Introduction to Computational Plasticity www.com This page intentionally left blank www.com Introduction to Computational Plasticity FIONN DUNNE AND NIK PETRINIC Department of Engineering Science Oxford University, UK 1 www.com 3 Great Clarendon Street, Oxford ox2 6dp Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press, 2005 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2005 Reprinted 2006 (with corrections) All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data (Data available) Library of Congress Cataloging in Publication Data (Data available) Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, King’s Lynn ISBN 0-19-856826-6 (Hbk) 978-0-19-856826-1 3 5 7 9 10 8 6 4 2 www.com To Hannah and Roberta, with love www.com This page intentionally left blank www.com Preface The intention of this book is to bridge the gap between undergraduate texts in engineering plasticity and the many excellent books in computational plasticity aimed at more senior graduate students, researchers, and practising engineers working in solid mechanics.
The book is in two parts. The first introduces microplasticity and covers continuum plasticity, the kinematics of large deformations and continuum mechanics, the finite element method, implicit and explicit integration of plasticity constitutive equations, and the implementation of the constitutive equations, and the associated material Jacobian, into finite element software. In particular, the implemen- tation into the commercial code ABAQUS is addressed (and to help, we provide a range of ABAQUS material model UMATs), together, importantly, with the tests necessary to verify the implementation. Our intention, wherever possible, is to develop a good physical feel for the plasticity models and equations described by considering, at every stage, the simplification of the equations to uniaxial conditions.
In addition, we hope to provide a reasonably physical understanding of some of the large deformation quantities (such as the continuum spin) and concepts (such as objectivity) which are often unfamiliar to many undergraduate engineering students who demand more than just a mathematical description. The second part of the book introduces a range of plasticity models including those for superplasticity, porous plasticity, creep, cyclic plasticity, and thermo-mechanical fatigue (TMF). We also describe a number of practical applications of the plasticity models introduced to demonstrate the reasonable maturity of continuum plasticity in engineering practice. We hope, above all, that this book will help all those—undergraduates, graduates, researchers, and practising engineers—who need to move on from knowledge of undergraduate plasticity to modern practice in computational plasticity.
Our aims have been to encourage development of understanding, and ease of passage to the more advanced texts on computational plasticity.com This page intentionally left blank www.com Contents Acknowledgements xii Notation xiii Part I. Microplasticity and continuum plasticity 1.3 Critical resolved shear stress 7 1.4 Dislocations 8 Further reading 10 2.6 Combined isotropic and kinematic hardening 36 2.7 Viscoplasticity and creep 38 Further reading 45 3. Kinematics of large deformations and continuum mechanics 47 3.2 The deformation gradient 48 3.3 Measures of strain 49 3.4 Interpretation of strain measures 52 3.5 Polar decomposition 57 www.6 Velocity gradient, rate of deformation, and continuum spin 60 3.7 Elastic–plastic coupling 66 3.8 Objective stress rates 69 3.9 Summary 81 Further reading 82 4. The finite element method for static and dynamic plasticity 83 4.3 Introduction to the finite element method 96 4.4 Finite element equilibrium equations 100 4.5 Integration of momentum balance and equilibrium equations 136 Further reading 142 5.
Implicit and explicit integration of von Mises plasticity 143 5.2 Implicit and explicit integration of constitutive equations 143 5.5 Implicit integration in viscoplasticity 161 5.6 Incrementally objective integration for large deformations 167 Further reading 168 6. Implementation of plasticity models into finite element code 169 6.3 Verification of implementations 171 6.4 Isotropic hardening plasticity implementation 172 6.5 Large deformation implementations 176 6.6 Elasto-viscoplasticity implementation 180 Part II.2 Some properties of superplastic alloys 185 www.com Contents xi 7.3 Constitutive equations for superplasticity 189 7.4 Multiaxial constitutive equations and applications 192 References 197 8.2 Finite element implementation of the porous material constitutive equations 201 8.3 Application to consolidation of Ti–MMCs 205 References 207 9. Creep in an aero-engine combustor material 209 9.2 Physically based constitutive equations 209 9.3 Multiaxial implementation into ABAQUS 212 References 217 Appendix 9. Cyclic plasticity, creep, and TMF 219 10.2 Constitutive equations for cyclic plasticity 219 10.3 Constitutive equations for C263 undergoing TMF 222 References 227 Appendix A: Elements of tensor algebra 229 Differentiation 231 The chain rule 232 Rotation 233 Appendix B: Fortran coding available via the OUP website 235 Index 239 www.com Acknowledgements The authors would like to express their sincere gratitude to Esteban Busso for reading a draft and providing many helpful comments and suggestions, to Paul Buckley for the provision of the figures in Chapter 1, and to Jinguo Lin for permission to use Figures 7.
The authors acknowledge, with gratitude, permission granted to reproduce the following figures: Figures 7.14: Elsevier Ltd, Oxford, UK.6: Institute of Materials Communications Ltd, London, UK.3: Elsevier Ltd, Oxford, UK.com Notation • Regular italic typeface (v, σ, .): scalars, scalar functions. • Bold italic typeface (P , v, A, σ , .): points, vectors, tensors, vector and tensor functions. • Helvetica bold italic typeface (C, c, I, .): fourth order tensors. Operations f (·) function of (·) det[·] determinant of [·] Tr[·] trace of [·] ln(·) logarithm of (·) [·] increment of [·] ∂ [·] partial derivative of [·] with respect to x ∂x ∇(·) = grad[·] gradient of [·] div[·] = tr[∇(·)] divergence of [·] x·y scalar product of vectors x⊗y dyadic product of vectors σ :ε double contraction of tensors √ |u| = √u · u norm of vector |A| = A : A norm of tensor Some commonly used notation C fourth-order tensor of material constants D rate of deformation tensor E Lagrangian strain tensor E Young’s modulus ε strain tensor www.com xiv Notation F deformation gradient f force vector field ρ density I second-order identity tensor J Jacobian K stiffness matrix M mass matrix ν Poisson’s ratio P material particle P material point ∈ Rn R rotation tensor R real set σ Cauchy stress tensor t time t surface traction vector u displacement vector field u̇ velocity vector field ü acceleration vector field W work www.
Microplasticity and continuum plasticity www.com This page intentionally left blank www.1 Introduction This chapter briefly introduces the origins of yield and plastic flow, and in particular, attempts to explain the usual assumptions in simple continuum plasticity of isotropy, incompressibility, and independence of hydrostatic stress. While short, we introduce grains, crystal slip, slip systems, resolved shear stress, and dislocations; the minimum knowledge of microplasticity for users of continuum plasticity. The origin of plasticity in crystalline materials is crystal slip. Metals are usually polycrystalline; that is, made up of many crystals in which atoms are stacked in a regular array.
A typical micrograph for a polycrystalline nickel-base superalloy is shown in Fig.1 in which the ‘crystal’ or grain boundaries can be seen. The grain size is about 100 µm. The grain boundaries demarcate regions of different crystallographic orientation. If we represent the crystallographic structure of a tiny region of a single grain by planes of atoms, as shown in Fig.2(a), we can then visualize plastic deformation taking place as shown in Fig.2(a) and (b); this is crystallographic slip.
Unlike elastic deformation, involving only the stretching of interatomic bonds, slip requires the breaking and re-forming of interatomic bonds and the motion of one plane of atoms relative to another. After shearing the crystal from configuration 1.2(a) to configuration 1.2(b), the structure is unchanged except at the extremities of the crystal. A number of very important phenomena in macroscopic plasticity become apparent from just two Figs 1.2: (1) plastic slip does not lead to volume change; this is the incompressibility condition of plasticity; (2) plastic slip is a shearing process; hydrostatic stress, at the macrolevel, can often be assumed not to influence slip; (3) in a polycrystal, plastic yielding is often an isotropic process. As we will see later, the incompressibility condition is very important in macro- scale plasticity and manifests itself at the heart of constitutive equations for plasticity.com 4 Microplasticity 100 mm Fig.1 Micrograph of polycrystal nickel-base alloy C263.2 Schematic representation of the crystallographic structure within a single grain undergoing slip.
However, not all plastic deformation processes are incompressible. A porous metal, for example, under compressive load may undergo plastic deformation during which the pores reduce in size. Consequently, there is a change of volume and a dependence on hydrostatic stress. However, the volume change does not originate from the plastic slip process itself, but from the pore closure.
The fact that plastic slip is a shearing process gives more information about the nature of yielding; in principle, it tells us that plastic deformation is independent of hydrostatic stress (pressure). For non-porous metals, this is one of the cornerstones of yield criteria. The von Mises criterion, for example, is one in which the initiation of macroscale yield is quite independent of hydrostatic stress. If we take a sample of the theoretical material shown schematically in Fig.2(a) and submerge it to an ever deeper depth in an imaginary sea of water, the hydrostatic stress becomes ever larger but causes no more than the atoms in the theoretical material to come closer together.
It will never in itself be able to generate the shearing necessary for crystallographic slip.1 shows a micrograph of a polycrystal. If we assume that there is no pre- ferred crystallographic orientation, but that the orientation changes randomly from one www.com Crystal slip 5 (a) (b) Fig.3 (a) Photograph of a single zinc crystal and (b) a schematic diagram representing single slip in a single crystal. grain to the next, and if our sample of material contains a sufficiently large num- ber of grains, we can get a reasonable physical feel that macroscale yielding of the material will be isotropic. This is a further cornerstone of the von Mises yield criterion.2 Crystal slip The evidence for crystal slip being the origin of plasticity comes from mechanical tests carried out on single crystals of metals.
The single crystal of zinc shown in Fig.3(a) is a few millimetres in width and has been loaded beyond yield in tension. The planes that can be seen are those on which slip has occurred resulting from many hundreds of dislocations running through the crystal and emerging at the edge. Each dislocation contributes just one Burger’s vector of relative displacement, but with many such dislocations, the displacements become large.3(b) shows schematically what is happening in Fig. The ends of the test sample have not been constrained in the lateral directions.
It can be seen that single slip in this case leads to the horizontal displacement of one end relative to the other.