A Principles of Hyperplasticity Part 1 pdf

Principles of Hyperplasticity G. T. Houlsby and A. M. Puzrin Principles of Hyperplasticity An Approach to Plasticity Theory Based on Thermodynamic Principles 123 G. T. Houlsby, MA, DSc, FREng, FICE Department of Engineering Science Parks Road Oxford OX1 3PJ UK A. M. Puzrin, DSc ETH Zurich Institute of Geotechnical Engineering CH 8093 Zurich Switzerland British Library Cataloguing in Publication Data Houlsby, G.T. Principles of hyperplasticity: an approach to plasticity theory based on thermodynamic principles 1. Plasticity 2. Thermodynamics I. Title II. Puzrin, A.M. 531.3'85 ISBN-13: 9781846282393 ISBN-10: 184628239X Library of Congress Control Number: 2006936877 ISBN 978-1-84628-239-3 e-ISBN 1-84628-240-3 Printed on acid-free paper © Springer-Verlag London Limited 2006 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 o f 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 infor- mation contained in this book and cannot accept an y le g al responsibilit y or liabilit y for an y 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 particu- lar relevance to those working in geomechanics. Structure of this book After some introductory material in Chapter 1, a presentation of classical plas- ticity theory is given in Chapter 2. This chapter does not contain new material, but sets out the background and the terminology for the later chapters. Simi- larly, in Chapter 3, we present basic thermodynamic concepts, taking this as far as the thermodynamics of continua. In Chapter 4, we set out the hyperplastic formulation, and this chapter forms the core of the book. Much emphasis is placed on the fact that this approach al- lows us to define plasticity models by specifying just two scalar functions. In Chapter 5, we describe some simple applications, and examine different forms of energy and dissipation functions to enable the reader to become familiar with the mathematical forms that these functions take for different cases. In Chapter 6, we discuss some of the more advanced approaches to plasticity theories, as a preparation for Chapters 7 and 8. In Chapter 7, we extend the ap- proach to the use of multiple internal variables. We find that multiple independ- ent dissipation mechanisms are related to multiple yield surfaces. Then we use this approach to develop significantly more complex models employing multiple vi Preface yield surfaces (comparable to the nested yield surface models frequently em- ployed in geotechnical engineering). In Chapter 8, we further extend the concept to an infinite number of internal variables. When the finite number of internal variables is replaced by a continu- ous field of variables, the resulting models allow smooth transitions between elastic and plastic behaviour. This is an important development conceptually, and introduces the need for functionals (as opposed to functions). However, the use of some mathematical techniques which may be unfamiliar to some readers is amply repaid by the benefits that follow. We term the use of a continuous field of internal variables “continuous hyperplasticity.” Chapters 9 and 10 are devoted to examples from geomechanics, addressing issues such as effective stress modelling, the treatment of the small-strain nonlinearity of soils, critical state soil mechanics, friction, dilation, and non- associated flow. Most of the book is concerned with rate-independent materials, but in Chap- ter 11 we briefly examine ways these ideas are extended to materials with some rate dependence. Ziegler (1983) devotes much attention to rate-dependent mate- rials, but Chapter 11 is focused on elastic-viscoplastic modelling of materials, concentrating on materials that are dominated by plastic behaviour, yet include some rate effects. The strength of the hyperplastic method is that the entire constitutive re- sponse of a material is expressed through two scalar functions (or functionals). In Chapter 12, we explore how this method can be extended to include other features of material behaviour such as thermal effects and conduction phenom- ena. The chapter focuses particularly on the behaviour of porous media. In Chapter 13, we re-express hyperplasticity theory in the terminology of con- vex analysis, which allows to be expressed many of the concepts encountered earlier more rigorously. The only reason we do not employ this approach earlier is to make the book as accessible as possible to a wide readership, by avoiding where possible unfamiliar mathematical methods. Appendix D provides an in- troduction to this subject. In particular, we find that convex analysis proves convenient for expressing models that include constraints, and it also provides a more rigorous link between dissipation and yield. Chapter 14 contains a number of disconnected applications of the approach to mechanics that we have described elsewhere in this book. The breadth of applica- tions illustrates that the hyperplasticity approach is a powerful unifying tech- nique for studying many problems of constitutive modelling in mechanics. In Chapter 15, we summarise the hyperplasticity approach in a very general form and point the way to future developments. In developing the ideas that are presented in this book, we have found that our understanding of the subject has at each stage been enabled by the identification of appropriate mathematical techniques to describe the phenomena of interest. Therefore, we include appendices on some of the mathematical techniques that we employ in this book; in particular, we describe the Frechet derivatives used in Preface vii the analysis of functionals, we discuss the Legendre transform, and we provide an introduction to convex analysis. There are occasions in the book where we find it convenient, for didactic pur- poses, to repeat material. We prefer to do this rather than require the reader to have to refer too much to material found in different chapters. In a book such as this, it is inevitable that a great deal of specialised terminol- ogy must be used. In some disciplines (notably medicine), it is customary to give technical terms long, obscure names of classical derivation. Engineers have an even more disconcerting habit of using commonplace words, which have every- day meanings, to express different and precise technical terms. Examples in this book are “stress”, “strain”, “elastic”, “plastic”, “yield”, “normal”, “function” and many other words. Where a specialised meaning is implied, we shall generally draw attention to this by showing a phrase in italics when it is first used and defined. Acknowledgments We wish to express our gratitude to many people who have influenced this book. It was the late Professor Peter Wroth who guided the first author toward the rigorous study of the interplay between plasticity theory and thermodynamics, and who was first his supervisor and later a much respected colleague. The roots of the ideas expressed here lie in the work of the late Professor Hans Ziegler, who was a considerate correspondent with the first author. Many of the ideas lay dormant for several years until a very successful collaboration with Professor Ian Collins, who brought to bear a number of mathematical techniques (notably the Legendre Transform) which are central to the developments described here. Professor Michael Sewell provided, in a brief conversation, the key to the treatment of part of the formulation: that of treating F and F as separate vari- ables. Professor Martin Brokate is thanked for very fruitful discussions at the Horton conference in 2002. Guy Houlsby (Oxford University) Alexander Puzrin (ETH Zurich) Contents 1 Introduction 1 1.1 Plasticity and Thermodynamics 1 1.1.1 Purpose of this Book 1 1.1.2 Advantages of Our Approach 2 1.1.3 Generality 2 1.1.4 Ziegler’s Orthogonality Condition 3 1.1.5 Constitutive Models 4 1.2 Context of this Book 4 1.3 Notation 5 1.4 Some Basic Continuum Mechanics 6 1.4.1 Small Deformations and Small Strains 6 1.4.2 Sign Convention 8 1.5 Equations of Continuum Mechanics 8 1.5.1 Equilibrium 9 1.5.2 Compatibility 9 1.5.3 Initial and Boundary Conditions 9 1.5.4 Work Conjugacy 10 1.5.5 Numbers of Variables and Equations 11 2 Classical Elasticity and Plasticity 13 2.1 Elasticity 13 2.2 Basic Concepts of Plasticity Theory 16 2.3 Incremental Stiffness in Plasticity Models 19 2.3.1 Perfect Plasticity 20 2.3.2 Hardening Plasticity 22 2.3.3 Isotropic Hardening 25 2.3.4 Kinematic Hardening 27 2.3.5 Discussion of Hardening Laws 28 2.4 Frictional Plasticity 28 x Contents 2.5 Restrictions on Plasticity Theories 30 2.5.1 Drucker's Stability Postulate 31 2.5.2 Il'iushin's Postulate of Plasticity 32 3 Thermodynamics 35 3.1 Classical Thermodynamics 35 3.1.1 Introduction 35 3.1.2 The First Law 36 3.1.3 The Second Law 38 3.2 Thermodynamics of Fluids 40 3.2.1 Energy Functions 42 3.2.2 An Example of an Internal Energy Function 43 3.2.3 Perfect Gases 44 3.3 Thermomechanics of Continua 47 3.3.1 Terminology 47 3.3.2 Thermoelasticity 48 3.3.3 Internal Variables and Dissipation 49 4 The Hyperplastic Formalism 53 4.1 Introduction 53 4.2 Internal Variables and Generalised Stress 53 4.3 Dissipation and Dissipative Generalised Stress 54 4.3.1 The Laws of Thermodynamics 54 4.3.2 Dissipation Function 55 4.3.3 Dissipative Generalised Stress 56 4.4 Yield Surface 56 4.4.1 Definition 56 4.4.2 The Flow Rule 57 4.4.3 Convexity 58 4.4.4 Uniqueness of the Yield Function 58 4.5 Transformations from Internal Variable to Generalised Stress 59 4.6 A Complete Formulation 59 4.7 Incremental Response 62 4.8 Isothermal and Adiabatic Conditions 66 4.9 Plastic Strains 67 4.10 Yield Surface in Stress Space 68 4.11 Conversions Between Potentials 69 4.11.1 Entropy and Temperature 69 4.11.2 Stress and Strain 70 4.11.3 Internal Variable and Generalised Stress 70 4.11.4 Dissipation Function to Yield Function 70 4.11.5 Yield Function to Dissipation Function 71 Contents xi 4.12 Constraints 71 4.12.1 Constraints on Strains 72 4.12.2 Constraints on Plastic Strain Rates 73 4.13 Advantages of Hyperplasticity 74 4.14 Summary 74 5 Elastic and Plastic Models in Hyperplasticity 77 5.1 Elasticity and Thermoelasticity 77 5.1.1 One-dimensional Elasticity 77 5.1.2 Isotropic Elasticity 78 5.1.3 Incompressible Elasticity 78 5.1.4 Isotropic Thermoelasticity 79 5.1.5 Hierarchy of Isotropic Elastic Models 80 5.2 Perfect Elastoplasticity 81 5.2.1 One-dimensional Elastoplasticity 81 5.2.2 Von Mises Elastoplasticity 83 5.2.3 Rigid-plastic Models 84 5.3 Frictional Plasticity and Non-associated Flow 84 5.3.1 A Two-dimensional Model 85 5.3.2 Dilation 86 5.3.3 The Drucker-Prager Model with Non-associated Flow 87 5.4 Strain Hardening 88 5.4.1 Theory of Strain-hardening Hyperplasticity 88 5.4.2 Isotropic Hardening 91 5.4.3 Kinematic Hardening 96 5.4.4 Mixed Hardening 101 5.5 Hierarchy of Plastic Models 102 6 Advanced Plasticity Theories 105 6.1 Developments of Classical Plasticity Theory 105 6.2 Bounding Surface Plasticity 105 6.3 Nested Surface Plasticity 107 6.4 Multiple Surface Plasticity 110 6.5 Remarks on the Intersection of Yield Surfaces 112 6.5.1 The Non-intersection Condition 112 6.5.2 Example of Intersecting Surfaces 112 6.5.3 What Occurs when the Surfaces Intersect? 115 6.6 Alternative Approaches to Material Non-linearity 117 6.7 Comparison of Advanced Plasticity Models 118 7 Multisurface Hyperplasticity 119 7.1 Motivation 119 7.2 Multiple Internal Variables 120 xii Contents 7.3 Kinematic Hardening with Multiple Yield Surfaces 121 7.3.1 Potential Functions 121 7.3.2 Link to Conventional Plasticity 121 7.3.3 Incremental Response 123 7.4 One-dimensional Example (the Iwan Model) 125 7.5 Multidimensional Example (von Mises Yield Surfaces) 128 7.6 Summary 131 8 Continuous Hyperplasticity 133 8.1 Generalised Thermodynamics and Rational Mechanics 133 8.2 Internal Functions 134 8.3 Energy and Dissipation Functionals 134 8.3.1 Energy Functional 134 8.3.2 Generalised Stress Function 135 8.3.3 Dissipation Functional 136 8.3.4 Dissipative Generalised Stress Function 136 8.4 Legendre Transformations of the Functionals 137 8.4.1 Legendre Transformations of the Energy Functional 137 8.4.2 Legendre Transformation of the Dissipation Functional 138 8.5 Incremental Response 138 8.6 Kinematic Hardening with Infinitely Many Yield Surfaces 142 8.6.1 Potential Functionals 142 8.6.2 Link to Conventional Plasticity 143 8.6.3 Incremental Response 145 8.7 Example: One-dimensional Continuous Hyperplastic Model 146 8.8 Calibration of Continuous Kinematic Hardening Models 148 8.9 Example: Calibration of the Weighting Function 148 8.9.1 Formulation of the One-dimensional Model 148 8.9.2 Analogy with the Extended Iwan’s Model 149 8.9.3 Model Calibration Using the Initial Loading Curve 150 8.9.4 Unloading Behaviour 151 8.10 Example: Calibration of the Plastic Modulus Function 151 8.10.1 Formulation of the Multidimensional von Mises Model 151 8.10.2 Model Calibration Using the Initial Loading Curve 154 8.10.3 Analogy with an Advanced Plasticity Model 155 8.11 Hierarchy of Multisurface and Continuous Models 155 9 Applications in Geomechanics: Elasticity and Small Strains 159 9.1 Special Features of Mechanical Behaviour of Soils 159 9.2 Sign Convention and Triaxial Variables 159 9.3 Effective Stresses 160 [...]... 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 a material derivative, a a a, i vi ˆ a any function of the internal variable aij deviatoric aij a, i of second ˆ ˆ ,a a order tensor aij , ij akk... 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 2 21 11. 2.4 A Continuum Model 223 xiv 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 ... Functions 19 0 10 .2 Towards Unified Soil Models 19 1 10 .2 .1 Small Strain Non-linearity: Hyperbolic Stress-strain Law 19 1 10 .2.2 Modified Forms of the Energy Functionals 19 3 10 .2.3 Combining Small-strain and Critical State Behaviour 19 5 10 .2.4 Examples 19 8 10 .2.5 Continuous Hyperplastic Modified Cam-Clay 203 10 .3 Frictional Behaviour and Non-associated Flow 204 10 .3 .1 The... 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. .. Potential Functions 17 8 9.5.3 Behaviour of the Model During Initial Proportional Loading 18 0 9.5.4 Behaviour of the Model During Proportional Cyclic Loading 18 4 9.5.5 Concluding Remarks 18 6 10 Applications in Geomechanics: Plasticity and Friction 18 7 10 .1 Critical State Models 18 7 10 .1. 1 Hyperplastic Formulation of Modified Cam-Clay 18 7 10 .1. 2 Non-uniqueness of. .. 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 2 31 Visco-hyperplastic Model for Undrained Behaviour of Clay 233 11 .5 .1 Formulation 233 11 .5.2 Incremental Response 234 11 .5.3... 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 3 01 15 .1 Summary of the Complete Formalism... 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... 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 (Almost) universally accepted as true Strong theoretical grounds for acceptance No experimental counterexamples . 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. similar variables o a initial value of a  a rate of a with time, {w w  aat  a material derivative, { ,  ii aaav ˆ a any function of the internal variable K ,  {K ˆˆ aa . Remarks 18 6 10 Applications in Geomechanics: Plasticity and Friction 18 7 10 .1 Critical State Models 18 7 10 .1. 1 Hyperplastic Formulation of Modified Cam-Clay 18 7 10 .1. 2 Non-uniqueness of the