (9.84)–(9.86) and subsequent integration of these equations along any straight effective stress path results in a normalized stress-strain

9.5 Applications in Geomechanics: Elasticity and Small Strains

183

curve identical to that given by Equation (9.98) used by Puzrin and Burland (1998) in their model. The model requires seven independent parameters: ae , aL , b, xL , m, n , and G * . The whole set of parameters can be derived from a single consolidated undrained triaxial test with local deformation measurements [Puzrin and Burland, (1998)]. Equation (9.101) is a rather complex, and perhaps unlikely, form of the expression for hˆ  K . However, this form was only chosen to result in the form of the non-linear stress-strain curve already used by Puzrin and Burland (1998) and given in Equation (9.98). They chose this particular form because it models extremely well (much better than other proposed curves) the rapid loss of stiffness of soils at very small strains. Other, simpler forms of hˆ  K could, however, be chosen, and the corresponding stress-strain curve readily obtained (see the next section). One purpose of expressing models within a thermodynamic context is that the expressions employed for the storage and dissipation of energy can be interpreted in physical terms, and sometimes are identified with particular physical processes. This approach has met with some success, for instance, in the plasticity of metals. Such progress is a long-term aim of the work presented here, as this offers the opportunity to unify theoretical and experimental observations. If, for instance, simpler forms of hˆ  K can be identified that result in good fits to experimental results, then it may be possible to provide physical explanations for the form of the function. This is a long-term aim, and no such explanation is at present offered here.

Undrained Loading An important case in soil mechanics is so-called “undrained” loading, in which the volume is constant. Because of the form of Equation (9.94), it can be seen that a proportional loading path produces a straight path in strain space. Conversely, straight strain paths will result in straight stress paths. One such example is undrained loading ( H v 0 ). For this case, it follows that ­ pc  p0c ½ y ­ 0 ½ y ­ J Hs ½ D® ¾  V  V0 ® ¾ ® ¾ , so that the undrained stress path ¯ q  q0 ¿ x ¯Hs ¿ x ¯3G * Hs ¿ is inclined at a constant slope 3G * J in  pc, q space. Puzrin and Burland (1998) suggest that the value of the parameter m J 3G * should be inferred directly from the slope of the effective stress path in an undrained test.

184

9 Small Strain Plasticity, Non-linearity, and Anisotropy

9.5.4 Behaviour of the Model During Proportional Cyclic Loading Consider proportional unloading, taking place after stress reversal at a stress state Vr reached during initial loading emanating from the initial stress state V 0 , and described by the following equation: § 2aa · ¨1  ¸  Vr  V0 ar ¹ ©

 V  V0

(9.102)

T

where aa  V  V0  U  Ka B  V  V0  U  Ka is the semi-diameter of the largest of the yield surfaces currently involved in reverse plastic loading and T

ar  V r  V 0 B  V r  V 0 is the semi-diameter of the largest of the yield surfaces involved in the initial monotonic loading. It follows from the initial ar loading conditions that  V r  V 0  V L  V0 . Recalling the definition aL  V  V0 y  V L  V0 , it follows that during the unloading,

y

ar  2aa aL

(9.103)

When aa  ae , it follows from (9.79) that yˆ g  V  V 0  Uˆ  K , K  0, K >0;1@ so that from (9.86), Oˆ  K 0 , and the stress-strain behaviour is elastic with crosscoupling described by Equation (9.91). When aa  >ae ; aL @ , we divide the field (9.88) of the yield surfaces ar  ae , so that yˆ g  V  V 0  Uˆ  K , K into three regions. Defining Kr aL  ae ar a  Kr , the first region corresponds to the range K > Kr ;1@ , and the yield surfaces yˆ g  V  V 0  Uˆ  K , K in this region were involved neither in initial loading, nor in subsequent reverse loading; therefore their back stresses are zero. In the second region, where K > Ka ; Kr @ , the yield surfaces g yˆ  V  V 0  Uˆ  K , K were involved in initial loading, but not in subsequent reverse loading; therefore, their back stresses can still be defined using expression (9.93): § a  K · § ar  a  K · Uˆ  K ¨ 1  ¸  Vr  V0 ¨ ¸  V L  V0 ar ¹ aL © ¹ ©

(9.104)

Finally, in the third region, where K >0; Ka @ , the yield surfaces yˆ g  V  V 0  Uˆ  K , K were involved in both initial loading and subsequent reverse loading, so that their back stresses are defined by substituting expression (9.102) in the following equation: Uˆ  K

 V  V0 

a  K § a  2a  a  K ·  Vr  V0 ¨ r a ¸  V L  V0 ar aL © ¹

(9.105)

9.5 Applications in Geomechanics: Elasticity and Small Strains

185

Substitution of Equations (9.102)–(9.105) in expression (9.83) yields the same expression as (9.95), but in this case x

y

Ka

K

³

³

r D § a  K · D § ar  a  K · y d  K  ¨ ¸ ¨ ¸ dK *ˆ * aL ¹ aL 3G h  K © 3G hˆ  K © ¹ 0 Ka

(9.106)

At the point of stress reversal, V r , the normalized strain is defined using Equation (9.96): Kr

xr

yr 

§

D

³ 3G*hˆ  K ¨© yr  0

a  K · ¸ dK aL ¹

(9.107)

yr aL  ae ar is itself a function of yr , the normalized stress at aL aL  ae the stress reversal. Then, using Equations (9.103) and (9.107), Equation (9.106) can be rewritten as

where Kr

K

x

y

D

§

³ 3G*hˆ  K ¨© y  0

a  K · ¸ dK aL ¹

(9.108)

yaL  ae xr  x yr  y , x , and y . where K aL  ae 2 2 As seen, Equation (9.108) produces the same logarithmic normalized stressstrain curve given in (9.96) and (9.98), subject to transformation described by the above expressions. The initial curve is enlarged in both coordinate directions by a factor of 2, rotated by 180°, and originates from the stress reversal point on the initial loading normalized stress-strain curve (Figure 8.3). This kind of unloading behaviour is consistent with Masing rules, and confirms the pure kinematic hardening nature of the model presented here. It can be further shown that as soon as the normalized stress y reaches the value  yr , the third region vanishes, and the normalized stress-strain behaviour is again described by the initial logarithmic normalized stress-strain curve given in (9.96) and (9.98). This curve is rotated by 180°, and initiates from the origin. Generally, it can be shown that for proportional cyclic loading, a cycle of a larger amplitude wipes out any memory of all preceding events of a smaller magnitude. If another stress reversal takes place with subsequent reloading, the initial portion of the reloading normalized stress-strain curve is again given by x  xr y  yr and y , which is again consisEquation (9.108), but with x 2 2 tent with Masing rules (see Chapter 8) and produces closed hysteretic loops for uniform proportional cyclic loading (Figure 8.3). Again, if the loading is continued beyond the largest previous stress reversal, reloading produces the initial normalized stress-strain curve (9.96).

186

9 Small Strain Plasticity, Non-linearity, and Anisotropy

Figure 9.5. Normalized stress-strain curves for uniform proportional cyclic loading

It can be shown that for transient and cyclic loading, the continuous hyperplastic formulation of the model produces stress-strain behaviour identical to that of a classical plasticity model with an infinite number of kinematic hardening elliptical yield surfaces, each one with the associated flow rule and Ziegler’s translation rule.

9.5.5 Concluding Remarks The purpose of this section has been to demonstrate that a model for the small strain behaviour of soils, previously presented by Puzrin and Burland (1998) within classical plasticity concepts, together with some additional rules for handling stress reversals, can be expressed within a rigorous thermomechanical formulation. Modelling of the small strain non-linearity of soils is one of the key challenges of current theoretical soil mechanics. The establishment of a theoretical framework within which such models can be achieved is regarded as an important step towards a fuller understanding of soil behaviour. A particular challenge is to combine the modelling of soil behaviour at small and large strains, where the latter has been very successfully achieved within the context of critical state soil mechanics. In the next chapter, we present a model which addresses this issue.

Chapter 10

Applications in Geomechanics: Plasticity and Friction

10.1 Critical State Models Critical State Soil Mechanics (Schofield and Wroth, 1968) is arguably the single most successful framework for understanding the behaviour of soils. In particular, within this approach, complete mathematical models have been formulated to describe the behaviour of soft clays. The most widely used of them is the “Modified Cam-Clay” of Roscoe and Burland (1968). Such models are highly successful in describing the principal features of soft clay behaviour: yield and relatively large strains under certain stress conditions and small, largely recoverable, strains under other conditions. The coupling between volume and strength changes is properly described. The basic critical state models of course have their deficiencies. They do not, for instance, describe the anisotropy developed under one-dimensional consolidation conditions. Nor do they fit well the behaviour of heavily overconsolidated clays. Most importantly, they do not describe a wide range of phenomena that occur due to non-linearities within the conventional yield surface. Treatment of these phenomena within the hyperplastic framework will be presented in the next section; in this section, we show how the classical Modified Cam-Clay plasticity model can be derived.

10.1.1 Hyperplastic Formulation of Modified Cam-Clay In the following, we use the triaxial effective stress variables  pc, q defined in Section 9.2. Because all stresses discussed are effective stresses, the mean effective stress will be written simply p rather than pc . Conjugate to the stresses are the strains  Hv , Hs . For the plastic strains, we use D p , D q , and the generalised

 stresses conjugate to them are  F p , Fq or  F p , Fq .



188

10 Applications in Geomechanics: Plasticity and Friction

g  p, q, Dv , Ds , and then it follows that

The Gibbs free energy is expressed as g

wg Hv  wp wg Hs  wq wg Fp  wD p

Fq



(10.1) (10.2) (10.3)

wg wDq

(10.4)

If the dissipation function is used, it is expressed in the form d d p, q, D p , Dq , D p , D q , and then





Fp

wd wD p

(10.5)

Fq

wd wD q

(10.6)

The dissipation function must be a homogeneous first-order function of the internal variable rates D p , D q .



y



Alternatively, if the yield y p, q, D p , Dq , F p , Fq 0 , then





surface

is

specified

in

the

form

D p

/

wy wF p

(10.7)

D q

/

wy wFq

(10.8)

where / is an undetermined multiplier. We use / for the plastic multiplier here rather than O, because the latter is usually used as one of the material parameters in the Modified Cam-Clay model. Formally, these equations, together with the condition that F p , Fq F p , Fq are all that is needed to specify completely the constitutive







behaviour of a plastic material. The Modified Cam-Clay model can be expressed conveniently within the hyperplastic approach by defining the following functions. The main parameters are defined in Figure 10.1, where V is the specific volume (total volume divided by volume of solids). The values of p, q, and px at the reference (zero) values of strain and plastic strain are po , 0, and pxo , respectively. The (constant) shear modulus is G. The following equations can be simplified by noting the definition px pxo exp D p  O  N .





10.1 Critical State Models

189

ic rit C al

O

st

M

ln(V)

e at

q

e lin

c pi ro ot Is

px

ln(p)

N

N C L

px

p

Figure 10.1. Definitions of modified Cam-clay parameters

The Gibbs free energy is chosen as g

§ § p · · q2 § Dp · Np ¨¨ log ¨ ¸  1 ¸¸   pD p  qDq   O  N pxo exp ¨ ¸ (10.9) ©ON¹ © © po ¹ ¹ 6G





The first two terms define the elastic behaviour (the unusual first term results in a bulk modulus proportional to pressure). The third term ensures that the internal variable plays the role of the plastic strain (see Table 5.2 in Section 5.5). The final term defines the hardening of the yield surface. From Equations (10.1) and (10.2), it follows that Hv



§ p · N log ¨ ¸  D p © po ¹ wg q   Dq wq 3G

wg wp

Hs

Fp



wg wD p Fq

§ Dp · p  pxo exp ¨ ¸ ©ON¹ wg  q wDq

(10.10) (10.11) (10.12) (10.13)

The dissipation function may be specified as § Dp · 2 2 2 pxo exp ¨ ¸ D p  M D q O  N © ¹ which leads [Equations (10.6)] to d

(10.14)

Fp

wd wD p

§ Dp · D v pxo exp ¨ ¸ 2 O  N © ¹ D p  M 2D q2

(10.15)

Fq

wd wD q

§ Dp · M 2D s pxo exp ¨ ¸ © O  N ¹ D 2p  M 2D q2

(10.16)

190

10 Applications in Geomechanics: Plasticity and Friction

These equations may be combined to obtain the yield function, which of course can alternatively be taken as the starting point:

y

F2p

Fq2

§ § Dp ··  2  ¨ pxo exp ¨ ¸ ¸¸ ¨ M © © O  N ¹¹

2

0

(10.17)

From this, it follows [Equation (10.8)] that D p

/

D q

/

wy wF p

wy wFq

2/F p 2/

M2

Fq

(10.18) (10.19)

The derivation of the Modified Cam-Clay model from the above functions is not pursued here, but it can readily be verified that the above equations do define incremental behaviour consistent with the usual formulation of Modified Cam-Clay. The only exception is that consolidation and swelling lines are considered straight in  ln p,lnV space rather than  ln p,V space. The result is that O and N have slightly different meanings from their usual ones and that the (variable) elastic bulk modulus is given by K p N rather than the more usual K pV N . Butterfield (1979) argues that the modified form is more satisfactory. Note that the above choice of the energy functions is not unique. This topic was addressed briefly by Collins and Houlsby (1997) and is discussed more fully in the following section.

10.1.2 Non-uniqueness of the Energy Functions Collins and Houlsby (1997) demonstrated that the modified Cam-Clay model can be derived from either of two different pairs of Gibbs free energy and dissipation functions. This raises the interesting concept that, because the same constitutive behaviour can be derived from different energy functions, then conversely the energy functions are not uniquely determined by the constitutive behaviour. The energy functions are not therefore objectively observable quantities. The case discussed by Collins and Houlsby is a special case of the following more general result. Consider a model specified by g g1  V, D and d d1  V, D, D . It follows that H wg1 wV , F wg1 wD , and F wd1 wD . Using F F gives wg1 wD  wd1 wD 0 . g g1  V, D  g 2  D and Now consider a model in which   d d1  V, D, D   wg 2 wD D . In this case, again H wg1 wV , but this time F wg1 wD  wg 2 wD and F wd1 wD  wg 2 wD . However, using F F again gives wg1 wD  wd1 wD 0 . Thus identical constitutive behaviour is given by the

10.2 Towards Unified Soil Models

191

two models. Of course, the models are acceptable only if both d1  V, D, D ! 0 and d1  V, D, D   wg 2 wD D ! 0 for all D . For typical forms of the dissipation function, it often proves possible to find a function g 2 that satisfies this condition. The particular model described above may alternatively be derived from the expressions, § § p · · q2 Np ¨¨ log ¨ ¸  1 ¸¸   pD p  qDq © © po ¹ ¹ 6G § Dp · 2 2 2 d pxo exp ¨ ¸ D p  D p  M D q O  N © ¹



g







(10.20) (10.21)

10.2 Towards Unified Soil Models One of the major limitations of the basic critical state models is that they do not describe a wide range of phenomena that occur due to the irreversibility of strains and non-linearities within the conventional yield surface which may take place at a very small strain level. In Sections 9.5 and 10.1, we demonstrated how both large-scale yielding and small strain plasticity can be formulated (separately) within a hyperplastic and continuous hyperplastic framework. This section presents a unified formulation, where both large and small strain plasticity are described within a single, unified, continuous hyperplastic model.

10.2.1 Small Strain Non-linearity: Hyperbolic Stress-strain Law In Section 9.5, we presented a continuous hyperplastic formulation of small strain non-linearity based on the Puzrin and Burland (1996) logarithmic function. It provides realistic fitting of the typical “S-shaped” curves of secant shear stiffness against the logarithm of shear strain observed for soils (see Figure 10.2b). For the purposes of this section, however, the simple hyperbolic form is sufficient to illustrate the principles involved. As in the logarithmic model, a one-dimensional hyperbolic model can be defined by two potential functionals: 1 1 ˆ h  K V2 ˆ@  V D ˆ  K dK  g > V, D (10.22)  Dˆ  K 2 dK ³ ³ 2E 2 0

ˆ º d ªD ¬ ¼

0

1

³ kK Dˆ  K dK

(10.23)

0

The dissipative generalised stress function Fˆ  K is obtained as Fˆ  K

wdˆ ˆ  K kK S D ˆ  K wD





(10.24)

192

10 Applications in Geomechanics: Plasticity and Friction

so that the field of yield functions is given by

 Fˆ  K 2  k2 K2

yˆ  K

(10.25)

0

The generalised stress is defined by wgˆ Fˆ  K  ˆ  K wD

ˆ  K V  hˆ  K D

(10.26)

and the strain is given by H 

1

wg wV

V ˆ  K dK  D E

³

(10.27)

0

Combining Equations (10.24), (10.26), and (10.27) allows the stress-strain curve for monotonic loading (from zero initial plastic strain) to be expressed as H

V  E

K*

³ 0

V  kK dK hˆ  K

(10.28)

where K * specifies the largest yield surface that has yet been activated, such that Fˆ  K * kK* Fˆ  K * V . Differentiation of Equation (10.28) twice with respect to V (using standard results for the differential of a definite integral in which the limits are themselves variable) leads to the important result d2H dV

2

1 ˆ kh  V k

(10.29)

so that the plastic modulus function hˆ  K is uniquely related to the second derivative of the initial “backbone” curve H  V (see Section 8.10). The hyperbolic stress-strain curve (Figure 10.2a) is given by (see Section 8.7) H

Vk  2  a V E k  V

2

V  a  1 V  E E k  V

(10.30)

where a E E50 , E is the initial stiffness, and E50 is the secant stiffness to 2 1 d 2 H 2  a  1 k V k 2 . This curve is generated by the function , so kh  V k dV2 E  k  V 3 that for the hyperbolic model, Equation (10.22) should be supplemented by hˆ  K

3

E 1  K

2  a  1

(10.31)

10.2 Towards Unified Soil Models

193

1 0.9

(a)

0.8 0.7

/c

0.6 0.5 0.4 0.3

1

0.2

a

0.1 0 0

2

4

6

8

10

1

2

HE /c 1 0.9

(b)

0.8

E secant / E

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -3

-2

-1

0

log10(H E / c )

Figure 10.2. (a) Hyperbolic stress strain curve (for a = 5); (b) typical stiffness - log strain curve for soil

10.2.2 Modified Forms of the Energy Functionals The continuous hyperplastic formulation presented in Chapter 8 considered a Gibbs free energy of the form, ˆ ij º g ª¬ Vij , D ¼

³ gˆ  Vij , Dˆ ij  K , K *  K dK

(10.32)

8

and a dissipation function of the form, ˆ ij , D ˆ ij º d ª Vij , D ¬ ¼

1

³ dˆ  Vij , Dˆ ij Dˆ ij  K , K *  K dK

(10.33)

0

The following result is derived from the Frechet differential of the energy functional: ˆ ij º dD ˆ g c ª¬Vij , D ¼ ij

³

8



ˆ ij  K , K wgˆ Vij , D ˆ ij  K wD

dDˆ *  K dK ij

(10.34)

194

10 Applications in Geomechanics: Plasticity and Friction

ˆ ij . By defining where g c indicates the derivative with respect to the function D ˆ ij  K *  K dK  sT Hij V ij  Fˆ ij  K D

³

g

(10.35)

8

an expression for the generalised stress function is obtained: Fˆ ij



wgˆ ˆ ij  K wD

(10.36)

Similarly, wdˆ

Fˆ ij

(10.37)

ˆ ij  K wD

In the cases considered below, g takes a more complicated form that can be written as 1



 





ˆ ij º g1 Vij  Vij Dij  g 2 Dij gˆ3 Dˆ ij  K , K dK  g 4 Dij g ª¬ Vij , Dij , D ³ ¼

(10.38)

0

1

³ Dˆ ij dK

where for convenience, a variable Dij

is also introduced. From the

0

above, it can be shown that 1

ˆ ij º dD ˆ g c ª¬Vij , Dij , D ¼ ij



³ g 2 Dij 0



ˆ ij  K , K wgˆ3 D ˆ ij  K wD

dDˆ dK ij

(10.39)

and wg wDij

1

wg wg ˆ ij  K dK  4 Vij  2 ³ gˆ3 D wDij wDij





(10.40)

0

If the time rate of change of the Gibbs free energy is written 1

g

Hij V ij  Fij Dij  ³ Fˆ ij  K Dˆ ij  K dK  sT

(10.41)

0

and the definition of Dij as a constraint equation,



1





1

ˆ ij  K dK Dij  Dˆ ij  K dK 0 c c1 Dij  ³ cˆ2 D ³ 0

0

(10.42)

10.2 Towards Unified Soil Models

195

then it is possible to define the generalised stresses in terms of the derivatives of an augmented energy expression g  /c c , where /c is a multiplier to be determined:



Fˆ ij

 g 2 Dij

Fij

Vij 



ˆ ij  K , K wgˆ3 D ˆ ij  K wD 1

/



ˆ ij  K wcˆ2 D c



ˆ ij  K wD

wg 2 wg wc ˆ ij  K dK  4  / c 1 gˆ3 D wDij ³ wDij wDij





(10.43)

(10.44)

0

The dissipation is considered of the form, ˆ ij , D ˆ ij º d ª Vij , D ¬ ¼

1

³ dˆ1  Vij , Dij , K dˆ2  Dˆ ij  K dK

(10.45)

0

From this, the Frechet differential leads to the result,



ˆ

wDˆwd2K

(10.46)

0

(10.47)

Fˆ ij  K dˆ1 Vij , Dij , K

ij

and it also follows that Fij

wd wD ij

Given particular forms of the functions, the above equations are sufficient to determine the constitutive behaviour.

10.2.3 Combining Small-strain and Critical State Behaviour As mentioned in earlier sections, a major criticism of critical state models is the fact that they describe the behaviour of soils inadequately at small strains. Coupled to this is poor performance with respect to modelling cyclic loading, and no modelling of the effects of immediate past history. To remedy this situation, the benefits of the simple hyperplastic framework for describing kinematic hardening of an infinite number of yield surfaces is combined with the Modified Cam-Clay model. The following expressions are suggested for the thermodynamic potentials: g

§ § p · · q2 Np ¨¨ log ¨ ¸  1 ¸¸   pD p  qDq © © po ¹ ¹ 6 gp





1  K 3  px N Dˆ 2p  3gpx Dˆ q2 ³ dK   O  N px 2  a  1 2 0 1

(10.48)

196

10 Applications in Geomechanics: Plasticity and Friction 1

d

³  px K

ˆ 2p  M 2 D ˆ q2 dK D

(10.49)

0

1

§ Dp · pxo exp ¨¨ ¸ have been introduced. O  N ¸¹ © 0 Note in the above that the stiffness factors for the kinematic hardening of the yield surfaces [the integral term in Equation (10.48)] have been made proportional to preconsolidation pressure rather than pressure. This has the advantage of avoiding elastic-plastic coupling, which alters the meaning of the internal variable (Collins and Houlsby, 1997). However, the presence of the preconsolidation pressure in these expressions considerably complicates the derivatives of the Gibbs free energy functional. In the following, a linearised version of the continuous hyperplastic Modified Cam-Clay is therefore explored, although the derivation from Equations (10.48) and (10.49) is set out in Section 10.2.6. This avoids some of the coupling terms and will serve as an example to illustrate the main features of the model. The suggested Gibbs free energy and dissipation expressions are

where the definitions D

g



³ Dˆ dK and px

2

3

1

2

2

hD p 1  K K Dˆ p  3GDˆ q p2 q 2   pD p  qDq  dK  2K 6G 2  a  1 2 2



³

(10.50)

0

1

d

³  hKD p

ˆ 2p  M 2 D ˆ q2 dK D

(10.51)

0

Comparison of Equation (10.50) with Equation (10.38) gives g1 g 2 1 , gˆ3

1  K 3 K Dˆ 2p  3GDˆ q2 2  a  1 2

, g 4

hD 2p



p2 q 2 ,  2K 6G

. From these, it follows that

2

wg1 wp

p  Dp K

(10.52)

wg1 wq

q  Dq 3G

(10.53)

Hv



Hs



wcˆ2 p wgˆ3  / cp ˆp ˆp wD wD

Fˆ p



Fˆ q

wcˆ2q wgˆ  3  / cq ˆq ˆq wD wD

3



K 1  K

2  a  1

ˆ p  / cp D

(10.54)

Dˆ q  / cq

(10.55)

3



3G 1  K 2  a  1

10.2 Towards Unified Soil Models

Fp

wc1 p wg 4  / cp wD p wD p

p

p  hD p  / cp

(10.56)

q  / cq

(10.57)

wcq wg 4  / cq wDq wDq

Fq

q

Fˆ p

wdˆ ˆ p wD

Fˆ q

wdˆ ˆ q wD

ˆ p D

 hKD p

ˆ 2p  M 2 D ˆ q2 D ˆ q M 2D

 hKD p

197

ˆ 2p  M 2 D ˆ q2 D

(10.58)

(10.59)

Therefore, the yield surface is Fˆ q2 2  hKD p yˆ Fˆ 2p  2 M





(10.60)

To derive the incremental constitutive behaviour, the above are differentiated further to give the following, which also use Fˆ Fˆ and F F 0 [see Equation (10.47)]: p  D p K q  D q 3G

H v

H s

Fˆ p

(10.61)

3

p  Fˆ q

K 1  K  ˆ p  hD p D 2  a  1

(10.62)

3

3G 1  K  ˆq D 2  a  1

q 

(10.63)

Finally, the derivatives of the yield surface and the consistency condition are required: ˆ p D

/

ˆ q D

/

wyˆ wFˆ p

wyˆ wFˆ q

2/Fˆ p 2/ M2

Fˆ q

2Fˆ q Fˆ q 2  2  hK Dˆ p Dˆ p yˆ 2Fˆ p Fˆ p  M2

(10.64)

(10.65)

198

10 Applications in Geomechanics: Plasticity and Friction

Although it would be attractive to carry out calculations using the functions of K directly, at least with the presently available software, it is necessary first to discretise these functions in terms of a finite set of values. The internal function ˆ i , i 1!n . The ˆ  K is therefore represented by a set of n internal variables D D result is that the field of an infinite number of yield surfaces is thus approximated by a finite number of yield surfaces. In the calculations presented below, the field is represented by 10 yield surfaces. For the purposes of numerical calculation, the continuous hyperplasticity models therefore have much in common with multisurface plasticity. There is, however, a significant difference in that the internal variables clearly play the role of approximating the underlying internal function. This opens up possibilities in the future of adopting more sophisticated numerical representations of the function. A detailed implementation of the above equations for numerical calculations is addressed in Section 10.2.5.

10.2.4 Examples Some example calculations illustrate the features of the model described above. All the following calculations use the constants G 2000 , K 2000 , h 200 , M 1 , and a 1.1 . The units are arbitrary, but note that for a model of a specific soil, the constants G, K, and h have the dimensions of stiffness, whilst M and a are dimensionless. Figure 10.3 shows the behaviour of the soil in an isotropic consolidation test, first loaded to p 100 , then unloaded to p 20 , and then reloaded. It can be seen that the unloading line is curved, as is the reloading line. When the preconsolidation pressure is reached, there is no sharp yield, but instead the reloading curve merges smoothly with the virgin consolidation line. This type of behaviour is a feature of many soils. The curvature of the unloading and reloading lines (and hence the openness of the hysteresis loop) is controlled by the form chosen 0

Volumetric strain v

1 2 3 4 5 6 7 8 9 0

20

40

60

80

100

120

140

160

Mean stress p Figure 10.3. Consolidation curve for linearised continuous hyperplastic modified Cam-Clay

10.2 Towards Unified Soil Models

199

for the kernel function in the expression for the Gibbs free energy [equivalent to h  K in Equation (10.22)]. In this particular model, the curvature is controlled by the value of the parameter a. As the curvature is reduced, the preconsolidation point on reloading becomes more sharply defined in the response. Figure 10.4 shows (bold line) the stress path for a sample which is first consolidated isotropically to p 100 and then sheared undrained. The undrained stress path is typical of a normally consolidated clay. Also shown in the figure are the positions of the 10 yield surfaces used in the calculation. Note that the yield surfaces overlap. There is a widely held misconception that multiple yield surfaces should be “nested,” i. e. non-overlapping. This condition is usually attributed to Prevost (1978), but there are no strong reasons why this needs to be the case, see action 6.5. Note two features of the field of yield surfaces. Firstly, the small surfaces are “dragged” by the stress point, and therefore provide coding of the past stress history. Secondly, note that the largest yield surface has never been engaged by the stress point. This latter effect is due to a subtlety of the interaction between the yield surfaces. As plastic volumetric strain occurs on the inner surfaces, the size of all yield surfaces increases. The result is that during isotropic consolidation, once a sufficient number of the inner surfaces have been engaged, the surfaces expand at a sufficient rate that the outer surfaces are never encountered by the stress point. 60

q 40

20

0 0

20

40

60

80

100

120 p

-20

-40

-60

Figure 10.4. Undrained stress path and field of yield surfaces

200

10 Applications in Geomechanics: Plasticity and Friction

Figure 10.5. Undrained stress-strain curves

Figure 10.5 shows the undrained deviator stress against deviator strain curve for the same test as in Figure 10.4, but continued with two unload-reload cycles. Not only is there hysteresis on unloading and reloading, but there is also some accumulation of shear strain during the cycles. One of the most important features of the model described here is its ability to capture the effects of recent stress history. Figure 10.6 gives the definitions of a series of stress points. In the following, we shall consider the results of four undrained tests from point B, which is at an overconsolidation ratio of 2. However, each of the tests will have been preceded by a different immediate past history, so that point B will have been approached by a drained loading from the direction of point C, A, D, or E, respectively, in the four tests. Such a group of tests was used by Atkinson et al. (1990) to demonstrate the importance of immediate past stress history.

q

E O

A

B

C

p

D Figure 10.6. Stress paths for clay with OCR = 2

10.2 Towards Unified Soil Models

201

60

60

q

q

(a)

(b)

40

40

20

20

0

0

0

20

40

60

80

100

120 p

0

-20

-20

-40

-40

-60

-60

60

20

40

60

80

100

120 p

60

q

(c)

40

(d)

q 40

20

20

0

0 0

20

40

60

80

100

120 p

0

-20

-20

-40

-40

-60

-60

20

40

60

80

100

120 p

Figure 10.7. Fields of yield surfaces after stress paths (a) OACB; (b) OACAB; (c) OACBDB; (d) OACBEB

Also shown in the figure is the position of the yield surface that would exist for a simple Modified Cam-Clay model. All stress histories lie within this surface, so that the simple model would predict purely elastic response for all four cases. Figure 10.7 shows the positions of the yield surfaces after the four different stress histories. Dragging of the surfaces behind the stress point is clear. Figure 10.8 shows the resulting undrained stress-strain curves when the sample is sheared from point B. The four responses are quite different. The sample that has just suffered the most severe stress reversal (c) shows the stiffest response, whilst the sample in which the stress path is a smooth continuation of the previous path (d) shows the most flexible behaviour. The two cases where the stress path turns through a 90° angle show an intermediate response. At large strain, the response for case (b) is softest because of the swelling that has occurred during the unloading to point A. The above behaviour is shown more clearly in Figure 10.9, which shows the same data in terms of the normalised secant stiffness Gs G , where Gs 'q 3'Hs , against the deviatoric strain (on a logarithmic scale). Each of the tests shows the characteristic “S-shaped” curve which is commonly observed. The high stiffness is maintained longest by the sample with the complete stress reversal. The much lower stiffness for the sample with a smoothly continuing stress path is apparent.

202

10 Applications in Geomechanics: Plasticity and Friction

Figure 10.8. Stress-strain curves for clay at OCR of 2 after different immediate past stress histories

The past stress history affects not only the stiffness of the sample. Figure 10.10 shows the undrained stress paths for the four tests. In this case, the most notable differences are between case (a), where the immediate past history involved a reduction in mean stress, and case (b), where it involved an increase in stress. In the first case, the subsequent undrained path first involves an increase in mean stress, and in the second it involves a reduction. Thus the model predicts that effective stress paths (and hence pore pressures) during undrained behaviour would depend on immediate past history. This behaviour is exactly as observed by Stallebrass and Taylor (1997).

Figure 10.9. Normalized stiffness against log(strain) for clay at OCR of 2 after different immediate past stress histories

10.2 Towards Unified Soil Models

203

40 35

q

(b)

(c)

(d)

(a)

30 25 20 15 10 5 0 0

20

40

60

p

80

Figure 10.10. Stress paths for clay at OCR of 2 after different immediate past stress histories. (The irregularities in the curves are a result of numerical discretisation)

The above examples illustrate that a model based on the continuous hyperplasticity approach can capture many of the important features of soil behaviour at small strains, whilst being consistent with the ideas of critical state soil mechanics for larger strains. Models that achieve this have been published before, and often use the multiple surface plasticity concept. The benefits of the continuous hyperplastic approach lie, however, in the extremely compact representation of the models. All the results presented in Figures 10.3–10.10 arise from a model which is specified entirely by two equations, (10.50) and (10.51), together with standard procedures. No additional ad hoc assumptions or rules are required.

10.2.5 Continuous Hyperplastic Modified Cam-Clay In the above, a linearised version of the continuous hyperplastic modified Camclay, derived from simplified Equations (10.50) and (10.51), was explored. For completeness, we now explore the full non-linear formulation derived from Equations (10.48) and (10.49): Hv

Fˆ p Fˆ q

§ p · q2 N log ¨ ¸   Dp 2 © po ¹ 6 gp q Hs  Dq 3gp 

(10.66) (10.67)

1  K 3 § px ˆ · ¨ D p ¸  / cp 2  a  1 © N ¹

(10.68)

1  K 3 ˆ 3gpDq  / cq 2  a  1

(10.69)



204

10 Applications in Geomechanics: Plasticity and Friction

1  K 3  px N Dˆ 2p  3gpx Dˆ q2 dK  p  / x cp 2  a  1 2O  N 0 1

Fp

p³

Fq

q  /cq

Fˆ p

px K

Fˆ q

px K

(10.70) (10.71)

ˆ p D

(10.72)

ˆ 2p  M 2 D ˆ q2 D ˆ q M 2D

(10.73)

ˆ 2p  M 2 D ˆ q2 D

from which the field of yield surfaces can be derived as Fˆ q2 2   px K yˆ Fˆ 2p  2 M

0

(10.74)

From the above, the following can be derived: 3 ˆ 2p  3gpx D ˆ q2 1  K  px N D  1  K 3 § px Dˆ · p³ dK  px  p¸ ¨ 2  a  1 2O  N 2  a  1 © N ¹ 0 1

Fˆ p

Fˆ q

q

1  K 3 ˆ 3gpDq 2  a  1

(10.75)

(10.76)

The derivation of the incremental form of the model is complicated by the integral term in Equation (10.75), but this can nevertheless be accommodated by a method similar to that adopted in Section 10.2.5. We have presented here one possible way of accommodating both small strain behaviour and critical state concepts in a single model based on hyperplasticity. There are many ways this can be achieved by hyperplasticity. For instance, alternative approaches within hyperplasticity are explored by Einav (2002) [see also Einav and Puzrin (2003, 2004a,b) and Einav et al. (2003a,b)] and by Likitlesuang (2003) [see also Likitlersuang and Houlsby (2006)].

10.3 Frictional Behaviour and Non-associated Flow In Chapter 5, we gave examples of plasticity models derived using the hyperplastic approach. Now we examine in more detail frictional plasticity (i. e. plasticity in which the dissipation is pressure-dependent), which is typical of the behaviour of geotechnical materials (particularly soils). As follows from Chapter 5, within the hyperplastic framework, the pressure-dependency of the dissipation function results in non-associated plastic flow.

10.3 Frictional Behaviour and Non-associated Flow

205

The yield surface considered here is the generalised Houlsby (1986) criterion, which is a generalisation of the criterion introduced by Matsuoka and Nakai (1974) for describing yielding in geotechnical materials. This case also serves as an example of the derivation of the yield function from the dissipation function and vice versa. It illustrates the use of constraints and the fact that the Legendre transformations between different functions are sometimes not straightforward. This particular example is most conveniently expressed in terms of principal stresses and strains. The example will be developed in this way, and the final results restated in tensorial form. In the following, we use single subscripts to identify the principal values of tensors. We start with an assumed form of the dissipation function:

d

8ª  k  PV1  k  PV2  D 1  D 2 2   k  PV2  k  PV3  D 2  D 3 2 « ¬ 9 (10.77) 2º   k  PV3  k  PV1  D 3  D 1 » ¼

where k is the cohesive component of shear strength and P is a frictional factor. Note that for P 0 this reduces to the dissipation function for the von Mises material studied earlier in Chapter 5. For convenience, we define the following additional variables:

f1

k  PV1

(10.78)

a1

D 2  D 3

(10.79)

Similar definitions to the above apply with appropriate cyclic permutation of the subscripts 1, 2, and 3. The dissipation function may then be rewritten d

8ª f1 f 2a32  f 2 f3a12  f3 f1a22 º ¼ 9¬

(10.80)

10.3.1 The Dissipation to Yield Surface Transformation For simplicity, we shall assume zero plastic dilation, which we introduce through the following constraint:



c D ij

Then the generalised stresses are wd wc F1 / wD 1 wD 1

D 1  D 2  D 3

0

F2

wd wc /  wD 2 wD 2

8 > f1 f2a3  f1 f3a2 @  / 9d 8 > f2 f3a1  f1 f2a3 @  / 9d

F3

wd wc / wD 3 wD 3

8 > f1 f3a2  f2 f3a1 @  / 9d

(10.81)

(10.82) (10.83) (10.84)

206

10 Applications in Geomechanics: Plasticity and Friction

where / is the Lagrangian multiplier associated with the constraint, which can be immediately determined as /  F1  F2  F3 3 . Introducing the definitions b1 f 2 f3a1 and k1 F2  F3 (each with cyclic permutations of the subscripts), Equations (10.82)–(10.84) can be rewritten 8 > 2b1  b2  b3 @ 9d 8 >2b2  b1  b3 @ 9d 8 >2b3  b1  b2 @ 9d

k1 k2 k3

They can be further simplified by introducing b

(10.85) (10.86) (10.87)

b1  b2  b3

3:

3d k1 b1  b 8 3d k2 b2  b 8 3d k3 b3  b 8

Since a1  a2  a3

(10.88) (10.89) (10.90)

0 , the following can be obtained: b1 f1  b2 f 2  b3 f3

0

(10.91)

Equations (10.88)–(10.91) can be used to obtain a solution for b: b 

3d  k1 f1  k2 f 2  k3 f3 8  f1  f2  f3

(10.92)

When Equations (10.88)–(10.90) are squared, multiplied by the corresponding f, and summed, the following is obtained: 9 2 2 d k1 f1  k22 f 2  k32 f3 64







(10.93)



b12 f1  b22 f 2  b32 f3  2b  b1 f1  b2 f 2  b3 f3  b 2  f1  f 2  f3

Note that the dissipation function can also be expressed in the form, 9 f1 f 2 f3d 2 8

b12 f1  b22 f 2  b32 f3

(10.94)

Substitution of Equations (10.91) and (10.94) in (10.93) gives 9 2 2 d k1 f1  k22 f 2  k32 f3 64



98  f1 f2 f3d2  b2  f1  f2  f3

(10.95)

10.3 Frictional Behaviour and Non-associated Flow

207

Substitution of Equation (10.92) in (10.95) allows elimination of the plastic strain rates to give the equation of the yield surface:





k12 f1  k22 f 2  k32 f3  8 f1 f 2 f3 

 k1 f1  k2 f2  k3 f3 2  f1  f2  f3

(10.96)

0

If the third term on the lift side of Equation (10.96) vanishes, a condition which is discussed below, this yield surface reduces to the Houlsby (1986) criterion, which can be written

 k12 f1  k22 f2  k32 f3  8 f1 f2 f3

(10.97)

0

or alternatively,

 F1  F2 2  F 2  F 3 2  F3  F1 2   8  k  PV1  k  PV2  k  PV2  k  PV3  k  PV3  k  PV1

(10.98)

0

The third term on the left side of Equation (10.96) vanishes as long as the back stress is the same for all generalised stress components, so that k1 V2  V3 F2  F3 , and similarly for cyclic permutations of the subscripts 1, 2, and 3. This will be true if the back stress is isotropic or zero and will be achieved by appropriate choice of the energy function. In particular, if kinematic hardening is not present, this term will disappear.

10.3.2 The Yield Surface to Dissipation Transformation Now as an alternative, assume as the starting point a yield function in the form of the generalised Houlsby (1986) criterion: y

 k12 f1  k22 f2  k32 f3  8 f1 f2 f3   k1 f1 f1 k2ff22  fk33 f3

2

0

(10.99)

in which the last term would be zero for an isotropic back stress. By differentiating (10.99) with respect to F, we obtain D 1 2O D 2 2O D 3 2O

 k1 f1  k2 f2  k3 f3 f  f  3 2  f1  f2  f3 k f  k f  k f k1 f1  k3 f3  1 1 2 2 3 3  f1  f3  f1  f2  f3 k f  k f  k f k2 f 2  k1 f1  1 1 2 2 3 3  f 2  f1  f1  f2  f3

k3 f3  k2 f 2 

where O is the plastic multiplier.

(10.100) (10.101) (10.102)

208

10 Applications in Geomechanics: Plasticity and Friction

Equations (10.100) are not independent, resulting in a constraint D 1  D 2  D 3 0 . They can (after some manipulation) be rewritten as a1 3 f1  k1  e 2O a 2 3 f 2  k2  e 2O a3 3 f3  k3  e 2O

(10.103) (10.104) (10.105)

where

k f  k f  k f e  11 2 2 3 3  f1  f2  f3

(10.106)

Rewriting Equations (10.103)–(10.105) , we obtain k1

a1 e 6Of1

(10.107)

k2

a 2 e 6O f 2

(10.108)

k3

a3 e 6O f 3

(10.109)

Substitution of Equations (10.106)–(10.109) in the yield surface Equation (10.99) gives (after considerable manipulation) the following expression for O: f1 f 2a32  f1 f 2a12  f1 f3a22 8 f1 f 2 f3

6O

(10.110)

From the definition of the dissipation function, together with the constraint D 1  D 2  D 3 0 already derived, it follows that d

F1D 1  F 2 D 2  F3D 3

1  k1a1  k2a2  k3a3 3

Substitution of Equations (10.107)–(10.110), and a1  a2  a3 gives the following expression for the dissipation function: d

8 f1 f 2a32  f1 f 2a12  f1 f3a22 9



which is of course identical to (10.80).



(10.111)

0 in (10.111)

(10.112)

10.4 Further Applications of Hyperplasticity in Geomechanics

209

10.3.3 Tensorial Form For convenience, define fij kGij  PVij . The dissipation and yield functions described above can then be written d y

8ª fii f jk D kl D lj  fij D ij 9 «¬





2º

»¼

(10.113)

fii F jk Fkj  3 fij F jk Fki  2 fij F ji Fkk



4 3

 f

ii

3  3 fii f jk fkj  2 fij f jk fki



2 § · 1 ¨ 3 fij f ji Fkl Flk  fij f ji  Fkk ¸ 0  fmm ¨¨   f 2 F F  3 f F 2  2 f F f F ¸¸ ii kl lk ij ji ii jj kl lk ¹ ©



(10.114)



Although the expression for the yield surface appears particularly complex in this form, it should be noted that the final bracketed term is zero for isotropic back stress.

10.4 Further Applications of Hyperplasticity in Geomechanics Collins and Hilder (2002) use the hyperplastic approach to develop families of models describing the elastic/plastic behaviour of cohesionless soils deforming under triaxial conditions. Once the form of the free energy and dissipation potential functions have been specified, the corresponding yield surfaces, flow rules, isotropic and kinematic hardening rules as well as the elasticity law are deduced in a systematic manner. The families contain the classical linear frictional (Coulomb type) models and the classical critical state models as special cases. The generalised models discussed here include non-associated flow rules, shear as well as volumetric hardening, anisotropic responses, and rotational yield loci. The various parameters needed to describe the models can be interpreted in terms of the ratio of plastic work which is dissipated, to that which is stored. Non-associated behaviour is found whenever this division between dissipated and stored work is not equal. Microlevel interpretations of stored plastic work are discussed. The models automatically satisfy the laws of thermodynamics, and there is no need to invoke any stability postulates. Some classical forms of the peak-strength/dilatancy relationship are established theoretically. Some representative drained and undrained paths are computed. Collins and Kelly (2002) and Collins (2003) propose a systematic, hyperplasticity based procedure for developing constitutive models for frictional materials possessing a critical state in a three-dimensional context. The models involve a number of parameters, which can be interpreted in terms of micromechanical energy storage and dissipative mechanisms. In most cases non-associated flow

210

10 Applications in Geomechanics: Plasticity and Friction

rules are predicted, and in some cases, the yield surfaces have concave segments. The procedure is more general than that traditionally used for materials with non-associated flow rules; in that plastic potentials are not needed and not presumed to exist. In illustration, examples of families of models are given in which the critical state surface is either the Drucker-Prager or the Matsuoka-Nakai cone. Einav and Puzrin (2003, 2004b) unify the hyperplasticity and continuous hyperplasticity formulations to facilitate derivation of new forms of thermodynamic models that exhibit both global continuous behaviour and the phenomenon of abrupt stiffness change within a single package. The concept allows for development of a technique for modelling different kinematic stiffness regions bounded within an outer isotropic hardening yield surface, as observed in clays. This feature is employed for derivation of a new continuous hyperplastic critical state model. It was demonstrated how the specification of two potential functionals allows derivation of constitutive models that satisfy the laws of thermodynamics, and at the same time account for many important aspects of soil behaviour. Houlsby and Mortara (2004) explore an approach to the behaviour of frictional materials based on continuous hyperplasticity. The approach draws on earlier work by Houlsby (1992). Collins and Muhunthan (2003) explore the relationship between stress– dilatancy, anisotropy, and plastic dissipation for geotechnical materials. Hyperplasticity is used to analyse the stress–dilatancy relation for shear deformations of frictional, granular materials. Central to this approach is the recognition that in general, deformations of granular materials can involve stored plastic work. It is recognised that in shear deformations, where the shape of the grains induces the dilatancy, the confining pressure is a constraint and dissipates no energy. Nevertheless, it is shown that, in such shear deformation, the total plastic work rate is equal to the rate of energy dissipation, and that this is given by Thurairajah’s classical observations. It is further shown that anisotropy is necessarily induced and that the stress–dilatancy relation is given by Taylor’s well-known formula. It is also demonstrated that the extant procedure for deducing yield conditions and flow rules from specified dissipation functions is invalid.

Chapter 11

Rate Effects

11.1 Theoretical Background 11.1.1 Preliminaries So far we have considered only materials that are rate-independent, that is to say, the response is the same irrespective of the strain rate. This feature of the material behaviour is due to the fact that the dissipation function is chosen to be a homogeneous first-order function of the internal variable rate. Mathematically, this can be expressed through Euler’s equation as

wd D ij wD ij

d

(11.1)

Several features of the behaviour of rate-independent materials follow from this special form of the dissipation function. In particular, the special case of the Legendre transform of a homogeneous first-order dissipation function (see Appendix C, Section C.6) gives rise to the existence of a yield surface. Many materials, whilst primarily rate-independent, do show some small dependence on the strain rate. Typically, the yield stress may be observed as increasing marginally with the strain rate. Creep under sustained stresses and relaxation of stress at fixed strain are processes that are also related to strain rate effects. Most geotechnical materials, for instance, exhibit these types of behaviour to a certain extent. This type of response is often modelled semi-empirically, frequently with different (and not always consistent) theories used to model the rate-dependence of strength and the processes of creep and relaxation. Properly, however, all these phenomena should be encompassed within a single approach that explains all rate-dependent processes. This can be achieved within the framework described in this book by considering dissipation functions that are not

212

11 Rate Effects

homogeneous, first-order functions of the internal variable rate, so that Equation (11.1) does not apply. As stated before, the formalism adopted in this book is based on the method used by Ziegler (1977). He describes principally materials that are rate-dependent (concentrating mainly on linear viscous materials, for which the dissipation function is quadratic), and devotes relatively little attention to the special case of rate-independence. Here, we have taken the inverse approach. Having started with the rate-independent case, we shall now examine briefly the implications of departures from it. This approach offers a different, and we believe useful, insight into the treatment of rate-dependent materials. Following the concepts behind Ziegler’s method, but adopting a slightly different terminology and sequence to the argument, first we define:

Q

d wd D ij wD ij

(11.2)

Comparing with Equation (11.1), we see that Q 1 for a rate-independent material. For any dissipation function d which is a homogeneous function of degree n in D ij , it follows from Euler’s theorem that Q is simply a constant equal to 1 n . One possibility is that we generalise the definition of the dissipative generalised stress to

Fij

Q

wd wD ij

(11.3)

wd D ij d , which can be compared with Equation (4.10). In wD ij this case, it follows once again that, by comparison with Equation (4.8), Fij  Fij D ij 0 . Then, following exactly the same argument as before, we so that Fij D ij



Q



adopt the constitutive hypothesis (equivalent to Ziegler’s orthogonality condition) that Fij Fij . This form of the equations corresponds exactly to Ziegler’s original approach (although expressed slightly differently). The presence of the factor Q in the above formalism is not too much of an inconvenience when the dissipation is a homogenous function of the internal variable rate, because in this case Q is simply a constant. However, if the dissipation is not a homogeneous function of the rates, as proves useful to describe materials with a weak rate-dependence, then the presence of the factor Q in Equation (11.3) is a significant inconvenience. Specifically in this case, d is said to be a pseudo-potential rather than a true potential, and it is not possible to take the Legendre transform of d to interchange Fij and D ij as dependent and independent variables.

11.1 Theoretical Background

213

11.1.2 The Force Potential and the Flow Potential If, however, d can be written in the form d

wz D ij , where z z xij , Dij , D ij , wD ij





then certain advantages, explored below, can be gained. First we can note that if d is homogeneous and first order in D ij , then z { d . Next, it is clear from Euler’s equation that if d is homogeneous and of order n, then z d n . Rather than (11.3), we prefer to adopt the following definition of the dissipative generalised stress:

wz wD ij

(11.4)

From Equation (11.4), it follows that Fij D ij

d as before, so that once again

Fij

 Fij  Fij D ij

0 , and we make the constitutive assumption Fij

Fij .

The principal advantage of using the function z is that, unlike the dissipation function d (which is a pseudo-potential), the function z serves as a true potential for Fij . The function z could properly be defined as the generalised stress potential, but for brevity, we shall simply refer to it as the force potential. Because it serves as a potential for the generalised stresses, a simple Legendre transformation can be made:



w xij , Dij , Fij



Fij D ij  z d  z

(11.5)

ww wFij

(11.6)

such that

D ij

The function w has a clear analogy with the yield function in the rateindependent case, but because z is not homogeneous and first order in the rates, the Legendre transform is no longer the degenerate special case; so although y 0 for the rate-independent case, in general, the condition w 0 does not apply. The function w could properly be called the internal variable rate potential, but again for brevity, we shall simply call it the flow potential. Note that the sum of the force and flow potentials is equal to the dissipation function z  w d . The potentials z and w have been defined elsewhere [see, for example, Maugin (1999)], but previous authors seem to have concentrated almost exclusively on the linear viscous case where both are quadratic. Confusingly, both z and w have been referred to in the literature as dissipation potentials, a terminology we deliberately avoid here. We shall now explore how the potential z can be defined if the dissipation function d is known. Since z is obtained by integration from d (see below), it is determined only to within an additive constant. To make the definition of z precise, we shall specify that z 0 when the internal variable rates are all zero.

214

11 Rate Effects

Since d 0 in this case, and w = d – z, it also follows that w = 0 when the rates are zero. As noted above, if d is a homogeneous function of order n in the internal variable rates, we can choose z d n , which is also of order n, so that wz D ij nz d . It follows that if d can be represented in the form, wD ij N

d

¦ dk

(11.7)

k 1

where each of the N functions dk is itself homogeneous and of order nk in the internal variable rates, then z can be chosen as N

z

d

¦ nkk

(11.8)

k 1

We call functions that can be expressed in the form of Equation (11.7) pseudo-homogeneous. For the one-dimensional case d d  D , then it is straightforward to redz dz d  D arrange d  D FD D as and then integrate to give dD dD D D o

z  D o

³ 0

d  D dD D

(11.9)

For more general cases, we can proceed as follows. Consider the definition Wo



z Wo D ij

³



d WD ij W

0

Differentiation with respect to Wo gives

Wo 1 yields the result

 D

dW

 D ij w  WoD ij

wz WoD ij

wz D ij

(11.10)



d WoD ij Wo

, and setting



 ij d D ij . This demonstrates that ij Fij D wD ij application of the definition (11.10) leads to a potential with the required property that Fij wz wD ij satisfies Fij D ij d . Thus by setting Wo 1 in (11.10), we obtain 1d

 ³

z D ij

0

 WD ij dW W

(11.11)

Each of the cases in Equations (11.8) and (11.9) are special cases of the relationship in (11.11).

11.1 Theoretical Background

215

A simple change of variable to x ln W gives an alternative expression to (11.11) which may be more convenient in some cases: 0

 ³ d  ex D ij dx

z D ij

(11.12)

f

Therefore, we have demonstrated that, if the dissipation exists as a function of the internal variable rates, then it is possible to derive the force potential by integration.

11.1.3 Incremental Response In the rate-independent case, we found that the incremental response could always be derived if the model was specified by one of the energy functions (u, f, g, or h) and the yield function y. Alternatively, if the dissipation function is defined then, although the incremental response can be derived by applying various ad hoc procedures for particular models, it has not proved possible to do this by a completely general and automated procedure. We find that there is a parallel position for the rate-dependent materials. If w is specified, then the incremental response can be obtained automatically, whereas if z is specified, then ad hoc procedures again have to be used for each model. Assuming that the model is specified by the Gibbs free energy g and the flow potential w, we can define the following differential relationships:

H ij



w2 g w2 g V kl  D kl wVij wVkl wVij wDkl

(11.13)

F ij



w2 g w2 g V kl  D kl wDij wVkl wDij wDkl

(11.14)

D ij

ww wFij

(11.6), bis

which can simply be combined to give the incremental stress-strain response

H ij



w2 g w 2 g ww V kl  wVij wVkl wVij wDkl wFij

(11.15)

Note that for a rate-dependent material, the time increment in a calculation has a real physical meaning, whilst for a rate-independent material, the time increment is artificial. For a real increment of time dt, (11.15) can be rewritten

dHij



w2 g w 2 g ww dVkl  dt wVij wVkl wVij wDkl wFij

(11.16)

216

11 Rate Effects

As well as the stresses and strains, it is necessary to update Dij and Fij because the differentials in Equation (11.15) are functions of these variables. The updating is achieved by using Equations (11.6) and (11.14).

11.2 Examples 11.2.1 One-dimensional Model with Additive Viscous Term The simplest of one-dimensional elastic-plastic models is defined by the functions: g



V2  VD 2E

d k D

(11.17) (11.18)

where E is the elastic stiffness and k is the strength. On the other hand, an equivalent viscoelastic model would be defined by (11.17) together with

d PD 2

(11.19)

where P is the viscosity. A simple elastic-viscoplastic model is obtained by combining the two dissipation functions:

d k D  PD 2

(11.20)

A schematic representation of the model represented by Equations (11.17) and (11.20) is given in Figure 11.1, in which it can be seen that the plastic and viscous elements act in parallel.

Figure 11.1. Schematic representation of elastic-viscoplastic model

11.2 Examples

217

Noting that the dissipation function is now non-homogeneous and using any of (11.8), (11.9), or (11.11), one can derive P z k D  D 2 2

(11.21)

The differentials of the potentials now give

wg V D wV E wg F  V wD wz F kS  D  PD wD H 

(11.22) (11.23) (11.24)

Noting that if D 0 , then S  D is undefined, we consider first the case when D z 0 . For this case, it follows that (since V F F ), Equation (11.24) gives V

 k  P D S  D

(11.25)

so that V ! k , and the signs of V and D must always be the same. A corollary is that if V d k , then D 0 and H V E , i. e. incrementally elastic behaviour occurs. Thus the elastic region is bounded by a “yield surface” in stress space V  k 0 . The complete response for both D 0 and D z 0 can therefore be expressed by rearranging (11.25) as

D

V k P

S V

(11.26)

are Macaulay brackets, x 0, x  0 , x x, x t 0 . The above result where can be obtained in a mathematically more rigorous and concise way by using the terminology of convex analysis; see Chapter 13 and Appendix D. It now follows that, differentiating (11.22) and substituting (11.26),

H

V k V  S V E P

(11.27)

Considering tests at a constant strain rate H , the viscoplastic part of the response described by (11.27) can be rearranged as

dV EV  dH PH

E

EkS  V PH

(11.28)

which integrates to

§ EH · V A exp ¨  ¸  PH  kS  D © PH ¹

(11.29)

218

11 Rate Effects

where A is a constant of integration. For the first loading, D ! 0 and V k at H k E , so that for this case, we can obtain the solution for the viscoplastic part of the curve as § § k  EH · · V PH ¨ 1  exp ¨ ¸¸  k © PH ¹ ¹ ©

(11.30)

or in normalised form, V k

§ k § EH · · ½ PH ­ ®1  exp ¨ ¨ 1  ¸ ¸ ¾  1 k ¯ k ¹ ¹¿ © PH ©

(11.31)

Figure 11.2 shows the normalised stress-strain curves for PH k 0.0, 0.1, 0.2, and 0.3 (the first is equivalent to the model without viscosity). It can be seen that this model provides a satisfactory starting point for describing rate-dependent plastic behaviour, in which there is a linear increase in strength with strain rate. We have obtained the above response from the specification of the force potential. Alternatively, we can subtract (11.21) from (11.20) to obtain the flow potential w d  z PD 2 2 . Substituting (11.26), but writing F instead of V, we can express the flow potential as a function of F: F k

w

2

(11.32)

2P

1.4 1.2

V k

1

PH k

0.8 0.6 0.4

0.3 0 .2 0 .1 0 .0

0.2 0 0

1

2

3

EH k

4

Figure 11.2. Normalised stress-strain curves for different constant strain rates

5

11.2 Examples

219

We could have taken Equation (11.32) as our starting point rather than (11.21). Differentiating (11.32) gives

D

ww wF

F  k S F P

(11.33)

which [using (11.23)], immediately leads to (11.26), so that the derivation of a response based on the flow potential w is briefer than the derivation using the force potential z.

11.2.2 A Non-linear Viscosity Model An alternative approach to modelling viscous effects is to modify (11.18) to n

d k D r1n

(11.34)

where the constant r, with the dimensions of strain rate, has been introduced to maintain the dimension of stress for the constant k. It immediately follows that k n 1n (11.35) D r z n So that instead of (11.24), we obtain wz n1 1n (11.36) F kS  D D r wD There is now no purely elastic region, and non-linear viscous behaviour occurs whenever the stress is non-zero. Noting that Equation (11.36) implies that the signs of V and D must be the same, it can be rearranged to 1

§ V · n1 D rS  V ¨ ¸ © k ¹

(11.37)

so that, differentiating (11.22) and substituting, we obtain 1

§ V · n1 V (11.38) H  rS  V ¨ ¸ E k © ¹ Considering again tests at constant strain rate H , (11.38) can be rearranged as

dV dH

1 · § § V · n 1 ¸ ¨ r E ¨1  S  V ¨ ¸ ¸ © k ¹ ¨ H ¸ © ¹

(11.39)

Equation (11.39) cannot be integrated analytically without recourse to special functions, but Figure 11.3 shows stress-strain curves for n 1.1 and different strain rates obtained by numerical integration (using forward differences with

220

11 Rate Effects

1.4 1.2

V k

1

PH k

0.8 0.6 0.4

4 3 2 1

0.2 0 0

1

2

3

EH k

4

5

Figure 11.3. Normalised stress-strain curves for non-linear viscosity model

an interval of EH k 0.1 ). The curves shown are for H r 1, 2, 3, and 4. A test at infinitesimal strain rate would simply give V 0 . Figure 11.4 shows the stress-strain curves (again produced by numerical integration) for H r 1 for n values of 2, 1.5, 1.2, and 1.1. It shows how the elasticplastic response is approached asymptotically, for a given strain rate, as n o 1 . Although the curves in Figures 11.2 and 11.3 are at first sight remarkably similar, there are a number of important differences in the character of the response. Firstly, the curves shown are not necessarily for comparable strain rates. Also, for the second case, infinitesimally slow straining results in zero stress, whilst in the first case, it gives the elastic-plastic response. Once straining is stopped, the stress in the first model relaxes to V k , but in the second model, it relaxes to V 0 . Clearly, the models represented by Equations (11.21) and (11.35) could both be expressed within a more general model defined by

k n z k1 D  2 D r1n n

(11.40)

in which the first model is obtained by setting n 2 and k2 Pr , and the second model simply by setting k1 0 . This serves as a good example of the way this approach to the formulation of constitutive models allows to be set them within a hierarchy, where simpler models are subsets of more complex models. The model defined by Equation (11.40) is used as the basis for a continuum model in Section 11.2.4.

11.2 Examples

221

1.2

PH k 1.1

1

V k

0.8

PH k

0.6

1.2

PH k 1.5

0.4

PH k

2

0.2 0 0

1

2

3

EH k

4

5

Figure 11.4. Stress-strain curves for different powers of n

11.2.3 Rate Process Theory We now consider a more sophisticated example of rate-dependence. Many ratedependent processes can be regarded as thermally activated processes. The resulting approach is known as rate process theory. We omit the details here, but the final result is that the rate x of the process depends on the driving force q in the following way: x

A sinh  Bq

(11.41)

where A and B are functions of the temperature. We shall consider fixed temperature here, and treat A and B as constants. Mitchell (1976) gives a useful discussion of the theory in the context of the mechanics of soils. Identifying x with D and q with F, we are expecting a relationship of the 1 § D · form F sinh 1 ¨ ¸ , so that the viscous dissipation is in the form B © A¹ D D · §  1 sinh ¨ ¸ . Finally, adding a plastic term, and changing the constants, FD B © A¹ we obtain a dissipation function of the form, § D · d k D  Pr D sinh 1 ¨ ¸ ©r¹

(11.42)

Note that this form of d cannot be decomposed as pseudo-homogeneous in the form of Equation (11.7) (at least without recourse to an infinite series). However,

222

11 Rate Effects

for a one-dimensional model, we can apply (11.9): effectively, we rearrange d FD as F d D and then integrate the equation dz dD F . Then we obtain § D · (11.43) F k sg  D  Pr sinh 1 ¨ ¸ ©r¹ and

§ · § D · z k D  Pr ¨ D sinh 1 ¨ ¸  r  D 2  r 2 ¸ (11.44) ©r¹ © ¹ Using (11.43) in place of (11.24), we follow exactly the same procedure as in the example in Section 11.2.1 and derive the following in place of Equation (11.26): § V k D r sinh ¨ ¨ Pr ©

· ¸¸ S  V ¹

(11.45)

From this, it follows that § V k V  r sinh ¨ ¨ Pr E ©

H

· ¸¸ S  V ¹

(11.46)

So that for a test at constant strain rate, · (11.47) ¸¸ S  V ¹ Equation (11.47) can be integrated to give, for the plastic part of the curve,

dV dH

E

§ V k Er sinh ¨ ¨ Pr H ©

­° § § EH  k · V 2Pr ° 1 § r · · r ½ SV  tanh 1 ®x tanh ¨ ¨ ¸ x  tanh ¨  ¸ ¸   ¾ k k r x 2 P H H © ¹ ¹ ©© ¹ ¿° ¯°

(11.48)

r2

. Results for this case are plotted in Figure 11.5 for H r 0, H 2 0.1, 1, and 10 when Pr k 0.04 . Rate process theory results in a shear strength which, for low values of H r , increases linearly with strain rate, whilst at high values of H r , it increases linearly with the logarithm of the strain rate. This represents quite realistically the behaviour of a number of materials, particularly geotechnical materials. Alternatively, if a formulation based on the flow potential is required, then we where x

1

can note that w d  z Pr





D 2  r 2  r , which can be expressed as

­° § F k w Pr 2 ®cosh ¨ ¨ © Pr ¯°

· ½° ¸¸  1¾ ¹ ¿°

(11.49)

Note that if this form is used, then the expression for D , and hence for the strain rate follow very simply by differentiation.

11.2 Examples

223

1.4 1.2

V k

1

H r 10 1 0.1 0.0

0.8 0.6 0.4 0.2 0 0

1

2

3

EH k

4

5

Figure 11.5. Stress-strain curves for rate process theory

11.2.4 A Continuum Model Consider, for example, an extension of the von Mises model in which the dissipation function d z k 2 D ijc D ijc is modified to



d k1 2 D ijc D ijc  k2 2n 2 r1n D ijc D ijc

n2



(11.50)

where the constant factors are introduced so that k1 and k2 retain simple meanings. By incorporating a constraint D kk 0 , the force potential can be derived as n2 k 2n 2 r1n z c k1 2 D ij D ij  2 D ij D ij  /D kk n





(11.51)

Applying (11.4), we obtain

Fij

§ k1 2 n 2  k2 2n 2 r1n  D ckl D ckl  ¨¨   c c © D kl D kl

Fijc

k

Fkk

3/

1

n1 2 2  k2 2n 2 r1n  D ckl D ckl 



2·

¸¸ D ijc  /Gij ¹ D ijc

c D mn c D mn

(11.52)

224

11 Rate Effects

The bracketed term in the second Equation (11.52) is always a positive scalar, so the direction of Fijc is the same as that of D ijc , and we can deduce that

k

Fcij

1

n1 2  k2 2n 2 r1n  D ckl D ckl 

2

S Fc ij

ij

(11.53)



[see Appendix 1.1 for the definition of the tensorial signum function Sij xij ]. It follows that



k1 2  k2 2n 2 r1n D ijc D ijc

Fijc Fijc



n1 2

It follows that if D ijc z 0 , then Fijc Fijc ! k1 2 . Conversely, if then D ijc 0 . Both cases can be expressed by rewriting (11.54) as

(11.54) Fijc Fijc d k1 2 ,

1

D ij

§ r¨ ¨ ©

c Fckl  k1 Fkl k2 2n /2

2 · n 1 ¸ Sij Fijc ¸ ¹



n  1 k2 2n 2 r1n

Now we note that w d  z

n



 D klc D klc n 2 .

(11.55)

To express

w w Fijc , we substitute (11.55) to obtain n n2§

w

r  n  1 k2 2 n

¨ ¨¨ ©

Fijc Fijc  k1 2 · n1 ¸ ¸¸ k2 2n 2 ¹

(11.56)

from which we can immediately derive (11.55) by differentiation. Note that although (11.56) appears somewhat complex, this is largely because a number of multiplying factors have been introduced so that the constants retain simple physical meanings. The underlying structure of the equation represents nothing more than raising the rate-independent flow potential w Fijc Fijc  k 2 to the power n  n  1 .

11.3 Models with Multiple Internal Variables Just as rate-independent models can be extended to include multiple internal variables and then (in continuous hyperplasticity) internal functions which represent infinite numbers of internal variables, the same generalisation can be achieved for rate-dependent models. As a stepping stone to the continuous models, we first consider multiple internal variables.

11.3 Models with Multiple Internal Variables

225

11.3.1 Multiple Internal Variables In the previous sections, we considered materials characterised by a single kinematic internal variable, Dij , which typically was a scalar or a second-order tensor. The kinematic internal variable can often be conveniently identified with the plastic (or in the case of rate dependence, viscoplastic) strain. In previous developments, we have considered how multiple internal variables correspond to multiple yield surfaces, and this observation carries over to the rate-dependent case. As for the rate-dependent case, the generalisation to multiple internal variables requires that the function for the Gibbs free energy (ignoring thermal ef1 N fects) g Vij , Dij is simply generalised to g Vij , Dij !Dij . The correspond-







ing differential Fij



wg wDij is replaced by

n Fij



wg n wDij

(11.57)

where n 1!N . The flow and force potentials, w Vij , Dij , Fij and z Vij , Dij , D ij , become 1  N 1 N 1  N 1 N w Vij , Dij !Dij , Fij !Fij and z g Vij , Dij !Dij , D ij !D ij . The corre-











wz wDij and D ij

sponding differentials Fij







ww wFij are replaced by

n Fij

wz g n wDij

(11.58)

n D ij

ww g n wFij

(11.59)

n Ziegler’s orthogonality condition is simply Fij

n Fij . If any of the N inter-

nal variables are scalars rather than tensors, then all that is necessary is to drop the subscripts from the appropriate variables.

11.3.2 Incremental Response Double differentiation of the energy function yields H ij



N w2 g w2 g m V kl  ¦ D m kl wVij wVkl m 1 wV wD ij

n F ij



N w2 g w2 g m V kl  D n  n m kl wDij wVkl m 1 wDij wD kl

¦

(11.60)

kl

(11.61)

226

11 Rate Effects

Figure 11.6. Multiple surface viscoplastic model

which can be combined with Equation (11.59) to give the incremental stress strain response, dHij



N w2 g w2 g ww g dVkl  ¦ dt m m wVij wVkl m 1 wVij wD kl wFkl

(11.62)

As well as the stresses and strains, it is necessary to update all D n and Fn ij ij because the differentials in Equation (11.62) are functions of these variables. The updating is achieved by using Equation (11.59) for the incremental updating of n n n Dij , together with (11.61) [and the orthogonality condition Fij Fij ].

11.3.3 Example An extension of the one-dimensional elastic-viscoplastic kinematic hardening model presented in Section 11.2.1 to multiple internal variables is defined by the functions, 

g

d

N V2 1 N  V ¦ Dn  ¦ Hn Dn2 2E 2n 1 n 1 N

N

n 1

n 1

¦ kn D n  ¦ PnD n2

(11.63)

(11.64)

A schematic representation of this model is given in Figure 11.6. One can easily derive N

z

¦ kn D n 

n 1

1 N ¦ Pn D n2 2n 1

(11.65)

11.3 Models with Multiple Internal Variables

227

And further, it is possible to derive an expression analogous to Equation (11.32): w

N

Fn  kn

n 1

2Pn

¦

2

(11.66)

Substitution of (11.63) and (11.66) in (11.62) now gives the incremental relationship, N F k 1 n n dV  S  Fn dt E Pn n 1

¦

dH

(11.67)

where Fn

Fn



wg wDn

V  H n Dn

(11.68)

and

dDn

ww dt wFn

Fn  kn Pn

S  Fn dt

(11.69)

For the particular case in which all hardening and viscous components are the same, H1 H 2 ... H N H and P1 P2 ... P N P , substituting Equation (11.68) in (11.67) gives N V  HD  k 1 n n dV  ¦ S  V  H Dn dt E P n 1

dH

(11.70)

Consider initial plastic loading, such that the first N* elements in Figure 11.6 are V  HDn  kn , for sliding, i. e. kN * d V  kN *1 . In this case, V  HDn  kn all

n d N* ,

V  HDn  kn

and

0,

for

all

n ! N* .

Noting

that

N

H wg wV V E  ¦ Dn , the plastic response in (11.70) can be rearranged as n 1

H 

H 1 N* V EN*  H H  ¦ knS  V  H Dn  V E EP P Pn 1

(11.71)

We consider again first monotonic loading with constant strain rate H c (so that H H c t ). Assuming again V t H Dn , so that S  V  H Dn 1 , integration of the differential Equation (9.71) gives N* § · 2 V ¨ H H  kn ¸ aN *  PH c aN *  C N * exp   EH  PH c aN * ¨ ¸ n 1 ¹ ©

¦

(11.72)

228

11 Rate Effects

where an E  En  H and CN * is the integration constant which takes a different value for each N * 1!N . The value of C1 is obtained from the initial condition for plastic loading ( V k1 ; H V E ):

C1 PH c a12 exp  k1  PH c a1

(11.73)

and each subsequent value of CN * can be found by obtaining the solution (from the previous curve segment) for H at V kN * . Thus the entire rate-dependent stress-strain curve can be constructed. When PH c o 0 , i. e. for very low viscosity or very slow loading, Equation (11.72) describes piecewise, linear, kinematic hardening plastic behaviour, as discussed by Puzrin and Houlsby (2001b). On unloading and subsequent reloading, Masing type hysteresis will be observed.

11.4 Continuous Models with Internal Functions We now generalise further from multiple internal variables to an infinite number of internal variables, following the same pattern as for rate-dependent materials. The internal function will be expressed in terms of the internal coordinate K, ˆ ij  K . Again we use the “hat” notation to so we write the internal function as D distinguish any variable that is a function of the internal coordinate, and for ˆ {D ˆ  K . brevity, we shall omit the specific dependence on K, thus D

11.4.1 Energy Potential Functional ˆ ij º . The free energy function will now become a free energy functional g ª¬Vij , D ¼ Note that again we use square brackets > @ to distinguish a functional from a function. We assume for the present that the functional can be written in the particular form, ˆ ij º g ª¬Vij , D ¼

³ gˆ  Vij , Dˆ ij , K *dK

(11.74)

8

where Y is the domain of K. Other more general forms of a functional are possible, but the form in Equation (11.74) proves of practical use and importance. It is convenient to introduce for generality the (non-negative) weighting function *  K within the integral in Equation (11.74), as this adds a useful element of flexibility to later aspects of the formulation. Alternatively, * can simply be taken as unity, and its role is simply absorbed within the function gˆ . In some

11.4 Continuous Models with Internal Functions

229

cases, it may be more convenient to consider the free energy as the sum of a function and a functional: ˆ ij º g ª¬Vij , D ¼







ˆ ij , K *dK g1 Vij  ³ gˆ 2 Vij , D

(11.75)

8

For simplicity, however, we shall first describe just the functional form of Equation (11.74) here. In any case, g1 in Equation (11.75) can be included with gˆ 2 within the integral simply by dividing by the constant ³ *dK . 8

The generalised stress, which is the quantity that is work-conjugate to the internal kinematic variable is defined by Equation (11.57). Corresponding therefore to the kinematic internal function is a generalised stress function Fˆ ij  K , Fˆ ij



wgˆ ˆ ij wD

(11.76)

Equation (11.76) can also be seen as a generalisation of (11.57) when the finite number of internal variables becomes infinite and is treated as a continuous function.

11.4.2 Force Potential Functional The dissipation functional in the form, ˆ ij , D ˆ ij º d ª¬Vij , D ¼

³ dˆ  Vij , Dˆ ij , Dˆ ij , K *dK t 0

(11.77)

8

ˆ ij , and is now allowed to be other than a homogeneous first-order function in D we define the force potential functional, ˆ ij , D ˆ ij º z ª¬Vij , D ¼

³ zˆ  Vij , Dˆ ij , Dˆ ij , K *dK

(11.78)

8

with the property

Fˆ ij

wzˆ g ˆ wD

(11.79)

ij

We shall adopt the strong condition Fˆ ij orthogonality.

Fˆ ij here as a generalisation of Ziegler’s

230

11 Rate Effects

11.4.3 Legendre Transformation of the Force Potential Functional For rate-independent materials, the Legendre-Fenchel transformation of the dissipation functional is a singular transformation producing the yield functional. For rate-dependent materials, we have a transformation of the force potential functional. This transformation is not singular and produces a flow potential:

³ wˆ  Vij , Dˆ ij , Fˆ ij , K *dK

ˆ ij , Fˆ ij º w ª¬Vij , D ¼

(11.80)

8

where



ˆ ij , Fˆ ij , K wˆ Vij , D



ˆ ij  zˆ Vij , Dˆ ij , Dˆ ij , K Fˆ ij D





(11.81)

together with the result that wwˆ wFˆ ij

ˆ ij D

(11.82)

11.4.4 Incremental Response Given the knowledge of the energy functional and the flow potential functional, it is possible to derive the entire incremental response for an elastic-viscoplastic material. This is of particular importance because non-linear material models are frequently implemented in finite element codes for which an incremental response is required. Double differentiation of the energy functional gives the rates of the variables:

H ij

§ w 2 gˆ w 2 gˆ  · ˆ ¸ * dK  V  D ¨ kl ³ ¨ wVij wVkl ˆ kl kl ¸ wV wD ij ¹ 8©

(11.83)

w 2 gˆ w 2 gˆ  ˆ V kl  D ˆ ij wVkl ˆ ij wD ˆ kl kl wD wD

(11.84)

Fˆ ij

Equations (11.83) to (11.84) are used together with Equation (11.82) to derive

°­ dHij °½ ® ˆ ¾ °¯dFij °¿

ª w 2 gˆ *d K « « 8 wVij wVkl « w 2 gˆ « « wD ˆ ij wVkl ¬

³

º w 2 gˆ wwˆ g *d K » ˆ mn wFˆ mn wVij wD » ­dVkl ½ 8 ¾ »® 2ˆ g w g wwˆ » ¯ dt ¿ ˆ ij wD ˆ mn wFˆ mn » wD ¼

³

(11.85)

Thus we can see that the entire constitutive response of the material [expressed through the incremental stress-strain relationships (11.85) and the evolution ˆ ij wwˆ g wFˆ ij dt for the internal variables] can be derived from equations dD





the original two thermodynamic functionals.

11.4 Continuous Models with Internal Functions

231

11.4.5 Example A rate-independent kinematic hardening model with (in effect) an infinite number of yield surfaces can be defined by the functions,



g

V2 hˆ 2 ˆ dK  ˆ *dK  V D* D 2E 2

³

³

(11.86)

³ kˆ Dˆ *dK

(11.87)

8

d

8

Y

There is considerable freedom in the way the model is specified. Any two of the functions hˆ , kˆ , and * can be specified, and the third treated as a free function that depends on the shape of the monotonic loading curve that is desired. Puzrin and Houlsby (2001b) show, for instance, that if one chooses * 1 and kˆ kK , then d 2H dV2 1 khˆ  V k , and alternatively, if one chooses hˆ H and kˆ kK , then d 2H dV2 *  V k  kH . For our purposes here, it is convenient to





consider * 1 and hˆ H , with kˆ as the free function. For this case, one can show that kˆ  K V , where K H   dH dV  1 E . After the backbone curve H f  V is specified for calibration of the model, these equations can be solved to give kˆ . For example, for the hyperbolic stress-strain curve H  k E  V  k  V , the solution gives kˆ k  k 1  EK H . In this formulation, it can be shown that the numerical value of the parameter H has no effect on the shape of the stress-strain curves predicted, and for convenience, we simply adopt H E , so that kˆ k  k 1  K . Different choices of H simply result in different scalings of the variable K. In this model, the domain of K is 0!f . Having chosen * 1 and hˆ H with kˆ as the free function, we can extend the model with multiple internal variables presented in the previous section to a case of internal functions by defining the following functionals:

g



f

f

³

³

H V2 ˆ dK  ˆ 2dK V D D 2E 2 0

d

(11.88)

0

f

f

0

0

2 ³ kˆ Dˆ dK  P ³ Dˆ dK

(11.89)

so that gˆ

H 2 V2 ˆ D ˆ  VD 2E 2 ˆ  PD ˆ 2 dˆ kˆ D 

(11.90) (11.91)

232

11 Rate Effects

P 2 ˆ  D ˆ zˆ kˆ D 2 2

Fˆ  kˆ

wˆ

(11.92)

(11.93)

2P

The differentials of the potentials now give

H 

f

wgˆ wV

V ˆ dK  D E ³

(11.94)

ˆ V  HD

(11.95)

0

wgˆ Fˆ  ˆ wD

Fˆ  kˆ

wwˆ ˆ  K D wFˆ

P

(11.96)

S  Fˆ

Using the orthogonality condition Fˆ Fˆ and substituting (11.95) and (11.96) in the differential of (11.94), we obtain the incremental stress-strain response, dV  dt dH E

f

³

ˆ  kˆ V  HD P

0

ˆ dK S V  HD

(11.97)

Consider again the case of initial loading, such that the stress reaches the value ˆ  kˆ ˆ  kˆ , for all K d K* , and of V kˆ K* . In this case, V  H D V  HD



ˆ  kˆ V  HD

f

0 , for all K ! K* . After substitution of

³ Dˆ dK

from Equation

0

(11.94), the plastic response in (11.97) can be rearranged as

PH

PV §V ·  H¨  H¸  E ©E ¹

K*

³  V  kˆ dK

(11.98)

0

When viscous effects are negligible ( P Ÿ 0 ), Equation (11.98) reproduces the static backbone curve H f  V , which should be specified for calibration of the K*

model. It follows therefore that H





 V E  1 H ³ V  kˆ dK

f  V and, Equa-

0

tion (11.98) can be rearranged as

H 

H H P

V H  f V E P

(11.99)

The parameter H here does not appear separately from the parameter P and cannot be calibrated independently against the experimental data. Therefore we adopt H E and calibrate the parameter P correspondingly. For any

11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay

233

stress-controlled loading [specified as V V t ], Equation (11.99) is a linear first-order differential equation and can be integrated to give t

H e

 Ht P

­ V  W H ½  f ¬ªV  W ¼º ¾ e H W P dW E P ¿

³ ¯® 0

(11.100)

For strain-controlled loading [specified as H H t ], Equation (11.99) is in general a non-linear first-order differential equation and cannot always be integrated analytically. For example, for a constant strain rate loading and the hyperbolic backbone curve H f  V  k E  V  k  V , Equation (11.99) can be transformed into an Abel equation of type 2, class A, which can be integrated only in a closed form using special functions. When viscous effects are neglected, Equation (11.99) describes smooth kinematic hardening elastoplastic behaviour (see Puzrin and Houlsby, 2001b). On unloading and subsequent reloading, Masing type hysteresis will be observed.

11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay 11.5.1 Formulation As discussed above in Section 11.2.3, many rate-dependent processes can be regarded as thermally activated processes. The resulting approach is known as rate process theory. In Section 11.2.3, we considered a single yield surface model with this type of rate-dependent behaviour. The flow potential of Equation 11.2.3 can be easily generalised for internal functions by the following functional: 1­ ˆ K § Fˆ  k  D ° w Pr 2 ®cosh ¨ ¨ Pr ° © 0¯

³

· ½° ¸  1¾ dK ¸ ¹ ¿°

(11.101)

where P is the viscosity and the constant r, with the dimensions of strain rate, has been introduced to maintain the dimension of stress for k. If k is a constant, then the resulting model involves rate-dependent perfect plasticity. Strain hardening or softening can be introduced by allowing the parameter k to be dependˆ . For example, a simple linear strain softening can be achieved by ent on D adopting the following expression: 1

ˆ k0  k1 D kD ³ ˆ dK

(11.102)

0

where k0 and k1 are parameters calibrated to fit experimental data. (Note that clearly the model is valid only for the limited range of plastic strains

234

11 Rate Effects

1

³ Dˆ dK  k0

k1 and only for monotonic loading. For non-monotonic loading,

0

hardening should be introduced via accumulated plastic strain using a conˆ could be straint, e. g. see Chapters 2 and 5.) A more complex dependency on D introduced, but the above is sufficient to illustrate the model here. The flow potential functional (11.101), together with the Gibbs free energy functional

g



1

1

³

³

V2 1 ˆ dK  hˆD ˆ 2dK V D 2E 2 0

(11.103)

0

define a continuous visco-hyperplastic model. Formulation of the model is completed by specifying the functional form of hˆ , which we will take as E hˆ  K 1  K 3 (11.104) 2  a  1 The rate-independent case with no strain softening (k1 0) produces a hyperbolic stress-strain curve H  V E    a  1 E V2  k0  V . The initial stiffness





is E and the secant stiffness to V k0 2 is E a .

11.5.2 Incremental Response ˆ V  hˆD ˆ , we obAfter differentiating (11.101) and substituting Fˆ Fˆ wgˆ wD tain the evolution equation for the internal function: § V  hˆD ˆ  k D ˆ K ¨ r sinh ¨ Pr ¨ © and the incremental response is given by ˆ dD

wwˆ g wFˆ

· ¸ ˆˆ ¸ S V  hD dt ¸ ¹





(11.105)

§ V  hˆD 1 ˆ  kD ˆ K · dV ¨ ¸ ˆˆ (11.106)  dt ³ r sinh ¨ ¸ S V  hD dK P E r ¨ ¸ 0 © ¹ Consider now the case of initial monotonic loading, such that the stress reaches 1 § · ˆ  kD ˆ K ˆ  K dK ¸ K* kK * . In this case, V  hˆD the value of V ¨ k0  k1 D ¨ ¸ 0 © ¹ ˆ  kD ˆ K 0 , for all K ! K* . Then, ˆ K, for all K d K* and V  hˆD V  hˆDˆ  k  D



dH



³

the incremental response is given by K* § V  hˆD ˆ  kD ˆ K · dV ¸ dK dH  dt r sinh ¨ ¨ ¸ E r P © ¹

³ 0

(11.107)

11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay

235

where K* V k for V k d 1 and K* 1 for V k ! 1 . Thus, for initial loading with a constant strain rate H c , the stress-strain response can be obtained by numerical integration of the following two equations:

ˆ  K dD dH for 0 d K  V k and

§ V  E 1  K 3 D ˆ  K  2a  2  kK · r ¸ sinh ¨ ¨ ¸ H c Pr © ¹

ˆ  K dD dH

(11.108)

0 for K t V k

§ min V k,1 ˆ · dD  K ¸ ¨ dV E 1  dK dH ¨ ¸ dH 0 © ¹

³

(11.109)

11.5.3 Comparison with Experimental Results We now compare this model with experimental data obtained by Vaid and Campanella (1977) for the behaviour of a natural clay. Figure 11.7a shows the data they obtained in undrained triaxial compression tests at different rates. Figure 11.7b shows the equivalent calculations using the above model with the parameters E V1c c 160 , a 1.5 , k0 V1c c 0.545 , k1 V1c c 0.3 , r 0.008 min 1, and P V1cc 2.8 min . All parameters involving stress have been normalised by dividc . The agreement between tests and theing by the preconsolidation pressure V1c ory is good; only the detail of the latter parts of the curves at low strain rates were

Figure 11.7. Comparison between (a) stress-strain curves for undrained tests from Vaid and Campanella (1977) and (b) the theoretical model

236

11 Rate Effects

Figure 11.8. Variation of undrained strength with strain rate: comparison between theory and data from Vaid and Campanella (1977)

not captured accurately. Figure 11.8 shows the variation of peak strength with strain rate as observed by Vaid and Campanella, with the theoretical results (open circles) superimposed. The figure shows the characteristic response of the rate process theory: a linear increase in strength with strain rate at low strain rate (appearing as almost a constant strength on the logarithmic plot) and a linear

Figure 11.9 Undrained stress-strain curves at different strain rates: (a) data from Vaid and Campanella (1977) and (b) theoretical curves

11.5 Visco-hyperplastic Model for Undrained Behaviour of Clay

237

Figure 11.10. Comparison of (a) creep data from Vaid and Campanella (1977) and (b) the theoretical curves

increase with the logarithm of the strain rate at high strain rates. The strain rate at which the transition occurs is characterised by the parameter r, and the intersection of the two straight line sections indicated on Figure 11.8 occurs at H r 2 . The slope of the section at high strain rate is approximately Pr loge 10 . Figure 11.9a shows detail from tests at two strain rates, together with a test in which the strain rate was suddenly changed from the lower to the higher rate at a strain of about 0.8%. Figure 11.9b shows the equivalent theoretical calculation. The model is clearly able to capture accurately the transition, which appears as the line connecting the two main curves. Figure 11.10a shows the results of constant stress creep tests, in which strain is plotted against time for different constant stress values, and Figure 11.10b

Figure 11.11. Creep data in terms of strain rate: (a) data from Vaid and Campanella (1977) and (b) the theoretical model

238

11 Rate Effects

Figure 11.12. Comparison of theoretical results for rupture life with data from Vaid and Campanella (1977)

shows the equivalent calculations. Apart from the detail of the curve at V V1c c 0.5 , the agreement is very close. Figures 11.11a and 11.11b show the same data presented in terms of strain rate against time. Above a certain stress level, the phenomenon of creep rupture occurs, and Figure 11.12 shows Vaid and Campanella’s data for the time to rupture against the stress level. The superimposed open circles show the theoretical calculations. No creep rupture is predicted at stress ratios lower than 0.485. It is remarkable that a model entirely encapsulated by the two potential functions, defined in Equations (11.101)–(11.103) and using only six material parameters (each of which can be given a clearly defined physical interpretation), is able to capture the diversity of behaviour shown in Figures 11.7–11.12.

11.5.4 Extension of the Model to Three Dimensions The above model may be extended to three dimensions by generalizing Equations (11.101)–(11.103) to ­

wg

· ½ 3 ˆ ij K ¸ ° Fˆ ijc Fˆ ijc  k D 2 ¸ ° ¸  1¾ dK Pr ¸ ° ¸ ° ¹ ¿

§ ¨ ˆ K Pr 2 °®cosh ¨ wd ¨ 0 0° ¨ ° ¨ © ¯ 1

1°

³

³



ˆ ij k D



1

k0  k1

3 ˆ ij D ˆ ij dK D 2

³ 0

g



Vii V jj 18K



Vijc Vijc 4G

1

1

³

³

ˆ ij dK  hˆD ˆ ij D ˆ ij dK  Vij D 0

0

(11.110)

(11.111)

(11.112)

11.6 Advantages of the Rate-dependent Formulation

239

where K and G are the bulk and shear moduli and the other parameters retain their original meanings (except that hˆ now bears the same relationship to G as it previously did to E). A prime is used to indicate the deviator of a tensor. Differentiation of (11.112) gives

Hij



wg wVij

Vkk Gij

Vijc

1

ˆ ij dK  D

³

(11.113)

· 3 ˆ ij K ¸ 3 ˆ c Fˆ ckl Fˆ ckl  k D F 2 ¸ 2 ij ¸ 3 Pr ¸ Fˆ ckl Fˆ ckl ¸ 2 ¹

(11.114)

9K



2G

0

Differentiation of (11.110) gives

ˆ ij D

wwˆ g wFˆ ij

§ ¨ ¨ r sinh ¨ ¨ ¨ ©



Further differentiation of (11.113) with respect to time and substitution of (11.114) then leads to the incremental stress-strain relationship.

11.6 Advantages of the Rate-dependent Formulation The benefits of extending the rate-independent framework to include rate dependency for materials that exhibit any significant rate effects are obvious. There is an additional benefit, however, which is worth a separate discussion. It is well known that when rate-independent behaviour is described as the limiting case of rate-dependent behaviour, significant simplifications in calculations can be achieved. Finite element calculations of purely plastic behaviour can, for instance, be efficiently carried out using a viscoplastic algorithm with an artificial very small viscosity; see, for instance, Owen and Hinton (1980). A comparison between the incremental response expressions (11.85) and similar expressions for the rate-independent case developed in Chapter 8 is a good illustration of the greater simplicity of the rate-dependent equations. The source of the higher complexity of the rate-independent expressions is the necessity to satisfy the consistency condition for yield. In other words, it is necessary to make sure that the stress state in plastic loading always stays on the yield surface. This may also cause significant numerical difficulties, requiring treatment by special procedures. The rate-dependent framework is free from these limitations.

Chapter 12

Behaviour of Porous Continua

12.1 Introduction In previous chapters, we have developed a theory for plastic materials in which the entire constitutive response is determined by specification of two potential functions. There are many other areas of continuum mechanics where similar approaches have been made. For instance, Ziegler develops theories for viscous materials. Many authors treat flow processes within a thermodynamic context and frequently use a dissipation function. The special features of rate-independent materials have been the reason for a slightly different emphasis here, from that in most treatments of the subject. We now explore how the hyperplasticity approach can be generalised and set within the context of a wider variety of types of material behaviour. In particular, we shall continue to emphasize the use of two potential functions and the use of Legendre transformations to obtain alternative formulations. When more complex materials are considered, there are two classes of dissipative behaviour. The first is associated with fluxes, for instance flow in a porous medium or the flow of electrical current. In these cases, the dissipation is associated with the spatial gradient of some variable (e. g. the hydraulic head for flow in a porous medium, the voltage for an electrical problem). Constitutive behaviour is usually described by a linear relationship between the flux and the spatial gradient. The second type of dissipation is associated with the temporal variation of internal variables. The plasticity problems treated in earlier chapters are of this character. Viscous behaviour can also be described in this way. Most texts that treat the thermodynamics of dissipative continua concentrate either on fluxes or on rates of change of internal variables. However, whilst the two problems have much in common, they also have important differences. Most obviously, one involves a spatial variation and the other a temporal variation. It is tempting to treat both in the same way, and many texts adopt this

242

12 Behaviour of Porous Continua

approach, using, for instance, “generalised forces” and “generalised fluxes”. Here, we adopt a slightly different approach, keeping separate those variables associated with fluxes and those associated with internal variables. In this way, the different ways that the two types of process appear in the relevant equations can be made clearer. Rather than considering the possibility of abstract, unspecified fluxes, we find it more useful to consider a concrete example. The case that we consider is a very important problem in geomechanics and other fields, namely, flow in a porous medium. This is a useful example because the flux itself has mass, which introduces a number of features to the problem that need careful treatment. The porous medium has to be treated as consisting of two phases, and there is a partition of the extensive quantities (e. g. internal energy, entropy) between the solid skeleton and fluid phases. In previous chapters, we adopted a small strain formulation. The problem of coupled fluid and skeleton behaviour cannot be treated rigorously within the small strain framework, because there is a coupling between strains, fluid flow, and density changes. In the small strain formulation, the density is treated as a constant. In the following therefore, it is necessary to move to a large strain formulation. There is a choice between adopting a Lagrangian approach, in which the problem is formulated in terms of initial coordinates, and an Eulerian approach, in which it is formulated within the current coordinates. We adopt the Eulerian approach for much of the following development because this allows a more direct interpretation of the variables. It will prove necessary, however, to transform to Lagrangian variables for part of the analysis. In the small strain approach, for convenience, all extensive quantities were defined per unit volume. Since the density was in effect constant, this is equivalent to using extensive quantities per unit mass, but avoids a factor of the density appearing throughout the equations. In large strain analysis, it is necessary to use extensive quantities per unit mass, as is usual in thermodynamics, and we adopt this approach below.

12.2 Thermomechanical Framework As mentioned above, we adopt here an Eulerian approach to describe a material undergoing large strain, i. e. the description of the material is based on the current coordinate system. In this, it will be necessary to distinguish between the time differential of a variable x at a particular point in space, which we shall denote by wx wt x , and the material or convective derivative, which represents the rate of change of an element of the material, which has a current velocity vi . We denote the material derivative by dx dt x x  x,i vi . From this definition, it follows that the chain rule applies to the convective derivative, e. g. the mate  xy . rial derivative of xy is xy

12.2 Thermomechanical Framework

243

12.2.1 Density Definitions, Velocities, and Balance Laws Consider a volume V fixed in space bounded by a surface S. The unit outward normal to the boundary is ni . The volume contains porous material with a skeleton material of density Us and with a porosity n (volume of voids divided by total volume). Thus the mass of skeleton per unit total volume is U 1  n Us . We should also note that U is the “dry density” in the terminology of soil mechanics. The velocity of the skeleton at any point is vi , so that the mass flux of the skeleton per unit area is Uvi , and the outward mass flux per unit area from V is Uvi ni . For conservation of mass, we can write that the rate of increase of mass within the volume, plus the outward mass flux is zero:

³ U dV  ³ Uvini dS

V

0

(12.1)

S

Applying Gauss’s divergence theorem1, we can write

³ U   Uvi ,i dV

0.

V

Then noting that V is arbitrary, we can write this in local form:

U   Uvi ,i

U  U,i vi  Uvi,i

U  Uvi,i

0

(12.2)

which establishes the link between the material rate of change of dry density and the dilatation rate. A comment is relevant here about the importance of the assumption that the volume V is arbitrary. This is only justified provided that V is large enough so that averaged values of stresses, strains, etc., over the volume element are meaningful. Such an element is said to be a “representative volume element”. In the context of the mechanics of granular materials, this will typically require that the element contains many thousands of particles. At the same time, the element must be sufficiently small so that changes of stresses, etc., across the element are small. This requirement conflicts with the first, and there are classes of problems for which both criteria cannot be satisfied simultaneously. Such problems (e. g. those involving strong localisation) are not amenable to treatment by conventional continuum mechanics. We now allow for the possibility of fluxes of a pore fluid. We shall consider a pore fluid, the amount of which is specified by the parameter w defined as mass of fluid per unit mass of skeleton material (i. e. the water content in the terminology of soil mechanics). Note that in the study of the mechanics of granular media, a wide variety of different quantities are used to define the amount of fluid in the porous medium. The flux of the fluid mass is mi per unit area relative to the skeleton. The total flux vector of the fluid is therefore 1

In the above terminology, Gauss’s divergence theorem states that for any variable x that is con-

³

³ x,i dV .

S

V

tinuous and differentiable in V, xni dS

244

12 Behaviour of Porous Continua

mi  Uwvi , and the outward flux of the fluid across the boundary S follows as mi  Uwvi ni . We note that the mass of fluid per unit volume of skeleton is Uw. It follows that

Uw nUw w

where U as

(12.3)

is the density of the fluid. The mass flux vector mi can also be written mi

Uw wi



Uw n viw  vi



(12.4)

where wi is the Darcy artificial seepage velocity and viw is the average absolute velocity of the fluid. Noting that the mass of the fluid is conserved, there is a balance equation analogous to (12.1) of the form:

³ wwt  Uw dV  ³ mi  Uwvi ni dS

V

0

(12.5)

S

which we can rewrite in local form by using the divergence theorem of Gauss to obtain the local conservation law:

U w  Uw  mi,i  U,i wvi  Uw,i vi  Uwvi,i

0

(12.6)

or

Uw  U w  mi,i  Uwvi,i

0

(12.7)

By virtue of the skeleton mass conservation, Equation (12.2), this becomes

Uw  mi,i

0

(12.8)

It is convenient to obtain a combined continuity equation for flow of the skeleton and pore fluid. First, we can note that U U s 1  n  Us n , so that we can rewrite the mass continuity equation as

U s 1  n  Us n  Us 1  n vi,i

0

(12.9)

By manipulation of (12.7), we can also obtain



U w n  Uw n  Uw wi

,i  nUw vi,i

0

(12.10)

Finally dividing (12.9) by Us and (12.10) by Uw and adding, we obtain vi ,i  wi ,i  wi

Uw ,i Uw

n

U w Uw

 1  n

U s Us

0

(12.11)

12.2 Thermomechanical Framework

245

If both the soil grains and the pore fluid are incompressible, then this reduces to the simple form vi,i  wi,i 0 . Introducing v w 1 Uw and v s 1 Us , the continuity equation can also be written vi,i  wi,i mi v,wi  Uwv w  Uv s , where

1 v v s  wv w U

(12.12)

12.2.2 Tractions, Stresses, Work, and Energy The tractions (forces per unit area) on the skeleton on the fraction 1  n of the boundary S are ti , and the pressure in the pore fluid is p which acts on a fraction n of the boundary. The work done per unit area by the surroundings against the tractions on S is therefore 1  n ti vi , and that done against the pore pressure is npni viw . There are also body forces arising from the gravitational field of strength g i . The work done per unit volume by the body forces on the skeleton is Uvi g i and on the fluid is Uwviw gi . The heat flux per unit area is qi , so that the outward heat flux from S per unit area is qi ni . As an extensive quantity, the kinetic energy of all the matter enclosed in volume V may be written as the sum of the kinetic energies of the skeleton and of the fluid: 2 1 1 2 U  vi dV  Uw viw dV 2 2

³

K

³

V



(12.13)

V

At this stage, we are neglecting tortuosity effects, which are due to the fact that the pore fluid must take a tortuous path between the skeleton particles, so that the average speed of the water particles is higher than the magnitude of the average velocity. We shall, however, show how the results can be modified later to take this into account. Now consider the rate of change in kinetic energy in the volume V, which can be written K

w § Uvi vi Uwviw viw  2 © 2

³ wt ¨¨

V

· ¸ dV ¸ ¹

§ Uwviw viw § Uv v ·  ¨ i i ¸ v j n j dS  ¨ ¨ 2 © 2 ¹ S S©

³

³

(12.14) · w ¸ v j n j dS ¸ ¹

The volume integral reflects changes in kinetic energy with time in the volume, and the surface integrals account for the kinetic energy brought into the volume due to the skeleton and pore fluid movement through the surface. Applying the theorem of Gauss and grouping the resulting terms,

246

12 Behaviour of Porous Continua

§ vi vi 2

³ Uvi  vi  vi, j v j dV  ³ U  Uv j, j  U, j v j ¨©

K

V



³

V

Uwviw



viw

 viw, j v wj

· ¸ dV ¹

dV

(12.15)

V



§

w w

w w w vi vi ³  U w  Uw  Uwv j, j  Uw, j v j  U, j wv j ¨¨ 2

©

V

· ¸ dV ¸ ¹

Recalling the mass balance equations for the skeleton and for the fluid, (12.2) and (12.6), respectively, we note that the second and fourth integrals vanish. We introduce also the definitions of the accelerations of the skeleton and fluid particles, respectively:

vi

ai aiw

 viw

vi  vi , j v j

(12.16)

viw  viw, j v wj

(12.17)

where the material derivative with respect to a fluid particle is denoted by  x  x,i viw x . The expression for the rate of change of kinetic energy becomes K

³  Uvi ai  Uwvi ai

w w

dV

V

³

V

(12.18)



Uai  Uwaiw vi dV  ³ mi aiw dV V

12.2.3 The First Law The First Law of Thermodynamics states that there is a variable, called specific internal energy, such that the rate of increase of internal energy in the volume plus the rate of change of the kinetic energy in this volume is equal to the sum of the rates of energy input at the boundaries plus the rate of work of the body forces in the volume. We attribute a specific internal energy us to the skeleton and uw to the pore fluid. The first law therefore becomes

³ wwt  Uu

s

V



³

V

³ S



 Uwuw dV 

Uai  Uwaiw

³  Uu vi  Uwu s

w w vi

nidS

S

vi dV  ³ miaiw dV

V w ª1  n ti vi  npni vi º dS  ¬ ¼

(12.19)

³  Uvi  Uwvi

w

V

gidV  ³  qini dS S

12.2 Thermomechanical Framework

247

We can note that the tractions and pore pressure are related to the stresses by 1  n t j  npn j Vij ni , so that 1  n t j v j Vij v j ni  nv wj pn j  pw j n j and we can rewrite the above as:

³ wwt  Uu

V



s







 Uwuw dV  ³ Uus vi  Uwuw viw ni dS S



 ³ Uai  Uwaiw vi dV  ³ mi aiw dV V

(12.20)

V

³  Vij v j  pwi  qi ni dS  ³ Uvi  Uwvi

w

S

gi dV

V

Applying the divergence theorem of Gauss, we obtain the local form:

Uus  Uwuw  Uus vi  Uwuw viw ,i  Vij v j  pwi  qi ,i   Uvi  Uwviw gi   Uai  Uwaiw vi  mi aiw w wt

(12.21)

and expanding the differentials in the first row of Equation (12.21), we obtain

Uus  Uwuw  Uusvi,i  Uwuw vi,i  uwmi ,i d Uus  Uwuw  U  w   us  U wuw  u,wi mi  Uwu dt  d dt

(12.22)

Uu s  Uwu w  u,wi mi The second row of Equation (12.21) may be transformed as follows:

 Vij v j  pwi  qi ,i  Uvi  Uwviw gi  Vij,i  Ug j v j  Vij v j,i  p,i wi  pwi,i  qi,i  Uwnviw gi  Vij,i  U1  w g j v j  Vij v j,i  Uw gi  p,i wi  pwi,i  qi,i

(12.23)

We can then decompose vi, j into its symmetrical and antisymmetric parts, identifying the former as the strain rate and the latter as the vorticity tensor:

dij Zij

vi, j  v j,i 1 v v j ,i 2  i, j

1 2

(12.24) (12.25)

248

12 Behaviour of Porous Continua

After substitution of (12.22)–(12.25) in (12.21), we can write Uu s  Uwu w  u,wi mi

 Vij,i  U1  w g j  Ua j  Uwawj v j  Vij Z ji Vij dij   Uw  g i  aiw  p,i wi  pwi,i  qi,i

(12.26)

12.2.4 Equations of Motion No change in internal energy should, however, be caused by either a rigid body translation or rotation, so that we can conclude that Vij ,i  U 1  w g j  Ua j  Uwawj v j 0 and Vij Z ji 0 for all v j and Z ji . These





are the virtual work forms of the direct and rotational equilibrium conditions. From the latter, it follows that the antisymmetric part of Vij must be zero, i. e. Vij is symmetrical. This condition is usually referred to as that of complementary shear stresses. From the former, it follows that Vij ,i  U 1  w g j  Ua j  Uwawj

0

(12.27)

which can be recognised as the equations of motion (or the static equilibrium equations in the case of zero acceleration). Equation (12.27) expresses the momentum balance for the porous medium considered as a whole and has been derived as part of a formulation rather than postulated. However, this equation is not sufficient to describe of the momentum balance of the pore fluid, which cannot be derived until some constitutive statement is made about interaction between the fluid and skeleton. The missing fluid balance equation will be derived later as part of the formulation. In view of Equation (12.27), Equation (12.26) reduces to

Uu s  Uwu w  u,wi mi

 





Vij dij  Uw g i  aiw  p,i wi  pwi,i  qi,i

(12.28)

12.2.5 The Second Law The Second Law of Thermodynamics can be stated in a number of different ways. We state it here in the form that there exists a function of state, the specific entropy s, such that the rate of entropy production is non-negative. We attribute ss to the skeleton and sw to the pore fluid, so that the specific entropy s of the whole medium is

s s s  wsw

(12.29)

12.2 Thermomechanical Framework

249

The flux of the entropy Ki is defined by Ki qi T . Unlike the case of the flux of the pore fluid, the total amount of entropy is not conserved. This is expressed by rewriting the fundamental inequality for the entropy in the form:

³ wwt  Us

s



 Uwsw dV 

V

³ Us vi  Uws s

w w vi

S

nidS t ³ §¨©  qTi ni ·¸¹ dS

(12.30)

S

The above equation states that the rate of increase of entropy within the volume plus the convection of entropy across the boundary is greater than or equal to the entropy flux (from heat flow) into the volume. The additional entropy production is due to dissipative processes. Applying the divergence theorem, we obtain w wt

Uss  Uwsw  §¨© Uss vi  Uwsw viw  qTi ·¸¹,i t 0

(12.31)

which can be written as d dt

Uss  Uwsw  Uss vi,i  Uwsw vi,i   swmi ,i  §¨© qTi ·¸¹,i

d dt

Uss  Uwsw  U ss  U wsw   s,wi mi  Uws w  §¨© qTi ·¸¹,i

Us s  Uws w  s,wi mi 

qi ,i T



qi T,i T2

(12.32)

Udt t0 T

where dt is the specific dissipation, corresponding to the irreversible part of entropy production and must always be non-negative. We use dt here to distinguish this quantity from d used in earlier chapters for just the “mechanical dissipution”. The quantity dt includes also the term usually referred to as “thermal dissipution”, which is due to the heat flux. The condition that the total dissipation be non-negative is slightly less restrictive than the earlier requirement that the mechanical dissipation be non-negative.

12.2.6 Combining the First and Second Laws Now we combine Equations (12.28) and (12.32) to obtain Uu s  Uwu w  Udt Vij dij  pwi ,i  UTs s  UwTs w  Ki T,i









 ª«Uw g i  aiw  p,i º» wi  u,wi  Ts,wi mi ¬ ¼

(12.33)

250

12 Behaviour of Porous Continua

which, by virtue of the continuity equation, can be written as

Uu s  Uwu w  Udt

 Vij  pGij dij  Upv s  Uwpvw  UTss  UwTsw







(12.34)



Ki T,i  g i  aiw  v w p,i mi  u,wi  Ts,wi  pv,wi mi Defining the total internal energy per mass of skeleton as u us  wuw , we obtain u  dt

1 Vij  pGij dij  pv s  wpv w  Ts s  wTs w  uw w U 1 1 1  g i  aiw  v w p,i mi  u,wi  Ts,wi  pv,wi mi  Ki T,i U U U











(12.35)



The left-hand side is clearly the sum of a stored term ( u ) and a dissipated term ( dt ). It is tempting therefore to identify it with the total energy input, but this would be incorrect, as that is represented by u itself. The right-hand side of (12.35) includes three types of term. The first involves the strain rate. The second type involves material differentials, and the third involves fluxes. The presence of the strain rate poses a problem within the Eulerian formulation because it is not possible to express the strain rate as a material derivative of any observable quantity. This problem can be avoided by adopting a Lagrangian formulation. We can rewrite 1 Vij  pGij dij U





1 Sij  p L Gij ' ij Uo





(12.36)

where Sij is the Piola-Kirchhoff stress tensor and 'ij is the Green-Lagrange strain defined by 2'ij Pki Pkj  Gij where Pij wxi wX j and xi and Xi are the current (Eulerian) and initial (Lagrangian) coordinates of a material point measured in a Cartesian system. The initial dry density is Uo . It can be shown 1 1 that Sij det Pij Pik Vkl Pjl and ' P d P . It is also necessary to introduce



the variable pL

ij

ki kl lj



p det Pij Pik1Pik1 which is the transformation of the pore pres-

sure to the Lagrangian coordinate system. We note that in Lagrangian coordinates, no distinction is necessary between the time and material derivatives so w' wt d' dt ' . that ' ij

ij

ij

ij

In principle, it would be possible to transform all other variables to Lagrangian coordinates, too, but this has the disadvantage that the physical meaning is lost. Since these transformations are not strictly necessary for the following argument, we shall leave the remaining terms in their Eulerian form.

12.2 Thermomechanical Framework

We can now write (12.35) as 1 u  dt Sij  p L Gij ' ij  pv s  wpv w  Ts s  wTs w  uw w Uo





251

(12.37)

1 1 1  g i  aiw  v w p,i mi  u,wi  Ts,wi  pv,wi mi  Ki T,i U U U









12.2.7 The Internal Energy Function Now we adopt the hypothesis that the internal energy is a function of the strains, the entropy, the water content, the extensive quantities v s , ss , vw , and sw , and certain internal variables Dij . We assume that the function can be decomposed in the form,



us  'ij , Dij , v s , s s  wuw  v w , s w

u u 'ij , Dij , v s , s s , w , v w , sw

(12.38)

so that wu  wu wu s wu s 'ij  D ij  v  s w'ij wDij wv s ws s

u 

wu wu w wu w w  v  s ww wv w ws w

(12.39)

wus  wus wus wus 'ij  D ij  s v s  s s s w'ij wDij wv ws uw w  w

wuw w wuw w  w v s wv w ws w

12.2.8 The Dissipation Function and Force Potential We also postulate that dissipation is a function of the same state variables, and also of D ij (rate of change of internal variables) and of the fluxes, i. e., dt



dt 'ij , Dij , v s , s s , w , v w , s w , D ij , mi , Ki



(12.40)

We can either derive the force potential z, from the dissipation, using the procedure described in Section 11.1.2, or we can assume the form of z and derive dt . In either case, we can write wz wz wz dt D ij  mi  Ki (12.41) wD ij wmi wKi

252

12 Behaviour of Porous Continua

where z, the force potential, is a function of the same variables as dt : z



z 'ij , Dij , v s , s s , w , v w , s w , D ij , mi , Ki



(12.42)

12.2.9 Constitutive Equations Substituting Equations (12.39) and (12.41) in (12.35) and collecting terms, we obtain § 1 § wus ·  wus Sij  p L Gij  0 ¨ ¸ 'ij  ¨  p  s ¨ ¨ Uo w'ij ¸¹ wv © ©



§ wuw  w¨ p  ¨ wv w ©



· w § wuw ¸ v  w ¨ T  w ¸ ¨ ws ¹ ©

· s § wus ¸ v  ¨ T  s ¸ ¨ ws ¹ ©

· w ¸ s ¸ ¹

§ wus wz · ¨  ¸ D ¨ wDij wD ij ¸ ij © ¹

(12.43)

§1 wz  ¨ g i  aiw  v w p,i  mi U w © 1  u,wi  Ts,wi  pv,wi mi U







· s ¸ s ¸ ¹

· § 1 wz · ¸ mi  ¨  T,i  ¸ Ki U wK i ¹ ¹ ©



Now Equation (12.43) should be satisfied for any combination of ' ij , D ij , v s , s s ,

v w , sw , Ki and mi , and since all these quantities are independent of each other, each term in Equation (12.43) has to be equal to zero independently. Because the internal energy function (12.38) is independent of ' ij , v s , s s , v w and s w , from the first two rows of (12.43), it follows that

1 Sij  pL Gij Uo



p T

p T



wus wv s wus ws s

wuw wv w wuw wsw

wus w'ij

(12.44) (12.45) (12.46) (12.47) (12.48)

12.2 Thermomechanical Framework

253

Considering now the spatial gradient of uw , we can obtain

u,wi

wuw wv

w

v,wi 

wuw ws

sw w ,i

 pv,wi  Ts,wi

(12.49)

so that the fifth row of (12.43) is identically zero. Equations (12.44)–(12.48) express an essential property of the internal energy function: that it is a potential for stresses and temperature. The basic form of these relationships are well known from hyperelasticity, but the particular expressions here deserve some comment. First, note that the intensive quantities of pore pressure and temperature each appear as a partial derivative of the internal energy of the skeleton and of the pore fluid. The fact that both derivatives are related to the same value of the intensive variable reflects an assumption of intimate mixing of the two phases. The temperature of the solids and fluid is assumed to be the same, and the pore pressure acts equally on the solids and the fluid. Later we will find it convenient to use separate variables for the two phases, but we shall assume that the values are equal (effectively making the assumption of intimate mixing explicit). Equation (12.44) embodies Terzaghi’s principle of effective stress for a porous medium. It demonstrates that (for the choice of kinematic variables we have made) the quantity that is work conjugate to the strain rate is not the total stress Vij but the effective stress Vij Vij  pGij (the positive sign appears because we have followed the tensile positive convention usual in continuum mechanics for the stresses, whilst the pore pressure is positive in compression). The corresponding definition of the Lagrangian effective stress is given by the following equation: Sij

Sij  p L Gij

(12.50)

Unfortunately, an argument similar to that used to develop Equations (12.44)–(12.48) above cannot be applied to the terms in the third and fourth rows of Equation (12.43) because the z function does depend on D ij , Ki , and mi . Assuming independence of the fluxes and of the rate of the internal variable, § wus § wz · wz · 0 , ¨ T,i  U  ¸ D only the weaker conditions ¨ ¸ Ki 0 , and ¨ wDij wD ij ¸ ij wKi ¹ © © ¹ § w wz · w ¨ v p,i  g i  ai  U ¸ mi 0 can be formally derived from Equation w mi ¹ © (12.43). However, at this stage, we restrict ourselves to analysis of models to which stricter conditions (than those described above) can be applied:





wu wz  wDij wD ij T,i

U

wz wKi

0

(12.51) (12.52)

254

12 Behaviour of Porous Continua

v w p,i  g i  aiw

U

wz wmi

(12.53)

Equation (12.51) is Ziegler’s orthogonality condition, defining viscoplastic constitutive behaviour, Equation (12.52) is (for an appropriate form of z) the Fourier heat conduction law, and Equations (12.53), are the missing equations of motion of the pore fluid, which becomes more obvious after they are rewritten as follows:

  pGij ,i  Uw g j  Uw awj  Uw U wwmz

0

(12.54)

j

where the third term in the left part can be identified with a “drag” force. When inertial effects can be neglected and with an appropriate choice of force potential z, Equation (12.53) becomes Darcy’s law for fluid flow (for a fluid of constant density Uw ).

12.2.10 Discussion Equations (12.44)–(12.48) and (12.51)–(12.53) represent a complete set of constitutive relationships describing a material, which is therefore defined entirely by specification of two scalar potential functions u and z in Equations (12.38) and (12.42), respectively. Equations (12.51)–(12.53) are sufficient, but not necessary, to ensure that the Laws of Thermodynamics are obeyed. Note that the entire formulation of constitutive behaviour is based on the following principles: x the mass conservation laws (12.1) and (12.5); x the First Law of Thermodynamics (12.19); x the Second Law of Thermodynamics (12.30) and the following assumptions: x the existence of the internal energy function (12.38) independent of rigid body translation and rotation; x the existence of the quasi-homogeneous dissipation function (12.40) satisfying Equation (12.41); x the two above functions should be related through Ziegler’s orthogonality condition (12.51) and the Onsager reciprocity relationships for the fluxes that follow directly from (12.52) and (12.53). Needless to say, the above formulation of constitutive behaviour is guaranteed to satisfy the Laws of Thermodynamics. We make no secret of the fact, however, that we have introduced some additional, more restrictive assumptions. It is for the reader to decide whether these restrictions reduce the scope for constitutive modelling to such an extent that the materials described are no

12.3 The Complete Formulation

255

longer realistic. We address below, however, some of the advantages that follow from adopting the more restrictive approach. It is our belief that a very wide variety of material response can be described within this framework. Furthermore, we are not aware of any specific counterexamples from the physical world that provide clear evidence that the restrictions imposed above are invalid.

12.3 The Complete Formulation We summarise the position we have arrived at as follows. The first step is to specify a constitutive model through internal energy and force potential functions:



us  'ij , Dij , v s , s s  wuw  v w , s w z  'ij , Dij , v s , s s , w , v w , s w , D ij , mi , Ki

u u 'ij , Dij , v s , s s , w , v w , sw

z

(12.55)

(12.56)

Using these functions, the differential relationships in Table 12.1 are applied. The variables involved in the solution (with the numbers of variables for vector and tensor quantities) are xi (3), vi (3), viw (3), ai (3), aiw (3), Pij (9), 'ij (6), Sij (6), Dij (6), U, w, v s , ss , vw , sw , p, pL, T, mi (3), and Ki (3); that is 57 variables in all. We supplement these with four further formal variables through T Ts Tw and p ps pw . The use of these variables allows the Legendre transform of the energy function to be carried out consistently. Specification of initial and boundary conditions completes the formulation.

12.3.1 Modifications to Account for Tortuosity As mentioned above, the effects of tortuosity have been neglected in these derivations. It is usual to account for such effects by introducing a tortuosity factor a (see, e. g. Coussy, 1995), which is the ratio between the average of the squared microscopic relative velocity of the fluid with respect to the skeleton, and the square of the average of the same quantity. It is straightforward to show that a is a factor always greater than or equal to unity. When this factor is included, the expression for the kinetic energy becomes:

K

1 1 § 2 U  vi dV  ³ Uw ¨ viw ³ 2 2 ©



V

V

2



  a  1 vir

2·

¸ dV ¹

(12.57)

where vir viw  vi is the macroscopic relative velocity of the fluid with respect to the skeleton.

256

12 Behaviour of Porous Continua

When the additional terms due to this change are followed through to Equation (12.37), the only change necessary is to replace aiw in the first term on the second line with a modified acceleration term aiwe , which is defined as  1 (12.58) aiwe aiw   a  1 vir  avir 2 Note that the term in aiw in the equation of motion (12.27) is not altered. In Table 10.1, the aiw term in the first equation of motion is unaltered, but that in the second equation of motion is modified to aiwe . Because there is now the additional variable a in the problem, a further equation is now required. This would be a constitutive relationship for the tortuosity factor a, which could, for instance, be expressed as a function of the porosity. Berryman (1980) suggests the expression a  n  1 2n for a matrix of spherical particles. In terms of the variables in Table 12.1, this would become a

Uwvw  1

2Uwv w .

12.4 Legendre-Fenchel Transforms In classical thermodynamics, in addition to the specific internal energy (u), three other energy functions are defined: the specific Helmholtz free energy (f), the specific enthalpy (h), and the specific Gibbs free energy (g). These functions are related to the specific internal energy (12.55) through a series of LegendreFenchel transformations (Appendix C), as shown in Table 12.2. As demonstrated in Table 12.2, all derived energy functions e can be also decomposed into the specific energy functions e s and e w attributed to the skeleton and fluid, respectively. Moreover, by the virtue of Equations (12.12) and (12.29), these parts can be transformed independently. Constitutive equations in the last row of Table 12.2 follow directly as properties of the corresponding Legendre-Fenchel transformations (Appendix C). The choice of formulation will depend on the application. For instance, the four forms of the energy potential in classical thermodynamics are adopted in different cases (e. g. isothermal problems, adiabatic problems, etc.). The dissipation function formulation can be also adjusted to the chosen energy formulation, simply by expressing the force potential z through corresponding variables:

   



zu 'ij , Dij , v s , s s , w , v w , sw , D ij , mi , Ki ½ ° ° f s s w w  z z 'ij , Dij , v , T , w , v , T , Dij , mi , Ki ° ° ¾ h s s w w  z z Sij , Dij , p , s , w , p , s , Dij , mi , Ki ° ° g s s w w  z z Sij , Dij , p , T , w , p , T , Dij , mi , Ki °° ¿ z

(12.59)

12.5 Small Strain Formulation

257

In principle, it would be possible to define the dissipation function in terms of a set of variables different from the energy function, but only in particular circumstances might this be useful.

12.5 Small Strain Formulation The displacement vector of a point with initial coordinates Xi is defined by ui xi  Xi , so that the deformation gradient can be expressed as Pij Gij  ui, j . The assumption of small strains is equivalent to ui , j  1 . Therefore,



det Pij # 1  ui,i and, after the higher order terms are neglected, the following simplifications can be applied to the formulation in Table 12.1: x 2'ij Pki Pkj  Gij # ui, j  u j ,i 2Hij , where Hij is the linearized strain tensor; x Sij det Pij Pik1Vkl Pjl1 # Vij ; x pL p det Pij Pik1Pik1 # p .

 

The linearized strain tensor can, in some circumstances, be decomposed into e  p elastic and plastic components Hij Hij  Hij , and the kinematic internal vari p able Dij can be associated with the plastic strain tensor Hij . Table 12.1 Summary of equations

Equation type Differential of free energy (skeleton)

Equation Sij  p L Gij s

 p wu wv

Uo wus w'ij

6

s

1

T wus ws s

Differentials of free energy (pore fluid)

 p wu

w

1

wv

w

1

s

Ziegler’s orthogonality condition

wu wDij  wz wD ij

Heat conduction law

T,i

Mass balance equations

U  Uvi,i

Second Law of Thermodynamics

1

T wus ws s

0

Uwz wKi

Uw  mi ,i

Number of equations

0

6 3 1

0

§ · 1 1 T ¨ s s  ws w  s,wi mi  Ki,i ¸ U U © ¹ wz wz wz D ij  mi  Ki t 0 wD ij wmi wKi

1 1

258

12 Behaviour of Porous Continua Table 12.1 (continued)

 Pjk Sik ,i  Uo 1  w g j

Equations of motion



Uo a j  wawj



3

Strain definition

v w p,i  g i  aiw Uwz wmi 2'ij Pki Pkj  Gij

Deformation gradient

Pij

wxi wX j

9

Skeleton velocity

vi

3

Skeleton acceleration

ai

xi vi

Fluid acceleration

aiw

3 6

3

viw  viw, j v wj



viw

3



3

Mass flux definition

mi

Density definition

1 U v s  wv w

1

Lagrangian pore pressure

pL



1

Uw

 vi

p det Pij Pik1Pik1

Total number of equations

57

12.6 Example The following example describes a conventional thermo-poro-elastoplastic model which can describe the small strain behaviour of a saturated, isotropic, frictional, granular material. The constitutive behaviour is completely defined by the following two potential functions. The first is the Gibbs free energy function g g s  wg w , which we assume has constant, linear and quadratic terms, written in the following form so that certain constants retain their usual meaning: gs

g 0s   p  p0 v0s   T  T0 s0s  

gw

 p  p0 2 2K s

v0s  3Ds  T  T0  p  p0 v0s  c sp

 T  T0 2 2T0

(12.60)

· 1 § 1 Vii V jj 1 Vijc Vijc   D  T  T0 Vkk  Vij Dij ¸¸ ¨¨ U0 © 3K 6 2G 2 ¹

g 0w   p  p0 v0w   T  T0 s0w 

 p  p0 2 2K w

v0w  3Dw  T  T0  p  p0 v0w  c w p

 T  T0 2

(12.61)

2T0

where initial values are denoted with the subscript “0” and the constants have the following physical meanings: x K s and K w are the isothermal bulk moduli of the skeleton particles and fluid, respectively; x 3Ds and 3Dw are the volumetric thermal expansion coefficients of the skeleton particles and fluid, respectively;

12.6 Example

259

x c sp and c w p are the mass heat capacity at constant pressure po of the skeleton particles and fluid, respectively; x K and G are the isothermal bulk and shear moduli of the skeleton matrix, respectively; x D is the linear thermal expansion coefficient of the skeleton matrix. The second function required is the force potential:

z

PVii vw / T D ijc D ijc  Ki Ki (12.62) 3E D ijc D ijc  D ii  mi mi  U0 U0 2Ukm 2UkK





where / is a Lagrangian multiplier associated with a dilation constraint. The constants have the following physical meanings: x km is the permeability coefficient, x kK is the thermal conductivity coefficient, x P and E are coefficients related to the effective angles of friction Ic and dilation < obtained in triaxial compression:

2 2 sin \

E

(12.63)

3  3  sin \ 2 2 sin Ic

PE

(12.64)

3  3  sin Ic

The following constitutive relationships can be derived from the above formulation using equations in the last column of Table 12.2:

Vii  3D  T  T0  Dii ; Hijc 3K

Hii

v s  v0s v0s

3Ds  T  T0 

v w  v0w

ss

s0s  c sp sw

 T  T0 T0

s0w  c w p

2G

T0

(12.65)

(12.66)

Ks p  p0

(12.67)

Kw

 3Ds  p  p0 v0s 

 T  T0

 Dijc

p  p0

3Dw  T  T0 

v0w

Vijc

DVkk U0

 3Dw  p  p0 v0w

(12.68) (12.69)

260

Table 12.2. Energy potentials for use in large strain continuum mechanics of porous media



s

s

w

u 'ij , Dij , v , s , w , v , s



us 'ij , Dij , v s , s s wu

w

v

w

,s

w

w

Helmholtz free energy





s

s

w

w

f 'ij , Dij , v , T , w , v , T





f s 'ij , Dij , v s , Ts



wf

w

v

w

w

,T

Sij

Sij  pL Gij

Fij

Uo

ps



Ts

Uo

wu w'ij

wus wDij

wus ; w wuw p  wv s wv w wus ; w wuw T ws s wsw

Sij

Sij  pL Gij

Fij

Uo

ps



whw pw , s w

ss

wf s wDij

w





g Sij , Dij , ps , Ts , w, pw , Tw





g s Sij , Dij , ps , Ts



wg

w

p

w

w

,T







h u  Sij 'ij Uo  pv

g

hs

us  Sij 'ij Uo  ps v s

gs

us  s s Ts  Sij 'ij Uo  ps v s

hw

uw  pw v w

gw

uw  s w Tw  pw v w

s

wf w'ij

Gibbs free energy

w





Uo

s

hs Sij , Dij , ps , s s

wf s ; w wf w p  wv w wv s wf s ; w wf w  s s  w wT wT 



s

h Sij , Dij , p , s , w , p , s



f u  sT f s us  s s Ts f w uw  sw Tw s



Enthalpy

u  sT  Sij 'ij Uo  pv

'ij

Uo

wh wSij

'ij

Uo

wg s wSij

Fij

Uo

whs wDij

Fij

Uo

wg s wDij

s

vs

Ts

whs ; w v wps whs ; w T ws s

whw wps whw

ws w

vs

ss

wg s ; w wg w v wps wpw s wg w wg w  w  s ; s wT wT

12 Behaviour of Porous Continua

Internal energy

Conclusions

261

Equations (12.65) represent the decomposition of the strain tensor into elastic and plastic components, where the elastic part is defined by conventional thermoelasticity. Equations (12.66) and (12.68) give thermoelastic relationships for the skeleton, and Equations (12.67) and (12.69) represent classical thermoelastic relationships for fluids. Evolution equations for plastic strains are obtained by defining the generalised wg s wz g Vij and the dissipative generalised stress Fij Uo , so stress Fij Uo wD ij wDij that

 PVii  3/E

Fijc

Fii

D ijc D ijc D ijc

(12.70) (12.71)

3/

By eliminating D ijc and / from Equations (12.70) and (12.71), we obtain the equation of the yield surface in the dissipative generalised stress space:



y Vij , Fij



Fijc Fijc   PVii  EFii 0

(12.72)

 y F (where O is a Lagrangian multiplier) follows from The flow rule D ij Ow ij the properties of the Legendre-Fenchel transformation y Fij D ij U0  z 0 relating the yield surface to the force potential. When Ziegler’s orthogonality condition Fij Fij is applied to the equations of the yield surface and flow rule, it becomes clear that the behaviour described is equivalent to that of a perfectly plastic model with the Drucker-Prager failure cone defined by an effective angle of internal friction Ic and a non-associated flow rule with the plastic potential cone defined by an angle of dilation <. Finally, from Equations (12.52) and (12.53), after neglecting inertial effects and substituting Ki qi T and mi wi v w , qi wi

kKT,i



km v w p,i  g i

(12.73)



(12.74)

Equation (12.73) is the isotropic Fourier heat conduction law, and for constant vw , Equation (12.74) becomes the isotropic version of Darcy’s law for fluid conduction.

12.7 Conclusions The theoretical framework presented in this chapter extends the applicability of the principles of hyperplasticity to problems involving:

262

x x x x x

12 Behaviour of Porous Continua

large strains, fluid flow in porous media, heat flow in porous media, viscous effects, inertial effects.

Apart from this generalization, the proposed framework places less stringent restrictions on the class of derived constitutive models in terms of the requirements of the Second Law of Thermodynamics. Within the framework of Chapter 4, the fact that the mechanical dissipation had to be non-negative resulted in a condition that is more stringent than the Second Law. Within the present framework, it is the total dissipation (including dissipation due to heat and fluid fluxes) that has to be non-negative, which is equivalent to the Second Law. As in the standard hyperplastic approach, the entire constitutive behaviour is completely defined by specification of two scalar potential functions. However, in the generalised framework, these functions also include properties related to different phases of the media and their interaction. The fluid and heat conduction laws are also built into these potentials, completing description of the constitutive behaviour of complex media.

Chapter 13

Convex Analysis and Hyperplasticity

13.1 Introduction So far we have avoided use of the terminology of convex analysis in the presentation of hyperplasticity. The reason has been to make this book as accessible as possible to engineers, many of whom will not be familiar with the mathematical techniques of convex analysis. However, this terminology is the most natural and rigorous for the description of many of the concepts we require. So in this chapter, we re-present hyperplasticity in a convex analytical framework. This allows us to treat certain issues more rigorously where, in previous chapters, we have glossed over some problems. In general, we follow the terminology employed by Han and Reddy (1999) in their book in which they make much use of convex analysis for conventional plasticity theory. We acknowledge, too, that the French school of plasticity has made much use of this approach for many years. The principal motivation for adopting the convex analytical approach is that it allows us to deal more rigorously with the relationships between the dissipation function and the yield function. It will be recalled that so far we have treated this relationship as a special degenerate case of the Legendre transform. In convex analysis, the Legendre transform is generalised to the Legendre-Fenchel transform (or Fenchel dual), and this allows more thorough treatment of the degenerate case. The alternative cases of elastic or elastic-plastic behaviour are also treated simply by convex analysis. The applicability of convex analysis to plasticity becomes so apparent that it seems highly likely that this will become the standard paradigm for expressing plasticity theory. Many of the concepts that have been given special names by plasticity theorists have parallels in the much more widely applied field of convex analysis. The advantages of expressing plasticity in this way are therefore twofold. Firstly, there is the extra rigour that is achieved; secondly, numerous

264

13 Convex Analysis and Hyperplasticity

standard mathematical results can be employed, some of which give useful, new insights into plasticity theory. Further advantages come in the treatment of constraints that arise in (a) extreme cases such as incompressible elasticity or (b) dilation constraints in plasticity. These are treated by using indicator functions, which are one of the most simple and powerful concepts in convex analysis. Indicators can also be used to express unilateral constraints, which arise, for instance, in materials that are able to sustain compression but not tension. In hyperplasticity, the indeterminacy of the form of the yield function can be resolved by the use of a canonical yield function which is closely related to the gauge function of convex analysis. A brief introduction to the concepts of convex analysis and the terminology used here is given in Appendix D, and familiarity with the material there is assumed in the following sections. It is strongly recommended that any reader unfamiliar with convex analysis should study Appendix D in detail before proceeding further with this chapter. Given that the notation of convex analysis is not entirely standardised, even a reader familiar with convex analysis may find it useful to study Appendix D, where our notation and terminology are defined.

13.2 Hyperplasticity Re-expressed in Convex Analytical Terms When potentials are not differentiable in the conventional sense, convex analysis serves as the framework for expressing constitutive behaviour, subject only to the limitation that the potentials must be convex. This does not prove too restrictive for our purposes. A complete exposition of hyperplasticity in convex analysis terminology would be lengthy, but suffice it to say (at least for simple examples) that each occurrence of a differential becomes a subdifferential. Thus instead of V wf wH we write Vwf  H . Thus the equations defining the “g formulation”, in which g  V, D and d  V, D, D are specified, may be expressed succinctly as

H  wV   g

(13.1)

F w D   g

(13.2)

Fw D d

(13.3)

F F

(13.4)

in which each of the variables may be a scalar or tensor, and the internal variables (and generalised stresses) may be a single variable, multiple variables, or an infinite number of variables.

13.3 Examples from Elasticity

265

13.3 Examples from Elasticity Before going on to examine more complex problems in plasticity, it is useful to gain some familiarity with the techniques of convex analysis by looking at some problems in elasticity. As an example of the way convex analysis can be used to express constraints, consider some simple variants on elasticity. Linear elasticity (in one dimension) is given by either of the expressions,

f

EH 2 2

(13.5a)

g

V2 2E

(13.5b)

or

Using derivations based on the subdifferential (which in this case includes simply the derivative, because both the above are smooth strictly convex functions) Vwf  H hence V EH , or Hw   g  V hence H V E . Now consider a rigid material, which can be considered as the limit E o f . The resulting f can be written in terms of the indicator function:

f

I^0`  H

(13.6)

which has the Fenchel dual  g 0 . The subdifferential of f gives V N^0`  H , which gives V > f, f @ for H 0 , and is otherwise empty, so that there is zero strain for any finite stress. Conversely, non-zero strain is impossible. The subdifferential of  g (just consisting of the derivative) gives H 0 directly, irrespective of the stress. In a comparable way, the limit E o 0 , i. e. an infinitely flexible material, is obtained from either f 0 or  g I^0`  V .

The above considerations become of more practical application as one moves to two and three dimensions. For instance, triaxial linear elasticity is given by

f

K Hv2 3GH2s  2 2

(13.7a)

p2 q 2  2K 6G

(13.7b)

or

g

Incompressible elasticity ( K o f ) is simply given by f

I^0`  v 

3GH2 2

(13.8a)

266

13 Convex Analysis and Hyperplasticity

or

g

q2 6G

(13.8b)

without the need to introduce a separate constraint. Note that whenever it is required to constrain a variable x to a zero value, where x is one of the arguments of an energy function, one simply adds the indicator function I^0`  x . In the dual form, the Fenchel dual does not depend on the variable conjugate to x. The above results can of course very simply be extended to full continuum models. Unilateral constraints can also be treated using convex analysis. A onedimensional material with zero stiffness in tension (i. e. a “cracking” material) can obtained from

f

Ec H

2

(13.9a)

2

or

g

I> f,0@  V 

V 2Ec

2

(13.9b)

where we recall that the Macaulay bracket is defined such that x x if x t 0 and x 0 if x  0 (where we use a tensile positive convention). Such a model might, for instance, be the starting point for modelling masonry materials, concrete, or soft rocks. Another case, rigid in tension and with zero stiffness in compression (in other words, the “light inextensible string” found in many elementary textbooks) is given by

f

I> f,0@  H

(13.10a)

g

I>0,f@  V

(13.10b)

or

In each of the above cases, elementary application of the subdifferential formulae gives the required constitutive behaviour, effectively applying the “constraints” (unilateral or bilateral) as required. Table 13.1 gives the forms of both f and g required to specify a number of different types of “elastic” materials, together with the derived stresses and strains. The table illustrates how the convex analytical framework can be used to express concisely the behaviour of materials with “corners” in the response, e. g. at the tension to compression transition.

Table 13.1. Some types of one-dimensional “elastic” materials

Model

Tension, compression moduli

Linear elastic

E, E

Rigid

f, f

Infinitely flexible Bilinear elastic Elastic-cracking (no tension)

0,0

Et , Ec 0, Ec

f H

g V

V2 2E

EH2 2 I^0`  H

Et H

2

2

Ec H

V EH

0 I^0`  V

0 E H  c 2

2

V

2

2Et



V

2 2

I> f,0@  V 

H

V 2Ec

V E

V > f, f @

H 0

V 0

H > f, f @

V Et H  Ec H

H

V  Ec H

­ V ½ H  N > f,0@  V  ® ¾ ¯ Ec ¿

2

2Ec

2

H  w   g  V

Vwf  H

2

2

V Et



V Ec

Et ,0

Rigid-cracking

0,f

I>0,f@  H

I> f,0@  V

V N>0,f@  H

­ V ½ H® ¾  N>0,f@  V ¯ Et ¿ H N> f,0@  V

Inextensible string

f,0

I> f,0@  H

I>0,f@  V

V N> f,0@  H

H N>0,f@  V

Et H 2

V

2Et

 I>0,f@  V

V Et H

13.3 Examples from Elasticity

Elastic string (no compression)

267

268

13 Convex Analysis and Hyperplasticity

13.4 The Yield Surface Revisited The dissipation function (which is in this case the same as the force potential) d d  D z  D is a first-order function of D , and the conjugate generalised stress is defined by F wd  D , which is the generalisation of F wd wD . The set X (capital F) of accessible stress states can be found by identifying the dissipation function as the support function of a convex set of F; hence applying Equation (D.21) from Appendix D,

&

^F

`

F, D d d  D , D

(13.11)

Note that here the notation , is used for an inner product, or more generally the action of a linear operator on a function. The indicator and gauge functions of X can be determined in the usual way. Note that the indicator is the dual of the support function, so it is the flow potential:

­ 0, F & I& F ® w F ¯f, F &

(13.12)

where D ww  F N &  F , which is the generalisation of D ww wF . It is useful at this stage to obtain the gauge function, J &  F inf ^P t 0 F P&`

(13.13)

The gauge may also be obtained directly as the polar of the dissipation: J& F

F, D   dom d d  D 0 zD sup

(13.14)

Furthermore, we define the canonical yield function (in the usual sense adopted in hyperplasticity) as y  F J &  F  1. Then, applying Equation (D.17), I &  F w  F I> f,0@ ¬ª y  F ¼º

(13.15)

So that applying the usual approach, we obtain any of the following: D ww  F wI X  F N X  F OwJ X  F Owy  F

(13.16)

where O t 0 [see Lemma 4.5 of Han and Reddy (1999)]. The above is the equivalent of the usual D Owy wF . Clearly, wy  F plays the role of wy wF , and O has its usual meaning. In particular, O 0 for a point within the yield surface (interior of X) and takes any value in the range >0, f @ for a point on the yield surface (boundary of X).

13.4 The Yield Surface Revisited

269

It can be seen, however, that the assumption made in developments in Chapter 4 that, because D ww wF Owy wF with O an arbitrary multiplier, one could deduce that w Oy was slightly too simplistic a step. Now we are in a position to address the process of obtaining either a yield surface from a dissipation function or vice versa. If we start with d z d  D , then we apply (13.7) to find the set of admissible states X, and then use (D.15), together with the definition of the canonical yield function: y  F inf ^P t 0 F P&`  1

^

inf P t 0 F P^Fc inf ^P t 0

`

Fc, D d d  D , D  dom d`  1

(13.17)

F, D d Pd  D , D  dom d`  1

so that y  F can in principle be determined directly from d  D . This is an important result. Then, D Owy  F . Conversely, if we first specify the yield surface y  F in the normal way, then X is easily obtained from & F y  F d 0 , and the dissipation function is then the support function of this set:

^

`

d  D V&  D sup^ F, D

F &` sup^ F, D

y  F d 0`

(13.18)

so that d  D can in principle be determined directly from y  F . This too is an important result, although it is more obvious than the transformation from dissipation to yield. It is not essential for (13.18), but there is a clear preference for expressing the yield surface in canonical form such that J &  F y  F  1 is a homogeneous first-order function of F, so that it can be interpreted as the gauge function of the set X. Note that the yield function is not itself positively homogeneous, but it is, however, expressible as a positively homogeneous function minus unity. If it is chosen this way, then y is dimensionless, so that O has the dimension of stress times strain rate. If y is expressed in canonical form, then the dissipation function can be expressed directly as the polar,

d  D

sup 0 zF&

F, D

 y  F  1

(13.19)

The results are summarised as follows: Option 1: start from the specified dissipation function d z d  D : Fwd  D

(13.20)

270

13 Convex Analysis and Hyperplasticity

y  F inf ^O t 0

F, D d Od  D , D `  1

F, D 1   dom d d  D 0 zD sup

(13.21)

Option 2: start from specified y  F : w  F I> f,0@  y  F

(13.22)

D ww  F Owy  F

(13.23)

d  D sup^ F, D

y  F d 0`

(13.24)

Note that if y is not expressed in canonical form, it cannot be readily converted to the gauge, and so the dissipation function cannot simply be obtained as the polar of the gauge. The function w (the flow potential) is the indicator of the set of admissible generalised stress states. If y  F is in canonical form such that J X  F y  F  1 is homogeneous of order one, then applying Option 2 to obtain d and then applying Option 1 to obtain y will return the original function. If this condition is not satisfied, then applying this procedure will give a different functional form of the yield function (the canonical form), but specifying the same yield surface. Thus if the yield surface is not originally defined in canonical form, it can be converted to canonical form by first applying Equation (13.18) and then (13.17) (although in specific instances there may often be more straightforward ways of achieving the same objective).

13.5 Examples from Plasticity We first consider how indicator functions can be used to introduce dilation constraints. A plastically incompressible cohesive material in triaxial space, with cohesive strength (maximum allowable shear stress) c, can be defined by

g

p2 q 2   qDs 2K 6G

(13.25)

d 2c D s

(13.26)

in which only a plastic shear strain is introduced. The canonical yield function can be obtained as

y

Fq 2c

1

(13.27)

13.5 Examples from Plasticity

271

Alternatively both the plastic strain components are introduced, but the volumetric component is constrained to zero. This approach proves more fruitful for further development. In the past, this has been achieved by imposing a separate constraint, but now we do so by introducing an indicator function into the dissipation:

g

p2 q2   pDv  qDs 2K 6G

(13.28)

d 2c D s  I^0`  D v

(13.29)

The yield function is unchanged for this case, and is again given by (13.23). This model is readily altered to frictional, non-dilative plasticity by changing the dissipation to

d M p D s  I^0`  D v

(13.30)

Note that we have introduced a Macaulay bracket on p which we did not use before, but strictly it is necessary to ensure that the dissipation cannot be negative. The corresponding canonical yield function is

y

Fq M p

(13.31)

1

The virtue of introducing the plastic volumetric strain is seen in that the model can now be further modified to include dilation by changing d to d

M p D s  I^0`  D v  B D s



(13.32)

The canonical yield function for this case becomes

y

Fq  BF p M p

(13.33)

1

This can be compared with the yield locus y

Fq  Mp  BF p

0 used in the

earlier example in Chapter 5. The above are some simple examples of the way in which expressions using convex analysis terminology can provide a succinct description of plasticity models for geotechnical materials. They may provide the starting point for using this approach in more sophisticated modelling.

Chapter 14

Further Topics in Hyperplasticity

14.1 Introduction In the previous chapters, we developed a technique for describing the behaviour of materials based on the use of two scalar functions. We placed great emphasis on the fact that, once these functions (or functionals) are known, then the entire response of the material can be determined. As we concentrated principally on the behaviour of rate-independent materials, we termed this approach “hyperplasticity”, although in Chapter 11, we showed that rate-dependent materials, too, can be described by this method. In this chapter, we extend the hyperplasticity approach to modelling problems in a number of different areas. The sections of this chapter are not connected, but represent different developments, each of which takes the hyperplasticity approach as the point of departure. They are intended to illustrate the fact that, once an engineer is familiar with this approach to constitutive modelling, it can provide a powerful technique capable of wide generality. In these extensions of the applications of hyperplasticity, the two most important features to be borne in mind are (a) the emphasis on using two scalar functions to define material behaviour (different choices of functions are available through the use of Legendre transforms), and (b) no ad hoc assumptions are required; once the functions are specified, the entire material behaviour follows. Two of the applications considered below (in Sections 14.3 and 14.5) involve non-dissipative systems, so that strictly we should call them applications of hyperelasticity, not hyperplasticity, but we use them here simply to demonstrate the continuity of our approach.

274

14 Further Topics in Hyperplasticity

14.2 Damage Mechanics So far, the only rate-independent dissipative materials we have considered are those in which the internal parameter plays the role of the plastic strain. In this section, we investigate some different forms of the free energy and dissipation expressions and discover that for these forms the internal variable plays the role of the damage parameter in conventional isotropic damage models. This development demonstrates the versatility and generality of the hyperplastic approach. Consider a Helmholtz free energy, f  H, D

1  D

EH 2 2

(14.1)

where D is a “damage parameter” such that 0 d D d 1. The factor 1  D is effectively applied as a reduction factor on the elastic stiffness E. The symbol D or ] is often used in the literature of “continuum damage mechanics” (CDM) for the damage parameter, but here we use D to emphasise that this is simply interpreted as an internal variable and is treated in the same way in the formulation as before. Note, however, that although the role of D is again associated with irreversible behaviour, it no longer has the physical interpretation of “plastic strain”. It is straightforward to derive

V

wf wH F 

1  D EH wf wD

EH 2 2

(14.2)

(14.3)

The first equation reveals that the usual value of the stress V EH for an elastic model is simply reduced by the factor 1  D . When D 0 (no damage), the model is simply elastic, and as D increases, the stress reduces by comparison with the undamaged case. It follows, too, that V

1  D EH  EHD F EHH

(14.4) (14.5)

It is also straightforward to show that the equivalent Gibbs free energy is V2 g  V, D f  VH . Note that this function cannot be decomposed into 2E 1  D the form g  V, D g1  V  g 2  D  VD that we have used previously for (uncoupled) plasticity models. It can, however, be interpreted as a type of coupled plasticity in which the reduction of stiffness is so extreme that, on a return to zero stress, there is no irreversible strain.

14.2 Damage Mechanics

275

We now introduce the force potential, which is equal to the dissipation function for this rate-independent case: z d

f1  D pos  D

(14.6)

where pos  x is as defined in Appendix D (Section D.10) on convex analysis. The function f1  D is chosen as a positive function. The subgradient of z gives F: Fwz  D

f1  D H  D

(14.7)

where H(x) is also defined in Appendix D, Section D.10. Now we can use F F to eliminate both the generalised forces and obtain EH 2  f1  D H  D 2

(14.8)

We now consider three cases. (a) D  0 , which cannot occur, since H  D is the empty set for this case, from which we deduce that the parameter D cannot decrease. (b) D 0 , in which case H  D > f,1@ . However, Equation (14.8) can be satisfied only by a non-negative element of H  D , since f1  D is positive and

EH2 2 is non-negative. Then, from (14.2), we obtain V H 1  D E , so that the material behaves as if it were “elastic” with a reduced stiffness 1  D E . Note, however, that there is no “plastic strain” in that the straight stressstrain relationship always passes through the origin. Since all non-negative EH 2 d f1  D , elements of H  D are less than or equal to unity, it follows that 2 so that the range of strain (and hence stress) is bounded for both negative and positive strains. EH 2 f1  D . Eliminating D between this 2 V· EH2 § f1 ¨ 1  ¸ , which is in general a non-linear equation and (14.2) gives 2 © EH ¹ stress-strain curve, antisymmetric about the origin.

(c) D ! 0 , in which case H  D 1 and

The pattern of behaviour defined above is illustrated in Figure 14.1. By obtaining the Fenchel dual of the force potential (the details of the process are omitted here), we obtain the flow potential, w  D, F I> f,1@  F  f1  D Then one can obtain D ww  F N> f,1@  F  f1  D , F  f1  D and D is indeterminate and non-negative for possible to obtain the canonical yield function y  D, F F

(14.9) so that D 0 for F f1  D . It is also f1  D  1 .

276

14 Further Topics in Hyperplasticity

Example 14.2.1 Consider the simple example f1  D Fr ln 1  D D f where Fr and Df are material constants. For D z 0 , F f1  D , and we can solve for D, giving





D Df 1  exp  F Fr Df 1  exp  EH2 2Fr When damage is occurring, D ! 0 , we obtain EHH F F

wf1  D wD

D

.

Fr D Df  D

(14.10)

which is the evolution equation for D. As stated above, “elastic” behaviour occurs on unloading and reloading with reduced stiffness; “yield” occurs when the maximum past value of f1  D EH2 2 is reached. The physical meaning of the constants in the model are as follows. The initial stiffness is E. The variable Df controls the value of the stiffness at large strain. For this case, both the secant and tangent stiffnesses are E 1  Df . 1 The value of Df must be 0 d Df d 1 . For D f ! | 0.6914 there 1  2exp  3 2 is a peak (and a minimum) in the initial stress-strain curve; otherwise, there is no peak. The variable Fr controls how rapidly the damage develops; lower values give more rapid development. For instance, the position of the peak (if present) is at a strain proportional to Fr E . For Df 1 , the peak is at H Fr E .

EH2/2 = f1(1 - V/EH)

V

E 1 1

(1 - D)E

H

Figure 14.1. Behaviour of a simple damage model

14.3 Elementary Structural Analysis

277

The generalisation of this type of model to a continuum is relatively straightforward. For instance, the factor 1 1  D can be applied to the Gibbs free energy for an elastic material, and when combined with the force potential or flow potential defined above, this will give a simple isotropic damage model. The model described here is essentially the same as that described by Holzapfel (2000), although expressed in different terminology. Note that, unlike the plasticity models, simple continuum models for damage involve a scalar internal variable, rather than a tensorial one. It is worth noting that the damage parameter D is dimensionless, so that the variables F and F here have the dimension of energy per unit volume (i. e. the same dimensions as f and g). Therefore, the terminology “generalised stress” is not appropriate for this application. Noting that F F EH2 2 , the physical interpretation of these variables is that they represent undamaged Helmholtz free energy. Note also that the generalisation of any elasticity model to an equivalent isotropic damage model involves only the simple procedure outlined above.

14.3 Elementary Structural Analysis 14.3.1 Pin-jointed Structures Much use of energy methods is made in structural analysis, for instance, in the analysis of stability of structures and bifurcation of solutions. It is useful to express energy methods for structures within the terminology we have developed for hyperplasticity, as this allows a consistency of treatment of different classes of problems. First of all, we confine our attention to pin-jointed structures. Consider a structure consisting of n elastic bars connected at m nodes. The lengths of each bar are li (i 1!n ), their unstressed lengths are loi , tensions are Ti , and axial stiffnesses are ki . Thus for each bar we can write Ti ki  li  loi . 1 2 The stored elastic energy in each bar is ki  li  loi , and the total elastic en2 ergy (internal energy or Helmholtz free energy) of the assembly is 1 n 2 u f ¦ ki li  loi . 2i 1 The position vector of each node is x j ( j 1!m ); and the external force applied to the node is Pj . The kinematic variables defining the problem are thus li





and x j , so that one would define f f x j , li , although the specific form noted above does not make f a function of x j . The variables x j are the equivalent of the strains encountered in previous chapters, whilst the li play the role of the internal variables D.

278

14 Further Topics in Hyperplasticity

We can express the compatibility of the structure through n compatibility equations expressing the length of each bar; each acts as a constraint on the kinematic variables: 1§ ¨ x a  x bi 2© i



ci



2

·  li2 ¸ 0 ¹

(14.11)

where ai and bi (1 d ai  bi d m) are the numbers of the nodes connected by bar i. There are other ways of expressing the compatibility relationships (e. g. ci x ai  xbi  li 0), and it is not necessary to use the specific form above. It is simply a convenient choice for bar structures. An elementary application of the hyperplasticity approach for a system with constraints establishes the equilibrium relationship, n wc wf  /i i wx j i 1 wx j

¦

Pj

(14.12)

where /i are a set of Lagrangian multipliers associated with each of the constraints. These are eliminated by noting that, for the internal variables lk (k 1!n ), there are n internal equilibrium relationships of the form, Fk

0

n wc wf  /i i wlk i 1 wlk

¦

(14.13)

Here the generalised stress is zero because there is no dissipation. Noting the specific forms of the energy and constraint expressions (in which each length appears only once), this immediately becomes (with no summation over k) 0 kk  lk  lok  / k lk

(14.14)

ki  li  loi

Ti with the tenli li sion coefficients (tension divided by length) of each bar. Furthermore, we can obtain the specific form of the expression for the applied forces: We can identify the Lagrangian multipliers /i

Pj

¦ /i  x a

i

j ai

¦ /i  x a

 x bi 

i

j bi

 x bi



(14.15)

This is, of course, simply a standard expression for the applied loads in tension coefficient form. An alternative way of expressing the above is first to form a modified energy n wf c wf c . expression f c f  ¦ /i ci , and then write Pj and 0 wlk wx j i 1

14.3 Elementary Structural Analysis

279

Example 14.3.1 A Simple Bar Structure Consider the single bar structure (in fact a mechanism!) shown in Figure 14.2.

V

(x,y)

H

P

(0,0) Figure 14.2. Bar mechanism

We can write f Then,

1 2 k  l  lo and the constraint c 2

H V

0

wf wc / wx wx wf wc / wy wy

wf wc / wl wl

1 2 x  y2  l2 2





0.

/x

(14.16)

/y

(14.17)

k  l  lo  /l

(14.18)

These equations can, of course, be manipulated to give the correct mechaniy x k  l  lo . k  l  lo , V cal behaviour of the bar: H l l

14.3.2 More General Structures The above results can now be extended to more general forms of structures. Consider any structure of n linearly elastic deformable elements, possibly connected by rigid components. Let the parameters (possibly vectorial) defining the kinematic state of each deformable element be li (i 1!n ). The value of this parameter when the component is unstressed is loi ; the force conjugate to the parameter li is Ti . If the elastic energy stored (internal energy or Helmholtz free energy) in the element is fi fi  li , then Ti wfi wli , and the (symmetrical)

280

14 Further Topics in Hyperplasticity

w 2 fi wli2 . Thus for each element, we can write

stiffness of the element, is k i

Ti

k i  li  loi . The total elastic energy of the assembly is f

n

¦ fi . i 1

The structure is subjected to applied loads at m points with position vectors x j ( j 1!m ), and the external force applied to each node is Pj (we allow the possibility that the coordinates include rotations and therefore the “forces” include moments). The kinematic variables defining the problem are thus li and x j , so that one would define f f x j , li , although the specific form does





not make f a function of x j . We can express the compatibility of the structure through n compatibility equations (kinematic constraints) expressing the relationships between the internal coordinates li and the external coordinates x j :

ci



ci x j , lk



0

(14.19)

An elementary application of hyperplasticity to this system with constraints establishes the equilibrium relationships, Pj

n wC wE  ¦ /i i wx j i 1 wx j

(14.20)

where /i are a set of Lagrangian multipliers associated with each of the constraints. These are eliminated by noting that, associated with the internal variables lk (k 1!n ), there are n internal equilibrium relationships of the form, 0

wC wE n  ¦ /i i wlk i 1 wlk

(14.21)

The above are sufficient to define the behaviour of the structure.

Example 14.3.2 A Spring-bar Linkage that Exhibits Instability Figure 14.3 shows a simple assembly of a rigid bar and a rotational spring. 1 2 The elastic energy (Helmholtz free energy) is f kT , and the constraint 2 equations can be expressed as c1 x  L sin T 0 and c2 y  L cos T 0 . Thus,

H V

0

wf wc wc  /1 1  /2 2 /1 wx wx wx wf wc1 wc2  /1  /2 /2 wy wy wy

wf wc wc  /1 1  /2 2 wT wT wT

kT  /1L cos T  / 2 L sin T

(14.22) (14.23) (14.24)

14.3 Elementary Structural Analysis

281

V

(x,y)

H

L T kT k

Figure 14.3. Bar-spring linkage which demonstrates instability

/1 0 and kT 0 kT  / 2 L sin T kT  VL sin T . The solution is T 0 or V  , which L sin T correctly describes the onset of instability at V  k L , and also for the subsequent behaviour. Now consider the special case H 0 . We obtain H

14.3.3 Assemblies of Rigid Elements It may seem surprising that an energy method should be applicable to systems consisting entirely of rigid elements, because they involve no storage of internal energy. But the generality of the hyperplasticity approach is sufficient that these too can be analysed successfully. This is most easily illustrated by examples.

Example 14.3.3 A Redundant Structure Consider the structure shown in Figure 14.4, which consists entirely of rigid bars. The internal energy (Helmholtz free energy) is identically zero, but the constraints are 1  x  L 2  y 2  2L2 0 2 1 2 c2 x  y 2  L2 0 2 1 c3  x  L 2  y 2  2L2 0 2

c1













(14.25) (14.26) (14.27)

282

14 Further Topics in Hyperplasticity

V H

(x,y)

L

(-L,0)

(0,0)

(L,0)

L

L

Figure 14.4. Redundant structure of rigid bars

These lead to

H

wc wc1 wc  / 2 2  /3 3 /1  x  L  / 2 x  /3  x  L wx wx wx wC wC wC V /1 1  / 2 2  /3 3 /1 y  / 2 y  /3 y wy wy wy

/1

(14.28) (14.29)

and Substituting and gives x 0 H L  /1  /3 y L V L  /1  / 2  /3 . These equations correctly describe the behaviour of the structure in terms of the unknown bar forces (expressed in terms of tension coefficients).

Example 14.3.4 A Mechanism Consider the mechanism shown in Figure 14.5. The internal energy is again identically zero. The compatibility relations (constraints) are 1 2 (14.30) c1 x1  y12  L2 0 2 1 (14.31) c2  x2  x1 2   y2  y1 2  L2 0 2 1 (14.32) c3  x2  L  y22  L2 0 2













14.3 Elementary Structural Analysis

V1

283

V2 H1

H2

(x1,y1)

(x2,y2)

(0,0)

(L,0)

L

L Figure 14.5. Mechanism consisting of rigid bars

These lead to

H1

/1

wc wc1 wc  / 2 2  /3 3 wx1 wx1 wx1

/1x1  / 2  x1  x2

(14.33)

V1

/1

wc wc1 wc  / 2 2  /3 3 wy1 wy1 wy1

/1 y1  /2  y1  y2

(14.34)

/ 2  x2  x1  /3  x2  L

(14.35)

/ 2  y2  y1  /3 y2

(14.36)

/1

H2

V2

wc wc1 wc  / 2 2  /3 3 wx2 wx2 wx2

/1

wc wc1 wc  / 2 2  /3 3 wy2 wy2 wy2

which correctly describes the behaviour of the structure as a function of its geometry. For instance, from the geometry shown in the diagram, we readily obtain H1  H 2 /1x1  /3  x2  L 0, which describes the inability of the mechanism to carry lateral load in this configuration. On the other hand, it can carry arbitrary vertical loads.

284

14 Further Topics in Hyperplasticity

14.4 Bending of Prismatic Beams Elementary beam theory, assuming that plane sections remain plane, gives the well-known relationship between axial stress V , applied bending moment M, and curvature 1 R for an elastic beam bent about one of its principal axes: V y

M I

E R

(14.37)

where y is the distance from the neutral axis, I is the second moment of area (equal to bd 3 12 for a rectangular beam of breadth b and depth d), and E is Young’s modulus. It follows that the moment-curvature relationship for an elasEbd 3 1 and the stress distribution is as in Figtic rectangular beam is M 12 R ure 14.6a. On the other hand, the fully plastic moment for a rectangular beam is readily V y bd 2 calculated as M p , where V y is the yield stress in pure tension. The 4 stress distribution is as in Figure 14.6c. The above relationships immediately suggest the following expressions for the free energy and dissipation per unit length of the beam: Ebd 3 § 1 · ¨  DR ¸ 24 © R ¹

f

d

V y bd 2 4

2

D R

(14.38)

(14.39)

where the internal variable D R is the plastic curvature of the beam.

Figure 14.6. Stress distributions in rectangular beam: (a) elastic; (b) elastic-plastic; (c) fully plastic

14.4 Bending of Prismatic Beams

285

B

1 0.8

A

M / Mp

0.6 0.4 0.2 0 0

2

4

6

8

10

(1 / R )(Ed / 2Vy ) Figure 14.7. Moment-curvature relationship for an elastic-plastic beam

Application of the standard derivations to these expressions shows that they express simple elastic-perfectly plastic behaviour, and the response of the beam would be as in the dotted line in Figure 14.7, with yield at point B. This, however, oversimplifies the response of the beam, as it yields progressively from the outV y bd 2 , point A in Figside (see Figure 14.6b). Initial yield occurs at M Me 6 ure 14.7, followed by progressive yielding which gives a hardening response that asymptotically approaches the fully plastic moment. The correct response of the beam can be described by considering the axial strain at any point in the beam H y R . The free energy and dissipation expressions that express the elastic plastic behaviour of any element of the beam are E f  H  D 2 and d V y D . Integrating the free energy and dissipation over 2 the area of the beam gives d2

f

³

d 2

E  Hˆ  Dˆ 2 b dy 2 d2

d

³

d 2

12

³

1 2

2

E § Kd ˆ ·¸ bd dK D ¨ 2© R ¹

12

ˆ b dy Vy D

³

1 2

12

³

fˆ dK

(14.40)

1 2

12

ˆ bd dK Vy D

³

dˆ dK

(14.41)

1 2

ˆ to emphasise that they are functions of the where we have written Hˆ and D internal coordinate, and furthermore have substituted K y d . The internal coordinate K has the physical interpretation of the dimensionless distance from the neutral axis.

286

14 Further Topics in Hyperplasticity

Application of the standard formulation results in

M

12

wf w 1 R

F 

F

§ Kd ˆ · E¨  D ¸ Kbd 2 dK © R ¹ 1 2

³

wfˆ ˆ wD wdˆ ˆ wD

(14.42)

§ Kd ˆ · E¨  D ¸ Kbd 2 © R ¹

(14.43)



(14.44)

ˆ bd V yS D

It is straightforward to show that these equations correctly describe the response shown in Figure 14.7. Furthermore, they also describe the unloading and reloading of the beam correctly. They can easily be extended to allow for axial compression of the beam by including the average axial extension of the beam Kd ˆ and then noting that the D Ha in the expression for the axial strain Hˆ Ha  R axial tension in the beam T is given by

T

wf wHa

12

Kd § ˆ ·¸ bd dK E ¨ Ha  D R © ¹ 1 2

³

(14.45)

14.5 Large Deformation Rubber Elasticity The following analysis is based on a stretch in three mutually orthogonal principal directions only, as is usual practice in rubber or polymer elasticity. However, we make no restriction to small strains. Consider a unit cube with initial density Uo (and therefore of mass Uo ). The cube is then stretched by factors Oi , i 1!3 in the three principal directions. The current volume of the cube becomes J O1O2O3 , its density is U Uo J , and its specific volume (volume per unit mass) is v 1 U J Uo . We could choose alternative variables to describe the stretch. For instance, we could adopt the Cauchy “small strain” definition, i. e. strain = change of length/initial length). In this case, Oi 1  Hi . Alternatively, we could use the Hencky “logarithmic” strain ei log Oi . The choice of Oi , Hi or ei as the kinematic variables is simply a matter of convenience, because they can be simply mapped to each other. The Cauchy stresses on the faces of the elementary cube are Vi . Now consider increments of stretch dOi . The work input will be O2 O3V1dO1  O3O1V2dO2  O1O 2V3dO3 , so that the work input per unit mass is

14.5 Large Deformation Rubber Elasticity

287

· O1O2 O3 § V1 V3 V2 ¨ dO1  dO 2  dO3 ¸ . Thus the quantity that is work conjuUo © O1 O2 O3 ¹ J Vi Vi gate to the increment of stretch dOi is (with no summation over i). Uo Oi UOi J Alternatively, note that dei dOi Oi , so that dW  V1de1  V2de2  V3de3 , Uo J Vi Vi and the quantity that is work conjugate to dei is . This is a convenient Uo U form and is a powerful argument in favour of using the Hencky strains. · V3 V2 J § V1 Finally, also note that dHi dOi leads to dW ¨ dH1  dH2  dH3 ¸ , Uo © O1 O2 O3 ¹ JVi Vi . In the so that the quantity that is work conjugate to dHi is Uo 1  Hi U 1  Hi following, however, we choose to work in terms of the principal stretches. We confine ourselves first to isothermal problems, and so we work in terms of the Helmholtz free energy. In rubber elasticity, it is common to write the energy (per unit mass) G in the form f O12  O 22  O32  3 , and to combine this with an incom2Uo pressibility constraint c O1O 2 O3  1 J  1 0 . Applying the usual formulation, J Vi wf wc / , which for the specific forms of the functions becomes (notUo Oi wOi wOi ing that the incompressibility constraint requires that J 1 ) dW





Vi Uo Oi

G 1 Oi  / Uo Oi

(14.46)

Now consider a uniaxial stretch with O1 O , O 2 O3 O 1 2 and V2 V3 0 . V1 G 1 For this case, we obtain O/ in the 1 direction and Uo O Uo O G 1 2 1 0 O  / 1 2 in both the 2- and 3- directions. We can eliminate the Uo O G 1· § Lagrangian multiplier /  and obtain, finally, V1 G ¨ O 2  ¸ . This is the Uo O O¹ © classical result in rubber elasticity, in which one can identify that G is the conventional small strain shear modulus. Now consider the case where we also include a bulk stiffness. In this case, we G K 2 O12  O 22  O32  3   log J , although might write (for instance) f 2Uo Uo





288

14 Further Topics in Hyperplasticity

there are a number of other possible forms for compressible large strain elasticity. On differentiation, this gives (with no summation over i)

J Vi Uo Oi

GOi K log J  Uo Uo Oi

(14.47)

or GOi2 K log J  J J

Vi

For uniaxial tension O1

O V2

V3

0 and O 2

(14.48)

O 3,

J 0 GO 22  K log J G  K log J O so that

K log J J

(14.49)

G  , and it follows that O V1

GO 2 K log J  J J

§ O2 1 · G¨  ¸ ¨ J O¸ © ¹

(14.50)

which is very similar to the incompressible result. Note that J cannot be expressed analytically in terms of O because of the transcendental nature of K log J G  . O J For small strain, it can be shown that the above equations reduce to

V1

2G  2H1  H2  H3  K  H1  H2  H3 3

(14.51)

which is, as expected, the usual small strain expression in terms of principal strains.

14.6 Fibre-reinforced Material Consider an elastic material as defined in Section 2.1. The free energy is f

3K

Hii H jj 6

 2G

Hijc Hijc

(14.52)

2

Now consider that embedded within this material is a (dilute) proportion of unidirectional fibres, with a small concentration c (volume of fibres divided by total volume). The fibres have a Young’s modulus Ef , and they are aligned in the direction with unit vector ni . We assume that the fibres are bonded to the parent (matrix) material and therefore undergo the same strain in the direction ni . The elongational strain in

14.6 Fibre-reinforced Material

289

this direction is Hij ni n j , so it is straightforward to see that the free energy of the composite material is f

3K

Hii H jj 6

 2G

Hijc Hijc 2



cEf Hij ni n j Hkl nk nl 2

(14.53)

and the stress is

Vij

K Hkk Gij  2GHijc  cEf Hkl nk nl ni n j

(14.54)

For a given strain, the material simply exhibits a stress that is augmented by a term that depends on the fibres. Now consider the possibility that the matrix material exhibits von Mises type plasticity, as described (in terms of the Gibbs free energy) in Section 5.2.2. The Helmholtz free energy becomes

f

3K

Hii H jj 6

 2G

 Hijc  Dij  Hijc  Dij  cEf H n n H 2

which we augment by the constraint Dkk ric strain. The dissipation is

2

ij i j kl nk nl

(14.55)

0 so that there is no plastic volumet-

d k 2 D ij D ij

(14.56)

 Fij 2G  Hijc  Dij  /Gij Fij k 2Sij  D ij

(14.57)

Now we obtain

Vij

K Hkk Gij  2G Hijc  Dij  cEf Hkl nk nl ni n j

(14.58) (14.59)

Equating Fij and Fij immediately gives / 0 , and eliminating the generalised stresses, we obtain an expression for the yield surface in strain space:







4G2 Hijc  Dij Hijc  Dij  2k2

0

(14.60)

In principle we can solve (14.57) for the strains in terms of the stresses and internal variables, and substitute this result in (14.60) to obtain the yield surface in terms of the stress and internal variables. In practice, this is most straightforwardly carried out by numerically inverting the expression:

Vij  2GDij

1 § · § · ¨ K Gij Gkl  2G ¨ Gik G jl  Gij Gkl ¸  cEf ni n j nk nl ¸ Hkl 3 © ¹ © ¹

(14.61)

The presence of the internal variables in the expression for the yield surface in terms of stresses means that the composite material exhibits strain hardening.

290

14 Further Topics in Hyperplasticity

14.7 Analysis of Axial and Lateral Pile Capacity The final example, drawn from geotechnical engineering, demonstrates how the continuous hyperplastic approach can be used to analyse simple systems involving soil-structure interaction. Unlike some previous applications, in this case we can give a physical meaning to the internal coordinate K. We consider a pile analysed by the “Winkler” method in which the reaction from the soil is determined by pointwise load-displacement relationships along the pile, but interactions among the relationships are not taken into account. The method is otherwise known as the use of “t-z” curves for vertical loading and “p-y” curves for lateral loading.

14.7.1 Rigid Pile under Vertical Loading Consider a rigid pile of length L and radius R in clay with shear modulus G and undrained shear strength c, under purely vertical loading. The resistance to movement is provided by shaft resistance on the side of pile and end bearing at the tip, see Figure 14.8. At each point along the pile, we assume a simple elasticplastic relationship (the “t-z” relationship) between the vertical displacement w and the shear stress W (these variables are usually called z and t, respectively, in the piling literature). We specify the response at any point along the pile by the following functions, in which it is convenient to work from the Helmholtz free energy:

f

G  w  D 2 2]R

(14.62)

d ac D

(14.63)

where ] and a are dimensionless factors (see Fleming et al., 1985) and D is the plastic vertical displacement of the pile shaft. The standard approach gives

W

wf ww

F 

wf wD

F

G w  D ]R G w  D W ]R

wd wD

acS  D

(14.64) (14.65) (14.66)

The yield function for the shear stress on the side of the pile can be expressed as y F  ac 0 .

14.7 Analysis of Axial and Lateral Pile Capacity

291

Figure 14.8. Tractions on a pile

The end-bearing stress V on the pile is also specified by an elastic-plastic relationship: f

2G  w  D L 2 SR 1  Q

(14.67)

N c c D L

(14.68)

d

where the stiffness and end-bearing factors follow the approach taken by Fleming et al., (1985). We distinguish D L , the plastic displacement of the pile base, from D (the equivalent for the pile shaft) because we have in effect two parallel plasticity mechanisms in which the onset of plastic strains occurs at a much smaller displacement on the shaft than at the base. The analogy is shown in Figure 14.9.

292

14 Further Topics in Hyperplasticity

Figure 14.9. Conceptual models of shaft and base of pile

14.7 Analysis of Axial and Lateral Pile Capacity

293

It follows that

wf ww

V FL



wf wD L

4G w  DL SR 1  Q

(14.69)

4G w  DL V SR 1  Q

(14.70)

wd wD L

FL

N c cS  D L

(14.71)

and the yield function at the pile tip can be derived as y FL  N c c 0. The entire pile response can now be obtained by integrating the energy and dissipation terms over the shaft of the pile and adding the end-bearing term summed over the end area. For convenience, we express the distance down the pile in terms of a dimensionless coordinate K z L , where z is the distance below ground level and L is the length of the pile. To emphasise that it is now ˆ . Again, we must distinguish here between a function of K, we rewrite D as D ˆ D 1 , the plastic displacement at the tip relevant to skin friction, and D L , the plastic displacement at the tip of the pile relevant to end bearing. The free energy and dissipation functions are 1

f

G

³ 2]R  w  Dˆ 0 1

2

 2SR LdK 

2GR

³ fdˆ K  1  Q  w  DL

2GR  w  D L 2 1 Q

(14.72)

2

0

1

d

³ ac Dˆ  2SR LdK   SR 0 1

2

Nc c D L (14.73)

³ dˆ dK   SR

2

Nc c D L

0

Application of the standard approach now results in the following expression for the vertical load V:

V

wf ww

1

4GR

G

³ ]R  w  Dˆ  2SR LdK  1  Q  w  DL

(14.74)

wfˆ Fˆ  ˆ wD

(14.75)

0

Fˆ

wdˆ ˆ wD

G  w  Dˆ  2SR L ]R



ˆ  2SR L acS D

(14.76)

294

14 Further Topics in Hyperplasticity

wf wD L

FL



FL

wd wD L

4GR w  DL 1 Q

(14.77)

 SR2 Nc cS D L

(14.78)

It is straightforward to show that these equations correctly define the mechanical behaviour of the pile, in that the total load is the integral of the shear stress over the surface of the pile, plus the end bearing; each of these terms is determined by a simple elastic-plastic relationship. Note that no additional difficulty is caused if the strength c is a function of depth.

14.7.2 Flexible Pile under Vertical Loading The only modification to the energy expressions necessary to accommodate a pile that is flexible rather than rigid is that there is now an additional free energy term due to elastic compression of the pile: p f

L

2 EA § wwˆ · ³ 2 ¨© wz ¹¸ dz 0

2

1

EA § wwˆ · ³ 2L ¨© wK ¹¸ dK

(14.79)

0

where we have written the vertical displacement w as wˆ wˆ  K to emphasise that it is now a function of the distance down the pile. The total free energy therefore becomes 1§

f

G EA § wwˆ · 2 ³ ¨¨ 2]R  wˆ  Dˆ  2SR L  2L ¨© wK ¸¹ 0©

2·

2GR 2 ¸ dK  wˆ 1  D L (14.80)  ¸ 1 Q ¹

and the dissipation expression is unchanged. We consider the possibility of the following externally applied loading on the pile: (a) a vertical load V at the top (as before) and (b) a distributed load vˆ per unit length along the pile. Later, we shall assume that the latter is zero, but it is convenient to introduce it at this stage. The work done by these external forces is the Frechet differential of the free energy with respect to the displacement wˆ . (Compare it with the usual equation V wf wH for a continuum, so that W VH  wf wH H ): 1

ˆ ˆ K W Vwˆ  0  ³ vwLd 0

ˆ f w , wˆ

(14.81)

14.7 Analysis of Axial and Lateral Pile Capacity

295

Carrying out the Frechet derivative, we obtain 1

1§

ˆ ˆ K Vwˆ  0  vwLd

G

³ ¨¨© ]R  wˆ  Dˆ wˆ  2SR L 

³ 0

0



EA § wwˆ · § wwˆ · · ¸ ¸ dK ¨ ¸¨ L © wK ¹ ¨© wK ¸¹ ¸¹ (14.82)

2GR  wˆ 1  D L wˆ 1 1 Q

Applying integration by parts to the second term in the integral on the righthand side, this becomes: 1

ˆ ˆ K Vwˆ  0  ³ vwLd 0

1§

G EA § w 2wˆ ·  · ˆ wˆ  2SR L  ˆ ¨ w  D ¨ 2 ¸ wˆ ¸ dK  ³ ¨ ]R ¨ wK ¸ ¸ L © ¹ ¹ 0© 1

(14.83)

ª EA § wwˆ ·  º 2GR « ¨  wˆ 1  DL wˆ 1 ¸ wˆ »  L wK  Q 1 ¹ ¼0 ¬ ©

Noting though that wˆ is arbitrary, we can simply compare coefficients in terms in wˆ . For 0  K  1 (note the strict inequalities), we compare the terms within the integrals to obtain vˆ 0

G EA § w 2wˆ ·  wˆ  Dˆ  2SR L  ¨¨ 2 ¸¸ L © wK ¹ ]R

(14.84)

G  wˆ  Dˆ and the com]R EA § wwˆ · pressive force at any point in the pile is P  ¨ ¸ , this simply represents L © wK ¹ 1 wP 2SRW . the differential equation for the vertical equilibrium of the pile L wK At the top of the pile, we compare terms in wˆ  0 and obtain Since the (upward) shear stress on the pile is W

V



EA § wwˆ · ¨ ¸ L © wK ¹K 0

(14.85)

which is simply the relationship between the applied vertical load and the compressive strain at the top of the pile. Similarly, at the base, we compare terms in wˆ 1 to obtain

0

EA § wwˆ · ¨ ¸ L © wK ¹K

 1

2GR  wˆ 1  DL 1 Q

(14.86)

which is the relationship between the end-bearing force and the compressive strain at the pile toe.

296

14 Further Topics in Hyperplasticity

Finally, if we express f and d in ways similar to those used for the rigid pile, 1

f

³ 0

ˆ K  2GR  wˆ 1  D 2 and d fd L 1 Q

1

ˆ K  SR  ³ dd

2

Nc c D L , we can obtain the

0

generalised forces:

wfˆ Fˆ  ˆ wD wdˆ ˆ wD

Fˆ FL



ˆ  2SR L acS D

(14.87) (14.88)

wf wD L

4GR  wˆ 1  D L 1 Q

(14.89)

wd wD L

 SR2 Nc cS D L

(14.90)



Fˆ L

G  wˆ  Dˆ  2SR L ]R

Combining the above equations yields the following results. For 0  K  1 ,

EA d 2wˆ L dK2

G  wˆ  Dˆ  2SR L acS Dˆ  2SR L ]R



(14.91)

which is the differential equation for the compression of the pile in terms of the mobilised shear stress on the pile shaft (which is in turn given by an elasticplastic expression). At K 1 ,



EA § dwˆ · ¨ ¸ L © dK ¹ K

1

4GR  wˆ 1  DL 1 Q

 SR2 Nc cS D L

(14.92)

which expresses the elastic-plastic response for the end bearing at the pile tip and its relationship to the strain in the base of the pile. The above equations could have been readily (and more easily) obtained by conventional methods of analysis of the mechanics of the pile, but our purpose here is to demonstrate the consistency and generality of the hyperplasticity approach. The important principle illustrated in this example is that we were able to define the correct mechanical behaviour from a free energy functional that contained differential as well as algebraic terms. Note that the above formalism for describing the pile problem is only one of ˆ as if it were a number of possible approaches. In particular, we chose to treat w ˆ an external kinematic variable. An alternative would be to treat w as an internal variable for 0  K d 1 and just the displacement wˆ  0 as an external variable. In this latter case, the orthogonality condition results in a vertical equilibrium equation at any point down the pile, and this provides an example which brings physical meaning to the interpretation of orthogonality as an “internal equilibrium” condition.

14.7 Analysis of Axial and Lateral Pile Capacity

297

14.7.3 Rigid Pile under Lateral Loading Now consider the rigid pile under lateral loading. We shall consider the resistance to movement provided solely by shaft resistance (i. e. we ignore any lateral shear resistance on the tip). At each point along the pile, we assume a simple elastic-plastic relationship (the “p-y” relationship) between the lateral displacement uˆ u  zT u  KLT and the normal stress V (called y and p, respectively, in piling literature). We specify the following functions:

f

k uˆ  Dˆ 2 4R ˆ d N cD

(14.93) (14.94)

h

where k is the “modulus of subgrade reaction” and N h is a lateral bearing ˆ now represents the lateral plastic discapacity factor. The internal variable D placement of the pile. The standard approach gives

wf wuˆ

V

wf wD wd F wD

F 

k uˆ  Dˆ 2R k uˆ  Dˆ V 2R

(14.95) (14.96)



ˆ N hcS D

(14.97)

The yield function for the stress on the side of the pile can be expressed as y F  N hc 0 . Now the entire pile response can be obtained by integrating the energy and dissipation terms over the length of the pile: L

f

³ 0

1

k u  KLT  Dˆ 2 2RdK 4R

³ 0

kL u  KLT  Dˆ 2 dK 2

1

d

³ 2RLNhc Dˆ dK 0

1

³ fdˆ K

(14.98)

0

1

ˆ K ³ dd

(14.99)

0

Application of the standard method results in

H

M

1

wf wu wf wT

³ kL u  KLT  Dˆ dK

(14.100)

0

1 2

³ kL Ku  KLT  Dˆ dK 0

(14.101)

298

14 Further Topics in Hyperplasticity

wfˆ ˆ Fˆ  kL  u  KLT  D ˆ wD wdˆ ˆ 2RLN h cS D Fˆ ˆ wD

(14.102)



(14.103)

It is straightforward to show that these equations correctly define the mechanical behaviour of the pile, in that the horizontal load is the integral of the lateral stress over the length of the pile, and the moment is the integral of the lateral stress times distance from the surface. The lateral stress is given by a simple elastic-perfectly plastic relationship.

14.7.4 Flexible Pile under Lateral Loading In the same way as for the axially loaded pile, the only change in the energy functions that needs to be considered for a flexible pile under lateral loading is to add an energy term that accounts for the bending stiffness: L

p f

³ 0

2

EI § w 2uˆ · ¨ ¸ dz 2 ¨© wz 2 ¸¹

1

2

EI § w 2uˆ · ¨ 2 ¸ dK 3¨ ¸ 0 2L © wK ¹

(14.104)

³

Thus the total free energy becomes: 2· 1 kL EI § w 2uˆ · ¸ 2 ˆK ¨ ˆ  (14.105) f uˆ  D d K fd ¨ ¸  3¨ 2¸ ¸ ¨ 2 wK 2 L © ¹ ¹ 0© 0 and the dissipation is still as given by Equation (14.99). Now consider that the pile is subjected to the following external loads: (a) a horizontal force H and moment M at the head of the pile and (b) distributed ˆ . The latter are again just introduced for convenlateral load hˆ and moment m ience and will both be set to zero later. As for the vertical pile, we can relate the external loads to the Frechet differential of the free energy: 1§

³

W

³

1§ § wuˆ · · M § wuˆ ·  ˆ ˆ  m ˆ ˆ ¨ ¸ ¸ dK  ¨ huL Hu  0  ¨¨ ¸¸ ¨ wK ¸ ¸ ¨ L © wK ¹ © ¹¹ K 0 0©

³

ˆ f u , uˆ

(14.106)

Carrying out the Frechet differentiation of the free energy, we obtain 1§ § wuˆ · · M § wuˆ · ˆ ˆ  m ˆ ¨ ¸ ¸ dK  ¨ huL Huˆ  0  ¨¨ ¸¸ ¨ wK ¸ ¸ ¨ L © wK ¹ © ¹¹ K 0 0©

³

1§

EI § w 2uˆ ·§ w 2uˆ · · ˆ uˆ  ¨ ¨ kL  uˆ  D ¸¨ 2 ¸ ¸ dK 3¨ 2 ¸¨ ¨ wK wK ¹¸ ¹¸ L © ¹© 0©

³

(14.107)

14.7 Analysis of Axial and Lateral Pile Capacity

299

Applying integration by parts (a) to the last term in the integral on the lefthand side of the equation and (b) twice to the last term in the integral on the right-hand side, we obtain M § wuˆ · Huˆ  0  ¨¨ ¸¸ L © wK ¹ K

1 § wuˆ · ˆ ˆ  m ˆ ¨ ¸ dK  ³ huL ¨ wK ¸ © ¹ 0 0

M § wuˆ · Huˆ  0  ¨¨ ¸¸ L © wK ¹ K

1

1 ˆ ·· § ˆ  § wm ˆ ¨ ˆ ˆ º  ³ ¨ huL ¸ uˆ ¸ dK ª¬mu ¼ 0 © wK ¹ ¹ 0 0© 1

1§

ª EI § w 2uˆ · § wuˆ · º EI § w 3uˆ · § wuˆ · · ˆ uˆ  ¨ ˆ ¨ ¸  D K  kL u d « 3 ¨ 2 ¸ ¨¨ ¸¸ » ¸  ¨ ¸ ³¨ 3¨ 3 ¸ ¨ wK ¸ ¸ ¨ ¸ wK L « © ¹ © ¹ © ¹ ¬ L © wK ¹ © wK ¹ »¼ 0 0 1

1§

1

ª EI § w3uˆ · º ª EI § w 2uˆ · § wuˆ · º EI § w 4 uˆ · · ³ ¨¨ kL uˆ  Dˆ uˆ  L3 ¨¨© wK4 ¸¸¹uˆ ¸¸ dK  «« L3 ¨¨© wK3 ¸¸¹ uˆ »»  «« L3 ¨¨© wK2 ¸¸¹ ¨¨© wK ¸¸¹»» ¹ ¬ ¼0 ¬ ¼0 0©

(14.108) Now, noting that uˆ is arbitrary, we follow the same procedure as before for ˆ 0 to obtain for 0  K  1 , the vertically loaded pile. First, we set hˆ 0 and m ˆ  0 kL  uˆ  D

EI § w 4 uˆ · ¨ ¸ L3 ¨© wK4 ¸¹

(14.109)

which is simply the differential equation relating bending of the pile to the applied lateral load. can be treated Next, at the top of the pile, we note that uˆ  0 and wuˆ wK



K 0

as independent variables, so that, by equating the coefficients in these terms separately we obtain, H

M L

EI § w3uˆ · ¨ ¸ L3 ¨© wK3 ¸¹K 

(14.110) 0

EI § w 2uˆ · ¨ ¸ L3 ¨© wK2 ¸¹K

(14.111) 0

which are effectively the boundary conditions at the top of the pile for shear force and bending moment.

300

14 Further Topics in Hyperplasticity



Finally, at the bottom of the pile, we treat uˆ 1 and wuˆ wK

K 1 as independ-

ent to obtain the conditions of zero shear force and bending moment at the base: 0 

0

EI § w 3uˆ · ¨ ¸ L3 ¨© wK3 ¸¹K

(14.112) 1

EI § w 2uˆ · ¨ ¸ L3 ¨© wK2 ¸¹K 1

(14.113)

The formulation is completed by

wfˆ ˆ Fˆ  kL  uˆ  D ˆ wD wdˆ ˆ Fˆ 2RLN h cS D ˆ wD

(14.114)



(14.115)

The response of the pile is thus represented by the differential equation for the bending of the pile as a function of the applied lateral loading, which in turn is determined by an elastic-plastic relationship: EI d 4 uˆ 3

4

L dK



ˆ 2RLN h cS D ˆ kL  uˆ  D

(14.116)

supplemented by the boundary conditions at the top and bottom of the pile. It is freely admitted that a conventional structural analysis approach to derivation of the governing equations for the pile would be slightly more straightforward, and certainly would involve simpler mathematics. Our purpose here is not, however, to find the simplest way of describing pile behaviour. Instead, it is to demonstrate the power and generality of the continuous hyperplastic approach. Solutions can be obtained by consistently applying the hyperplastic formulation in cases where the functionals describing the system behaviour are rather more complex than we have encountered hitherto. In particular, they include functions of differentials of continuous kinematic variables. The application of hyperplasticity principles to modelling foundation behaviour is finding a number of applications. Einav (2005) presents an analysis of the behaviour of piles which follows a rather similar approach to the analysis presented above. Houlsby et al. (2005) use a similar approach for the analysis of shallow foundations under cyclic loading. Finally the “force resultant” models for shallow foundation behaviour, originally expressed in terms of conventional plasticity theory [Martin and Houlsby (2001), Houlsby and Cassidy (2002)], can be expressed in the hyperplasticity framework, enabling the extension to continuous hyperplasticity, e. g. Lam and Houlsby (2005).

Chapter 15

Concluding Remarks

15.1 Summary of the Complete Formalism In conclusion, we restate the complete hyperplastic formalism in a concise form. To make it sufficiently general, we accommodate (a) infinite numbers of internal variables, (b) total dissipation, and (c) rate effects. For brevity, we use a symbolic rather than subscript notation for tensors. Assume that the local state of the material is completely defined by knowledge of (a) the strain H, (b) the entropy s, and (c) certain internal variables D. The form of H is tensorial, s is scalar, and we make no restrictions at this stage on the form of D (for instance, they could be scalars or tensors, possibly infinite in number). Then, the constitutive behaviour of the material will be completely defined by specifying two thermodynamic potential function(al)s of state. The first potential is the specific internal energy, a function(al) of state u u > H, s, D @ , which is a property satisfying the first law of thermodynamics:

u V : H  ’.q

(15.1)

where V is the stress tensor that is work-conjugate to the strain rate and q is the heat flux vector. The second potential is the force potential z z > H, s, D, K, D @ , a function(al) of state, change of internal variable D , and entropy flux K q T ( T is the nonnegative thermodynamic temperature) such that

d

wz .K  F, D wK

(15.2)

where F, D denotes the action of the linear operator F on D (F is the Frechet derivative of z with respect to D ). The quantity d is the specific total dissipation (the first part is usually called the thermal dissipation, and the second part the

302

15 Concluding Remarks

mechanical dissipation). It satisfies the Second Law of Thermodynamics in the form,

d t0 T

s  ’.K

(15.3)

which can also be written:

Ts  ’.q  ’T.K d t 0

(15.4)

Adding Equations (15.1) and (15.4), we obtain

u  d V : H  Ts  ’T.K

(15.5)

Assuming that u u > H, s, D @ is Frechet-differentiable and denoting F as the linear operator that is minus the Frechet derivative of u with respect to D, we obtain

u

wu wu : H  s  F, D wH ws

(15.6)

Substitution of Equations (15.6) and (15.2) in (15.5) yields

· § wu · § wu · § wz ¨  V ¸ : H  ¨  T ¸ s  ¨  ’T ¸ .K  F, D  F, D © wH ¹ © ws ¹ © wK ¹

0

(15.7)

Finally, assuming that

’T 

wz wK

(15.8)

and Ziegler’s orthogonality condition in the form that the following two linear operators are identical: F F

(15.9)

from Equation (15.7), we obtain (for the mutually independent processes of straining and change of entropy):

V T

wu wH wu ws

(15.10) (15.11)

Equations (15.8)–(15.11) are sufficient to establish the constitutive behaviour of a material defined by u u > H, s, D @ and z z > H, s, D, K, D @ . In particular, the entire response arises from the fact that these functionals serve as potentials for various dependent variables. It is sometimes convenient to supplement the above potentials by constraints, either in terms of certain variables or of their rates. For cases involving either unilateral or bilateral constraints, the formalism of convex analysis allows

15.3 Some Future Directions

303

a more rigorous expression of the mathematical structure of the theories. Convex analysis is also an appropriate framework for expressing the behaviour of rate-independent (plastic) materials.

15.2 Legendre-Fenchel Transforms Starting from the above formulation in terms of the internal energy u and the force potential z, it is possible to carry out a series of Legendre-Fenchel transforms to interchange the roles of dependent and independent variables. In this book, we have made much use of these transforms. The internal energy can be transformed to the enthalpy h, Gibbs free energy g, or Helmholtz free energy f; the four forms of the energy function form a closed chain of transforms. Quite independently, the force potential z can be transformed to the flow potential w, and the two potentials satisfy the relationship z  w d . Use of the flow potential proves more straightforward for the derivation of incremental response. For the special case of rate-independent materials, which have been our principal concern, z { d and w { Oy 0 , where y is the yield surface in generalised stress space. In this case, the dissipation is a homogeneous first-order function of the rates of the internal variable.

15.3 Some Future Directions We believe that the formulation we have adopted in this book offers enormous scope for modelling elastic-plastic materials. Although we make no claim that the orthogonality principle is universally true, it appears that a very wide variety of materials (and indeed structural systems) conform to the principle. The great advantage is that these materials can be described entirely in terms of just two potential functions. The compactness of the resulting formulation surely means that this should be the preferred way of describing such materials. In this context, the French usage of the term “standard material” is appropriate, but we feel that we have taken the approach further than the French school. They have usually observed that the potentials can be defined, but have not, we believe, exploited their application fully. Here we have gone much further, regarding the potentials (and their various transforms) as fundamental, and using them as the starting point to define particular materials. An approach that is rooted in the use of potentials offers excellent scope for further development. The existence of potentials is intimately connected to extremum principles (see, for instance, Sewell, 1987), and to issues such as uniqueness of response. It is anticipated that further mathematical work may well lead to theorems about the behaviour of hyperplastic materials which go beyond the

304

15 Concluding Remarks

current (rather weak) theorems on the plasticity of materials with non-associated flow (Palmer, 1966). Continuous hyperplasticity, involving the use of functionals to describe fields of infinite numbers of yield surfaces, is clearly in its infancy. Modern computing techniques, particularly the use of symbolic algebra, offer the possibility of major developments in this area. As shown in Section 14.7, the use of this technique to describe complex issues such as soil-structure interaction is very promising. Finally, we have shown in Chapter 13 how plasticity theory (and more particularly hyperplasticity) is naturally expressed within the language of convex analysis. This approach has already been adopted by some authors (notably Han and Reddy, 1999), but we expect that it will become the standard language for plasticity theory in due course. The yield surface will be thought of in terms of either the indicator or the gauge function of a convex set, and the dissipation function as the corresponding support function. The fact that the indicator and support functions are Fenchel duals and that the support and gauge functions are polars may be usefully exploited to move between different descriptions. Use of convex analysis will allow to be brought a whole range of mathematical techniques to bear on plasticity theory, by using a language that is extensively used elsewhere, rather than just the ad hoc terminology of plasticity theorists.

15.4 Concluding Remarks In this book, we have attempted to present a unified approach to the constitutive modelling of dissipative materials, based on thermodynamic principles. Although we have concentrated on rate-independent “plastic” materials, we have also considered a wider class of problems. In the approach we term “hyperplasticity,” once two scalar functions are known (essentially defining the stored energy and the rate of dissipation), then we can derive the entire constitutive response without recourse to any additional ad hoc assumptions. This gives the structure of our theories an appealing simplicity, and also opens the door for development of theorems about the behaviour of such materials. The approach also allows materials to be simply classified and put into hierarchies. We make no extravagant claims of generality for our approach, and freely accept that, based on Ziegler’s concept of orthogonality, we impose a somewhat tighter restriction than the Laws of Thermodynamics necessitate. Nevertheless, we have found that our approach can be used successfully in describing a very wide class of materials, and leave it for the reader to judge whether these mathematical models realistically represent the response of materials of importance in engineering.

Appendix A Functions, Functionals and their Derivatives

A.1 Functions and Functionals The usage of variables and functions in this book will be familiar to most readers, but some of the techniques used for analysis of functionals may be unfamiliar. As a preamble to the description of functionals, first we describe the terminology for the simpler cases. The following is not intended as a rigorous or comprehensive introduction but simply a clarification of the terminology used here. A variable is a symbol used to represent an unspecified value of a set. The value of the variable is one particular member of the set and the range is the set itself. For example, the variable x might represent a real number. In this case, the range is the set of real numbers from  f to f, and in a particular instance, the value of x may be the number 3.81. A function of one variable associates a value from one set (the range of the function) with each value from another set (the domain of the function). For example, the function y 3x 2 expresses y as a function of the variable x. If the domain of the function (i. e. the range of its argument x) is the set of real numbers, then the resulting range of the function y will be the set of non-negative real numbers. The variable y is called the dependent variable or value of the function, and x is the independent variable or the argument of the function. If the specific form of a function is not given, then it is written in the form y f  x , or y y  x . The concept is simply extended to functions of several arguments, e. g. y y  x1 , x2 , x3 . The arguments may be vectorial or tensorial rather than scalar, as may be the dependent variable. A functional is loosely defined as a “function of a function”. It is a function in which one or more of the arguments is itself a function. The range of the functional may be either another function or just a variable. A functional is distinguished here from a function by use of square brackets > @ . Thus, for instance, z z > y @ , where y y  x .

306

Appendix A Functions, Functionals and their Derivatives

It is sometimes necessary to make a careful distinction between the function f itself and its value at a particular value of x, which we shall denote f  x . In this regard, we follow the more usual convention, but note that in some subjects, it is common to use f  x for the function and f for its value.

A.2 Some Special Functions Throughout this book, we need to make much use of some special functions, principally related to absolute values and certain singularities. We denote the absolute value by x x, x t 0 ; x x, x  0 . We use the following notation for the Macaulay bracket: x x, x t 0 ; x 0, x  0 . We also need the derivatives of the above functions. They may be loosely defined as follows, although more formal definitions are provided by the subdifferential, which is introduced in Appendix D on convex analysis. For the time being, we define a modified signum function as S  x 1, x  0 , S  x 1, x ! 0 , and S  x as undefined for x 0 , but within the range 1 d S  0 d 1 . Note that this does not correspond to the conventional signum function, usually denoted by “sgn” or “sg”, and which is usually defined by sgn  0 0 . 1 We define the Heaviside step function as H  x  S  x  1 , so that 2 H  x 0, x  0 , H  x 1, x ! 0 , and H  x is undefined for x 0 , but within the range 0 d H  0 d 1 . We define the Dirac impulse function G  x through the relationship b

³ G  x  x0 f  x dx f  x0 ,

a

a d x0 d b . It can be proven straightforwardly that

Hc  x G  x , where f c  x denotes the differential of f  x . The quantity that plays the same role as the absolute value for a symmetrical second-order tensor is xkl xkl , and we define the derivative of this with respect to xij as a generalised tensorial signum function, which we write xij Sij xij . Like the signum function of a scalar, this quantity is undexkl xkl fined when xkl xkl 0 (which necessarily occurs only when xij 0 ), but we require Sij  0 Sij  0 d 1 . It is emphasised again that some of the above definitions can be set in a much more satisfactory formalism by the use of convex analysis; see Appendix D.



A.3 Derivatives and Differentials

307

A.3 Derivatives and Differentials The derivative of a function is the instantaneous rate of change of the function df { f c x with respect to one of its arguments. The definition of the derivative dx of the function f  x with respect to x should be familiar: f c x

lim

f  x  Gx  f  x

Gx o0

Gx

(A.1)

The above is of course the fundamental definition from which many familiar expressions for derivatives of particular functions are obtained. When the function has more than one argument, partial derivatives are defined by obtaining the derivative with respect to one argument whilst considering the others as constants. Thus if f f  x, y the partial derivative wf { f x  x, y with respect to x is defined as: wx f x  x, y

lim

Gx o0

f  x  Gx, y  f  x, y Gx

(A.2)

In the main text, the differential of an integral is required. If b x F x f  x, t dt , then application of the basic definitions results in a x

³

b x

F c x

³

a x

wf db da dt  f  x, b  f  x, a wx dx dx

(A.3)

The differential of a function f  x is defined by df f c  x dx , where dx is an independent variable. (Note the formal distinction, often ignored, between a differential and a derivative). The total differential of a function of more than one argument, for example f  x, y , is defined in the following way: df

wf wf dx  dy wx wy

(A.4)

wf dx is a partial differential. wx The concept of the differential of a function can be extended to that of a functional by using either the Gateaux or Frechet differential, and these are developed as follows.

where each of the terms of the form

308

Appendix A Functions, Functionals and their Derivatives

In the classical calculus of variations, the variation of a functional f >u@ is defined in terms of a variation Gu of its argument function: Gf >u, Gu@ { lim

f >u  HGu@  f >u@

(A.5)

H

Ho0

where H is a scalar. A more precise statement defining Gf is based on a choice of norm in the space of f:

lim Gf >u, Gu@ 

f >u  HGu@  f >u@ H

Ho0

(A.6)

0

For sufficiently well-behaved functionals, Gf will be a linear functional of its argument Gu, so that Gf >u, DGu@ DGf >u, Gu @ , for all scalars D, and Gf >u, Gu1  Gu2 @ Gf >u, Gu1 @  Gf >u, Gu2 @ , for all Gu1 and Gu2 . In this case, the functional Gf may be presented as the operation of a linear operator f c>u @ on the function Gu, which we write in the following way: Gf >u, Gu @

f c>u @, Gu

f u >u @, Gu

(A.7)

Note that the above expression does not represent simply the inner product of f c>u @ and Gu, although in certain simple cases, it does take this form. The linear operator f c>u @ above is known as the Gateaux derivative of the functional f. An alternative basic definition for the generalised derivative of f (the Frechet derivative) requires that f c be that linear operator satisfying lim

Gu o0

f >u  Gu @  f >u @  f c>u @, Gu Gu

0

(A.8)

where the norms are defined in some appropriate way. It can be shown that the Gateaux and Frechet definitions are equivalent when the linear operator f c is continuous in the function u, and so we shall simply refer to f c in this book as the Frechet derivative. It is not essential to retain the variational notation in the definition of the Frechet derivative; therefore, a variation Gu can be replaced by any fixed v. Furthermore, when this variation is replaced by the differential du, the resulting functional df >u, du@ f c>u@, du will be referred to as the Frechet differential. Frechet derivatives are used in this book to define Legendre transformations of functionals (see Appendix C), and Frechet differentials are used in deriving the incremental response of material behaviour.

A.4 Selected Results

309

A.4 Selected Results A.4.1 Frechet Derivatives of Integrals Although the Frechet derivative is defined for general functionals, here we are interested principally in functionals of the form, f >uˆ @

³ fˆ uˆ  K , K wˆ  K dK

(A.9)

8

where Y is the domain of K and fˆ is a continuously differentiable function of the variable uˆ , which is in turn a function of K. Here and in the main text, we adopt the convention that any quantity which is a function of K is denoted by a ‘hat’ notation, e. g. wˆ . Then, according to definitions (A.7) and (A.8), one can show that the Frechet differential of the functional (A.9) is given by

df >uˆ, duˆ @

wfˆ  uˆ, K ³ wuˆ duˆ  K wˆ  K dK

f c>uˆ @, duˆ

(A.10)

8

Now consider a more general case in which there are several independent variables: f >uˆ1 !uˆN @ ³ fˆ  uˆ1  K !uˆN  K , K wˆ  K dK , where fˆ is a continu8

ously differentiable function of functional variables uˆi , i 1!N . When the variable uˆ in (A.7) and (A.8) is identified as the full N-dimensional space of functions uˆi  K , then the Frechet differential is

df >uˆ, duˆ @

N

§ wfˆ  uˆ1 !uˆN , K

³ ¦ ¨¨

f c>uˆ @, duˆ

wui

8 i 1©

· duˆi  K ¸ wˆ  K dK ¸ ¹

(A.11)

For the definition of partial Legendre transformations, the variable uˆ in definition (A.7) and (A.8) is identified as an n-dimensional subspace of the full Ndimensional space of functions uˆi  K . In this case, the corresponding Frechet derivative is given by the following operator f c , which is linear in any integrable functions vˆi  K :

f c>uˆ @, vˆ

n

§ wfˆ  uˆ1 !uˆN , K

³ ¦ ¨¨

8 i 1©

wuˆi

· vˆi  K ¸ wˆ  K dK ¸ ¹

(A.12)

310

Appendix A Functions, Functionals and their Derivatives

A.4.2 Frechet Derivatives of Integrals Containing Differential Terms Now consider the case where the integral contains differential terms. It is b duˆ ˆ dK with straightforward to show that the Frechet differential of f >u@ dK

³

b

b

Frechet differential of f >uˆ @ b

a 2ˆ

d u

³ dK2 dK with respect to the function uˆ is given by

b

d 2vˆ

ˆ

dv b ³ dK dK >vˆ@a . Similarly, the

f c>uˆ @, vˆ

respect to the function uˆ is given by

a

a

ª dvˆ º f c>uˆ @, vˆ dK « » . 2 dK ¬ dK ¼ a a It is straightforward to show that the Frechet differential of 2 b 1 § duˆ · f >uˆ @ ³ ¨ ¸ dK with respect to the function uˆ is given by 2 dK ¹ a ©

³

b

f c>uˆ @, vˆ

duˆ dvˆ

³ dK dK dK . Integrating by parts, we obtain

a

b

f c>uˆ @, vˆ

b

ª duˆ º d 2uˆ ˆ ˆ K v  vd « dK » ¬ ¼ a a dK2

(A.13)

³

The latter form proves convenient in some applications. 2

b

Similarly, the Frechet differential of f >uˆ @ b

function uˆ is given by f c>uˆ @, vˆ

1 § d 2uˆ · ³ 2 ¨¨© dK2 ¸¸¹ dK with respect to the a

2ˆ

2ˆ

d ud v

³ dK2 dK2 dK . Integrating by parts twice, we a

obtain b

f c>uˆ @, vˆ

b

ª d 2uˆ dvˆ º ª d 3uˆ º b d 4 uˆ ˆ K « 2 »  « 3 vˆ »  ³ 4 vd ¬« dK dK »¼a «¬ dK »¼ a a dK

(A.14)

where again this latter form again proves convenient in some applications.

Appendix B Tensors

B.1 Tensor Definitions and Identities A second-order Cartesian tensor a is also written aij or in matrix form for ª a11 a12 three dimensions as ««a21 a22 «¬a31 a32

a13 º a23 »» . We use the summation convention over a33 »¼ 3

a repeated index; thus, for instance, akk

a11  a22  a33 and aij a jk

¦ aij a jk . j 1

Alternatively, we may write aij a jk defined by Gij transpose aij

j ; Gij

1, i

2

a . The unit tensor, or Kronecker G is

0, i z j . A tensor is symmetric if it is equal to its

a ji and is antisymmetric, or skew-symmetric, if aij

inverse of a tensor or matrix is defined by aij a jk1

a ji . The

Gik , and a tensor or matrix is

orthogonal if its inverse is equal to its transpose aij a jk

Gik .

The tensor has principal values a1 , a2 , a3 , which are the eigenvalues of the matrix, and are the solutions of the cubic equation, a3  a2 I1  aI 2  I3

(B.1)

0

where I1 , I 2 , I3 are the invariants of the tensor, which are I1 I2

aii

tr  a a1  a2  a3

1 1 2 tr a 2  tr  a aij a ji  aii a jj 2 2 a1a2  a2a3  a3a1





 

(B.2)



(B.3)

312

Appendix B Tensors

I3

1 2aij a jk aki  3aij a ji akk  aii a jj akk 6 1 3 2tr a 3  3tr a 2 tr  a  tr  a det  a a1a2a3 6





 





(B.4)

Note that some authors define I 2 with the opposite sign, but we prefer the notation used here; otherwise, J 2 (see below), which plays a major role in the analysis of shear behaviour, is always negative. The traces of the powers may alternatively be chosen as defining the three invariants, tr  a aii



tr a 2



tr a 3

aij a ji

aij a jk aki

a1  a2  a3 a12  a22  a32

a13  a23  a33

(B.5)

I1 2I 2  I12

(B.6)

3I 3  3I 2 I1  I13

(B.7)

The deviator of a tensor is defined as follows: aijc

so that I1c

1 aij  akk Gij 3

(B.8)

tr  a c 0 . The second and third invariants of the deviator are also

often required and may be written in a variety of forms: I 2c

1 1 aij a ji  aii a jj 2 6 1§ 1 1 2 · 2 2 ¨ tr a  tr  a ¸ I 2  I1 2© 3 3 ¹ 1 2 a1  a22  a32  a1a2  a2a3  a3a1 3 1  a1  a2 2   a2  a3 2   a3  a1 2 6 J2

1 aijc acji 2









I3c

1 c aijc acjk aki 3

J3

(B.9)



1§ 2 · ¨ aij a jk aki  aij a ji akk  aii a jj akk ¸ 3© 9 ¹

1§ 1 2 1 2 3 3· 3 2 ¨ tr a  tr a tr  a  tr  a ¸ I3  I 2 I1  I1 3© 3 9 3 27 ¹ 1  2a1  a2  a3  2a2  a3  a1  2a3  a1  a2 27 1 2 a13  a23  a33 27



^







3 a12  a2  a3  a22  a3  a1  a32  a1  a2  12a1a2a3

`

(B.10)

B.2 Mixed Invariants

313

B.2 Mixed Invariants The four mixed invariants of two tensors can be written as: tr  ab aij b ji



tr a 2 b

aij a jk bki

a1b1  a2b2  a3b3 a12b1  a22b2  a32b3

 aijb jkbki a1b12  a2b22  a3b32 tr  a 2 b2 aij a jk bkl bli a12b12  a22b22  a32b32 tr ab2

(B.11) (B.12) (B.13) (B.14)

where the forms expressed in terms of the principal values only apply if the principal axes coincide for the two tensors. Thus for two tensors, there are 10 invariants, three for each tensor alone and four mixed invariants.

B.2.1 Differentials of Invariants of Tensors Since the various potentials used in this book are most often written in terms of invariants and then are differentiated to obtain the constitutive behaviour, it is convenient to note the differentials of tensors and their invariants given in Table B.1.

314

Appendix B Tensors

Table B.1. Differentials of functions of tensors and their invariants

f

df daij

akl

Gki Glj

c akl

1 Gki Glj  Gij Gkl 3

I1

Gij

I2

a ji  I1Gij

I3

a jk aki  a ji I1  I 2 Gij

J2

acji

1 a ji  I1Gij 3

J3

c  J 2 Gij acjk aki

tr  a

Gij

 tr  a 3

2a ji

tr a 2

3a jk aki

tr  ab

b ji

 tr  b2a tr  a 2 b2

2a jk bki

tr a 2 b

b jk bki 2a jk bkl bli

Appendix C Legendre Transformations

C.1 Introduction The Legendre transformation is one of the most useful in applied mathematics, although its role is not always explicitly recognised. Well-known examples include the relation between the Lagrangian and Hamiltonian functions in analytical mechanics, between strain energy and complementary energy in elasticity theory, between the various potentials that occur in thermodynamics, and between the physical and hodograph planes occurring in the theories of the flow of compressible fluids and perfectly plastic solids. The Legendre transformation plays a central role in the general theory of complementary variational and extremum principles. Sewell (1987) presents a comprehensive account of the theory from this viewpoint with particular emphasis on singular points. These transformations have also been widely employed in rate formulations of elastic/plastic materials to transfer between stress-rate and deformation-rate potentials, e. g. Hill (1959, 1978, 1987); Sewell (1987). These applications are rather different from those used in this book. We review therefore those basic properties of the transformation that are needed in the main text.

C.2 Geometrical Representation in (n + 1)-dimensional Space A function Z X(xi ) , i 1!n , defines a surface * in  n  1 -dimensional  Z , xi space. However, the same surface can be regarded as the envelope of tangent hyperplanes. One way of describing the Legendre transformation is that it allows one to construct the functional representation that describes Z in terms of these tangent hyperplanes. This relationship is a well-known duality in geometry. The gradients of the function X  xi are denoted by yi : yi

wX wxi

(C.1)

316

Appendix C Legendre Transformations

*

Z

Tangent hyperplane P(X, xi)

X Q(-Y, 0i) -Y xi Figure C.1. Representation of * in (n + 1)-dimensional space

so that the normal to * in the  n  1 -dimensional space is  1, yi . If the tangent hyperplane at the point P  X , xi on * cuts the Z axis at Q  Y ,0i , the vector  X  Y , xi lies in the tangent hyperplane (Figure C.1), and hence is orthogonal to the normal to * at P. Forming the scalar product of these two vectors therefore leads to

X  xi  Y  yi xi yi

(C.2)

The function Z Y  yi defines the family of enveloping tangent hyperplanes and hence is the required dual description of the surface *. The form of this function can be found by eliminating the n variables xi from the  n  1 equations in (C.1) and (C.2). This can be achieved locally, provided that (C.1) can be inverted and solved for the xi 's, i. e. provided the Hessian matrix wyi w 2 X , is non-singular. Points at which the determinant of the Hessian wx j wxi wx j matrix vanishes are singularities of the transformation (Sewell, 1987). Differentiating (C.2) at a non-singular point with respect to yi gives

wX wx j wY  wx j wyi wyi

yj

wx j wyi

 xi

(C.3)

C.3 Geometrical Representation in n-dimensional Space

317

which, by virtue of (C.1) reduces to xi

wY wyi

(C.4)

Relations (C.1)–(C.3) define the Legendre transformation. This transformation is self-dual because, if the function Z Y  yi is used to define a surface *c “pointwise” in  Z , yi space, then Z X  xi describes the same surface *c “planewise” because  1, xi define the normal to *c and X is the intercept of the tangent plane with the Z axis from (C.2). The transformation is not in general straightforward to perform analytically. An exception is when X(xi ) is a quadratic form, X(xi ) 12 Aij xi x j , where Aij is a non-singular, symmetrical matrix. Hence, the dual variables are wX yi Aij x j , so that xi Aij1 y j , and the Legendre dual is also a quadratic wxi form: Y  yi xi yi  X  xi Aij1 yi y j  12 Aij1 yi y j

The transformation in general is succinctly written wX yi wxi X  xi  Y  yi xi yi xi

wY wyi

1 A1 y y 2 ij i j

(C.5)

(C.1)bis (C.2)bis (C.4)bis

The choice of the sign of the dual function is somewhat arbitrary, and Y is sometimes written instead of Y. The choice is usually governed by physical considerations.

C.3 Geometrical Representation in n-dimensional Space An alternative geometrical visualisation in n-dimensional space is also valuable in gaining understanding of formal results. For fixed C but variable xi , the relation I  xi , yi { X  xi  xi yi  C 0

(C.6)

defines a family of hyperplanes in n-dimensional yi space. These hyperplanes envelope a surface in this space, the equation of which is obtained by eliminating the xi between (C.4) and wI wX (C.7)  yi 0 wxi wxi

318

Appendix C Legendre Transformations

On comparison with (C.1) and (C.2), it follows that the equation of this surface is Y  yi C , so that the hyperplanes defined by (C.4) envelope the level surfaces of the dual function Y. Dually, the hyperplanes defined by (C.8)

\  xi , yi { Y  yi  xi yi  C

envelope the level surfaces of X  xi in xi space. These level surfaces are, of course, the “cross sections” of the  n  1 -dimensional surfaces Z  X  xi and Z  Y  yi discussed above.

C.4 Homogeneous Functions Of particular importance in applications in continuum mechanics are cases where the function Z X  xi is homogeneous of degree p in the xi 's, so that X  Oxi OpX  xi for any scalar O. From Euler's theorem for such functions, it follows that pX  xi xi

wX wxi

xi yi

(C.9)

wY wyi

(C.10)

so that from (C.2), qY  yi xi yi

yi

1 1 1 , so that the Legendre dual Y  yi is necessarily homogeneous  p q p . of degree q p 1 In the example above, p 2 , so that X and Y are both homogeneous of degree two. A familiar example of this situation is in linear elasticity where the elastic strain energy E Hij and the complementary energy C Vij are both quadratic

where





functions of their argument and satisfy the fundamental relation,

 

E Hij  C Vij

Vij Hij

(C.11)

Another case of particular importance in rate-independent plasticity theory occurs when X is homogeneous and of degree one, so that X  xi xi yi , in which case the dual function Y  yi is identically zero from (C.2). There is a simple geometric interpretation of this far-reaching result. Since X  Oxi OX  xi , the  n  1 -dimensional surface Z X  xi is a hypercone with its vertex at the origin. Hence, all tangent hyperplanes meet the Z axis at Z 0 , so that Y  yi 0 for all yi . This special case is pursued further later,

C.5 Partial Legendre Transformations

319

and the terminology of convex analysis will prove particular useful in its treatment (see Appendix D).

C.5 Partial Legendre Transformations Now suppose that the functions depend on two families of variables, X  xi , Di say, where xi and Di are n- and m-dimensional vectors, respectively. We can perform the Legendre transformation with respect to the xi variables as above and obtain the dual function Y  yi , Di . The variables Di play a passive role in this transformation and are treated as constant parameters. Hence, the three basic equations are now X  xi , Di  Y  yi , Di xi yi wX wxi

yi

wY wyi

and xi

(C.12) (C.13)

If the derivatives of X with respect to the passive variables Di are denoted by Ei , then it follows from (C.12) that Ei

wX wDi



wY wDi

(C.14)

It is also possible, in general, to perform a Legendre transformation on X  xi , Di with respect to the Di variables and construct a second dual function V  xi , Ei with the properties, X  xi , Di  V  xi , Ei Di Ei

(C.15)

where wX , Di wDi

Ei

wV wEi

(C.16)

and furthermore: yi

wX wxi



wV wxi

(C.17)

since now the xi 's are the passive variables. This process can be continued. A Legendre transformation of Y  yi , Di with respect to the Di variables produces a fourth function W  yi , Ei . The same function is obtained by transforming V  xi , Ei with respect to the xi variables. A closed chain of transformation is hence produced as shown in Figure C.2, where the basic differential relations are summarised. The best known example

320

Appendix C Legendre Transformations

X ( xi , D i ) wX , Ei wxi

yi

X Y

wX wDi

xi yi

VX

Y ( yi , D i ) xi

wY , Ei wyi

V ( xi , Ei )

wY  wDi

Y W

D i Ei

X Y W V

0

yi W V

 D i Ei



wV , Di wxi

wV wEi

 xi yi

W ( yi , Ei ) xi



wW , Di wxi



wW wEi

Figure C.2. Chain of four partial Legendre transformations

of such a closed chain of transformations is in classical thermodynamics, where the four functions are the internal energy u  s, v , the Helmholtz free energy f  T, v , the Gibbs free energy g  T, p , and the enthalpy h  s, p , where T , s, v, and p are the temperature, entropy, specific volume, and pressure respectively, e. g. Callen (1960). Other examples are given by Sewell (1987).

C.6 The Singular Transformation When X is homogeneous of order one in xi , so that OX  xi , Di X  Oxi , Di , the value of yi wX / wxi is unaffected by the transformation xi o Oxi , and so the mapping from xi o yi is f o 1 . Furthermore, since xi yi

(C.18)

X  xi , Di

the dual function Y  yi , Di is identically zero, as already noted above, and so dY

wY wY dyi  dD i wyi wDi

0

(C.19)

But also from (C.13), xi dyi  yi dxi

wX wX dxi  dD i wxi wDi

(C.20)

C.7 Legendre Transformations of Functionals

321

which by virtue of (C.1) reduces to xi dyi 

wX d Di wDi

(C.21)

0

Hence, by comparing (C.19) with (C.21), it follows that xi

O

wY wyi

and  wX wDi

O

wY wDi

(C.22)

where O is an undetermined scalar, reflecting the non-unique nature of this singular transformation. The above development is classical in the sense that all the functions are assumed to be sufficiently smooth for all derivatives to exist. In practice, the surfaces encountered in plasticity theory, on occasion, contain flats, edges, and corners. Such surfaces and the functions defining them can be included in the general theory using some of the concepts of convex analysis. In particular, the commonly defined derivative is replaced by the concept of a “subdifferential”, and the simple Legendre transformation is generalised to the “Legendre-Fenchel transformation” or “Fenchel dual”. For simplicity of presentation, we have so far used the classical notation, and convex analysis is introduced in Appendix D. Treatments of the mechanics of elastic/plastic materials that use convex analysis notation may be found in Maugin (1992), Reddy and Martin (1994), and notably Han and Reddy (1999). Because our main concern here is to exhibit the overall structure of the theory as it affects the developments of constitutive laws, we have not highlighted the behaviour of any convexity properties of the various functions under the transformations. These considerations are very important for questions of uniqueness, stability, and the proof of extremum principles, which are beyond the scope of this book, but are fruitful areas for future research. Some of these aspects of Legendre transformations are considered at length in the book by Sewell (1987).

C.7 Legendre Transformations of Functionals C.7.1 Integral Functional of a Single Function Consider a functional, X > xˆ @

³ Xˆ  xˆ  K , K wˆ  K dK

(C.23)

8

where Y is the domain of K and Xˆ is a continuously differentiable function of a functional variable xˆ .

322

Appendix C Legendre Transformations

If yˆ  K

wXˆ  xˆ  K , K , then the Legendre transform of the function Xˆ is wxˆ  K

Yˆ  yˆ  K , K xˆ  K yˆ  K  Xˆ  xˆ  K , K

(C.24)

It

follows from the standard properties of the transform that wYˆ  yˆ  K , K xˆ  K . wyˆ  K The functional defined by (C.25) Y > yˆ @ ³ Yˆ  yˆ  K , K wˆ  K dK ³ xˆ  K yˆ  K wˆ  K dK  X > xˆ @ 8

8

may then be considered the Legendre transform of the original functional, and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions.

C.7.2 Integral Functional of Multiple Functions A case of interest in the present work is a Legendre transform of a functional of the form, X > xˆ, uˆ @

³ Xˆ  xˆ  K , uˆ  K , K wˆ  K dK

(C.26)

8

where Xˆ is a continuously differentiable function of the variables xˆ  K and uˆ  K . wXˆ  xˆ  K , uˆ  K , K , the Legendre transform of the function Denoting yˆ  K wxˆ  K Xˆ with respect to the variable xˆ  K is defined as

Yˆ  yˆ , uˆ, K xˆ  K yˆ  K  Xˆ  xˆ  K , uˆ  K , K From the standard properties of the transform, it follows that wYˆ  yˆ  K , uˆ  K , K xˆ  K w yˆ  K

wYˆ  yˆ  K , uˆ  K , K w uˆ  K



wXˆ  xˆ  K , uˆ  K , K w uˆ  K

(C.27)

(C.28) (C.29)

Then, the Legendre transformation of functional (C.26) in function xˆ , where function uˆ is a passive variable, is given by the functional, Y > yˆ , uˆ @

³ Yˆ  yˆ  K , uˆ  K , K wˆ  K dK ³ xˆ  K yˆ  K wˆ  K dK  X > xˆ, uˆ @

8

8

(C.30)

C.7 Legendre Transformations of Functionals

323

and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions. When xˆ is not a function but a variable, denoted x, all above equations are valid, except that Equation (C.30) may be rewritten as Y > y , uˆ @

³ Yˆ  y, uˆ  K , K wˆ  K dK

xy  X > x, uˆ @

(C.31)

8

where y

³

wXˆ  x, u  K , K

8

wx

(C.32)

wˆ  K dK

When the function Xˆ is a continuously differentiable function of the function xˆ (or variable x) and any finite number N of functions uˆi , the same Equations (C.27)–(C.30) are still valid, except that Equation (C.29) unfolds into N equations: wYˆ  yˆ , uˆ1 !uˆN , K wXˆ  xˆ, uˆ1 ! yˆ N , uˆ, K (C.33)  i 1! N , w uˆi w uˆi

C.7.3 The Singular Transformation An important case in rate-independent plasticity theory occurs when functional Xˆ  xˆ, uˆ, K in (C.26) is homogeneous of degree one in, say, xˆ  K :

Xˆ  Oxˆ  K , uˆ  K , K O Xˆ  xˆ  K , uˆ  K , K From Euler’s theorem, it follows that wXˆ  xˆ  K , uˆ  K , K Xˆ  xˆ  K , uˆ  K , K xˆ  K wxˆ  K

yˆ  K xˆ  K

(C.34)

(C.35)

Then the Legendre transformation of the function Xˆ  xˆ  K , uˆ  K , K with respect to uˆ  K , when other variables and functions are passive, is defined by Equation (C.27), so that after substitution of (C.35), we obtain (C.36) Yˆ  yˆ  K , uˆ  K , K xˆ  K yˆ  K  Xˆ  xˆ  K , uˆ  K , K { 0 The properties of this transformation are wYˆ  yˆ  K , xˆ  K , K xˆ  K Oˆ  K wyˆ  K

wXˆ  xˆ  K , uˆ  K , K wuˆ  K

wYˆ  yˆ  K , uˆ  K , K Oˆ  K wuˆ  K

(C.37) (C.38)

324

Appendix C Legendre Transformations

where Oˆ  K is an undetermined scalar, reflecting the non-unique nature of this singular transformation. Then the Legendre transformation of functional (C.26) in function xˆ , when function uˆ is a passive variable, is given by the functional,

Y > yˆ , uˆ @

³ Yˆ  yˆ  K ,uˆ  K , K wˆ  K dK ³ xˆ  K yˆ  K wˆ  K dK  X > xˆ,uˆ@ { 0

8

(C.39)

8

and using definitions of Appendix A, it can be confirmed that this definition satisfies the appropriate differential conditions.

Appendix D Convex Analysis

D.1 Introduction The terminology of convex analysis allows a number of the issues relating to hyperplastic materials to be expressed succinctly. In particular, through the definition of the subdifferential, it allows rigorous treatment of functions with singularities of various sorts. These arise, for instance, in the treatment of the yield function. A brief summary of some basic concepts of convex analysis is given here. The terminology is based chiefly on that of Han and Reddy (1999). A more detailed introduction to the subject is given by Rockafellar (1970). No attempt is made to provide rigorous, comprehensive definitions here. For a fuller treatment, reference should be made to the above texts. Although it is currently used by only a minority of those studying plasticity, it seems likely that in time convex analysis will become the standard paradigm for expressing plasticity theory.

D.2 Some Terminology of Sets We use brackets ^ ` to indicate a set, so that ^0, 1, 3.5` is simply a set containing the numbers 0, 1 and 3.5. A closed set containing a range of numbers is denoted by > , @ , thus >a, b @ ^x a d x d b` , where the meaning of the contents of the final bracket is “x, such that a d x d b ”. We use ‡ to denote the null (empty) set. In the following, C is a subset in a normed vector space V (in simple terms a space in which a measure of distance is defined), usually with the dimension of Rn (with n finite), but possibly infinite dimensional. The notation , is used for an inner product, or more generally the action of a linear operator on a function. The space V c is the space dual to V under the inner product x*, x , so

326

Appendix D Convex Analysis

that x V and x* V c . More generally, V c is termed the topological dual space of V (the space of linear functionals on V). The operation of summation of two sets, illustrated in Figure D.1a, is defined by

C1  C2

^x1  x2

x1  C1 , x2  C2 `

(D.1)

The operation of scalar multiplication of a set, illustrated in Figure D.1b is defined by

OC

^Ox

x  C`

(D.2)

It is also convenient to define the operation of multiplication of a set C by a set S of scalars:

SC

^Ox

O  S, x  C`

(D.3)

The definitions of the interior and boundary of a set are intuitively simple concepts, but their formal definitions depend first on the definition of distance. In Rn , the Euclidian distance is defined as d  x, y

x y

x  y, x  y

12

(D.4)

and we define the open ball of radius r and centered at xo as

B  xo , r

^x

d  x, xo  r`

(D.5)

The interior of C is then defined as

int C

^x

H ! 0, B  x, H  C`

(D.6)

x2 C C

x1

(a)

(b)

Figure D.1. (a) Summation of sets; (b) scalar multiple of a set

D.3 Convex Sets and Functions

327

This means that there exists some H (possibly very small) so that a ball of radius H is entirely contained in C. The closure of C is defined as the intersection of all sets obtained by adding a ball of non-zero radius to C:

cl C ^C  B  0, H H ! 0,`

(D.7)

Finally, the boundary of C is that part of the closure of C that is not interior:

bdy C cl C \ int C

(D.8)

D.3 Convex Sets and Functions A set C is convex if and only if

1  O x  Oy  C , x, y  C , 0  O  1

(D.9)

where, for instance, x, y  C means “for all x and y belonging to C”. Simple examples of convex and non-convex sets in two-dimensional space are given in Figure D.2. A function f whose domain is a convex subset C of V and whose range is real or rf is convex if and only if

f  1  O x  Oy d 1  O f  x  Of  y , x, y  C , 0  O  1

(D.10)

This is illustrated for a function of a single variable in Figure D.3. Convexity requires that NP d NQ for all N between X and Y. This property has to be true for all pairs of X,Y within the domain of the function. A function is strictly convex if d can be replaced by < in (D.10) for all x z y . The effective domain of a function is defined as the part of the domain for which the function is not; thus, dom  f  x ^x V f  x  f` .

Figure D.2. Non-convex and convex functions

328

Appendix D Convex Analysis

z = f(x)

z

Q P

x + (1-O)y = Ox + (1-O)y

X

N (1-O)

x

Y O

Figure D.3. Graph of a convex function of one variable

D.4 Subdifferentials and Subgradients The concept of the subdifferential of a convex function is a generalisation of the concept of differentiation. It allows the process of differentiation to be extended to convex functions that are not smooth (i. e. continuous and differentiable in the conventional sense to any required degree). If V is a vector space and V c is its dual under the inner product , , then x* V c is said to be a subgradient of the function f  x , x V , if and only if f  y  f  x t x*,  y  x , y . The subdifferential, denoted by wf  x , is the subset of V c consisting of all vectors x * satisfying the definition of the subgradient:

wf  x

^x *  V c

`

f  y  f  x t x*,  y  x , y

(D.11)

For a function of one variable, the subdifferential is the set of the slopes of lines passing through a point on the graph of the function, but lying entirely on or below the graph. The concept is illustrated in Figure D.4. The concept of the subdifferential allows us to define “derivatives” of nondifferentiable functions. For example, the subdifferential of w x is the signum function, which we now define as a set-valued function:

S  x ww  x

x0 ­ ^1` , ° ®> 1, 1@, x 0 ° ^1` , x!0 ¯

(D.12)

D.5 Functions Defined for Convex Sets

329

w w = f(x)

P

x Figure D.4. Subgradients of a function at a non-smooth point

Thus at a point x, wf  x may be a set consisting of a single number equal to wf wx , or a set of numbers, or (in the case of a non-convex function) may be empty.

D.5 Functions Defined for Convex Sets The indicator function of a set C is a convex function defined by

­ 0, x  C IC  x ® ¯f, x  C

(D.13)

so that the indicator function is simply zero for any x that is a member of the set and f elsewhere. Although this appears at first sight to be a rather curious function, it proves to have many applications. In particular, it plays an important role in plasticity in that it is closely related to the yield function. The normal cone NC  x of a convex set C is the set-valued function defined by

NC  x

^x *  V c

x*,  y  x d 0, y  C

`

(D.14)

330

Appendix D Convex Analysis

It is straightforward to show that NC  x ^0` if x  int C (the point is in the interior of the set), that NC  x can be identified geometrically with the cone of normals to C at x if x  bdy C (the point is on the boundary of the set), and further that NC  x is empty if x  C (the point is outside the set). Furthermore, the subdifferential of an indicator function of any convex set is the normal cone of that set: wIC  x NC  x . Another important function defined for a convex set is the gauge function or Minkowski function, defined for a set C as JC  x inf ^P t 0 x PC`

(D.15)

where inf ^x` denotes the infimum, or lowest value of a set. In other words, JC  x is the smallest positive factor by which the set can be scaled, and x is a member of the scaled set. The meaning is most easily understood for sets that contain the origin (which proves to be the case for all sets of interest in hyperplasticity). In the following, we shall therefore assume that C is convex and contains the origin. It is straightforward to see in this case that JC  x 1 for any point on the boundary of the set, is less than unity for a point inside the set, and is greater than unity for a point outside the set. At the origin, JC  0 0 . In the context of (hyper)plasticity, it is immediately obvious that the gauge may be related to the conventional yield function. If the set C is the set of (generalised) stresses F that are accessible for any given state of the internal variables (the elastic region), then the yield function is a function conventionally taken as zero at the boundary of this set (the yield surface), negative within, and positive without. One possible expression for the yield function would therefore be y  F JC  F  1 . Other functions could of course be chosen as the yield function, but this is perhaps the most rational choice; so we follow Han and Reddy (1999) in calling this the canonical yield function. To emphasise the case where the yield surface is written in this way, we shall give it the special notation y  F JC  F  1 . The gauge function is always homogeneous of order one in its argument x, so that JC  Ox OJC  x . (In the language of convex analysis, such functions are simply referred to as positively homogeneous.) The canonical yield function is therefore conveniently written in the form of a positively homogeneous function of the (generalised) stresses, minus unity. It is also clear that the gauge of the set of accessible generalised stresses contains exactly the same information as the yield function, canonical or otherwise, and there may be benefits from specifying the gauge rather than the yield function. It is useful to note that at the boundary of C, the normal cone can also be written NC  x wIC  x OwJC  x , 0 d O d f . This proves to be a convenient form that allows the normal cone to be expressed in terms of the subdifferential of the gauge function and therefore (in hyperplasticity), of the canonical yield function.

D.6 Legendre-Fenchel Transformation

331

It is straightforward to see that the definition (D.15) can be inverted. Given a positively homogeneous function f  x , one can define a set C, such that f  x is the gauge function of C:

C

^x

`

f x d 1

(D.16)

which has the property that JC  x f  x . It is worth noting, too, that the indicator function of a set containing the origin can always be expressed in the following form, and this proves useful in the application of convex analysis to (hyper)plasticity: IC  x I> f,0@  JC  x  1

(D.17)

The function I> f,0@  x is simply zero for all non-positive values of x and f for positive values.

D.6 Legendre-Fenchel Transformation The Legendre-Fenchel transformation (often simply called the Fenchel dual, or conjugate function) is a generalisation of the concept of the Legendre transformation. If f  x is a convex function defined for all x V , its Legendre-Fenchel transformation is f *  x * , where x* V c is defined by

f *  x * sup^ x*, x  f  x `

(D.18)

xV

where sup means the supremum, or highest value for any x V . xV

It is straightforward to show the Fenchel dual is a generalisation of the Legendre transform. We use the notation that if x* wf  x and f *  x * is the Fenchel dual of f  x , then x wf *  x * . Some useful Fenchel duals, together with their subdifferentials are given in Table D.1. The dual of the sum of two functions involves the process of infimal convolution:

f

† g  x

inf ^ f  x  y  g  y `

(D.19)

yV

Table D.1 also includes some special functions that are defined in Section D.10.

332

Appendix D Convex Analysis

Table D.1. Some Fenchel duals and their subdifferentials

f x

wf  x

f *  x *

I^0`  x I> f,0@  x

N^0`  x N> f,0@  x

I> 1,1@  x

wf *  x *

0

^0`

I>0,f@  x *

N>0,f@  x *

N> 1,1@  x

x*

S x *

I>0,1@  x

N>0,1@  x

x*

H x *

pos  x

Hx

I> f,0@  x * 1

N> f,0@  x * 1

I^0`  x * I^1`  x *

N^0`  x * N^1`  x *

x2 2

^0` ^1` ^x`

x *2 2

^x *`

xn , n ! 1

^nxn1`

 n  1 §¨

exp x

x * log x * x *

f x  g x

^expx` wf  x  wg  x

 f * † g *  x *

f  ax

awf  ax

f * x * a

1

x

n n 1

x*· ¸ © n ¹

­°§ x * ·1 n 1 ½° ®¨ ¸ ¾ n ¯°© ¹ ¿°

^log x *` 1 wf *  x * a a

D.7 The Support Function The support function is also defined for a convex set. For a convex set C in V, if x* V c , then the support function is defined by VC  x * sup^ x, x *

x  C`

(D.20)

Note that although C is a set of values of the variable x, the argument of the support function is the variable x * conjugate to x. It can be shown that the support function is the Fenchel dual of the indicator function. The support function is always homogeneous of order one in x * , i. e. it is positively homogeneous. It follows that any homogeneous order-one function defines a set in dual space. In hyperplasticity, one can observe that the dissipation function is homogeneous and order one in the internal variable rates. It can thus be interpreted as a support function, and the set it defines in the dual space of generalised stresses is the set of accessible generalised stress states (the elastic region). The Fenchel dual of the dissipation function is the indicator function for this set of accessible states, which is zero throughout the set. We can identify this indicator function

D.7 The Support Function

333

with the Legendre transform w Oy of the dissipation function introduced in Chapter 4. Equation (D.20) can be inverted to obtain the set C from the support function. If f *  x * is a homogeneous first-order function in x * , then the set defined by solving the system of inequalities,

C

^x

`

x, x * d f *  x * , x *

(D.21)

satisfies the condition that VC  x * f *  x * . Application of (D.21), with f *  x * as the dissipation function, allows the set of accessible (generalised) stresses to be derived from the dissipation function in a systematic manner; hence the elastic region can be derived from the dissipation function. The subdifferential of the support function defines a set called the maximal responsive map (see Han and Reddy, 1999, although we depart from their notation here): PC  x * wVC  x *

(D.22)

The normal cone and the maximal responsive map are inverse in the sense that x PC  x * œ x*  NC  x

(D.23)

It also follows [see Lemma 4.2 of Han and Reddy (1999)] that C is simply related to the support function by the subdifferential at the origin, i. e. the maximal responsive map at the origin. Thus C PC  0 wVC  0

(D.24)

Both the gauge and support functions are positively homogeneous. Defining the domain of the support function S dom  VC  x * , it can be shown (see Han and Reddy, 1999) that JC  x

sup

x*, x

0 z x *S VC  x *

(D.25)

VCD . The process is and, defining the domain of the gauge function

and JC  x is called the polar of VC  x * , written JC symmetrical so that VC G dom  JC  x ,

JCD

VC  x *

sup

x*, x

0 z xG JC  x

(D.26)

Further, we have the following inequality: VC  x * JC  x t x*, x , x*  S, x  G

(D.27)

334

Appendix D Convex Analysis

and the equality holds for x wVC  x * : VC  x * JC  x

x*, x , x wVC  x * , x*  S

(D.28)

Application of (D.25), together with y  F JC  F  1 , allows the canonical yield function to be determined directly from the dissipation function.

D.8 Further Results in Convex Analysis Whilst the above are the most important results needed in Chapter 13, it is worth noting some further relationships between convex sets and functions. The polar Cq of a convex set C is defined as

^x *

Cq

VC  x * d 1`

(D.29)

in other words, it is the set for which the support function of C is the gauge. The polar f q of a non-negative convex function f, which is zero at the origin, is defined by

f q  x * inf ^P t 0

x, x * d 1  Pf  x , x`

(D.30)

and it can be shown that Equations (D.25) and (D.26) can be derived from this more general relationship. Finally, the indicator and the gauge of a convex set are said to be obverse to each other, where the obverse g of a function f is defined by

g  x inf ^O ! 0



 f O  x d 1`



(D.31)

and the operation  f O  x Of O 1x is called right scalar multiplication [note that  f 0  x I^0`  x if f  x z f and  f 0  x

f  x if f  x f ].

D.9 Summary of Results for Plasticity Theory In summary, we have the following concepts from convex analysis which are of relevance in plasticity theory: x A convex set C in V. x The indicator function IC  x of the set. x The gauge function JC  x of the set. x The support function VC  x * , which is the Fenchel dual of the indicator, and is also the polar of the gauge function. x The normal cone N C  x wIC  x which is a set in V c which is the subdifferential of the indicator function.

D.9 Summary of Results for Plasticity Theory

335

x The maximal responsive map, which is the subdifferential of the support function PC  x * wVC  x * . x The set C is the subdifferential of the support function at the origin C PC  0 wVC  0 . The relationships among these quantities are illustrated in Figure D.5 for the simple one-dimensional set C > a, b @ . The roles of these concepts in hyperplasticity are explored in more detail in Chapter 11, but Table D.2 gives the correspondences among some concepts in conventional plasticity theory and in the convex analytical approach. Table D.2 Correspondences between conventional plasticity theory and the convex analytical approach

Conventional plasticity theory

Convex analytical approach to (hyper)plasticity

Elastic region

A convex set in (generalised) stress space.

Yield surface

The indicator function or (for some purposes) the gauge function, or equivalently the canonical yield function.

Plastic potential and flow rule

Gauge function (or equivalently the canonical yield function) and the normal cone.

Plastic work

Support function (equal to the dissipation function). NB: For models in which energy can be stored through plastic straining, this is not equal to plastic work.

(No equivalents in conventional theory)

The indicator of the elastic region and the support function (dissipation) are Fenchel duals. The gauge function of the elastic region and the support function (dissipation) are polars.

336

Appendix D Convex Analysis

C

> a, b@

Cq

ICq  x *

J C x

V C x *

I C x

x*  wI C  x

> 1 a , 1 b@

x  wV C  x * P C x *

N C x

Figure D.5. Relationships among functions of a convex set in one dimension

D.10 Some Special Functions We have already introduced the signum function (D.12), which we treat as a setvalued function. Closely related is the Heaviside step function:

H x

x0 ­ ^0` , 1 °  S  x  ^1` ®>0, 1@, x 0 2 ° ^1` , x ! 0 ¯

(D.32)

D.10 Some Special Functions

337

It is also useful to define a closely related set-valued function:

H x

x0 ­ ‡, ° ®> f, 1@, x 0 ° ^1` , x !0 ¯

(D.33)

H  x , which can also be written as N>0,f@  x , is useful because it is the subdifferential of the positive values of x, defined as

­f, x  0 pos  x ® ¯ x, x t 0

(D.34)

Note that careful distinction is needed among pos  x , the absolute value x , and the Macaulay bracket x . Note that all of these functions are convex.

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Vaid, Y.P. and Campanella, R.G. (1977) Time dependent behaviour of undisturbed clay, J. Geotech. Eng. Div., ASCE, Vol. 103, No. 7, 693709 Valanis, K.C. (1975) On the foundations of endochronic theory of viscoplasticity, Arch. of Mech., Vol. 27, 857868 Whittle, A.J. (1993) Evaluation of a constitutive model for overconsolidated clays, Géotechnique, Vol. 43, No. 2, 289313 Ziegler, H. (1959) A modification of Prager’s hardening rule, Q. Appl. Mech., Vol. 17, 5565 Ziegler, H. (1977, 1983) An introduction to thermomechanics, North Holland, Amsterdam (2nd ed. 1983) Zienciewicz, O.C. (1977) The finite element method, 3rd ed., McGraw-Hill, London Zytynski, M., Randolph, M.F., Nova, R., and Wroth, C.P. (1978) On modelling the unloadingreloading behaviour of soils, Int. J. Num. Anal. Methods Geomechanics, Vol. 2, 8793

Index

A adiabatic 43–46, 49, 66, 67, 79, 256 advanced plasticity models 118 anisotropic elasticity 172 anisotropy 120, 163, 164, 169, 172, 210, 340 associated flow rule 18, 21, 28, 29, 32, 58, 69, 74, 83, 86, 96, 100, 108–114, 118, 130, 155, 176–179, 186, 209, 210, 261, 342 B back stress 27, 68, 89, 97, 108, 123, 130, 149, 153, 179, 180, 184, 207, 209 back stress function 144 backbone curve 192, 232, 233 bar structure 279 bending moment 299, 300 bending stiffness 298 body force 9, 245, 246 bounding surface plasicity 105–110, 118 bulk modulus 14, 44, 79, 162, 190 C canonical yield function 17, 58, 268– 271, 275, 330, 334, 335 Cauchy small strain tensor 7 Cauchy stress 8, 11, 286 classical thermodynamics 35, 36, 47, 59, 256, 320 classical thermodynamics of fluids 40, 43, 48 Clausius-Duhem inequality 38, 161

cohesionless soils 209 cohesive material 270, 340 compatibility 8–11, 278, 280, 282 complementary energy 48, 77, 164, 168, 170, 172, 318 compliance matrix 22, 23, 49, 66, 171 conjugate variables 57 conservation of energy 37 consistency condition 20–25, 63, 91, 97, 109, 111, 115, 124, 145, 179, 197, 239 constitutive behaviour 10, 21, 42, 59, 69, 74, 75, 88, 91–93, 96–98, 102, 124, 125, 148, 149, 155, 180, 190, 195, 197, 254, 258, 262, 266, 302, 313 constitutive models 1–4, 8, 11, 19, 62, 65, 74, 79, 115, 117, 156, 182, 210, 220, 255, 256, 259, 262, 273, 304, 340– 343 constraints 1, 71, 73, 84, 205, 264–266, 270, 278–282, 302 continuous field of yield surfaces 119, 151, 155, 340 continuous hyperplastic model 142, 177, 191 continuous hyperplasticity 133, 146, 155, 203, 210, 224, 300, 341 continuous material memory 119 continuum mechanics 6, 8–11, 48, 243, 253, 260, 318, 342 contraction 18, 87 convective derivative 9, 242 convex analysis 4, 17, 217, 263–266, 271, 275, 304, 306, 321, 325, 330, 331, 334 convex function 265, 328–331, 334 convex sets 327, 334

346

Index

convexity 58, 321 coupled materials 32, 176 creep 211, 237, 238 creep rupture 238 critical state 28, 87, 186, 187, 191, 195, 203, 204, 209, 210, 340 cross-coupling 112, 180, 184 D damage mechanics 274 damage parameter 274, 277 Darcy’s law for fluid flow 254 decoupled materials 103 deformation gradient tensor 6 degenerate transform 57, 263 density 9, 47, 172, 242–244, 254, 286 deviatoric stress 78, 84, 94, 99, 129, 154, 159, 182, 200, 201 differential 10, 11, 20, 47, 57, 59, 68, 74, 88, 120, 138, 150, 181, 193, 216, 217, 225–227, 232, 233, 247, 250, 255, 264, 295, 296, 299, 300, 306–310, 313, 322– 324, 328 dilation 29, 73, 86, 205, 259, 261, 270, 271, 340 Dirac impulse function 306 displacement 6–11, 31, 114, 179, 257, 290–297 displacement gradient tensor 6 dissipation 3, 5, 30, 37, 40, 41, 50, 53– 62, 66–75, 82–103, 121–131, 136, 139, 143–145, 149, 153, 161, 176, 178, 183, 188–196, 204–217, 221, 223, 229, 241, 249, 251, 256, 257, 262, 268–271, 275, 278, 284, 285, 289, 293, 294, 297, 298, 301–304, 332–335, 339 dissipation functional 136, 138, 143, 144, 152, 178, 230 dissipative coupling 176 dissipative generalised stress 3, 54, 56, 75, 91, 94, 96, 99, 125, 129, 152, 178, 213, 261 dissipative generalised stress function 136, 143, 148, 191 dissipative materials 48, 54, 274, 304, 342 Drucker’s stability postulate 4 Drucker-Prager model 87, 261 dry density 243, 250 dummy subscripts 21

E effective angle of friction 259 effective stresses 159–162, 187 elastic material 13, 15, 133, 288, 339 elastic strains 16, 19, 20, 176 elasticity 11–16, 20, 77–80, 111, 162– 164, 209, 265, 277, 286, 288, 339, 340 elastic-viscoplastic model 216 elliptical yield surfaces 178, 186 end bearing 293–296 endochronic theory 117, 118, 343 energy functional 134, 193, 230 energy function 42, 45, 47, 49, 68, 80, 120, 137, 190, 215, 256, 298 entropy 36, 38, 40–46, 55, 65–70, 74, 75, 80, 248–251, 302, 320 entropy flux 54, 249, 301 equation of state 36, 45, 46 equations of motion 248, 254 equilibrium 8–11, 32, 36, 41, 51, 248, 278, 280, 295, 296 Euclidian distance 326 Euler’s theorem 56, 136, 212, 323 Euler-Almansi tensor 7 Eulerian formulation 7, 242, 250 evolution equations 19, 50, 53, 65, 142, 230 extensive quantities 40, 47, 242, 251 extremum principles 2, 303, 321 F Fenchel dual 265, 266, 275, 304, 321, 331–335 fibre-reinforced material 288 finite element 2, 62, 112, 138, 172, 230, 343 First Law 36, 38, 246 flexible pile 294, 298 flow potential 213, 215, 218, 219, 222, 224, 230, 233, 270, 277, 303 flow potential functional 230, 234 flow rule 17, 22, 24, 29, 57, 63, 83–86, 89, 94, 99, 101, 103, 111, 122, 143– 146, 179, 210, 261, 335 fluid 40, 242–248, 253–262, 315 fluxes 241–243, 250–254, 262 force potential 213, 215, 218, 219, 223, 225, 251–256, 259, 261, 268, 275, 277, 303 force potential functional 229, 230

Index

Fourier heat conduction law 261 Frechet derivative/differential 144, 193, 195, 295, 298, 302, 308–310 free energy functional 134, 143, 144, 196, 234, 296 friction 28, 29, 32, 37, 74, 84–86, 159, 205, 209, 210, 261, 271, 340 frictional material 28–32, 69, 90, 103, 204, 210 G gas constant 45 Gateaux derivative/differential 307, 308 gauge function 264, 268, 269, 304, 330– 335 Gauss’s divergence theorem 243 generalised fluxes 242 generalised forces 242, 296 generalised failure criterion 207 generalised signum function 82 generalised stress 53–58, 65, 68, 70, 73, 75, 82, 85–93, 96, 97, 123, 124, 130, 135, 138, 144, 150, 192, 195, 205, 207, 213, 229, 261, 268, 270, 278, 289, 303, 332 generalised stress function 135, 137, 144, 145, 194, 229 generalised tensorial signum function 83, 306 generalised thermodynamics 1, 3, 54, 133, 155, 341 geotechnical materials 2, 28, 74, 142, 160, 205, 210, 222, 271, 339 gravitational acceleration vector 9 Green-Lagrange strain tensor 7 H hardening laws 28 hardening modulus 22, 24, 109, 110, 148 hardening parameters 18, 19 hardening plasticity 19, 22, 24, 342 heat capacity 259 heat engine 39, 40 heat flow/flux 36, 39, 41, 44, 50, 66, 74, 161, 245, 249, 262, 301 heat supply 37, 40, 41, 54 Heaviside step function 306, 336 Hessian 70, 71, 316 hierarchy of models 15, 80, 102, 220

347

homogeneous first-order function 56, 58, 70, 73, 75, 121, 188, 229, 269, 303, 333 homogeneous function 88, 212, 214, 269, 318, 331 Hooke’s law 95, 100, 130, 155 hyperbolic stress-strain law 191, 192 hyperelastic material 14, 15, 48 hyperelasticity 15, 20, 253, 273 hypoelastic material 13, 15 hypoelasticity 15, 20 hypoplasticity 117 hysteretic behaviour 28, 107, 110, 111, 162, 200, 233 I Il'iushin's postulate of plasticity 32 image point 106–110 incompressibility condition 72, 94, 98– 102, 128, 130, 152, 155 incompressibility constraint 79, 287 incompressible elasticity 78, 81 incremental response 48, 62, 68, 69, 74, 75, 90, 92, 96, 123, 124, 138, 145, 150, 215, 230, 234, 239, 303, 308 incremental strain vector 107, 108 incremental stress vector 106, 107, 116 incremental stress-strain relationship 2, 19–21, 64, 112, 142, 239 indicator function 264–266, 270, 271, 329–335 inertial effects 261, 262 initial and boundary conditions 8, 255 initial stiffness 147, 162, 192, 234, 276 intensive quantities 40, 253 internal coordinate 134, 137, 228, 285, 290 internal energy 36–46, 49, 54, 55, 66, 78, 246–255, 279–282, 303, 320 internal function 103, 121, 134–137, 155, 179, 198, 228–234, 342 internal variables 1, 10, 33, 49, 53, 54, 71, 74, 84, 103, 120–125, 131–135, 142, 155, 173, 198, 224, 225, 228–230, 241, 242, 251, 264, 278, 280, 289, 301, 330 intrinsic time 117 invariants of the tensor 311 irrecoverable behaviour 15 irreversible behaviour 50, 51, 117, 274

348

Index

isentropic 43–46, 67 isothermal 14, 43, 46–49, 66, 74, 79, 102, 258, 259, 287 isotropic elasticity 78, 83 isotropic hardening 25–28, 92–95, 101, 103, 210, 341 isotropic thermoelasticity 49, 79 Iwan model 125–127, 149, 150 K kinematic hardening 27, 28, 97–103, 112–115, 119, 121, 123, 130, 142, 147, 151, 155, 156, 185, 186, 196, 207, 209, 228, 231, 233, 342 kinematic internal variable 53, 103, 120, 175, 225, 257 kinetic energy 245, 246, 255 L Lagrangian formulation 7, 242, 250 Lagrangian multiplier 72, 87, 206, 261, 278, 280, 287 large displacement theory 9 large strain analysis 5, 242 Laws of Thermodynamics 15, 162, 210 Legendre transform 4, 42, 43, 46–49, 56, 57, 68, 69, 72, 73, 82, 88, 89, 122, 123, 137, 143, 144, 167, 205, 212, 213, 255, 263, 273, 309, 315–324, 331, 333 Legendre-Fenchel transformation 82, 230, 256, 261, 321, 331 limiting strain 182 linear elastic region 100, 119, 179, 181 linear elasticity 13, 14, 78, 265, 318 linear hardening 27, 96–98, 127, 128, 341 link to conventional plasticity 102, 121 loading history 110, 159, 172 loading surface 106, 107, 118 logarithmic stress-strain curve 180, 191

mean stress 84, 159, 162, 169, 172, 202 mechanical dissipation 50, 55, 75, 86, 136, 161, 262, 302 mechanical power 36, 37 memory of stress reversals 120 micromechanical energy 209 Minkowski function 330 mixed invariants 313 Modified Cam-Clay model 162 modulus coupling 176 modulus of subgrade reaction 297 multiple internal variables 53, 120, 131, 135, 224–228, 231 multiple stress reversals 177 multiple surface models 111, 118, 125, 142 multisurface hyperplasticity 119 N nested surface models 111, 118 non-associated plastic flow 2, 32, 204 non-dilative plasticity 271 non-dissipative materials 48 non-intersection condition 112–117 non-linear elasticity 1, 165 non-linear viscous behaviour 219 non-uniqueness 190 normal cone 329, 330, 333–335 normality 18, 31, 103, 123, 144 notation 5 O one-dimensional elastoplasticity 81 Onsager reciprocity relationships 254 orthogonality condition 53, 56, 63, 226, 232, 296 overconsolidated clays 177, 187, 342, 343 overconsolidation ratio 172, 175, 200, 341

M

P

Macaulay brackets 92, 116, 179, 217 mapping rule 106, 118 Masing rules 28, 147, 151, 185 mass balance equations 244, 246, 254 mass flux 243 material derivative 242, 246, 250 Maxwell’s relations 43

partial derivative/differential 307 partial Legendre transformations 319 passive variables 59, 122, 319 perfect gas 35, 36, 41, 44–46 perfect plasticity 18–23, 32, 81, 103, 233 permeability coefficient 259 pile capacity 290

Index

pin-jointed structures 277 Piola-Kirchhoff stress tensor 250 plastic moduli 148 plastic modulus function 151, 178, 192 plastic multiplier 18, 20, 23, 65, 108, 111, 207 plastic potential 2, 17–22, 29, 32, 33, 58, 86, 111, 122, 210, 261 plastic strain 16–29, 32, 33, 49, 57, 67, 73, 82–86, 89, 90, 93–99, 105–114, 117, 121–123, 126, 129, 131, 138, 144, 149, 150, 154, 156, 172–177, 180, 188, 189, 192, 207, 234, 261, 271, 274, 275, 291, 335 plastic strain increments 16, 18, 69, 173 plastic strain rate tensor 88, 121 plastic work 19, 24, 29, 30, 90, 209, 210, 335 plasticity theory 1–5, 16, 18, 28, 33, 35, 57, 58, 62, 89, 107, 117, 143, 177, 263, 264, 300, 304, 321, 323, 334, 335 Poisson's ratio 162 polar function 269, 270, 304, 333–335 pore fluid 160, 161, 243–249, 253, 254, 257 pore water pressure 160 porosity 243, 256 porous continua 241, 339 porous medium 241–243, 248, 253 potential functionals 142, 148, 151, 177, 210 potential functions 2, 74, 88, 89, 93, 98, 102, 121, 125, 128, 156, 209, 238, 241, 254, 258, 262, 303, 341 potentials 2, 59, 74, 122, 173, 176, 213, 217, 232, 262, 264, 302, 303, 313, 315, 340 power input 37, 160, 161 Prager’s translation rule 28, 114 preconsolidation pressure 162, 172– 175, 196, 198, 235 pressure 36, 40–47, 74, 160–163, 166, 172, 175, 196, 202, 204, 210, 245, 247, 250, 253, 258, 259, 320, 341 principal stretches 287 prismatic beams 284 property 36, 38, 42, 46, 54, 57, 74, 88, 229, 253, 301, 331 proportional loading 131, 155, 180–183

349

Q quadratic functions 78, 318 quasi-homogeneous dissipation function 254 R rate effects 211, 239 rate process theory 221, 223, 233, 236 rate-dependent materials 212, 215, 221, 228, 230, 239, 273 rate-dependent models 224 rate-independent materials 1, 3, 51, 117, 136, 230, 303 rates of the plastic strains 87 rational mechanics 49, 133, 155 rational thermodynamics 2, 3 redundant structure 281 reservoir 38–40 reversibility 40, 117, 191 reversible materials 40 reversible processes 41, 49, 341 rigid pile 290, 296, 297 rigid-plastic materials 84 rubber elasticity 286, 287 S saturated granular materials 160 secant shear stiffness 177, 191 Second Law 3, 38, 54, 161, 248 shear modulus 14, 79, 162, 188, 290, 341 sign convention 8 simple shear 16, 26, 94, 99, 102 singular transformation 58, 71, 73, 138, 230, 320, 321, 324 skin friction 293 sliding element 97, 98, 126–128, 134 slip stress 97, 98, 126, 134, 149 small deformations 6–8 small displacement 7, 8 small strain analysis 5–8, 47 small strain region 179 small strain stiffness 162 small strains 6–8, 50, 162, 183, 203, 257, 286 soil skeleton 161, 242–250, 253–261 soils 2, 28, 32, 33, 74, 107, 112, 118, 119, 159, 162, 163, 172, 174, 183, 186, 191, 195, 198, 204, 221, 339–343

350

Index

source of heat 37 specific enthalpy 42, 256 specific entropy 40, 45, 248 specific Gibbs free energy 42, 93, 96, 98, 101, 102, 121, 142, 175, 177, 256 specific Gibbs free energy functional 179 specific heat 44–46 specific heat at constant pressure 45 specific heat at constant volume 45, 46 specific Helmholtz free energy 42, 175, 256 specific internal energy 40, 74, 246, 256, 301 specific volume 36, 40–44, 47, 188, 286, 320 S-shaped curve 177 standard material 5, 303 state variables 35–37, 42, 49, 251 stiffness 44, 46, 67, 109–112, 147, 151, 156, 162, 163, 166, 168–174, 177, 178, 182, 183, 192, 193, 196, 198, 201, 202, 210, 234, 266, 274–277, 280, 287, 291, 340, 342 stiffness matrix 20–25, 32, 48, 66, 170– 172, 178 strain contours 169, 170 strain decomposition 20 strain energy potential 14 strain hardening 19, 24, 89, 93, 118, 123, 144, 289 strain-hardening hyperplasticity 88 strains 7–16, 23, 28, 31, 32, 66, 71, 72, 79, 85, 87, 117, 159, 160, 163, 164, 167–170, 177, 186, 187, 191, 195, 203, 205, 226, 233, 243, 251, 262, 266, 277, 287–289 strain-softening behaviour 31 strength parameters 148 stress history 111, 112, 118, 200, 202 stress reversal 98, 114, 127, 151, 184– 186, 201 stress tensor 9, 47, 154, 163, 175, 301 stress–dilatancy relation 210 stress-induced anisotropy 169, 172 stretch 286, 287 structural analysis 277, 300 structural anisotropy 172 St-Venant model 97, 98 subdifferential 82, 264–266, 321, 328– 335, 337

subgradient 275, 328 subscript notation 5, 301 support function 268, 269, 304, 332– 335 surroundings 36–38, 245 T tangent modulus 98, 127, 128, 150 temperature 32, 36–46, 54, 55, 65, 66, 69, 70, 75, 79, 80, 88, 133, 221, 253, 301, 320 temperature gradient 41, 55 Terzaghi’s principle of effective stress 253 thermal conductivity coefficient 259 thermal dissipation 41, 50, 54, 75, 301 thermal expansion 44, 46, 49, 79, 80 thermal expansion coefficient 259 thermally activated processes 221, 233 thermodynamic closed system 35 thermodynamic efficiency 39 thermodynamic equilibrium 36, 51 thermodynamic process 6, 7, 20, 21, 26, 36–46, 50, 51, 75, 80, 81, 110, 148, 183, 211, 221, 222, 233, 242, 249, 269, 275, 302, 319, 331, 333 thermodynamics 1, 2, 4, 5, 15, 18, 31, 35, 36, 40, 42, 47–51, 54, 66, 133, 137, 162, 242, 256, 304, 341 thermodynamics of fluids 40, 47 thermodynamics with internal variables 1, 49 thermoelasticity 48, 79, 80, 261, 340 thermomechanics of continua 47 Third Law 36 tortuosity 245, 255, 256 total differential 307 tractions 10, 245, 247 triaxial test 159, 160, 183, 339 true stress space 68, 69, 86, 92, 95, 97, 100, 123, 130, 144, 179 U unchanged system 37–39 uncoupled materials 32 undamaged Helmholtz free energy 277 unified soil models 191 uniqueness 58, 190, 303, 321 unsaturated granular material 161, 340

Index

V velocity 9, 160, 243–245, 255, 258 virgin consolidation line 198 visco-hyperplastic model 233, 234 viscous materials 212 voids ratio 162 volumetric behaviour 29, 78, 100, 129, 153, 165–172, 182, 209, 271, 289 volumetric thermal expansion coefficients 258 von Mises yield surface 16, 26, 27, 83, 95, 100, 101, 130, 155 W weighting function 143, 148–151, 177, 228 Winkler method 290

351

work conjugacy 10, 160 work hardening 19, 24, 341 Y yield stress 16, 26, 211, 284 yield surface 2, 16–33, 57–59, 68–70, 83–91, 95, 96, 100, 103–131, 142–147, 150–155, 177–180, 184, 188–192, 195– 201, 204–210, 217, 225, 231, 233, 239, 261, 268–270, 289, 304, 330 Z Zeroth Law 36 Ziegler’s orthogonality condition 3, 75, 225, 254, 257, 261, 302 Ziegler’s translation rule 28, 114, 186