Non-Linear Mechanics of Materials
SOLID MECHANICS AND ITS APPLICATIONS
Volume 167 Series Editor:
G.M.L. GLADWELL Department of Civil Engineering University of Waterloo Waterloo, Ontario, Canada N2L 3GI
Aims and Scope of the Series The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.
For other titles published in this series, go to www.springer.com/series/6557
Jacques Besson Georges Cailletaud Jean-Louis Chaboche Samuel Forest
Non-Linear Mechanics of Materials In cooperation with Marc Blétry
Jacques Besson Centre des Matériaux CNRS UMR 7633 MINES ParisTech BP 87, 91003 Evry Cedex France
[email protected]
Samuel Forest Centre des Matériaux CNRS UMR 7633 MINES ParisTech BP 87, 91003 Evry Cedex France
[email protected]
Georges Cailletaud Centre des Matériaux CNRS UMR 7633 MINES ParisTech BP 87, 91003 Evry Cedex France
[email protected]
Marc Blétry Institut de Chimie et des Matériaux Paris-Est CNRS UMR 7182 Université Paris XII 2-8, rue H. Dunant 94320 Thiais France
[email protected]
Jean-Louis Chaboche Dépt. Matériaux et Structures Métallique Office National d’Études et de Recherches Aérospatiales (ONERA) 29 avenue de la Division Leclerc 92322 Chatillon CX France
[email protected]
ISBN 978-90-481-3355-0 e-ISBN 978-90-481-3356-7 DOI 10.1007/978-90-481-3356-7 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009940116 c Springer Science+Business Media B.V. 2010 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Foreword
Invited by Jacques Besson, Georges Cailletaud, Jean-Louis Chaboche and Samuel Forest to introduce their book “Nonlinear mechanics of materials”, I am pleased to deliver an affirmative response to the honor, the confidence and the friendship that they thus show me. • Pleased at an emotional level first, because the authors belong to a scientific community that has blossomed in France for the last twenty years and that has brought much to all of us. Moreover, two of them have been my students and are now friends and colleagues. This community of researchers in solid mechanics, first built on young people, is now mature and reaches out across France and overseas. The strong French participation in mechanical conferences and publications in great scientific journals testify to their contributions. So does this book. • Pleased at a scientific level, as it is a modern presentation of materials mechanics, from microscopic to mesoscopic scale, from physical to numerical aspects, from phenomenology to homogenization, from field calculations to localization. Everything is in it, or almost! “Nonlinear” is the central point and everybody knows that there are many pitfalls. The authors managed to avoid them thanks to an exhaustive but well delimited approach. • Pleased, lastly, on a technical level because through the very design of the book, the writers have chosen to address an informed but not expert audience. The elementary notions have indeed been left aside, but to the benefit of a clear synthesis of the developments made in the last twenty years. However, the basic tools called “General concepts” are rigorously described in a concise manner, in spite of the scope of the subjects: virtual power, thermodynamics, solving of nonlinear systems, integration of differential equations, finite element method. The authors, through their education and their research at the Centre des Matériaux of Mines ParisTech and at ONERA use, with the same virtuosity, constitutive equations and numerical calculations; that is the originality and the appeal of this book. Plasticity and viscoplasticity are explained up to multimechanism models; damage up to the problem posed by its deactivation in compression,
vi
Non-Linear Mechanics of Materials
so hard to define in 3D; the behavior of heterogeneous materials up to nonlinear self-consistent schemes; finite strain (in an accessible manner!) up to Cosserat continuum; structural analysis written for constitutive equations “programmers”, who will find here good “seedlings”; and, at last, the extreme limit of the behavior represented by localization phenomena. More physics, more numerics reflect well the current trends in Mechanics. No doubt that PhD students, researchers in solid mechanics, engineers close to calculation codes related to optimal use of materials will find in this book matter to deepen their knowledge, many “ready to use” tools and almost 300 references. Jacques, Georges, JeanLouis, Samuel, thank you for sharing with us your knowledge. Cachan, 7th May 01
Jean Lemaitre Emeritus Professor at Paris-6 University LMT-Cachan
Preface
“Everything should be made as simple as possible, but no simpler”, said a famous swiss-german author. Several tens years after this quote, a lot remains to be done in the realm of materials. In their daily practice, engineers use too often rather naive models. However, significant progress have been made in a recent period. The 80’s have seen the boom of many models, the 90’s the appearance of robust algorithms allowing their use in structural calculations. At the dawn of the new millenary, we are then prepared for a significant overhaul of the design methods in materials mechanics. The remaining is a problem of data, and this is why characterizing is more than ever important. It is mandatory to complete the handbooks, by going way further than the usual information such as yield stress, creep or crack behaviour, by including data on the actual cyclic behaviour, the evolution of the behaviour with microsctucture in use, the mechanical response at the scale of phases, etc. One should not neglect education, as the domain of nonlinear mechanics holds specific problems. The authors consider that there is a great risk to see insufficiently qualified engineers handling more and more complex numerical tools. It appears to them that, currently, all conditions are met so that this risk takes shape, with the exclusive celebration of a “systemic” approach that leads (there is only 24 hours per day) to reducing the fundamental teaching and, without a doubt, the technical basis of the young graduates. However, it goes without saying that it is always necessary to revise the way things are set forth, renew and adapt the teaching methods. That is why we have decided to associate with the description of the models, the methods that are necessary to use them. It is then justified to regroup in one book a summary of models that have become classic and an opening towards burning issues. The first version of the present work appeared during a course at IPSI (Institut pour la Promotion des Sciences de l’Ingénieur), in September 1997, with the same title. The conversion from handout to book that followed turned out to be a long and painfull experience, especially near the end, when one must finish the next day and that, clearly, there will remain so many things to write. . . Of course, one should study the optimizing methods and identification process, experimental data analysis, to bring the model up to an operational state. Of course, it would also be necessary to deal with lifetime prediction methods. Of course, one
viii
Non-Linear Mechanics of Materials
should open wider one’s arms and treat the multi-physics coupling problems. So, may the readers that might be disappointed by such deficiencies forgive us, better still, may them write to us, so that the most blatant omissions may be corrected in the future if—as we hope—this book is only the first step of a greater project that should lead to a more up-to-date, and more immediately usable presentation of the models and methods in materials mechanics. The discussion about the existence of God(s) was not dealt with in this book. To be frank, this would have been out of this series topics, and it might have been difficult to homogenize the notations. In a first approximation, the reader should consider the results as independent of the answer to that question. However, when stepping back to contemplate the work, and give to one’s keyboard new tasks, beyond the usual thanks to those who live with us every day, the one who brought us to existence, to our previous and coming masters, comes the need to be silent for a few minutes and, to put some humanity into the forthcoming equations, suggest the reader other readings, as new conjugate directions in space. J. Besson, G. Cailletaud, J.-L. Chaboche, S. Forest—juin 2001 The first creature resembled a lion, the second was like a calf, the third had a face like that of a human being, and the fourth looked like an eagle in flight. (Revelation 4,7) Avec soulagement, avec humiliation, avec terreur, il comprit que lui aussi était une apparence, qu’un autre était en train de le rêver. Jorge Luis Borges (Fictions, Les ruines circulaires) Die Eswelt hat Zusammenhang im Raum und in der Zeit. Die Duwelt hat in Raum und Zeit keinen Zusammenhang. Martin Buber (Ich und Du) J’avance en poésie comme un cheval de trait tel celui-là de jadis dans les labours de fond qui avait l’oreille dressée à se saisir réel les frais matins d’été dans les mondes brumeux. Gaston Miron (L’homme rapaillé)
Contents
1 Introduction 1.1
1.2
1
Model construction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Models for understanding . . . . . . . . . . . . . . . . . . .
2
1.1.2
Models for designing . . . . . . . . . . . . . . . . . . . . . .
3
Applications to models . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 General concepts
7
2.1
Formulation of the constitutive equations . . . . . . . . . . . . . . . .
7
2.2
Principle of virtual power . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3
Thermodynamics of irreversible processes . . . . . . . . . . . . . . .
10
2.3.1
First and second principles of thermodynamics . . . . . . . .
10
2.3.2
Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3.3
Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.3.4
Linear thermoelasticity . . . . . . . . . . . . . . . . . . . . .
12
2.3.5
Nonlinear behavior . . . . . . . . . . . . . . . . . . . . . . .
14
Main class of constitutive equations . . . . . . . . . . . . . . . . . .
17
2.4.1
Basic building blocks . . . . . . . . . . . . . . . . . . . . . .
17
2.4.2
One-dimensional plasticity . . . . . . . . . . . . . . . . . . .
18
2.4.3
One-dimensional viscoelasticity . . . . . . . . . . . . . . . .
22
2.4.4
Study of a combined model . . . . . . . . . . . . . . . . . .
24
2.4.5
One-dimensional viscoplasticity . . . . . . . . . . . . . . . .
25
2.4
x
Non-Linear Mechanics of Materials
2.5
2.6
2.7
2.8
2.4.6
Temperature influence . . . . . . . . . . . . . . . . . . . . .
29
2.4.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Yield criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.5.1
Available tools . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.5.2
Criteria without hydrostatic pressure . . . . . . . . . . . . . .
31
2.5.3
Criteria involving hydrostatic pressure . . . . . . . . . . . . .
33
2.5.4
Anisotropic criteria . . . . . . . . . . . . . . . . . . . . . . .
36
Numerical methods for nonlinear equations . . . . . . . . . . . . . .
37
2.6.1
Newton-type methods/modified Newton . . . . . . . . . . . .
38
2.6.2
One unknown case, order of convergence . . . . . . . . . . .
39
2.6.3
BFGS method (Broyden–Fletcher–Goldfarb–Shanno) . . . . .
41
2.6.4
Iterative method—conjugate gradient . . . . . . . . . . . . .
43
2.6.5
Riks method . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.6.6
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Numerical solution of differential equations . . . . . . . . . . . . . .
48
2.7.1
General overview . . . . . . . . . . . . . . . . . . . . . . . .
48
2.7.2
Runge–Kutta method . . . . . . . . . . . . . . . . . . . . . .
49
2.7.3
θ -methods . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.7.4
Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Finite element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.8.1
Spatial discretization . . . . . . . . . . . . . . . . . . . . . .
54
2.8.2
Discrete integration method . . . . . . . . . . . . . . . . . .
55
2.8.3
Discretization of fields of unknowns . . . . . . . . . . . . . .
55
2.8.4
Application to mechanics . . . . . . . . . . . . . . . . . . . .
57
2.8.5
Finite element discretization of Greenberg’s principle . . . . .
58
2.8.6
Another presentation of the finite element discretization . . .
61
2.8.7
Assembly through example . . . . . . . . . . . . . . . . . . .
62
2.8.8
Principle of resolution . . . . . . . . . . . . . . . . . . . . .
64
2.8.9
Mechanical behavior in the finite element method . . . . . . .
65
Contents
3 3D plasticity and viscoplasticity 3.1
xi
67
Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
3.1.1
Strain decomposition . . . . . . . . . . . . . . . . . . . . . .
67
3.1.2
Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.1.3
Flow rules . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.1.4
Hardening rules . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.1.5
Generalized standard materials . . . . . . . . . . . . . . . . .
69
Formulation of the constitutive equations . . . . . . . . . . . . . . . .
70
3.2.1
State variables definition . . . . . . . . . . . . . . . . . . . .
70
3.2.2
Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.2.3
From viscoplasticity to plasticity . . . . . . . . . . . . . . . .
72
3.2.4
Comments on the formulation of the plastic constitutive equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Flow direction associated to the classical criteria . . . . . . . . . . . .
76
3.3.1
Von Mises criterion . . . . . . . . . . . . . . . . . . . . . . .
76
3.3.2
Tresca criterion . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.3.3
Drucker–Prager criterion . . . . . . . . . . . . . . . . . . . .
77
Expression of some particular constitutive equations in plasticity . . .
78
3.4.1
Prandtl–Reuss model . . . . . . . . . . . . . . . . . . . . . .
78
3.4.2
Hencky–Mises model . . . . . . . . . . . . . . . . . . . . . .
79
3.4.3
Prager model . . . . . . . . . . . . . . . . . . . . . . . . . .
79
Flow under prescribed strain rate . . . . . . . . . . . . . . . . . . . .
80
3.5.1
Case of an elastic–perfectly plastic material . . . . . . . . . .
80
3.5.2
Case of a material with hardening . . . . . . . . . . . . . . .
81
3.6
Non-associated plasticity . . . . . . . . . . . . . . . . . . . . . . . .
81
3.7
Non-linear hardening . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.7.1
Kinematics and isotropic hardening . . . . . . . . . . . . . .
83
3.7.2
Dissipated energy, stored energy . . . . . . . . . . . . . . . .
84
3.7.3
Typical results . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.2
3.3
3.4
3.5
xii
Non-Linear Mechanics of Materials
3.8
3.9
Some classical extensions . . . . . . . . . . . . . . . . . . . . . . . .
92
3.8.1
Multikinematic . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.8.2
Modification of the dynamic recovery term . . . . . . . . . .
92
3.8.3
Other models for progressive deformation . . . . . . . . . . .
95
3.8.4
Memory effect . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.8.5
Hardening followed by softening . . . . . . . . . . . . . . . .
95
3.8.6
Non-proportional loading . . . . . . . . . . . . . . . . . . . .
95
3.8.7
Anisotropic plastic behavior . . . . . . . . . . . . . . . . . .
96
Hardening and recovery in viscoplasticity . . . . . . . . . . . . . . .
97
3.9.1
Kinematic and isotropic hardening . . . . . . . . . . . . . . .
97
3.9.2
Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.9.3
Strain hardening . . . . . . . . . . . . . . . . . . . . . . . . 100
3.10 Multimechanism models . . . . . . . . . . . . . . . . . . . . . . . . 101 3.10.1 General formulation . . . . . . . . . . . . . . . . . . . . . . 101 3.10.2 Multimechanism–multicriteria models . . . . . . . . . . . . . 102 3.10.3 A single crystal model . . . . . . . . . . . . . . . . . . . . . 103 3.10.4 Two mechanisms and two criteria models (2M2C) . . . . . . 108 3.10.5 Simultaneous plastic and viscoplastic flows . . . . . . . . . . 110 3.10.6 Two mechanisms and one-criterion models (2M1C) . . . . . . 112 3.10.7 Compressible materials . . . . . . . . . . . . . . . . . . . . . 115 3.11 Behavior of porous materials . . . . . . . . . . . . . . . . . . . . . . 116 3.11.1 Presentation of some models . . . . . . . . . . . . . . . . . . 117 3.11.2 Yield criterion . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.11.3 Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.11.4 (Visco)plastic flow . . . . . . . . . . . . . . . . . . . . . . . 120 3.11.5 Evolution of porosity . . . . . . . . . . . . . . . . . . . . . . 122 3.11.6 Elastic behavior . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.11.7 Effective plastic deformation, hardening . . . . . . . . . . . . 122 3.11.8 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Contents
xiii
3.11.9 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 Introduction to damage mechanics
127
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.2
Notions and general concepts . . . . . . . . . . . . . . . . . . . . . . 128
4.3
4.4
4.5
4.6
4.2.1
Various kinds of damages . . . . . . . . . . . . . . . . . . . 128
4.2.2
Distinction between deformation, damage, propagation . . . . 129
4.2.3
Damage definitions and measures . . . . . . . . . . . . . . . 131
4.2.4
Energy dissipated by damage . . . . . . . . . . . . . . . . . . 136
Damage variables and state laws . . . . . . . . . . . . . . . . . . . . 138 4.3.1
Tensorial nature of the damage variables . . . . . . . . . . . . 138
4.3.2
Some possible choices for elasticity laws . . . . . . . . . . . 139
4.3.3
Effective stress concept . . . . . . . . . . . . . . . . . . . . . 140
State and dissipative couplings . . . . . . . . . . . . . . . . . . . . . 147 4.4.1
Various forms of state coupling . . . . . . . . . . . . . . . . 147
4.4.2
Coupling of dissipations . . . . . . . . . . . . . . . . . . . . 149
4.4.3
Some possibilities for the elastic limit criterion . . . . . . . . 150
4.4.4
General approach for plasticity/damage coupling . . . . . . . 152
4.4.5
Advantages and drawbacks of both types of equivalence . . . 155
4.4.6
Asymptotic behavior near fracture . . . . . . . . . . . . . . . 157
Damage deactivation . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.5.1
Deactivation in the isotropic damage case . . . . . . . . . . . 163
4.5.2
Difficulties associated with anisotropy . . . . . . . . . . . . . 164
4.5.3
A deactivation criterion that preserves continuity . . . . . . . 165
4.5.4
Consequences for plasticity coupled to damage . . . . . . . . 168
Damage evolution laws . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.6.1
Rate-independent normality law . . . . . . . . . . . . . . . . 169
4.6.2
Form of the non-damage criterion . . . . . . . . . . . . . . . 170
4.6.3
Consistency condition . . . . . . . . . . . . . . . . . . . . . 173
xiv
Non-Linear Mechanics of Materials
4.7
4.6.4
Tangent operator . . . . . . . . . . . . . . . . . . . . . . . . 175
4.6.5
Rate-dependent laws . . . . . . . . . . . . . . . . . . . . . . 177
Examples of damage models in brittle materials . . . . . . . . . . . . 181 4.7.1
Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.7.2
Application to composites . . . . . . . . . . . . . . . . . . . 184
4.7.3
Modeling the ceramic–ceramic composites . . . . . . . . . . 187
5 Elements of microstructural mechanics 5.1
5.2
5.3
5.4
195
Characteristic lengths and scales in microstructural mechanics . . . . 195 5.1.1
Objectives of heterogeneous materials mechanics and homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.1.2
Microstructure/RVE/structure . . . . . . . . . . . . . . . . . 204
5.1.3
Spatial averages, ensemble averages; equivalent homogeneous medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.1.4
Local behavior of phases: status of phenomenology in the mechanics of heterogeneous materials . . . . . . . . . . . . . . 208
Some homogenization techniques . . . . . . . . . . . . . . . . . . . 210 5.2.1
Averaging procedures . . . . . . . . . . . . . . . . . . . . . 210
5.2.2
Micromechanical problem; boundary conditions . . . . . . . 211
5.2.3
Hill–Mandel lemma . . . . . . . . . . . . . . . . . . . . . . 212
5.2.4
Periodic case: use of multiscale asymptotic expansions . . . . 213
Application to linear elastic heterogeneous materials . . . . . . . . . 214 5.3.1
Stress-strain concentration problem; effective moduli . . . . . 214
5.3.2
Variational formulations; bounds . . . . . . . . . . . . . . . . 216
5.3.3
Case of a macro-heterogeneous medium . . . . . . . . . . . . 219
5.3.4
Statistical methods . . . . . . . . . . . . . . . . . . . . . . . 221
5.3.5
Self-consistent and generalized self-consistent schemes . . . . 227
Some examples, applications and extensions . . . . . . . . . . . . . . 229 5.4.1
Hill’s lens representation of bounds . . . . . . . . . . . . . . 229
Contents
5.5
5.6
5.7
5.8
5.4.2
Eshelby problems: elastic inclusion and elastic inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.4.3
Hashin–Shtrikman bounds (case of a locally and globally isotropic two-phase material) . . . . . . . . . . . . . . . . . . . . . . . 233
5.4.4
Self-consistent model . . . . . . . . . . . . . . . . . . . . . . 234
5.4.5
Dilute distribution . . . . . . . . . . . . . . . . . . . . . . . 235
Homogenization in thermoelasticity . . . . . . . . . . . . . . . . . . 235 5.5.1
Given eigenstrain field; residual stresses . . . . . . . . . . . . 235
5.5.2
Auxiliary problems in thermoelasticity; coupled thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Nonlinear homogenization . . . . . . . . . . . . . . . . . . . . . . . 241 5.6.1
Hill’s method in elastoplasticity . . . . . . . . . . . . . . . . 242
5.6.2
Approximations of the self-consistent scheme: Kröner and Berveiller–Zaoui models . . . . . . . . . . . . . . . . . . . . 243
5.6.3
Influence of the heterogeneity of plastic strain surrounding the inclusion on the quality of the self-consistent estimate in elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.6.4
Identification of the stress concentration law . . . . . . . . . . 247
5.6.5
Polycrystal behavior . . . . . . . . . . . . . . . . . . . . . . 254
Computation of RVE . . . . . . . . . . . . . . . . . . . . . . . . . . 258 5.7.1
Representative volume element size . . . . . . . . . . . . . . 260
5.7.2
A definition of the RVE size . . . . . . . . . . . . . . . . . . 264
5.7.3
RVE size for bulk copper polycrystals . . . . . . . . . . . . . 265
5.7.4
RVE size for thin polycrystalline copper sheets . . . . . . . . 268
5.7.5
Elastoplastic behavior of polycrystalline aggregates . . . . . . 274
Homogenization of “coarse grain structures” . . . . . . . . . . . . . . 275 5.8.1
An example of inhomogeneous average loading of a unit cell . 276
5.8.2
Generalized Hill–Mandel condition . . . . . . . . . . . . . . 278
6 Inelastic constitutive laws at finite deformation 6.1
xv
279
Geometry and kinematics of continuum . . . . . . . . . . . . . . . . 279
xvi
Non-Linear Mechanics of Materials
6.2
6.3
6.4
6.5
6.1.1
Observer and change of observer . . . . . . . . . . . . . . . . 279
6.1.2
Objective tensors . . . . . . . . . . . . . . . . . . . . . . . . 280
6.1.3
Position of the material body . . . . . . . . . . . . . . . . . . 280
6.1.4
Local placement and metrics . . . . . . . . . . . . . . . . . . 281
6.1.5
Rates; strain-rate . . . . . . . . . . . . . . . . . . . . . . . . 283
6.1.6
Objective derivatives . . . . . . . . . . . . . . . . . . . . . . 284
6.1.7
Strain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Sthenics and statics of the continuum . . . . . . . . . . . . . . . . . . 290 6.2.1
The method of virtual power methods; principle of objectivity of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.2.2
Lagrangian formulation of equilibrium . . . . . . . . . . . . 292
6.2.3
Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . 294
Constitutive laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 6.3.1
Formulation of constitutive laws . . . . . . . . . . . . . . . . 295
6.3.2
Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
6.3.3
Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 299
6.3.4
Plastic and viscoplastic fluids . . . . . . . . . . . . . . . . . 299
6.3.5
Elastoviscoplasticity . . . . . . . . . . . . . . . . . . . . . . 300
Application: Simple glide . . . . . . . . . . . . . . . . . . . . . . . . 308 6.4.1
Rotation of material fibres . . . . . . . . . . . . . . . . . . . 308
6.4.2
Analysis in elasticity and elastoplasticity . . . . . . . . . . . 309
6.4.3
Single crystal plasticity . . . . . . . . . . . . . . . . . . . . . 314
Finite deformations of generalized continua . . . . . . . . . . . . . . 319 6.5.1
Kinematics of Cosserat continuum . . . . . . . . . . . . . . . 319
6.5.2
Sthenics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.5.3
Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 324
6.5.4
Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . 330
7 Nonlinear structural analysis
333
Contents
xvii
7.1
The material object . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
7.2
Examples of implementations of particular models . . . . . . . . . . 336
7.3
7.2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
7.2.2
Prandtl–Reuss law . . . . . . . . . . . . . . . . . . . . . . . 336
7.2.3
Multikinematic law . . . . . . . . . . . . . . . . . . . . . . . 343
7.2.4
Porous materials . . . . . . . . . . . . . . . . . . . . . . . . 350
Specificities related to finite elements . . . . . . . . . . . . . . . . . 357 7.3.1
The “volume element” element . . . . . . . . . . . . . . . . 357
7.3.2
Treating incompressibility . . . . . . . . . . . . . . . . . . . 358
7.3.3
Plane stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
7.3.4
Periodic structures . . . . . . . . . . . . . . . . . . . . . . . 364
7.3.5
Large deformations . . . . . . . . . . . . . . . . . . . . . . . 364
7.3.6
Cosserat elements . . . . . . . . . . . . . . . . . . . . . . . . 369
8 Strain localization phenomena 8.1
8.2
371
Bifurcation modes in elastoplasticity . . . . . . . . . . . . . . . . . . 371 8.1.1
Formulation of the boundary value problem . . . . . . . . . . 371
8.1.2
Loss of uniqueness, general bifurcation modes . . . . . . . . 373
8.1.3
Well-posedness of the rate boundary value problem for the linear comparison solid . . . . . . . . . . . . . . . . . . . . . 376
8.1.4
Existence of velocity gradient discontinuities . . . . . . . . . 377
8.1.5
Bifurcation analysis in elastoplasticity . . . . . . . . . . . . . 380
8.1.6
Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
8.1.7
Localization criteria . . . . . . . . . . . . . . . . . . . . . . 386
8.1.8
Numerical simulation of some localization modes in elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Regularization methods . . . . . . . . . . . . . . . . . . . . . . . . . 395 8.2.1
Mesh dependence . . . . . . . . . . . . . . . . . . . . . . . . 395
8.2.2
Cosserat continuum at small deformation . . . . . . . . . . . 400
xviii
Non-Linear Mechanics of Materials
8.2.3
Elastoplastic Cosserat continuum and strain localization phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
Appendix: Notation used
407
A.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Contracted product: ., :, ::, etc. . . . . . . . . . . . . . . . . . 407 Dyadic product: ⊗, ⊗ , etc. . . . . . . . . . . . . . . . . . . . 407 Special tensors . . . . . . . . . . . . . . . . . . . . . . . . . 408 A.2 Vectors, matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Contracted products . . . . . . . . . . . . . . . . . . . . . . 408 Dyadic product . . . . . . . . . . . . . . . . . . . . . . . . . 408 A.3 Voigt notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Bibliography
411
Index
431
Chapter 1
Introduction
1.1. Model construction Throughout history, mankind has continuously developed its knowledge of the environment and its ability to use the resources of that environment for practical purposes, at the same time putting its imprint on it. During this development, there is usually no need for quantification of the elements utilized. But when the need arises to analyze and transmit its experience to a newer generation, or as systems increase in complexity, it becomes necessary to quantitatively describe this environment, whether natural or artificial, to identify its main characteristics, to find simplified representations of its variety of attributes, and to determine which components react in similar ways to given external stimuli. It is then necessary to build models, which consist of sets of algorithms and numbers, and which will continue to become more and more abstract. The field of materials science and structural engineering is not immune to this evolution. Where research and engineering meet, they share models for understanding and models for design, the former being useful for testing ideas for mechanisms, and the latter for building simplified representations of complex systems. This book addresses mechanical models by considering various scales, from the structure of the mechanical system, down to the microstructure of the material. In this framework, models for understanding will often be intended for the development of new grades of materials, while model for designing will be used for the dimensioning of mechanical components. In fact, improving performance rests on the ability to develop knowledge about materials while enhancing the attributes of the materials themselves. Thus, the knowledge of constitutive equations of materials [LEM85b] allows an optimal design involving the proper material at the right place. Using a smart design is not only elegant, but exhibits two important aspects: J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_1,
2
Non-Linear Mechanics of Materials
• there is an improvement in security. Having a good knowledge of the physical phenomena is better than applying a large security coefficient, which in fact stands for “ignorance coefficient”; moreover, in some cases, using larger amounts of matter can be detrimental: for example, increasing wall thickness of a pressure vessel can indeed decrease purely mechanical stresses, but also be detrimental if there are thermal gradients in the wall; • the result is a better performance from an ecological point of view. Thus, saving a few tenths of a gram on each can leads to significant savings of raw material, if one considers the multiple of billions that are made each year; likewise, decreasing weight of cars or planes reduces fuel consumption.
1.1.1. Models for understanding When one comprehends a model, one has captured a part of reality. One can check whether the material or the mechanical system behaves according to the predictions, or not. That is why a class of deductive models is currently being developed [FRA91]. They try to take into account the microstructure of the material in order to determine its macroscopic properties. Thus, a metal will be considered as a polycrystal, aggregate of grains of different crystallographic orientations with a well-characterized individual behavior; a composite will be represented by its matrix and its fibres; a concrete by its matrix and gravel. This approach aims to better predict the average global behavior through a representation of the materials heterogeneities: the model remains valid when the volume fraction of the components evolve. It may use simplified methods to account for microstructure, but calls more and more on microstructure calculations, in which a representative material element is discretized so that its response is evaluated by a finite element analysis. It is then rather powerful, due to its very principle, but it requires hard work to carry out, so that its use is still devoted to prospective understanding of local material behavior, having in view the improvement of the macroscopic mechanical properties. Conversely, the inductive approach will simply try to characterize in a global manner the behavior of a representative material element, disregarding the fine structure of the material. The task is to determine the cause–effect relationships between input and output of the studied process. Indeed, after an averaging process, different microscopical mechanisms can lead to identical global responses. However, using this approach outside of its initial domain of identification can be dangerous. Nonetheless, most of the time, this method is the only one appropriate for industrial design. These two approaches are phenomenological: building the model calls on a set of equations that must then be adjusted to a given material thanks to an experimental database made of mechanical experiments and, possibly, microstructural observations.
Introduction
3
1.1.2. Models for designing Good knowledge and good use of materials involve then three domains of activity. 1. Developing the new material itself (this domain does not exist in the case of geomaterials): the challenge is to promote new microstructures that improve intrinsic performance. 2. Characterizing the effective properties: the point here is to improve the knowledge of an existing material (the physical mechanisms related to deformation processes; the macroscopical mechanical effects), then to reduce uncertainties and improve reliability of studied models. 3. Developing numerical models: improvement of the component calculations, by introducing better algorithms or numerical methods, allows larger models to be treated, for instance 3D instead of 2D. Point (1) is the domain of chemists and metallurgists, point (2), the one of mechanics of materials (solid materials for our applications) and point (3), the one of computational mechanics. The following chapters are situated at the intersection between (2) and (3). In addition, the models developed relate to various environments and industrial operations that are shown in Fig. 1.1. The phase of design (Fig. 1.1a) involves a synthetic approach to the problem, which is in fact solved by an inverse method, i.e.: “what shape should a component have, which material should it be made of in order to fulfill requirements?”. Since there are many external elements, sometimes nonscientific, there is generally no other solution than to choose simple descriptions of the materials, and to apply codes or simplified rules. In most cases, this approach is sufficient. There sometimes remain nonstraightforward cases (high security components, . . .) that implies introduction of a justification procedure (Fig. 1.1b). Unlike the previous one, this approach is analytical, as geometry, loading, material, etc. are set. The point is just to characterize resistance through a direct calculation. This procedure can be used during construction, or much later on, in order to extend lifetime: a system (for instance a nuclear power plant) whose initial guaranteed lifetime is 30 years, by using simplified dimensioning methods, can have its lifetime extended for ten years or so thanks to more precise methods. It is necessary to use the most advanced models in the case where expertise is required, (Fig. 1.1c) as such an operation comes after an unexpected problem. It is important in this case to confront the model used with the corresponding real-world physical phenomena. Optimization (Fig. 1.1d) tends to diversify the model, thanks to powerful new computers that may perform dozens of numerical simulations on the studied structure.
4
Non-Linear Mechanics of Materials
Figure 1.1. Industrial operations where materials behavior is a key point
In the framework of these applications, the chosen model is going to depend on the domains of application. Without going into details, it is easy to imagine that, for instance, a steel at room temperature could be considered as linear elastic for the calculation of the deflection of a mechanical structure, viscoelastic for vibration damping, rigid-perfectly plastic for a limit loading calculation, elastoviscoplastic for the study of residual stresses; a polymer will be considered solid for shock problems, and fluid to study its stability over long periods of time.
1.2. Applications to models Using nonlinear constitutive equations has not yet become common practice in industry. For this purpose it is necessary to have adequate models and parameters characterizing the material and structural analysis code that will support those models. However, the 1980s was a period of important development for modeling. During this time there simultaneously bloomed macroscopical models answering to a wealth of different behaviors, the beginning of the use of ever more complex models, involving homogenization methods, and the power of computers allowing to use an ever growing number of variables to represent the material. Accordingly, the expression mechanics
Introduction
5
of materials emerged to refer to the study of the volume element of the mechanical engineer. The 1990s allowed a certain clarification in the models, but also and especially the appearance of robust integration methods, that are mandatory for involving very nonlinear models in routine calculations. These methods have been implemented in finite element analysis codes. All that is left to do now, is to build material databases and to save CPU time, in order to cope with nonlinear problems of 105 to 106 nodes in a couple of hours. The present book intends to provide a state of the art picture in terms of constitutive equations and associated numerical methods. Structural analysis is an academic discipline, where material modeling has (too) little room. This work does not address, of course, mesh generation problems, general methods to solve linear systems, . . . However, the reader will find here a description of the numerical methods available for implementing constitutive equations in structural analysis, like integration methods. On the other hand, the identification step of the parameters of the model has not been considered here. This void should be filled in the future. The reader will then find successively: – a short outline of basic concepts needed for understanding the book; this part contains an elementary presentation of mechanics and thermodynamics, an introduction to the nonlinear behavior of materials under one-dimensional loadings, the description of the main plasticity criteria, and a “survival kit” on numerical methods and finite element analysis; – a presentation of the models of plasticity and viscoplasticity in small strain, gathering classical models and more advanced configurations, notably on multimechanism models; – some elements of damage mechanics, including a discussion of the coupling between damage and plasticity, and of the activation–deactivation phenomenon; – a brief presentation of heterogeneous material mechanics and homogenization techniques, applied to elasticity, thermoelasticity, and nonlinear rheologies, which mentions a pragmatic solution to the scale transition problem; – a generalization of the previous models to finite deformation, with a few illustrations, and an application to generalized continuum media; – the implementation of the presented models in a finite element code, where the generic character of the method is emphasized; – the description of strain localization phenomena, with some comments on regularization methods, that prevent them from occurring.
6
Non-Linear Mechanics of Materials
This work calls on many handbooks for source material, not all of which, due to their number and breadth of coverage can be mentioned here. Having in mind the keyword materials, the reader is referred to the books of Friedel [FRI64] and Jaoul [JAO85], MacClintock and Argon [MCC66], Argon [ARG75], Ashby and Jones [ASH80]. Concerning solid mechanics, the books to take into consideration are those of Hill [HIL89], Germain [GER73b], Mandel [MAN66, MAN78], Salençon et al. [SAL84, HAL87], but also books on geomaterials [DAR87, COU91]. The constitutive equations are presented in the classical work of Lubliner [LUB90]. At last, we should mention the book of Simo and Hughes [SIM97], which constitutes an important contribution in terms of numerical treatment, and more recent books of Doghri [DOG00] or Doltsiris [DOL00]. But, among all those that have been cited, we give a particular mention to books issued from the French school of mechanics and materials [LEM85b, FRA91, FRA93], a group to which the present book claims to belong.
Chapter 2
General concepts
2.1. Formulation of the constitutive equations The constitutive equations presented in this section are generally expressed as ordinary differential equations (ODE), so that the mechanical response depends on the present load and on its history (hereditary behavior). Two strategies can be considered to account for this property. The first one introduces a functional dependency between the variables. In the second one, it is assumed that the history can be “summarised” in a series of internal variables, which allow the material to have the memory of the past history, and to have a full description of its present state. Except for the case of linear viscoelasticity, the models developed in the framework of the second approach are easier to manage than those coming from the first approach. The other important assumptions which are classically used in the development of constitutive equations are the following: 1. the local state principle, which assumes that the behavior in a given point M depends only on the variables defined in this point, and not from the surrounding; 2. the principle of material simplicity, which states that the mechanical behavior depends only on the first gradient of the transformation tensor; 3. the objectivity principle, expressing that the behavior does not depend on the observer; a consequence is that time cannot be used explicitly in the constitutive equations. All these assumptions will be revisited in Chap. 6. For the case of homogeneous isotropic materials, they can be summarized by a relation between stresses and strains, J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_2,
8
Non-Linear Mechanics of Materials
such as: t σ (t) = F−∞ (ε(τ ))
(2.1)
2.2. Principle of virtual power The present section gives a “minimal overview” of the principle of virtual power. The reader is invited to go to Chap. 6 on finite transformations for a comprehensive description. The principle tells us that, for any field of virtual velocity, v (x, t), the sum of the power of internal and external forces is equal to zero, for any domain D, with an external surface ∂D = S in a solid (Fig. 2.1).
Figure 2.1. Domain D, surface ∂D = S in a solid
The virtual power of the internal forces can be evaluated by means of the double dot product of the real stress tensor by the virtual strain-rate tensor: σ∼ : ε∼˙ dV (2.2) P (i) = − D
A classical series of manipulations can be made on this equation: • due to the symmetry of the stress tensor σ∼ , the strain rate can be replaced by the symmetrical part of the velocity gradient, then by the velocity gradient; • the resulting equation can be integrated by parts, and the divergence theorem can be applied, resulting in a partition of the internal power into a volumetric term and a surface term; • denoting by n the outgoing normal at a given point of ∂D, the following new expression can be obtained:
General concepts
P (i) =
D
(divσ∼ ).v dV −
∂D
(σ∼ .n) : v dS
9
(2.3)
The power of external forces consists in a volumetric term, for long distance actions, and a second one, representing surface forces: (e) P = f .v dV + F .v dS (2.4) D
∂D
The application of the principle leads to (divσ∼ + f ).v dV − D
∂D
(σ∼ .n − F ).v dS = 0
(2.5)
Since no assumption has been made on the velocity field, it can now be taken equal to zero successively on the external surface, or in any domain inside the body; it follows that divσ∼ + f = 0 σ∼ .n = F
∀x ∈ D
(2.6)
∀x ∈ ∂D
(2.7) (2.8)
The equilibrium of the body and the boundary conditions are obtained by choosing for D the solid itself. The same calculation scheme can be applied by using a statically admissible (SA) stress tensor field σ∼ ∗ , and a kinematically admissible (KA) velocity field v (a priori they are not linked by a constitutive equation), which verify respectively (2.9) and (2.10), and (2.11): divσ∼ ∗ + f = 0 in σij∗ nj vi
= Fi
on
= vid
on
(2.9)
Sfi Sdi
(2.10) (2.11)
where Sfi are the surfaces where the component i of the surface forces F is imposed, Sdi are the surfaces where the component i of the velocity is imposed, with Sdi ∩ Sfi = 0 It follows that
Sdi ∪ Sfi = S
∀i = 1, 2, 3
σ∼ ∗ : ∼˙ dV =
f .u˙ dV +
S
F .u˙ dS
(2.12)
10
Non-Linear Mechanics of Materials
2.3. Thermodynamics of irreversible processes The purpose of this section is to provide a few comments on the balance of energy, taking into account the effect of a mechanical strain, and the interactions between temperature, elastic strain and plastic strain. The development uses the small strain formalism; additional terms needed for a finite strain framework are shown in Chap. 6.
2.3.1. First and second principles of thermodynamics After mass conservation and equilibrium equations, the third fundamental conservation law in Continuum Mechanics expresses conservation of energy: this is the first principle of thermodynamics. After this principle, for any time during the thermomechanical process, the convective derivative of the total energy of a system is the sum ˙ of the power of the external applied forces, P (e) , and of the rate of received heat, Q. The energy is the sum of the internal energy and of the kinetic energy, which will not be considered here. The principle can be written by introducing the notation E for the internal energy on a domain D, and the specific internal energy e: de dE ˙ ρ dV = P (e) + Q (2.13) = dt D dt If the real fields are used as SA and KA fields, the power of the external forces can be written as P (e) =
D
σ∼ : ε∼˙ dV
(2.14)
˙ is the sum of two different contributions. The first one corresponds The term Q to the surface density of the received heat rate, which is defined as a function of the vector q and of the normal n, (n is the outgoing normal to the surface ∂D of the domain considered). The second, r, represents a volumetric heat coming from sources like chemical reactions, phase transformation or induction heating: ˙ = rdV − q.ndS = (r − div q) dV (2.15) Q D
∂D
D
The variation of the specific internal energy is then the sum of the rate of specific energy due to internal forces and of the rate of specific heat received: ρ
de = σ∼ : ε∼˙ + r − div q dt
(2.16)
General concepts
11
The second principle provides an upper bound of the rate of heat that can be taken by the volume D at a temperature T , and can be written as a function of the entropy S or of the specific entropy s: q.n r dS ≥ dV − dS dt D T ∂D T hence
ρ
D
q ds r − + div dV ≥ 0 dt T T
(2.17)
(2.18)
Helmoltz free energy is denoted by ψ. Using the equality e = ψ + T s, together with (2.16) and (2.18), the following relation, called Clausius’ inequality, holds: σ∼ : ε∼˙ − ρ
1 dψ − ρs T˙ − q . grad(T ) ≥ 0 dt T
(2.19)
2.3.2. Dissipation The method of local state assumes that the thermodynamical state of the continuum at a given point and a given time is completely given by the definition of a set of state variables. The free energy defines a potential, depending on temperature and on the state variables αI , which characterizes the mechanical system. It is then possible to express the derivative of ψ as dψ ∂ψ ˙ ∂ψ α˙I = T + dt ∂T ∂αI
(2.20)
Using this expression in (2.19), the following expressions are successively found: s=− σ∼ : ε∼˙ − ρ
∂ψ ∂T
(2.21)
∂ψ 1 α˙I − q . grad(T ) ≥ 0 ∂αI T
(2.22)
Two types of volumetric contributions can be seen in (2.22), a mechanical “intrinsic” dissipation, φ1 , and a purely thermal dissipation, φ2 : ∂ψ α˙I ∂αI
(2.23)
1 q . grad(T ) T
(2.24)
φ1 = σ∼ : ε∼˙ − ρ φ2 = −
12
Non-Linear Mechanics of Materials
2.3.3. Heat equation Classically, intrinsic and thermal dissipation are assumed to be uncoupled, so that one has to check the positivity of each of them. A positive thermal dissipation is obtained with Fourier’s conduction law. For the case of an isotropic material, it can be written: q = −k(T , αI ) grad(T )
(2.25)
The scalar function k, whose values are strictly positive, describes the conduction (units: W/m/K). The expression enforces the positivity of φ2 . If the conduction is anisotropic, it will be described by a non-negative quadratic form: φ2 =
1 grad T .k∼ .grad T T
(2.26)
Combining Fourier’s law with the relation (2.16), and using the expression of the derivative of ψ with respect to the temperature and the state variables, the so-called heat equation is obtained: ∂ψ ∂ 2ψ ˙ α˙ I −T (2.27) div k grad(T ) = ρCε T − r − σ∼ : ε∼˙ + ρ ∂αI ∂T ∂αI The equation provides a law for the evaluation of the temperature evolution in presence of mechanical deformation. One has defined a specific heat at constant strain, Cε = T ∂s/∂T (unit: J/kg/K). The left-hand side of the equation is proportional to the Laplacian T for the case of an isotropic conduction with a constant k coefficient. Since the unit of ρ is kg/m3 , the product ρCε , which characterizes the amount of heat trapped in the material, thus producing the temperature elevation, is in N/m2 /K. The product ρCε T˙ is then consistent with the volumetric term r, which is expressed in J/m3 , and with the mechanical terms σ∼ : ε∼˙ (Pa/s). The shape of the present expression does not depend on the mechanical constitutive equations. The differences between the materials will simply be taken into account by the shape of the free energy ψ and by the nature of state variables. The dissipated mechanical energy is represented by the term: φ1 = σ∼ : ε∼˙ − ρ (∂ψ/∂αI ) α˙ I , which takes into account the energy input, and the stored part of this energy (the storage can be temporary or not). A thermomechanical coupling can be exhibited at the level of the state variables if the cross derivative ∂ 2 ψ/(∂T ∂αI ) is non-zero.
2.3.4. Linear thermoelasticity The strain partition involves only an elastic and a thermal part: the elastic strain is taken equal to zero for a reference state σ∼ I , and the thermal dilatation is equal to
General concepts
13
zero for a reference temperature T I . For the isotropic case, a linear thermoelasticity law can simply be obtained by choosing the following expression for the free energy, where K characterizes the compressibility of the material (3K = 3λ + 2μ, λ and μ are the Lamé constants) and α the linear coefficient of thermal expansion: 1 ψ = σ∼ I : ε∼ + λ (trace(ε∼ ))2 + με∼ : ε∼ − 3Kα trace(ε∼ )(T − T I ) 2 1 ρCε − (T − T I )2 2 TI
(2.28)
The state variable is the elastic strain. Since the thermoelastic behavior is a reversible process, the intrinsic dissipation, φ1 = (σ∼ − ρ∂ψ/∂ε∼ ) : ε∼˙ , must be equal to zero for any infinitesimal increment of the thermoelastic strain. As a consequence, the partial derivative of ψ provides the elastic constitutive equations: ∂ψ = σ∼ I + λ trace(ε∼ )I∼ + 2 μ ε∼ − 3Kα(T − T I )I∼ ∂ε∼ σ∼ − σ∼ I = λ trace(ε∼ ) − 3α(T − T I ) I∼ + 2με∼ σ∼ = ρ
(2.29) (2.30)
The variation of the specific entropy can also be obtained: ρs = −ρ
ρCε ∂ψ = 3Kαtrace(ε∼ ) + I (T − T I ) ∂T T
(2.31)
This expression is consistent with the definition of Cε in the vicinity of T I , Cε = T I ∂s/∂T . One can easily illustrate the difference between the adiabatic and the isothermal evolutions on this simple case. Assuming that s = 0 in (2.31), the variation of temperature produced by a volume change V /V = trace(ε∼ ) can then be expressed: (T − T I ) = −
3KαT I trace(ε∼ ) ρCε
(2.32)
Introducing this expression in the constitutive equation (2.30), one can compare the responses: – isothermal: σ∼ − σ∼ I = λtrace(ε∼ )I∼ + 2με∼ 9K 2 α 2 T I trace(ε∼ )I∼ + 2με∼ – adiabatic: σ∼ − σ∼ I = λ + ρCε
(2.33) (2.34)
The shear modulus is left unchanged for the two cases (isothermal and adiabatic) but the first Lamé coefficient λ is larger for an adiabatic evolution. On the other hand, (2.31) shows that, for an adiabatic loading, a positive volume change (for instance in pure tension) will produce a decrease of the temperature, and vice-versa. The order of magnitude of these perturbations is small. For instance, typical values for steels are
14
Non-Linear Mechanics of Materials
(E = 200 GPa, ν = 0.3, ρ = 8000 kg/m3 , Cε = 500 J/kg/K, α = 15 × 10−6 ), such that: T 3Kα V V =− = −1.875 (2.35) T ρCε V V For an anisotropic body, the thermoelastic law can be written as: σ∼ = σ∼ I + : (ε∼ − ε∼ th ) ∼ ∼
(2.36)
where is a fourth-rank tensor characterizing the elastic moduli, and ε∼ th a second∼ ∼ (T − T I ), ε∼ the second-rank rank tensor characterizing thermal dilatation ε∼ th = α ∼ elastic strain tensor.
2.3.5. Nonlinear behavior Expression of the dissipation In a small perturbation framework, the total strain is partitioned into a thermoelastic (elastic and thermal dilatation) ε∼ e , and an inelastic strain ε∼ p : ε∼ = ε∼ e + ε∼ p
(2.37)
Note the slight notation change, since the term ε∼ e stands for the sum of the elastic and the thermal parts. Another modification is introduced in order to select between: • the variables αI that characterize the hardening in the material (id est the evolution of the properties by those mechanisms which preserve the continuum, namely local stress redistribution, variation of the dislocation density, . . .). The notation αI will still be in use for them; • variables characterizing damage, corresponding to material degradation related to porosity growth or microcrack propagation. These variables will be named dJ . The most general set of state variables includes then (ε∼ e , αI , dJ ), and the intrinsic dissipation is: ∂ψ ∂ψ ∂ψ ˙ α˙ I − ρ (2.38) dJ φ1 = σ∼ − ρ e : ε∼˙ e + σ∼ : ε∼˙ p − ρ ∂ε∼ ∂αI ∂dJ Since thermoelasticity is a thermoelastic process: φ1 = σ∼ : ε∼˙ p − ρ
∂ψ ∂ψ ˙ α˙ I − ρ dJ ∂αI ∂dJ
(2.39)
General concepts
15
The complete definition of the model is not made unless a constitutive equation is chosen. This will be made later. The second and third terms in the right-hand side describe the free energy change in the representative material element at constant temperature. This is the amount of energy which is stored in the material due to its hardening, or which is used to break open cracks or defects. This energy must be substracted from the plastic power to get the amount of energy transformed into heat. Having a positive value of 1 is equivalent to having an irreversible process that produces heat. The increase of temperature is given by: div k grad(T ) = ρCε T˙ − r − φ1 − φis (2.40) where φis represents the sum of the so called “isentropic” terms, deduced from the second term in parenthesis in the right-hand side of (2.27). Its origin is the temperature dependence of the thermodynamical forces.
Approximated evaluation of the dissipation In the literature, there is generally no specific connection between the expression of the constitutive equation and the intrinsic dissipation term, and this dissipation is simply evaluated as a fraction β of the plastic power (with 0 ≤ β ≤ 1). This is only an approximation, as explained by (2.39). The value classically chosen for β is larger than 0.9, specially for the case of large strains. This has to be related to the fact that, in this type of regime, the hardening is saturated, so that the material is no longer able to store energy by means of any hardening process. The case of an adiabatic loading can be introduced in this relation, by mentioning that the term div(k grad(T )) becomes zero. An estimation of the temperature evolution can be written (assuming no volumetric heat source for the sake of brevity): ρCε T˙ = φ1
(2.41)
The temperature increase will be significant for the case of rather rapid loadings, preventing any heat leak from happening. Consequently, unless thermoplastic coupling must be taken into account, for instance for forging problems, it will produce temperature increase for any type of external load (e.g., tension or compression).
Generalized standard model The thermodynamical forces AI (resp. yJ ) associated to the state variables αI (resp. dJ ) are defined by: ∂ψ ∂ψ yJ = −ρ (2.42) AI = ρ ∂αI ∂dJ
16
Non-Linear Mechanics of Materials
where the components of the stress tensor and the variables AI and yJ are put together in the vector Z. The vector z collects plastic strain and the state variables. The intrinsic dissipation is now: (2.43) 1 = σ∼ : ε∼˙ p − AI α˙ I + yJ d˙J = Z z˙ with:
z = ε∼ p , −αI , dJ
Z = {σ∼ , AI , yJ }
(2.44)
A model will be “generalized standard” if and only if there exists a potential (Z ) such that: ∂ z˙ = (2.45) ∂Z If is a convex function of Z, and contains the origin, the intrinsic dissipation is automatically positive, since, for any point Z, all the points of the domain defined by the equipotential surface = Cte are located on the same side of the tangent plane, defined by the normal ∂/∂Z , and: φ1 = Z
∂ ∂Z
(2.46)
Using the Legendre–Fenchel’s transform, one can introduce ∗ , depending on the time derivative of the state variables: ∗ (z˙ ) = max (Z z˙ − (Z)) Z
(2.47)
The variables Z can then be obtained as the partial derivative of ∗ with respect to z˙ . The main assumption in the generalized standard models is that the information needed to predict the nonlinear behavior of the material can be captured in two potentials: • free energy, defining the relationships between the state variables and the associated thermodynamical forces, and the constitutive equations of reversible phenomena; • the dissipation potential, providing information on the irreversibility of the process.
State coupling, dissipation coupling A state coupling [MAR89] will result from the presence in the free energy of terms involving the product of two state variables. In the partial derivative, there is an additional term in the thermodynamical force. This type of coupling will be considered for instance between damage and elasticity (see Chap. 4). It introduces a symmetry
General concepts
17
of the interactions, for instance for two variables A1 and A2 , thermodynamical forces associated with α1 and α2 : ∂A2 ∂ 2φ ∂A1 = = ∂α2 ∂α1 ∂α1 ∂α2
(2.48)
On the other hand, a dissipation coupling is introduced each time there are several potential functions K , such as: z˙ =
∂K K
∂Z
(2.49)
This last type of coupling can be seen in “multisurface” models, or in crystal plasticity.
2.4. Main class of constitutive equations 2.4.1. Basic building blocks The nature of the qualitative response of materials to a few simple tests allows us to define several classes. These basic material behaviors can be illustrated by elementary mechanical systems. There are only three types, elasticity, time independent plasticity, and viscosity. The corresponding elements are: 1. The spring, characterizing linear elasticity, for which strain is totally reversible after unloading, and providing a one-to-one relation between stress and strain components (Fig. 2.2a). 2. The dash-pot, representing linear (Fig. 2.2b) or nonlinear (Fig. 2.2c) viscosity. The viscosity is said to be “pure” if a one-to-one relation between load and strain rate is introduced. If the relation is linear, the corresponding model is Newton’s law. 3. The frictional device, used to illustrate a basic behavior where permanent strain appears if the load is large enough. If the corresponding threshold does not depend on the subsequent loading, the behavior is perfectly plastic. If, in addition, the elastic strain before permanent strain is neglected, the model is simply rigid-plastic. All these elements can be combined together in order to form rheological models. They represent mechanical systems that can help to define various models. This type of construction has nothing to do with the deformation mechanisms, but it can be useful to better understand the nature of the stress–strain relations in the various
18
Non-Linear Mechanics of Materials
Figure 2.2. The basic “building blocks”
constitutive equations. New models can be developed easily by considering the combination of several elements. This introduction offers the opportunity to present in a simple framework most of the concepts used for tridimensional loading cases. The mechanical response of the systems will be considered in three different planes, allowing us to illustrate the behavior for various loading types: – hardening, or monotonic increase of the stress or the strain (strain–stress plane, ε–σ ); – creep, a test performed under constant stress (time–strain plane, t–ε); – relaxation, a test performed under constant strain (time–stress plane, t–σ ).
2.4.2. One-dimensional plasticity Elastic perfectly plastic models The system shown in Fig. 2.3a produces an elastic perfectly plastic behavior, as modeled in Fig. 2.3c. The maximum absolute value of the stress that can be applied to the system is σy . The model is characterized by a loading function f , depending on the stress σ only, and defined by: f (σ ) = |σ | − σy (2.50)
General concepts
19
Figure 2.3. Systems built as series or parallel sets of friction device and spring
The elasticity domain corresponds to the negative values of f , and the behavior of the system can be summarized by the following equations: – elastic behavior if: – elastic unloading if: – plastic flow if:
f <0 f =0 f =0
and f˙ < 0 and f˙ = 0
(˙ε = ε˙ e = σ˙ /E) (˙ε = ε˙ e = σ˙ /E) (˙ε = ε˙ p )
For the elastic regime, the plastic strain rate is zero, meanwhile the elastic strain rate becomes zero during plastic flow. As a consequence, the expression of the plastic strain rate cannot take the stress as a critical variable. The total strain rate must be chosen as a driving variable. The model does not involve any hardening, since the stress level no longer changes when stress reaches the limit of the elastic domain. There is no stored energy during the deformation, and the heat dissipation is equal to the plastic power. The model can reach infinite strain under constant load, leading to breakdown of the system by excessive strain.
Prager model When the elements are associated in parallel, as shown in Fig. 2.3b, the resulting behavior is illustrated in Fig. 2.3d. In such a case, a linear hardening is observed. The hardening is said to be “kinematic”, since it depends on the current value of the
20
Non-Linear Mechanics of Materials
plastic strain only. Under the present form, the model is rigid-plastic. It becomes elasto-plastic if a second spring is added, in series with respect to the first assembly. The shape of the curve in the plane σ –εp is related to the stress level in the friction device, X. Its expression is obtained after writing that, during plastic flow, the stress in the spring is X = H ε p . On the other hand, plastic flow appears only if the absolute value of the stress in the friction device, |σ − H ε p |, is equal to σy . For a given strain, this stress X represents an internal stress that characterizes the new neutral state in the material. This second example gives the opportunity to consider a more complete model. The loading function depends now on the applied stress and on the internal stress. It is written: (2.51) f (σ, X) = |σ − X| − σy Plastic flow occurs if and only if f = 0 and f˙ = 0. This leads to the following condition: ∂f ˙ ∂f σ˙ + X=0 (2.52) ∂σ ∂X Thus: sign(σ − X) σ˙ + sign(σ − X) X˙ = 0 ˙ and finally: ε˙ p = σ˙ /H σ˙ = X,
(2.53) (2.54)
In this case, there is a stress evolution during plastic flow, such that it can be used as a control variable. One can also express the plastic strain rate as a function of the total strain rate, by using the strain partition assumption, combined with the expression of the plastic strain rate. In this framework, one can return to the perfectly plastic case by choosing H = 0: E ε˙ (2.55) ε˙ p = E+H The amount of dissipated energy for one cycle is exactly the same as for the previous assembly. As a matter of fact, a fraction of the energy is now temporarily stored in the material (here in the spring), and totally recovered after unloading. This introduces the concept of reversible hardening, for those types of hardenings which do not involve any dissipation for a loading cycle. Most of the hardening rules developed later in this chapter will introduce an energy dissipation (see Sect. 3.7.2).
General expression for one-dimensional elastoplasticity In the general case, the “loading–unloading” conditions are the following; – elastic behavior if: f (σ, Ai ) < 0 – elastic unloading if: f (σ, Ai ) = 0 – plastic flow if: f (σ, Ai ) = 0
and f˙(σ, Ai ) < 0 and f˙(σ, Ai ) = 0
(˙ε = σ˙ /E) (˙ε = σ˙ /E) (˙ε = σ˙ /E + ε˙ p )
General concepts
21
The plastic modulus H generally depends on plastic strain and on the hardening variables. Its value for a given point defined by (σ, Ai ) can be found by enforcing the condition that the current point remains at the border of the elastic domain during plastic flow. The resulting equation is the “consistency condition”: f˙(σ, Ai ) = 0
(2.56)
These expressions may appear too complex to define one-dimensional plastic flow (and they are!), nevertheless this framework is useful, since nothing but the same tools will be used for characterizing three-dimensional loadings. In the previously defined examples, the yield domain is either fixed or mobile, but for each case, its size is kept constant. No hardening variable is associated to the perfectly plastic case. Linear kinematic hardening involves a variable X which depends on the current value of the plastic strain. This variable will become tensorial in the 3D case. The corresponding type of hardening is called kinematic hardening (Fig. 2.4b). In the special case illustrated by the rheological model, the evolution of the variable X is linear with respect to the plastic strain, so that the model is called linear kinematic (Prager, 1958). Instead of being translated and keeping a constant size, the elastic domain can have a constant location (classically, the origin of the deviatoric stress space), and a changing size. This other case describes a material which presents isotropic hardening (Taylor and Quinney, 1931). The hardening variable in f is the size of the elastic domain, denoted R: (2.57) f (σ, X, R) = |σ | − R − σy The evolution of this variable does not depend on the sign of the plastic strain rate. It will then be related to the accumulated plastic strain rate, p, whose rate is defined by the absolute value of the plastic strain rate: p˙ = |˙ε p |. There is no difference between p and ε p for a monotonic loading. In such a case, checking the consistency condition is equivalent to enforcing the current value of the applied stress to be on the boundary of the yield domain. For kinematic hardening, one gets σ = X + σy , and for isotropic hardening, σ = R + σy . As a consequence, the shape of the tension curve is exactly determined by the shape of the equations governing the evolution of the hardening variables. The two models previously considered provide linear stress-strain curves after the initial yield, with either a null or a constant plastic modulus. Most of the time, one has to consider curves which have decreasing values as a function of the strain (for instance Ramberg–Osgood’s model, with two material parameters, K and m), or an exponential function (this last formulation is often more satisfactory, since it introduces a saturating value, the ultimate stress σu on the material element; again there are two material parameters, σu and b): σ = σy + K(ε p )m
(2.58)
σ = σu + (σy − σu ) exp(−b ε ) p
(2.59)
Let us note that the plastic behavior is often defined by a series of values of the couple (plastic strain–stress), without any reference to an explicit form of the constitutive
22
Non-Linear Mechanics of Materials
equation. The hardening type considered in this case will be isotropic. This type of hardening is predominant for rather large strains (more than 10%). Nevertheless one has to keep in mind that kinematic hardening plays an important role during unloadings, even for large strains, and that it is predominant for small strains and cyclic loadings. This is for instance the only way to successfully represent the Bauschinger effect: the yield limit under a compressive (resp. tensile) loading applied after a tensile (resp. compressive) prehardening is smaller than the reference yield limit for a compression (resp. tension) loading. Isotropic hardening is nevertheless often promoted, since its numerical implementation is easier, compared to kinematic hardening.
Figure 2.4. Illustration of the two main hardening types
2.4.3. One-dimensional viscoelasticity A first rheological model Maxwell’s model is formed by gathering a dashpot and a spring as a serial assembly (Fig. 2.5a), meanwhile Voigt’s model puts the same elements in parallel (Fig. 2.5b). The constitutive equations for each of these models are respectively: −Maxwell: ε˙ = σ˙ /E0 + σ/η −Voigt: σ = H ε + η˙ε or
ε˙ = (σ − H ε)/η
(2.60) (2.61)
Voigt’s model is not able to produce any instantaneous response (no elasticity). Its relaxation function is then not continuous but piecewise derivable, with a finite jump at the loading time: when applied at t = 0, a strain jump will produce an infinite stress. The applicability of the model for modeling relaxation is then limited, but it can be used if the strain is applied progressively. In fact, most of the time, an additional spring is added, leading to Kelvin–Voigt’s model (see next paragraph). Under a
General concepts
23
Figure 2.5. Elementary behavior of Maxwell’s and Voigt’s models
constant stress σ0 , the strain tends to an asymptotic value, σ0 /H , the amount of creep is then limited (Fig. 2.5c). On the other hand, if the strain is slowly increased until ε0 , then fixed at this last value, the asymptotic stress will be H ε0 . The stress does not disappear completely. On the contrary, for Maxwell’s model, the creep rate is constant (secondary creep, Fig. 2.5c), and the stress tends to zero during a relaxation test (Fig. 2.5d). When models and loadings are both so simple, the mechanical response can be obtained easily by integration of the differential equations. The expressions are then respectively, for Maxwell’s model: −creep for a stress σ0 : ε = σ0 /E0 + σ0 t / η −relaxation for a strain ε0 : σ = E0 ε0 exp[−t/τ ]
(2.62) (2.63)
and for Voigt’s model: −creep for a stress σ0 :
ε = (σ0 / H )(1 − exp[−t/τ ])
(2.64)
The unit of the constants τ = η/E0 and τ = η/H is second; τ stands for the so-called Maxwell relaxation time.
24
Non-Linear Mechanics of Materials
Figure 2.6. Example of a combined model
2.4.4. Study of a combined model As shown in Fig. 2.6a, Kelvin–Voigt’s model provides the following mechanical responses for t > 0, under a creep loading at a stress σ0 (denoting by τf the characteristic time τf = η/H ), and under a relaxation loading at a strain ε0 (denoting by τr the characteristic time τr = η/(H + E0 )): ε(t) = C(t) σ0 = σ (t) = E(t) ε0 =
1 1 + (1 − exp[−t/τf ]) σ0 E0 H
H E0 + exp[−t/τr ] E0 ε0 H + E0 H + E0
(2.65) (2.66)
The relaxation time τr is shorter than the corresponding time for creep loading, τf . The material evolution toward an asymptotic state is then quicker in relaxation than in creep. Zener’s model (Fig. 2.6b) can be considered as another Kelvin–Voigt’s model, after introducing the variable change 1/E1 = 1/E0 + 1/H , and E2 = E0 + H . The two models are exactly equivalent under both creep and relaxation loading. This model is sometimes used for representing concrete behavior. The models previously described can be improved: • a generalized Kelvin–Voigt’s model is obtained by constructing a series assembly of several sets of dashpot-springs (H, η); this type of approach is used for network polymers; • a generalized Maxwell’s model is obtained by considering a parallel assembly of several sets of dashpot-springs (E2 , η). This model is generally in good agreement with the behavior of thermoplastic polymers.
General concepts
25
2.4.5. One-dimensional viscoplasticity A simple rheological model
Figure 2.7. Generalized Bingham’s model
In Fig. 2.7a, it is shown how a viscoplastic model can be derived from a plastic model, by adding a dashpot. The resulting model is the so-called generalized Bingham’s model. The original Bingham’s model involves neither the spring assembled in series (E → ∞, no instantaneous elastic behavior, this is a rigid-viscoplastic model) nor the spring in parallel (H = 0, no hardening). The elastic strain is characterized by the spring (E), the viscoplastic strain, denoted by ε vp , is illustrated by the parallel assembly of the friction device and the dashpot. The equations of the model are obtained by combining all the elementary subsets: X = H ε vp
σv = η ε˙ vp
σp ≤ σy
(2.67)
where X, σv and σp are respectively the stresses in the spring (H ), in the dashpot and in the friction device, and: (2.68) σ = X + σv + σp An elastic domain is then present in this viscoplastic model. The border of the domain is reached when |σp | = σy . Three regimes can then be observed, according to the viscoplastic strain rate, that can be either zero, positive, or negative: (a) ε˙ vp = 0 (b) ε˙ vp > 0 (c)
ε˙ vp < 0
|σp | = |σ − H ε vp | ≤ σy σp = σ − H ε vp − η ε˙ vp = σy
(2.69) (2.70)
σp = σ − H ε vp − η ε˙ vp = −σy
(2.71)
Case (a) corresponds to the interior of the elastic domain (|σp | < σy ) or to a state of elastic unloading (|σp | = σy and |σ˙ p | ≤ 0). For the two other cases, a viscoplastic flow is present, with (|σp | = σy and |σ˙ p | = 0). Using the definition x = max(x, 0), the three cases can be summarized by the unique expression:
η ε˙ vp = |σ − X| − σy sign(σ − X) (2.72)
26
Non-Linear Mechanics of Materials
or:
f sign(σ − X) with f (σ, X) = |σ − X| − σy (2.73) η If compared to the initial plastic model, the nature of the present one is totally different, since the current point representing the state of stress can be in the area defined by f > 0. On the other hand, the viscoplastic strain rate is time dependent. It can be non-zero without any stress increment, for instance. This new type of behavior explains why, in Fig. 2.7b, the tensile curve is no longer unique (the larger the strain rate, the higher the viscous stress σv and the higher the tension curve). This explains also why, during unloading at prescribed stress, the current stress point does not meet instantaneously the elastic domain (one can have a positive viscoplastic flow for decreasing stress values). Last but not least, this type of model allows the user to model creep tests and relaxation tests. ε˙ vp =
For a creep test (Fig. 2.8), assuming that stress is applied instantaneously at time t = 0 (with a step between 0 and σo > σy ), starting from a reference state where all the strains are equal to zero, the model predicts that the viscoplastic strain is given by an exponential function of the time t, with a characteristic time τf = η/H (Fig. 2.8a): σo − σy t ε vp = (2.74) 1 − exp − H τf Figure 2.8b shows, in the stress-viscoplastic strain plane, the evolutions of the internal stress X and of the viscoplastic threshold X + σy . When the threshold reaches the level of the applied stress σo , the creep strain rate becomes zero.
Figure 2.8. Creep with a Bingham’s model
For a relaxation test, the mechanical response to a strain step (from 0 to εo such as Eεo > σy ) involves now a characteristic relaxation time τr = η/(E + H ): Eεo t t E + (2.75) 1 − exp − H + E exp − σ = σy E+H τr E+H τr Figure 2.9a shows the path followed by the current stress point during the relaxation (slope −E, since ε˙ vp + σ˙ /E = 0). On the other hand, Fig. 2.9b shows the path
General concepts
27
followed during a fading memory test, when recovery is present. The test consists of the following steps: OA, load from 0 to εmax ; AB, constant strain εmax ; BC, unloading from εmax to 0; CD, constant strain equal to zero. Depending on the initial loading level, the viscoplastic strain increment after unloading (branch BC of the loading path) can be either negative or equal to zero, nevertheless, the viscoplastic strain will never reach zero, unless the initial yield σy is zero itself. In fact, the asymptotic steady state is obtained as the intersection of the segment of slope −E joining C and O, and of the lower yield, −σy + H εp . If there is no initial threshold, the strain partition is no longer relevant: besides, the resulting model is Kelvin–Voigt’s model, a viscoelastic one, and there is no residual stress in the fading memory test.
Figure 2.9. Typical behavior of Bingham’s model under prescribed strain (relaxation, then fading memory test)
A few classical viscoplastic models In the previous example, the viscoplastic strain rate is proportional to a given “active” stress, obtained as the difference between the applied stress and the threshold. This active stress represents the distance between the current point and the border of the elastic domain. This is then nothing but the value taken by the function f . The linear relationship can be also replaced by a more general one, by introducing the viscosity function φ, which provides then for a tensile test: ε˙ vp = φ(f )
(2.76)
For a model presenting both kinematic and isotropic hardening, this relation can be inverted to get, for a tensile test: σ = σy + X + R + φ −1 (˙ε vp ) = σy + X + R + σv
(2.77)
28
Non-Linear Mechanics of Materials
The tension curve is determined by the evolution of the threshold, as in the plastic case (through the X and R variables), but also by a viscosity function, which determines the value of the viscous stress σv . For obvious experimental reasons, it is assumed that φ(0) = 0, and that φ is a monotonic, always increasing function. For the limiting case where σv becomes zero, a time independent plastic model is recovered. In a viscoplastic framework, one can have two ways to introduce hardening. As previously shown, the variables X and R, defined for plastic models, are still a good choice for a viscoplastic one. A second opportunity is to change the shape of the viscous stress. The models using the first approach are said to have an additive hardening, meanwhile those which use the second choice have a product hardening. A combined approach is also possible. Finally, opposite to the plastic case, a model in which the elastic domain is reduced to the origin (σ = 0), is admissible, even without hardening. In fact, this is nothing but the most simple model, Norton’s model (with two material parameters, K and n): ε˙
vp
=
|σ | K
n sign(σ )
(2.78)
It can be generalized to produce a model with a threshold, but still without hardening, and also to produce more complex models, using again X and R (additive hardening, [LEM85b])
ε˙
vp
ε˙ vp
|σ | − σy n = sign(σ ) K |σ − X| − R − σy n = sign(σ − X) K
(2.79) (2.80)
Many other solutions can be found in the literature, for instance, a hyperbolic sinus is used by Sellars and Teggart instead of a power function (no hardening, material parameters A and K): |σ | ε˙ vp = A sinh sign(σ ) (2.81) K Product type hardening is obtained when the function φ does not depend only on f . For instance, Lemaitre’s rule can be written with three material parameters, K, m and n (all of them positive): ε˙
vp
=
|σ | K
n
p −n/m sign(σ ) with p˙ = |˙εvp |
(2.82)
General concepts
29
2.4.6. Temperature influence All the material parameters previously defined are likely to depend on temperature. The relationships can be defined by tables, for instance each time they are identified for a series of constant temperature tests. For those cases where the physical mechanisms are well defined, one can write precise explicit expressions for the temperature evolution. The most popular rule is Arrhenius’. This rule is valid for creep tests. It introduces a thermal activation energy, Q, and R, the constant of perfect gas (note that the ratio Q/R is homogeneous to a temperature), and assumes that, for a given load, the higher the temperature, the higher the viscoplastic strain rate: ε˙ vp = ε˙ o exp(−Q/RT )f (σ )
(2.83)
Time–temperature equivalences can then be founded on this rule, so that researchers often use laboratory tests at temperatures higher than the target temperature in the applications, in order to collect in a short time a series of data valid for longer time periods. The approach must be taken with care. For materials presenting aging, new physical phenomena can appear, so that the extrapolation is no longer valid. Anyway, this approach cannot generally be extended to large temperature ranges.
2.4.7. Summary The very general expressions presented in this section summarize the nature of the models for elasticity, plasticity, viscoplasticity. For the last two, an elastic domain has been defined (possibly reduced to the origin for the viscoplastic model) and hardening variables have been introduced. They have thus a lot of points in common. But they also have a big difference: plastic flow is time independent, viscoplastic flow is delayed: dεvp = g(σ, . . . )dt (2.84) dε p = g(σ, . . . )dσ This last remark will have important consequences for the expression of the elasto(visco-)plastic tangent moduli. For the moment, only very naive forms of the constitutive equations have been considered. More realistic expressions will be given later on, directly in the threedimensional form.
2.5. Yield criteria The one-dimensional expressions delivered in the preceding section for the inelastic models exhibit an elastic domain, which is an area in the space formed by the stress
30
Non-Linear Mechanics of Materials
and the hardening variables in which there is no plastic flow. For the one-dimensional case, this domain is restricted to a segment that can be transformed by translation or expansion (sometimes, this is only one point). On the other hand, several models are able to describe the presence of a maximum stress level, limiting the admissible state for a material element. The aim of the present section is then to extend these concepts for 3D loadings. It starts with a short review of the tools available in the framework of continuum mechanics, then the main classes of criteria are shown. Each material will have its relevant model built from the combination of a given flow rule, taken from the previous section, and of a criterion chosen in the following list.
2.5.1. Available tools For the one-dimensional case, the elastic domain was characterized by two critical stress values, one in tension and one in compression, for which plastic flow may happen. For Prager’s model, the initial yield is defined by the segment [−σy , σy ], and the current location for a strain ε p is [−σy + X, σy + X], with X = H ε p . It is described by the loading function f (defined from R2 into R, f : (σ, X) → f (σ, X)). Under multiaxial loadings, f will be a function of the stress tensor σ∼ and of the ten= H ε∼ p , (from R12 in R), such as: if f (σ∼ , X ) < 0, the stress state is elastic, sor X ∼ ∼ ) = 0, the current point is at the boundary of the domain, meanwhile the if f (σ∼ , X ∼ ) > 0 defines the outside of the domain. In the general case, f will condition f (σ∼ , X ∼ operate on stresses and on tensorial or scalar hardening variables, thus a set (σ∼ , Ai ). The present section will not consider hardening: only the restrictions of the function f to the stress space is described. Experiments show that, for most materials, the initial elastic domain is convex (this is specially true for metals and alloys whose deformation mechanisms are crystallographic slip). The loading function is then convex with respect to σ∼ , so that, for any real λ between 0 and 1, and for any couple of stresses (σ∼ 1 , σ∼ 2 ) belonging to the boundary: (2.85) f (λ σ∼ 1 + (1 − λ) σ∼ 2 ) λ f (σ∼ 1 ) + (1 − λ)f (σ∼ 2 ) Material symmetries have to be respected by the plasticity criteria. In particular, for isotropic materials, f must be a symmetric function defined by means of eigenstresses, or a function of the stress invariants given by the characteristic polynomial: I1 = Tr (σ∼ ) = σii I2 = (1/2) Tr σ∼ 2 = (1/2) σij σj i I3 = (1/3) Tr σ∼ 3 = (1/3) σij σj k σki
(2.86) (2.87) (2.88)
It is experimentally observed that plastic flow does not depend on hydrostatic pressure. As a consequence, the deviatoric stress and its invariants are chosen instead of
General concepts
31
the stress tensor itself in the expressions of the plasticity criteria. The deviatoric stress tensor ∼s is obtained by removing the hydrostatic pressure from σ∼ . s = σ∼ − (I1 /3) I∼ J1 = Tr ∼s = 0 J2 = (1/2) Tr ∼s 2 = (1/2) sij sj i J3 = (1/3) Tr ∼s 3 = (1/3) sij sj k ski
∼
(2.89) (2.90) (2.91) (2.92)
It is often convenient, for identification purpose, to define criteria by means of expressions homogeneous to stresses. This leads to writing f as a function of J instead of J2 : as shown by (2.93), J is nothing but the absolute value of the applied stress for a tensile loading, and can be expressed with respect to the eigenstresses σ1 , σ2 , σ3 for 3D cases: 1/2 1/2 J = (3/2)sij sj i = (1/2) (σ1 −σ2 )2 +(σ2 −σ3 )2 +(σ3 −σ1 )2 = |σ | (2.93) The preceding value is related to the so-called octahedral shear stress. The octahedral planes are those whose normal vector is {1, 1, 1} in the principal stress space. The normal and the tangential components of the stress vector for this plane can be expressed as a function of the three eigenstresses σ1 , σ2 , σ3 : √ σoct = (1/3) I1 τoct = ( 2/3) J (2.94) The value of J is then nothing but octahedral shear. The plane with a normal (1, 1, 1) will then play a specific role for the representation of the criteria. The reason is that all the stress states which differ only by a spherical tensor (they are equivalent with respect to a criterion which does not introduce hydrostatic pressure as a critical variable) have the same projection on this plane. This plane is shown in Fig. 2.10. Its equation is σ1 + σ2 + σ3 = −I1 /3. The projections of the three principal axes determine sectors of 2π/3.
2.5.2. Criteria without hydrostatic pressure Von Mises criterion Since the trace of the stress tensor does not play an active role, a simple idea consists in using the second invariant of the deviatoric stress tensor, or J . This produces an ellipsoid in the space of the symmetric stress tensors (since J has a quadratic expression with respect to the components sij ). Denoting by σy the initial yield stress in tension, the expression is then: (2.95) f (σ∼ ) = J − σy
32
Non-Linear Mechanics of Materials
show the points equivalent to a simple tension, show the points equivalent to a simple compression (for instance an equibiaxial loading, since the state σ1 = σ2 = σ is equivalent to σ3 = −σ ), is a shear state
Figure 2.10. Definition of the deviatoric plane
Tresca criterion Von Mises criterion involves the maximum shear in each principal plane, (σi − σj ). On the contrary, Tresca criterion takes only the largest. As in the preceding case, hydrostatic pressure does not modify the value of the criterion. Instead of having a regular surface like von Mises, the Tresca criterion is only piecewise linear. f (σ∼ ) = max |σi − σj | − σy i,j
(2.96)
Comparison of Tresca and von Mises criteria Several subspaces are convenient to compare von Mises and Tresca criteria. The limit surfaces defined by each of them are drawn in given planes. The most popular choices are: • the tension–shear plane (Fig. 2.11a), when the only non-zero components are σ = σ11 and τ = σ12 ; the expressions of the criteria are then: − von Mises: − Tresca:
1/2 − σy f (σ, τ ) = σ 2 + 3τ 2 2 2 1/2 f (σ, τ ) = σ + 4τ − σy
(2.97) (2.98)
• the plane of two eigenstresses (σ1 , σ2 ) (Fig. 2.11b), when the third one σ3 is zero:
General concepts
1/2 − von Mises: f (σ1 , σ2 ) = σ12 + σ22 − σ1 σ2 − σy − Tresca: f (σ1 , σ2 ) = σ2 − σy if 0 σ1 σ2 f (σ1 , σ2 ) = σ1 − σy if 0 σ2 σ1 f (σ1 , σ2 ) = σ1 − σ2 − σy if σ2 0 σ1 (symmetry wrt σ1 = σ2 axis)
33
(2.99) (2.100) (2.101) (2.102) (2.103)
• in the deviatoric plane (Fig. 2.10), the von Mises criterion is represented by a circle (this is consistent with the interpretation in terms of octahedral shear); Tresca’s criterion is a hexagon; • in the principal space, each criterion is represented by a cylinder whose axis is (1, 1, 1), and the section is the curve defined in the deviatoric plane.
Figure 2.11. Comparison of Tresca (dashed √ lines) and von Mises (continuous lines) criteria (a) in tension–shear (von Mises: τm = σy / 3, Tresca: τt = σy /2), (b) in biaxial tension
2.5.3. Criteria involving hydrostatic pressure This type of criterion is needed to represent the behavior of powders, soils, and also for models that take damage into account. Classically, a compressive hydrostatic pressure makes plastic flow more difficult. One of the consequences of this formulation is the dissymmetry between tension and compression. This section shows a few examples, and Sect. 3.11 will complete the description for porous materials.
Drucker–Prager criterion This is a generalization of the von Mises criterion, formed by means of a linear combination of the second invariant of the stress deviator and of the trace of the stress
34
Non-Linear Mechanics of Materials
tensor. It is still represented by a circle in the deviatoric plane, nevertheless, the radius of this circle depends on the location on the axis (σ1 = σ2 = σ3 ) (see Fig. 2.12a): f (σ∼ ) = (1 − α)J + αI1 − σy
(2.104)
The yield stress under simple tension is still σy , and the yield stress in simple compression is −σy /(1 − 2 α). The parameter α depends on the material. It has to be between 0 and 1/2. For α = 0, this model is equivalent to von Mises (Fig. 2.12b).
Figure 2.12. Drucker–Prager criterion, (a) in eigenstress space, (b) in the plane I1 − J
Mohr–Coulomb criterion The Mohr–Coulomb model is an extension of the Tresca criterion, involving, like the latter, the maximum shear stress, but also a pressure dependent stress, chosen as the centre of the Mohr circle corresponding to the maximum shear: f (σ∼ ) = σ1 − σ3 + (σ1 + σ3 ) sin φ − 2C cos φ
(with σ3 σ2 σ1 )
(2.105)
The idea behind the criterion is the presence of internal friction in the material, so that the ultimate shear that can be applied to the material (Tt in Fig. 2.13a) is larger and larger if the normal compression stress is high. The admissible limit forms an intrinsic curve in Mohr’s plane. The expression (2.105) is obtained with the assumption of a linear friction coefficient: |Tt | < − tan(φ) Tn + C
(2.106)
The material parameter C is called cohesion, and illustrates the shear stress which can be applied on the material under zero mean stress. The angle φ denotes the internal friction in the material. If C is zero and not φ, the material is pulverulent. If φ is zero and not C, as for the Tresca criterion, the material is purely cohesive.
General concepts
35
Another expression of the criterion can be written as a function of the coefficient of passive earth pressure, Kp , and of the compressive yield stress, Rp : f (σ∼ ) = Kp σ1 − σ3 − Rp with 1 + sin φ 2 C cos φ Rp = Kp = 1 − sin φ 1 − sin φ
(2.107) (2.108)
In the deviatoric plane (Fig. 2.13b), the criterion is represented by an irregular hexagon, characterized by the following values (with p = (−1/3)I1 ): √ 6(C cos φ − p sin φ)/(3 + sin φ) √ σc = 2 6(−C cos φ + p sin φ)/(3 − sin φ) σt = 2
(2.109) (2.110)
Figure 2.13. Illustration of the Mohr–Coulomb criterion, (a) in the Mohr plane, (b) in the deviatoric plane
“Closed” criteria The two preceding criteria predict that the material cannot meet the yield limit under triaxial compression. This behavior is generally not true for real materials sensitive to hydrostatic pressure. More complex models have then been developed to correctly simulate stress states coming from operations like high isostatic pressure experiments for instance. The limit curve is decomposed into two parts, that are pasted for a critical negative value of the hydrostatic pressure. The most popular of these models is the cap model, which starts from Drucker–Prager, and introduces an elliptic “cap” for the negative values of the hydrostatic pressure, or the Cam–clay model (developed at Cambridge, used mainly for clays), whose limit curve is defined by two ellipses in the plane (I1 − J ).
36
Non-Linear Mechanics of Materials
2.5.4. Anisotropic criteria Literature shows that experimental yield surfaces of metallic alloys determined after plastic deformation exhibit an expansion, a translation and a distortion. The two first events are taken into account by isotropic and kinematic hardening. This is not the case of the last one, which is not described by classical models. Kinematic hardening and distortion induce anisotropy. On the other hand, materials like composites are fundamentally anisotropic, due to the local material geometry (e.g., long fibers composites). These two types of anisotropy are generally well represented by heterogeneous material models, (see Chap. 5), nevertheless they are still not too manageable in an industrial environment. This is why other solutions are investigated here to represent anisotropic materials. The most general approach consists in defining the criterion as a function of the components of the stress tensor in a given manifold. The chosen form must be intrinsic, in order to have a result independent of manifold change. A guide to build this type of model is provided by the invariant theory (see for instance [BOE78]). The most popular criterion is a quadratic form, used to generalize the von Mises criterion, through the following expression: : σ∼ )1/2 JB (σ∼ ) = (σ∼ : B ∼
(2.111)
∼
where B is a fourth-order tensor. Taking the tensor J∼ instead of a general B (J such ∼ ∼ ∼ ∼
∼
∼
∼
as ∼s = J∼ : σ∼ (s∼ deviatoric stress tensor)) will lead to the classical von Mises criterion. ∼
The number of independent components of the tensor can be reduced according to the symmetries of the material. In addition to the classical symmetries Bij kl = Bij lk = Bj ikl = Bklij , one has to enforce the condition Bjj kl = 0 in order to ensure plastic : σ∼ ). There are still incompressibility (the plastic strain rate is proportional to B ∼ ∼
15 free material parameters (as for a symmetric 5 × 5 matrix). If the material has three perpendicular planes of symmetry, the coupling terms between axial and shear components (such as B1112 ), are equal to zero, so that there are only six independent terms. In the symmetry frame, one finds the classical expression: f (σ∼ ) = (F (σ11 − σ22 )2 + G(σ22 − σ33 )2 + H (σ33 − σ11 )2 1/2
2 2 2 + 2Lσ12 + 2Mσ23 + 2N σ13 )
− σy = fH (σ∼ )
(2.112)
are: Writing the fourth-order tensor like a 6 × 6 matrix, the terms in B ∼ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
F +H −F −H 0 0 0
−F G+F −G 0 0 0
−H −G H +G 0 0 0
0 0 0 2L 0 0
0 0 0 0 2M 0
0 0 0 0 0 2N
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
∼
(2.113)
General concepts
37
A simple manipulation allows us to check that the same criterion can be expressed as : a function of the components of the stress deviator associated to σ∼ , JB (σ∼ ) = (s∼ : B ∼ are: s )1/2 , where the components of B ∼ ∼ ⎛ 2F − G + 2H 0 0 ⎜ 0 2F + 2G − H 0 ⎜ ⎜ 0 0 −F + 2G + 2H ⎜ ⎜ 0 0 0 ⎜ ⎝ 0 0 0 0 0 0
∼
∼
0 0 0 0 0 0 2L 0 0 2M 0 0
0 0 0 0 0 2N
⎞ ⎟ ⎟ ⎟ ⎟ (2.114) ⎟ ⎟ ⎠
For a transverse isotropy around the axis x3 , only three independent coefficients are left, since F = G, L = M, N = F + 2H . A full isotropy will induce also F = H , L = N, N = 3F , and leads to the tensor J∼ and von Mises criterion, as already mentioned.
∼
For the case of asymmetric yield in tension and compression, one may consider an expression involving a linear expression of the stress components, e.g., for Tsaï criterion: f (σ∼ ) = fH (σ∼ ) + Q(σ22 − σ33 ) + P (σ11 − σ33 ) (2.115) The type of generalization shown for the case of criteria expressed in terms of invariants can also be applied to criteria expressed in terms of eigenstresses. A classical case in geotechnical applications deals with transversely isotropic materials: they have a criterion expressed in terms of eigenstresses, plus N and T , which are respectively the normal stress and the shear stress for a facet perpendicular to the axis defining slices schistosity, defined by the normed vector n: 1/2 T = σ∼ .n2 − N 2 (2.116) N = n.σ∼ .n For instance, Coulomb criterion is written: f (σ∼ ) = max(Kp max σi − min σi − Rc , T + N tan φ − C ) 1 + sin φ 2C cos φ Rc = Kp = 1 − sin φ 1 − sin φ
(2.117) (2.118)
where φ denotes the friction angle in the slice planes, C the cohesion, φ the friction angle for the slip of two slices, C the cohesion.
2.6. Numerical methods for nonlinear equations We present here general methods to solve systems of nonlinear equations that can be written as: {R}({U }) = {0} (2.119)
38
Non-Linear Mechanics of Materials
where {R} depends on {U }. The linear case is encountered when {R}({U }) can be written [K].{U } + {A}, where [K] does not depend on {U }. One should note that an equation written in the form {R}({U }) = {A} can be rewritten in the general form of (2.119). In the framework of the finite element method, these methods will be used simultaneously to solve both the “global” problem and to integrate the constitutive equations (“local” problem).
2.6.1. Newton-type methods/modified Newton Newton method is iterative and consists in linearizing the previous equation (k: iteration number): ∂{R} .({U } − {U }k ) (2.120) {R}({U }) = {R}({U }k ) + ∂{U } {U }={U }k We will note: [K]({U }) =
∂{R} ∂{U }
(2.121)
Kij ({U }) =
∂Ri ∂Uj
(2.122)
or
One gets after resolution of the linearized problem (Fig. 2.14a): {U }k+1 = {U }k − [K]−1 k .{R}k
(2.123)
In the case of systems with a large number of variables, the calculation cost of [K]−1 may become very high compared to the calculation cost of {R}({U }) and the
Figure 2.14. (a) Newton method, (b) Modified Newton method
General concepts
39
product [K]−1 .{R}k . It sometimes is interesting to keep the inverse matrix obtained for the first iteration and to use it for the following iterations without recalculating the tangent matrix and its inverse (see Fig. 2.14b). In this case, the convergence is slower in terms of iterations number but can be faster in terms of calculation time. One get, then: {U }k+1 = {U }k − [K]−1 0 .{R}k
(2.124)
It is possible to use other variants of the modified-Newton method. Thus, by inverting the matrix [K] during the first two increments, and by keeping it constant afterwards, one gets: {U }1
= {U }0 − [K]−1 0 .{R}0
{U }2
= {U }1 − [K]−1 1 .{R}1 .. .
{U }k+1
(2.125)
= {U }k − [K]−1 1 .{R}k
If one has to calculate several increments for which a nonlinear system must be solved (as for the finite element method), one can keep the [K]−1 matrix that was calculated for the first increment. However, it is clear that this solution will not be valid in the case of contact, finite strain, etc.
2.6.2. One unknown case, order of convergence Fixed point method The equation to solve here is: f (x) = 0
(2.126)
where x is scalar. A first method consists in writing this equation as: x = g (x)
(2.127)
whose solution is called a fixed point. One gets the solution in an iterative way by choosing an initial value x0 : xn+1 = g (xn )
(2.128)
Let s be the solution of x = g(x). If there exists an interval around s such that |g | ≤ K < 1 then the series xn converges towards s. To prove this, we notice first of all that there always exists t verifying t ∈ [x, s] such that (mean value theorem): g (x) − g (s) = g (t) (x − s)
(2.129)
40
Non-Linear Mechanics of Materials
As g(s) = s and xn = g(xn−1 ), one gets: |xn − s| = |g (xn−1 ) − g (s) | ≤ |g (t) ||xn−1 − s| ≤ K|xn−1 − s| ≤ · · · ≤ K n |x0 − s|
(2.130)
As K < 1, limn→∞ |xn − s| = 0.
Iterative methods: order of convergence Let n be the error on xn
x n = s + n
(2.131)
By taking the Taylor expansion of xn+1 one gets 1 xn+1 = g (xn ) = g(s) + g (s) (xn − s) + g (s) (xn − s)2 2 1 = g(s) + g (s)n + g (s)n2 2
(2.132)
And then
1 (2.133) xn+1 − g(s) = xn+1 − s = n+1 = g (s)n + g (s)n2 2 The order of an iterative method gives a measurement of its rate of convergence. At first-order it comes: (2.134) n+1 ≈ g (s)n and at second-order: n+1 ≈
1 g (s)n2 2
(2.135)
Application to Newton method For the Newton method, the Taylor expansion of xn about 0 is used to find xn+1 : f (xn+1 ) = f (xn ) + (xn+1 − xn ) f (xn ) = 0 Then: xn+1 = xn −
f (xn ) f (xn )
(2.136)
(2.137)
This is equivalent to the fixed point method with: g(x) = x −
f (x) f (x)
(2.138)
General concepts
One gets
f (x)f (x) f (x)2
(2.139)
f (x) f (x)f (x) f (x)f (x)2 + − 2 f (x) f (x)3 f (x)2
(2.140)
g (x) = and g (x) =
41
One notes that:
g (s) = 0
(2.141)
The Newton method is then of the second-order. Moreover, there exists an interval around s such that |g (s)| < 1. Newton method always converges for an initial value x0 close enough to the solution.
Application to the modified Newton method In the case of the modified Newton method, one rewrites (2.136) as: f (xn+1 ) = f (xn ) + (xn+1 − xn ) K = 0
(2.142)
where K is a constant. We then have: xn+1 = xn −
f (xn ) K
(2.143)
g(x) = x −
f (x) K
(2.144)
Let and
f (x) K As g (s) = 0, this method is of first-order. It converges for K such that: g (x) = 1 −
−1 < 1 −
f (s) <1 K
(2.145)
(2.146)
K being a derivative evaluated at iteration i, let K = f (xi ), K and f (s) must then be of the same sign and f (s) < 2 K.
2.6.3. BFGS method (Broyden–Fletcher–Goldfarb–Shanno) An alternative to Newton method [BAT82, MAT79, PRE88] consists in updating the inverse of [K] in order to obtain, at iteration k, a secant approximation of the matrix at iteration k − 1. If one defines the increment of the unknown {U } as: {δ}k = {U }k − {U }k−1
(2.147)
42
Non-Linear Mechanics of Materials
and the increment of the deviation from the solution: {γ }k = {R}k − {R}k−1
(2.148)
the updated matrix [K]k must verify: {γ }k = [K]k .{δ}k
(2.149)
BFGS method consists of three steps: 1. Evaluation of an increment of vector {U }: {U } = −[K]−1 k−1 .{R}k−1
(2.150)
{U } gives the direction of the actual increment. 2. Seeking a solution in direction {U } in the form: {U }k = {U }k−1 + β {U }
(2.151)
where β is a scalar. β is calculated to minimize the scalar product: − {U }.{R}k
(2.152)
However, one can skip this step and consider β = 1. {δ}k and {γ }k can then be calculated. 3. Evaluation of the corrected matrix [K]−1 k . In BFGS method, the updated matrix has to be written as: −1 T (2.153) [K]−1 k = [A]k .[K]k−1 .[A]k where [A] is a matrix expressed as: [A]k = [I ] + {v}k ⊗ {w}k where {v}k and {w}k are vectors calculated from known quantities: 1/2 {δ}k .{γ }k [K]−1 {v}k = − k−1 .{δ}k − {γ }k −1 {δ}k .[K]k−1 .{δ}k 1 {w}k = {δ}k {δ}k .{γ }k
(2.154)
(2.155) (2.156)
Direction {U } is obtained by the product: {U } = − ([I ] + {w}k−1 ⊗ {v}k−1 ) · · · ([I ] + {w}1 ⊗ {v}1 ) .[K]−1 0 . ([I ] + {v}1 ⊗ {w}1 ) · · · ([I ] + {v}k−1 ⊗ {w}k−1 ) .{R}k−1
(2.157)
It is not necessary to calculate [K]−1 k . It is sufficient to multiply {R}k−1 by [I ] + {v}k−1 ⊗ {w}k−1 · · · [I ] + {v}1 ⊗ {w}1 , then the result by [K]−1 0 and to multiply in inverse order [I ] + {w}1 ⊗ {v}1 ) · · · [I ] + {w}k−1 ⊗ {v}k−1 to get {U }.
General concepts
43
2.6.4. Iterative method—conjugate gradient This method [LUE84] was developed to solve systems of the form: V ({U }) =
1 {U }.[K].{U } − {U }.{F } 2
(2.158)
The derivative of the previous equation is: ∂V = [K].{U } − {F } = {r} ∂{U }
(2.159)
where {r} is the residual or “gradient” of (2.158). The method uses the following approximation of the solution: {d}i = {r}i + βi {d}i−1 {U }i+1 = {U }i + αi {d}i = {U }i + αi {r}i + αi βi {d}i−1
(2.160)
where {d}i is the direction of research. One searches for αi and βi in order to minimize V ({u}), so that: ∂V ∂V =0 =0 (2.161) ∂αi ∂βi with: ∂V ∂V ∂{U }i+1 = . = ([K].{U }i+1 − {F }) .{d}i = {r}i+1 .{d}i = 0 (2.162) ∂αi ∂{U }i+1 ∂αi and ∂V ∂V ∂{U }i+1 = . = {r}i+1 . (αi {d}i−1 ) = αi {r}i+1 .{d}i−1 = 0 ∂βi ∂{U }i+1 ∂βi
(2.163)
The problem is equivalent to finding αi and βi such that {r}i+1 .{d}i = 0 {r}i+1 .{d}i−1 = 0
(2.164) (2.165)
with {r}i+1 = −{F } + [K].{U }i+1 = −{F } + [K]. ({U }i + αi {r}i + αi βi {d}i−1 ) = −{F } + [K].{U }i + αi [K]. ({r}i + βi {d}i−1 ) = {r}i + αi [K]. ({r}i + βi {d}i−1 ) = {r}i + αi [K].{d}i
(2.166)
44
Non-Linear Mechanics of Materials
Noting that, by construction, {r}i .{d}i−1 = 0, one gets {r}i+1 .{d}i−1 = αi {d}i−1 .[K].({r}i + βi {d}i−1 ) = 0
(2.167)
Then
{d}i−1 .[K].{r}i (2.168) {d}i−1 .[K].{d}i−1 βi corresponds to the coefficient of optimal descent. From βi , it is possible to calculate {d}i . Using relations {r}i+1 = {r}i + αi [K].{d}i and {r}i+1 .{d}i = 0, one gets: βi = −
αi = −
{r}i .{d}i {d}i .[K].{d}i
(2.169)
This corresponds to an orthogonalization, in the sense of the scalar product associated to [K], of the new gradient with respect to the previous direction. The main advantage of the conjugate gradient method is that it is not necessary to calculate the matrix [K]−1 . The main computational effort lies in the matrix–vector product. The other phases of the calculation are mere scalar products and linear combinations of vectors. In a finite element context, it is not necessary to store the global matrix [K] but it is then necessary to recalculate the elementary matrices for each iteration.
2.6.5. Riks method This method [RIK79], because of its formulation, applies particularly well to mechanics.
Control of the problem The most natural control of the finite element problem consists in applying external loads ({Fe }). The system to solve is then: {Fi } ({U }) = {Fe }
(2.170)
This kind of control is suitable for the case where load is a monotonic function of displacement. Should a maximum load (plasticity) or a structure instability (plate buckling) exist, load ceases to be monotonic and control is achieved imposing displacement where the load is applied. This boils down to decreasing the number of degrees of freedom of the system. Reactions corresponding to these degrees of freedom are then free. In the case where there exists a snap-back point on the load–displacement (e.g., buckling), one must exert a mixed (load and displacement) control. Figure 2.15 sums up the three situations.
General concepts
45
Figure 2.15. Type of response load–displacement: (a) Increasing load. Single U –F couple in all cases. (b) Load–displacement curve with maximum loading. Several solutions in displacement for a given load. (c) Load–displacement curve with snap-back in displacement. Several solutions in load for a given displacement
Control of the load–displacement response The equilibrium equation is written: λ{Fe } = {Fi } ({U })
(2.171)
{Fe } is the reference loads vector and is known. λ is the scalar load parameter. From now on, there is an additional unknown. The path ({U }, λ) can be parameterized by the arc length s such that λ = λ(s)
{U } = {U }(s)
The unitary vector tangent to the curve writes: 1 d{U } {t} =
m ds 1 dλ m ds
with m =
d{U } d{U } dλ dλ . + ds ds ds ds
(2.172)
(2.173)
The derivative of the equilibrium equation (2.171) with respect to s leads to: dλ d{U } {Fe } − [K]. = {0} ds ds
(2.174)
with [K] = d{Fi }/d{U }. We will write
Then, one gets:
{UF } = [K −1 ].{Fe }
(2.175)
d{U } dλ = {UF } ds ds
(2.176)
46
Non-Linear Mechanics of Materials
Imposed arc length The mixed-control technique with imposed arc length consists in realizing increments in ({U }, λ) space such that the increment has a given length s: {U }. {U } + ( λ)2 = ( s)2
(2.177)
by writing {U } ≈ s λ ≈ s
dλ {UF } ds
(2.178)
dλ ds
(2.179)
the previous equation becomes:
dλ ds
2
{UF }.{UF } +
dλ ds
2 =1
(2.180)
Then, one gets for the first iteration: ± s λ1 = √ 1 + {UF }.{UF }
(2.181)
{U }1 = λ1 {UF }
(2.182)
The sign of λ1 is chosen to maximize the scalar product λp λ1 + {U }p . {U }1
(2.183)
where ( λp , {U }p ) are the solution increments of the previous step. We assume that there is a certain continuity of the displacement direction from one increment to the other. For the following iterations, we note that δ{UR }i = [K]−1 . λi {Fe } − {Fi }i
(2.184)
δ{UR }i corresponds to the displacement increment calculated for a simple load control. The iteration solution is written {U }i+1 = {U }i + δ{UR }i + δλ{UF }
(2.185)
2 {U }i+1 . {U }i+1 + λi+1 = ( s)2
(2.186)
with the condition:
General concepts
47
δλ{UF } corresponds to a change in the applied load. Furthermore, we have λi+1 = λi + δλ We get then a quadratic equation in δλ (taking into account that ( λi )2 = ( s)2 ): a (δλ)2 + bδλ + c = 0
(2.187) {U }i . {U }i
with a = 1 + {UF }.{UF } b = 2 {v}i .{UF } + λi
+
(2.188)
c = 2{U }i .δ{UR }i + δ{UR }i .δ{UR }i with {v}i = {U }i + δ{UR }i There are then two roots. One chooses the root that allows minimizing the angle between two successive evaluations of {U }. Then, we maximize: cos θ i =
{ {U }i , λi }.{ {U }i+1 , λi+1 } ( s)2
(2.189)
2.6.6. Convergence Various ways exist to test the convergence of the iterative resolution. • We try to solve {R} = {0}. The resolution process is stopped when {R} is small enough. {R}n < R (2.190) where R is the required precision, with: 1/n n Ri {R}n =
(2.191)
i
The norm · ∞ is often used, {R}∞ = max |Ri | i
(2.192)
• In numerous cases, equation {R} = {0} can be written {R}i − {R}e = {0} where {R}e is imposed (external load). A relative error can then be defined: {R}i − {R}e < r {R}e
(2.193)
where r is the required relative precision. In mechanics {R}i corresponds to internal forces and {R}e to external forces. In some cases (for example thermal cooling) {R}e = {0}; it is then impossible to define a relative error and one must use the absolute error {R}i − {R}e < R .
48
Non-Linear Mechanics of Materials
• The search for a solution can be stopped when {U } becomes stable; i.e.: {U }k+1 − {U }k n < U
(2.194)
Note: In some cases, vectors {R} and {U } are not homogeneous. For example in the coupled thermo-mechanical case, {U } contains displacements and temperatures, and {R} forces and thermal flows. One must then normalize with predetermined quantities the various components of {R} and {U } before testing the convergence.
2.7. Numerical solution of differential equations 2.7.1. General overview In this section first-order differential equations (DE) are dealt with. Higher order DE can generally be rewritten as first-order ones. For example: dy d 2y + q(t) = r(t) dt dt 2
(2.195)
can be rewritten: dy = z(t) dt dz = r(t) − q(t)z(t) dt
(2.196) (2.197)
where z is a new variable. In a more general way, a first-order DE can be written: v˙i = fi (t, v1 , . . . , vn )
i = 1...n
(2.198)
or {v} ˙ = {f }(t, {v})
(2.199)
where {v} is a vector of size n. It is necessary to add to these equations initial conditions written as: (2.200) {v}(t = t0 ) = {v}0 In this section, we introduce two integration methods: (a) explicit Runge–Kutta type methods [PRE88, TOU93], (b) implicit middle point methods (or θ-method). We want to integrate over a time interval t from initial time t. The easiest method is Euler’s method which is entirely explicit and very easy to formulate as it relies only on the vector {v} given at initial time t: {v}(t + t) = {v}(t) + t{v}(t, ˙ {v}) However, this method is neither precise nor stable.
(2.201)
General concepts
49
2.7.2. Runge–Kutta method Basic method By using a Taylor series, one gets: {v}(t + t) = {v}(t) + {v}(t) t ˙ + O( t 2 ) Euler’s integration has a second-order precision middle point (t + t/2). Let
O( t 2 ).
One can however test the
{δv1 } = t{v}(t) ˙ and
(2.202)
(2.203)
1 t , {v}(t) + {δv1 } ˙ t+ {δv2 } = t{v} 2 2 t = t {v}(t) ˙ + {v}(t) ¨ 2 = {δv1 } +
t 2 {v}(t) ¨ 2
(2.204)
The second-order Taylor series gives: {v}(t + t) = {v}(t) + {v}(t) t ˙ + {v}(t) ¨
t 2 + O( t 3 ) 2
(2.205)
2 /2 which is equal to {δv } − {δv }. It follows One can eliminate the term {v}(t) t ¨ 2 1 that: (2.206) {v}(t + t) = {v}(t) + {δv2 } + O( t 3 )
This formula shows that the first-order term in t has been eliminated: one gets, then, a second-order method. This procedure can be generalized. The fourth-order Runge–Kutta method writes: {δv1 } = t{v} ˙ (t, {v}) 1 t {δv2 } = t{v} , {v} + {δv1 } ˙ t+ 2 2 1 t , {v} + {δv2 } ˙ t+ (2.207) {δv3 } = t{v} 2 2 ˙ (t + t, {v} + {δv3 }) {δv4 } = t{v} 1 1 1 1 {v}(t + t) = {v}(t) + {δv1 } + {δv2 } + {δv3 } + {δv4 } + O( t 5 ) 6 3 3 6 This method requires then four evaluations of the function {v} ˙ whereas the secondorder method requires only two. It will then be more efficient if it allows a time step twice as large for the same precision. However high order does not always mean high precision. This is why it is appropriate to use an adaptive time step method.
50
Non-Linear Mechanics of Materials
Adaptive time step method The aim of such a method is to get a predetermined precision while minimizing the calculation time. The method is designed to use a large time step where the function {v} ˙ varies little and to reduce it when {v} ˙ varies a lot. Let t be the time increment over which the integration has to be made. It is divided into n δtk increments so that: t = k=n k=1 δtk . In the case of the fourth-order method, the technique consists in using one time step of magnitude δt and two time steps of magnitude δt/2. Thus, one needs 11 evaluations of {v} ˙ (both integrations share the same starting point) that must be compared to the eight evaluations needed to integrate using only two time steps δt/2 (as one gets the same precision). Extra cost is then 11/8 = 1.375. Let {v}(t + δt) be the exact solution, then: {v}(t + δt) = {v1 } + (δt)5 {φ} + O(δt 6 )
(2.208)
{v}(t + δt) = {v2 } + 2(δt/2) {φ} + O(δt ) 5
6
(2.209)
where {v1 } is the result obtained with a time step δt and {v2 } the result obtained with two time steps δt/2. At fifth-order, φ remains constant over the increment. The fifthorder Taylor series indicates that: {φ} 5!1 {v (5) }. The difference between both estimations is an indicator of the error: {E} = {v2 } − {v1 }
(2.210)
This difference is what must be kept smaller than a given precision by adjusting δt. Ignoring the (δt)6 terms, (2.208) and (2.209) can be solved. It follows that: 1 {E} + O(δt 6 ) (2.211) 15 We get then a fifth-order estimate. However, it is not always possible to control integration error. One can use {E} to modify the time step. Let {E 0 } be the desired precision. Here, the precision is a vector. If Ei < Ei0 , ∀i, then we can increase the next time step, if ∃i, Ei > Ei0 , the current time step must be decreased. In both cases, correction factor (α) is given by: 1/5 E0 δti+1 α= = min i (2.212) i Ei δti {v}(t + δt) = {v2 } +
because the error is in δt 5 for the fourth-order Runge–Kutta method. We take, of course, the most penalizing factor. {E 0 } is still to be chosen. We can choose a fraction of the value of {v}: (2.213) Ei0 = |vi | In cases where vi is close to 0 one can use the following trick: dvi Ei0 = |vi | + δt dt
(2.214)
General concepts
51
Precision {E 0 } must be obtained over the full increment t and not over the local ones δtk . The smaller δt is compared to t, the smaller the imposed precision must be. In such a case, one can use the following approximation: dvi 0 (2.215) Ei = δt dt Then, the local precision depends on δt, and the exponent of the formula (2.212) is not exact anymore: it must be (1/4) instead of (1/5). Considering that these two exponents differ little and that error estimates are not exact, we use, in a practical approach, the following formula to update the local time step: ⎧ E0 ⎪ ⎨ S mini | Ei |1/5 if δt increases i α= (2.216) 0 ⎪ ⎩ S min | Ei |1/4 if δt decreases i Ei where S is a security factor slightly smaller than 1.
2.7.3. θ -methods θ -methods are implicit methods. There exist two possible formulations. Let { v} be the increment of the vector {v} over the time step t; there are two ways to evaluate { v} (with 0 ≤ θ ≤ 1): t{v}(t ˙ + θ t) type 1 (2.217) { v} = t ((1 − θ ){v}(t) ˙ + θ {v}(t ˙ + t)) type 2 θ = 0 corresponds to the explicit Euler method. If θ = 0 the previous equations must be solved in order to find v that satisfies them.
Type 1 method (generalized middle point) { v} is obtained by solving the following nonlinear system of equations: {R} = { v} − t{v}(t ˙ + θ t) = {0}
(2.218)
One uses a Newton method for which one has to evaluate the Jacobian [J ] of the previous equation: ∂{v} ˙ ∂{R} [J ] = = [1] − t (2.219) ∂{ v} ∂{ v} t+θ t
52
Non-Linear Mechanics of Materials
Type 2 method (generalized trapezoidal rule) { v} is obtained by solving the following nonlinear system of equations: {R} = { v} − t ((1 − θ ){v}(t) ˙ + θ {v}(t ˙ + t)) = {0}
(2.220)
As previously, one uses the Newton method for which one has to evaluate the Jacobian: ∂{v} ˙ ∂{R} = [1] − t θ (2.221) [J ] = ∂{ v} ∂ v t+ t
2.7.4. Comment Which integrating method? θ -methods require calculation of the Jacobian, which cannot always be evaluated. In that case, one must use Runge–Kutta type methods. Calculation of [J ] can be quite heavy. However one can use an approximation: the convergence towards a solution is then slower.
Calculation of the consistent tangent matrix with a θ -method For the small strain case where one postulates the additive separation of strains, i.e., ε∼ = ε∼ e + ε∼ p + ε∼ th
(2.222)
The θ -method allows also, after convergence, calculation of the consistent tangent matrix of the behavior. One can rewrite the system of equations to solve as: {R} = {R0 } #
with {R} =
#
and {R0 } =
ε∼ e + ε∼ p vi − t v˙i ε∼ − ε∼ th 0
(2.223) $ (2.224) $
{R0 } is imposed. One can then apply an infinitesimal variation to {R0 }. $ # δ ε∼ {δR0 } = 0
(2.225)
(2.226)
General concepts
53
and we get {δR} = [J ].{δv} = {δR0 }
(2.227)
{δv} = [J ]−1 .{δR0 }
(2.228)
δ ε∼ e = J∼ e : δ ε∼
(2.229)
then
It follows that ∼
where [J∼ e ] is the part of [J ]−1 corresponding to the upper left part: ∼
−1
[J ]
= [J ] =
[J∼ e ] ∼
] [J12
] [J ] [J21 22
(2.230)
: ε∼ e we get: As σ∼ = ∼ ∼
: δ ε∼ e δ σ∼ = ∼ ∼
(2.231)
The tangent operator of the behavior is then: = : J∼ e L ∼c ∼ ∼
∼
(2.232)
∼
It is important to note that we will get from L a “consistent tangent matrix” [SIM85a] ∼c ∼
(i.e., consistent with the integration scheme), estimated from the incremental form of the differential system. The resulting form is different from the one resulting from rate formulation, as we will see below.
Consistent tangent matrix calculation for Runge–Kutta type integration In the framework of the Runge–Kutta method, it is not possible to get directly the tangent matrix of the behavior. One can however calculate it by perturbating the solution. Let ε∼ be the strain increment leading to the stress increment σ∼ . The tangent matrix is calculated by perturbation of ε∼ . Using Voigt notation, one gets: Lc ij =
σi ({ ε∼ } + δεj {bj }) − σi ({ ε∼ }) δεj
(2.233)
where {bj } is the vector such that: j
bi = δij
(2.234)
54
Non-Linear Mechanics of Materials
2.8. Finite element The aim of this section is to recall some notions of the finite element method. The role of material behavior in the framework of this method will be particularly emphasized. For a more comprehensive presentation, one can refer to [BAT82, HUG87, BAT91, ZIE00].
2.8.1. Spatial discretization Space is discretized by sets of nodes, edges, faces (3D) and elements. A point in space is located using its spatial coordinates in physical space x = (x, y, z) and by its coordinates in the reference space of the element to which it belongs η = (η, ζ, ξ ) (Fig. 2.16). Nodal coordinates of an element are given by x i , i = 1 . . . n. Coordinates x are linked to coordinates η by the relation (sum from 1 to n over i): x = N i (η) x i
(2.235)
where N i are shape functions such that: N i (ηj ) = δij
and
N i (η) = 1,
∀η
(2.236)
i
One will find in the books given as references the expression of the shape functions of each element. Furthermore, one defines the Jacobian matrix of the transformation from the reference space to the physical space η → x: J∼ =
∂x ∂η
Figure 2.16. Spatial discretization (• = node)
(2.237)
General concepts
Then: Jij =
∂(N k xik ) ∂N k ∂xi = = xik ∂ηj ∂ηj ∂ηj
55
(2.238)
The Jacobian is: J = det(J∼ )
(2.239)
2.8.2. Discrete integration method To perform spatial integrations (volume, surfaces, lines), a Gauss method is generally used which consists in replacing a continuous integration by a discrete sum such as (written here in the one-dimensional case): 1 f (x)dx = w i f (x i ) (2.240) −1
where the x i are the predetermined positions where the function f is evaluated (Gauss points) and the w i are weights associated to each of them. A discrete integration with n Gauss points allows us to exactly evaluate a polynomial of order 2n − 1. In the case of higher-order polynomials or non-polynomial functions, Gauss integration gives an approximation of the actual integral. The method is then easily extended to 2D and 3D cases for the case of square reference elements (2D) or brick elements (3D) for which positions and weight of Gauss points can be calculated from the 1D case. For the triangular elements (2D) and tetrahedron (3D), positions and weight must be calculated specifically. An integral over a finite element volume Ve can be written as follows, in order to integrate over the reference element Vr : f ( x )dx = f (η)J dη = f (ηi )(J w i ) (2.241) Ve
Vr
where ηi is the position of the various Gauss points in the reference space (Fig. 2.17). v i will be the volume of the Gauss point i, with: v i = J wi
(2.242)
2.8.3. Discretization of fields of unknowns Fields of unknowns (displacement, temperature, pressure, etc.) must also be discretized in order to allow problem solving. These fields can be defined:
56
Non-Linear Mechanics of Materials
Figure 2.17. Gauss points in parent element: 8 nodes square element, 6 nodes triangle element: (• = node, + = Gauss point)
at the nodes: displacement, pressure, temperature for each element: pressure when treating incompressibility case, plane strain. by group of elements: periodical elements used in the framework of homogenization plane elements including torsion/flexion effects in a third direction In general, one uses fields defined at the nodes. These are interpolated using the interpolation function of the elements. They are then continuous. In the case of a displacement field, we have: u(η) = N k (η)uk
or ui (η) = N k (η)uki
(2.243)
where uk is the displacement at the node k. In the same way, for thermal problems we will have (T : temperature): (2.244) T (η) = N i (η)T i Gradients can then be simply calculated: (grad u)ij =
∂N n ∂ηk ∂ui ∂ui ∂ηk = = uni ∂xj ∂ηk ∂xj ∂ηk ∂xj
(2.245)
(grad T )i =
∂N n ∂ηk ∂T ∂T ∂ηk = = Tn ∂xi ∂ηk ∂xi ∂ηk ∂xi
(2.246)
∼
By writing second-order tensors as vectors (Voigt notation), one can rewrite the previous equations as matrix-vector products: % & grad u = [B].{ue } (2.247) ∼
grad T = [A].{T e }
(2.248)
General concepts
57
where {ue } and {T e } represent vectors of variables relative to the considered element: ⎧ 1 ⎫ ⎪ ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ 1 ⎫ ⎪ 1 ⎪ ⎪ ⎪ u ⎪ ⎪ 3 ⎪ ⎪ ⎪ ⎨ T ⎪ ⎬ ⎨ ⎬ . .. e e . {u } = (2.249) {T } = . . ⎪ ⎪ ⎪ ⎭ ⎩ n ⎪ ⎪ ⎪ ⎪ ⎪ n T ⎪ u1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪ ⎩ 2n ⎪ ⎭ u3 Processing incompressibility (7.3.2), plain stress (7.3.3) and periodic conditions (7.3.4) will provide examples of fields of unknowns defined per element or per group of elements. Isoparametric elements: we call isoparametric elements, the elements for which unknowns and coordinates are interpolated in the same way: interpolation function and shape functions are then identical.
2.8.4. Application to mechanics Equilibrium formulation The discretized displacement field sought by the finite element method verifies the displacement boundary conditions: it is then kinematically admissible. However the stress field associated to the displacement field is not necessarily statically admissible. Then, the problem reduces to finding the displacement field that satisfies the PVP: the associated stress field will be statistically admissible.
Greenberg’s minimum principle We assume here that material behaviors can be written as (elasticity and plasticity cases): : ε∼˙ (2.250) σ∼˙ = L ∼ ∼
If u˙ is a kinematically admissible field, it is associated with a strain field ε∼˙ and a : ε∼˙ . Generally the field σ∼˙ is not statically admissible. One shows stress field σ∼˙ = L ∼ ∼
that the solution field u˙ minimizes the functional B(u) ˙ defined by: 3 1 B= f˙.u˙ d − σ∼˙ : ε∼˙ d − T˙id u˙ i dS 2 Sfi i=1
(2.251)
58
Non-Linear Mechanics of Materials
Actually, it suffices to calculate the difference B( u˙ ) − B( u˙ ): B( u˙ ) − B( u˙ ) =
1 2
(σ∼˙ : ε∼˙ − σ∼˙ : ε∼˙ )d
f˙.( u˙ − u˙ )d −
−
3
i=1
Sfi
T˙id (u˙ i − u˙ i )dS
(2.252)
The field σ∼ being a solution, it is statically admissible. In addition, ε∼ and ε∼ are statically admissible. By applying the PVP, we get that:
σ∼˙ : (˙ε∼ − ε∼˙ )d =
f˙.( u˙ − u˙ )d +
3 i=1
where 1 B( u˙ ) − B( u˙ ) = 2
Sf
T˙id (u˙ i − u˙ i )dS
(2.253)
(σ∼˙ : ε∼˙ + σ∼˙ : ε∼˙ − 2σ∼˙ : ε∼˙ )dV
(2.254)
σ∼˙ and σ∼˙ are then replaced by using the constitutive equation (2.250): σ∼˙ = L : ε∼˙ and ∼ ∼
: ε are elastic, or both of them elastoplastic, the σ∼˙ = L ˙ . If both tensors L and L ∼ ∼ ∼ ∼ ∼
∼
∼
expression can be written in a simple form, which is positive, provided that the tensors are positive definite: 1 B( u˙ ) − B( u˙ ) = (˙ε − ε∼˙ ) : L : (˙ε∼ − ε∼˙ )dV (2.255) ∼ 2 ∼ ∼ The positivity is still verified if one of the two tensors is elastic and the second one elastoplastic. This demonstrates that the solution field u˙ minimizes the functional B and is unique. Should the elastoplastic tensor not be positive, the proposition is not exact anymore.
2.8.5. Finite element discretization of Greenberg’s principle We introduce here the matrix [B] that enables strain-rate calculation from nodal displacements (cf. Sect. 2.8.3): {˙ε∼ } = [B].{u˙ e } (2.256) We get that σ∼˙ = L : ε∼˙ or {σ∼˙ } = [L ].[B].{u˙ e }. The functional B is then calculated ∼ ∼ ∼
element by element:
∼
B=
e
Be
(2.257)
General concepts
59
with 1 B ( u˙ ) = 2
e
−
Ve
Ve
* + .[B].{u˙ e }dV {u˙ e }.[B]T . L ∼ ∼
f˙.([N ].{u˙ e })dV −
3 i=1
Sfe,i
T˙id ({N }.{u˙ ei })dS
* + 1 e T = {u˙ }. .[B]dV .{u˙ e } [B] . L ∼ 2 ∼ Ve 3 − [N]T .f˙dV .{u˙ e } − Ve
i=1
Sfe,i
where [N] is the shape functions matrix given by: ⎡ 1 N 0 0 . . . Nn 1 [N ] = ⎣ 0 N 0 ... 0 0 0 N1 . . . 0
T˙id {N }dS .{u˙ ei }
⎤ 0 0 ⎦ Nn
0 Nn 0
(2.258)
(2.259)
{N } is the shape functions vector: ⎧ 1⎫ ⎪ ⎨N ⎪ ⎬ .. {N } = . ⎪ ⎩ n⎪ ⎭ N
(2.260)
and {uei } is the vector formed by the components i of the nodal displacement of the element: ⎧ 1⎫ ⎪ ⎨ui ⎪ ⎬ {uei } = ... (2.261) ⎪ ⎩ n⎪ ⎭ ui Each element contributes to the external {Fee } and internal {Fie } forces and to the stiffness [K e ]: e
F˙e =
[N ] .f˙dV T
Ve
+
i=1
[K e ] =
Ve
3 Sfe,i
T˙id {N }dS
* + .[B]dV [B]T . L ∼
(2.262)
(2.263)
∼
and
F˙ie = [K e ].{u˙ e } =
Ve
* + [B]T . L .[B].{u˙ e }dV = ∼ ∼
Ve
[B]T .{σ∼˙ }
(2.264)
60
Non-Linear Mechanics of Materials
The global vector of external and internal forces as well as the stiffness matrix are then obtained by assembly [HUG87]:
(2.265) F˙i = A F˙ie e
˙ ˙ (2.266) Fe = A Fe e (2.267) [K] = A[K ] where A represents the assembly operation. One wants to minimize B( u˙ ), i.e., to find {u} ˙ such that: dB = {0} (2.268) d{u} ˙ where {u} ˙ is the vector containing all the unknowns. The system to solve can be reformulated as:
(2.269) [K].{u} ˙ = F˙e We will call {Fe } the vectors of the external forces corresponding to the loads applied on the structure, {Fi } the vector of the internal forces and [K] the global stiffness matrix. Until now, the problem has been written using a rate formulation but it will be solved in an incremental form expressed as: { Fi } = [Ke ].{ u} = { Fe }
(2.270)
in the small strain framework. The small strain case corresponds to the situation where the displacements are small compared to the size (L) of the structures u L (small displacements) and where the strains are small ε 1. The change from the rate to the incremental formulation is the basis of the finite element treatment of the small strain case. The possible nonlinearities in the resolution of the problem result from the nonlinearity of the material behavior only. One notices that the internal forces can also be calculated as follows: [B]T .{σ∼ }dV {Fie } =
(2.271)
Ve
{Fi }, {Fe } and [K] are calculated element by element and then assembled. The internal forces of an element are calculated by Gauss integration: {Fie } = [B]T .{σ∼ }dV = [Bg ]T .{σ∼ g }(Jg wg ) (2.272) Ve
g
The elementary stiffness matrix is assessed in the following way: * + * + [K e ] = [B]T . L [Bg ]T . L .[B]dV = .[Bg ](Jg wg ) ∼ ∼ Ve
∼
g
∼
(2.273)
Once the vectors {Fi } and {Fe } have been assembled, their size, nd , corresponds to the number of degrees of freedom of the problem. The matrix [K] has a size nd × nd . {Fi }k is the reaction associated with the degree of freedom k.
General concepts
61
2.8.6. Another presentation of the finite element discretization One can get the principle of the finite-element discretization from the principle of virtual power. The discretization of the power of external forces leads naturally to (2.262) for each element. The discretized displacement field verifies the boundary conditions in displacement: it is then kinematically admissible. However, the stress field associated with the displacement field is not necessarily statically admissible. The problem boils down to finding the displacement field u (discretized by {u}) that allows the PVP to be satisfied: the associated stress field will then be statically admissible. For any ˙ the power of internal stress is given by: virtual field u, ˙ discretized by {u}, σ∼ ( u ) : ε∼˙ ( u˙ )dV wi = = {σ∼ ({ue })}.[B].{u˙ e }dV e
=
Ve
e
Ve
[B] .{σ∼ ({u })}dV T
e
.{u˙ e }
˙ ≡ {Fi ({u})}.{u}
(2.274)
With each degree of freedom is associated an internal reaction. All of them are gathered in the vector {Fi ({u})}, a definition of which is given by the previous equation. It must be verified for any kinematically admissible field {u}. ˙ It follows that {Fi } is equal to: {Fi } = A({Fie }) with {Fie } =
Ve
[B]T .{σ∼ ({ue })}dV
(2.275)
The problem to solve with respect to {u} is then: {Fi ({u})} = {Fe }
(2.276)
It is a nonlinear system that can be solved by Newton’s method. It is suitable to calculate the Jacobian matrix: [K] =
∂{Fie ({ue })} ∂{Fi ({u})} =A ≡ A[K e ] ∂{u} ∂{ue }
(2.277)
Let us calculate [K e ]: ∂{Fie ({ue })} ∂{ue } * + } ∂{ε∼ } T ∂{σ ∼ . = .[B]dV [B] . [B]T . L = ∼c e} ∂{ε } ∂{u ∼ Ve Ve ∼
[K e ] =
(2.278)
We find, of course, the same result as in (2.263). However, it appears very clearly that the stiffness matrix [K] is only used for the iterative resolution of (2.276). One can
62
Non-Linear Mechanics of Materials
then use an approximation of it, either by an approximation of L , or by using the ∼c BFGS method to solve (2.276).
∼
Note: In (2.276), {Fe } can, in some cases, depend on {u}. Corrective terms must then be applied to the matrix [K].
2.8.7. Assembly through example In order to better understand the assembly operation, let us consider a simple example. A mesh containing three linear elements A, B, C and seven nodes 1. . . 7 is represented in Fig. 2.18. It is a planar mechanical problem; there are 14 degrees of freedom (7 × 2) corresponding to the node displacements. In each element, the nodes have a local numbering written in italic. The vector of the unknown {u} is then: {u} = {u1x , u1y , u2x , u2y , u3x , u3y , u4x , u4y , u5x , u5y , u6x , u6y , u7x , u7y }
(2.279)
For the element A, the vector of the local unknown {uA } is: A1 A2 A2 A3 A3 1 1 2 2 4 4 {uA } = {uA1 x , uy , ux , uy , ux , uy } = {ux , uy , ux , uy , ux , uy }
(2.280)
The vector of the internal forces associated with the element A is: {FiA } = {FxA1 , FyA1 , FxA2 , FyA2 , FxA3 , FyA3 }
(2.281)
Figure 2.18. Example of meshes (mesh: A, B and C) showing the global numbering of the nodes and the local one (in italic)
General concepts
63
The elementary stiffness matrix [K A ] is a 6 × 6 full matrix. The assembly of the vectors {FiA }, {FiB } and {FiC } gives the vector {Fi } such that:
{Fi } =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
Fx1
=
FxA1
Fy1
=
FyA1
Fx2
=
FxA2 + FxB1
Fy2
=
FyA2 + FyB1
Fx3
=
FxB2
Fy3
=
FyB2
Fx4
=
FxA3 + FxB4 + FxC1
Fy4
=
Fx5
=
Fy5 Fx6 Fy6 Fx7 Fy7
=
FyA3 + FyB4 + FyC1 ⎪ ⎪ ⎪ ⎪ ⎪ B3 C2 ⎪ ⎪ Fx + Fx ⎪ ⎪ ⎪ ⎪ B3 C2 ⎪ Fy + Fy ⎪ ⎪ ⎪ ⎪ C4 ⎪ Fx ⎪ ⎪ ⎪ ⎪ ⎪ C4 ⎪ Fy ⎪ ⎪ ⎪ ⎪ C3 ⎪ Fx ⎪ ⎪ ⎪ ⎭ C3 Fy
= = = =
(2.282)
The assembly of the elementary matrices [K A ], [K B ] and [K C ] gives a matrix whose non–null terms correspond to couples of columns and lines whose number represents nodes belonging to the same element, or nodes that have been connected by the user:
[K] =
(2.283)
The various symbols , , , , , , indicate that the corresponding terms of the matrix [K] are the sum of the terms coming from the various elementary matrices. For example: T B C = K6,7 + K4,1 K10,7
(2.284)
64
Non-Linear Mechanics of Materials
2.8.8. Principle of resolution It is practically impossible, even for a monotonic loading, to solve (2.276) for any loading. Because of the nonlinearities, the iterative search process of the solution does not converge. The solution is then obtained using a progressive incremental loading. At time t, the solution {u}t is known. One applies then an increment t, and one searches iteratively the corresponding increment of the solution {u}. The chart of the algorithm is given in Fig. 2.19.
Figure 2.19. Iterative search of increment {u} for a time step t
Note: For each increment, one searches iteratively the increment {u}. It is generally initialized with {u} = {0}. However, it is possible, particularly in the case of monotonic loading, to initialize {u} for increment j > 1 to
initial j
{u}increment =
tj solution j −1 {u}increment tj −1
(2.285)
General concepts
65
2.8.9. Mechanical behavior in the finite element method The mechanical behavior in finite element method must then, for a strain increment ε∼ , give the corresponding stress increment σ∼ along with the consistent tangent matrix of the mechanical behavior = L ∼c ∼
∂ σ∼ ∂ ε∼
(2.286)
The equation to solve is written: {Fi } =
[B]T .{σ }dV ≡ {Fe }
(2.287)
is Only the stress tensor σ∼ is necessary for the equilibrium formulation. The tensor L ∼c ∼
only used for the calculation of the elementary stiffness matrix [K e ] that is involved in the iterative solving of the previous system. It is then not necessary to exactly calculate L : an estimate can be sufficient. For example, one can use the elasticity matrix for ∼ ∼ ∼
L and use a BFGS type method to correct the global matrix. ∼
∼
∼
It is also appropriate not to forget that the material behavior is characterized by a set of variables (V) whose values must be calculated at the end of each increment V t+ t . The interface between finite element and mechanical behavior can then be summed up [FOE97, BES98b]:
The software description of the mechanical behavior must not depend on the FEM. Reciprocally the element formulation/implementation must not depend on the behavior type (for example elasticity, plasticity, creep, etc.). There must just be correspondence between the element and the behavior formulation: small strain, finite strain, thermal problem, etc.
Chapter 3
3D plasticity and viscoplasticity
3.1. Generality The multiplicity of constitutive equations, and in particular the wide diversity of criteria and evolution laws in elastoplasticity as well as in viscoplasticity, reflects the variety of real materials. It would be impossible to establish an exhaustive list of models, all the more because researchers still propose new versions every month. As a consequence, this chapter will be dedicated to a more modest task, namely to present the general framework used to write such models, by instancing the topic with the most classical constitutive equations. We will limit the presentation a to small strain framework (small displacements and small strain gradients). The text tries to find an equilibrium between a presentation of theoretical concepts and a presentation of models that actually work for real materials. Readers interested in a more mathematical presentation of plasticity could consult [DUV72, MOR83, HAL87, SIM97], while for the cyclic loading we could suggest [MRÓ95]. First of all, we sum-up the general concepts that have been introduced in the previous chapters. Except for Sect. 3.8, the text deals with initially isotropic plastic materials.
3.1.1. Strain decomposition The symmetric strain tensor ε∼ is the sum of several contributions: • An elastic part, ε∼ e , function of the variation of the stress tensor σ∼ between current and initial state (stress in the reference state σ∼ I ; in a large number of applications it is the zero stress state, but it can be present as, for example, in the case of geotechnical applications). In linear elasticity, the relation introduces J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_3,
68
Non-Linear Mechanics of Materials
the compliance tensor (4th-order tensor), inverse of the elastic rigidity tensor : ∼ ∼
−1 ε∼ e = : (σ∼ − σ∼ I ) ∼ ∼
(3.1)
• The thermal expansion, ε∼ th , function of the current temperature T and the tem, which can depend perature of the reference state TI . It is written as a tensor α ∼ on temperature, and which is spherical for isotropic and cubic materials. ε∼ th = (T − TI )α ∼
(3.2)
• A non-elastic part, ε∼ ne , itself resolved in a plastic, ε∼ p , and a viscoplastic, ε∼ vp , contribution (governed by elastoplatic and elastoviscoplastic constitutive equations). • Possibly a part, ε∼ cp , due to phase transformations during strain. Hence:
−1 ε∼ = : (σ∼ − σ∼ I ) + ε∼ th + ε∼ p + ε∼ vp + ε∼ cp ∼ ∼
(3.3)
The decomposition of the non-elastic part of the strain expresses that, during a transformation of the materials, several mechanisms can be active and produce an energy dissipation, and that, in the considered timescale, the viscosity of some mechanisms can be neglected (time independent plasticity ε∼ p ) whereas, for the others, real time is a critical variable in the rate definition (viscoplastic strain ε∼ vp ).
3.1.2. Criteria Each of the mechanisms responsible for the inelastic behavior is characterized by several parameters, called hardening variables, summarizing at a given time the material state and the influence of the past thermomechanical loading. As noticed in the previous chapter, the yield surface is defined in the stress-hardening variable (and temperature) space. For given temperature and hardening, it is a part of the vectorial space of dimension 6 of symmetrical tensors of second order, De = {σ∼ /f (σ∼ , AI , T ) 0}, the condition f (σ∼ , AI , T ) = 0 defining the limit of the yield surface.
3.1.3. Flow rules These are the rules that are going to allow us to define plastic or viscoplastic strain rate out of the yield surface. The study of rheological models has shown the nature of the equations involved to describe the flow rate intensity. It is linked to the total stress
3D plasticity and viscoplasticity
69
or strain rate for a plastic model, and to the current stress state and internal variables for a viscoplastic model. To generalize previous results to the 3D case, it is important to take into account the flow direction. This direction must be defined by a tensor in the 6 dimension vector space of symmetric tensors of second order.
3.1.4. Hardening rules In any real transformation, mechanical energy transmitted to the material is only partly returned by it. The other part is dissipated under one of the following forms: • increase of temperature (specific heat), • phase transformation (latent heat), • heat production given up to the surrounding environment, • internal structural modification of the material (dislocation movement, slip at grain boundaries, creation of new cracks. . .). This modification (or rearrangement) of the microstructure of the material during a transformation leads to a new state in which mechanical properties can evolve or not. Strain can leave the yield surface De unchanged (perfect plasticity, no hardening), or lead to a smaller yield surface (softening, negative hardening) or to a larger one (hardening, positive hardening). Expressing the effect of hardening is a very complex task, which closely depends on the class of material. Some materials even present hardening and then softening behavior during cyclic loading, for example. In addition, the nature of the hardening can be modified by complex loading paths or by aging of the material. Hardening models are then made of rules that characterize the evolution of hardening variables during an inelastic straining process. As seen for the one-dimensional case, the main types of hardening are the isotropic and the kinematic hardening.
3.1.5. Generalized standard materials Building a theory for plasticity or viscoplasticity involves, generally, the definition of a yield function, a flow rule and a hardening function. In the case of a generalized standard material, these three entities are defined by only one potential. It is then a particular class of materials, where flow and hardening are fully defined once the yield function is given. A systematic description involves “generalized strain” variables αI , associated to “generalized stresses”, AI (seen as fluxes). The description of this
70
Non-Linear Mechanics of Materials
class of materials [HAL75], interesting from a theoretical point of view, because the existence of the potential makes it possible to demonstrate theorems of existence and uniqueness of solutions, is suggested by the thermodynamics of irreversible processes. The internal dissipation evaluated by the thermodynamical approach is written: φ1 = σ∼ : ε∼˙ p − AI α˙ I
(3.4)
In this formalism, the variables AI are the scalar or tensorial hardening variables associated with the state variables αI , that are scalar or tensorial, accordingly. They are defined provided the shape of the specific free energy ψ has been set: AI = ρ
∂ψ ∂αI
(3.5)
We note that the stress is a particular flux variable, which is associated to the opposite of the plastic strain. Then, we will consider the vector Z constituted by the stress tensor components and the variables AI ’s, and the vector z, constituted by the plastic strain components and the variables (−αI ), such that: φ1 = Z˙z
(3.6)
This dissipation positivity can be ensured a priori if we admit the existence of a potential , real-valued function of the flux variables, defining the evolution of the state variables: ∂ (3.7) z˙ = ∂Z Indeed, it suffices that the equipotential defined by in the flux variables space are convex surfaces and that they contain the origin. An important particular case is encountered when depends on the flux variables through the threshold function f . In that case, the state variables rates can be written as: z˙ =
∂ ∂f ∂f ∂Z
(3.8)
In the right-hand side of the previous equation, the first term is scalar, and corresponds to the flow intensity; the second term has the dimension of the flux variable space, it gives the flow direction, which is then defined by the normal to the yield surface. The (visco)plastic flow direction is given by ∂f /∂σ∼ , and the evolution direction of the hardening variables by ∂f /∂AI .
3.2. Formulation of the constitutive equations 3.2.1. State variables definition Hereafter, the gradient of f with respect to σ will be denoted by n , n = ∂f/∂σ∼ . We ∼ ∼ describe first the class of the generalized standard models, defined in the previous
3D plasticity and viscoplasticity
71
paragraph, that are built upon the definition of the yield function only. The above formulation provides naturally the nature of the hardening variables to use for a given form of the yield function. We cover here, as an example, the case of kinematic and isotropic linear hardenings coupled with the von Mises criterion. Introducing for kinematic hardening the scalar R to model isotropic hardening and the tensor X ∼ (which is a deviatoric tensor), the yield function is written: 1/2 3 (s , R) = J (σ − X ) − R − σ = − X ) : (s − X ) − R − σy (3.9) f (σ∼ , X y ∼ ∼ ∼ ∼ ∼ 2 ∼ ∼ The tensorial variable α , associated with the hardening variable X , is then noth∼ ∼ ing but the plastic strain itself, whereas the variable p, associated with the hardening variable R, is the accumulated plastic strain: ˙ =− α ∼
∂ ∂f ∂ ∂f = = ε∼˙ p ∂f ∂X ∂f ∂σ ∼ ∼
p˙ = −
∂ ∂ ∂f = ∂f ∂R ∂f
(3.10)
The variable p is called accumulated plastic strain, because it measures the length of the strain path (see Sect. 3.3.1): 2 p p 1/2 2 ∂ ∂ 1/2 ∂ ε∼˙ : ε∼˙ n n (3.11) = : = ∼ ∼ 3 3 ∂f ∂f ∂f Under uniaxial loading, the plastic strain rate tensor is diagonal (˙ε p , −(1/2)˙ε p , −(1/2)˙ε p ) and the calculation of p˙ gives: p˙ = |˙ε p |.
3.2.2. Viscoplasticity The previous formalism applies directly; the function constitutes a viscoplastic potential, since it includes enough information to fully characterize the flow intensity and the flow direction. The derivative ∂/∂f is the viscosity function. Choosing a viscoplasticity model boils down to choosing a function f and a form of the potential. As a consequence, the one-dimensional Norton’s model can be simply generalized to 3D by adopting the von Mises criterion, defined by the stress dependent function f , such as f = J (σ∼ ), and the following function as a potential: J (σ∼ ) n+1 K (3.12) = n+1 K
Then: ε∼˙
vp
=
J K
n
∂J ∂σ∼
(3.13)
72
Non-Linear Mechanics of Materials
The first term of the previous equation gives the flow intensity. It is actually equal to (|σ |/K)n for a simple tension load. The partial derivative of J with respect to σ∼ is given by: ∂J ∂J ∂s∼ 3s 1 3s = : = ∼ : (I∼ − I∼ ⊗ I∼ ) = ∼ (3.14) ∂σ∼ ∂s∼ ∂σ∼ 2J 3 2J ∼ As already seen, for such a kind of model, the yield stress is always zero, and the yield surface is reduced to the origin. This case would be of no interest for a plasticity model independent of the time. A Bingham model can be recovered by choosing a quadratic form for the function : η J (σ∼ ) − σy 2 (3.15) = 2 η
3.2.3. From viscoplasticity to plasticity Figure 3.1a shows the form of the viscoplastic potential , an increasing monotonic function of f , such that (0) = 0, which illustrates the fact that the flow intensity depends on the “altitude” of the current operating point, and that, geometrically, the direction of the inelastic strain rate vector is normal to the equipotential surfaces. When the function becomes more and more nonlinear (for instance by choosing a power function whose exponent n tends towards infinity), the projection of the equipotential upon the (σ∼ , AI ) space narrows around the surface f = 0. That way, one defines a region of space where the potential is zero, and another where it varies quickly. At the limit, becomes confused with the function indicating the yield surface (Fig. 3.1b), and the flow intensity cannot be defined anymore as ∂/∂f . This effect results from the fact that in viscoplasticity the flow rate is defined by the “excess of stress”, or distance between the current operating point and the yield surface, and that this excess of stress remains null in plasticity, as the operating point stays on the yield surface during flow (f (σ∼ , AI ) = 0). This illustrates the difference in nature between viscoplasticity and plasticity theories. The viscoplastic framework offers many opportunities to write a model by choosing the viscosity function, whereas, in the plasticity framework (for the conditions studied until now), the expression of the yield surface fully determines the flow intensity. It is then necessary to introduce an additional element to handle the problem in plasticity. We assume then that we maximize the intrinsic dissipation 1 . As this maximization must be constrained, to express the fact that f remains negative or null, we write [LUE84]: F(Z) = Z˙z − λ˙ f (3.16)
3D plasticity and viscoplasticity
73
Figure 3.1. Comparison between plasticity and viscoplasticity theories, (a) viscoplastic potential, (b) plastic model obtained as limit behavior
by introducing the plastic multiplier λ˙ to express the constraint. An extremum can be reached only if the partial derivative ∂F/∂Z, is nil, which gives: z˙ = λ˙
∂f ∂Z
(3.17)
Thus: ε∼˙ p = λ˙
∂f ˙ = λn ∼ ∂σ∼
α˙ I = −λ˙
∂f ∂AI
(3.18)
The extremum is a maximum iff the function f is convex. The propositions (i) and (ii) hereafter are then equivalent: (i) Z being real maximizes the intrinsic dissipation; (ii) the flow direction z˙ is normal to the surface defined by f ; the domain defined by f is convex. The normality rule is then preserved. In plasticity, the scalar equation provided by the consistency condition f˙ = 0 will replace the equation that vanishes from the viscoplastic formalism due to the singularity of the indicator function of f in f = 0. ˙ The plastic formalism consists in replacing ∂/∂f by the plastic multiplier λ.
3.2.4. Comments on the formulation of the plastic constitutive equations Yield condition In 3D, the yield condition is written:
74
Non-Linear Mechanics of Materials
−elastic unloading if:
−1 (˙ε∼ = : σ∼˙ ) ∼ ∼ f (σ∼ , AI ) = 0 and f˙(σ∼ , AI ) < 0
−plastic flow if:
f (σ∼ , AI ) = 0 and f˙(σ∼ , AI ) = 0
−yield surface interior: f (σ∼ , AI ) < 0
−1 (˙ε∼ = : σ∼˙ + ε∼˙ p ) ∼ ∼
(3.19) −1
(˙ε∼ = ∼ ∼
: σ∼˙ ) (3.20) (3.21)
The condition to verify during flow is f˙(σ∼ , AI ) = 0, and not just n : σ∼˙ > 0 ∼ as too often indicated mistakenly. The difference is perceptible when the function f depends on external parameters such as temperature, chemical composition, . . . whose evolution can be imposed independently from the plasticity problem. Noting all these parameters PJ , the consistency condition f˙ = 0 is written (with YI = ∂f/∂AI ): ⏐ ⏐ ∂AI ⏐ ∂AI ⏐ ⏐ ⏐ P˙J ˙ ˙ ˙ : σ∼˙ + YI AI = 0 with AI = α˙ K + (3.22) f =n ∼ ∂αK ⏐PJ ∂PJ ⏐αI In (3.22), the non-diagonal terms of ∂AI /∂αK introduce a possible state coupling, as indicated in Sect. 2.3.5. By replacing internal variable rates with their expression (in (3.18)), we get: ⏐ ⏐ ⏐ ∂AI ⏐ ⏐ P˙J with H = YI ∂AI ⏐ YK : σ ˙ + (3.23) H λ˙ = n ⏐ ∼ ∼ ∂PJ αI ∂αK ⏐PJ where the numerator includes, in addition to the traditional term in σ∼˙ , a second driving term for plastic flow, which depends on the evolution of external parameters, and where H is the plastic modulus.
Perfectly plastic behavior The principle of the “maximal plastic work”, proposed by von Mises, and then by Hill in 1951, states that: “The work of the real stress σ∼ associated with the real plastic strain rate ε∼˙ p is greater than the work of any other admissible tensor σ∼ ∗ (i.e., not violating the plasticity criterion) associated with ε∼˙ p ”. (σ∼ − σ∼ ∗ ) : ε∼˙ p 0
(3.24)
This principle can actually be demonstrated in the case of metals that deform by slip and follow the Schmid law. It is not exhibited by all materials, in particular soils. It constitutes a particular case of the maximization of the intrinsic dissipation hypothesis enunciated in the previous paragraph, when there is no hardening, i.e., when the material is perfectly plastic. For the record, we give here the consequences
3D plasticity and viscoplasticity
75
that derive classically from this principle. They are expressed simply in terms of stress and plastic strain, but they can also be read (for a hardening material) in terms of generalized stress and generalized plastic strain. If one chooses σ∼ ∗ on the yield surface, one verifies that, if σ∼ is in the yield surface, = 0∼. If the yield surface does not contain any corner, the principle of maximal ∼ work can be applied from a starting point σ∼ on the yield surface, by choosing a point σ∼ ∗ infinitely close, on the yield surface as well. σ∼ ∗ is deduced from σ∼ thanks to a unit tensor ∼t ∗ belonging to the plane tangent to the surface in σ∼ (σ∼ ∗ = σ∼ + kt∼∗ , with k > 0). The same operation can be made, by considering σ∼ ∗ = σ∼ − kt∼∗ as a starting point, which leads to the following inequalities: ε˙ p
kt∼∗ : ε∼˙ p 0 and − kt∼∗ : ε∼˙ p 0 so that: ∼t ∗ : ε∼˙ p = 0
(3.25) (3.26)
The plastic strain rate is collinear to the normal to the yield surface, n . The scalar λ˙ ∼ defines the flow intensity. It is always positive, as, by choosing now σ∼ ∗ on the normal is collinear to n and has the to the point σ∼ , inside the yield surface, (σ∼ − σ∼ ∗ ) = kn ∼ ∼ same direction (k > 0) so that the expression (σ∼ − σ∼ ∗ ) : ε∼˙ p 0 becomes: : λ˙ n 0 hence: λ˙ 0 kn ∼ ∼
(3.27)
There exists then a set of conditions between λ˙ and f , known as Karush–Kuhn– Tucker conditions: ˙ =0 λ˙ 0 f 0 λf (3.28) In the case of materials conforming to the principle of maximal work, the yield surface plays simultaneously the role of a pseudo-plastic potential, and determines the plastic flow by allowing computation of the scalar multiplier. If the surface is not regular and has a corner at the point σ∼ , there exists here a cone of normals, in which lies the direction of the plastic strain increment. By applying again the principle of maximal work from a stress state σ∼ on the yield surface, and by considering σ∼ ∗ inside the yield surface, the normality rule allows us to write the following expression, which states that the surface must be entirely on the same side of the tangent plane in σ∼ (Fig. 3.2b): 0 (σ∼ − σ∼ ∗ ) : n ∼
(3.29)
The yield surface is then convex. This is also true for the function f . During plastic flow, the point representing stress state has to stay on the fixed yield surface. The plastic multiplier is undetermined; the plastic load condition and the consistency condition become respectively (in the absence of external parameters): for f (σ∼ ) = 0 and f˙(σ∼ ) = 0 : during flow: n : σ∼˙ = 0 ∼
ε∼˙ p = λ˙
∂f ˙ = λn ∼ ∂σ∼
(3.30) (3.31)
76
Non-Linear Mechanics of Materials
Figure 3.2. Consequences of the principle of maximal work (a) illustration of the normality rule, (b) convexity of f
3.3. Flow direction associated to the classical criteria The flow directions are first calculated for a perfectly plastic material. The modifications due to hardening are indicated in the next paragraph.
3.3.1. Von Mises criterion The yield function is written f (σ∼ ) = J (σ∼ ) − σy , so that the normal n is: ∼ = n ∼ With:
∂f ∂J ∂J ∂s∼ = = : ∂σ∼ ∂σ∼ ∂s∼ ∂σ∼
∂s∼ 1 = J∼ = I∼ − I∼ ⊗ I∼ ∂σ∼ 3 ∼ ∼
(3.32)
(3.33)
one gets:
J :σ 3s 3 ∼∼ ∼ = ∼ (3.34) 2 J 2J In the case of the von Mises criterion, the flow direction is given by the deviator of the stress tensor. n = ∼
This expression can be simplified for a tension loading along direction 1: ⎛ ⎞ 1 0 0 2σ ⎝ 0 −1/2 0 ⎠ s= J = |σ | ∼ 3 0 0 −1/2 ⎛ ⎞ 1 0 0 0 ⎠ sign(σ ) = ⎝ 0 −1/2 n ∼ 0 0 −1/2
(3.35)
3D plasticity and viscoplasticity
77
3.3.2. Tresca criterion The flow rule is defined by sectors in the eigenstress space. For example, for the σ1 > σ2 > σ3 case, the yield function is written: f (σ∼ ) = |σ1 − σ3 | − σy , so that, for all the stress states that verify this inequality, as the material does not deform along axis 2 (shear type strain), the plastic strain rate has the same components: ⎛
si σ1 > σ2 > σ3 :
⎞ 1 0 0 ε∼˙ p = λ˙ ⎝ 0 0 0 ⎠ 0 0 −1
(3.36)
The definition of the normal poses a specific problem for the stress states corresponding to singular points: for instance, in tension, when σ1 > σ2 = σ3 = 0, the criterion is written indifferently f (σ∼ ) = |σ1 − σ2 | − σy or f (σ∼ ) = |σ1 − σ3 | − σy . One can then define two multipliers, each of them referring to a form of the criterion. If these two multipliers are chosen equal, the model gives the same form as von Mises criterion in tension. However, as soon as the stress state differs from the strict equality between the components σ2 and σ3 , it is one of the two shear regimes that gains the upper hand; ⎛
if σ1 > σ2 = σ3 = 0 :
⎞ ⎛ ⎞ 1 0 0 1 0 0 ε∼˙ p = λ˙ ⎝ 0 0 0 ⎠ + μ˙ ⎝ 0 −1 0 ⎠ 0 0 −1 0 0 0
(3.37)
3.3.3. Drucker–Prager criterion The yield function is written f (σ∼ ) = (1 − α)J (σ∼ ) + αI1 (σ∼ ) − σy . As a result, the has a spherical component. The plastic strain evaluated with such a criterion normal n ∼ does not reduce to a deviatoric shape, and generates a volume increase, whatever the applied load is: 3 ∼s + αI∼ 2J Tr ε∼˙ p = 3α λ˙
= (1 − α) n ∼
(3.38) (3.39)
78
Non-Linear Mechanics of Materials
3.4. Expression of some particular constitutive equations in plasticity 3.4.1. Prandtl–Reuss model This model is obtained by using the von Mises criterion and an isotropic hardening rule. The yield function is: f (σ∼ , R) = J (σ∼ ) − σy − R(p)
(3.40)
The isotropic hardening is described by the function R(p). In the case of an uniaxial loading in tension, where only the component σ11 = σ is not zero, the equality f (σ∼ , R) = 0 boils down to: σ = σy + R(p)
(3.41)
The curve described by (σy +R(p)) is nothing but the hardening curve for monotop nous uniaxial loading, the tensile strain ε11 = ε p being equal in that case to the accumulated plastic strain. The plastic modulus can be assessed as the slope of that curve: dR dR (3.42) H = p = σ = σy + R(ε p ) dε dp R(p) can be defined point per point, by an exponential or a power function, as we saw in the chapter on uniaxial plasticity. Whatever the form chosen for R, the consistency condition allows us to find the plastic multiplier (λ˙ = p): ˙ ∂f ∂f ˙ : σ∼˙ + : σ∼˙ − H p˙ = 0 R = 0 is written n ∼ ∂σ∼ ∂R n : σ˙ and λ˙ = ∼ ∼ H
(3.43) (3.44)
The Prandtl–Reuss model allows us to determine the direction and intensity of plastic flow: n : σ˙ 3s ˙ = ∼ ∼ n with n = ∼ ε∼˙ p = λn (3.45) ∼ ∼ H ∼ 2J In the particular case of a tension load, this general expression reduces to the usual uniaxial form: n11 = sign(σ )
n : σ∼˙ = σ˙ sign(σ ) and λ˙ = p˙ = ε˙ 11 ∼ n11 σ˙ σ˙ so that ε˙ p = n11 = H H p
(3.46) (3.47)
3D plasticity and viscoplasticity
79
3.4.2. Hencky–Mises model The previous equations must be integrated to get the plastic strain path. Generally, the normal n rotates during the loading, so that there is a coupling between the com∼ ponents of the plastic strain tensor. The coupling disappears when the hypothesis of simple loading is valid, i.e., when the external loading in terms of stress increases proportionally to a single scalar parameter k, starting from a virgin initial state. When the stress varies between 0 and σ∼ M , k goes from 0 to 1, and: σ∼ = kσ∼ M
σ∼˙ = k σ∼˙ M
s = ks∼M
∼
J = kJM
(3.48)
The normal has then a fixed direction during loading: n = ∼
3 ∼s M 2 JM
(3.49)
The plastic strain tensor is now defined if the equivalent plastic strain is known: t t ˙ pdt ˙ = p= λdt (3.50) 0
0
This will come by integrating the following rate equation: : σ∼˙ n 3 sM σ∼ M k˙ JM k˙ ∼ = = : H 2 JM H H
(3.51)
The upper bound of the integral is k = 1, meanwhile the lower bound is ke , the value of k for which the initial yield surface (ke JM = σy ) is reached: ε∼ p = pn ∼
with p =
J (σ∼ ) − σy JM (k − ke ) = H H
(3.52)
3.4.3. Prager model The Prager model is obtained by using the von Mises criterion and a linear-kinematic hardening rule. One has to introduce a hardening variable X , associated with plastic ∼ p . This variable is deviatoric, the yield function is = (2/3)H ε strain, written: X ∼ ∼ written simply: ) = J (σ∼ − X ) − σy (3.53) f (σ∼ , X ∼ ∼ The consistency condition is: ∂f ∂f ˙ : σ∼˙ + :X = 0 or ∼ ∂σ∼ ∂X ∼
˙ =0 n : σ∼˙ − n :X ∼ ∼ ∼
(3.54)
80
Non-Linear Mechanics of Materials
In the previous equations, J and n are defined by: ∼ )= J (σ∼ − X ∼
1/2 3 (s∼ − X ) : (s − X ) ∼ ∼ ∼ 2
3 ∼s − X ∼ 2 J (σ∼ − X ) ∼
(3.55)
2 H λ˙ n = H λ˙ hence λ˙ = (n : σ∼˙ )/H ∼ ∼ 3
(3.56)
and n = ∼
One gets then: ˙ =n: n : σ∼˙ = n :X ∼ ∼ ∼ ∼
The plastic multiplier has the same formal expression as in the isotropic hardening case; however, one should note that the definition of n is modified, and that H is ∼ constant. With uniaxial loading, σ = σ11 is the only non-zero component of the stress tensor, and by posing X = (3/2)X11 , the yield function and the consistency condition are written: |σ − X| = σy
σ˙ = X˙ = H ε˙ p
(3.57)
3.5. Flow under prescribed strain rate 3.5.1. Case of an elastic–perfectly plastic material The plastic multiplier is undetermined for an elastic–perfectly plastic material loaded under prescribed stress rate. This is because, the plastic modulus being zero, there exists an infinity of equivalent positions in plastic strain for a given admissible stress state, such that J (σ∼ ) = σy : thereby, in tension σ11 = σ0 , all the diagonal tensors (ε p , (−1/2)ε p , (−1/2)ε p ) are possible solutions. Imposing the total strain rate changes, of course, this result. The plastic multiplier could then be determined by combining the elastic constitutive equation written in terms of rate and the consistency condition: : (˙ε∼ − ε∼˙ p ) σ∼˙ = ∼ ∼
and n : σ∼˙ = 0 ∼
(3.58)
The scalar product of the two members of the first relation by n allows an “inversion” ∼ of the equation: ˙ n : σ∼˙ = n : : (˙ε∼ − ε∼˙ p ) = n : : ε∼˙ − n : : λn ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ n : : ε ˙ ∼ ∼ ∼ ∼ so that: λ˙ = n : :n ∼ ∼ ∼ ∼
(3.59) (3.60)
3D plasticity and viscoplasticity
81
In the special case of isotropic elasticity and of the von Mises criterion, one gets successively the following simplifications: 3s (3.61)
ij kl = λδij δkl + μ(δik δj l + δil δj k ), n = ∼ ∼ 2J nij ij kl = 2μnkl nij ij kl nkl = 3μ nij ij kl ε˙ kl = 2μnkl ε˙ kl (3.62) 2 λ˙ = n : ε˙ (3.63) 3∼ ∼ For an uniaxial loading, with ε˙ = ε˙ 11 , this last expression reduces to: λ˙ = ε˙ sign(σ )
which gives again ε˙ p = ε˙
(3.64)
3.5.2. Case of a material with hardening As indicated by both examples in the previous paragraph, the consistency condition is always written in the same form for the common constitutive equations of isotropic materials. By comparison with the case of perfectly plastic materials, only the consistency condition is going to change; it is then necessary to start from: : (˙ε∼ − ε∼˙ p ) σ∼˙ = ∼ ∼
and: n : σ∼˙ = H p˙ ∼
(3.65)
After multiplication of the two members of the first relation by n , we get: ∼ λ˙ =
n : : ε∼˙ ∼ ∼ ∼
(3.66)
H +n : :n ∼ ∼ ∼ ∼
This allows us to define the tangent behavior in elastoplasticity:
:n ) ⊗ (n : ) ( ∼ ∼ p ∼ ∼ ∼ σ∼˙ = : (˙ε∼ − ε∼˙ ) = − ∼ : ε∼˙ ∼ ∼ H +n : :n ∼ ∼ ∼ ∼ ∼
(3.67)
∼
The tangent elastoplatic operator is perfectly symmetric. In the case of isotropic elasticity and von Mises material, the multiplier becomes: λ˙ =
2μn : ε∼˙ ∼ H + 3μ
(3.68)
3.6. Non-associated plasticity The models that have been described until now use the same function as the limit of the yield surface, for the determination of the flow direction and to evaluate the evolution of the hardening variables. One can actually distinguish three kinds of models,
82
Non-Linear Mechanics of Materials
depending on what is used for each of these points. threshold flow (1)
f
(2)
f
(3)
f
∂f ε∼˙ p = λ˙ ∂σ∼ ∂f p ε∼˙ = λ˙ ∂σ∼ ˙ ε∼˙ p = λP ∼
hardening ∂f α˙ I = −λ˙ ∂AI
(3.69)
˙ I α˙ I = −λM
(3.70)
˙ I α˙ I = −λM
(3.71)
The model (1) is generalized standard, (2) is simply associated (the function f is not used anymore for the hardening but still for the flow), the form (3), the most general, characterizes a non-associated model. In this last case, the plastic multiplier is calculated as previously, thanks to the = ∂f/∂σ∼ , and in consistency condition, which becomes, (preserving the notation n ∼ the absence of external parameters): n : σ∼˙ + ∼ Successively: λ˙ =
n : σ∼˙ ∼ H
∂f ˙ AI = 0 ∂AI
with H =
(3.72)
∂f ∂AI MI ∂AI ∂αI
(3.73)
I
λ˙ =
n : : ε∼˙ ∼ ∼ ∼
(3.74)
H +n : : P∼ ∼ ∼ ∼
σ∼˙ = − ∼ ∼
( : P∼ ) ⊗ (n : ) ∼ ∼ ∼ ∼
∼
H +n : : P∼ ∼ ∼
: ε∼˙
(3.75)
∼
Now, we notice that the tangent elastoplastic operator does not respect anymore the main symmetry.
3.7. Non-linear hardening The models treated here are traditional phenomenological ones. On one hand, materials obeying the current criteria (von Mises, Hill) are considered. On the other hand, crystallographic models are also described, since they present interesting characteristics, in particular they do not presuppose the shape of the macroscopic criterion of plasticity. They are now rather classical in the academic environment, but they are not implemented yet in the main commercial codes. To try to be more exhaustive, it would be necessary to review the abundant literature dealing with constitutive equations. In
3D plasticity and viscoplasticity
83
fact, we will restrict ourselves to mentioning a few key papers. The reader is invited to refer to the cycles of conferences that are focused on the subject, like “International Conference on Plasticity” or the Euromech-Mecamat conferences.
3.7.1. Kinematic and isotropic hardening After two or three decades of development of new constitutive models, the form that now dominates brings into play a yield surface whose translation and expansion are that defines the center (kinematic governed by the evolution of a tensorial variable X ∼ hardening), and a scalar variable that defines the radius (isotropic hardening). In this framework, it is of course necessary to go beyond linear evolution for both types of hardening. We will write the threshold function as indicated in (3.76) if we wish to follow the generalized normality hypothesis, or more simply under the classical form (3.77), with—in this case—an associated (but not generalized standard) (visco)plastic formalism, if the first framework is considered too restrictive. , R) = J (σ∼ − X ) − σy − R + f (σ∼ , X ∼ ∼ 1/2 3 X : X 2∼ ∼ f (σ∼ , X , R) = J (σ∼ − X ) − σy − R ∼ ∼
D 2 R2 J (X ) + ∼ 2C 2Q
)= with J (X ∼
(3.76) (3.77)
Following the formalism of the previous chapter, we get: ∂f 3D X λ˙ = n − ∼ ∂X 2C ∼ ∼ R ∂f ˙ = 1− λ˙ r˙ = −λ ∂R Q
˙ = −λ˙ α ∼
(3.78) (3.79)
The variables that represent the hardening are α and r. Their indication allows one ∼ and R, if one chooses a quadratic form for the free to get the hardening variables X ∼ energy: 1 1 ψ = · · · + bQr 2 + Cα :α 2 3 ∼ ∼ 2 X = Cα R = bQr ∼ 3 ∼
(3.80) (3.81)
As in the previously studied cases, the plastic multiplier is equal to the accumulated plastic strain rate. The evolution of the hardening variables in the case where the
84
Non-Linear Mechanics of Materials
coefficients are constant, is written simply as a function of p: ˙ ˙ = 2 C ε˙ p − DXp˙ X ∼ ∼ 3 ∼ R˙ = b(Q − R)p˙ thus
(3.82) R = Q(1 − exp(−bp))
(3.83)
However, the accumulated strain is no longer the state variable that characterizes isotropic hardening. The adequate variable must saturate when the accumulated plastic strain increases. In such a model, the coefficients depending on the material are the initial yield stress, σy , two coefficients to represent the evolution of the isotropic hardening, b and Q, and two coefficients to represent the evolution of kinematic hardening, C and D. This nonlinear kinematic hardening rule was proposed by Chaboche [CHA77] and Armstrong and Frederick [ARM66].
3.7.2. Dissipated energy, stored energy The intrinsic dissipation evaluated with the kind of model described in the previous paragraph is written, in the case where we adopt the formalism of the generalized standard model: ˙ :α φ1 = σ∼ : ε∼˙ p − R r˙ − X ∼ ∼ 2 R D 2 −R+ :n + J (X ) λ˙ −X = σ∼ : n ∼ ∼ ∼ ∼ Q C
(3.84) (3.85)
with: −X :n = J (σ∼ − X ) σ∼ : n ∼ ∼ ∼ ∼ hence:
R2 D 2 ) − R + (X ) λ˙ + J φ1 = J (σ∼ − X ∼ ∼ Q C R2 D 2 ) λ˙ + J (X = f + σY + ∼ 2Q 2C
(3.86)
(3.87) (3.88)
In this expression: – f λ˙ corresponds to the viscous dissipation; it is positive if the model is viscoplastic, and zero for a plastic model; – σY λ˙ is the friction related dissipation due to the initial threshold; – both quadratic terms correspond to dissipations linked to the nonlinearity of the hardening; these last two terms disappear for linear expressions, and the calculated dissipation is then the one of the corresponding perfectly plastic model.
3D plasticity and viscoplasticity
85
In the energy balance, the stored energy part that contributes to the free energy variation is: ˙ :α ψ˙ = R r˙ + X ∼ ∼
3D −bp −bp : n − X p˙ = Q(1 − e )e p˙ + X ∼ ∼ 2C ∼
(3.89) (3.90)
These formulae clearly show that the hardening corresponds to the capability of the material to store energy. When the hardening variables reach limiting values, the derivative of the free energy becomes zero and the material does not undergo any further hardening. Moreover, since p increases for any loading type, the energy stored by the isotropic hardening mechanism is not retrievable. On the contrary, the energy stored in the linear kinematic hardening component can be fully recovered when plastic strain comes back to zero: the energy storage is only temporary in that case (let us think the spring-friction device system to have a physical illustration of this observation). Dissipation is found if kinematic hardening is nonlinear, and the dissipation clearly relates to the value of the material parameter D which controls the nonlinearity. The reader concerned with these problems could refer to the publications that tried to quantify these various terms from experimental results [CHR92, CHR98, CHA93c]. The classical yield function case, without quadratic terms, is subject to only small modifications of the previous results. Indeed, it corresponds to the case (2) in Sect. 3.6, with an associated model but not a generalized standard one. The only change concerns the amount of dissipated energy, the dissipation due to nonlinearity being twice as much as the previous one.
3.7.3. Typical results Tension In tension along axis 1, the accumulated plastic strain p is equal to the plastic strain p ε11 , denoted by ε p hereafter. The second invariant of the stress tensor is equal to its component σ11 , denoted by σ hereafter. For a model comprising both an isotropic and a kinematic hardening, the tension curve is then modeled by: σ = σy + Q(1 − exp(−bε p )) +
C (1 − exp(−Dε p )) D
(3.91)
The maximum value of the stress reached is then σy + Q + (C/D), which identifies with the ultimate stress. Equation (3.91) shows that it is impossible to distinguish between isotropic and kinematic hardening during a uniaxial tensile test. This is illustrated in Fig. 3.3. If the coefficient D is zero, the result is nothing but a Prager’s
86
Non-Linear Mechanics of Materials
Figure 3.3. An example of identical responses in tension for isotropic and kinematic hardening (E = 200000)
model, the contribution of the kinematic hardening being then simply Cε p . The difference is important, as there is no longer a limit stress in tension. What follows will show that this difference also has consequences on cyclic loading.
Loading under prescribed symmetrical strain Prescribing a strain loading with symmetrical bounds in tension and compression is the basic experiment used to characterize cyclic plasticity. The curve (plastic strain– stress) presents an hysteresis. By collecting the coordinates of the loop extremities, one can build the cyclic hardening curve in the plane ε p –σ . After a transitory rounded shape, the isotropic hardening rule leads to a cycle shaped like a parallelogram, with two horizontal edges (Fig. 3.4a). If Q is positive, there is a cyclic hardening (as observed on the figure), otherwise, there is a cyclic softening. The linear kinematic hardening rule leads also to a parallelogram, from the second cycle on (Fig. 3.4b). The nonlinear kinematic hardening rule leads to a more realistic shape (Fig. 3.4c), all the more if it is coupled with an evolution of the isotropic hardening (Fig. 3.5). Along a branch of the hysteresis curve, one can integrate the nonlinear kinematic variable, to find (with η = 1 for a branch in tension and η = −1 for one in compresp sion), from a starting point (ε0 –X0 ): X=η
C C p + X0 − η exp(−ηD(ε p − ε0 )) D D
(3.92)
3D plasticity and viscoplasticity
Figure 3.4. 200000)
87
Elementary response to a symmetrical loading under prescribed strain (E =
88
Non-Linear Mechanics of Materials
Figure 3.5. Loading under symmetrical prescribed strain with nonlinear isotropic and nonlinear kinematic hardening (E = 200000)
The use of the same formula leads for the cyclic curve to an analytic description: σ = 2(σy + Q) + 2
C tanh(Dε p /2) D
(3.93)
Loading under non-symmetrical prescribed strain The various models differ in their responses to a loading under non-symmetrical strain. The isotropic hardening model stabilizes on the same cycle as in the symmetrical loading case, and remains symmetrical in stress. On the contrary, the linear kinematic hardening develops a mean stress (Fig. 3.6a), as the parallelogram of the equilibrium cycle leans on invariant lines whose slope is the elastoplastic modulus in the ε–σ plane. At last, there is a mean stress relaxation for the nonlinear kinematic hardening model (Fig. 3.6b). Finally, one should note that, in the case of the superposition of a linear kinematic model and a nonlinear kinematic one, the former has the upper hand, and there subsists a mean stress (Fig. 3.6c). The real behavior of materials is often intermediate (partial relaxation of the mean stress), so that modeling involves more complex models.
Non-symmetrical imposed stress (1D ratcheting) Under a non-symmetric tension-compression loading σmin –σmax , the mechanical response can (i) become elastic after a phase of elastoplastic behavior (elastic shakedown), (ii) present a progressive deformation that stops on an open cycle (plastic
3D plasticity and viscoplasticity
Figure 3.6. Response obtained in non-symmetrical imposed strain (E = 200000)
89
90
Non-Linear Mechanics of Materials
Figure 3.7. Response obtained for a prescribed non-symmetrical stress path (1D ratcheting) (E = 200000)
shakedown), (iii) present a progressive unbounded strain that leads to ruin (ratcheting). For the case of linear kinematic hardening, the plastic strain that occurs during the first tension step is such that σmax = σy + Cε p . There will then be plastic shakedown if σmax − σmin < 2σy , otherwise there will be elastic shakedown (Fig. 3.7a). In the case of nonlinear kinematic hardening, one necessarily gets σmax < σy + C/D and σmin > −σy − C/D, in order not to exceed the limit load predicted by the model. The plastic strain produced by the first tension is such that σmax = σy + (C/D)(1 − exp(Dε p )). There will then be elastic shakedown if σmax − σmin < 2σy , and, otherwise, ratcheting (see Fig. 3.7b), with a regular ratcheting step at each cycle,
3D plasticity and viscoplasticity
91
Figure 3.8. Imposed tension and alternate shear (tensile-shear ratcheting) (E = 200000)
leading to a plastic strain accumulation per cycle δε p :
(C/D)2 − (σmin + σy )2 1 p ln δε = D (C/D)2 − (σmax − σy )2
(3.94)
Imposed tension and alternate shear (tensile-shear ratcheting) Another useful case in practical applications is the tensile-shear ratcheting, for instance when a material element undergoes a constant tensile stress σ0 and an alternate symmetrical loading between τm and −τm . The linear kinematic hardening model gives rise to an elastic shakedown as soon as (σ0 )2 + 3(τm )2 < (σy )2 , and, otherwise,
92
Non-Linear Mechanics of Materials
to a plastic shakedown (Fig. 3.8a). The nonlinear kinematic model gives rise to an elastic shakedown only if:
σ y σ0 σy + (C/D)
2 + 3(τm )2 < (σy )2
(3.95)
and to ratcheting for all the other cases (Fig. 3.8b). The progressive axial strain can be related to the amplitude of shear plastic strain: 4 δε p = √ 3
(σ0 )2 (σy + (C/D))2 − (σ0 )2
1/2 p
ε12
(3.96)
None of the two assumptions of hardening evolution is really satisfying, since linear kinematic hardening denies ratcheting, which actually exists, and nonlinear kinematic hardening overestimates it.
3.8. Some classical extensions 3.8.1. Multikinematic A more acute description of the curvature of the experimental tensile curve, and a better description of the stress redistribution effects, in particular for small plastic strain, is obtained in the framework of the kinematic hardening model by superimposing sev; each of them is independent, and the new model thus defined only eral variables X ∼ introduces a flexibility for the identification, thanks to a sort of development involving exponentials whose coefficients Di have various orders of magnitude: X = ∼
X = ∼i
i
2 Ci α i 3 ∼
˙ = ε∼˙ p − with α ∼i
3Di Xi p˙ 2Ci ∼
(3.97)
3.8.2. Modification of the dynamic recovery term The proposition of the form of the dynamic recovery term, which geometrical construction was initially proposed by Mróz [MRÓ67], has been widely commented on since then. There exist many changes, which leave the model in the generic form: ˙ =m p˙ α ∼ ∼
with m =n − ∼ ∼
3D :X 2C ∼∼ ∼
(3.98)
3D plasticity and viscoplasticity
93
Intensity change in (3.98) is simply proportional to In the case of an intensity change, the tensor ∼ ∼
identity. The simplest change consists in choosing = (φ∞ + (1 − φ∞ ) exp(−ωp)) I∼ ∼ ∼
∼
(3.99)
which allows the form of the kinematic hardening to be transformed between the initial cycle (saturation rate D, asymptote C/D) and the stabilized cycle (saturation rate φ∞ D, asymptote C/(φ∞ D)) [MAR89]. In the presence of a loading under non-symmetrical prescribed strain, experience shows that a stress redistribution occurs for large amplitudes, but not at all, or just a little, for lower ones. This effect can be represented by superimposing a large number of kinematics, as indicated in the previous paragraph, but it can also be pertinent to use a form that suppresses the dynamic recovery term for low amplitudes: according to such a rule, the hardening is linear for small amplitudes, and nonlinear for larger amplitudes [CHA91]. An approach of this kind has also been used by other authors [OHN93a, OHN93b].
= ∼ ∼
DJ (X ) − ωC ∼ 1−ω
m1
1 m I DJ (X ) 2 ∼∼ ∼
(3.100)
We recover a classical model when m1 = m2 and ω = 0. Otherwise, the dynamic recovery term is present only when the second invariant of X exceeds ωC/D, ω being ∼ comprised between 0 and 1.
Fading memory The properties of multiaxial ratcheting are linked to the flow direction, which, in the case of linear kinematic hardening, has a zero component along 11 direction, for a shear loading cycle with an imposed axial stress, since the component 11 of the kinematic variable increases until it reaches the level of the imposed stress. This is not the case with nonlinear kinematic hardening, due to the dynamic recovery term, which is not collinear to the plastic flow direction. The fading memory model [BUR87] , and allows regulation of the 2D projects the dynamic recovery term on the normal n ∼ ratcheting, by considering kinematic hardening couples, one with an ordinary dynamic recovery term, the other one with a fading memory term. It is rigorously identical to the nonlinear kinematic model for proportional loading. =n ⊗n ∼ ∼ ∼ ∼
(3.101)
94
Non-Linear Mechanics of Materials
Figure 3.9. Demonstration of the effect of the fading memory term on tensile-shear ratcheting: (a) Variation of the steady state according to η, (b) Reverse flow direction in case of unloading
Figure 3.9 illustrates this kind of model through a simple example. In Fig. 3.9a, a stress controlled loading is applied, with a constant axial value of 100 MPa, and a shear value oscillating between +150 MPa and −150 MPa. The kinematic hardening is split into two elements, the first one containing a classical dynamic recovery term, the other one a fading memory term, each of them having the same material parameter D = 300, the sum of the two coefficients C being constant (60000 MPa). All these configurations produce the same model under uniaxal loading. η is the fraction of each hardening kind: η = 1 corresponds to the classical hardening, and, for example, η = 0.75 characterizes a partition like C = 45000 for classical hardening and C = 15000 for fading memory. A ratcheting effect is found for η = 1, whereas the progressive deformation stops (farther or closer) in all other cases. In addition, Fig. 3.9b shows the axial strain evolution when, after 20 ratcheting cycles, one brings the axial stress back to 0. There is, in each case, a light backward ratcheting, the final axial strain remaining positive. It should be noted that, in the literature, some authors, instead of “splitting” the variables as indicated here, settle for a “mixed” dynamic recovery term, including a classical and a fading part, in a single hardening variable. This rule is obviously not identical and seems harder to identify.
3D plasticity and viscoplasticity
95
3.8.3. Other models for progressive deformation A detailed bibliography is devoted to the study of the progressive deformation of the material element, under uniaxial and multiaxial loading, and also on structural effects. By restricting to the material element, we will underline among others [OHN93a, OHN93b, CHA94, HAS94a, MAC95], for the uniaxial case, and [GUI92, HAS94b, DEL95, JIA94, TAH99], for the tensile-shear or biaxial ratcheting.
3.8.4. Memory effect Some materials, such as the austenitic stainless steels, show strong “memory effects” of the maximum plastic strain, i.e., after a rather large strain, the behavior that follows (for example in low amplitude cyclic loading) presents a strong hardening in comparison with the reference behavior at the same loading level. This effect can be modeled thanks to a yield surface in the plastic strains space, which keeps then in memory the highest loading value reached in its history, for example for the equivalent von Mises strain. The memorization process can be total or progressive. The variable thus obtained can be used in the models described in the previous section, for example by acting over the asymptote of the isotropic hardening variable [CHA79b].
3.8.5. Hardening followed by softening Some alloys present under cyclic loading a hardening followed by softening. This kind of behavior can be represented simply by modifying the form of the isotropic hardening rule. Two exponential functions of the accumulated plastic strain are used for that purpose, one increasing, with a high b coefficient (fast variation), the other one, decreasing, with a low b coefficient (slow variation).
3.8.6. Non-proportional loading In non-proportional loading, some materials present additional hardenings, which greatly exceed the level obtained for a proportional loading of equivalent range. The more complex the loading path, the larger the incremental hardening is. This effect can be explained at the microstructure scale, notably because more complex load paths involve a larger number of mechanisms (slip system, twinning) that interfere with each other. A possible modeling consists in increasing the asymptote of the isotropic hardening, thanks to a non-proportionality parameter [BEN87]. A large number of
96
Non-Linear Mechanics of Materials
“angles” can be considered to measure the non-proportionality degree, from the simple value σ∼ : σ∼˙ /J (σ∼ )J (σ∼˙ ) to more complex values evaluated from kinematic internal stresses [CAL97]. We will also show that this kind of effect is naturally present in the crystallographic models (see Sect. 5.6.5), in which multiaxial non-proportional loading activates a large number of slip systems, and thus induces more hardening than proportional loading.
3.8.7. Anisotropic plastic behavior Once the initial shape of the yield surface is characterized, the evolution of the yield surface under hardening remains to be worked out. Anisotropy can be written naturally on the kinematic hardening, by replacing the coefficients C and D by fourth-rank tensors, and by choosing a criterion of the following form [CHA96a]: , R) = JB (σ∼ − X )−R−σ −y f (σ∼ , X ∼ ∼
(3.102)
with: JB (σ∼ − X ) = ((σ∼ − X ):B : (σ∼ − X )) ∼ ∼ ∼ ∼
1/2
∼
and: X =C :α ∼ ∼ ∼ ∼
˙ = ε∼˙ p − D :X p˙ α ∼ ∼ ∼
(3.103) (3.104)
∼
Other, more complex, models [DEL96] involve several superimposed kinematic variables. One then has to use four fourth-rank tensors, M ,N , Q, R that must all ∼ ∼ ∼ ∼
∼
∼ ∼
∼
fulfill the condition cited in Sect. 2.5.4 in order to respect incompressibility. The evolution of internal stress involves three superposed variables 2 (1) : ε∼˙ p − Q : X −X p˙ α ˙ = Y ∗N ∼ ∼ ∼ ∼ ∼ 3 ∼ ∼ JR (α ) m0 α ∼ N :R : ∼ (3.105) − λr sinh ∼ ∼ α0 J ) ∼ ∼ R (α ∼ 2 (2) p (1) α ˙∼ (1) = Y ∗ N : ε ˙ − Q : X − X p˙ (3.106) ∼ ∼ ∼ 3 ∼∼ ∼ ∼ 2 (2) : ε˙ p − Q : X p˙ (3.107) α ˙ (2) = Y ∗ N ∼ ∼ ∼ 3 ∼∼ ∼ ∼ =pα with: X ∼ ∼
(1) (1) X = p1 α ∼ ∼
(2) (2) X = p2 α ∼ ∼
(3.108)
In its original version, this model includes a recovery term in (3.105), which will be discussed in the next paragraph, and a viscoplastic formulation. The viscoplastic strain follows the normality rule with a flow rule in sinh to account for the strong nonlinearity of the creep rate as a function of the stress: : (s∼ − X ) JM (σ∼ − X ∼ ) n ∼ 3 M ∼ (3.109) with v˙ = ε˙ 0 sinh ε∼˙ p = v˙ ∼ 2 JM (σ − X) N0
3D plasticity and viscoplasticity
97
There exists also some propositions to model anisotropy induced by strain; for example, the Baltov–Sawczuck model [BAL65] is obtained by posing respectively for : f and B ∼ ∼
, R) = JB (σ∼ − X ) − R − σy f (σ∼ , X ∼ ∼
with B = I∼ − (1/3)I∼ ⊗ I∼ + Aε∼ p ⊗ ε∼ p (3.110) ∼ ∼
∼
The two criteria quoted above use only expressions of degree 2 in stress. In fact, there is a large number of candidate invariants to describe the anisotropic behavior, depending on the symmetries of the material element [BOE78]. For instance, in the case of single crystals with cubic symmetry, propositions of criteria using unvariants of high order have been made, such as [NOU92, NOU95]: ⎫1/12 ⎧
3 2 ⎬ ⎨ 3 −R (3.111) (I1 + 2a4 I4 + 3a8 I8 − (a6 I6 )4 f = ⎭ ⎩ 2 with: 2 2 2 + S22 + S33 I1 = S11 I2 = S11 S22 + S22 S33 + S33 S11
(3.112) (3.113)
2 2 2 I4 = S12 + S23 + S31
(3.114)
I8
(3.115)
=
4 S12
4 + S23
4 + S31
in the crystallographic where the Sij stand for the components of the deviator of σ∼ −X ∼ frame.
3.9. Hardening and recovery in viscoplasticity 3.9.1. Kinematic and isotropic hardening All the models that have been described above can of course be reformulated in the framework of viscoplasticity. One gets then an additive hardening model ([LEM85b]). As indicated by formula (2.76), this is equivalent to introducing a function (f ) that will play the role of viscoplastic potential. Choosing for a power function leads to introducing two new coefficients, K and n: f n+1 K (3.116) (f ) = n+1 K The value of the viscoplastic flow intensity attached to that expression is: f n p˙ = K
(3.117)
98
Non-Linear Mechanics of Materials
This last model has been largely developed by Chaboche ([CHA89]). It allows expressing the stress deviator as a function of internal variables and thus identifying the internal and viscous stresses defined previously. + (σy + R + K(p) ˙ (1/n) )n s=X ∼ ∼
∼
with n = ∼
∂f ∂σ∼
(3.118)
The previous expression contains in particular Norton’s model, which is obtained by canceling the internal variables and the initial threshold. It is only able to predict secondary creep, and therefore should be used with caution. Nevertheless, it is generally verified at very high temperature. At constant viscous stress, keeping σy non-zero leads to a constant threshold viscoplastic model. The form including isotropic and kinematic hardenings at the same time is of course preferable at intermediate temperatures. At high temperature, a microstructure rearrangement due to thermal activation may occur, and produce a softening while straining. It follows a decrease of internal stresses, which can be formally accounted for by adding a recovery potential to the ordinary viscoplastic one. The implementation of such terms is generally carried out on the kinematic hardening term. Expression (3.120) shows, for example, how to combine recovery and linear kinematic hardening: K (σ, Ai ) = n+1 ˙ = ε∼˙ p − α ∼
J (σ∼ − X ) − σy
∼
n+1
K 3D 3 X p˙ − 2C ∼ 2C
J (X ) ∼ M
m
3M + 2C(m + 1)
J (X ) ∼ M
m+1
(3.119) X ∼ J (X ) ∼
(3.120)
With m = 1 this expression matches exactly the “dislocation–creep” (Orowan) proposed by metallurgists. Other recovery forms are, of course, conceivable, in particular on the isotropic variable. The recovery produces a decrease of the asymptotic value of the considered variable. Thus, under uniaxial loading, the kinematic variable X does not tend anymore towards C/D for low stresses. This effect is important in identification at high temperature as, in its absence, the kinematic hardening models lead to a limited creep for all the stress levels below σy + R + C/D. Recovery leads to an asymptotic value of X that tends towards 0 for stress values just above the initial limit of elasticity, which, in general, is more in agreement with experimental data. Figure 3.10 shows the rate obtained in creep for a given stress: – ε˙ init is the initial creep rate, assuming that the initial loading is fast enough to prevent viscoplastic strain during the transient;
3D plasticity and viscoplasticity
99
Figure 3.10. Viscoplastic strain rate obtained at the beginning of creep (˙εinit ), or at steady states, without (˙εso ) and with (˙εsr ) recovery
– ε˙ so is the stabilized viscoplastic strain rate when there is no recovery: in this case, creep stops for stresses between 100 and 300 MPa; – ε˙ sr is the stabilized viscoplastic strain rate in the presence of recovery: there exists then an intermediate regime between 100 and 300 MPa, for which the asymptotic value Xs of X is not C/D = 200 MPa anymore, but such that: m Xs =0 (3.121) (C − DXs )˙εsr − M This equation combined with
ε˙ sr
=
σo − Xs − σ y K
n (3.122)
gives the full solution of the problem. Many authors have exploited the “hierarchichal” structure of the equations of this paragraph. One can cite the approaches using imbricated kinematic hardening [DEL87, ROU85a], or for instance the SUVIC model. The latter, developed at first to represent the behavior of the salt [AUB99], presents a classical structure with kinematic and isotropic variables, but with particularly sophisticated evolutions:
) − R − R0 n f (σ∼ − B ∼ (3.123) v˙ = A K + K0 with, for example:
R R − xrR p R˙ = A3 1 − v˙ − A4 R C
and
R = R0
v˙ sinh ˙0
1/n m (3.124)
100
Non-Linear Mechanics of Materials
The other hardening variables (B and K) obey equations of the same kind. ∼
3.9.2. Aging The description of the material properties from a general viewpoint exceeds the scope of this book. In many cases, it would have to include phase transformations, displacive or diffusive ones, whose modeling implies either a micro–macro approach or supplementary strain components (see for example the classical review [BER97]). However, it is easy to represent the modifications of the mechanical properties due to changes in the precipitates. One could refer, for example, to [CAI79, CHA95] for dissolution-precipitation phenomena, or [MAR89, CAI00] for coalescence. In this last case, one can just consider a scalar aging variable, a, which tends towards an asymptotic value with time, according to a thermally activated process (with two coefficients a∞ and τ depending on the temperature), and which intervenes in the definition of the hardening variables: a∞ − a (3.125) a˙ = τ σy = R0 (T ) + R0∗ (T )(1 − a) (3.126) Thus, in (3.126), the initial yield stress involves a term R0∗ (T ) that disappears with aging, leading to a softening of the material. With an appropriate choice of a∞ and τ as a function of the temperature, one can also represent a hardening effect related to cooling periods, reactivated by overheating periods [CHA95].
3.9.3. Strain hardening As for Norton’s model, strain hardening rule is a viscoplastic model that does not introduce any threshold, so that it cannot be used in cyclic loading. It is suitable to represent properly the initial creep (or primary creep) or the relaxation (but, classically, not both at the same time). The model is said to introduce a product hardening ([LEM85b]); the hardening comes from evolution of the viscous stress with the accumulated strain, to get, for example (using von Mises criterion): J (σ∼ ) n −n/m p p n (3.127) ε∼˙ = ∼ K In the case of uniaxial loading, this equation can be explicitly integrated for creep (3.128), or relaxation, by neglecting for the latter the viscoplastic strain change during relaxation. m/(m+n) m + n J (σ∼ ) n εp = t (3.128) m K
3D plasticity and viscoplasticity
101
3.10. Multimechanism models 3.10.1. General formulation This kind of models should be used each time inelastic strain originates from several distinct mechanisms. This section is limited to the case where it is possible to keep treating elasticity in a classical way, and where only the inelastic strain is made of several contributions. Various subsets can be considered in the group of variables Z (see Sect. 2.3.5), let: {Z } = Zs (3.129) s
{Z s },
In each subset a distinction can be made between elementary stresses σ∼ s for the mechanisms and a set of hardening variables YJss . These subsets are the critical variables in the free energy, so that: ψ= ψs (Z sJs ) (3.130) s
Then, there remains the choice to involve each of these mechanisms in some elementary potentials s , each one being attached to a particular mechanism through its own criterion f s (3.131), or to a single potential , each mechanism contributing to a common global criterion f (3.133): ε∼˙ p =
∂s s
∂σ∼
=
∂s ∂f s ∂s s = ns : B ∼ s ∂σ s ∼ ∂f ∂f ∼ ∼ s s
∂f s ∂σ∼ s s and B = ∼ ∂σ∼ s ∂σ∼ ∼ ∂ ∂ ∂f ∂ s ε∼˙ p = = = n : Bs ∂σ∼ ∂f ∂σ∼ ∂f s ∼ ∼∼ s = with n ∼
s = with n ∼
(3.131)
(3.132) (3.133)
∂f ∂σ∼ s s and B = ∼ ∂σ∼ s ∂σ∼ ∼
(3.134)
s represents in some cases a part of the concentration law that proThe tensor B ∼ ∼
vides the local stress as a function of the global stress for heterogeneous materials. We s are identity tensors. will only describe here the approaches in which the tensors B ∼ ∼
The first method can apply either by considering mechanisms of the same nature (all plastic or all viscoplastic) or by gathering the contributions. Koiter [KOI60] and Mandel [MAN65] authored the original contributions regarding that kind of approach in plasticity. It is first used, in Sect. 3.10.2, in which all the mechanisms are identical in nature. The approach is logically applied to build single crystal models (Sect. 3.10.3):
102
Non-Linear Mechanics of Materials
this is because each slip system represents a mechanism, and the elementary stress seen by each mechanism is the macroscopic stress (no scale transition). Then, one shows other applications of multimechanism/multicriteria with two mechanisms, by considering each of the three cases: plastic–plastic, viscoplastic– viscoplastic, plastic–viscoplastic (Sect. 3.10.5). On the contrary, if a single potential model is considered, the response will be entirely plastic or viscoplastic, as indicated in Sect. 3.10.6.
3.10.2. Multimechanism–multicriteria models Their formulation goes along the same lines as ordinary models, with a plastic or viscoplastic formulation. By limiting, for simplicity sake, to an associated form, the starting point is a series of potentials s , s = 1..S, depending on σ∼ and on flux variables AI , through the yield function f s . In viscoplasticity, the problem is immediately solved for a generalized standard material: ε∼˙ p =
∂s s
α˙ I =
∂σ∼
∂s s
∂AI
with v˙ s =
= v˙ s
∂f s ∂σ∼
(3.135)
= v˙ s
∂f s ∂AI
(3.136)
∂r ∂f s
(3.137)
One would keep a standard model form by choosing for αI evolution an expression like α˙ I = v˙ s MIs instead of (3.136). The dissipation is written: ˙ φ1 = σ∼ : ε˙ p − AI α ∼I
(3.138)
In plasticity, the previous equations do not close the problem, since v˙ s is not explicitly determined. A vector of plastic multipliers replaces (3.137). It must be determined by solving the linear system formed by the S consistency conditions f˙r = 0: r : σ∼˙ + YIr A˙ I f˙r = n ∼
r with n = ∼
∂f r ∂f r and YIr = ∂σ∼ ∂AI
(3.139)
the rate of change of the hardening variables can be formulated as: ∂AI ∂AI s s A˙ I = α˙ K = − Y λ˙ ∂αK ∂αK K
(3.140)
The combination of (3.139) and (3.140) leads then to: r n : σ∼˙ − Hrs λ˙ s = 0 ∼
with Hrs = YIr
∂AI s Y ∂αK K
(3.141)
3D plasticity and viscoplasticity
103
In that case, the hardening is governed by a symmetrical interaction matrix,1 whose components Hrs characterize the hardening brought by the mechanism s on the mechanism r. Knowing that: r r n : σ∼˙ = n : : (˙ε∼ − ε∼˙ p ) ∼ ∼ ∼ ∼
(3.142)
a relation can also be established in terms of total strain rate, giving access to the s each member of equation 3.142, and by ustangent matrix, by multiplying by n ∼ ing (3.141): r s r : :n + Hrs )λ˙ s = n : : ε∼˙ (3.143) (n ∼ ∼ ∼ ∼ ∼ ∼
∼
The system includes only those equations that are formed with the activated mechanisms. Assuming the hypothesis of intrinsic maximum dissipation, the determination of the set of active slip systems is unique: the combination of active systems must ˙ in (3.138). Except for either maximize the dissipation, or minimize the sum AI α ∼i the associated model, the previous matrix is not symmetrical, as one gets a term like r : : Ns. n ∼ ∼ ∼ ∼
3.10.3. A single crystal model Model presentation The model of the present section is associated, but not generalized standard. It is an application to the single crystal of the nonlinear hardening model presented in Sect. 3.7. It introduces isotropic r r and kinematic x r hardening variables, respectively associated to the state variables ρ r and kinematic α r , for each slip system r. By choosing a quadratic form of each state variable for the inelastic part of the free energy and by introducing a coupling on the isotropic variables, one gets: ρψ =
1 s 2 1 α + Q hrs ρ r ρ s c 2 s 2 r s
and: x r = cα r
r r = bQ
hrs ρ s
(3.144)
(3.145)
s
The matrix of component hrs is an interaction matrix, which represents the selfhardening on each system and the latent hardening [KOC66, FRA80] between the systems. The general form of the model corresponds to a single crystal that deforms by a slip mechanism on S systems r, in planes of normal nr along a direction lr . The 1 As ∂A /∂α = ∂A /∂α = ∂ 2 /∂α ∂α I K K I I K
104
Non-Linear Mechanics of Materials
r , allows us to calculate the shear stress on the system r from the orientation tensor, m ∼ stress tensor. Let us denote by τ0 the initial critical shear stress, and assume:
f r = |τ r − x r | − r r − τ0
(3.146)
1 r with τ r = σ∼ : m = σ∼ : (lr ⊗ nr + nr ⊗ lr ) ∼ 2
(3.147)
One gets a viscoplastic formulation by using the viscoplastic potential [RIC70, MAN72]:
K f r n+1 = r (f r ) = (3.148) n+1 r K r Hence: ∂ r r = γ˙ : m ∼ ∂σ∼ r
r n ∂ f r = with v˙ = ∂f r K r r r and γ˙ = v˙ sign(τ − x r )
ε∼˙ p =
(3.149) (3.150) (3.151)
The model is complete once the evolution of the state variables is defined: α˙ r = (sign(τ r − x r ) − dα r )v˙ r
(3.152)
ρ˙ = (1 − bρ )v˙
(3.153)
r
r
r
Combining (3.145) and (3.153) leads to: rr = Q
hrs 1 − exp(−bv s )
(3.154)
s
Beyond τ0 , the model includes two coefficients, K and n, characterizing the viscosity, and two others, c and d for kinematic hardening, while b, Q and the matrix hrs define isotropic hardening. The dissipation is written: φ1 = σ∼ : ε∼˙ p − =
s
x s α˙ s −
r s ρ˙ s
s
τ s γ˙ s − x s (sign(τ s − x s ) − dα s )v˙ s − r s (1 − bρ s )v˙ s
(3.155)
(3.156)
s
d s 2 s s s = x + br ρ v˙ s f + τ0 + c s
(3.157)
The four terms in the right-hand side of (3.157) correspond directly to the viscous dissipation (which would be zero for a plasticity model), and to the dissipation related to
3D plasticity and viscoplasticity
105
friction, and to nonlinearities of the isotropic and kinematic hardening. The dissipation is always positive for materials with hardening. For a plastic material, we would recover (3.141), with: Hrs = (c sign(τ s − x s ) − dx s )δrs + Qbhrs exp(−bv s )
(3.158)
The choice of active systems is still an issue in that case. The question was originally handled by Taylor [TAY38a], with no kinematic variable, for the case of an “isotropic” interaction matrix (all terms equal to 1), which leads to a single value for all the variables r s , which evolves linearly with the slip-rates. Among all the possible configurations, the author suggests choosing the one that minimizes the sum of the rates over all the active systems. This hypothesis is found here easily by calculating the dissipation (see (3.155)) with the same hypothesis (no x s , b = 0). Indeed, denoting by r the common value of all the critical shear stresses, we get: v˙ s (3.159) φ1 = σ∼ : ε∼˙ p − r s
This “minimization of the internal work” is equivalent to a “maximization of the plastic work” [BIS51], as shown by handling the problem with nonlinear programming techniques [CHI69]. Maximizing the dissipation estimated in (3.155) provides a generalization of the previous methods. For a general model, the active systems selection does not necessarily lead to a minimal sum s v˙ s . To conclude, one should note that another selection method, based on a decomposition of the singular matrix [GOL83] has been proposed [ANA96]. It leads to minimizing the norm of the slip-rate under the form s (v˙ s )1/2 .
Examples of application Working with a crystallographic approach and the Schmid criterion leads to a piecewise linear global yield surface, built by using the expression of the criteria on each system. Figure 3.11 shows that, depending on the active slip system family, the resulting shape of the criterion varies. The plots displayed are obtained for the case of a face-centered cubic material, assuming that only the octahedral systems are active, and then that the cubic systems are also active, with the same initial critical shear stress r0 . It is also worth noting that the outcome is different depending on the coordinate system considered to apply the tensile-shear loading. Let N and M be two orthogonal unit vectors; the stress tensor is supposed to be of the form: σ∼ = σ N ⊗ N + τ (N ⊗ M + M ⊗ N)
(3.160)
The resolved shear stress τ s on the system s defined by the plan ns and the direction ls is in that case: ! " (3.161) τ s = σ (N.ns )(N.ls ) + τ (N.ns )(M.ls ) + (N.ls )(M.ns )
106
Non-Linear Mechanics of Materials
Figure 3.11. Initial yield surface of a single crystal under tension–shear loading, as a function of the active slip systems and of the direction of loading, (a) N : 001; M : 010, (b) N : 001; M : 110
As an illustration, it is shown that this expression provides a different result if, after choosing for N the (001) direction, shear is applied either along a (010) direction (Fig. 3.11a) or (110) direction (Fig. 3.11b) [NOU94]. Neglecting this result has led to many mistakes in the analysis of tensile–shear results of single crystals in the recent literature. This influence of crystallography has various consequences, that are illustrated in Fig. 3.12. Figure 3.12a shows the response in the stress plane σ11 –σ12 for loadings under prescribed strain; the path is linear in the ε11 –ε12 plane. Keeping the same value for ε11 , a low variation in ε12 produces a drastic change in the flow direction, as the representative point in the stress plane is “attracted” by the multiple points of the yield surface. Since the small strain formalism is used, this effect is just the result of the normality rule. Two plots are shown, in Fig. 3.12b with octahedral slip only, and in Fig. 3.12c with the octahedral and the cube slip families. The response of the material is different: due to the larger amount of slip systems, the stress level is lower in the second case; the flow direction is also modified. To conclude this section, Fig. 3.13 shows typical results obtained in tension along several crystallographic directions for a FCC material.
Remark The aim of the previous section was to underline the similarity between the equations that can be used for describing single crystal behavior and the macroscopic constitutive equations. It differs from the classical literature on the same subject, in which,
3D plasticity and viscoplasticity
107
Figure 3.12. Biaxial loading under prescribed strain, (a) influence of a low modification max /ε max = 0, 525; lower path: ε max /ε max = 0, 475) on the loading path (upper path: ε12 11 12 11 (b) and (c) influence of crystallography for a given loading ((b) with octahedral systems, (c) with max max octahedral and cubic systems (paths: ε12 /ε11 = 2))
after Taylor’s work [TAY38b], the main steps are a formulation of multi-mechanisms models leading to a description of multiple slip [MAN65, HIL66], definition of the internal variables and of viscoplastic potential [RIC70, MAN72], a finite-transformation formulation [RIC71, HIL72, TEO76, ASA77, ASA83b, ASA83a], and then use in numerical simulation [PIE85, ASA85]. There are two “lineages” of models, either formulated phenomenologically, or referring explicitly to the dislocation densities. The usual phenomenological models do not include kinematic hardening, and consider either a time independent plastic behavior, that introduces hardening in the re-
108
Non-Linear Mechanics of Materials
Figure 3.13. Tensile curves for various crystallographic directions, with only the octahedral systems activated. The elastic characteristics are C11 = 250 GPa; C12 = 200 GPa; C44 = 100 GPa
solved critical shear only, or a viscoplastic behavior, without any threshold and with a product type hardening, such as [PIE85]: # r #1/m #τ # and γ˙ s = v˙ s sign(τ s ) v˙ r = ## r ## τc hrs v˙ s with hrs = (q + (1 − q)δrs ) H τ˙cr =
(3.162) (3.163)
s
This kind of model is generally acceptable to represent properly the behavior in monotonic loading and texture changes. Time independent plastic behavior can be reached by choosing high values for the coefficient m. More elaborated models are proposed in the viscoplastic framework [KAL92], or to represent twinning [STA98]. Some authors are inclined towards a description involving explicitly dislocation densities [FRA85, TEO93, CUI92, BUS96, TAB97, EST98, NEM98]. There exists a profuse literature on the subject, which cannot be addressed in this section. The reader will find complementary information regarding finite transformations in Chap. 6, and could consult classical reviews on the matter [HAV92, TEO97].
3.10.4. Two mechanisms and two criteria models (2M2C) An interesting particular case among the models considered until now is the two mechanisms one, each of the two obeying a von Mises criterion with isotropic and kinematic
3D plasticity and viscoplasticity
109
hardening. One writes then the potential as a sum: = 1 (f 1 ) + 2 (f 2 )
(3.164)
I f I = J (σ∼ − X ) − R I − R0I ∼
(3.165)
with, for I = 1, 2: The inelastic strain is then obtained as: ∂1 1 ∂2 2 n + n ∂f 1 ∼ ∂f 2 ∼ ∂f I I = with n ∼ ∂σ∼
ε∼˙ p =
(3.166) (3.167)
The expressions of Sect. 3.7.1 can be reused here. However, experience ([SAI95]) shows that it is necessary to introduce coupling in the free energy between the state (the variables associated to XI s). One chooses then (no implicit sum over variables α ∼I I and J ): 2 1 1 I J ρψ = CI J α : α + b Q (3.168) rI I I ∼ ∼ 3 2 I
J
I
One could also consider a coupling between the isotropic variables, or even an isotropic-kinematic “mixed” coupling. This will not be treated here. One gets then: I X = ∼
2 J CI J α ∼ 3
R I = bI QI r I
(3.169)
J
and, with straightforward notation, the evolution equations: 3DI I ∂I I ˙ = n − X α ∼ ∼ 2CI I ∼ ∂f I R I ∂I r˙ = 1 − QI ∂f I I
(3.170) (3.171)
The formulation of the model where both mechanisms are viscoplastic is straightforward, as ∂I /∂f I is then perfectly determined. It derives from the choice of a viscosity function among those in Sect. 2.4.5. If, on the contrary, both mechanisms are plastic, the model formulation generates a system such as the one presented in Sect. 3.10.2, that has to be solved, by examining three cases, namely zero, one or two active mechanisms. In this last case, we get: $ $ % % 1 : σ˙ − M n2 : σ˙ 2 : σ˙ − M n1 : σ˙ M22 n M11 n 12 ∼ 21 ∼ ∼ ∼ 1 2 ∼ ∼ ∼ ∼ λ˙ = (3.172) λ˙ = M11 M22 − M12 M21 M11 M22 − M12 M21
110
Non-Linear Mechanics of Materials
The model properties are going to depend on the denominator (indetermination if it is zero). One gets (without summing over I and J ): I I :n + bI (QI − R I ) MI I = CI I − DI X ∼ ∼ CI J I 2 I J J :n − DJ n :X MI J = CI J n ∼ ∼ ∼ 3 CJ J ∼
(3.173) (3.174)
3.10.5. Simultaneous plastic and viscoplastic flows Plastic formulation generates instantaneous responses, the viscoplastic one, delayed responses. The processing of all responses of a material by a viscoplastic model has been called a “unified model”. Nevertheless, it is sometimes useful, for some materials, to introduce two deformation regimes, one fast, the other one slow, for example (i) for the materials that possess simultaneously strong hardening and creep abilities, such as the austenitic stainless steels, and in which Portevin-Le Chatelier effect occurs, producing jerky flow and a decreasing viscous stress versus strain rate, (ii) for the materials in which instantaneous strain (plasticity) and creep (viscoplasticity) occur in disconnected stress ranges, as for example Inco718 type Nickel-based alloys. The simultaneous use of two mechanisms provides the proper rate effect on the tensile curve in the first case, and allows us to perform a modular identification in the second case, each model being active in a different zone of stress and time. The models used introduce then a plastic threshold function as well as a viscoplastic potential. The total strain is then the sum of elastic, plastic, and viscoplastic strains: p + vn ˙ ∼v ε∼˙ = ε∼˙ e + λ˙ n ∼
(3.175)
The most common models use only a Norton’s law in viscoplasticity, and a linear isotropic or kinematic hardening in plasticity. Nevertheless, experience shows that it is often necessary to involve a coupling between plastic and viscoplastic evolution laws ([CON89]). As an example, one can consider the following model (note in this section the change of notation Cvp = (2/3)C12 ): v ) − R v − Rov f v = J (σ∼ − X ∼
= (2/3)Cv α + Cvp α X ∼ ∼ ∼ v
v
p
p f p = J (σ∼ − X ) − R p − Rop ∼
X = (2/3)Cp α + Cvp α ∼ ∼ ∼ p
p
R = bp Qp r R = bv Qv r 3Dp 3Dv v v v p p ˙ = ε∼˙ − ˙ = ε∼˙ − α X v˙ X p λ˙ α ∼ ∼ 2Cv ∼ 2Cp ∼ Rv Rp v p v˙ r˙ = 1 − p˙ r˙ = 1 − Qv Qp v
v
p
p
v
(3.176) (3.177) (3.178) (3.179) (3.180)
3D plasticity and viscoplasticity
111
Figure 3.14. Influence of the coupling term in a creep–tension test (creep 555 h at 140 MPa, then tension at ε˙ = 10−4 s −1 ); the reference is obtained in tension at ε˙ = 10−4 s −1
p ˙ v : n σ∼˙ − Cvp α ∼ ∼ Hp
v n f v˙ = K
λ˙ =
p p with Hp = Cp − Dp X :n + bp (Qp − R p ) ∼ ∼
(3.181) (3.182)
This model involves five coefficients for the plastic law (Rop , Qp , bp , Cp , Dp ), seven for the viscoplastic law (Rov , Qv , bv , Cv , Dv , K, n), and a coefficient characterizing the coupling between them (Cvp ). The resulting models are then identified thanks to cyclic plasticity experiments, creep experiments, and “mixed” experiments displaying the coupling between plasticity and creep. Figure 3.14 illustrates the effect of the coupling parameter Cvp during a creep loading followed by a plastic reloading. Without coupling (Cvp = 0), the creep period does not produce any hardening in the plastic model, and leads then to a reloading curve that remains very far from the reference curve. Experience shows usually that for such a loading path, the curve obtained in a creep-plasticity test meets the reference curve, which justifies the coupling [GOO84, OHA83]. It is worth noting that the partitioning between plastic and viscoplastic flow is done naturally, as a function of the strain rate, as the viscoplastic strain rate influences the calculation of the plastic multiplier. When the strain rate is large, the viscoplastic strain has no time to build up: the stress rate dependent term in 3.181 is much bigger than the corrective term in α ˙ v , so that the plastic strain rate is large, and the regime ∼ is dominated by plasticity. On the contrary, for small strain rates, the corrective term can be as large as the driving term, and the plastic multiplier can be deactivated. The
112
Non-Linear Mechanics of Materials
flow regime can then become purely viscoplastic. In tension, the effect of the loading rate, simulated with an elementary model leads to an ordinary viscous effect at low rate, and to a saturation towards an instantaneous plasticity model at high rate, which matches the experiment (Fig. 3.15a). In each of these tests, the inelastic strain rate is mainly viscoplastic at low rate, and purely plastic at high rate (Fig. 3.15b). With the coefficients used to simulate tension, the creep tests show primary or secondary creep, depending on the load level, but also a significant viscoplastic flow below the apparent yield stress (Fig. 3.15c,d). The “dependence of the yield stress on strain rate” is then reproduced, that is the apparent onset of plastic flow occurs for larger strain rates later than for low strain rates. Depending on the relative values of the material parameters corresponding to plastic self-hardening, viscoplastic self-hardening and cross hardening, the model can account for inverse strain rate effects in some range of loading, as illustrated in Fig. 3.16. In tension, the chosen set predicts an ordinary effect of the rate at low loading rate, a maximum of stress at intermediate loading rate, and a stress drop for higher rates. Such a behavior is found for materials in which exists an impurity-dislocation interaction effect [BLA87]. The same kind of model can also be written by summing two strains of the same nature, plastic or viscoplastic.
3.10.6. Two mechanisms and one-criterion models (2M1C) After having formulated independently several mechanisms, they can be combined in only one criterion. Such a model is then either in a time independent plastic framework or in a viscoplastic framework. The two criteria of equation 3.176 are then replaced by one, for example [ZAR87]: 1/2 1 2 2 2 ) + J (σ − X ) −R f = J (σ∼ − X ∼ ∼ ∼
(3.183)
A coupling is still introduced between the kinematic variables: 1 1 2 X = (2/3) C α + C α 11 12 ∼ ∼ ∼
2 2 1 = (2/3) C α + C α X 22 12 ∼ ∼ ∼
(3.184)
The plastic formulation of that model has been studied in detail [ZAR87]; one can also write a viscoplastic formulation [CAI95]. In both cases, there is only one isotropic variable left, so that the two relations of formula (3.180) are replaced by one, the driving term P˙ being, depending on the case, the (unique) plastic multiplier or the derivative of the viscoplastic potential with
3D plasticity and viscoplasticity
113
Figure 3.15. Illustration of the model in tension and creep. (a) Influence of the strain rate on the tensile curve, (b) Fraction of the time independent plastic strain in the inelastic strain as a function of the loading rate, (c) Creep at various loads, (d) Presence of creep for a stress level lower than the apparent yield stress (600 h creep)
114
Non-Linear Mechanics of Materials
Figure 3.16. Description of the inverse strain rate effect in viscoplasticity
Figure 3.17. Comparison between: (a) a model with two mechanisms and two criteria, (b) a model with two mechanisms and one criterion
3D plasticity and viscoplasticity
respect to f , v: ˙
R ˙ r˙ = 1 − P Q
115
(3.185)
The isotropic variable R is written: R = bQr = Q(1 − exp(−bP ))
(3.186)
n f P˙ = v˙ = K
(3.187)
In viscoplasticity:
In time independent plasticity: $
1 + J n2 ) : σ˙ (J1 n 2∼ ∼ ∼ P˙ = λ˙ = hR + hX1 + hX2
% (3.188)
with: hR = b (Q − R) R 2 3D1 1 1 1 2 : C n − X J n + C J n hX1 = 11 1 12 2 ∼ ∼ 3 ∼ 2C11 ∼ 2 3D2 2 2 1 2 : C n − X J n + C J n hX2 = 12 1 22 2 ∼ ∼ 3 ∼ 2C22 ∼
(3.189) (3.190) (3.191)
i , (i = 1, 2) in (3.188) and (3.191) matches the classical form ni = The notation n ∼ ∼ i i ), the expression of the normal becoming then (with J = (3/2)(s∼ − X )/J (σ∼ − X i ∼ ∼ i )): J (σ∼ − X ∼ 1 + J n2 J1 n 2∼ ∼ = (3.192) n ∼ 2 2 (J1 + J2 )1/2
For that kind of model, the plastic multiplier (or v˙ in viscoplasticity) does not match anymore the equivalent strain rate. The detailed study of the model shows that it allows us to control to some extent the amount of ratcheting, for example for a non2 ) is not symmetrical one-dimensional loading. When the determinant (C11 C22 − C12 zero, the ratcheting stops after a transient period showing progressive deformation. It is worth noting that the mechanical steady state can present an open cycle (Fig. 3.18).
3.10.7. Compressible materials The models which are sensitive to hydrostatic pressure constitute a very particular category, handled in detail in the next section. In fact, the present approach can be
116
Non-Linear Mechanics of Materials
Figure 3.18. Illustration of the transient ratcheting followed by plastic shakedown obtained with a 2M1C model, under a non-symmetrical tension–compression loading
related to this new category of models. The elliptic model can be taken as an example. Assuming for instance that the model involves both isotropic and kinematic hardening, the chosen criterion will then be: 1/2 2 2 ) + I (σ − X ) −R (3.193) f = J (σ∼ − X ∼ ∼ ∼ This form might be considered as a 2M1C type model, as studied previously, in which the first criterion would impact the deviatoric part and the second one, the spherical , and the part. The kinematic variable of the first criterion is x∼ , deviatoric part of X ∼ . one of the second criterion is Xll , trace of X ∼
3.11. Behavior of porous materials In this part, the behavior of plastic and viscoplastic porous materials is introduced. They are characterized by the presence of cavities, which, as functions of the applied loading, can grow or collapse. The constitutive models introduced in this part can be used to model forming and ductile fracture. The porosity η is defined by the ratio: # volume of the cavities ## η= (3.194) # zero stress total volume reference temperature
It is then the relative ratio of voids in the relaxed-stresses state at reference temperature. In particular, a purely elastic volume change does not induce any modification of the porosity. We will call the healthy matter surrounding the cavities a matrix.
3D plasticity and viscoplasticity
117
3.11.1. Presentation of some models This class of materials is characterized by a yield surface depending on the porosity. We will distinguish the materials having a plastic flow direction associated to this surface from the non-associated materials. We will consider only the associated case (one could refer to [DES89, KAN82, NEM80, VER83] for the non-associated case). The formulation of the various models depends on the definition of a scalar effective stress σ taking into account the hydrostatic pressure. The effective stress is explicitly or implicitly defined by the following equation: ψ∗ (σ∼ , η, σ ) = 0
(3.195)
We define also the following quantities: σM = k1 σ11 + k2 σ22 + k3 σ33 & 3 σE = s:H :s 2 ∼ ∼∼ ∼
(3.196) (3.197)
in order to be able to take into account the material anisotropy by using Hill’s tensor H [DOE95, PRA98]. In the isotropic case, H = I∼ , so that σE is equal to von Mises ∼ ∼ ∼
∼
∼
stress and k1 = k2 = k3 = 1 so that σM = tr(σ∼ ). In the case where the coefficients k1 , k2 , k3 differ, (3.196) is valid only in a particular material reference frame; it is for example the case for materials containing ellipsoidal cavities [GOL94, GOL97]. The choice of the function ψ∗ determines the model to use. Among those, one often uses 2 Green ψ∗ = 3CσE2 + F σM − σ2 σ2 q2 σ M Gurson ψ∗ = E2 + 2η q1 cosh − 1 + q12 η2 2σ σ σE σM + σ1 ηD exp Rousselier ψ∗ = − σ 1−η (1 − η)σ1 2 σ 2 σM + pc − pc2 − σ2 modified Cam-clay ψ∗ = E2 + 3 m
(3.198) (3.199) (3.200) (3.201)
The Gurson and Rousselier models are used to model ductile fracture. The Green and Cam-clay models serve for forming. The shape of the surfaces ψ∗ = 0 in the plane σM –σE , at given σ , is represented in Fig. 3.19. The following remarks can be made on the various models. Green [GRE72, SHI76, ABO88, BES92] This model was initially presented as a phenomenologic extension to von Mises criterion incorporating hydrostatic pressure. The simplest extension consists in defining a quadratic surface in the
118
Non-Linear Mechanics of Materials
Figure 3.19. Comparison between the plasticity surfaces of Green (dotted line coinciding with Gurson model for σM = 0 and σE = 0), Rousselier (in dotted line the symmetrized model (3.206)), Cam-clay (for σ = 0) (dotted line showing the critical state line σE = −mσM /3)
σM –σE plane. It is generally used for hot forming. C and F are functions of the porosity; C(0) = 1 and F (0) = 0 give a plastically incompressible material when porosity becomes zero. One can also note that some recent homogenization models [PON98] make it possible to find an elliptic form for the plasticity criterion. Gurson [GUR77, TVE84, BEC88, TVE90b] This model was initially established in the framework of micromechanics. The plasticity surface is obtained by analysing of a hollow sphere using an upper bound method [GUR77]. The matrix is assumed to be isotropic, rigid (i.e., no elasticity) and perfectly plastic. The Gurson analysis leads to a plasticity surface defined by: σE2 σM − 1 + η2 ψ∗ = 2 + 2η cosh 2σ σ
(3.202)
The comparison with finite element computations over some cavity/matrix elementary cells [KOP88] as well as the comparison with experimental data led up to a phenomenological modification of the model resulting in (3.199). In this
3D plasticity and viscoplasticity
119
equation, η is a function of the porosity η which generally is written: η si η < ηc η = (3.203) ηc + δ(η − ηc ) si η > ηc where ηc is the porosity from which cavities coalescence starts. For η < ηc the damage is entirely ruled by the cavities growth. δ > 1 is a parameter that enables us to reproduce the deleterious effect of coalescence on mechanical resistance. The phenomenological form of equation (3.203) is the most used one, as it is the simplest. However, nothing forbids using another function, either for physical reasons, or for numerical reasons [MAT94]. q1 and q2 are parameters which are in general: q1 = 1.5
q2 = 1
(3.204)
Fracture (i.e., ψ∗ = 0 for σ∼ = 0∼) takes place for η (η) =
1 q1
Some authors suggest that ηc , δ, or even q1 should depend on the triaxiality of the stress and on the initial porosity η0 [ZHA95]. Rousselier [ROU87] This model was also proposed in the framework of ductile fracture. It includes two coefficients σ1 and D. Based on the ductile fracture model proposed by Rice & Tracey [RIC69], one often uses σ1 =
2 Rm 3
(3.205)
D is close to 2. In this model, fracture occurs when η = 1; in order to get a more realistic model one considers that beyond a critical porosity ηc the material is broken. There is then a stress drop. This model is particular in that it does not have a vertical tangent for σE = 0 in the σM –σE plane. Equation (3.200) indicates that the criterion is not symmetric with respect to σM (i.e., ψ∗ (σE , σM , σ ) = ψ∗ (σE , −σM , σ )). However, one can symmetrize it by replacing (3.200) by: σE |σM | ψ∗ = (3.206) + σ1 ηD exp − σ 1−η 3(1 − η)σ1 The surface ψ∗ = 0 presents then a vertex for σM = 0. In general, this model is used to describe ductile fracture with σM > 0. Modified Cam-clay [SCH65, FAV99] This model is generally used for cold forming of powder materials. pc is a decreasing function of the porosity. It is generally considered that m is constant. This model does not apply to compression state (trσ∼ < 0). It is generally not used at low porosity values.
120
Non-Linear Mechanics of Materials
3.11.2. Yield criterion From the definition of σ , the plasticity criterion is written: f (σ∼ , η, R) = σ − R = 0
(3.207)
where R is the yield strength of the non-damaged material. A more classical presentation considers the equation ψ(σ∼ , η, R) = 0 as the plasticity criterion. The presentation adopted here allows treating all the models identically as well as an immediate extension to viscoplasticity.
3.11.3. Viscoplasticity For a viscoplastic material, the yield condition is written: f (σ∼ , η, R) = σ − R ≥ 0
(3.208)
The difference σ − R defines a creep effective stress that permits us to define the strain-rate from the creep law of the non-damaged material [PRA98, STE97].
3.11.4. (Visco)plastic flow Applying the normality rule, the (visco)plastic strain-rate tensor is given by:2 ε∼˙ p = (1 − η)λ˙
∂f ∂σ = (1 − η)λ˙ ∂σ∼ ∂σ∼
(3.209)
λ˙ is the plastic multiplier, which is obtained either by the consistency condition, in plasticity: ∂σ ∂σ : σ∼˙ + η˙ − R˙ = 0 (3.210) ψ˙ ∗ = ∂σ∼ ∂η or by applying the creep law of the non-damaged material: λ˙ = φ(σ − R, . . . )
(3.211)
In the case where σ only depends on σM and σE we get: ∂σ ∂σ ∂σM ∂σ ∂σE = + ∂σ∼ ∂σM ∂σ∼ ∂σE ∂σ∼
(3.212)
2 Writing the proportionality coefficient as (1 − η)λ ˙ is a convention often used. We could have written ˙ . ∼˙ p = λn ∼
3D plasticity and viscoplasticity
121
Figure 3.20. Normal to the surface σ = constant in the σM –σE plane indicating the volume change and shear components
in the isotropic case: = n ∼
∂σ ∂σ 3 ∼s ∂σ = + I ∂σ∼ ∂σE 2 σE ∂σM ∼
(3.213)
In particular, the trace of the tensor n is 3∂σ /∂σM and is not zero, which corresponds ∼ to a volume change. The vector (∂σ /∂σM , ∂σ /∂σE ) corresponds to the normal to the surfaces σ = constant in the σM –σE plane. The component ∂σ /∂σM corresponds then to the volume change and ∂σ /∂σE to the shear strain (Fig. 3.20). Remark: σ can be defined implicitly as for the Gurson’s model case. To calculate , one notes that calculating ∂σ /∂σ∼ is done at constant porosity for ψ∗ = 0 and n ∼ ψ˙ ∗ = 0, then: δψ∗ = 0 = hence:
∂ψ∗ ∂ψ∗ δσ + : δσ∼ ∂σ ∂σ∼
∂σ ∂ψ∗ −1 ∂ψ∗ =− ∂σ∼ ∂σ ∂σ∼
(3.214)
(3.215)
Remarks: For Green and Gurson models, there is no volume change at σM = 0 (pure shear). In the case of the Rousselier model (3.200) ∂σ /∂σM is always positive, which in particular implies damage growth under pure shear. In the case of the Camclay model, the points corresponding to pure shear strain are located on the line σE = − m3 σM (critical state line [SCH65], Fig. 3.19). In the case of the Rousselier model, the shear component in plastic strain is never zero, even for a purely hydrostatic loading. It is not the case for the other models.
122
Non-Linear Mechanics of Materials
3.11.5. Evolution of porosity Evolution of porosity is obtained from mass conservation. Let Vm be the volume occupied by the matrix at σ∼ = 0∼ and Vt the total volume. As the quantity of matrix is constant, V˙m = 0. In addition:
Then, after derivation:
Vm = (1 − η)Vt
(3.216)
−ηV ˙ t + (1 − η)V˙t = 0
(3.217)
hence: η˙ = (1 − η)
V˙t = (1 − η) Tr ε∼˙ p Vt
(3.218)
This equation integrates easily and we get: η = 1 − (1 − η0 ) exp − Tr ε∼ p
(3.219)
where η0 is the initial porosity. The evolution of the porosity is then, in that case, entirely controlled by plastic deformation.
3.11.6. Elastic behavior The elastic behavior of porous materials is classically given by: (η, . . . ) : ε∼ e σ∼ = ∼ ∼
(3.220)
can depend on porosity. However, there is no coupling between damThe tensor ∼ ∼
age kinetic and elastic behavior as in the case of damageable behavior presented in Sect. 4.7. Regarding ductile fracture, porosity remains generally very low so that the with porosity is often neglected. dependency of ∼ ∼
3.11.7. Effective plastic deformation, hardening ˙ One considers then The effective plastic deformation, p, is defined such that p˙ = λ. that this quantity is representative of matrix hardening. In the case of the Green and Gurson models, it is possible to show that macroscopic and microscopic plastic power are even: (3.221) σ∼ : ε∼˙ p = (1 − η)σ p˙
3D plasticity and viscoplasticity
123
In the case of the isotropic Green model, it is possible to express p˙ as a function of the plastic strain-rates tensor as: 1 p2 1/2 2 p p 1 (3.222) ε∼˙ : ε∼˙ + ε˙ kk p˙ = 1 − η 3C 9F In the' case of the isotropic Rousselier model, p˙ is equal to von Mises invariant: p˙ = ˙λ = 2 ε˙ p : ε˙ p . One should note that it is not the case for other models. This 3∼ ∼ particularity is due to the form of the plasticity criterion. The Cam-clay model is a special case in that one often assumes R = 0 to be true so the material breaks as soon as it reaches a tensile stress state. To model a measure of resistance in shear and tension, one can use a value of R such as: R = constant pc . The variation of p corresponds to an isotropic hardening via R which is generally a function of p. Then, the plasticity criterion is written: f (σ∼ , η, R) = σ (σ∼ , η) − R(p, T , . . . )
(3.223)
R can depend on other parameters such as temperature but not on porosity. Indeed, it is a quantity relative to the matrix only. Kinematic hardening: We will note some extensions of the models for porous materials (mainly in the framework of the Gurson model) to the case of kinematic hardening [MEA85, BEC86, LEB95, ARN97]. These extensions are not used much at the moment.
3.11.8. Nucleation In general, materials contain inclusions. During deformation, the inclusion/matrix interface, or the inclusion itself, can break. Then new cavities appear: it is the nucleation process. To account for this phenomenon, one assumes that porosity variation is the sum of a growth term and a production term related to nucleation. Equation (3.218) is then replaced by: (3.224) η˙ = (1 − η) Tr ε∼˙ p + η˙ g where the term η˙ g is the porosity increase due to the nucleation. This term is purely phenomenological, but can be adjusted using measurements of damage kinetics at inclusions [DEV97b, GUI98]. One distinguishes three kinds of nucleation: Plastic strain driven nucleation: In this case, η˙ g is given by: η˙ g = A(p, . . . )p˙ where A is a coefficient that can depend on the variables of the model.
(3.225)
124
Non-Linear Mechanics of Materials
Stress driven nucleation: In this case, η˙ g is given by: η˙ g = Aσ˙ + B σ˙ kk
(3.226)
A and B, again, are coefficients that can depend on the variables of the model. Mixed nucleation: To conclude, it is possible that both stress and strain contribute to the nucleation kinetic. One defines then an effective stress σc :
η˙ g is then given by:
σc = ασ + βσkk + Hp
(3.227)
η˙ g = C(σc , . . . )σ˙ c
(3.228)
where α, β, H and C are coefficients. Generally, strain controlled nucleation is used. The case where only the effective stress σ plays a role in the stress controlled nucleation law is equivalent, in the plastic case, to strain driven nucleation as σ = R(p) is a function of the plastic strain. The term σkk in the stress driven nucleation laws can be problematic as it can decrease (during fracture, stresses become zero). A solution consists in not taking into account σkk when σ˙ kk < 0.
3.11.9. Example To give an example, let us consider the Gurson model. The parameters of the model are: q1 = 1.5 q2 = 1.0 ( η if η < ηc η = ηc + δ(η − ηc ) if η > ηc
(3.229) with ηc = 0.002, δ = 8.0 (3.230)
The initial porosity is η0 = 0.001. These values are representative of a carbon steel containing MnS inclusions, assumed to have a low cohesion with the matrix and which, as a consequence, act like cavities. The hardening law is given by: R(p) = R0 + Q(1 − e−bp ) with R0 = 300 MPa, Q = 100 MPa, b = 40
(3.231)
Figure 3.21 represents the stress-strain curves for a tensile test (Fig. 3.21a) and a biaxial tensile test (Fig. 3.21b). In both cases, the stress decreases because of damage. The porosity variation rate, in the case of Gurson’s model, is: η˙ =
2 σM ) 3(1 − η)2 q1 q2 η σ2 sinh ( q2σ 2 σM 2σE2 + q1 q2 η σM σ sinh ( q2σ )
p˙
(3.232)
3D plasticity and viscoplasticity
125
Figure 3.21. Stress-strain curves for a Gurson’s material: (a) uniaxial tension, (b) biaxial tension. The dotted curves represent the response of the von-Mises non-damageable material
Figure 3.22. Porosity-strain curves for a Gurson material in the uniaxial and the biaxial tension cases. The dotted curve indicates the value of the coefficient ηc = 0.002
126
Non-Linear Mechanics of Materials
For low porosity (η 1), the effective stress σ is close to σE and then: 3 q2 σ M η˙ q1 q2 η sinh p˙ (3.233) 2 2σE ' In this case, p˙ is close to von Mises equivalent strain: p˙ (2/3)˙ε∼ p : ε∼˙ p . One recognizes in the previous equation, the stress triaxiality ratio τ = σM /(3σE ). Damage rate due to cavities growth increases with damage. Figure 3.22 displays the evolution of porosity as a function of plastic strain p in the case of uniaxial tension (τ = 1/3) and biaxial tension (τ = 2/3).
Chapter 4
Introduction to damage mechanics
4.1. Introduction In the past twenty years the field of Continuum Damage Mechanics (CDM) has grown considerably, from the concepts initially introduced by Kachanov [KAC58] and then Rabotnov [RAB69]. Damage is considered as a deterioration process of matter, consecutive but not identical to irreversible strain. The defects that appear correspond to accumulation and localization of dislocations (in metals). They have an irreversible character much more pronounced than plastic deformation. Moreover, many brittle materials deform by damaging. Following the Russian school’s work, damage mechanics initially developed in Europe, essentially for applications to creep of metallic materials. The English school, notably with Leckie and Hayhurst’s works [HAY72, LEC74], made a remarkable contribution at the beginning of the 1970s, with some Polishx [CHR76], Swedish [HUL79] or Japanese [MUR80] contributions. But it was in France that basic concepts of damage mechanics were formulated at the theoretical level, in particular through thermodynamic formalism [CHA77, GER83, LEM96]. It was also in France that researchers systematically looked for other kinds of damage (fatigue, ductile fracture) and other kinds of materials [LEM78, CHA74, LAD83, LEM84, MAZ86]. It was only at the beginning of the 1980s that damage mechanics became recognized (actually rediscovered) in the US, with the works of Krajcinovic [KRA81, KRA84], Ortiz [ORT85a], Ju [JU89], then of Talreja [TAL85], Chow [CHO87], Voyiadjis [VOY93] and many others. This scientific area has since been booming throughout the world, in terms of basic research (all the problems are far from being solved) and in terms of applications. Fracture mechanics considers the crack itself, by modifying the boundary conditions of the structure. On the contrary, damage mechanics brings into play only the J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_4,
128
Non-Linear Mechanics of Materials
defects through a homogenization concept and describes their evolution macroscopically, by staying in the framework of continuum mechanics. The references [HUL79, CHA81, MUR83a, KRA84, LEM85b, CHA88, LEM96, SKR98] are relatively exhaustive and general presentations of damage mechanics. In this chapter, we will address several aspects. The first part introduces the elementary notions and the general concepts, by distinguishing irreversible strain, damage and macroscopic propagation. We make precise the kind of definition usable for damage variables. To conclude, some remarks are made on the significance of the dissipated energies by damage, by referring to the thermodynamic framework presented in Sect. 4.2.4. The theoretical bases of damage mechanics are developed in greater detail in the following sections, with regard to both the state laws and the associated evolution and dissipations, by taking into account the coupling intervening between mechanical behavior and damage. The effective stress concept is discussed according to its two main versions (Sect. 4.3). Moreover one specifies the mathematical nature of damage variables, scalar or tensorial, in connection with initial and/or induced anisotropies. Various forms of state coupling and associated dissipative coupling are introduced and discussed, corresponding to various theoretical schemes used in the literature (Sect. 4.4). The essential notion of damage deactivation (associated to the potential closing of microcracks) is detailed (Sect. 4.5). We show the modeling difficulties associated with this effect and indicate an approach providing a sensible compromise.The general form of damage evolution laws is then presented (Sect. 4.6). In the time-independent theory, we show through examples how the consistency conditions and the tangent operator associated to the various theories are written. Standard or pseudo-standard formulations are finally proposed in the framework of a time-dependent theory (elastoviscoplastic/viscodamageable). In the last part some applications are presented. Although many works exist on the damage mechanics of the metallic materials, we will restrict ourselves to the brittle materials case. Section 4.7 shows a short example relative to concrete and presents a hierarchal approach for the modeling of composite materials, by examining the case of the ceramic/ceramic woven-composites.
4.2. Notions and general concepts 4.2.1. Various kinds of damages As was mentioned in the introduction, material damage is the result of relatively large irreversible defects, much more than plastic deformations are. In the metallic materials, there are for example: – microcavities of ductile fracture, which are created by decohesion at the inter-
Introduction to damage mechanics
129
faces between inclusions and matrix, increase in number and in volume, then coalesce to form a macroscopical defect leading to the fracture of the tensile sample or to trigger the ductile tear of the structure [PIN81, FRA93]; – fatigue microcracks, that are initiated generally in persistent slip band, close to the free surface of the sample or of the piece (phase I fatigue). They progress by shear (at 45◦ of the local principal stress), on the surface and/or towards the interior, then more or less perpendicular to the macroscopic principal stress (phase II fatigue or mode I propagation). One of these microcracks soon becomes the main crack, that must then be considered as macroscopic and treated in the framework of fracture mechanics [CHA81]; – intercrystalline slip and cavities that appear at the grain boundaries while damaging by creep at high temperature. These cavities are created by accumulation of dislocations on the boundaries and by locking of these dislocations on the point defects (precipitate, inclusions, for instance). The diffusion mechanisms of the vacancies can also play a significant role at the highest temperatures. The cavities increase in number and in size, with independent kinetics, then coalesce and break some grain boundaries, which causes a brittle fracture, accompanied by a small macroscopic deformation. In some other materials, the damage processes are different but can be considered in an homogenized manner: – in concrete, the defects appear by debonding between the aggregates and the cement and provoke a micro-cracking relatively spread out, at least in a first phase. It is similar for rocks; – for composites, several scales of defects (and analysis) can play a role: debonding at the fiber-matrix interface, accompanied by friction and wear, microcracking of the matrix (transverse cracking, parallel to the fibers, for example), fiber fracture. They could be homogenized at the scale of each ply of a laminate composite structure. On the contrary, the delamination (crack growing at the interface between layers of different orientations) could not be homogenized and will have to be treated as a discrete defect, either by fracture mechanics or by an interface damage mechanics.
4.2.2. Distinction between deformation, damage, propagation It is important to distinguish these three aspects, that can correspond to three successive states of the material:
130
Non-Linear Mechanics of Materials
Figure 4.1. Irreversible deformation in plasticity and in brittle damage
– irreversible strains (Fig. 4.1a) are described in the framework of continuum mechanics in a way that allows developing the plasticity and viscoplasticity laws introduced in Chap. 3. Their irreversibility (at the macroscopic scale) is not total, as one can redeform plastically the material to give it back its original shape. On the contrary, we assume that the damage is a definitive deterioration. The cavities, microvoids, decohesions and microcracks are important enough so that a load in operation (at temperature applications) cannot rejoin the broken bonds. Thus we will consider that damage is always accompanied by a (definitive) decrease of the mechanical resistance of the material, even if this decrease is sometimes difficult to measure experimentally. Figure 4.1b indicates the typical (and ideal) situation of the quasi-elastic brittle material, in which the nonlinearity of the behavior is entirely due to damage, that is being noticeable by the decrease of elastic modulus (upon unloading). On the contrary irreversible deformation is negligible; – the notion of continuous damage relates to the possibility to use continuum mechanics for the modeling of damageable structures. In order to work on a representative volume element (RVE), with strain and stress variables defined in average, it is necessary that defects be numerous and small enough. Figure 4.2a schematically indicates the notion of material RVE associated with damage mechanics. The damage variable, D, is assumed to be initially zero when there is no decohesion of matter; – on the contrary, fracture mechanics generally considers only one macroscopic crack, clearly identified in geometry and dimension, a crack that propagates through the structure whose material is considered to be a continuum (generally
Introduction to damage mechanics
131
Figure 4.2. Damage mechanics’ and fracture mechanics’ RVE
suppress a non-damaged one). Figure 4.2 shows the correspondence between both situations, with the use of the external load and displacements applied to the piece, instead of the average stress and strain applied to the material RVE. It also indicates the correspondence between the thermodynamic force Y associated with the diffuse damage D and the energy release rate G associated with the unitary increase of the crack a. Obviously, the limitation of damage mechanics is linked to the macroscopical localization, for which diffuse damage gathers in a single main crack (Fig. 4.3 for the case of fatigue of metallic materials). This situation will correspond to the definition of the initiation (of a main macroscopic crack) that will be used in this book. It corresponds also to the final situation, with D = Dc , where Dc will often be considered as equal to 1.
4.2.3. Damage definitions and measures It is desirable to have, in conjunction with the definition of variables, a measurement method, even if it is sometimes difficult to use practically. Four kinds of measurement can be considered: – measurement by remaining lifetime: it is the one that interests the engineer in order to calculate the component lifetimes. By reference to a reference test (with a given load) whose lifetime until fracture is known, the damage induced by any prior load is measured by loading with the same reference load. The ratio of the
132
Non-Linear Mechanics of Materials
Figure 4.3. The two definitions of crack initiation in fatigue
lifetimes gives a damage measurement, using the underlying definition of the Palmgren–Miner rule (linear damage hypothesis); – microstructural measurements: it is the physical measurement of actual defects. It allows us to quantify cavity volume fraction (ductile damage), the surface density of broken grain boundaries (creep), the number of microcracks or the crack area summation per unit volume (fatigue). These measurements are necessarily destructive and difficult to integrate directly into a mechanical model. The notion of net stress described hereafter refers however to that kind of measurement; – measurement using physical parameters. One can measure the evolution of parameters such as density [JON77], resistivity [CAI80], acoustic emission. Once the effects of deformation alone have been taken into account, the rest of the evolution is characterized by a damage parameter. A model is, of course, necessary in order to pass to a mechanical damage. – measurements of mechanical behavior evolution. These are the most adapted to mechanical modeling. We will distinguish two methods, based either on the notion of net section (and suppress stress), or on the notion of effective stress, which are both introduced below for the uniaxial case. The net stress is the average stress applied over the resisting section (or net section) of the damaged sample. As shown Fig. 4.4, this occurs on the current geometry of the sample, after taking into account the macroscopical deformations. The net stress σ ∗
Introduction to damage mechanics
133
Figure 4.4. Apparent stress, true stress and neat stress
is deduced from Cauchy stress σ via an average section decrease factor ω due to voids and cracks. One will write: S σ (4.1) σ∗ = ∗σ = S 1−ω Such an approach was used by Murakami [MUR80, MUR83a] for example, to directly generalize to the anisotropic case the notion of creep damage in polycrystals. The effective stress, on the contrary, takes into account stress concentrations in the vicinity of the defects, and is based on the measured macroscopic behavior of the damaged material [CHA77]. By analogy with the previous case, one defines an effective area S˜ and the effective stress is then, in uniaxial: σ˜ =
σ S σ = ˜ 1−D S
(4.2)
where D represents the macroscopical effect of deterioration of the mechanical behavior (for example of the elasticity modulus). This notion generalizes in multiaxial, either by the hypothesis of strain equivalence [CHA77, LEM78], or by the hypothesis of energy equivalence [COR79]. The first hypothesis enunciates: “The effective stress tensor σ∼˜ should be applied to a non-damaged element of material (all other things being the same) so that it deforms like (same strain tensor) a damaged element submitted to the actual current stress tensor σ∼ (Fig. 4.5).” The measurement of the mechanical behavior evolution can be conducted through the measurement of various parameters (the choice depends on the kind of application): – elastic modulus evolution, used for ductile plastic damage [LEM85a], but also for fatigue [PLU86], especially in composites, as shown on Fig. 4.6a [CHA86]. The measured damage is then: D =1−
E˜ E
(4.3)
134
Non-Linear Mechanics of Materials
Figure 4.5. Definition of the effective stress
Figure 4.6. Damage measurements by effective stress
– plastic or viscoplastic behavior [LEM78, CHA78]. Figure 4.6b shows the example of IN100 superalloy used in jet engine blades and the AU2GN aluminium light alloy, Concorde’s structural alloy. Knowing the exponent N of the sec-
Introduction to damage mechanics
135
Figure 4.7. Fatigue damage evolution curves as measured on three materials and corresponding evolution of the normalized stress, from [PLU86]
ondary creep law (Norton’s law), one finds the damage thanks to measurement of the strain-rate in secondary creep ε˙ s , and in tertiary creep, ε˙ : D =1−
ε˙ s ε˙
1/N (4.4)
– cyclic behavior, for the Low-Cycle-Fatigue of metallic materials [CHA74]. Figure 4.7, from [PLU86], shows that this method applies to materials with unstable cyclic behavior and is well correlated with the measurement of the modulus evolution.
– one can also use the ultrasonic wave velocity, instead of direct measurement of the elastic modulus. This technic has been developed with success to measure up to the nine parameters of an orthotropic composite [BAS89].
As we shall see below, the effective stress is used in the constitutive equations instead of Cauchy stress in order to describe the impact of the damage on the macroscopic material behavior. As a first approximation it is the same quantity that intervenes, whatever the kind of studied behavior is (elastic, plastic, viscoplastic).
136
Non-Linear Mechanics of Materials
4.2.4. Energy dissipated by damage The thermodynamic aspects of damage mechanics will be developed further and used in the following sections. For the moment, we limit ourselves to introducing the consequences of the choices made in the thermodynamic framework, consequences relative to state laws. The state potential, supposedly the Helmholtz free energy (Sect. 4.2), involves damage in the elastic term ψe and in the hardening term ψp . Here, we assume firstly that damage does not play a role in ψp . Secondly, we consider for the moment only one variable D, held scalar to simplify the notation, and we note neither the presence of the temperature nor that of the thermal dilatation. It turns out then: ρψ = ρ(ψe + ψp ) =
1 e ˜ ε : (D) : ε∼ e + ρψp (αj ) 2 ∼ ∼∼
(4.5)
˜ (D) represents the elastic stiffness tensor of the damaged material. The therwhere ∼ ∼
modynamic force associated to damage D is defined by: ˜ ∼ 1 e ∂ ∂ψ ∼ = − ε∼ : : εe y=− ∂D 2 ∂D ∼
(4.6)
We can demonstrate that it is equal to half the variation of the elastic energy generated by a damage increase at constant stress and temperature: 1 dWe y= (4.7) 2 dD σ,T For that, it is sufficient to define the variation of elastic energy as dWe = σ∼ : dε∼ e , to calculate σ∼ thanks to the state law and to differentiate: σ∼ = ρ
∂ψ ˜ : εe = ∼ ∼ ∂ε∼ e ∼
˜ : dεe + dσ∼ = ∼ ∼ ∼
˜ ∂ ∼ ∼
∂D
: ε∼ e dD
(4.8)
(4.9)
By multiplying on the left by ε∼ e , we get ˜ : dεe + ε e : ε∼ e : dσ∼ = ε∼ e : ∼ ∼ ∼ ∼
˜ ∂ ∼ ∼
∂D
: ε∼ e dD = σ∼ : dε∼ e − 2ydD
(4.10)
At constant stress, one gets the expected result. y is then exactly like the elastic energy release rate for crack growth, the global parameter of fracture mechanics,
Introduction to damage mechanics
137
generally noted G, and which has been used in the classical fracture theories since Griffith’s work. In fact, there is a complete analogy between fracture mechanics and damage mechanics, in its thermodynamic framework, with the following correspondences: fracture mechanics
damage mechanics
Structure
⇔
V.E.R.
external loads
⇔
stress applied on RVE
displacements of the point of load application
⇔
average strain on RVE
crack (length, surface)
⇔
damage variable (diffuse)
elastic energy release rate G
⇔
thermodynamic force y (elastic energy release rate)
Figure 4.8 illustrates the partition of the dissipated energies during a tensile test. The curve OA B represents the hardening evolution during plastic flow (OAB). Parts AB and BC correspond respectively to plastic flow and to the increase in elastic strain during the damaging process (here sketched at constant stress). The total dissipated
Figure 4.8. Scheme of the dissipation during plastic flow and damage
138
Non-Linear Mechanics of Materials
energy is divided between (1) the energy stored in the material (hardening), (2) the energy dissipated as heat, (3) the energy released during the damaging process, possibly converted into heat. Variable y, the thermodynamic force associated with damage, can be used to define the damage evolution criterion and finally a fracture criterion for the volume element. It is an energetic criterion: the total energy dissipated by damage reaches a constant value at fracture.
4.3. Damage variables and state laws 4.3.1. Tensorial nature of the damage variables The simplest damage variables are of course scalar. Unfortunately, the defects that constitute damage, cavities, interface decohesion, microcracks, are generally oriented by the load that creates them. Thus, for example, the fatigue microcracks orientate perpendicularly to the largest principal stress. The creep cavities can also be oriented: in this case they grow preferentially on the grain boundaries orthogonal to the largest principal stress and they will break first. It is then necessary to use tensorial variables to describe the directional character of damage. A first way to describe the directional character is to use a scalar function with vectorial support (on R3 ), for each RVE around a material point, that associates to each direction n of the space a probability density of defect p(a 3 /V , n) having a direction perpendicular to n. More precisely, p(a 3 /V , n) is associated to the relative crack density a 3 /V , where a is the crack radius, assumed circular (disk) as a first approximate. The use of such a description in the framework of a mechanical constitutive equation would imply a discretization of the unit sphere (sphere of the directions around a material point), while staying within the framework of a micromechanical approach. Reasonings on material symmetries, due particularly to [LEC80], show that a series decomposition of that probability density can only display even moments, hence, by limiting to the fourth order 3 3 a a 1 p , n dV (4.11) ω= V V V V 3 3 a 1 a = (4.12) p , n n ⊗ ndV ω ∼ V V V V 3 3 a 1 a = (4.13) p , n n ⊗ n ⊗ n ⊗ ndV ω ∼ V V V V ∼ that are then possible candidates for damage internal variables.
Introduction to damage mechanics
139
4.3.2. Some possible choices for elasticity laws The above enumeration applies to damages corresponding to a given mechanism. Of course, the number of variables can be increased when various physical mechanisms develop defects of different natures. To render the elastic behavior of damaged material, depending on the desired precision and on the kind of application, one could then use: 1. One scalar variable: damage is considered isotropic, without privileged orientation. Thus, for the initially anisotropic material, one could write the elastic potential with only one scalar damage variable D: ρψ =
1 : ε∼ e (1 − D)ε∼ e : ∼ 2 ∼
(4.14)
where is the tensor of the initial elastic stiffnesses of the non-damaged ma∼ ∼ terial. This formulation is the one used since the origin of damage mechanics. It is the simplest approach, very easy to use and still very common, thanks particularly to Lemaître’s works [LEM96]. It gave rise to many developments for damage process of any nature and on many types of materials. One of its peculiarities is to define the thermodynamic force associated with damage by: Y = −ρ
1 ∂ψ : ε∼ e = ε∼ e : ∼ ∂D 2 ∼
(4.15)
i.e., the elastic energy of the effective non-damaged material. 2. Two scalar variables: It is more general to render the isotropic property of damage (i.e., conservation of the initial isotropy) by using two scalars, and D, instead of one [LAD83, JU89]. The first one is associated to the fraction of energy that contains the hydrostatic strain, the second one to the rest of the energy (see also the discussion based on micromechanics concepts by Kachanov [KAC93]): ρψ =
1 e2 e e + μ(1 − D)εij εij λ(1 − )εkk 2
(4.16)
3. Scalar variables associated with predefined material directions. This applies especially to composites, in which microcracks are often oriented by the structure and the orientation of its constituents (plies, various reinforcements, strand, fibers, matrix). In this case, there are as many scalar variables as there are privileged directions of the constituents. For example, for an elementary unidirectional ply in a composite laminate, which will be isolated as being a RVE, one should get a scalar variable d2 , corresponding to transverse cracks, parallel to the fibers that define direction 1. In general, one could limit oneself to three scalar variables, as for example in the woven composites, in which,
140
Non-Linear Mechanics of Materials
in addition to d2 we also have d1 , towards orthogonal strands, and possibly d3 , representing the effect of longitudinal microcracks, in the plane of the woven ply. The number of variables indicated here concerns, of course, only one cracking mechanism. In some theories ([LAD94] for example), there are two variables instead of one for each direction, the first associated to the variation of transverse Young’s modulus, the other associated to the evolution of the shear modulus. 4. A second-rank tensor: it is the most used kind of variables to render the damage induced anisotropy in an initially-isotropic material. It is the minimum complexity for an anisotropic theory. The validity of such a tensor can be demonstrated from the geometrical point of view of the decrease of resistant section, or net section. We sum-up below (Sect. 4.3.3.1) the approach of Murakami and Ohno [MUR80] that leads to the notion of net-stress tensor, applicable to the case of grain boundary cavitation in polycrystals submitted to creep. Different reasonings can also lead to the justification of a second-rank tensor [VAK71, KAC80]. However, a second-rank tensor only is not sufficient to describe completely the anisotropy induced by damage on the (elastic) behavior. It is then necessary to indicate how this second-rank tensor acts on the fourth-rank tensor characterizing the behavior (elastic stiffnesses or compliances). It is the function of damage effect tensors, themselves fourth-rank tensors. 5. A fourth-rank tensor: it is the lowest-order damage-variable that makes it possible to write the degradation accompanied by an induced anisotropy (the material being initially isotropic or anisotropic) as an operator acting on the fourthrank tensor representative of the elastic behavior. Used for the first time by Chaboche [CHA79a] this fourth-rank damage-variable is introduced naturally through the effective stress concept based on the strain equivalence principle (Sect. 4.3.3.2). In this case, the damage state variable, a fourth-rank tensor, directly plays the role of damage effect tensor as mentioned above for a theory based on a second-rank tensor. Some theories have been developed since then by using directly the elasticity tensor as a state variable [ORT85a, SIM87]. Thanks to micromechanics approaches it has been shown that it is necessary to use fourth-rank tensors in order to more accurately describe the damage induced anisotropy [LUB93, KAC93].
4.3.3. Effective stress concept In this section we consider the various ways to generalize to the anisotropic and 3D case the notions introduced for the uniaxial case in Sect. 4.2.3. We will study the “net” stress tensor used by Murakami and Ohno [MUR80] and then two ways to generalize the notion of “effective stress”.
Introduction to damage mechanics
141
Figure 4.9. Net section in the case of grain boundary cavitation
1. “Net stress” tensor It has been developed to take into account the grain boundaries cavitation in polycrystals undergoing creep. The main effect is considered to be due to the decrease of resistant section (or net section). Damage induced by cavitation on the boundary “k”, perpendicular to direction ν (k) (Fig. 4.9), is defined by: d = ∼
3 dS (k) ν (k) ⊗ ν (k) Sg (V ) g
(4.17)
(k)
where dSg is the broken area of the boundary “k” and Sg (V ) the total area of is then the grain boundaries in the REV of volume V . The current damage ∼ simply defined by summing over the directions and integrating over the volume: 3 = ∼ Sg (V ) N
k=1 V
ν (k) ⊗ ν (k) dSg(k)
(4.18)
is a second-rank tensor. Hence, it has three principal direcIt is obvious that ∼ tions ni and can be resolved into = ∼
3
i ni ⊗ ni
(4.19)
i=1
i represents the surface density of cavities on a plane perpendicular to ni . This creates a decrease of resistance, characterized by an increase of the stress tensor, as shown in Fig. 4.10, from [MUR80]. In these conditions, the force vector acting on the surface Sν is St = σ∼ .(Sν) where t is the corresponding stress vector, σ∼ is the Cauchy tensor acting on the RVE. The effective surface A∗ B ∗ C ∗ is submitted to the same force vector and thus S ∗ t ∗ = St
σ∼ ∗ .(S ∗ ν ∗ ) = σ∼ .(Sν) = σ∼ .(1∼ − )−1 .(S ∗ ν ∗ ) ∼
(4.20)
142
Non-Linear Mechanics of Materials
Figure 4.10. Net section and net stress tensor
where σ∼ ∗ is not necessarily symmetric. In practice, it is symmetrized and, denoting by σ∼ ∗ the symmetrized tensor: σ∼ ∗ =
1 + .σ σ∼ . ∼ ∼ ∼ 2
= (1∼ − )−1 ∼ ∼
(4.21)
2. Effective stress tensor based on strain equivalence Let us recall the definition [CHA77]: the effective stress tensor σ∼˜ is the one that should be applied to the RVE of undamaged material in order to get the same strain tensor as the one observed on the damaged RVE undergoing the current stress tensor σ∼ . In elasticity, the law of damaged material is given by (4.8), whereas the one of the non-damaged material is written as (2.36) from Chap. 2, without initial stress. By eliminating ε∼ e between both, we get −1 σ∼˜ = M : σ∼ ∼ ∼
(4.22)
is the damage effect operator, a fourth-rank tensor, that can be written where M ∼ ∼
˜ , respectively elasticity tensors of the undamaged and as a function of and ∼ ∼ ∼
damaged materials:
∼
˜ : −1 M = ∼ ∼ ∼ ∼
∼
∼
(4.23)
Introduction to damage mechanics
143
To get an expression similar to the one used in the uniaxial case (4.2) we let ∗ , where I is the fourth-rank unit tensor: = I∼ − D M ∼ ∼ ∼ ∼
∼
∼
∼
I∼ = ∼
1 1∼ ⊗ 1∼ + 1∼ ⊗ 1∼ 2
(4.24)
∗ is then the fourth-rank damage tensor, which can be measured thanks to the D ∼ ∼
˜: measurement of ∼ ∼
∗ ˜ : −1 D = I∼ − ∼ ∼ ∼ ∼
∼
∼
(4.25)
∼
The above choices boil down to assuming that the elastic stiffness tensor of the damage RVE is written ˜ = (I − D ∗ ) : (4.26) ∼ ∼ ∼ ∼ ∼
∼
∼
∼
In fact this tensor is generally not symmetrical, so that, if D is assumed to be a ∼ ∼
state variable, we will prefer to use:
˜ = 1 (I − D ) : + : (I − D T ) ∼ ∼ ∼ ∼ ∼ 2 ∼∼ ∼∼ ∼ ∼ ∼ ∼ ∼
(4.27)
The relation between both variables can be solved numerically (6 × 6 linear ∗: system) to find D from D ∼ ∼ ∼
∼
−1 ∗ D + :D : = 2D ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
(4.28)
will In practice, the theory will generally be used the other way around, as D ∼ ∼
be known thanks to a damage growth equation (and its integration under the previous loading undergone by the RVE). We will then use:
The effective stress tensor can be defined by (4.22) even if we do not use a fourth-rank damage tensor, and even whatever the chosen expression for the damaged stiffness tensor is. 3. Effective stress tensor based on energy equivalence We use the definition given by [COR79]: the elastic energy of damaged material under stress σ∼ and strain ε∼ e is the same as the elastic energy of the effective
144
Non-Linear Mechanics of Materials
(undamaged) material undergoing effective stress σ∼˜ and effective elastic strain ε∼˜ e . Then, we have: 1 1 1 σ∼˜ : ε∼˜ e = ε∼˜ e : : ε∼˜ e = σ∼˜ : S∼ : σ∼˜ ∼ 2 2 2 ∼ ∼ 1 e ˜ 1 1 e e : ε∼ = σ∼ : S∼˜ : σ∼ = W˜ e = σ∼ : ε∼ = ε∼ : ∼ 2 2 2 ∼ ∼
We =
(4.29) (4.30)
˜ and S˜ are respectively the stiffness and the compliance of damaged where ∼ ∼ ∼
∼
material. The effective stress and strain (different from the current strain) are necessarily: −1 T σ∼˜ = M : σ∼ ε∼˜ e = M : ε∼ e (4.31) ∼ ∼ ∼
∼
is still called the damage effect tensor. Of course, we have likewise: where M ∼ ∼
˜ = M : : MT ∼ ∼ ∼ ∼ ∼
∼
∼
−T −1 and S∼˜ = M : S∼ : M ∼ ∼
∼
∼
∼
∼
(4.32)
∼
This changes somewhat the meaning of the damage parameter with respect to , one would the previous case. If one wished to keep a fourth-rank tensor D ∼ ∼
write: ˜ = (I − D ) : : (I − D T ) ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
(4.33)
immediately symmetric, but where D acts now in a quadratic way. The quadratic ∼ ∼
form of (4.33) prevents this approach from being identified with any evolution ˜ (which was the case of the previous approach). of ∼ ∼
Actually, we will use it rather with a second-order damage tensor, d∼ , expressing (d ). Several possibilities are given below. directly the damage effect tensor M ∼ ∼ ∼
Let us note that the order of definition of the operators is necessarily different from the one that intervenes for the strain equivalence. Indeed, we have:
4. Some possible forms for the damaged elasticity law The matrix forms are given in the principal frame of damage, using Voigt’s convention, which orders the second-rank tensor ε∼ , in a column vector (ε11 , ε22 , ε33 , 2ε23 , 2ε31 , 2ε12 ). In some cases one gives the intrinsic form by using the tensorial products ⊗⊗⊗.
Introduction to damage mechanics
145
• For scalar damage variables used for composites, the additive form will be mainly used: m ˜ =− δi K (4.34) ∼ ∼ ∼i ∼
∼
∼
i=1
where δi , (i = 1, . . . , m) are scalar damage variables, oriented by the constituents. If the composite is globally orthotropic, m = 3, the fourth-rank tensors K , characteristic of the material, are easily written in the material ∼i ∼
principal axes. They include a very small number of material parameters. An example will be given for the SiC/SiC composites in Sect. 4.7.3. • For the second-rank-tensors damage variables, the multiplicative form used in the energy equivalence [COR79] is formulated without introducing any material coefficient. The “diagonal” form is the most used, being written in the principal frame of damage (with ui = 1 − di ): ⎡
u1 ⎢0 ⎢ ⎢0 ⎢ M (d ) = ⎢ ∼ ∼ ∼ ⎢ ⎣
0 u2 0
0 0 u3
⎤ √
u2 u3
√
u3 u1
√ u1 u2
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(4.35)
whose intrinsic form requires the definition of special tensorial operators (square root for example), which will not be explained here. This operator, applied with (4.32b) in the framework of energy equivalence, gives the compliance of damaged material (initially isotropic elastic with E and ν, Young’s modulus and Poisson’s ratio), as: ⎡
1 u21
⎢ ⎢sym ⎢ ⎢ 1 ⎢sym S∼˜ = ⎢ E⎢ ∼ ⎢ ⎢ ⎣
−ν u1 u2 1 u22
sym
−ν u1 u3 −ν u2 u3 1 u23
⎤
1+ν u2 u3
1+ν u3 u1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(4.36)
1+ν u1 u2
Let us note that this form does not adapt well to real problems, as can be seen by comparison to elementary solutions of micromechanics. For example, for a set of microcracks perpendicular to axis 1 (Fig. 4.11a), hence corresponding to a damage tensor with principal values d2 = d3 = 0, the
146
Non-Linear Mechanics of Materials
Figure 4.11. Two arrangements of microcracks for brittle materials; (a) perpendicular to tensile axis; (b) parallel to compression axis (transverse isotropy)
micromechanics gives the compliance [KAC93]: ⎡
E E˜
⎢−ν ⎢ ⎢−ν 1 ⎢ S∼˜ = ⎢ E⎢ ∼ ⎢ ⎣
−ν 1 −ν
⎤
−ν −ν 1
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
1+ν G ˜ G
(4.37)
G ˜ G
where 16 a 3 (1 − ν 2 ) E˜ = E/ 1 + 3 V
3 ˜ = G/ 1 + 16 a 1 − ν G 3 V 2−ν
with the notation of Sect. 4.3.1 for a and V . Clearly, both relations (4.36) and (4.37) cannot be identified, if only because of the non-diagonal term S˜12 , equal to −ν/E in one case and to −ν/(E(1 − d1 )) in the other case. • Still for the use of a second-rank tensor, but with the strain equivalence, one could use a damaged stiffness tensor of the form: ˜ (d ) = − [D (d ) : K ]s ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
(4.38)
Introduction to damage mechanics
147
where K is a fourth-rank tensor, characteristic of the material (which de∼ ∼
(d ) is a dampends in particular on the initial material symmetries) and D ∼ ∼ ∼
age effect tensor that we will express, for example:
1−ξ (d ) = ξ 1∼ ⊗ d∼ s + 1∼ ⊗ d∼ + 1∼ ⊗ d∼ s D ∼ ∼ 2 ∼
(4.39)
, where ξ is a material dependent parameter. As characteristic tensor K ∼ ∼
one can choose the initial elastic stiffness tensor, K = , and we recover ∼ ∼ ∼
∼
the usual relation (4.27). One can also choose ξ small and K = E I∼ , in ∼ order to bring the solution back to a micromechanical one.
∼
∼
• To conclude, with a fourth-rank tensor one could either use a relation such as (4.38) above, with D now a state variable, or consider the variation of ∼ ∼
the elasticity modulus as a state variable.
4.4. State and dissipative couplings Here we address the construction of the damage laws themselves, at least their general formulation. Of course, we have to ensure a certain consistency between the elastic constitutive law, damage growth and couplings between plasticity and damage.
4.4.1. Various forms of state coupling In what precedes, we have mainly considered the coupling between the damage and the elasticity law of the material. The notions of effective stress and the expressions given in Sect. 4.3.3 to introduce damage effect, apply to elastic behavior. A first question arises concerning the state laws that, in the framework of mechanical behavior, bring into play at the same time elasticity and hardening laws or rather the relationships between hardening state variables and their associated thermodynamic forces, generally used in the yield criteria: should damage intervene or not in the free energy term ψp associated with hardening? One can distinguish three cases: • (i) the initial theory does not consider this coupling [CHA77]. As a result, it provides an easy interpretation of the thermodynamic force associated with damage. As seen in Sect. 4.2.4, it is an elastic energy release rate, similar to the one used in linear fracture mechanics, where the localized plasticity at the crack tip is neglected (small scale yielding). This theory is usable even in the presence of plasticity and hardening. It simply means that there is no coupling between hardening and damage;
148
Non-Linear Mechanics of Materials
• (ii) the theory proposed by Cordebois and Sidoroff [COR79] introduces a state variable, β, scalar measure of the damage accumulation, which is going to play, for damage, a role similar to the one played by the isotropic hardening variable r for plasticity. The free energy is then written ψ = ψe (ε∼ e , D) + ψp (αj ) + ψd (β)
(4.40)
where the αj refer to all the hardening variables and where D is used as scalar variable (temperature is omitted for the sake of simplicity). With Y and B the thermodynamic forces associated with D and β: Y = −ρ
∂ψe ∂D
B=ρ
∂ψd ∂β
(4.41)
We will see below how variables B and β are used to construct the damage growth law. As in the previous case, there is no coupling between hardening and damage. Let us note that this approach modifies the interpretation of the energy dissipated by damage (by comparison with (i)) as, in that case, instead ˙ the intrinsic dissipation includes two terms: of Y D, int = Y D˙ − B β˙
(4.42)
• (iii) Independent of variable β above, one can consider the coupling between damage and hardening, with: ψ = ψe (ε∼ e , D) + ψp (αj , D)
(4.43)
In this case we change the meaning of the thermodynamic force associated with D, which includes now two terms: Y = −ρ
∂ψp ∂ψe ∂ψ = Ye + Yp = −ρ −ρ ∂D ∂D ∂D
(4.44)
By considering the isotropic damage, with an identical factor for both terms ψe and ψp , we can write: ψ = (1 − D) ψeo (ε∼ e ) + ψpo (αj )
with ρψeo (ε∼ e ) =
1 εe : : εe 2 ∼ ∼∼ ∼
(4.45)
We get the state laws: σ∼ = ρ
∂ψe = (1 − D) : ε∼ e ∼ ∂ε∼ e ∼
Aj = ρ
∂ψpo ∂ψ = (1 − D)ρ ∂αj ∂αj
(4.46)
and the corresponding effective stresses, still for the isotropic case: σ∼˜ =
σ∼
1−D
=ρ
∂ψeo ∂ε∼ e
A˜ j =
∂ψpo Aj =ρ 1−D ∂αj
(4.47)
Introduction to damage mechanics
149
Such an approach has been followed by Saanouni [SAA88] and by Ju [JU89]. It is currently the most used one as it offers a more “symmetrical” construction (see Sect. 4.6.3). We will see below in Sect. 4.4.6 that it allows naturally, in elastoplasticity with hardening and damage, canceling the stress when one reaches the fracture of the RVE (D = 1). This is useful in the framework of the local approaches in which one tries to describe the complete deterioration situation.
4.4.2. Coupling of dissipations In the case of evolution laws as well, there are various possibilities, whether one considers that the plasticity and damage mechanisms are identical or not: • (i) the first approach, that of Lemaître [LEM85a, LEM96], consists in considering only one mechanism, governed by plasticity, with only one dissipation potential (presented here with only one scalar damage variable): φ ∗ = F (σ∼ , Aj , Y ; D) = Fp (σ∼ , Aj ; D) + Fd (Y ; D)
(4.48)
˙ with a normality rule involving only one plastic multiplier λ: ε∼˙p = λ˙
∂Fp ∂F = λ˙ ∂σ∼ ∂σ∼
α˙ j = −λ˙
∂Fp ∂F = −λ˙ ∂Aj ∂Aj
∂Fd ∂F = λ˙ D˙ = λ˙ ∂Y ∂Y
(4.49)
(4.50)
Where mechanisms are time-independent, the multiplier λ˙ is determined by the consistency condition over plasticity. In viscoplasticity λ˙ is given as a known function of the σ∼ , Aj [BEN89a]. However, in this approach, the distinction between plasticity and damage mechanisms appears only through the partitioning between Fp ≡ fp and Fd ≡ fd , the former corresponding to the yield criterion, the latter to the damage criterion provided with a threshold. One can see that damage increases only if there is plastic flow. Likewise, beyond the damage initiation threshold, there cannot be plastic flow without an increase in damage. This appears then as a relatively strong limitation; • (ii) the second approach consists in considering separately the two mechanisms and the two associated criteria. There are then two independent dissipation potentials and two independent multipliers: Fp (σ∼ , Aj ; D)
fp (σ∼ , Aj ; D) ≤ 0
(4.51)
150
Non-Linear Mechanics of Materials
Fd (Y, B; D, β) ε∼˙ p = λ˙ p
∂Fp ∂σ∼
∂Fd D˙ = λ˙ d ∂Y
fd (Y, B; D, β) ≤ 0 α˙ j = −λ˙ p β˙ = −λ˙ d
∂Fp ∂Aj
∂Fd ∂B
(4.52) (4.53) (4.54)
Here D and β are parameters for the two potentials. For example, D can play role in fp through the effective stress concept, as we shall discuss in Sect. 4.4.3. The two independent multipliers λ˙ p and λ˙ d , respectively for plasticity and damage, are determined by the two consistency conditions, associated with both criteria f˙p = fp = 0, f˙d = fd = 0, at least when there is no time dependency. When there exists a time dependence, both multipliers can be given as known functions of the corresponding variables (see Sect. 4.6.5). A similar approach is followed by most of the authors in this area of damage mechanics [COR79, JU89, DRA76, CHO87, HAN92, ZHU95, CHA78, CHA96a]. It has the advantage of allowing us to build independent laws between plasticity and damage (but coupled through effective stress). In particular, the case of brittle damage can be described without significant plastic strain (for example for concretes, ceramic composites or even for creep damage of metals, especially brittle ones). To the contrary, there can be large plastic strains that do not cause damage, or very little, as for example pure shear in metallic alloys. This disassociation of the two kinds of mechanisms is exemplified in Fig. 4.12 (von Mises equivalent stress σeq as a function of the hydrostatic pressure σH ) where one clearly sees the possibility of reaching independently both surfaces (fp = 0 and fd = 0) and of making them progress, still in the framework of a rate-independent theory (and for the “isotropic” changes). Such a theory shows, in macroscopic modeling, that plastic strain and damage mechanisms are different: even if, at least for metallic materials, plastic strains are involved in both cases, it is not at the same scale, as plastic strains leading to damage have a much more localized character, with a macroscopic plastic strain that can remain low.
4.4.3. Some possibilities for the elastic limit criterion Damage influences plastic flow either through the elastic domain, or through the hardening law. Although this is not necessary, the effective stress concept provides a restrictive choice that reduces the number of possibilities and of parameters depending on the material, while offering satisfactory results in most studied cases, qualitatively as well as quantitatively. It is the way we choose to follow, but there still remains a significant number of possible variants.
Introduction to damage mechanics
151
Figure 4.12. Coupling of a plasticity and a damage criteria
The elastic limit criterion, and thus—at least for the “associated” formulations that we use—also the yield criterion, is going to depend on damage through the effective stress, by replacing σ∼ by σ∼˜ in the criterion of the non-damaged material. However we are going to get three additional possibilities when using a criterion that combines ) and isotropic (variable R) hardening. kinematic (variable X ∼ • (i) We admit that only effective stress is replaced, but that hardening variables are not enhanced by damage in the criterion. It is the choice made by Benallal [BEN89a], and then by Lemaître [LEM96]. The elastic limit criterion is then written − R − σy ≤ 0 , R) = J σ∼˜ − X (4.55) f = f (σ∼˜ , X ∼ ∼ where σ∼˜ is defined by one of the choices mentioned in Sect. 4.3.3, for example by relation (4.22). σy represents the elastic limit of the material. In Lemaître’s approach [LEM96], limited to isotropic damage, we get σ∼ f =J (4.56) −X − R − σy ≤ 0 ∼ 1−D In the case of non-isotropic damage, this approach has the distinctive feature . However, close to that the elastic domain does not remain centered about X ∼ fracture (D = 1) σ∼ tends towards 0 as it should (see Exercise 3, Sect. 4.4.6). • (ii) In addition to effective stress, one uses the effective kinematic hardening ˜ , deduced from X thanks to the same operation, similar to (4.22): variable, X ∼ ∼ ˜ = M −1 : X X ∼ ∼ ∼ ∼
(4.57)
152
Non-Linear Mechanics of Materials
by using the elastic limit criterion: ˜ , R) = J (σ˜ − X ˜ ) − R − σy ≤ 0 f = f (σ∼˜ , X ∼ ∼ ∼
(4.58)
Initially suggested in [LEM85b], it has became the most standard form. In this case, the elastic domain remains centered about X in the actual stress space. ∼ Moreover, by the choice of the state law, in which damage plays a role also in the kinematic hardening term (point (iii) in Sect. 4.4.1), it is possible to cancel stresses (all the components for the particular case of isotropic damage) when damage reaches the RVE fracture condition, D = 1. This is considered in Exercises 1 and 2, Sect. 4.4.6. This technique was used for the first time by Saanouni [SAA88] in the case of isotropic damage and of the strain equivalence hypothesis. It is the one developed in Sect. 4.4.4 below. ˜ , one uses also an effective value R˜ for the isotropic • (iii) As well as σ∼˜ and X ∼ hardening variable. This approach was used in the first applications [CHA77, CHA78], for the case with isotropic damage. One writes then: R˜ =
R 1−D
(4.59)
and the criterion is written ˜ , R) ˜ = J (σ˜ − X ˜ ) − R˜ − σy ≤ 0 f = f (σ∼˜ , X ∼ ∼ ∼
(4.60)
However, as it is hard to generalize this relation in the case of anisotropic damage, we will not consider this hypothesis.
4.4.4. General approach for plasticity/damage coupling We consider plasticity (or viscoplasticity) with kinematic and isotropic hardening. To simplify equations, we do not include hardening recovery phenomena although they could just as well be built in the same theoretical framework. The expressions are given for the isothermal case but their anisothermal generalization does not pose any problem. At the thermodynamic level we admit the existence of a state potential, the free energy, under the form (iii) of Sect. 4.4.1. For dissipation, we use the pseudo-standard GSM theory, in which we assume the existence of several independent dissipation potentials and several multipliers (as already mentioned for choice (ii) of Sect. 4.4.2). The initial medium is assumed anisotropic, described by Hill’s criterion for plasticity. Damage is also assumed anisotropic, (the tensorial nature of the damage variables will be studied in detail in Sect. 4.6: they are written as d with no more precision here).
Introduction to damage mechanics
153
1. State equations State equations are given by the free energy thermodynamic potential: 1 1 e ˜ e ε∼ : α i : C˜ (d) : α (d) : ε + + ρψr (r, d) ∼ ∼ ∼i 2 2 ∼ ∼∼ i ∼ i (4.61) are the kinematic-hardening state-variable (second-rank tensor). where the α i ∼ Their number is indeterminate. We are using only one isotropic hardening variable r. We get ∂ψ ˜ (d) : ε e (4.62) σ∼ = ρ e = ∼ ∼ ∂ε∼ ∼ ρψ = ρ(ψe + ψp ) =
X =ρ ∼i
∂ψ ˜ (d) : α i =C ∼ ∼i ∂α ∼ ∼i
R=ρ
∂ψr ∂ψ =ρ ∂r ∂r
(4.63)
The thermodynamic forces associated with damage will be defined later. To simplify the equation, we will assume that all the C are proportional to one ∼i ∼
another, i.e.: C = ci C . Thus, one can define the damage effect operator M ∼ ∼i ∼ ∼
∼
∼
for the elasticity law on one hand, and the damage effect operator N for the ∼ ∼
kinematic hardening law on the other hand: −1 : σ∼ σ∼˜ = M ∼ ∼
˜ = N −1 : Xi X ∼i ∼ ∼
(4.64)
˜ : C −1 N =C ∼ ∼ ∼
(4.65)
∼
Strain equivalence yields: ˜ : −1 = M ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
On the contrary, using energy equivalence, we can choose to take M = N , ∼ ∼ ∼
adding similar relationships for the effective “strains”: T :ε ε∼˜ = M ∼ ∼ ∼
T : εp ε∼˜ p = M ∼ ∼ ∼
T : εe ε∼˜ e = M ∼ ∼ ∼
∼
(4.66)
T :α ˜ =M α ∼i ∼ ∼i ∼
˜ and C˜ . We For the moment, we do not specify the relationships that define ∼ ∼ ∼
∼
will see in Sect. 4.4.5 the respective advantages and disadvantages of the two approaches (strain or energy equivalence). 2. Evolution equations in plasticity Plasticity/viscoplasticity evolution equations involve an elasticity domain f ≤ 0 (that can be reduced to a point in some viscoplasticity theories). We admit
154
Non-Linear Mechanics of Materials
Hill’s criterion, defined by the fourth-rank operator H , characterizing the non∼ ∼
damaged material and its symmetries.
1/2 ˜ H − R − k = (σ˜ − X ˜ ) : H : (σ˜ − X ˜) f = σ∼˜ − X −R−k ∼ ∼ ∼ ∼ ∼ ∼ ∼ ˜ ˜ = X with X ∼
∼i
(4.67) (4.68)
i
One chooses as potential associated to plasticity the following additive expression, where the second and third terms are introduced in order to get the dynamic recovery terms for kinematic and isotropic hardening: Fp = f +
1 ˜ ˜ + 1 g R2 γi X :Q:X ∼i ∼ i ∼ 2 2c ∼
(4.69)
i
Using (4.64), the generalized normality rule gives then: ε∼˙ = λ˙ p
˙ = −λ˙ α ∼i
−T : H : (σ˜ − X ˜) M ∼ ∼ ∼ ∼ ∂Fp ∼ −T ∼ ˙ =λ = λ˙ M :n ∼ ∼ ˜ H ∂σ∼ ∼ σ∼˜ − X ∼
N ∼ ∂Fp = λ˙ ∼ ∂X ∼i
−T
−T =N ∼ ∼
(4.70)
˜) :H : (σ∼˜ − X ∼ ∼
∼ −T ˜ λ˙ − γi N :Q:X ∼i ∼ ∼ ˜ H ∼ σ∼˜ − X ∼ ∼
T p ˜ λ˙ : M : ε ˙ − γ Q : X i ∼i ∼ ∼ ∼ ∼
∼
r˙ = −λ˙
∂Fp g = λ˙ − R λ˙ ∂R c
(4.71)
(4.72)
where we used n as the “unit” direction (in a space transformed by Hill’s crite∼ rion): n = ∼
˜) H : (σ∼˜ − X ∼ ∼ ∼
˜ σ∼˜ − X ∼ H
1/2 −1 n : n =1 −1 = n : H H ∼ ∼ ∼ ∼ ∼
(4.73)
In rate-independent plasticity, the multiplier λ˙ will be determined by the consistency condition f = f˙ = 0, possibly coupled with an analogous condition for the damage criterion (this determination will be studied in Sect. 4.6.3). In viscoplasticity, λ˙ can be written, for instance, as a power function:
1/2 f n ˙λ = ε˙ p : M : H −1 : M T : ε˙ p = (4.74) ∼ ∼ ∼ ∼ ∼ K ∼ ∼ ∼
Introduction to damage mechanics
155
3. No damage case Negligible damage leads to the evolution equations of viscoplasticity with kinematic and isotropic hardening, written for the case of the initially isotropic material [NOU90, CHA96a]: : (σ∼ − X ) H ∼ ∼
ε∼˙ p = λ˙ r˙ = λ˙ −
˙ = ε∼˙ p − γi Q : X α λ˙ ∼i ∼i
∼
σ∼ − X ∼ H
g ˙ Rλ c
(4.75)
∼ ∼
n
1 f −1 p 2 : ε ˙ = λ˙ = ε∼˙ p : H ∼ ∼ K ∼
(4.76)
and α , we get, in isothermal and, taking into account the relation between X ∼i ∼i conditions: ˙ = C i : α˙ = C i : ε˙ p − γi C i : Q : Xi λ˙ X ∼i ∼ ∼i ∼ ∼ ∼ ∼ ∼
∼
R˙ = (c − gR)λ˙
∼ ∼
∼
(4.77)
In the case of the isotropic material, taking the following expressions for the various operators: 3 d 3 1 = I∼ = I − 1⊗1 H ∼ 2∼ 2 ∼∼ 3 ∼ ∼ ∼
= C ∼ ∼
2 d I 3 ∼∼
Q= ∼ ∼
1 d I ci ∼∼
(4.78)
we get the classical laws, already mentioned in Chap. 3, Sect. 3.7. Note: one can write the criterion (4.67) in a slightly different but equivalent −1 : X ) : H ˜ : (σ − M : N −1 : X ) where H ˜ = : N form, using (σ∼ − M ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
−T : H : M −1 . With the energy equivalence, or if C and are collinear M ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
˜ : (σ − X ), where H ˜ is suppress Hill’s = N ), we get (σ∼ − X ) : H (then M ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
operator of the damaged material.
∼
∼
4.4.5. Advantages and drawbacks of both types of equivalence In what precedes, the notion of effective stress plays a crucial part in defining the plasticity/viscoplasticity coupled with damage without introducing new material parameters. On the contrary, we could have used operators determined independently for Hill’s criterion and for the dynamic recovery term of kinematic hardening, . . .(i.e., ˜ , etc). However, it is subject to some constraints or limitations, which we can ˜ ,Q H ∼ ∼
∼ ∼
briefly evoke, considering only the kinematic hardening:
156
Non-Linear Mechanics of Materials
1. Energy equivalence allows us to write the effective stiffness as an automatically symmetrical product ˜ = M : : MT (4.79) ∼ ∼ ∼ ∼ ∼
∼
∼
∼
where M = I∼ − D is the damage effect tensor. Therefore, one can choose the ∼ ∼ ∼
∼
∼
same kind of relationship for each of the fourth-rank operators involved in the different equations, for instance: ˜ = M : C : MT C ∼ ∼ ∼ ∼ ∼
∼
∼
˜ = M −T : H : M −1 H ∼ ∼ ∼ ∼ ∼
∼
∼
∼
˜ = M −T : Q : M −1 Q ∼ ∼ ∼ ∼
∼
∼ ∼
∼
(4.80)
∼
and remain consistent with effective variables (with M =N ): ∼ ∼ ∼
−1 : σ∼ σ∼˜ = M ∼ ∼
∼
˜ = M −1 : Xi X ∼i ∼ ∼
(4.81)
T ˜ =M α :α ∼i ∼ ∼i
(4.82)
∼
T ε∼˜ = M : ε∼ ∼ ∼
∼
The plasticity criterion and the dissipation potential are then written ˜ − k = σ − X ˜ − k f = σ∼˜ − X ∼ H ∼ H ∼ Fp = f +
1 ˜ : Xi γi X :Q ∼i ∼ ∼ 2 ∼
(4.83) (4.84)
i
and the normality rules provide: ε∼˙ p = λ˙
˜ : (σ − X) H ∼ ∼ ∼ ∼
σ∼ − X ∼ H˜
˙ = λn ∼
˜ : Xi λ˙ ˙ = ε∼˙ p − γi Q α ∼ ∼i ∼ ∼
(4.85)
where n has the same expression (4.73) as previously. Actually, through en∼ ergy equivalence, one could just as well write the dissipation potential and the corresponding normality rules in an “effective space”: ˜ −k f = σ∼˜ − X ∼ H
Fp = f +
1 ˜ ˜ γi X :Q:X ∼i ∼i ∼ 2 ∼
(4.86)
i
˜) H : (σ∼˜ − X ∼ ∼ ∂Fp ˙ = λ˙ ∼ = λn ε∼˙˜ p = λ˙ ∼ ˜ ∂ σ∼˜ σ∼˜ − X ∼ H
(4.87)
˙˜ = −λ˙ ∂Fp = ε˙˜ p − γi Q : X ˜ λ˙ α ∼i ∼i ∼ ∼ ∂X i ∼ ∼
(4.88)
Introduction to damage mechanics
157
T :ε ˙˜ = M T : α˙ , similar to (4.66), with the following relations ε∼˙˜ p = M ˙ p and α ∼ ∼ ∼i ∼ ∼i ∼
∼
and then passing in real space by applying the inverse relations of (4.66). One would then benefit from the dissipation equality in both spaces: ˙˜ p − ˜ : α˙˜ i ≥ 0 ˙ int = σ∼ : ε∼˙ p − X : α = σ ˜ : ε (4.89) X i ∼ ∼i ∼i ∼ ∼ ∼ i
i
Unfortunately relations such as (4.66) cannot be derived without incorporating the terms due to the variations of M and then this notion of effective space ∼ ∼
does not bring much (ε∼˙˜ p in (4.87) is not the derivative of ε∼˜ p ). Moreover, we have already mentioned this fact, energy equivalence hypothesis is in itself quite restrictive regarding anisotropy effects induced by damage, in elasticity (see point 2 of Sect. 4.3.3) as in plasticity/viscoplasticity. 2. Strain equivalence is a bit more tricky to carry out, because of the non-natural ˜ and C˜ . For instance: symmetrization of tensors ∼ ∼ ∼
∼
˜ = − = − D : ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
s
∼
∼
∼
∼
∼
∼
s
˜ = C − C = C − D : C C ∼ ∼ ∼ ∼ ∼ ∼
(4.90) (4.91)
−1 : σ and X ˜ = N −1 : Xi that involve lead to effective stresses σ∼˜ = M ∼i ∼ ∼ ∼ ∼ ∼
∼
two damage effect operators, M and N , in principle different from one another ∼ ∼ ∼
∼
(see (4.65)). However, this hypothesis does not bring any restriction concerning the effect of damage on the elastic behavior. Moreover, in the particular case and are collinear, for example if C = (c/ l) , we retrieve where the tensors C ∼ ∼ ∼ ∼ ∼
∼
∼
∼
M ≡ N and the relationships above (4.87) and (4.88) of energy equivalence ∼ ∼ ∼
∼
T : ε ˙˜ = M T : α˙ are not the derivatives of ε˜ p = ε p (where ε∼˙˜ p = M ˙ p and α ∼ ∼ ∼i ∼ ∼i ∼ ∼
and α˜ i = α ). ∼i
∼
∼
4.4.6. Asymptotic behavior near fracture In this section we consider the evolution of stress and hardening variables near the RVE fracture, when D tends towards 1. As an illustrative exercise, we check that, with approaches used, the (actual, Cauchy) stress tends necessarily towards 0. This point is important for numerical reasons, especially for applications of damage mechanics with local approaches to fracture. We consider rate-independent plasticity and use isotropic damage versions to simplify explanations. In an anisotropic case, only some stress components would cancel,
158
Non-Linear Mechanics of Materials
depending on the chosen fracture criterion. Of course, this demonstration is valid only as long as we do not take into account the possible bifurcation and localization phenomena that occur in structural analysis. We only examine here the “fundamental” solution of the isolated RVE. We conduct three analyses in parallel, by strain equivalence, energy equivalence and using a law without coupling on the hardening terms (approach followed by and C are Lemaître [LEM96]). To simplify, we assume that elasticity operators ∼ ∼ ∼
∼
collinear and we use only one nonlinear kinematic hardening variable. There is no isotropic hardening. In the framework of rate-independent plasticity we use an elasticity domain and a dissipation potential of the form:
1/2 ˜ − k = (σ˜ − X ˜ ) : H : (σ˜ − X ˜) −k f = σ∼˜ − X ∼ H ∼ ∼ ∼ ∼ ∼
(4.92)
1 ˜ ˜ : C −1 : X Fp = f + γ X ∼ 2 ∼ ∼∼
(4.93)
∼
−1 for the dynamic recovery term (compare with (4.67) where we have chosen Q = C ∼ ∼ ∼
∼
and (4.69) in Sect. 4.4.4). • Exercise 1: strain equivalence: The effective stresses are given by: −1 σ∼˜ = : (ε∼ − ε∼ p ) = M : σ∼ ∼ ∼ ∼
∼
˜ = C : α = M −1 : X X ∼ ∼ ∼ ∼ ∼ ∼
∼
(4.94)
where the damage effect operator is given by (4.65): ˜ : −1 = C˜ : C −1 = M ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
(4.95)
From the potential Fp we deduce the rates: ε∼˙ = λ˙ p
˙ = −λ˙ α ∼
−T : H : (σ˜ − X ˜) M ∼ ∼ ∼ ∼ ∂Fp ∼ ∼ ˙ ˙ −T : n =λ = λM ∼ ∼ ˜ H ∂σ∼ ∼ σ∼˜ − X ∼
∂Fp −T −1 ˜ ˙ = ε∼˙ p − γ M :C :X λ ∼ ∼ ∼ ∂X ∼ ∼ ∼ −T −T :α : n − γα = ε∼˙ p − γ M λ˙ = M λ˙ ∼ ∼ ∼ ∼ ∼ ∼
∼
(4.96)
(4.97)
−1 : n = 1). Let us assume gives the unit direction of plastic flow (n : H n ∼ ∼ ∼ ∼ ∼
a monotonic loading. When plastic strain and damage increase indefinitely,
Introduction to damage mechanics
159
Figure 4.13. Damage-plasticity coupling under tension (X and σ ) Parameters: E = 200000, C = 100000, k = 200, γ = 500, εpR = 0.02
→ 1/γ (whatever damage evolution λ˙ → ∞ and stabilization occurs for α ∼ ˜ = C : α , the norm of the ). As a result, as X is, written in the operator M ∼ ∼ ∼ ∼ ∼
∼
˜ actually tends towards a scalar value such as C/γ , as for the effective tensor X ∼ case without damage (Fig. 4.13). Let us examine the rate independent criterion. During plastic flow, we have necessarily: ˜ = n : (σ˜ − X ˜)=k σ∼˜ − X ∼ H ∼ ∼ ∼
(4.98)
i.e., ˜ + kH −1 : n σ∼˜ = X ∼ ∼ ∼ ∼
−1 or σ∼ = X + kM :H :n ∼ ∼ ∼ ∼ ∼
∼
(4.99)
by (1 − D)I∼ . We apply now the isotropic damage assumption, substituting M ∼ ∼
We find: −1 σ∼ = X + k(1 − D) H :n ∼ ∼ ∼ ∼
∼
(4.100)
Approaching fracture, when D tends towards 1, we do get the foretold result: ˜ and X ˜ tends towards C/γ . the norm of X tends towards 0, as X = (1 − D)X ∼ ∼ ∼ ∼ As a result the norm of σ∼ tends also towards 0. This evolution is schematized for uniaxial tension in Fig. 4.13, for both, effective and actual stresses.
160
Non-Linear Mechanics of Materials
• Exercise 2: energy equivalence: The relationships (4.94) are completed by: T : ε∼ e ε∼˜ e = M ∼ ∼
T ε∼˜ p = M : ε∼ p ∼ ∼
T ˜ =M :α α ∼ ∼ ∼ ∼
(4.101)
but we no longer have (4.95) as above, and the state laws are: ˜ : εe σ∼ = ∼ ∼
˜ :α X =C ∼ ∼ ∼
(4.102)
σ∼˜ = : ε∼˜ e ∼
˜ = C : α˜ X ∼ ∼ ∼
(4.103)
∼
∼
∼
∼
Let’s note that the damage effect operator, still a fourth-rank tensor, does not have the same role anymore, and should rather be considered as homogeneous to the “square root” of the one used in strain equivalence. The rate equations are now: ˙ −T : n (4.104) ε∼˙ p = λM ∼ ∼ ∼
−T −1 ˜ ˙ ˙ = ε∼˙ p − γ M :C :X α λ ∼ ∼ ∼ ∼ ∼ ∼ −T ˜ λ˙ : n − γα = ε∼˙ p − γ α λ˙ = M ∼ ∼ ∼ ∼ ∼
(4.105)
˜ whose norm tends towards 1/γ when plastic strain and damage inNow, it is α ∼ ˜ tends towards C/γ . Of course, relations (4.99) crease. Moreover, the norm of X ∼ remain valid and, in the isotropic damage case, X and σ∼ cancel when D → 1. ∼ The evolutions during a tensile test are analogous to the ones observed in the previous case. Curves of Fig. 4.13, in thin lines, were drawn assuming an evolution of damage as a function of plastic strain, given by D = (εp /εpR )2 and by proceeding, for the energy equivalence case, to a step by step integration, given in Exercise 4. • Exercise 3: theory without damage-hardening coupling In the case of the theory used by Lemaître [LEM96], the effective stress notion is not used for the kinematic hardening variable (see Sect. 4.4.3(i)). We show here that it gives globally analogous results, illustrated in Fig. 4.14. We will only consider the isotropic case, with relation (4.56) for the elasticity domain. The evolution law is then given by [LEM96]: ε∼˙ p =
λ˙ n 1−D∼
˙ = (1 − D)˙ε∼ p − γ α α λ˙ − γα λ˙ = n ∼ ∼ ∼ ∼
(4.106) (4.107)
Introduction to damage mechanics
161
Figure 4.14. Plasticity-damage coupling in tension (X and σ ); parameters: E = 200000, C = 50000, k = 200, γ = 200, εpR = 0.02
When D → 1 and plastic strain increases indefinitely, the norm of variable α ∼ tends towards 1/γ but in that case X = 23 Cα also has a norm that tends towards ∼ ∼ C/γ . However, the flow criterion applies with: 2 2 (4.108) kn kn + and σ = (1 − D) X + σ∼˜ = X ∼ ∼ ∼ 3 ∼ 3 ∼ So σ∼ still tends towards 0 when D tends towards 1, as shown in Fig. 4.14. Let us note that the stress results incurred by this last approach are very close to the one provided by the first one, with hardening-damage coupling. • Exercise 4: integration We want to get the response of the three models above in monotonic tension. In all three cases we assume that damage evolves as D = (εp /εpR )2 . In the strain equivalence case, integration of the kinematic hardening law is entirely analytic. In tension, (4.97) provides: α˙ = (1 − γ α)˙εp
(4.109)
1 1 − exp(−γ εp ) γ
(4.110)
After integration we find: α=
162
Non-Linear Mechanics of Materials
C X = (1 − D)Cα = γ
1−
εp εpR
2
1 − exp(−γ εp )
σ = X + k(1 − D)
(4.111) (4.112)
With the other approaches, integration is not analytical, as we have, respectively:
√ (4.113) α˙ = 1 − γ 1 − Dα ε˙ p and α˙ = (1 − D) (1 − γ α) ε˙ p We find an incremental approximation by setting D piecewise constant. For example, for exercise 2, between εp = εpi and εp = εpi+1 we integrate analytically, with γi = γ (1 − (εp2 i + εp2 i+1 )/(2 εp2 R ))1/2 . We find then easily (αi being the value at the beginning of the increment, coming from the previous calculation): 1 1 αi+1 = − − αi exp −γi (εpi+1 − εpi ) (4.114) γi γi Curves in Figs. 4.13 and 4.14 were plotted this way.
4.5. Damage deactivation In the previous sections we developed damage theories, in elasticity as well as in plasticity, by considering only the active damages, i.e., the effect of “opened” microcracks on the mechanical behavior. It is clear that, under the load applied to the material’s RVE, during unloading or cycles for example, some microcracks can close up and their impact on the behavior lowered or even canceled. In such a case we refer to deactivated damage. We address in this section the possible descriptions of these deactivation phenomena. We will remain at the phenomenological scale of a macroscopic damage mechanics approach. We mention the initial theory, proposed by Ladevèze and Lemaître [LAD84], limited to the isotropic damage case. Then we show some of the problems that arise when one takes into account deactivation in the anisotropic damage case. Figure 4.15 illustrates the deactivation problem in the case of the elastic damageable material (without residual strain), with a higher strength in compression, as in the case of concrete. Elastic states a, c, f, d, e, i, j are fully described by the state laws. On the contrary, states b, g, h, k are dissipative, damage evolves, inducing a nonlinearity of the stress-strain response. The deactivation problem is essentially due to change in elastic behavior (change of elastic modulus). This is illustrated in the figure by the transition from c (elastic unloading after tension, with a damaged elastic modulus) to d, elasticity in compression, this occurring at constant damage. Damage deactivation
Introduction to damage mechanics
Figure 4.15. compression
163
Scheme of the elastic behavior with damage and unilateral effect in tension-
induces an elasticity modulus increase, which recovers its initial value, as shown on the figure. The main difficulty of modeling is the bilinear character of elasticity law.
4.5.1. Deactivation in the isotropic damage case The first approach to damage deactivation was proposed by Ladevèze and Lemaître [LAD84]. It applies (with a scalar variable D) in the case of the elastic material, initially isotropic, and that remains isotropic while being damaged. The basic idea is that a tension stress (positive principal stress, for instance σ1 > 0) opens the microcracks, hence the damage D applies entirely, giving the following effective principal stress: σ1 (4.115) if σ1 > 0 σ˜ 1 = 1−D To the contrary, if the principal stress is negative, the damage effect is partly deactivated. Then, by introducing a material parameter 0 ≤ h ≤ 1, generally set to 0.2, it comes: σ1 (4.116) if σ1 < 0 σ˜ 1 = 1 − hD This criterion is generalized for the 3D case by using the decomposition of a stress tensor into positive and negative parts: σ∼ = σ∼ + + σ∼ −
(4.117)
164
Non-Linear Mechanics of Materials
This operation is perfectly well defined in the principal stress direction. It can be written in a general form using spectral decomposition, with the principal unit vectors of σ∼ (p 1 , p 2 , p 3 ): 3 σi p i ⊗ p i (4.118) σ∼ = i=1
σ∼ + =
3
σ∼ − =
H (σi )σi pi ⊗ p i
i=1
3
H (−σi )σi p i ⊗ p i
(4.119)
i=1
where H ( ) is Heaviside’s function. Moreover, we have to decompose the trace of the stress tensor: Tr(σ∼ ) = Tr(σ∼ ) − − Tr(σ∼ ) = σkk − −σkk . We will not detail here the thermodynamic potential used by Ladevèze and Lemaitre [LAD84], nor the bilinear elasticity law that follows. For more details, one can refer to [LAD83, LAD84, LEM96]. Such an approach allows us to build a bilinear-type unilateral elasticity law that respects symmetries of the elasticity operator and ensures continuity of stress-strain relationships whatever the loading is. For application to concrete, it has been generalized with two scalar damage variables, Dt and Dc , varying respectively with tensile and compression paths (see for example [MAZ89]).
4.5.2. Difficulties associated with anisotropy In the anisotropic case, whether it be for initial or for induced anisotropy, the decomposition of a stress tensor into positive and negative parts does not authorize the corresponding decoupling of energies. In particular, one cannot write the elastic energy under a form similar to: ρψe∗ =
1 + ˜+ 1 − σ∼ : S∼ : σ∼ + + σ∼ − : S∼˜ : σ∼ − 2 2 ∼ ∼
(4.120)
where σ∼ = σ∼ + + σ∼ − would be the spectral decomposition already mentioned, ψe∗ + − the complementary free energy and S∼˜ and S∼˜ the compliance tensors associated ∼
∼
respectively with active and inactive damage. In fact, in (4.120) some crossed terms ± would be missing, such as σ∼ − : S∼˜ : σ∼ + , necessary, for example, to ensure that we ∼
would recover the non-damaged case. ρψe∗ =
1 σ :S:σ 2 ∼ ∼∼ ∼
(4.121)
Some theories developed to take into account damage deactivation phenomenon in the anisotropic case lead to anomalies that must be considered unacceptable [CHA92a]:
Introduction to damage mechanics
165
• either the compliance (or stiffness) tensor can become non-symmetrical in some situations [RAM90], which shows that it does not derive from a potential, • or some (multiaxial and non-proportional) loading can generate discontinuities in the stress-strain response. It is the case for models proposed by Ju [JU89] or by Krajcinovic [KRA81]. Without going into the details of the corresponding theories, Table 4.1 [CHA93b] exemplifies these difficulties. We consider the case where principal directions of damage and strain coincide. We will only consider a biaxial loading. Krajcinovic and Fonseka’s theory affects the elastic stiffnesses via the values ω1 , ω2 , associated with microcracks alongside the orthogonal axis 1 and 2 (part (A), for ε1 > 0, ε2 > 0). When one of the principal strains becomes negative, for example ε1 < 0, the microcrack ω1 is expected to close. The theory assumes then that the corresponding term disappears (ω1 = 0) from the equations of elastic behavior. Unfortunately, we notice then, in part (B) the discontinuity of the term C12 , hence the discontinuity of σ11 when the sign of ε1 changes, as long as ε2 is not zero. The same kind of difficulties arise for the two other theories indicated in the table. To the contrary, the last formulation [CHA92b, CHA93b] preserves continuity. Indeed, only the coefficient in front of ε1 = 0 is affected by deactivation (which occurs precisely for ε1 = 0). This theory is introduced below.
4.5.3. A deactivation criterion that preserves continuity 1. General modeling framework ˜ (d ) when The difficulty resides in the choice of a general multiaxial writing of ∼ ∼ ∼
d∼ has privileged directions. The proposed formulation has two peculiarities: – the unilateral condition is written with special coordinates, linked to the current material (anisotropic because of damage), called damage principal frame; – in this frame the damage unilateral character affects only the diagonal component of stiffness (respectively compliance) corresponding to the damage principal direction, for which the sign of the associated normal strain (respectively normal stress) changes. In what follows, in elasticity, we assume that total strain ε∼ is the elastic strain (no residual strain). The detailed formulation, with decomposition of the stiffness and a deformation criterion, is introduced as follows: • let us assume that we know the first “principal direction of damage”, represented by the unit vector p, i.e., the direction of maximum damaging
166
Non-Linear Mechanics of Materials
(A)
ε1 > 0
ε2 > 0
Vectors
λ + 2μ + 2(C1 + C2 )ω12
λ + C1 (ω12 + ω22 )
(Krajcinovic–Fonseka)
λ + C1 (ω12 + ω22 )
λ + 2μ + 2(C1 + C2 )ω22
2nd-rank tensors
h+2 11 + λ(1 − δ)
h+2 12 + λ(1 − δ)
(Ramtani)
h+2 12 + λ(1 − δ)
h+2 22 + λ(1 − δ)
(λ + 2μ)(1 − d)
λ(1 − d)
λ(1 − d)
(λ + 2μ)(1 − d)
New formulation:
(λ + 2μ)(1 − d1 )2
λ(1 − d1 )(1 − d2 )
2nd-rank tensors
λ(1 − d1 )(1 − d2 )
(λ + 2μ)(1 − d2 )2
ε1 < 0
ε2 > 0
Vectors
λ + 2μ
λ + C1 ω22
(Krajcinovic–Fonseka)
λ + C1 ω22
λ + 2μ + 2(C1 + C2 )ω22
2nd-rank tensors
h−2 11 + λ(1 − δ)
h+2 12 + λ(1 − δ)
(Ramtani)
h−2 12 + λ(1 − δ)
h+2 22 + λ(1 − δ)
4th-rank tensor (Ju)
λ + 2μ
λ
λ
(λ + 2μ)(1 − d)
New formulation:
λ + 2μ
λ(1 − d1 )(1 − d2 )
2nd-rank tensors
λ(1 − d1 )(1 − d2 )
(λ + 2μ)(1 − d2 )2
4th-rank tensor (Ju)
(Cordebois–Sidoroff) (B)
(Cordebois–Sidoroff)
Table 4.1. Application of various damage deactivation criteria in biaxial deformation: illustration of discontinuities with classical models
effect. Let us consider that the material behavior for an “active damage” ˜ (d ), known (all microcracks open) is described by elastic-stiffness tensor ∼ ∼ for each damage state, and relation (4.62);
∼
• the damage is active (effective for the considered direction) if the normal strain εn = p.ε∼ .p is positive; • to the contrary, if εn is negative the corresponding damage becomes “in-
Introduction to damage mechanics
167
active” (for the direction p). Using Voigt’s notation to express the matrix ˜ 1111 must be replaced by 1111 (p is direcof the elasticity operator, tion 1) without affecting the non-diagonal terms, nor the shearing terms. The effective stiffness tensor can be written: eff ˜ (d ) + (1 − H (εn )) = ∼ ∼ ∼ ∼
∼
˜ (d ) : (p ⊗ p) p ⊗ p ⊗ p ⊗ p − (p ⊗ p) : ∼ ∼ ∼ ∼
(4.122)
∼
H being Heaviside’s function; • actually, using the fourth-rank projection tensor P∼ = p ⊗ p ⊗ p ⊗ p
(4.123)
∼
we can write (4.122) in a shorter way:
eff ˜ (d ) + H Tr −P : ε P : − ˜ (d ) : P = ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
(4.124)
∼
where Tr() is the trace operator; • in the case where three “principal directions of damage” have been defined (perpendicular directions), it is straightforward to generalize (4.124): eff ˜ (d ) + η = ∼ ∼ ∼ ∼
∼
3
˜ (d ) : P i (4.125) H Tr −P∼ i : ε∼ P∼ i : − ∼ ∼ ∼ ∼ ∼
i=1
∼
∼
∼
∼
where the P∼ i are the projection operators p i ⊗ p i ⊗ p i ⊗ p i . η is a ∼
material phenomenological coefficient that allows describing a potential residual effect of damage in a compression type situation (as h in relationship (4.116)). Note that the unilateral condition described in (4.125) affects only the principal diagonal terms (1111 , 2222 , 3333 ) of the elastic stiffness tensor written in the chosen frame. The fact that the non-diagonal terms are not affected by the deactivation condition is the key that allows ensuring simultaneously the symmetry of the elasticity tensor and the continuity of the stress-strain response whatever the applied loading is. Then, writing the material elastic behavior in the thermodynamic framework is made without a problem. It suffices to use free energy under the form: ρψ =
1 ε : eff : ε∼ 2 ∼ ∼∼
(4.126)
168
Non-Linear Mechanics of Materials
2. Choosing the “damage principal directions” The formulation can be adapted to several kinds of damage description and the definition of the “damage principal directions” remains free. Three possibilities can be considered: • if damage is described by a second-rank tensor, we will take naturally the principal directions of this tensor; • if damage is described by a fourth-rank tensor we could use its orthotropic ˜ (D ). But these operators do not remain principal directions or that of ∼ ∼ ∼
∼
orthotropic so we suggest introducing the “damage principal directions” as being the ones that minimize Young’s modulus in an uniaxial tensile test (which, we assume, is conducted in every direction of space); • a third proposition consists in using directly the principal strain directions, as in classical theories [ORT85a, JU89]. At any loading point we write ˜ (d ) in the principal frame of ε and, if the sign of one principal then ∼ ∼ ∼ ∼
strain changes, the corresponding diagonal term is modified according to relation (4.125). The behavior of this formulation is illustrated in biaxial conditions in the last row of Table 4.1, in which one uses the formulation of Cordebois and Sidoroff [COR79]). The part (B) show how, when ε1 becomes negative, only the term 1111 is changed, by canceling the effect of d1 . Of course, this preserves the stress response continuity.
4.5.4. Consequences for plasticity coupled to damage One can wish to simultaneously model plasticity (or viscoplasticity) coupled to damage, and the elastoviscoplatic behavior in the situation of deactivated damage. This situation occurs for example in Low Cycle Fatigue of metals or when metallic matrix composites are cyclically loaded. The main difficulty is then to simultaneously take into account two aspects: • strain at deactivation and plastic strain are not necessarily equal. In other words, deactivation does not necessarily take place because the stress sign changes. In some cases, this closing strain can be considered constant, in fact it depends on the production process of the material (associated with manufacturing residual stresses). Sometimes it varies, but not in direct correlation with plastic strain; • deactivation mentioned in the previous sections dealt with the elastic behavior. Now, it must act also upon plastic or viscoplastic behavior, upon yield strength criterion, hardening law, . . .
Introduction to damage mechanics
169
This area of modeling is still a wide open research topic. Some propositions and some attempts have been made, but problems and sticking points remain. Thus, we will not study here this delicate matter. Some elements will be given in Sect. 4.7 on the application to brittle materials. Note that some recent works, based on a micromechanical approach, lead to a different deactivation rule, less restraining, but accompanied by an elastic energy storage (see for example references [AND86, BOU96, CHA02]).
4.6. Damage evolution laws We address here the general forms of criteria and evolution equations usually used to describe damage growth. We can sort them in three categories: (i) the ones coming from a purely-standard thermodynamic approach (standard generalized materials, SGM), such as the one used by Lemaître [LEM96], (ii) the ones that follow from a “quasi-standard” formulation, in which one differentiates, for dissipation, plastic potential from damage potential and the corresponding multipliers, (iii) the ones, finally, that do not refer to a dissipation potential. Below, we distinguish between rate-independent and rate-dependent models: the former are used for ductile fracture of metallic materials, or for brittle materials, concrete and composites whereas the latter are associated with creep and viscoplasticity of metals. Some applications will be studied in Sect. 4.7. The purely standard approach (i) does not need any particular development. It is simply defined with what was introduced in Sect. 4.4.2(i). One can refer to [LEM96] for its use, in particular concerning the determination of the common multiplier λ˙ p = λ˙ d .
4.6.1. Rate-independent normality law In what follows, unless otherwise stated, we will be considering a damage variable d∼ , second-rank tensor (associated thermodynamic force y ). In some applications we ∼ will limit to a scalar variable D (associated force Y ). We follow the pseudo-standard formalism already described in Sect. 4.4.2(ii), with two independent criteria:
(4.127) fd y , B; d∼ , β ≤ 0 fp σ∼ , Aj ; d∼ ≤ 0 ∼
that demarcate respectively, in stress space and in the space of damage thermodynamic forces, the elasticity (non-plasticity) and the non-damage domains. These functions can be parametrized with the state variables d∼ and β. They also depend on temperature, even though it is not explicitly indicated. We consider also two independent
170
Non-Linear Mechanics of Materials
pseudo-potentials:
Fd y , B; d∼ , β
Fp σ∼ , Aj ; d∼
∼
(4.128)
The normality rule is posed assuming that plasticity occurs in a “quasi associated” way, i.e., the potential Fp is the sum of fp and a term uniquely related to hardening (to produce the dynamic recovery terms) while damage evolves in a purely “associated” way: Fd ≡ fd (4.129) Fp = fp + q(Aj ; d∼ ) With the normality laws (4.53) and (4.54), involving two independent multipliers λ˙ p and λ˙ d , we can write: ε∼˙ p = λ˙ p
∂fp ∂σ∼
α˙ j = −λ˙ p
∂fd d∼˙ = λ˙ d ∂y
∂fp ∂q − λ˙ p ∂Aj ∂Aj
β˙ = −λ˙ d
∼
∂fd ∂B
(4.130)
(4.131)
The multipliers λ˙ p and λ˙ d will have to be found by solving the consistency conditions fp = f˙p = 0 and fd = f˙d = 0. Recalling Fig. 4.12, they are determined independently when plasticity operates alone (fd < 0 → λ˙ d = 0) or when purely brittle damage occurs without plastic strain (fp < 0 → λ˙ p = 0). Should both processes occur simultaneously, one will have to consider the possibility of coupling between both. Note also that multipliers λ˙ p and λ˙ d must be positive. Otherwise, one considers that there is unloading, either for the plasticity criterion or for the damage criterion (and one writes respectively λ˙ p = 0 or λ˙ d = 0).
4.6.2. Form of the non-damage criterion 1. Criterion with the variables B and β The most common expression, copied from the plasticity case, uses a norm of the damage thermodynamic force. For example, if y is a second-rank tensor, ∼ one could take [CHO87, CHO89, ZHU95]: 1/2 − B − kd ≤ 0 fd = y J − B − kd = y : J∼ : y ∼
∼
∼
∼
(4.132)
in this section, J∼ is a fourth-rank tensor depending on the material. In this form ∼
of criterion, proposed initially by Cordebois and Sidoroff [COR79, COR82], and used by many authors [CHO87, ZHU95], the thermodynamic force B, associated with the accumulated damage β, describes the size increase (assumed
Introduction to damage mechanics
171
isotropic) of the non-damaging domain, whose initial size is given by kd . The normality rules (4.131) give in this case: d∼˙ = λ˙ d
J∼ : y ∼
∼
β˙ = λ˙ d
y J
(4.133)
∼
We can see that β plays the role of accumulated damage (like p in plasticity is the accumulated plastic strain). Here we have: 1 2 β˙ = λ˙ d = d∼˙ : J∼ −1 : d∼˙
(4.134)
∼
The damage multiplier is easily expressed in cases where there is no plasticity (elasticity-damage coupling only). The consistency condition f˙d = 0 gives then: (y : J∼ : y˙ ) 1 ∼ ∼ ∼ λ˙ d = (4.135) B (β) y J ∼
Of course, the expression as a function of ε∼˙ is harder to extract, as it depends on ˜ , which is delicate for the relationship between damage and elasticity operator ∼ ∼
an anisotropic damage. Still for the purely elastic case, assuming that y = ρ ∼
∂ψ and using the state law σ∼ = ρ ∂ ε , one can get the damage multiplier: ∼ λ˙ d =
∂ 2ψ 1 ∂ψ ∂ 2ψ 1 ∂ψ : J∼ : : ε∼˙ = Jij kl ε˙ rs H ∂d∼ ∼ ∂d∼ ⊗ ∂ε∼ H ∂dij ∂dkl ∂εrs
∂ψ ∂d ∼
(4.136)
with H = B (β) y J + ρy : J∼ : ∂ 2 /∂d ⊗ ∂d : J∼ : y / y J . ∼
∼
∼
∼
∼
∼
For an isotropic damage, variable y reduces to the scalar Y and operator J∼ is ∼
chosen to be identity. We get then a simpler expression: λ˙ d =
(ε∼ : : ε∼˙ ) ∼ σ˜ : ε˙ Y˙ ∼ = = ∼ ∼ B (β) B (β) B (β)
∼
(4.137)
Determining multiplier λ˙ d in the plasticity coupling case is addressed in Sect. 4.6.3. Note that this theory using variables B and β is not neutral vis-a-vis the dissipation associated with damage. B being generally positive, relation (4.41) shows that free energy increases with β, the accumulated damage variable, whereas it decreases with damage playing a role in elasticity and hardening terms. Then, intrinsic dissipation associated with damage processes holds two terms: d = y : d∼˙ − B β˙ ∼
(4.138)
172
Non-Linear Mechanics of Materials
The first one is dissipated by damage, the second corresponds to this energy storage (by β). However, damage remains positive. By choosing (4.132) above, we easily find: y : J : y
∼ ∼ ∼ ∼ − B λ˙ d = y J − B λ˙ d = kd λ˙ d ≥ 0 (4.139) d = ∼ y J ∼
and we notice that it is the initial kd that dissipates to heat while damage grows. 2. Criteria parameterized with state variables As much as energy stored by hardening can seem natural, the one stored by damage seems more questionable. Therefore, one could prefer a theory that does not introduce the variable β as state variable, but memorizes continuously the (maximal) size reached by the non-damage criterion. It is the procedure used by Ladevèze [LAD83], Allix [ALL90], Simo and Ju [SIM87] and many others. The criterion is written fd = (y ) − ω ≤ 0 ∼
(4.140)
and the evolution is submitted to the conditions: ∂fd d∼˙ = λ˙ d ∂y
λ˙ d = ω˙
(4.141)
∼
which is equivalent to considering that ω, instead of being associated with the accumulated damage, is the “memory” of the maximal size reached by the criterion. Indeed, if fd < 0 there is no evolution. On the contrary, if λ˙ d > 0, i.e., if there is damage growth, the consistency condition imposes: (4.142) ω(t) = max ωo , max (y (τ ) ) τ ≤t
∼
More generally, the size of the non-damage criterion can be made dependent on the damage variables themselves, considered as parameters: fd = (y ) − ω(d∼ ) ≤ 0 ∼
(4.143)
Function (y ) can be chosen similarly to (4.132), with y J , norm of y . We ∼ ∼ ∼ will see some examples related to composites in Sect. 4.7. 3. Effect of deactivation The effect of partial or full damage deactivation (microcracks closing) evoked in Sect. 4.5 for the elastic behavior, may induce a change of damage evolution itself. In principle this effect is introduced naturally through the thermodynamic force y , whose expression is then automatically modified. In this way, we arrive ∼
Introduction to damage mechanics
173
at the criteria by which tensile (σ∼ = σ∼ + ) and compressive (σ∼ = σ∼ − ) type damaging loading may be distinguished. Some models of this kind have been proposed for applications to concrete and rocks [ORT85a, SIM87, JU89]. We can also mention the simplified approach based on two scalar variables, Dt and Dc , one evolving with positive and the other one with negative stresses [MAZ86, MAZ89]. In addition, in damage criteria we will have to eliminate situations in which dissipation would become negative. This can happen in some theories based on scalar variables δα (composites, for example) for which the associated forces can become negative (in this case dissipation would be yα δ˙α < 0). In damage criteria one uses then yα instead of yα .
4.6.3. Consistency condition It is now possible to study entirely the consistency condition of the rate-independent models when plasticity and damage play a role simultaneously. We address only the situation in which the free energy corresponding to hardening depends also on damage, as in Sect. 4.4.1(iii). We consider only the strain equivalence hypothesis to define the effective stress. We show that, in this case, determinations of plasticity and damage multipliers are decoupled, at least for total-strain controlled loadings, which are the most important ones regarding numerical applications (“displacement based” finite element codes). Otherwise, if only the elastic part of free energy is affected by damage, the same situation of total strain control leads to a coupling in the determination of both multipliers λ˙ p and λ˙ d . We then have to solve a linear system, and more complex expressions. We make the following simplifying assumptions: • kinematic hardening only is considered, with one variable; • damage is assumed entirely active. The same conclusions would apply if it were not the case, but with lengthy calculations; • damage itself is assumed isotropic, the thermodynamic force being denoted by Y ; the damage effect operator is then M = (1 − D)I∼ . ∼ ∼
∼
In the framework of the simplifying assumptions enunciated above, free energy writes: ) (4.144) ψ = (1 − D) ψeo (ε∼ − ε∼ p ) + ψpo (α ∼ and the state laws: σ∼ = ρ
∂ψ ˜ : (ε − εp ) = (1 − D) : (ε − εp ) = ∼ ∼ ∼ ∼ ∼ ∼ ∂ε∼ ∼ ∼
(4.145)
174
Non-Linear Mechanics of Materials
X =ρ ∼
∂ψ ˜ : α = (1 − D)C : α =C ∼ ∼ ∼ ∼ ∂α ∼ ∼ ∼
(4.146)
so that:
σ∼ : (ε∼ − ε∼ p ) = ∼ 1−D ∼ ∼ X ˜ = M −1 : X = ∼ = C : α X ∼ ∼ ∼ ∼ ∼ 1−D ∼ ∼ The thermodynamic force associated with damage holds two terms: Y = Ye + Yp = ρ ψeo (ε∼ − ε∼ p ) + ψpo (α ) ∼ −1 σ∼˜ = M : σ∼ = ∼
=
(4.147) (4.148)
1 1 : (ε∼ − ε∼ p ) + α :C:α (ε − εp ) : ∼ 2 ∼ ∼ 2 ∼ ∼∼ ∼ ∼
(4.149)
The elastic limit criterion and the non-damage surface have the following form: ˜ − kp ≤ 0 fp = σ∼˜ − X ∼ H
fd = Y − ω(D) ≤ 0
(4.150)
and the normality rules yield: ε∼˙ p = λ˙ p
λ˙ p ∂fp −T n = λ˙ p M :n = ∼ ∼ ∂σ∼ 1−D∼ ∼
D˙ = λ˙ d
∂fp = λ˙ d ∂Y
(4.151)
Assuming that loading is total-strain controlled, the consistency condition over plasticity is written simply, without involving D˙ nor λ˙ d :
∂fp ˙ ˙˜ = n : σ˙˜ − X˙˜ f˙p = 0 = : σ∼˜ − X (4.152) ∼ ∼ ∼ ∂ σ∼˜ ˙ =n : : (˙ε∼ − ε∼˙ p ) − n :C :α (4.153) ∼ ∼ ∼ ∼ ∼ ∼
=n : : ε∼˙ − ∼ ∼
n : :n ∼ ∼ ∼
∼
∼
1−D
∼
λ˙ p −
:C :n n ∼ ∼ ∼ ∼
1−D
λ˙ p − γ
:X n ∼ ∼
(1 − D)2
λ˙ p
(4.154)
which immediately yields: 1 n : : ε˙ (4.155) h ∼ ∼∼ ∼ n : ( +C ):n ∼ ∼ ∼ n:X ∼ ∼ ∼ h= −γ ∼ ∼ 2 (4.156) 1−D (1 − D) Then, we determine the multiplier λ˙ d thanks to the consistency condition on the damage criterion, with: λ˙ p =
˜ : α˙ (4.157) ˙ = σ∼˜ : (˙ε∼ − ε∼˙ p ) + X : (˙ε∼ − ε∼˙ p ) + α :C :α Y˙ = Y˙e + Y˙p = (ε∼ − ε∼ p ) : ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
We get f˙d = Y˙ − ω (D)D˙ = σ∼˜ : ε∼˙ −
˜):n (σ∼˜ − X ∼ ∼ ˙ λp − ω (D)λ˙ d = 0 1−D
(4.158)
Introduction to damage mechanics
175
taking into account the plasticity criterion (4.150a): σ∼˜ : ε∼˙ −
kp λ˙ p − ω (D)λ˙ d = 0 1−D
(4.159)
We can then easily write λ˙ d :
1 1 : ε∼˙ = σ− kp + γ α n: :C :α ∼ ∼ ∼ ∼ ∼ ω (D) (1 − D)ω (D) ∼ h ∼ ∼ (4.160) The simplicity of expressions (4.155) and (4.160) is obvious. It is not the case when the hardening-damage coupling is built differently, because damage rate, and therefore λ˙ d , is involved in the plasticity consistency conditions (4.154). Note that we find a form similar to (4.137) in the case without plasticity, i.e., posing kp = γ = 0 in (4.160). λ˙ d =
σ∼˜ : ε∼˙ −
kp ˙ 1−D λp
4.6.4. Tangent operator In structural calculations, for the case of rate-independent laws, we often need the tangent operator with a stress-strain behavior defined by: σ∼˙ = L : ε∼˙ ∼ ∼
(4.161)
The expression of this tangent operator is often quite difficult. Below, we limit to three exercises with the same conditions as previously, with a scalar damage variable. For the elastic case, we show first how the chosen formulation, based on the thermodynamic framework, leads to a symmetric tangent operator (principal symmetry of a fourth-rank tensor). However, this is not always true, as it is shown in Exercise 2. At last, in the coupled elastoplastic case (Exercise 3), we show how the tangent operator is never symmetric, even for the simplest linear kinematic hardening law. Using a more complex hardening cannot modify this result. 1. Exercise 1: Elasticity coupled with damage—standard approach. The state laws are written, for the stress and for the damage thermodynamic force (with ε∼ = ε∼ e ): 1 Y = ε∼ : : ε∼ (4.162) ∼ 2 ∼ We saw in Sect. 4.6.3 that the consistency condition of the non-damage criterion (4.158) allowed us, in the case where λ˙ p = 0, to write the damage evolution equation as: Y˙ (4.163) D˙ = λ˙ d = ω (D)
176
Non-Linear Mechanics of Materials
˜ = Deriving state laws (4.162) and taking into account the damage isotropy, ∼ ∼
(1 − D) , we get: ∼ ∼
: ε∼˙ − : ε∼ D˙ σ∼˙ = (1 − D) ∼ ∼ ∼
: ε∼˙ Y˙ = ε∼ : ∼
∼
∼
(4.164)
Combining (4.163) and (4.164) we find: : ε∼˙ − σ∼˙ = (1 − D) ∼
: ε∼ )(ε∼ : : ε∼˙ ) ( ∼ ∼ ∼
∼
(4.165)
ω (D)
∼
which leads to the tangent operator of (4.161): L = (1 − D) − ∼ ∼ ∼
∼
1 ω (D)
˜ − : (ε∼ ⊗ ε∼ ) : = ∼ ∼ ∼ ∼
∼
∼
1
σ˜ ⊗ σ∼˜
ω (D) ∼
(4.166)
Obviously, this operator follows principal symmetry. 2. Exercise 2: Elasticity coupled with damage—non-standard approach. The nonstandard character comes from the use of a damage criterion based on strain instead of thermodynamic force associated with damage. Using for example Mazars’ criterion [MAZ82], in the particular case where all the principal strains are positive, one writes: fd = ε∼ − ω(D) − kd ≤ 0
(4.167)
Damage law is then written D˙ = λ˙ d =
1 ω (D)
ε∼ : ε∼˙ ε∼
(4.168)
Constitutive law derived from (4.164) yields: : ε∼˙ − σ∼˙ = (1 − D) ∼ ∼
: ε∼ )(ε∼ : ε∼˙ ) ( ∼ ∼
(4.169)
ω (D) ε∼
hence the tangent operator: L = (1 − D) − ∼ ∼ ∼
∼
1 ω (D) ε ∼
˜ − : (ε∼ ⊗ ε∼ ) = ∼ ∼ ∼
∼
1
σ˜ ⊗ ε∼
ω (D) ε ∼ ∼
(4.170)
which is not symmetric anymore. 3. Exercise 3: Elastoplasticity coupled with damage. We use the approach in which there is a state coupling between hardening and damage. For the sake of simplicity we only consider linear kinematic hardening (γ = 0). The state laws are given by (4.145) for elastic behavior and by (4.149) for Y . We saw
Introduction to damage mechanics
177
that the plastic multiplier was easily expressed by (4.155) whereas the damage multiplier was given by (4.160). Then, the derived constitutive law yields: σ∼˙ = (1 − D) : (˙ε∼ − ε∼˙ p ) − : (ε∼ − ε∼ p )D˙ ∼ ∼ ∼
˜ : ε˙ − : nλ˙ p − σ˜ λ˙ d = ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
kp ˜ : ε˙ − σ∼˜ ⊗ σ∼˜ : ε˙ − : nλ˙ p − σ˜ λ˙ p = ∼ ∼ ω (D) ∼ ∼∼ ∼ (1 − D)ω (D) ∼ ∼ ˜ : ε˙ − σ∼˜ ⊗ σ∼˜ : ε˙ = ∼ ∼ ω (D) ∼ ∼
kp σ∼˜ 1 n − : n − : : ε ˙ h ∼∼ ∼ (1 − D)ω (D) ∼ ∼∼ ∼
(4.171)
so that the tangent operator is written kp ˜ − σ∼˜ ⊗ σ∼˜ − 1 : (n ⊗ n) : + L (σ˜ ⊗ n) : (4.172) = ∼ ∼ ∼ ∼ ∼ ∼ ω (D) h ∼ h(1 − D)ω (D) ∼ ∼ ∼∼ ∼ ∼ ∼ With the first two terms we find the elastic case seen above. The third one is symmetric and corresponds to pure plasticity without damage. It is the last term, associated with damage/plasticity coupling, that is necessarily non-symmetric.
4.6.5. Rate-dependent laws Actually, rate dependent laws are easier both to formulate and to use, at least if one accepts the quasi-standard formalism mentioned above. Below, we address three ways to define damage law, giving for each of them the example of creep damage of metals. It is first necessary to specify the difficulty we want to solve, exemplified by creep damage. It is to be able to distinguish between viscoplasticity laws (rate dependent version of plasticity) and damage laws themselves, at least from two points of view: • the form of the multiaxial criterion itself. Indeed, we know well the big differences that exist, in stress space, between the form of equipotentials in viscoplasticity of metals, obeying almost von-Mises’ criterion, with quasi-identity between tensile and compressive behavior (for example), and the form of isochronous surfaces (surfaces of equal time to creep fracture), • the nonlinearity of the viscosity functions, associated on the one hand with viscoplasticity law, on the other hand with damage law. By restricting to power functions, we actually know that the exponents differ significantly. In what follows, although it is not always necessary (some applications involving a scalar isotropic damage), we use notation appropriate for a second-rank damage
178
Non-Linear Mechanics of Materials
tensor d∼ and the associated thermodynamic force y . Moreover, we recall that all the ∼ functions and criteria are likely to depend strongly on temperature (thermally activated phenomena), even if T is not explicitly specified. 1. Standard approach It consists in considering only one dissipation potential for all of the viscoplasticity and damage mechanisms. As for the rate-independent case one can use elasticity domain fp ≤ 0 and non-damage domain fd ≤ 0, respectively in the generalized stress space (σ∼ , Aj ) and the space of thermodynamic force associated with damage (here, without losing any generality we do not consider variables B and β of Sect. 1 in paragraph 4.6.2): fp = fp (σ∼ , Aj ; αj , d∼ )
fd = fd (y ; ε∼ e , d∼ ) ∼
(4.173)
However, unlike the rate-independent case, it is possible to go out of these domains (fp > 0 or fd > 0), causing viscoplasticity and damage. In these criteria, ε∼ e , αj and d∼ can intervene as parameters. We could build the potential in two different ways, as indicated below. • Laws without multiplicative function: the dissipation potential is the sum of two nonlinear functions, Fp and Fd , built from fp and fd , for example power functions: φ ∗ = Fp (σ∼ , Aj ; αj , d∼ ) + Fd (y ; ε∼ e , d∼ ) ∼ n+1 r+1 fp fd A K + = n+1 K r +1 A
(4.174) (4.175)
The rates of state variables derive then from this potential through the generalized normality rule: n ∂Fp fp ∂fp ∂φ ∗ ε∼˙ p = = = (4.176) ∂σ∼ ∂σ∼ K ∂σ∼ n ∂Fp fp ∂fp ∂φ ∗ =− =− (4.177) α˙ j = − ∂Aj ∂Aj K ∂Aj r ∗ ˙d = ∂φ = ∂Fd = fd ∂fd (4.178) ∼ ∂y ∂y A ∂y ∼
∼
∼
Proceeding like this, we distinguish the effects of nonlinearities held in power functions (with, for example, exponents r and n different). However this approach is restrictive regarding the form of the multiaxial damage criterion. Practically it is rather difficult to rebuild in the space of forces y a criterion that gives the shape of the isochronous surfaces ob∼ served in creep. The only way is then to assume that fd is linear in y and ∼
Introduction to damage mechanics
179
depends also on the elastic strains, playing the role of parameters. This allows introducing indirectly the stress thanks to the elasticity constitutive law. In this case, the choice for the expression of criterion fd in stress space is free. This approach is purely standard. However it uses state variables as parameters, which is quite artificial. The use of the hardening variables αj is necessary in order to introduce the nonlinear kinematic hardening laws, by adding to fp a term quadratic in Aj and by subtracting the corresponding quadratic term in αj , which is identical through the state laws (see [LEM85b]). • Laws with a single multiplicative function. We use the approach considered by Benallal [BEN89a] and then by Lemaître [LEM96]. We write again the dissipation potential as the sum of two terms: φ ∗ = Fp (σ∼ , Aj ; d∼ ) + Fd (y ; d∼ ) ∼
(4.179)
but the generalized normality rule involves now a single multiplier for both the viscoplasticity and damage mechanisms: λ˙ = λ˙ (σ∼ , Aj , y , d∼ )
(4.180)
∼
∂Fp ∂φ ∗ = λ˙ ε∼˙ p = λ˙ ∂σ∼ ∂σ∼
∂Fp ∂φ ∗ α˙ j = −λ˙ = −λ˙ ∂Aj ∂Aj
(4.181)
∗
∂φ ∂Fd = λ˙ d∼˙ = λ˙ ∂y ∂y ∼
(4.182)
∼
We can now, exactly as for rate-independent plasticity, consider Fp as the sum of fp and a term quadratic in Aj , as in (4.69), and Fd as identical to fd . Thus, this approach is less artificial than the first one. It is, however, much more restrictive, as we cannot distinguish anymore between the viscosity and damage nonlinearities, since they have the same nonlinear function λ˙ as multiplier. In particular, we cannot have two different exponents anymore. 2. Pseudo-standard approach In this case, we assume the existence of two independent dissipation potentials, one for each mechanism, viscoplasticity and damage, and of two independent viscosity functions: Fp = Fp (σ∼ , Aj ; d∼ )
Fd = Fd (y ; d∼ )
(4.183)
λ˙ p = λ˙ p (σ∼ , Aj ; d∼ )
λ˙ d = λ˙ d (y ; d∼ )
(4.184)
∼
∼
so that the generalized normality rule is now written ε∼˙ p = λ˙ p
∂Fp ∂σ∼
α˙ j = −λ˙ p
∂Fp ∂Aj
(4.185)
180
Non-Linear Mechanics of Materials
∂Fd d∼˙ = λ˙ d ∂y
(4.186)
∼
Here too, as for the law described just above, one can choose Fp as the sum of fp and a term quadratic in Aj (in order to introduce the nonlinear kinematic hardening recovery terms) and Fd identical to fd . To retrieve the case of the first laws without multiplier functions, one can choose both the viscoplasticity and damage multipliers as two power functions: n r fp fd λ˙ p = (4.187) λ˙ d = K A Therefore, we see that this approach combines the advantages of the two kinds of laws discussed above and avoids introducing the state variables ε∼ e and αj quite artificially. If we want to, we can even express directly λ˙ d as a function of the stress (instead of the forces y ), which leads to Hayhurst’s criterion [HAY72] ∼ in creep. It is clear, however, that, by construction, dissipation remains always positive: int = σ∼ : ε∼˙ − Aj α˙ j + y : d∼˙ (4.188) ⎛
∼
j
⎞ ∂Fp ∂Fp ⎠ + λ˙ d y : ∂Fd ≥ 0 = λ˙ p ⎝σ∼ : + Aj ∼ ∂σ∼ ∂Aj ∂y
(4.189)
∼
j
By definition, if both potential functions Fp and Fd are positive, convex and cancel at the origin, the two factors in front of λ˙ p ≥ 0 and λ˙ d ≥ 0 are positive. Of course, dissipation was positive also in both approaches discussed just above. 3. Damage as a fourth-rank tensor The quasi-standard approach is almost the only one that allows introducing a damage criterion, based on the stresses and an anisotropic damage described by a fourth-rank tensor, and ensuring at the same time that the Second Principle is verified. For this, it suffices to choose:
1/2 − kd Fd = fd (Y∼ ) = Y∼ Q − kd = Y∼ :: (Q : Y∼ ) ∼
∼
∼ ∼
∼
∼
(4.190)
where Q is a fourth-rank tensor depending on the material and kd a thresh∼ ∼
old, possibly zero (as in classical creep laws). The symbol :: denotes tensorial summation over four indices. In that case, we have: ) λ˙ d = λ˙ d (σ∼ ; D ∼ ∼
(4.191)
Introduction to damage mechanics
Q : Y∼ ∼ ∂Fd ∼ ˙ ˙ ˙ = λd = λd ∼ D ∼ ∂Y∼ Y∼ Q ∼ ∼
181
(4.192)
∼
We can even choose the linear version [CHA78, LEM85b], in which tensor Q ∼ ∼
gives the direction of damage rate, in the principal frame of effective stresses. We set then: (4.193) Fd = Tr Q : Y∼ − kd = Q :: Y∼ − kd = Qij kl Yij kl − kd ∼ ∼
∼ ∼
∼
so that:
∼
˙ = λ˙ d H (Q :: Y )Q D ∼ ∼ ∼ ∼
∼
∼
(4.194)
∼ ∼
where H is Heaviside’s function. Of course, we still verify the Second Principle as: ˙ = λ˙ d H (Q :: Y )Q :: Y = λ˙ d Q :: Y ≥ 0 int = Y∼ :: D (4.195) ∼ ∼ ∼ ∼ ∼
∼
∼ ∼
∼
∼ ∼
∼
∼ ∼
∼
4.7. Examples of damage models in brittle materials By brittle materials we mean those that do not deform much before damage and ultimate fracture of the RVE (or of the constituent). Aside from the case of glasses and ceramics, not covered herein, we find that kind of behavior in concrete and composites. Quite often, the brittle character can be observed at the macroscopical level, even when locally the matrix is able to deform (for some organic matrix there can be a large deformation, but highly localized so that the macroscopic behavior remains brittle). One of the essential characteristics is that the macroscopic behavior is going to be described by an elasticity law coupled to damage. In this section we address, very shortly, some uses of Continuum Damage Mechanics applied to that kind of situation.
4.7.1. Concrete Concrete is mainly made of aggregates of various dimensions, cement and water. Problems due to hygrometry and aging will not be addressed here. In fact, concrete is a composite, which we could consider as globally isotropic. 1. Damage and deformation mechanisms It is hard to separate deformation, damage and fracture phenomena as the initial microcracks and cavities, which exist prior to any loading, generate permanent strain that develops through brittle fracture mechanisms.
182
Non-Linear Mechanics of Materials
• This phenomenon is however not significant below a given value of the load. Deformation in this first phase results from quasi-reversible atomic movements and can be considered elastic with a low viscosity. • Brittle fracture by paste-grains decohesion being the main phenomenon of permanent strain and fracture, it is strongly influenced by the nature of the load. Elasticity limit or ultimate strength are 10 to 15 times higher in compression than in tension, which explains why concrete is mainly used in compression. Beyond the load corresponding to elasticity limit (or proportionality limit), the microcracks at the interface between paste and the biggest grains start to grow, generating at the macroscopic level permanent strain that superimpose to elastic deformations. • For even higher load levels, in a third phase, microcracks reach the paste due to decohesion, running parallel to the stress when under compression. Slip appears in crystalline grains, contributing also to permanent strain, which mainly occurs at constant volume. Microcracking damage becomes then strongly anisotropic. • The fourth phase is fracture: macroscopic cracks appear and propagate quickly, the stress necessary to induce new strain decreases, specific volume increases and final fracture occurs when cracks meet by localization, creating a discontinuity surface that crosses over the whole volume. 2. Elasticity coupled to damage Let us take a very simple and schematic example, by particularizing the equations to uniaxial tension. This example is sufficient to exemplify the possibilities of damage mechanics in the case of elasticity/damage coupling and to render quite accurately concrete behavior. We write the isotropic elasticity law and assume that damage is isotropic, described by a single scalar variable D. We assume also that damage D evolves with elastic strain, by introducing a varying threshold εD : s ε dε when ε = εD and dε = dεD > 0 (4.196) dD = εo dD = 0
when ε < εD or dε ≤ 0
(4.197)
Such an evolution law, rate-independent, is completely consistent with criteria that have been built in Sect. 4.6. Taking for initial conditions D = εD = 0 and integrating until fracture of the RVE, D = 1, we get: s+1 s+1 ε ε D= σ = Eε 1 − (4.198) εR εR where εR = ((s + 1)εos )1/(s+1) is the fracture strain. Figure 4.16a illustrates the stress evolution that we can obtain (for s = 2) with such an elasticity model
Introduction to damage mechanics
183
coupled to damage. During a possible unloading–reloading, we have a linear elastic behavior, with an elasticity modulus lower than initially. At reloading the behavior remains linear elastic until the new threshold (the maximum strain reached during previous loading) is reached. Results obtained by this very simple model are comparable to experimental results, Fig. 4.16b, obtained from various concretes loaded in uniaxial compression, classical loading mode for concrete study. Transposing the above model to the compression case, of course, does not pose any problem (it is sufficient to change signs).
Figure 4.16. (a) Stress-strain curve for damageable elastic theory (b) examples of application to concrete in compression mode (according to [KRA81])
184
Non-Linear Mechanics of Materials
Note that the notion of an evolutive damage threshold (here εD ) is consistent with a damage criterion based on the thermodynamic approach of Sect. 4.6, for instance with (4.150b), fd = Y − ω(D) ≤ 0. Indeed, in the case of isotropic damage, and limiting to the uniaxial case, we find: Y =ρ
∂ψ 1 = Eε 2 ∂D 2
(4.199)
so that, taking into account (4.198), we can identify the function ω(D) as: ω(D) =
1 2 2/(s+1) Eε D 2 R
(4.200)
4.7.2. Application to composites 1. The different scales Before dealing with composites damage, it is good to recall what are the different scales and what are the main mechanisms coming into play. Thus, we will have: • structure scale, or composite piece, often a laminate. Some very macroscopic approaches consider the laminate as a macroscopically homogeneous material. These approaches are now outdated. For structural analysis one uses the finite element method, in a plate, shell or 3D scheme, but dissociating kinematics (displacement discretization) and behavior aspects, handled ply-by-ply. • elementary ply scale (Fig. 4.17) is the one at which we are when building macroscopic constitutive laws (this scale is sometimes called mesoscopic). It is the uni-directional plies in organic or metallic matrix composites or
Figure 4.17. Various constituents of stratified composites
Introduction to damage mechanics
185
sometimes tissues (taffeta for SiC/SiC, satin for C/PMR15. . .). The composite (strand + matrix or fibre + matrix) is then considered a macroscopically continuous medium, even when damage occurs (transverse cracks, for example). In this framework, modeling interfacial phenomena between plies should be treated in a specific way: interfacial layer [LAD83, LAD94, LAD95, ALL90] or interfacial decohesion models [TVE90a, POI96, CHA97]. • micro scale is the one where elementary constituent behaviors are taken into account (matrix, fibres, matrix/fibre interface, . . .). In some cases, one analyses the behavior of woven composites through a two step micromacro technique (fibre + matrix → strand, then strand + matrix → woven composite). • an even finer scale would be the one of the phenomena occurring at matrix/fibre interfaces (diffusion zones, various interphases, . . .), out of the scope of the present book. 2. A hierarchical approach In this context, works conducted at ONERA, [SAA92, MAI97a, MAI97b], lead to building a hierarchical approach to the constitutive laws of various composites (Fig. 4.18): • base configuration is woven ceramic matrix composites (CMCs), SiC/SiC for instance, in which the behavior is mainly elastic, with a nonlinearity induced by damage (drop of elastic characteristics) but also very clear
Figure 4.18. A hierarchical approach of composites behavior
186
Non-Linear Mechanics of Materials
damage deactivation effects when going to compression mode. The models we obtain will be studied in the next section; • in the C/SiC or C/C case, irreversible deformations induced by damage and plasticity effects had to be introduced; • for C/PMR15 composites, used in SNECMA engines (cases, fan blade), were developed viscoplasticity and/or viscoelasticity damageable models, which makes it possible to describe anisotropies, hysteresis effects during load/unload cycles, relaxation or creep phenomena, and partial or full recovery; • these macroscopic models are also used for organic matrix composites (OMC) used under the form of uni-directional plies, and different but close versions are built for SiC/Ti metallic matrix composites (MMC) using approaches with micro-macro transitions. The approach as a whole is a part of continuum thermodynamics with internal variables, and more specifically of continuum damage mechanics. For plasticity and viscoplasticity problems, one uses the notions of kinematic hardening in a rather classical scheme (classical for metals). Concerning damage, the most elaborate versions take into account the following ingredients: • scalar damage variables, associated with microcracks oriented by the most resistant constituents (fibres, strands); • tensorial variables (second-rank) in order to describe microcracks whose orientation is related to applied stress directions and their history; • models describing potential damage induced by compression (split type microcracks); • damage deactivation phenomena are rendered via a criterion that respects both the symmetry of the elasticity operator and the continuity of the stress-strain responses whatever the multiaxial loading undergone by the composite. Of course, these models are determined from (macroscopic) experiments, generally in tension-compression mode at 0◦ and 45◦ (or 90◦ ). They can be validated on multiaxial experiments under simple or complex loading, as it will be shown at the end of next section. From all these works on the various classes of composites, it turns out that there are currently three kinds of difficulties or insufficiencies in macroscopic models: • damage deactivation turns out to be insufficient for some shear loadings, because of the restrictive condition of “continuity of the stress-strain response”. Ongoing research aims to “widen” this deactivation condition, based on micromechanical analysis of the phenomena of microcracks closing and of storage/dissipation of associated energies;
Introduction to damage mechanics
187
• load/unload hysteresis phenomena and associated residual strains, in systems without viscosity or plasticity of the matrix. This point is not addressed here. It implies taking into account fibre/matrix/interface/matrixcrack interaction, involving interfacial decohesion/friction mechanisms. This is clearly still an area of research; • fracture criteria, that can be of two natures: either involving critical values as materials parameters, or rather systematically looking for instabilities/bifurcations conditions associated with structural analysis itself.
4.7.3. Modeling the ceramic–ceramic composites Here, we adapt the theories introduced in Sects. 4.4, 4.5 and 4.6 to the case of brittle composite materials. We admit, however, the possibility of damage induced irreversible deformation (not introduced previously) and some plastic deformation. 1. State laws and damage effect Internal variables are either strains, or damage variables. We will distinguish more specifically ε∼ c , strain tensor at the damage deactivation point, ε∼ p , plastic strain tensor (due to inelastic phenomena inside the matrix), δi , (i = 1, 2, 3), three scalar damage variables, representing microcracks whose directions are given by the initial principal anisotropy axis of the material, i.e., by the directions of the strengthening constituents (fibres, strands, . . .), d∼ , the tensorial damage variable (of second order), representing microcracks whose directions are linked to the one of the loading that induces them. Free energy is assumed as: ψ=
1 1 eff (ε∼ − ε∼ p ) : : (ε∼ − ε∼ p ) + (ε∼ − ε∼ c ) : : (ε∼ − ε∼ c ) ∼o ∼ 2 2 ∼ ∼
(4.201)
From there, we deduce the stress tensor: σ∼ =
∂ψ eff : (ε∼ − ε∼ c ) + = + : (ε∼ c − ε∼ p ) o ∼ ∼ ∼o ∂ε∼ ∼ ∼ ∼
(4.202)
We note the term corresponding to the undamaged material and the one, affected by damage, that holds also the closing strain, with unilateral condition. The : (ε∼ − ε∼ p ). In pure stress state corresponding to ε∼ = ε∼ c is given by σc = ∼o ∼ tension, it corresponds either to a negative value (Fig. 4.20a), or to a positive one (Fig. 4.20b). Plastic strain ε∼ p is defined by reference to the undamaged initial ), with a configuration relaxed from the deactivation state behavior (stiffness ∼o ∼
ε∼ c , σc , as exemplified in Fig. 4.20a. The effect of damage is introduced by
188
Non-Linear Mechanics of Materials
Figure 4.19. Tension–compression of a SiC/SiC composite in the 0◦ direction. (a) Experiment, (b) calculation
Figure 4.20. Scheme of tension–compression deactivation for two closing positions and corresponding definition of plastic strain eff = ˜ = o + ˜ , given by the following equation: considering ∼ ∼ ∼ ∼ ∼
∼
∼
˜ =− ∼ ∼
∼
3 δi A : K − D (d ) : K ∼i ∼o ∼ ∼ ∼o i=1
∼
∼
∼
s
∼
(4.203)
s
is a material-dependent fourthin which all damages are considered active. K ∼o ∼
= or K = diag( ). For scalar rank tensor. Simplest choices can be K ∼o ∼o ∼o ∼o ∼
∼
∼
∼
is written from the directions p i , q i , r i associated with damage δi , operator A ∼i ∼
strengthening elements. For example, for a woven composite, p i is the vector
Introduction to damage mechanics
189
perpendicular to strands i, in the plane of the tissue, q i is the vector giving the of direction of the strand, r i the vector perpendicular to the plane. We take A ∼i ∼
the form:
A = α11 p i ⊗ p i ⊗ p i ⊗ p i + α12 (p i ⊗ p i ) ⊗ (q i ⊗ q i ) s ∼i ∼ + α13 (p i ⊗ p i ) ⊗ (r i ⊗ r i ) s + β12 (p i ⊗ q i )s ⊗ (p i ⊗ q i )s (4.204) + β13 (p i ⊗ r i )s ⊗ (p i ⊗ r i )s where α11 , α12 , α13 , β12 , β13 are coefficients depending on the material. In the particular case of the orthotropic tissue, i.e., the orthogonal strands, the formulation simplifies as we only have three vectors p 1 , p 2 , p 3 perpendicular to one another. For the tensorial variable, the damage effect tensor is given by (4.39) (see Sect. 4.3.3). 2. Damage deactivation eff given by: When one or another damage is deactivated, we use ∼ ∼
eff ˜ + η1 = ∼ ∼ ∼
∼
+ η2
3
δi H (−εi )P∼ i : A :K : P∼ i ∼i ∼o ∼
i=1 3 i=1
∼
∼
∼
s
H (−εni )N : D (d ) : K :N i o ∼ ∼ ∼ ∼ ∼i ∼
∼
∼
s
∼
(4.205)
where H is Heaviside’s function and where ni are the principal directions of damage. εi = p i .ε∼ .p i and εni = ni .ε∼ .ni are the normal strains to principal damage, respectively scalar and tensorial. The fourth order projection operators, , are defined as p i and ni through relationships such as P∼ i = p i ⊗ P∼ i and N ∼i ∼ ∼ ∼ pi ⊗ pi ⊗ pi . The deactivation criterion is written in the axis of the microcracks or in the principal directions of the second-rank tensor. It corresponds to the sign change of the associated normal strain, as it has been explained in Sect. 4.5.3. Here, this principle is generalized to the case where there are two kinds of damage, scalar and tensorial. Stress is obtained simply by deriving (4.202). The thermodynamic forces associated with damage variables, scalar and tensorial, are a little complex to write because of deactivation effects: yi = −
∂ψ 1 = ε∼ : A : K − η H (−ε )P : A : K : P 1 i ∼i i :ε ∼i ∼o ∼i ∼o ∼ s s ∼ ∂δi 2 ∼ ∼ ∼ ∼ ∼ ∼ (4.206)
190
Non-Linear Mechanics of Materials
∂ψ ∂d∼ ξ 1−ξ = (ε∼ Tr σ∼˜ + σ∼˜ Tr ε∼ ) + (σ∼˜ .ε∼ )s 4 2 3
1−ξ
ξ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ε Tr σ˜ + σ˜ Tr ε + σ∼˜ j .ε∼ j − η2 H (−εnj ) s 2 ∼j ∼ j ∼ j ∼j 2
y=− ∼
j =1
(4.207) with σ∼˜ = K : ε∼ , ε∼ ∗j = N : ε∼ = εn∗j nj ⊗ nj , σ∼˜ ∗j = K : ε∼ ∗j ∼o ∼o ∼j ∼
∼
∼
3. Damage evolution equations The evolution equations follow directly from the general formalism given in Sect. 4.6.2-2. The criteria associated with scalar damage are linked multiple criteria, defined in the space of the thermodynamic forces yi , of the form: ⎞ ⎛ 3 3 f i = gi ⎝ aij yj ⎠ − ro + bij δj ≤ 0 i = 1, 2, 3 (4.208) j =1
j =1
where a11 = a22 = a33 = b11 = b22 = b33 = 1. The coupling coefficients such as a12 allow setting the form of the non-damage surface in subspace y1 > 0, y2 > 0, whereas b21 allows setting the evolution of criterion f2 = 0 when damage δ1 evolves (δ2 = 0) or the other way around. When the forces yi are negative, we chose not to involve them in the criterion (McCauley’s symbol). The functions gi describe the damage evolution kinetics when ro defines the initial threshold. For the tensorial variable we use a specific form: 1 +
f =g χ y :Q:y ∼
∼ ∼
∼
+
2
+ (1 − χ) Tr y ∼
− νo − ζ Tr d∼ − (1 − ζ ) Tr (y .d∼ ) ≤ 0
(4.209)
∼
that involves the combination of two invariants. In the first one, for χ = 1, initial anisotropy of the composite intervenes (through tensor Q). Moreover, y + ∼ ∼
∼
is the positive part of tensor y , defined by the spectral decomposition indicated ∼ in Sect. 4.5.1 (see (4.117)). The second one, (χ = 0) is introduced in order to get as particular case the one of a material in which damage would evolve isotropically. Damage evolution equations are then given by: δ˙i =
j =1
3μ˙ j
∂fj ∂yi
∂f d˙ = μ˙ ∂y ∼
(4.210)
Introduction to damage mechanics
191
Figure 4.21. Scheme of damage under biaxial loading: (a) experiment and evolution with inverse coupling (scalar variables), (b) isotropic evolution, (c) LMT model, (d) decoupled evolution (tensorial variable)
in which the scalar multipliers follow consistency conditions such as: μ˙ j = 0 if fj < 0 or f˙j < 0 μ˙ j = 0 if fj = f˙j = 0
(4.211) (4.212)
We note that, thanks to the linearity of free energy as a function of damage variables, thermodynamic forces are independent from them and the multipliers of scalar and tensorial criteria are then determined in a decoupled way. A critical point concerns the coupling between directions in the case of nonproportional multiaxial loading. Let us assume, for example, a damaging uniaxial loading in direction 1, followed by an unloading and a new loading in direction 2, perpendicular to 1. Figure 4.21 schematically indicates what the various criteria give: • the few existing experiments ([SIQ93, MAI96, MAI97b]), seem to indicate that the non-damaging domain is smaller in direction 2 after damaging in direction 1 (than it would have been without prior damage). • a model such as the one used by LMT–Cachan [LAD94] gives a new nondamaging surface, wider in direction 2 than in direction 1 (the opposite of experiment); • the tensorial model (see (4.209)), with χ = 1 and ζ = 1 gives an identical extension in both directions: indeed, the threshold evolves in a purely scalar way, of isotropic type (isotropic evolution criterion);
192
Non-Linear Mechanics of Materials
• the multiple coupled criteria, to the contrary, allow getting closer to experiment, with a decrease of the non-damaging threshold in direction 2. It is sufficient for this to choose coupling terms such that b21 in expression (4.208) will be negative; • finally, the tensorial criterion of (4.209) with χ = 1 and ζ = 0 allows no threshold evolution in direction 2 while damage and corresponding threshold increase in direction 1. To sum up (and in the ad hoc principal space) the threshold function evolves with y1 d1 + y2 d2 . Then, after damage d1 provoked by y1 > 0 but y2 and d2 remaining zero, the threshold in direction 2, for y2 = 0, remains equal to its initial value. We have then a criterion with evolutions decoupled between directions. This criterion is used below to model biaxial tests in internal pressure/tension over SiC/SiC composites. 4. Examples of use We give here a few results originally from a very comprehensive experimental study on a SiC-SiC composite elaborated by Snecma. These works include tension–compression tests (monotonic or cyclic with increasing amplitude), as well as tension–torsion on tubes and internal pressure–tension on tubes. The model is determined from tension–compression experiments in direction 0 and 45◦ . Figure 4.19 shows a comparison for direction 0. The agreement is good, for axial deformation as well as for the transverse deformation. We note the damage deactivation effect, extremely strong on that kind of material, with an almost full recovery of the initial elastic modulus in compression. In this case, the deactivation strain εc was zero.
Figure 4.22. Tension–torsion test on SiC/SiC—Tension response
Introduction to damage mechanics
193
Figures 4.22 and 4.23 [MAI96, MAI97b] show results on thin tubes in tension– torsion, for monotonic proportional loading, with various load ratios. The model predicts very accurately the multiaxiality effects, although it was not determined from these experiments. We note in particular the transverse strains in Fig. 4.22. The biaxial internal pressure–tension experiment on a tube from Fig. 4.24
Figure 4.23. Tension–torsion test on SiC/SiC—Shear response
Figure 4.24. Prediction of Young’s modulus evolution (directions 1 and 2) on SiC/SiC for an incremental loading sequence in tension followed by an incremental loading sequence in internal pressure
194
Non-Linear Mechanics of Materials
[MAI97b] corresponds to a cyclic tension loading (with increasing maximum) followed by an internal pressure loading, also at increasing amplitude (the end effect is almost entirely compensated). We observe on the bottom part of the figure an excellent agreement between measured and calculated elasticity modulus). We note that the stress level in direction 2 (hoop stress) that provokes the onset of the decrease of modulus E22 is barely higher than the threshold in axial tension (approximately 90 MPa), that led to the decrease of E11 (approximately 80 MPa). It is the choice of term with Tr(y .d∼ ) in (4.209), with ζ = 0, that ∼ allows no evolution of damage threshold in direction 2 while damage increase by tension in direction 1 (the classical criteria, with ζ = 1, would give here 160 MPa instead of 90 MPa).
Chapter 5
Elements of microstructural mechanics
5.1. Characteristic lengths and scales in microstructural mechanics 5.1.1. Objectives of heterogeneous materials mechanics and homogenization All materials have a scale below which they cannot be considered homogeneous anymore, i.e., at this scale, the studied physical property varies from one point to another. However well-suited constitutive laws, nonlinear if needed, allow us in many cases to treat such a material as a homogeneous continuum and to calculate a structure made of this material based on this homogeneous representation. The process of building and identifying such constitutive laws relies on exploiting a set of experimental mechanical tests (as numerous as possible and involving loading paths similar to the one encountered in the final structure) but takes only a limited advantage of knowledge of material’s microstructure. The objectives of heterogeneous materials mechanics are the following: • deriving constitutive laws with a greater prediction power than the purely phenomenological laws previously identified; in particular, they must still be relevant if the composition of the material is slightly modified; • predicting the behavior of a mixture of components, even before processing the material; • optimizing a microstructure for a given macroscopic property. Special methods must be developed because taking explicitly all the heterogeneities into account is still generally impossible, and in any case not advisable, when modeling a structure. J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_5,
196
Non-Linear Mechanics of Materials
Definition of a representative volume element We start by enumerating some examples of heterogeneous materials in order to bring to light the notion of Representative Volume Element (RVE). From a mechanical point of view, the recovered single-phase metallic single-crystal (Cu, Al. . .) is the archetype of the medium that can be rightfully considered homogeneous at every scale above ten nanometers! However, a few percent of monotonous deformation or a few cycles of deformation are sufficient to create subboundaries (loss of orientation homogeneity) or dislocation structures (walls, . . .) over distances of the order of one micron and beyond. These heterogeneities significantly determine the subsequent behavior of the material. The heterogeneity of the material is then not necessarily an initial datum but can appear or develop during deformation. This is the notion of evolving microstructure. In the framework of this book, the case of evolving microstructures is not addressed. Only the influence of initial and constant heterogeneities on the behavior of the material is considered. The case of the polycrystal is an exception, as relative orientations of the grains can vary during deformation (Sect. 5.6.5). Figure 5.1 shows that even a single crystal can be heterogeneous and hold two phases with different properties, the hardest one as cuboidal inclusions, the other playing the role of matrix. The whole remains a single crystal because both phases are crystallographically coherent. Structural composite materials often contain two phases, such as the metallic matrix composites, with fibre or particle reinforcement (SiC–Ti, Fig. 5.2). The second
Figure 5.1. Dual phase microstructure of the nickel-based single-crystal superalloy SC16 (size of the precipitates about 0.5 μm, photo BAM, Berlin)
Elements of microstructural mechanics
197
Figure 5.2. Composite with long fibres SiC-Ti (fibre diameter 500 μm, from ONERA)
Figure 5.3. Cellular material: structure of a nickel foam
phase can be. . . empty!, like in open- (Fig. 5.3) or closed-cells metallic foams. In the three previous cases, the matrix itself is a heterogeneous material: a polycrystal. For example, a tantalum polycrystal is made of only one chemical component but is an aggregate of crystals of various orientations. We then call all the grains of a given volume having a similar orientation, a “phase”. In such a case, the number of phases tends towards infinity. One should note that it is not anymore an inclusionmatrix type of morphology but juxtaposed phases more or less randomly distributed
198
Non-Linear Mechanics of Materials
Figure 5.4. Polycrystalline structure of a zinc coating (we see as well the solidification dendrites inside the grains, grain size 200 μm, after [PAR04])
Figure 5.5. Direct and inverse {111} pole figures of an aluminium alloy sheet (from [ACH93])
according to a crystallographic and morphological texture (Fig. 5.5). As to the zinc layer deposited on galvanized steel sheets, it has a plane polycrystalline structure (Fig. 5.4). In many other cases a second phase grows inside each grain with a complex morphology (Figs. 5.6 and 5.7). The austenoferritic steels with bi-percolated structure and orientation relationships between phases constitute a case of extreme complexity (Fig. 5.8).
Elements of microstructural mechanics
199
Figure 5.6. Polycrystalline structure of a Cu–Zn–Al shape memory alloy; we see in the grains the martensite auto-adaptive variants (grain size 500 μm, from A.F. Gourgues, Mines ParisTech)
Figure 5.7. Dual phase structure inside a grain of the Ti6242Si alloy (width of the picture: 250 μm, from [JOU99])
200
Non-Linear Mechanics of Materials
Figure 5.8. Austenoferritic steel with bipercoled dual phase structure; cleavage cracks (bottom) (from [DEV97a])
Last of all, we will mention the case of ductile concretes (Fig. 5.9), which have the classical heterogeneous granular structure of concrete but also dispersed short metallic fibres, the whole being a compound of a brittle and a ductile material. The most simple heterogeneities distribution remains the periodic distribution that rationalizes quite accurately many composites (Fig. 5.2). Various levels of heterogeneities can coexist at different scales. It is the case of the woven composite in Fig. 5.10. In this case it is possible to observe what is called the scale-separability of heterogeneities (which the fibre and the ply are). The separability allows us to distinguish the heterogeneity levels, which permits successive applications of homogenizing methods to build the final behavior from the knowledge of the fibre behavior (Fig. 5.11).
Elements of microstructural mechanics
201
Figure 5.9. Deformation of a concrete reinforced by short metallic fibres (up: tensile test specimen with gauge, bottom: CT test specimen, from Mines ParisTech)
Figure 5.10. Monolayer woven SiC/SiC composite: surface and profile (from [MUN94])
202
Non-Linear Mechanics of Materials
Figure 5.11. Mechanical behavior and damage of the fiber alone (top) and of the plies in a woven composite (bottom, note the fracture of the fibers, of the matrix and even the delamination of the composite, from [BAX94])
Cermets (ceramic-metal) are gradient materials, i.e., with a composition gradient in one direction (Fig. 5.12). We see that in all cases (except the last one), it is possible to define a smallest volume holding, a priori, all the statistical (or deterministic, in the periodic case) information characterizing the distribution and the morphology of the heterogeneities of the material. Any larger volume can be deduced through successive translations or additional realizations of the known statistical characteristics. Each one deserves then to be called a representative volume element V . The existence of the RVE is intimately connected to the notion of separability of scales. When it is not the case (percolated or gradient materials), the existence of the RVE is not guaranteed anymore. In what follows we assume that such a RVE can be chosen, this definition depending only on the identity of phases and their distribution and not yet on the final structure or component using this material. There exist also characteristic lengths associated with the deformation mechanisms
Elements of microstructural mechanics
203
Figure 5.12. Gradual structure TAILLANT with 10/20/30% of cobalt, seen of the tip and section (from [FAV95])
Figure 5.13. Twinning in two zinc grains (c-axis perpendicular to the sheet, after [PAR04])
of the materials, in the case of nonlinear behavior. For the grains of a polycrystal, we will retain crystallographic slip according to particular slip systems [FRA91]. This mechanism can in some cases be challenged by twinning (as in the case of zinc, Fig. 5.13): twins introduce a new substructure at the grains scale, that can lead to a reconsideration of the definition of the pertinent scale in the micro-macro transition.
204
Non-Linear Mechanics of Materials
5.1.2. Microstructure/RVE/structure The triplet microstructure/RVE/structure must now be taken into account simultaneously for the sake of final application. It involves the knowledge of a few characteristic lengths: • d, the size of the heterogeneities (or of the largest heterogeneity); • l, the size of the RVE (possibly of the minimum RVE); • L, the size of the final structure to be calculated by means of the homogenized model; • Lexp , the size of the test specimen used to characterize the effective behavior (in order to check if the predictions are correct, for example); • Lw , the size of the fluctuations of the loading applied to the structure, which is a more pertinent parameter than L. A homogenization scheme is possible only when some combinations of ordering relationships exist between these quantities. The following relation must always be verified: (5.1) d < l < L, Lw It is not sufficient, though, to ensure the efficiency of a given homogenization scheme. In most parts of this section, we will require that dl
(5.2)
In the case of random materials (polycrystals. . .), this condition is often necessary to ensure the representative character of V . To determine the effective properties of the material, it will be necessary to apply boundary conditions on a volume V . Effective properties are expected to be independent of the kind of boundary conditions chosen. It can only be the case if (5.2) is fulfilled. In general, we will require that l L, Lw
(5.3)
because, as we will show, if the loading applied onto the structure varies slowly on the RVE, the homogenized solution is close to the real solution. However, this condition is not always fulfilled in structural modeling, in particular when the geometry of the components is complex or when the size of the representative cell is not negligible. In such a case, there still exist methods to estimate effective properties; we will discuss them in Sect. 5.8.
Elements of microstructural mechanics
205
The effective properties of the heterogeneous material can be determined experimentally quite easily if (5.4) l Lexp This condition can be softened in l < Lexp . However, this condition is not always attainable. For example, if we consider concrete used to make bridges, the RVE that contains the main heterogeneities can be more than one meter long. A standard test specimen is typically 15 cm in length for 30 cm in diameter. This is the scope of the problem addressed in Sect. 5.3.3. In some cases, the size of the largest heterogeneities is of the order of magnitude of one of the dimensions of the test specimen itself. In conclusion, we will assume in Sects. 5.2 to 5.5 (except 5.3.3), that d l Lexp , L, Lw
(5.5)
In the last section, we will consider the case where d < l < Lexp , L, Lw
5.1.3. Spatial averages, ensemble averages; equivalent homogeneous medium A way of modeling a structure made of heterogeneous materials without treating all of the heterogeneities individually consists in trying to replace the heterogeneous medium by an equivalent homogeneous medium endowed with so-called effective properties. Let us consider, for example, a structure S for which we would know, like the old Laplace, the position of every heterogeneity and for which the exact solution would then be known. How can we associate to this reference solution more regular fields that still fulfill equilibrium conditions? A first technique would consist in partitioning the structure into juxtaposed RVEs to which we attribute the mean value of stress and strain, for example. The field we get is however not differentiable. Or we can attribute to each point M of the structure the average value of the fields in a volume centered around M: it is the “moving window” cited in [HUE85] (Fig. 5.14). However, it is generally impossible to ensure that: .S = .V S
(5.6)
where the brackets represent the spatial average over the indicated volume. Then, it is not completely satisfying. We will therefore prefer to consider the mathematical expectation of the considered values. For this, the structure S is seen as a particular realization of the heterogeneous material and we retain only the average values in one point M on various realizations.
206
Non-Linear Mechanics of Materials
Figure 5.14. Between structure and microstructure: the notion of RVE
We define then the effective quantities such as strain n E (M) = ε = ε (M, α)f (α)dα ε∼ (M, α)/n ∼ ∼ ∼ S
(5.7)
α=1
where S represents the space of samples α with the probability density f , which we will approach experimentally for example through the rhs in the case of static uniformity. This is the set average approach, as opposed to the spatial average approach ε∼ V . By applying definition (5.7) to the phase (r) indicator function fr (x) (which is 1 if x is in r and 0 otherwise), we define the probability of finding the phase r at x: Pr (x) = fr = fr (x, α)p(α)dα, S
along with the probability of finding simultaneously x in r and x in s: Prs (x, x ) = fr (x)fs (x ). The heterogeneous medium is then called statistically uniform if the probabilities such as Pr do not depend on translations. Pr becomes a constant and Prs depends only on x − x . The statistical uniformity condition is in fact necessary to define a RVE. It is, of course, automatically fulfilled for periodic structures. Equality (5.6) is verified if the loading is almost uniform in a volume V (i.e., l Lw ), so that the notions of spatial average and ensemble average coincide. That
Elements of microstructural mechanics
207
is why the ergodic hypothesis is often made, assuming the equality between space and ensemble average. Assuming the ergodic hypothesis implies that all the statistical information is available in one realization of the RVE. This implies, actually, that heterogeneities are small enough (i.e., d l). When these hypotheses are fulfilled (d l, but also l Lw ), we can actually show, at least in the linear case, the existence and uniqueness of a homogeneous equivalent medium (HEM) for both random and periodic materials [SAB92]. The word homogenization is often restricted to this kind of situation. When these hypotheses are not fulfilled, there exist methods to evaluate effective properties, but not necessarily in a unique way. Then arises the question of assimilating a heterogeneous material to an effective homogeneous medium. In [HUE85], one will find more systematic conditions than the one mentioned here, for such an assimilation to become possible. These conditions are often hard to interpret. Balance equations (mass, energy, momentum, angular momentum, entropy) being known locally, one can wonder what is going to be the form of balance equations of the HEM. For example, from div σ∼ + ρf − ρ v˙ = 0 in the heterogeneous medium, applying mathematical expectation leads to: div σ∼ + ρf − ρ v˙ = ρv .grad v + ρ v˙ − ρ f ∼
where a is the difference a − a. We can see that the form of the global momentum balance resembles but is a priori different from the initial form. Under the statistical homogeneity hypothesis and low loading gradients, the balance law is stable through mathematical expectation. In the case of coupled thermoelasticity, we will see, thanks to asymptotic methods, that the form of the balance equations is preserved by homogenization, at least at first order. In some cases it is then possible to recognize in the averaged balance equations the form of the balance equations of a known continuum medium, which allows us to model the heterogeneous medium by a homogeneous equivalent one. One should note that the global form of the balance equations is not necessarily the same as the local ones. We can mention the case of filtration in a rigid porous medium for which the local level is governed by Stokes’ laws whereas the global level is governed by Darcy’s. It is then necessary to think first about the form of the global balance equations before even wondering about the constitutive laws. Constitutive laws are themselves not always stable through homogenization: For example, a mixture of Maxwell-type viscoelastic linear phases can give at a global level a viscoelastic medium with a continuous spectrum. Applying the expectation operator on the energy balance equation leads to a complex expression in which one can recognize an energy balance equation between
208
Non-Linear Mechanics of Materials
macroscopic variables if
σ∼ : ε∼˙ = σ∼ : ε∼˙
(5.8)
called Hill-Mandel condition. This condition is sometimes called Hill’s macro-homogeneity condition, insofar as they make it possible to define an HEM. In this book (except for the last Sect. 5.8), we will assume that the macro-homogeneity conditions are fulfilled so that the form of the governing balance equations is the same locally and globally.
5.1.4. Local behavior of phases: status of phenomenology in the mechanics of heterogeneous materials Heterogeneous materials mechanics claims to take advantage of materials science, but it cannot completely avoid mere phenomenology. The latter is simply pushed down to the behavior of the local level, to describe the behavior of each phase. The work on phenomenology is all the more delicate, which can, in some cases, ruin hopes attached to homogenization. Indeed, mechanical behavior of a constituent inside a heterogeneous material is not always the same as the one observed when it is isolated. When, generally, macroscopic mixing does not affect the properties of the constituents (large size components), it is not the same for a much thinner mixing. It is obvious that elastic properties of a Ni3 Al alloy cannot be obtained by homogenizing elastic properties of nickel and aluminium! There exists, in general, a distance d0 between the heterogeneities below which it is necessary to question the relevance of the local behavior that has been assumed. This distance can actually be very small (nm) for elastic properties but larger in the nonlinear case (1 μm). In the extreme example of Ni3 Al above, there is no point in homogenizing atoms; it is necessary to consider a new phase (other techniques of solid state physics allow one to predict in some cases elastic properties of alloys). For distances d d0 , the elastic behavior is in general less surprising but local damage phenomena can occur prematurely. In the nonlinear case, things are very different. Elastoplastic behavior of the grains of a polycrystal has actually very little in common with the (generally softer) behavior of the isolated single crystal, except when the grains are very large. Data related to a single crystal are not sufficient to predict the behavior of the polycrystal, because of the size effects associated with the presence of grain boundaries. In what follows, when talking about the mechanical behavior of a constituent, we mean its in-situ behavior, inside the aggregate. This leads to the problem of the experimental measurement of the local behavior of phases, which represents one of the main branches of heterogeneous materials mechanics, called micromechanics. As it is almost impossible to load homogeneously a
Elements of microstructural mechanics
209
single constituent, we always deal with a structural (or, better, a microstructural) problem. Measurement of local strain fields is a current experimental challenge. [ALL94, ZIE96, BOR01, DOU03] present a collection of standard or under-development methods, from grid methods to measurement of residual stress by X-ray, neutron diffraction or laser interferometry. The micro- (or even nano-)hardness is a simple and useful tool but still difficult to interpret. The key difficulties associated with local behavior measurement can be summarized in two words: surface, interface. The currently available measurements provide surface data and give access to the behavior at the surface of each phase material. However, the nonlinear behavior of materials is often different at the surface and in the bulk. It is the case, in particular, of metallic polycrystals. In addition to the zero traction condition that holds at the surface, the outlet of dislocations at the surface can make the surface grain softer than the bulk one. It is one of the key notions to understand fatigue of materials. Generally, the altered boundary layer is of the order of a few grains. The currently available homogenization schemes concern only the bulk behavior. Taking surface effects into account is a necessary and promising research topic. It follows that, on the one hand, a part of the prediction of the homogenization model relies often on an approximative knowledge of the local behavior and, on the other hand, that comparing the prediction of the local strain heterogeneities provided by these models with experimental results is generally not satisfying (however, it is generally not possible to know whether this gap is due to the quality of the homogenization scheme or to the relevance of the comparison itself (comparing values predicted for the bulk to surface observations)). The knowledge of interface or interphase behavior between components is often also very limited. We will distinguish the passive interfaces from the active ones (concrete during setting, for example, or corrosion or humidity influence). The passive ones can damage. In this chapter, however, interfaces will be assumed perfect: no displacement discontinuity (neither crack, nor slip) and transmission of load without loss (no fluid at interfaces). The whole toolbox of linear and nonlinear constitutive laws can then be used to describe heterogeneous local behavior. They are not, contrary to what is commonly believed, necessarily more simple than the macroscopic phenomenological laws that can describe the behavior of the aggregate. Besides, they must imperatively be validated under complex multiaxial or cyclic loading, because the stress and strain states inside a heterogeneity can be more severe than the macroscopic stress and strain in the structure. Phenomenology has then an essential role to play at that level, in conjunction with a sophisticated experimental study. The development of heterogeneous materials mechanics was then accompanied by a growth of the library of nonlinear models of behavior. We will only mention the increasing use of non local models (resort to generalized continuum) in order to describe the behavior of some constituents and the associated scale effects. Taking into account the latter (influence of the size of the constituents) is an important future development of heterogeneous materials me-
210
Non-Linear Mechanics of Materials
chanics that is not addressed in the present book (see for example [FOR00, CAI03, FOR05, FOR06a]).
5.2. Some homogenization techniques The following methods are presented under the small strain assumptions and for the isothermal case. Moreover, we assume that the RVE does not contain any hole or rigid component.
5.2.1. Averaging procedures We saw in Sect. 5.1.3 that spatial or ensemble averaging some physical quantities can be useful to predict the overall response of heterogeneous materials. In fact, it is important to be able to link volume integrals over V to surface integrals over its boundary ∂V where boundary conditions are prescribed. This is possible by means of the divergence or Gauss theorem. Spatial averaging of a compatible strain field ε∼ is written as 1 1 ε∼ dV = u dV ei ⊗ ej ε∼ = V V V V (i,j ) 1 = u nj ) dSei ⊗ ej V ∂V (i 1 } { = u ⊗ n dS V ∂V
(5.9) (5.10) (5.11)
where the braces refer to the symmetric part. The repeated indices are summed over. The parentheses around indices indicate symmetrization with respect to these indices. A Cartesian system of coordinates is used. The spatial average of a statically admissible stress field σ∼ ∗ , i.e., such that div σ∼ ∗ = 0 (noted SA in this chapter), is written as 1 1 σ∼ ∗ dV = σ ∗ dV ei ⊗ ej (5.12) σ∼ ∗ = V V V V ij 1 = σ ∗ δj )k dV ei ⊗ ej (5.13) V V (ik 1 = σ ∗ xj ),k dV ei ⊗ ej (5.14) V V (ik 1 = (σ ∗ xj ) ),k dV ei ⊗ ej (5.15) V V (ik
Elements of microstructural mechanics
1 σ ∗ nk xj ) dSei ⊗ ej V ∂V (ik 1 { ∗ = (σ .n) ⊗ x } dS V ∂V ∼
=
211
(5.16) (5.17)
For these calculations, Gauss’ theorem has been applied for the case where the force and displacement fields are continuous in the bulk V , which means that interfaces are perfect. Volume forces have been excluded for simplicity. If discontinuity surfaces and the volume forces f exist in the investigated volume, we get: 1 1 } } { { ε∼ = u ⊗ n dS + [[u ]] ⊗ n dS (5.18) V ∂V V 1 σ∼ = V ∗
{
∂V
1 + V
1 (σ∼ .n) ⊗ x dS + V }
∗
{
∗
{
}
(f − ρ v˙ ) ⊗ x dV
V
}
[[σ∼ .n]] ⊗ x dS
(5.19)
The quantity [[a]] denotes the jump (a + −a − ) of a through with associated normal n. The average of the work of internal forces is written as: 1 1 1 σ∼ ∗ : ε∼ = (σ∼ ∗ .n).udS + (f − ρ v˙ ) ⊗ u dV + [[(σ ∗ .n).u ]]dS V ∂V V V V ∼ (5.20)
5.2.2. Micromechanical problem; boundary conditions We consider the following boundary value problem with initial values, where the unknowns are the displacement field u and, possibly the fields of internal variables obeying some evolution laws: ⎧ εij = u(i,j ) ⎪ ⎪ ⎨ constitutive equations (5.21) σij,j = 0 ∀x ∈ V ⎪ ⎪ ⎩ boundary conditions and initial values where V is the chosen RVE. We assume that the local constitutive laws are known. This problem is in fact ill-posed for two reasons. First, in the case of a random material, the geometry of V is known only statistically and, second, we do not know precisely to which boundary conditions such a volume inside the macroscopicallyloaded heterogeneous material is subjected. To solve the first problem, we can either be satisfied with only one realization of V (V must then be large enough), which,
212
Non-Linear Mechanics of Materials
according to ergodic hypothesis, contains all the necessary statistical information, or, if this description is too demanding, consider successive realizations. At least three kinds of boundary conditions are then proposed: • Kinematic uniform boundary conditions (KUBC problem): .x u=E ∼
∀x ∈ ∂V
(5.22)
where E is a symmetric tensor that does not depend on x. ∼ • Static uniform boundary conditions (SUBC problem): σ∼ .n = .n ∼
∀x ∈ ∂V
(5.23)
where is a symmetric tensor that does not depend on x. ∼ • periodicity conditions (PER problem): when the medium is periodic, the element V is known in its smallest geometrical details and its shape makes it possible to tile space by translation. We look then for a displacement field of the form: .x + v ∀x ∈ V (5.24) u=E ∼ where v is periodic, i.e., v takes the same values on opposite sides of V ; moreover we impose that the stress vector σ∼ .n takes opposite values on opposite is prescribed but there exists also a dual sides. Here the constant strain tensor E ∼ formulation of this periodic problem. We can then prove the existence and uniqueness of the solution of these three boundary problems, at least in the linear case. In every case, if a solution u exists, then it follows from the calculations of Sect. 5.2.1 that: ε∼ = E ∼
(5.25)
in the case KUBC at the boundary and also in the periodic case, and σ∼ = ∼
(5.26)
for dual conditions SUBC.
5.2.3. Hill–Mandel lemma Let σ∼ ∗ and ε∼ be two fields, respectively of statically admissible stresses (div σ∼ ∗ = 0) and compatible strains ε∼ = { gradu } (noted KA in this chapter), not necessarily linked by the constitutive law. Then, if σ∼ ∗ verifies the SUBC conditions at the boundary or
Elements of microstructural mechanics
213
if ε∼ verifies the KUBC conditions of homogeneous strains at the boundary, or if both σ∼ ∗ and ε∼ verify the periodicity conditions, then: σ∼ ∗ : ε∼ = σ∼ ∗ : ε∼
(5.27)
The proof uses the same tools as in Sect. 5.2.1. By applying this lemma to the solution field itself for each kind of boundary conditions, we see that the three kinds of conditions make it possible to satisfy automatically Hill’s macro-homogeneity condition (5.8): σ∼ : ε∼ = σ∼ : ε∼
(5.28)
If some hypothesis formulated in Sect. 5.1.4 concerning V is not fulfilled, the previous equality is not a priori verified for any kind of boundary condition. Equation (5.28) prompts us to define the macroscopic stress tensor: = σ∼ ∼
(5.29)
in the case of homogeneous strain at the boundary and if E corresponds to the macro∼ scopic strain, and = ε∼ (5.30) E ∼ for the dual conditions SUBC, where corresponds to the macroscopic stresses. ∼
5.2.4. Periodic case: use of multiscale asymptotic expansions In the case of the periodic or quasi-periodic structures, we can define a parameter ε = l/Lω , which is small in the case of the slowly varying macroscopic fields. We consider then a basic cell Y = [0, 1]3 that permits us to build the microstructure of the material by successive translations, but without giving the variable y ∈ Y a precise unit (mm, μm. . .). We just say that the real cell is small, of the kind Y ε = εY . The actual coordinate is then approached by x = εy. It is as if the unit cell Y were a microscopic image on which we had forgotten to put the scale or the magnification, and we would merely say, instead, that it is small. Inside Y ε reign the fields uε and σ∼ ε , and the method consists in replacing them by their limit when ε tends towards 0. For that purpose, thanks to the so-called multiscale asymptotic expansion method, we look for an expansion of the quantities as: uε (x) = u0 (x, y) + εu1 (x, y) + ε 2 u2 (x, y) + ε 3 u3 (x, y) + · · ·
(5.31)
σ∼ ε (x) = σ∼ 0 (x, y) + εσ 1 (x, y) + ε 2 σ∼ 2 (x, y) + ε 3 σ∼ 3 (x, y) + · · ·
(5.32)
The functions that have been introduced (ui for example) depend on the global variable x and on the local variable y = x/ε. They are supposed to be of the same order
214
Non-Linear Mechanics of Materials
of magnitude and to be periodic with respect to the variable y. A function f ε (x) is associated with the function f (x, y), whose derivative is calculated with respect to x as follows: d ε ∂ ∂ f = f + 1/ε f (5.33) f ε (x) = f (x, y) dx ∂x ∂y The main interest of this method compared to Hill–Mandel’s approach (already presented above) appears essentially when the form of the balance and constitutive equations of the effective medium is not known a priori. The systematic application of this method allows us to deduce the form of these equations from the knowledge of local equations. An example will be given in the framework of thermoelasticity.
5.3. Application to linear elastic heterogeneous materials The local behavior of each component is assumed linear elastic in the isothermal and small strain framework.
5.3.1. Stress-strain concentration problem; effective moduli Concentration tensors Each of the three micromechanical problems (KUBC, SUBC, PER) admits a unique solution in terms of stresses and strains depending linearly on E (or ). There exists ∼ ∼ such that: then a concentration tensor field A ∼ ∼
ε∼ (x) = A (x) : E ∼ ∼
∀x ∈ V and ∀E ∼
∼
(5.34)
for the problem KUBC, and a tensor field B such that: ∼ ∼
σ∼ = B (x) : ∼ ∼ ∼
∀x ∈ V and ∀ ∼
(5.35)
for the dual problem SUBC. Taking into account (5.25) and (5.26), the concentration tensors are such that: A = B = I∼ (5.36) ∼ ∼ ∼
∼
∼
where the components of the identity tensor acting on the symmetrical tensors are: Iij kl =
1 (δik δj l + δil δj k ) 2
Direct definition of the effective moduli
Elements of microstructural mechanics
215
Let ∼c(x) and ∼s (x) be the fields of elasticity moduli and compliances of the hetero∼
∼
geneous material:
σ∼ (x) = ∼c(x) : ε∼ (x) ∼
∀x ∈ V
(5.37)
It follows, for the KUBC problem, that := σ∼ = c∼ : A :E = c∼ : A :E ∼ ∼ ∼ ∼ ∼ ∼
∼
(5.38)
∼
∼
and for the SUBC problem: E := ε∼ = s∼ : σ∼ = s∼ : B : = s∼ : B : ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
(5.39)
These properties allow us to define the effective elasticity moduli and effective compliances: h = c∼ : A S∼ h = s∼ : B (5.40) C ∼ ∼ ∼E ∼
∼
∼
∼
∼
∼
These expressions show that the effective tensors cannot be obtained from a simple “mixture law” but are modulated by the concentration tensors. Energetic definition We can also define the effective moduli from the average of twice the elastic energy: t σ∼ : ε∼ = ε∼ : ∼c : ε∼ = E : A : ∼c : A :E (5.41) ∼ ∼ ∼ ∼ ∼
∼
∼
∼
for the KUBC problem, and for the dual one: t σ∼ : ε∼ = σ∼ : ∼s : σ∼ = : B : ∼s : B : ∼ ∼ ∼ ∼
(5.42)
h t C = A : ∼c : A ∼ ∼ ∼ E
(5.43)
∼
∼
so that
∼
∼
∼
∼
∼
∼
t S∼ h = B : ∼s : B ∼ ∼ ∼
∼
∼
∼
We can show, thanks to the Hill–Mandel lemma and property (5.36) that both definitions (direct and energetic) are equivalent and that: h h C =C ∼ E ∼E ∼
∼
S∼ h = S∼ h
∼
(5.44)
∼
The definitions (5.43) show the symmetry of the homogenized tensors. h = (S h )−1 . We can only Generally, there is no reason why we should expect C ∼E ∼
∼
∼
establish the following property (which will be proved in Sect. 5.3.3): h (S∼ h )−1 ≤ C ∼E ∼
∼
(5.45)
216
Non-Linear Mechanics of Materials
in the sense of quadratic forms. Macrohomogeneity of the heterogeneous material corresponds to the case where the effective properties do not depend on the chosen boundary conditions. It is in this case only that we use the concept of RVE in its strict meaning. In the periodic case, macrohomogeneity is ensured. In the statistically uniform random case, the result is still verified and it is possible to build a RVE. The technical proof of this result can be h = (S∼ h )−1 and the indices can be suppressed. found in [SAB92]. In that case, C ∼ ∼E
∼
5.3.2. Variational formulations; bounds The theorem of minimum potential energy specifies that the solution field u of the KUBC problem realizes the minimum of W (u ) − (u ) among all the kinematically admissible displacement fields u (i.e., continuous and such that u (x) = E .x over ∼ ∂V ). The work of internal forces is: 1 ε(u) : c∼ : ε∼ (u)dV (5.46) W (u) = 2 V∼ ∼ and the work of the given external-forces (u) is zero in the case of the KUBC problem: (W (u ) − (u )) (5.47) W (u) − (u) = min u C.A.
And, according to the Hill–Mandel lemma, W (u) − (u) =
V V V : Ch : E σ∼ : ε∼ = σ∼ : ε∼ = E ∼ 2 2 2 ∼ ∼∼
Hence V V h :C :E ≤ ε∼ : c∼ : ε∼ E ∼ ∼ ∼ 2 2 ∼ ∼
(5.48)
The theorem of minimum complementary energy stipulates that the stress field solution of the SUBC problem realizes the minimum of W ∗ (σ∼ ∗ ) − ∗ (σ∼ ∗ ) over all the stress fields statistically admissible where 1 ∗ ∗ σ ∗ : s : σ ∗ dV (5.49) W (σ∼ ) = 2 V ∼ ∼∼ ∼ and ∗ the work of the displacements given in the field σ∼ ∗ : ∗ (σ∼ ∗ ) =
∂V
ud .(σ∼ ∗ n)dS =
∂V
(E .x).(σ∼ ∗ .n)dS = E : ∼ ∼
V
σ∼ ∗ dV
(5.50)
Elements of microstructural mechanics
217
It follows that 1 W (σ∼ ) − (σ∼ ) ≥ 2 ∗
Hence
∗
∗
∗
σ∼ : ε∼ dV − E : σ∼ dV ∼ V V 1 σ∼ : ε∼ − E ≥V : σ ∼ ∼ 2 V : Ch : E ≥− E ∼ 2 ∼ ∼∼
(5.51) (5.52) (5.53)
1 ∗ 1 h ∗ ∗ σ : ∼s : σ∼ − E − E :C :E = min : σ∼ ∼ ∼ σ ∗ S.A. 2 ∼ 2 ∼ ∼∼ ∼ ∼
(5.54)
1 1 h :C :E ≥ σ∼ ∗ : E − σ∼ ∗ : ∼s : σ∼ ∗ E ∼ ∼ ∼ ∼ 2 2 ∼ ∼
(5.55)
i.e.,
The following inequalities are deduced: 1 1 1 σ∼ ∗ : E − σ∼ ∗ : ∼s : σ∼ ∗ ≤ E : Ch : E ≤ ε∼ : ∼c : ε∼ ∼ ∼ 2 2 ∼ ∼∼ 2 ∼ ∼ ∗ ∀(u , σ∼ ) KA and SA respectively.
(5.56)
The stress based approach (solving the SUBC problem) gives: 1 1 1 ε∼ : − ε∼ : ∼c : ε∼ ≤ : S∼ h : ≤ σ∼ ∗ : ∼s : σ∼ ∗ ∼ ∼ ∼ 2 2 2 ∼ ∼ ∼
(5.57)
The previous inequalities provide the mean, by choosing judiciously the admissible fields, to bound the quadratic forms associated with the homogenized moduli. The quality of the resulting bounds depends on the available quantity of information concerning the microstructure of the heterogeneous material. An optimal bound, for given information, corresponds to the case where no narrowing of the bounds is possible without adding a new information about the distribution of heterogeneities. We have then exhausted the available statistical information. Zeroth order bounds M The rhs of (5.48) can be upper-bounded by introducing C , the tensor of the ∼ ∼
moduli of the stiffest component. Then we choose some admissible constant testfields. There is only ε∼ = E left. It follows that ∼ 1 1 E : Ch : E ≤ E : CM : E ∼ ∼ 2 ∼ ∼∼ 2 ∼ ∼∼
∀E ∼
(5.58)
218
Non-Linear Mechanics of Materials
m −1 m By upper-bounding the rhs of (5.57) by means of S∼ M = (C ) , where C represents ∼ ∼ ∼
∼
∼
the moduli of the softest component, and by introducing some constant test-fields (necessarily σ∼ ∗ = ), we get: ∼ 1 1 ≤ : SM : : Sh : ∼ ∼ 2 ∼ ∼∼ 2 ∼ ∼∼
∀ ∼
(5.59)
We will note more concisely the resulting inequalities as follows: m h M C ≤C ≤C ∼ ∼ ∼ ∼
∼
(5.60)
∼
These bounds are optimal when only the nature of the components is known. M , to test every In the anisotropic case, we will pay attention, while building C ∼ ∼
possible orientation, if no preferred orientation of the phase is known. It follows in M is necessarily isotropic. that case that C ∼ ∼
First order bounds When, in addition, the volume fractions of the components are known, we get the Voigt and Reuss bounds. For that purpose, we use in (5.48) the constant test fields ε∼ = ε∼ 0 and σ∼ ∗ = σ∼ 0 . , hence: In the case of the KUBC problem, we have necessarily ε∼ 0 = E ∼ h ≤ c∼ C ∼ ∼
∼
(Voigt)
(5.61)
With the fields σ∼ 0 , we try to maximize the lhs of (5.48), this maximum is reached for: σ∼ 0 = s∼−1 : E ∼
(5.62)
∼
hence
h s∼−1 ≤ C ∼ ∼
(Reuss)
∼
(5.63)
The bounding is then written h ≤ c∼ c∼−1 −1 ≤ C ∼ ∼
∼
∼
(5.64)
These bounds are optimal if the only available information concerns the volume fraction of the components. To establish the bounds of higher order based on variational methods, more elaborate admissible fields than the constant ones must be generated. The so-called polarization field method allows us to build such admissible fields (see [BOR01]). The second order bounds are presented as an example (Sect. 5.4). The setting of the successive bounds can also be deduced from the statistical approach presented in Sect. 5.3.4.
Elements of microstructural mechanics
219
5.3.3. Case of a macro-heterogeneous medium In the case of concrete, for example, the volumes V experimentally loaded can be smaller than the minimum RVE. It suffices to think of the experimental devices that would be necessary to characterize the concrete of a bridge (heterogeneities of the order of 1 meter)! It is possible, however, to assess the effective properties of the heterogeneous medium by considering ensemble averages over various experiments. app under For a given specimen, one can experimentally determine apparent moduli C ∼E app
∼
KUBC strain loading, apparent compliances S∼ under SUBC stress loading. In that app
case, we have a priori C ∼E now inequality (5.45).
∼
∼
= (S∼ )−1 . On the other hand, we are going to prove app
∼
We consider the solution (u1 , ε∼ 1 , σ∼ 1 ) of the KUBC problem with E given and the ∼ given. According to the theorem solution (u2 , ε∼ 2 , σ∼ 2 ) of the SUBC problem with ∼ of complementary energy for the KUBC problem, and since σ∼ 2 is SA for this problem, W ∗ (σ∼ 1 ) − ∗ (σ∼ 1 ) ≤ W ∗ (σ∼ 2 ) − ∗ (σ∼ 2 ) with
1 σ∼ 1 : ∼s : σ∼ 1 dV − u1 .(σ∼ 1 .n)dS 2 V ∼ ∂V V = σ∼ 1 : ε∼ 1 − V E : σ∼ 1 ∼ 2 V = − σ∼ 1 : ε∼ 1 2 V app :C :E =− E ∼ 2 ∼ ∼∼ E
(5.65)
W ∗ (σ∼ 1 ) − ∗ (σ∼ 1 ) =
and
1 W (σ∼ 2 ) − (σ∼ 2 ) = σ 2 : s : σ 2 dV − u1 .(σ∼ 2 .n)dS 2 V ∼ ∼∼ ∼ ∂V V app = :S : −VE : σ∼ 2 ∼ ∼ 2 ∼ ∼∼
V app = :S : −VE : ∼ ∼ ∼ 2 ∼ ∼∼
∗
(5.66)
∗
(5.67)
Hence, 1 1 app app E : − :S : ≤ E :C :E (5.68) ∼ ∼ ∼ ∼ 2 ∼ ∼∼
2 ∼ ∼∼ E Let us fix E and look for the maximum of the lhs. This maximum is obtained for ∼ = (S∼ )−1 : E ∼ ∼ app
∼
220
Non-Linear Mechanics of Materials
Hence
= (S∼ )−1 ≤ C ∼E
app
app
C ∼
∼
app
app
∼
(5.69)
∼
app
Let now C and C be the apparent moduli of a volume V obtained respec∼ EV ∼ V ∼
∼
tively with KUBC boundary conditions (E ) and SUBC boundary conditions ( ). Let ∼ ∼ app app and C of volumes V us assume that we only have access to the moduli C α ⊂ V, ∼ Eα ∼ α ∼
∼
app
app
of lower size. We are going to see that it is then possible to bound C and C . ∼ EV ∼ V ∼
∼
For this, we consider a partition {Vα , α = 1, n} of V such that ∪Vα = V without intersection. It is actually a batch of identical specimens cut from V . We determine or uniform at the the apparent moduli of each specimen under loading condition E ∼ ∼ boundary. We consider then the problem KUBC on V (the same imposed strain E ). Let ε∼ α be ∼ the solution of the problem KUBC on Vα . We then build ε∼ on V as the union of these fields, i.e., ε∼ (x) = ε∼ α (x) if x ∈ Vα . It is a kinematically admissible field because the displacements u are continuous due to the deformation condition imposed on the boundary. We can then apply the potential energy theorem on ε∼ (exact solution of the problem KUBC over V ) and ε∼ : W (u) − (u) ≤ W (u ) − (u )
(5.70)
ε∼ : ∼c : ε∼ ≤ ε∼ : ∼c : ε∼
(5.71)
which gives:
∼
∼
where the average is made over V . Yet app
ε∼ : ∼c : ε∼ = σ∼ : ε∼ = E :C :E ∼ ∼ ∼ EV ∼
∼
and ε∼ : ∼c : ε∼ =
Vα
∼
V
α
=
∼
Vα V
α
ε∼ α : ∼c : ε∼ α Vα app
E :C :E ∼ ∼ ∼ Eα ∼
app
=E :C ∼ ∼E ∼
:E ∼
where the statistical apparent moduli are defined by app
CE
=
Vα α
We deduce:
V
app
∼
app
≤C C ∼ EV ∼E ∼
app
C ∼ Eα
∼
(5.72)
(5.73)
Elements of microstructural mechanics
221
The complementary energy theorem applies then to the problem SUBC and gives: app
C ∼
∼
app
≤C ∼ V
(5.74)
∼
We then get the following chain of inequalities app
app
app
app
≤C ≤C ≤C ∼ V ∼ EV ∼E
C ∼
∼
∼
∼
(5.75)
∼
If V is a RVE, the apparent moduli over V coincide and then: app
C ∼
∼
app
h ≤C ≤C ∼ ∼E ∼
(5.76)
∼
app
where the CE, are determined from n experiments on specimens smaller than V . One can find more results on this subject in [HUE90] and application to various materials in [KAN03, KAN06].
5.3.4. Statistical methods A systematic theory developed by [KRÖ81] is outlined here. It enables us to set narrower bounds thanks to the available information in terms of correlation functions and introduces the notion of perfect disorder. Because of the rather high technicality of this method, it is possible to address it only in a second reading.
Integral equation of heterogeneous elasticity We consider a heterogeneous elastic solid whose local behavior is given by the random stiffness ∼c(r): ∼
σ∼ (r) = ∼c(r) : ε∼ (r) ∼
at r (the method would apply also to a non-local form σ∼ (r) =
(5.77)
c(r, r ∼ ∼
)
: ε∼ (r )dr ).
We choose a reference homogeneous medium described by the tensor of elasticity 0 and we define moduli C ∼ ∼
0 δc∼ = ∼c − C . ∼ ∼
∼
∼
The equilibrium equations are then written 0 div C : ε : ε δc =0 + div ∼ ∼ ∼ ∼ ∼
∼
(5.78)
(5.79)
222
Non-Linear Mechanics of Materials
If we consider δc∼ : ε∼ as a polarizing function, i.e. div δc∼ : ε∼ as a volume force, the ∼
∼
displacement field, solution of (5.79), is given by: 0 Gij (r, r )(δc∼ : ε∼ )j k,k dr ui (r) = ui (r) +
(5.80)
∼
0 where u0 is the solution of the problem associated with C , and the Green tensor G ∼ ∼ ∼
associated with the reference medium is defined by Cij0 kl Gkm,lj (r, r ) + δ(r, r )δim = 0
Gim (r, r ) = 0
∀r, r ∈
(5.81)
∀r ∈ ∂ , ∀r ∈
(5.82)
for displacements prescribed on ∂ (see [KRÖ76] for surface forces). Equation (5.80) can be written Gij,k (r, r )δcj kmn (r )um,n (r )dr (5.83) ui (r) = u0i (r) −
By defining the Green operator associated with the reference medium inj k (r, r ) = Gij,k n (r, r )(in)(j k)
(5.84)
we can deduce the strain field: 0 − εin = εin
inj k δcj kpq εpq dr
(5.85)
By using symbolic notation, the previous integral equation can be written: : δc∼) ∗ ε∼ = ε∼ 0 ε∼ + ( ∼ ∼
∼
(5.86)
which is called the Lippmann–Schwinger equation. A discussion about convergence of the integral and singularity of the Green operator for the case of finite or infinite solids can be found in [ZAO93a, BOR01].
Equation of the effective moduli The effective elasticity moduli characterizing the homogeneous equivalent medium can be defined as eff : ε∼ σ∼ = c∼ : ε∼ = C ∼ ∼
∼
ε∼ = s∼ : σ∼ = S∼ eff : σ∼ ∼
∼
Elements of microstructural mechanics
223
We adopt the ergodic hypothesis in order to consider only volume averages. We recall the energetic definition of the effective properties: eff : ε∼ ε∼ : ∼c : ε∼ = ε∼ : C ∼ ∼
∼
σ∼ : ∼s : σ∼ = σ∼ : S∼ eff : σ∼ ∼
∼
Both definitions coincide if Hill-Mandel conditions are fulfilled σ∼ : ε∼ = σ∼ : ε∼
(5.87)
In particular (5.87) is verified in the case of homogeneous boundary conditions ud = ε∼ .r
∀r ∈ ∂
T d = σ∼ .n
∀r ∈ ∂
or Equation (5.86) can be symbolically inverted ε∼ = (I∼ + : δc∼)−1 ∗ ε∼ 0 ∼ ∼
∼
ε∼ = (I∼ + : δc∼)−1 ∗ ε∼ 0 ∼ ∼
Hence
∼
: δc∼)−1 ∗ (I∼ + : δc∼)−1 −1 : ε∼ . ε∼ = (I∼ + ∼ ∼ ∼
∼
∼
∼
It follows that eff C = c∼ : (I∼ + : δc∼)−1 ∗ (I∼ + : δc∼)−1 −1 ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
A similar analysis exists for effective compliances. We introduce the fourth-rank tensor: 0 0 0 =C −C : :C ∼ ∼ ∼ ∼ ∼ (as σ∼ =
σ0 ∼
+C : ∼ 0
∼
∼
∼ ∼
∼
:C : δs∼ : ∼ 0
∼
∼
∼
∼
σ dr
∼
−C : δs∼) and we find ∼ 0
∼
∼
∼
: δs∼)−1 ∗ (I∼ + : δs∼)−1 −1 S∼ eff = s∼ : (I∼ + ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
0 eff By posing C =C , we get an implicit integral equation: ∼ ∼ ∼
∼
: δc∼)−1 = 0 δc∼ : (I∼ + ∼ ∼
∼
∼
∼
∼
224
Non-Linear Mechanics of Materials
Bounds Kröner [KRÖ77] has obtained bounds for the effective elastic behavior of heterogeneous materials. He introduces the projection operator P Pf = f = f − f By applying it to (5.86): ε∼ + (P : δc∼) ∗ ε∼ + (P : δc∼) ∗ ε∼ = 0 ∼ ∼ ∼
∼
∼
∼
and multiplying by δc∼ and then averaging, Kröner gets (dropping the stars), ∼
)−1 : δc∼ : P : δc∼ : ε∼ δc∼ : ε∼ = −(I∼ + δc∼ : P ∼ ∼ ∼
∼
On the other hand
∼
∼
∼
eff : ε∼ − σ∼ : ε∼ δc∼ : ε∼ = C ∼ ∼
Hence
∼
∼
eff = c∼ − (I∼ + δc : P )−1 : δc∼ : P : δc∼ C ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
(5.88)
Kröner gives a more handy form: eff C = c∼ : B ∼ ∼ ∼
∼
∼
with B = (I∼ + P : δc∼)−1 ∼ ∼ ∼
∼
∼
∼
(5.89)
0 (the property C : P : δc∼ = 0 has been used). To obtain the preceding form, the ∼ ∼ ∼
∼
∼
following expansion in series was used B = I∼ − P : δc∼ + P : δc∼ : P : δc∼ + · · · ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
(5.90)
whose convergence is ensured for example when c∼−1 : ∼c < 1 (inequality in the ∼
∼
sense of quadratic forms). This means that this convergence may not be ensured for every heterogeneous material [KRÖ76], however the convergence of the series can be proved for any statistically-uniform isotropic finite medium, without pores or rigid inclusions. We saw that the principle of minimum energy gives: eff : ε∼ ≤ ε∼ : c∼ : ε∼ ε∼ : C ∼ ∼
∼
σ∼ : S∼ eff : σ∼ ≤ σ∼ : ∼s : σ∼ ∼
ε
where ∼ and
σ ∼
∼
are admissible test-fields such that ε∼ = ε∼ and σ∼ = σ∼
Elements of microstructural mechanics
225
The most general test functions take the form (I∼ + P : τ∼ ) : ε∼ where τ∼ is arbitrary ∼ ∼
∼
∼
∼
(see [KRÖ77]). A particularly useful set of test functions, B : ε∼ , is obtained by ∼m ∼
truncating the series (5.90) after the m-th term (m = 0, 1, 2, . . .):
B = I∼ − P : δc∼ + (P : δc∼)2 + · · · + (−1)m (P : δc∼)m ∼m ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
(i.e., B = I∼ − P : δc∼ : B ). The test function B : ε∼ = B : ε∼ provides the ∼m ∼ ∼ m−1 ∼∞ ∼ ∼
∼
∼
∼
∼
∼
∼
eff exact solution C if the B are formed with the same Green operator as B . ∼m ∼ ∼ ∼
∼
∼
By using the relation T B :c:B = c∼ : B ∼m ∼ ∼m ∼ n−1 ∼
∼
∼
∼
∼
with n = 2m + 1, we get the upper bounds eff 0 (n) C ≤C + δc∼ : B =C ∼ n−1 ∼ ∼ ∼ ∼
∼
∼
∼
∼
Equation (5.89) provides an expansion in series of the effective moduli eff 0 C =C + δc∼ − c∼ : : ∼c + c∼ : : δc∼ : : ∼c ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
: δc∼) : : (δc∼ : : ∼c ) − (c∼ : ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
: δc∼) : : δc∼ : (δc∼ : : ∼c ) + · · · + (c∼ : ∼ ∼ ∼ ∼ ∼
∼
∼
∼
(5.91)
∼
(n) as a result the bounds C are obtained by truncating the series after n = 2m + 1 ∼ ∼
terms. The result holds whatever the chosen reference medium is. This is why these (n) bounds are called odd upper-bounds of odd order. Kröner showed that the C for n ∼ ∼
0 (0) (0) even are also upper bounds as long as C ≤C , where C represents the 0 order ∼ ∼ ∼ ∼
∼
∼
(−n) bounds already encountered in the previous section. Likewise lower bounds C ∼ ∼
can be obtained using (5.3.4).
Optimal bounds and estimates for effective properties of random materials (±n) are not always optimal bounds based on the statistical The previous bounds C ∼ ∼
information available about the structure of the material. If no information other than (±0) the nature of the components is available, bounds of order 0 C are optimal. The ∼ ∼
(±1) C are the Voigt–Reuss bounds and are optimal if only the volume fractions of the ∼ ∼
components are known. Statistical information can be given in terms of correlation functions of the random variables ∼c up to order n. In the series (5.91) the term c∼ : ∼
∼
226
Non-Linear Mechanics of Materials
: ∼c holds the two-point correlation function c∼(x) : ∼c(x ) [KRÖ81]. Likewise, ∼ ∼
∼
∼
∼
the higher order terms involve higher order correlation functions. Optimal bounds of order n ≥ 2 can be obtained by assuming that all the correlation functions are known up to and including order n. Only the knowledge of the infinite set of correlation eff . functions leads to the exact value of C ∼ ∼
C ∼
(2 )
= c∼ − c∼ : : ∼c ∼ ∼
∼
∼
∼
∼
0 (0) gives the optimal bound of order 2 if the Green operator is formed with C =C . ∼ ∼ ∼
The optimal bound of order 3 is
∼
(3 ) C = c∼ − c∼ : : ∼c : c∼ : : ∼c + ∼c : : δc∼ : : ∼c −1 : c∼ : : ∼c ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
(actually, this expression does not depend on the choice of the reference medium). Statistical information must describe on the one hand the material topology, i.e., the distribution of the geometric domains and, on the other hand, tell how to “fill” them with material parameters. If both distributions are uncorrelated, the medium is called statistically decoupled. This leads to important simplifications in the theory, to the extent that the correlation functions of ∼c can be factorized in a part hold∼
ing the moments of the material parameters and a function describing the topology [KRÖ81]. If the topological structure is decorrelated in that shape and size of the domains are completely or partially irregular, the statistically decoupled medium is moreover called statistically disordered. In this case, it is sufficient to use Green functions of the infinite medium, so that the Green operator can be decomposed in two parts: (r, r ) = E + F∼ ∼ ∼ ∼
with
∼
ˇ δ(r, r ) and =E E ∼ ∼ ∼
∼
∼
F∼ = F∼ˇ /r − r 3 ∼
∼
ˇ is constant and Fˇ depends only on angles. With the hypothesis of perfect where E ∼ ∼ ∼
∼
or gradual disorder, only the part E of Green’s operator contributes to the effective ∼ ∼
moduli. Kröner [KRÖ81] suggests a precise definition of a statistically disordered medium of degree n (gradual disorder) c∼ : (F∼ : ∼c )p = 0 0 < p < n ∼
∼
∼
0 eff =C . where the Green operator is formed with C ∼ ∼ ∼
∼
These conditions impose, for a perfectly disordered material (gradual disorder of infinite degree) eff = c∼ : (I∼ + E : δc∼)−1 : (I∼ + E : δc∼)−1 −1 C ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
(5.92)
Elements of microstructural mechanics
227
0 eff (the Green operator is formed with C =C ). ∼ ∼ ∼
∼
The optimal bounds of 2nd and 3rd degree disordered materials have been obtained by Kröner [KRÖ77]. When the phase distribution is isotropic at 2nd order and the material is disordered of degree 2
(2 ) C = c∼ − c∼ : E : ∼c ∼ ∼ ∼
∼
∼
∼
∼
0 (0) must be formed with C =C . If, in addition, one assumes that isotropy is where E ∼ ∼ ∼ ∼
∼
∼
verified at infinity order but the disorder is of degree 2, we get the Hashin–Shtrikman bounds, which will be calculated in a particular case in Sect. 5.4.3: H−S = c∼ : (I∼ + E : δc∼)−1 : (I∼ + E : δc∼)−1 −1 C ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
0 (0) (the Green operator is formed with C =C ). ∼ ∼ ∼
∼
For a material of 3rd degree in disorder and isotropy,
(3 ) C = c∼ − c∼ : E : ∼c : c∼ : E : ∼c + ∼c : E : δc∼ : E : ∼c −1 : c∼ : E : ∼c ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
∼
0 (1) =C ). (the Green operator is formed with C ∼ ∼ ∼
∼
5.3.5. Self-consistent and generalized self-consistent schemes In a polycrystalline aggregate, we shall call phase the set of all grains having a crystalline orientation ± d . Let f be its volume fraction. The shape of the grains can be needles (case of flattened grains) but if these needles are isotropically distributed, the geometry attached to the phase can be regarded as a sphere. The mechanical state of phase can be idealized as being the response of a spherical inclusion in an infinite isotropic matrix whose mechanical characteristics are the unknowns of the problem (homogeneous equivalent medium). The matrix is homogeneously loaded at infinity by = ε∼ E ∼ where ε∼ =
f ε∼
ε∼ is the (uniform, as we will see in Sect. 5.4.2) strain in the inclusion . This method of estimating effective behavior is called a self-consistent model. This approach is used in other areas of physics. The stress state in the heterogeneous inclusion can be calculated based on Eshelby’s methods recalled in Sect. 5.4.2 and in
228
Non-Linear Mechanics of Materials
[FRA91, FRA98]: eff σ∼ = +C : (I∼ − S∼ −1 ) : (ε∼ − E ) ∼ ∼ ∼ ∼
∼
∼
is the stress at infinite distance from the inclusion and S∼ is Eshelby’s tensor, pre∼ ∼
sented in Sect. 5.4.2. We can note that
eff−1 S∼ : C = ∼ ∼
∼
dV ∼
(5.93)
∼
0 eff being formed with C =C (see previous section). By writing σ∼ = ∼c : ε∼ and ∼ ∼ ∼ ∼
∼
∼
∼
dropping the index , we get: eff−1 ε∼ = (I∼ + S∼ : C : δc∼)−1 : E ∼ ∼ ∼
∼
(5.94)
∼
∼
This result could have been obtained directly from Lippmann–Schwinger’s equation (5.86) for a spherical inclusion: : δc∼)−1 : E ε∼ = (I∼ + E ∼ ∼ ∼
∼
(5.95)
∼
0 eff formed with C =C ). (E ∼ ∼ ∼ ∼
∼
∼
eff = σ∼ = C :E , we get From (5.95) and using ∼ ∼ ∼ ∼
C ∼
=C ∼
SC
∼
eff
= c∼ : (I∼ + E : δc∼)−1 ∼ ∼
∼
∼
∼
(5.96)
∼
As (5.95) imposes that (I∼ +E : δc∼)−1 = I∼ , it follows that (5.94) and (5.92) are iden∼ ∼
∼
∼
∼
tical. This means that the effective properties obtained according to the self-consistent model are those of a perfectly disordered material. Using (5.92), the self-consistent moduli can be obtained based on an iterative proeff (0) (1) is replaced by C (or C ), the result is C (or C ) cedure. At the first iteration, C ∼ ∼ ∼ ∼2 ∼3 ∼
∼
∼
and we continue thanks to the recurrence formula
∼
C = c∼ : (I∼ + E : δc∼n−2 )−1 : (I∼ + E : δc∼n−2 )−1 −1 ∼ n−2 ∼ n−2 ∼n ∼
∼
∼
∼
∼
∼
∼
∼
∼
(5.97)
0 where E is formed with C =C . ∼ n−2 ∼ ∼ n−2 ∼
∼
∼
We can build the bounds C in the same way. The C are bounds for materials ∼ −n ∼ ±n ∼
∼
with gradual disorder of degree n. They are optimal for n = 0, 1, 2, 3 and presumably SC (resp. C ) converges, it converges towards C also for n > 3. If the series C ∼n ∼ −n ∼ ∼
∼
∼
SC (resp. S∼ SC−1 = C ) that are the effective moduli of the perfectly disordered material ∼ ∼
[KRÖ77].
∼
Elements of microstructural mechanics
229
Figure 5.15. Self-consistent and generalized self-consistent models
The statistical theory developed by Kröner is well adapted to the case of random materials with a cellular microstructure. In contrast, the effective properties of random materials having a matrix-inclusion structure have been studied more recently. Hervé and Zaoui [HER90] propose a generalized self-consistent scheme for the matrix-inclusion configuration, also called a three-phase model. Let us consider a composite sphere containing a spherical core of material 1 and a concentric shell of material 2 (external and internal radii are b and a, c = a 3 /b3 is the volume fraction of inclusion). The composite sphere is embedded in an infinite matrix made of the homogeneous equivalent medium. The closure condition is then that the average strain in the composite sphere is equal to the strain ε∼ 0 imposed at infinite distance: = ε∼ ε∼ 0 = E ∼ where the average only concerns the composite assembly. Moreover, it can be proved that a statistical analysis such as in the previous section can be drawn for the pattern of the composite sphere assembly. Then, it is shown that the three-phase model provides the effective moduli of a perfectly disordered assembly of composite spheres.
5.4. Some examples, applications and extensions 5.4.1. Hill’s lens representation of bounds We consider a globally-isotropic linear-elastic heterogeneous-material. Each phase is also isotropic. For the description of isotropic fourth-rank tensors, we introduce
230
Non-Linear Mechanics of Materials
the tensors J∼ and K in addition to the unit tensor I∼ on symmetric tensors already ∼ ∼
encountered:
∼
∼
I∼ = J∼ + K ∼
(5.98)
Kij kl = 1/3δij δkl
(5.99)
Iij kl = 1/2(δik δj l + δil δj k )
(5.100)
∼
∼
∼
We recall that: Note the following properties: J∼ : ε∼ = ε∼ dev ∼
and = aJ∼ + bK A ∼ ∼ ∼
⇒
∼
∼
K : ε∼ = ∼ ∼
trε∼ 1 3 ∼
(5.101)
−1 A = 1/aJ∼ + 1/bK ∼ ∼ ∼
∼
∼
(5.102)
By assumption, 1 1 K + J ∼ h 3k ∼ 2μh ∼∼ ∼ ∼ ∼ ∼ where the shear and compressibility moduli intervene. We saw that: h = 3k h K + 2μh J∼ C ∼ ∼
S∼ h =
h E : (c∼ − C ):E ≥0 ∼ ∼ ∼ ∼
∼
(5.103)
∀E (Voigt) ∼
: (s∼ − S∼ h ) : ≥ 0 ∀ (Reuss) ∼ ∼ ∼ ∼
∼
With dev = 2μE + k(trE )1 c:E ∼ ∼ ∼ ∼
∼ ∼
s: = ∼
∼ ∼
This yields
1 dev tr + ∼ 1∼ 2μ ∼ 9k
μ−1 −1 ≤ μh ≤ μ k
−1 −1
≤ k ≤ k h
(5.104) (5.105)
This allows us to build the “Hill lens representation of bounds” for a two-phase material (Fig. 5.16). 1 3μ
What can be said about the effective Young’s modulus E h ? By noting that E1 = 1 + 9k , we find 1 −1 −1 1 −1 −1 h −1 μ + k E ≤ E ≤ (5.106) 3 9
These relations allow us to criticize a priori extensive use of “laws of mixture” in many situations. At most, a law of mixture consisting in estimating global values by averaging over local properties corresponds to one of the bounds of some effective parameters. Moreover, Voigt–Reuss bounds involve arithmetic and harmonic averages!
Elements of microstructural mechanics
231
Figure 5.16. Hill’s lens representation of bounds: bounding and estimation of effective properties of an isotropic mix of two isotropic and incompressible phases (μ1 = 70000 MPa and μ2 = 4000 MPa)
5.4.2. Eshelby problems: elastic inclusion and elastic inhomogeneity When we introduced the self-consistent scheme in elasticity, we were brought back to solving the problem of an elastic inclusion embedded in an infinite matrix with different elastic characteristics and loaded at infinity. It is the elastic inhomogeneity problem considered by Eshelby in 1957. He first solved the so-called inclusion problem for which an inclusion of the same material as the matrix undergoes an eigenstrain (thermal dilatation, phase transformation, see Sect. 5.5). These are two fundamental problems of the mechanics of heterogeneous media used, on the one hand, in the study of some metallurgical properties [TAN70, PIN76] and, on the other hand, in many homogenization models. Resolution methods of these problems are presented, for example, in [MUR87]. Solving Eshelby’s problem shows that the deformation in an ellipsoidal inclusion, undergoing an eigenstrain or having an initial strain ε∼ L , is homogeneous and is written ε∼ I = S∼ esh : ε∼ L
(5.107)
∼
where S∼ esh is called Eshelby’s tensor. Stresses are then: ∼
σ∼ I = ∼c : (ε∼ I − ε∼ L ) = ∼c : (S∼ esh − I∼ ) : ε∼ L ∼
∼
∼
∼
(5.108)
232
Non-Linear Mechanics of Materials
In the case where the matrix is isotropic (the only case for which Green’s functions are explicit) and for a spherical inclusion, one finds: + βJ∼ S∼ = αK ∼
(5.109)
3k 1+ν = 3k + 4μ 3(1 − ν)
(5.110)
2(4 − 5ν) 6(k + 2μ) = 5(3k + 4μ) 15(1 − ν)
(5.111)
∼
∼
where α= β=
∼
Whence,
α−β (5.112) (Tr ε∼ L )1∼ + βε∼ L 3 Applying the superposition theorem then allows us to calculate the stress state of an inclusion undergoing an eigenstrain ε∼ L , in an infinite matrix having itself an eigenstrain L , and subjected at infinity to a strain E , elasticity being assumed homogeneous. E ∼ ∼ ε∼ I =
Strain inside the inclusion is: L L L )+E + (E −E ) ε∼ = S∼ esh : (ε∼ L − E ∼ ∼ ∼ ∼
(5.113)
† L +L : (E − ε∼ L ) σ∼ = ∼ ∼ ∼
(5.114)
L = ∼c : (E −E ). ∼ ∼ ∼
(5.115)
† = ∼c : (I∼ − S∼ esh ) L ∼
(5.116)
∼
hence ∼
with ∼
The tensor ∼
∼
∼
∼
called the elastic accommodation tensor, is used very often in several homogenization models. Problem of homogeneous inclusion The problem of the elastic inhomogeneity (which is a priori distinct from the previous one), i.e., an inclusion made of a material different from the surrounding matrix and undergoing at infinity a loading E ∼ or , can be solved based on the previous results. For that purpose, a fictitious eigen∼ strain can be introduced. The solution can be found in [FRA91, FRA98]. One shows again that inside an ellipsoidal inclusion, the deformation is homogeneous. In the case of a sphere and assuming isotropic elasticity, (μ, k) for the inclusion and (μ0 , k0 ) for the matrix, one finds: ε∼ I =
Tr E 1 1 ∼ E dev 1+ 1 + α0 δk/k0 3 ∼ 1 + β0 δμ/μ0 ∼
(5.117)
Elements of microstructural mechanics
233
where the coefficients α0 and β0 are the same as in (5.110) and (5.111) using the elasticity coefficients of the matrix and δk = k − k0
δμ = μ − μ0 .
5.4.3. Hashin–Shtrikman bounds (case of a locally and globally isotropic twophase material) The
bounds
C ∼ (±2) ∼
of
(5.97)
(statistical
methods)
are
called
Hashin– 0
equal reShtrikman bounds. They are obtained by taking the reference medium C ∼ ∼
M m = C and C = C ) (cf. 5.3.2). It is spectively to the zero-order bounds (C ∼ +0 ∼ ∼ −0 ∼ ∼
∼
∼
∼
then possible to show that Hashin–Shtrikman bounds can also be obtained by solving M m M successively the problem of an inhomogeneity C (resp. C ) inside the matrix C ∼ ∼ ∼ ∼
∼
m m M M m (resp. C ) and then an inclusion C (resp. C ) in the matrix C (resp. C ). ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
We consider the case of spherical and isotropic inclusions: μ1 > μ2 , k1 > k2 . We determine now the upper Hashin–Shtrikman bound, by following the previous calculation scheme above. The total strain is E = ε∼ = f1 ε∼ 1 + f2 ε∼ 2 ∼ where ε∼ 1 and ε∼ 2 are obtained from the solutions of the auxiliary problems of an elastic at infinity. inhomogeneity embedded in a matrix subjected to the strain E ∼0 ε∼ 1 = E ∼0
234
Non-Linear Mechanics of Materials
ε∼ 2 = hence
E = f1 + ∼
1 1 1 + α1 k2k−k 1
1 − f1 1 1 + α1 k2k−k 1
Tr E 1 ∼0 1+ E dev 3 ∼ 1 + β1 μ2μ−μ1 ∼ 0 1
Tr E 1 dev ∼0 1 + f1 + E ∼0 1 3 ∼ 1 + β1 μ2μ−μ 1
which gives the adequate E to be imposed. We substitute then these relations in ∼0 = c∼ : ε∼ = f1 c∼ : ε∼ 1 + f2 ∼c : ε∼ 2 ∼ ∼
∼1
∼2
which we identify to the target global relation: dev = k H S+ (Tr E )1 + 2μH S+ E ∼ ∼ ∼ ∼
We get then the bulk modulus:
k
H S+
= f 1 k1 +
and the shear modulus:
μ
H S+
= f1 μ1 +
f 2 k2 1 1 + α1 k2k−k 1
f2 μ2 1 1 + β1 μ2k−μ 1
f1 +
f2 1 1 + α1 k2k−k 1
f1 +
−1
f2
−1
1 1 + β1 μ2μ−μ 1
In the case of two incompressible phases, the Hashin–Shtrikman shear modulus has a simpler expression: μ2 − μ1 μH S+ = μ1 + f2 1 1 + f1 25 μ2μ−μ 1 These results, in the incompressible case, are drawn on Hill’s lens representation of Fig. 5.16, which leads to a narrower gap than between Voigt and Reuss bounds. This means that additional statistical information has been used, namely local and global isotropy.
5.4.4. Self-consistent model To evaluate the self-consistent estimate in the case of a two-phase isotropic medium, we consider the auxiliary problems of an inhomogeneity ∼c1 embedded in an infinite ∼
at matrix endowed with the elastic moduli we are looking for and submitted to E ∼ infinity, and the same problem considering an inclusion ∼c2 . ∼
Elements of microstructural mechanics
235
It follows that μAC and k AC are solutions of the nonlinear system: 1=
f1
+
AC 1 + α AC k1k−k AC
f1
1=
+
AC 1 + β AC μ1μ−μ AC
f2 AC 1 + α AC k2k−k AC
f2 1 + β AC μ2μ−μ AC
AC
(5.118)
(5.119)
This system is solved numerically but an explicit solution can be obtained in the incompressible case. This last result is drawn on Hill’s lens representation. It can be checked that the curve lies between Hashin–Shtrikman bounds.
5.4.5. Dilute distribution 0 in which a low concentration of inclusions is If the material is made of a matrix C ∼ ∼
dispersed, it suffices to consider, in the case of a dual phase for example, the auxiliary problem associated with (1 − f0 ) 1.
5.5. Homogenization in thermoelasticity 5.5.1. Given eigenstrain field; residual stresses We come back here to the concept of eigenstrain introduced in Sect. 5.2, namely ε∼ L such that at each point: ε∼ = ε∼ e + ε∼ L What is the response of a heterogeneous medium undergoing eigenstrains, such as thermal strain, phase transformation induced strain or even plastic strain (if there is no additional evolution)? For the SUBC problem for example: S∼ h = s∼ : B ∼ ∼
∼
∼
according to Sect. 5.3.1. It follows that e E = s∼ : B : σ∼ = s∼ : B : σ∼ ∼ ∼ ∼ ∼
∼
∼
∼
because sij kl Bkl(mn) is, for m, n given, a compatible strain field. Then, e T T = B : ∼s : σ∼ = B : ε∼ e E ∼ ∼ ∼ ∼
∼
∼
236
Non-Linear Mechanics of Materials
We see that the global elastic deformation is not the average of the local elastic deformations. Moreover, e T T T E = B : (ε∼ − ε∼ L ) = B : ε∼ − B : ε∼ L ∼ ∼ ∼ ∼ ∼
∼
∼
T : ε∼ − B : ε∼ L = B ∼ ∼ T
∼
∼
=E − B : ε∼ ∼ ∼ T
L
∼
hence the definition of the global eigenstrain: L T E = B : ε∼ L ∼ ∼ ∼
Residual stresses We call residual stress field any field σ∼ r (x) such that σ∼ r = 0 div σ∼ r = 0 on V σ∼ r .n = 0 on ∂V If we add a macroscopic stress to a residual stress field, the resulting field σ∼ can be ∼ decomposed into σ∼ = σ∼ + σ∼ r with σ∼ = σ∼ = . The residual strain is defined by ∼ ε∼ r = ε∼ L + ∼s : σ∼ r = ε∼ L + ∼s : (σ∼ − σ∼ ) ∼
∼
= ε∼ L + ε∼ e − ∼s : σ∼ = ε∼ − ∼s : σ∼
∼
∼
Let us calculate the stored elastic energy: 1 1 1 1 σ : s : σ = σ∼ : ∼s : σ∼ + σ∼ : ∼s : σ∼ r + σ∼ r : ∼s : σ∼ r 2 ∼ ∼∼ ∼ 2 2 2 ∼ ∼ ∼ Yet, residual strains are compatible fields, hence: L σ∼ : ε∼ r = σ∼ : ε∼ r = :E ∼ ∼
= σ∼ : ε∼ − σ∼ : ∼s : σ∼ = :E − σ∼ : ∼s : σ∼ ∼ ∼ ∼
e = : Sh : . :E hence σ∼ : ∼s : σ∼ = ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
Elements of microstructural mechanics
237
Moreover σ∼ r is statically admissible and ∼s : σ∼ is compatible (because built from ∼
a statically admissible field), then
σ∼ : ∼s : σ∼ r = σ∼ r : s∼ : σ∼ = 0 ∼
∼
and it follows that 1 1 1 σ : s : σ = : Sh : + σ∼ r : ∼s : σ∼ r ∼ 2 ∼ ∼∼ ∼ 2 ∼ ∼∼ 2 ∼ Thus the total elastic strain energy contains in addition to the strain energy (1/2) : ∼ S∼ h : a stored energy that remains when the macroscopic load is removed. ∼ ∼
Application to thermoelasticity T where α is the tensor of thermal expansion. Here ε∼ L = ε∼ th = α ∼ ∼ At the macroscopic level, thermal strain is defined as th h = T α E ∼ ∼
According to the above analysis, we have h T = B :α α ∼ ∼ ∼ ∼
which is not a trivial “law of mixture”. In the case of a dual phase material, the effective thermal dilatation tensor is directly linked to the effective moduli. In the locally and globally isotropic case, we find: 1/k h − 1/k α h = α + (α2 − α1 ) 1/k2 − 1/k1 If moreover k1 = k2 , the result α h = α is retrieved.
5.5.2. Auxiliary problems in thermoelasticity; coupled thermoelasticity Starting from the local equations of thermomechanics, we are going to show how to obtain the balance and constitutive equations of the effective medium, using the method of asymptotic expansions for periodic materials. More advanced mathematical methods allow one to extend these results to non-periodic materials. In passing, we highlight the auxiliary problems to solve in order to determine the effective thermal and mechanical quantities.
238
Non-Linear Mechanics of Materials
We pursue the asymptotic development introduced in Sect. 5.2.4. We denote by
ε T ε du 1 du e(uε ) = + 2 dx dx the deformation associated with uε . With (5.31) and (5.33), we get e(uε ) = ε −1 ey (u0 ) + ex (u0 ) + ey (u1 ) + ε(ex (u1 ) + ey (u2 )) + ε 2 (ex (u2 ) + ey (u3 )) + · · ·
(5.120)
where we have written ex (.) = (∂./∂x + (∂./∂x)T )/2. The calculation of the divergence of the stress gives similarly (according to (5.33)): div σ∼ ε = ε −1 div y σ∼ 0 + div x σ∼ 0 + div y σ∼ 1 + ε(div x σ∼ 1 + div y σ∼ 2 ) + ε 2 (div x σ∼ 2 + div y σ∼ 3 )
(5.121)
In the general thermomechanical case one takes also into account the temperature field and heat flux vector: θ ε (x) = θ 0 (x, y) + εθ 1 (x, y) + ε 2 θ 2 (x, y) + ε 3 θ 3 (x, y) + · · ·
(5.122)
q ε (x) = q 0 (x, y) + εq 1 (x, y) + ε 2 q 2 (x, y) + ε 3 q 3 (x, y) + · · ·
(5.123)
We write the temperature gradient as a function of the terms of the expansion: g ε = grad θ ε = ε −1 grady θ 0 + gradx θ 0 + grady θ 1 + ε(gradx θ 1 + grady θ 2 ) + ε 2 (gradx θ 2 + grady θ 3 ) + ε 3 gradx θ 3 + · · ·
(5.124)
The local equations of the linearized thermomechanical problem are Hooke’s law, Fourier’s law, the mechanical equilibrium equations and the heat equation: ε ) σ∼ ε = ∼cε : (e(uε ) − τ ε α ∼
(5.125)
ε ε q ε = −λ .g ∼
(5.126)
∼
div σ∼ + f = 0 ε
(5.127)
−divq − e(u˙ ) : c∼ : α = β θ˙ ∼ ε
ε
ε
ε
ε
∼
(5.128)
where τ ε = θ ε − θ ref . To formulate this linearized version of the heat equation, we assume that the temperature is never too far from a temperature T0 and we define then β ε = ρ ε Cpε /T0
ε λ = k∼ ε /T0 ∼
Elements of microstructural mechanics
239
where ρ ε Cpε is the specific heat, and k∼ ε the thermal conductivity tensor of the material. To simplify, we do not mention here the boundary conditions and the initial conditions necessary to solve the thermoelastic and conduction problems. We substitute then the development introduced in the balance and constitutive equations and we order the terms according to the powers of ε. The results for the two first orders are: order −1
ey (u0 ) = 0
(5.129)
div y σ∼ = 0
(5.130)
grady θ 0 = 0
(5.131)
divy q 0 = 0
(5.132)
σ∼ 0 = ∼c(y) : (ex (u0 ) + ey (u1 ) − α τ 0) ∼
(5.133)
.(gradx θ 0 + grad1 θ 1 ) q 0 = −λ ∼
(5.134)
div x σ∼ 0 + div y σ∼ 1 + f = 0
(5.135)
0
order 0 ∼
= β θ˙ 0 −divx q 0 − divy q 1 − (ex (u˙ 0 ) + ey (u˙ 1 )) : ∼c : α ∼ ∼
(5.136)
ε (x) = α (y). We then have to solve where τ 0 = θ 0 − θ ref . We took for example α ∼ ∼ the four following auxiliary problems, whose unknowns are u0 and u1 :
Problem 1: ey (u0 ) = 0. Problem 2: grady θ 0 = 0. Problem 3: σ∼ 0 = ∼c(y) : (ex (u0 ) + ey (u1 ) − α τ 0) ∼ ∼
div y σ∼ 0 = 0
(5.137) (5.138)
Problem 4: q 0 = −λ .(gradx θ 0 + grad1 θ 1 ) ∼ divy q 0 = 0
(5.139) (5.140)
The fields we are looking for being periodic in y, problems 1 and 2 give immediately: u0 (x, y) = U 0 (x)
θ 0 (x, y) = 0 (x)
(5.141)
Problems 3 and 4 are independent and correspond respectively to an elasticity problem with a given eigenstrain field and to an uncoupled thermal conduction problem. They
240
Non-Linear Mechanics of Materials
must be solved on the unit cell Y . Solving them will provide, on the one hand, the effective elastic properties and, on the other hand, the effective conduction properties. Actually, we recognize the elementary problems for the case of periodic homogenization in isothermal elasticity and pure thermostatics. A stronger coupling arises only when solving problems associated with higher orders. Problems 3 and 4 are linear X, X , X with respect to the data U 0 and θ 0 . There exist then concentration tensors ∼ such that: X(y) : ex (U 0 ) + τ 0 X (y) (5.142) u1 (x, y) = U 1 (x) + − ∼
θ 1 (x, y) = 1 (x) + X (y).gradx 0
(5.143)
We note that I∼ + ey (X −) (the gradient acts on the first index only) is nothing but the ∼
∼
encountered in the case of homogenization in linear elasticconcentration tensor A ∼ ∼
ity (5.34). In what follows,ˆwill be the application of the operator grady to the various ˆ. concentration tensors: A = 1∼ + X ∼ ∼ ∼
∼
∼
Effective constitutive equations ˆ − α )), Starting from the first terms of the expansion: σ∼ 0 = ∼c : (A : ex (U 0 ) + τ 0 (X ∼ ∼ ∼ one defines the effective flux and stresses by:
∼
∼
= lim σ∼ ε = σ∼ 0 and Q = lim q ε = q 0 ∼ ε→0
ε→0
(5.144)
where the average is made with respect to the variable y over the unit cell Y . One obtains the effective laws as: ˆ )(0 − θ ref ) = c∼ : A : ex (U 0 ) − c∼ : (α −X ∼ ∼ ∼ ∼
(5.145)
ˆ ).grad 0 Q = −λ .(1 + X ∼ ∼ ∼
(5.146)
∼
∼
∼
At the global level Fourier’s and Hooke’s laws are recovered. The effective properties are averages of the local quantities, weighted by concentration tensors.
Effective balance of momentum equation We take the average of (5.135): + f = 0 div x σ∼ 0 + div y σ 1 + f = div ∼ The form of the local and global balance equations is then the same.
(5.147)
Elements of microstructural mechanics
241
Effective heat equation We take the average of (5.136): T ˆ : c : α ˙0 : ∼c : α = β + X −divQ − ex (U˙ 0 ) : A ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
(5.148)
The heat equation at global level has then the same form as at the local level, as long as we have the following property: T hom hom ˆ ) A : ∼c : α =C :α = c∼ : (α −X ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
according to (5.145) and (5.148). At first sight, this property is not obvious, but can be ˆ and Aij KL (KL being proved using the Hill–Mandel lemma. Indeed, we note that X ∼ fixed) are compatible strain fields since they are solutions of problem 3 over Y when one imposes respectively that 0 and (ex (U 0 )) = eK ⊗ eL be constant. It follows ˆ − α ) on the one hand, and AT : c on the other hand are self-equilibrated that ∼c : (X ∼ ∼ KL ∼ ∼ ∼
∼
∼
stress fields. It is then justified to apply the Hill–Mandel lemma (5.27), which leads to: ˆ ) = AT : c : (α − Xˆ ) = AT : c : (α − Xˆ ) c∼ : (α −X ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
T T ˆ = A : ∼c : α − A : ∼c : X ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
T T ˆ = A : ∼c : α − A : ∼c : X ∼ ∼ ∼ ∼ ∼
: ∼c : α = A ∼ ∼ T
∼
∼
∼
(5.149)
ˆ = 0 of the concentration tensors have been = 1∼ and X where the properties A ∼ ∼ used.
∼
∼
Applying the asymptotic development method in homogenization has allowed us to show that the form of the equations of coupled linear thermoelasticity is preserved through homogenization, at least at the first order considered here. It provides also the expression of the effective quantities and the formulation of the auxiliary problems to solve in order to calculate them. The validity of the results obtained in the case of non-periodic microstructures is shown in [FRA83].
5.6. Nonlinear homogenization Generalizing the treatments above to local nonlinear behaviors (elastoplastic, viscoplastic, elastoviscoplastic) is still an active field of research. In the case of random materials there exist several extensions of the self-consistent approach. In the
242
Non-Linear Mechanics of Materials
elastoplastic and viscoplastic cases, the results obtained in elasticity are generalized by using the secant, tangent or affine moduli of the laws of interest. We propose here to study a few aspects of nonlinear homogenization, in the case of the self-consistent approach, and to emphasize simple and explicit formulations. This will allow us eventually to perform structural computations by means of such homogenization models. For a more complete account of the nonlinear homogenization theory, the reader can refer to [SUQ97]. We will pay attention to the fact that homogenization methods proposed here and the very notion of homogenization make sense only when the initial and boundary value problem posed on the RVE admits a unique solution. Notably, in the case of local loss of ellipticity inside the RVE, due for example to strain localization (see Chap. 7), it is necessary to re-examine the meaning of the proposed averaging operations. This problem is still open at the present day.
5.6.1. Hill’s method in elastoplasticity Hill [HIL65] has developed a self-consistent scheme for the elastoplastic behavior of polycrystalline materials. For each phase, we consider the problem of an ellipsoidal elastoplatic inclusion embedded in an infinite elastoplastic matrix representing the homogeneous equivalent medium. The local behavior of each phase takes the linear incremental (multi-branch) form : ε∼˙ σ∼˙ = L ∼ ∼
In order to keep using Green’s methods, Hill neglects the perturbations of the tangent operator in the vicinity of the inclusion and assumes that the elastoplastic behavior of eff that links the matrix is described by the uniform tensor of instantaneous moduli L ∼ ∼
stresses and strain rates at infinity. ˙ = Leff : E˙ ∼ ∼ ∼ ∼
The solution of the problem is then formally identical to the elastic case presented in Sect. 5.3 and involves the Green operator. In particular, the stress rate in the inclusion is homogeneous and is written ˙ + L : (E˙ − ε˙ ) σ∼˙ = ∼ ∼ ∼ ∼
(5.150)
∼
where
eff =L : (S∼ −1 − I∼ ) L ∼ ∼ ∼
∼
∼
∼
eff according is called an accommodation tensor. Eshelby’s tensor is calculated using L ∼ ∼
to (5.93). It follows that
eff
˙ +L ) : ε∼˙ = (L +L ):E (L ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
Elements of microstructural mechanics
and
243
−1 eff
˙ : (L +L ) : (L +L ):E σ∼˙ = L ∼ ∼ ∼ ∼ ∼ ∼ ∼
Hence
∼
∼
∼
∼
eff
−1 eff
= L : (L +L ) : (L +L ) L ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
(5.151)
∼
is the solution of the implicit integral equation that must be solved step by step along eff and = S : L−1 , we see that (5.151) is = L −L the loading path. With δL ∼ ∼ ∼ ∼ ∼ ∼ ∼
identical to (5.96).
∼
∼
∼
∼
∼
This scheme can be adapted to aggregates of linear viscoplastic or viscoelastic materials. A strictly self-consistent processing for elastoviscoplastic materials is more intricate [BOR01]. Implementing the above results requires complex numerical treatments (solving integral equations), which hampers implementation of these models in finite element codes for structural computations. That is why additional simplifications are desirable as long as the interphase accommodation (elastoplastic or viscoelastoplastic) is fulfilled (on that matter, see [ZAO93b]). Moreover, the assessment of Hill’s approximation of a tangent uniform operator in the matrix has still to be done.
5.6.2. Approximations of the self-consistent scheme: Kröner and Berveiller–Zaoui models We consider again the heterogeneous elastoplastic case considered by Hill, but we . It follows that restrict to homogeneous elasticity described by the moduli C ∼ ∼
˙ p = ˙ε p E ∼ ∼ We take the decomposition of the macroscopic strain as ˙ = E˙ e + E˙ p E ∼ ∼ ∼ As a consequence, the concentration rule (5.150) can be written ˙ = Leff : (S −1 − I ) : C −1 : ( ˙ − σ˙ ) + Leff : (S −1 − I )(E˙ p − ε˙ p ) σ∼˙ − ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
which gives ˙ + (I + Leff : (S −1 − I ) : C −1 )−1 : Leff : (S −1 − I ) : (E˙ p − ε˙ p ) σ∼˙ = ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
∼
∼
∼
∼
Some approximations are possible. The most simple consists in using in the previous formula eff =C L ∼ ∼ ∼
∼
244
Non-Linear Mechanics of Materials
which authorizes only elastic accommodation of the matrix. That is why we call the tensor † =C : (I∼ − S∼ ) (5.152) L ∼ ∼ ∼
∼
∼
∼
the elastic accommodation tensor. Within this approximation, the stress concentration rule becomes ˙ + L† : (E˙ p − ε˙ p ) σ∼˙ = ∼ ∼ ∼ ∼ ∼
This hypothesis was made by [KRÖ61] in the isotropic case for elastoplastic polycrystals. It has been established, however, that this model is not very different from the results of Taylor’s analysis that consists in considering that strain is uniform in the whole aggregate (Voigt type hypothesis) [ZAO85]. In contrast, [BER79] introduce an isotropic approximation of the effective moduli in the following way: eff =L L ∼ ∼ ∼
with
L = 3k K + 2μ J∼ ∼ ∼ ∼
k
∼
∼
∼
μ
and are approximate tangent moduli to be specified. The associated Eshelby tensor is written S∼ = α K + β J∼ ∼ ∼
∼
with α =
3k 3k + 4μ
∼
β =
6(k + 2μ ) 5(3k + 4μ )
˙ p and ε˙ p have a zero trace, we get As E ∼ ∼ ˙ + 2μ σ∼˙ = ∼
μ (1 − β ) β μ + (1 − β )μ
˙ p − ε˙ p ) (E ∼ ∼
(5.153)
which can also be written ˙ + 2μ σ∼˙ = ∼
μ (7 − 5ν ) (E˙ p − ε∼˙ p ) μ (7 − 5ν ) + 2μ(4 − 5ν ) ∼
after noticing that β =
2(4 − 5ν ) 15(1 − ν )
By posing μ = μ and β = β, we get Kröner’s result [KRÖ61] written in rate form ˙ + 2μ(1 − β)(E˙ p − ε˙ p ) σ∼˙ = ∼ ∼ ∼
(5.154)
Elements of microstructural mechanics
245
5.6.3. Influence of the heterogeneity of plastic strain surrounding the inclusion on the quality of the self-consistent estimate in elastoplasticity We propose here to check the quality of the approximate concentration law (5.153) based on finite element analysis of a simple situation. We consider an elastoplastic dual-phase material, globally and locally isotropic. The behavior of the phases is described by: plasticity criterion F (σ∼ , R) = J2 (σ∼ ) − R
(5.155)
R = R0 + Hp
(5.156)
linear isotropic hardening
flow rule ε∼˙ p = pn ˙∼
n = ∼
3 ∼s 2J
(5.157)
The flow rule can be written: dε∼ p =
3 ∼s dJ 2J H
(5.158)
For a monotonic radial loading path, we have: ε∼ p = hs∼ with h =
p 3 2 R0 + Hp
(5.159)
The tensor of the secant moduli is obtained as follows: σ∼ = (3kK + 2μJ∼ ) : ε∼ − 2μhJ∼ : σ∼ ∼ ∼
∼
∼
σ∼ = (K + (1 + 2μh)J∼ )−1 : (3kK + 2μJ∼ ) : ε∼ ∼ ∼ ∼
which can be written σ∼ =
∼
L ∼ ∼
∼
∼
: ε∼ with
L = 3k K + 2μ J∼ ∼ ∼ ∼
∼
∼
k = k and μ =
μ 1 + 2μh
(5.160)
As an approximation of the behavior of the homogeneous equivalent medium, we use then the previous secant moduli, taking: p Ep : E ∼ h= ∼ (5.161) s:s ∼ ∼
246
Non-Linear Mechanics of Materials
for non-vanishing deviatoric stress ∼s . In these conditions, the concentration law (5.153) can be replaced by: + 2μ σ∼ = ∼
μ (1 − β ) (E p − ε∼ p ) β μ + (1 − β )μ ∼
(5.162)
for which we give also the following closed form to be implemented in numerical codes: + 2μ(1 − β) σ∼ = ∼
1+ν 1 + 6μh 7−5ν
1+
2(13−5ν) 2 2 8(1+ν) 15(1−ν) μh + μ h 15(1−ν)
with β=
p (E − ε∼ p ) ∼
(5.163)
2(4 − 5ν) . 15(1 − ν)
We now have the complete set of equations retained for the effective behavior of the HEM: ⎧ ˙ + 2μ μ (1−β ) (E˙ p − ε˙ p ) σ∼˙ i = ⎪ β μ+(1−β )μ ∼ ⎪ ∼ ∼i ⎪ ⎪ p si ⎨ ε∼˙ i = p˙ 32 J (∼σ i ) F (σ∼ i , Ri ) = J (σ∼ i ) − Ri (M) ∼ ⎪ ˙ p = i fi ε˙ p ⎪ E ⎪ ∼ ∼i ⎪ ⎩ = (3kK + 2μJ ) : E e ∼ ∼ ∼ ∼ ∼
∼
We try then to compare the stress σ∼ i calculated according to the previous equations to the stress that reigns in an inclusion of phase i embedded in an infinite matrix whose behavior is given by (M). For that purpose, we use a finite element analysis with the following mesh (axisymmetric case):
where the inclusion was separated from the matrix for the sake of clarity. The behavior (M) is attributed to the matrix whereas the constitutive equations of the inclusion
Elements of microstructural mechanics
247
medium are given by (5.155) to (5.157). The values of the hardening modulus chosen for the example are: H1 = 1000 MPa and H2 = 20000 MPa (moreover, E = 70000 MPa, ν = 0.33 and R01 = R02 = 130 MPa). Two tensile tests in direction 2 are calculated successively. We compare then strain and stress in the inclusion to the direct prediction from directly solving (M). Let us insist on the fact that direct application of (M) necessitates that h be uniform in the matrix. In contrast, in the finite element computation, strain and stress are heterogeneous around the inclusion so that h is not homogeneous. We recall that Hill’s treatment presented above neglects this heterogeneity. We check first that the self-consistency condition is fulfilled in the following sense = f1 ε∼ 1 + f2 ε∼ 2 E ∼
(5.164)
E being applied at infinity. Numerically, this relation is satisfied to within 0.6% (after ∼ a global strain of 3%). Figure 5.17 shows that the prediction of the approximate estimation of Berveiller–Zaoui is not far from the finite element result. This indicates that the strain heterogeneity surrounding the inclusion changes very little the quality of the estimation and justifies a posteriori Hill’s hypothesis, at least in the example considered here. However, if instead of Berveiller–Zaoui concentration law (5.162), we use Kröner’s relation (5.154), Fig. 5.18 shows that the stress heterogeneity is greatly overestimated. This confirms the fact that using elastic accommodation only does not provide a satisfying approximation of the self-consistent scheme in the nonlinear regime.
5.6.4. Identification of the stress concentration law We present a class of estimations of the effective nonlinear behavior of heterogeneous elastoviscoplastic materials. These estimations are based upon the choice of an explicit formulation of the concentration laws estimating the average stress within the phases depending on global stress and internal variables. These concentration rules involve parameters that we determine by an inverse method from structural analyses on the considered RVE. This method presents a systematic character and has the advantage of providing a set of explicit constitutive equations easily implementable for structural computations. It is essentially applied here to the self-consistent-type estimations introduced previously.
248
Non-Linear Mechanics of Materials
Figure 5.17. Comparing the approximate self-consistent Berveiller–Zaoui model to the finite element calculation of an inclusion of each phase embedded in an infinite matrix endowed with estimated effective behavior
Introducing the method in the elastic case We consider the following scheme: an assembly of various phases (composite spheres. . .) is placed as inclusion inside an infinite matrix whose behavior is unknown. The assembly is assumed to be representative of the microstructure of a random-type material. Its morphology can be very complicated. The matrix represents the homogeneous equivalent medium we are looking for or, at least, an estimate of it. The form of the constitutive laws is assumed to be fixed but the parameters are not known and will ∞ are applied be determined via an inverse method. Uniform boundary conditions E ∼ at infinity. We will say that the obtained model fulfills the self-consistency condition if, at any time: ∞ = ε∼ (5.165) E ∼ averaging over the composite assembly. Since the parameters characterizing the behavior of the HEM are unknown, we start from an initial set (Voigt bound, for example) and we try to minimize the functional: L(A) =
t
N(ε∼ 0t − ε∼ t )
(5.166)
Elements of microstructural mechanics
249
Figure 5.18. Comparing the approximate self-consistent Kröner model to the finite element calculation of an inclusion of each phase embedded in an infinite matrix endowed with estimated effective behavior
where N is a norm and t some instants throughout the studied experiment. In what follows, we will assume that the solution of this optimization problem exists. Its uniqueness is generally not ensured. We exemplify the procedure with the case of a linear elastic dual-phase material, locally and globally isotropic. We try first to find the self-consistent estimate of the effective behavior of this medium using the inverse method presented above. The exact solution is given by (5.118) and (5.119). We solve the axisymmetric problem of a spherical inclusion in the quasi-infinite HEM with the finite element method. Two calculations are necessary, whether the inclusion is made of phase 1 or 2. The procedure is then the following: we choose initial parameters E 0 , ν 0 for the matrix. ∞ at infinity for both calculations. Then, we calculate the average value We apply E ∼ of the deformation in each inclusion and we calculate the cost function: ∞ L(E 0 , ν 0 ) = E − (f1 ε∼ 1 + (1 − f1 )ε∼ 2 )2 ∼
(5.167)
An optimization method (such as the conjugate gradient method) is then used to minimize L with respect to (E 0 , ν 0 ). Each evaluation of the cost-function involves two finite element computations with new values of (E 0 , ν 0 ). The convergence of (E 0 , ν 0 ) towards the self-consistent solution is illustrated in Fig. 5.19, for the following values
250
Non-Linear Mechanics of Materials
Figure 5.19. Determination of the self-consistent moduli for an isotropic dual phase material by an inverse method: trajectory of the parameters E 0 and ν 0 starting from various initial values
of the parameters: E1 = 355000 MPa, ν1 = 0.2, f1 = 0.4, E2 = 70000 MPa, ν2 = 0.33.
Application to elastoplasticity: β models In the model described by (5.154), the accommodation term for a given phase involves the difference between the global plastic strain and the plastic strain of the given phase. The scalar accommodation factor decreases with plastic strain, which reduces the stress heterogeneity between the various phases. We can then attach to each component an “internal stress”, 2μ(1 − β)ε p , which depends nonlinearly on the current state of plastic deformation through the term β. The same result can be found by keeping a scalar factor constant, and by replacing plastic deformation with an internal variable β , having a nonlinear evolution in each phase. The concentration laws ∼ (5.154) and (5.153) can then be considered as members of the following family of explicit concentration laws: σ∼ i = + 2μ(1 − β) (B − βi ) ∼ ∼ ∼
(5.168)
Elements of microstructural mechanics
with = β = B ∼
∼
(5.169)
fi β i ∼
i
251
From a phenomenological point of view, this amounts to modifying the way the nonlinearity is taken into account. It is transferred to the tensorial part of the accommodation relation and requires a variable β i in each phase i, having the status of a ∼ nonlinear kinematic hardening variable. Accordingly, it is clear that the very form of the chosen rule is of phenomenological nature and should be improved if it fails at describing some features of elastoplastic accommodation. The form initially proposed for a crystalline aggregate [CAI87] involves for each grain all its slip systems, through their rate, γ˙s : p | γ˙s | (5.170) β˙ i = ε∼˙ i − Dβ i ∼
∼
s
Another solution consists in choosing the equivalent of the plastic strain in the sense of von Mises for the considered phase. [PIL90b, CAI94]: 2 p p β˙ i = ε∼˙ i − D J (˙ε∼ i ) β i ∼ ∼ 3
(5.171)
If the parameter D is zero in (5.170) or (5.171), Kröner’s rule (5.154) is retrieved. Therefore, it generalizes this rule in the sense that the parameter D can be determined in order to fulfill the self-consistency condition (5.165) at best. The concentration relation (5.168) can contain other terms, for instance several terms in β , or a combination ∼ of a nonlinear term and a linear one [PIL94], which can be solved in: p p 2 p β˙ i = ε∼˙ i − D(β i − δε∼ i ) J (˙ε∼ i ) ∼ ∼ 3
(5.172)
by introducing a second coefficient δ, to be identified. If necessary, we can also introduce a rate-dependent term, as in the relations cited in Sect. 3.9:
J2 (β i ) m β i 2 p p ∼ ∼ β˙ i = ε∼˙ i − D J2 (˙ε∼ i )β i − (5.173) ∼ ∼ 3 M J2 (β i ) ∼
with two additional coefficients m and M. The minimization procedure already used in elasticity can be used in the elastoplastic case to identify the previous coefficients for the case of (5.171). At each iteration of the minimization procedure, two axisymmetric tensile tests are computed on the inclusion-matrix mesh of Sect. 5.6.3. The cost function is then built from several values along the obtained mean stress-strain curves. With the example of the behavior of the elastoplastic isotropic dual phase material treated in Sect. 5.6.3, we find D = 55. With this value, the response is the same as with the Berveiller–Zaoui model on a monotonic radial path (Fig. 5.17). The advantage, however, of the present
252
Non-Linear Mechanics of Materials
Figure 5.20. Comparing the prediction of the self-consistent estimation with accommodation variables to the finite element simulation of an inclusion of each phase in an infinite matrix endowed with the HEM properties for a tension-compression cycle
formulation of the effective behavior is that it can be used for more complex global loading conditions, in particular in the case of low cycle fatigue. Figure 5.20 shows results of the modeling of a cycle E22 = ±0.8% thanks to the estimation proposed above. These results are compared to the corresponding finite element computation of ∞ . It the inclusion in its HEM matrix subjected at infinity to the same loading cycle E ∼ turns out that the estimation remains satisfactory.
Extension to complex loading and elastoviscoplasticity The previous approach can be applied to more complex global loading and local constitutive behavior. First, the case of a non-proportional biaxial loading is illustrated, as indicated in Fig. 5.21. The response in stress is given in Fig. 5.22. In each case, the response is compared to the finite element simulation to check that the self-consistent status of the proposed estimation is approximately preserved. If the behavior of the phases is elastoviscoplastic, the class of localization rules described above can still be used if the parameter D is re-identified. In the case of the isotropic dual phase material, we replace, in the effective constitutive laws (M), the
Elements of microstructural mechanics
253
Figure 5.21. Non-proportional biaxial loading imposed at infinity and average strain in inclusions obtained by finite element simulations
Figure 5.22. Comparing the prediction of the self-consistent estimation with accommodation variables to the finite element simulation of an inclusion of each phase in an infinite matrix endowed with the HEM properties, for a cycle of non-proportional biaxial loading
254
Non-Linear Mechanics of Materials
expression of the plastic multiplier by: J (σ∼ i ) − R ni p˙ i = Ki
(5.174)
where K and n are viscosity parameters. The case of two hardening viscoplastic phases is considered, with K1 = 10 MPa1/n1
and
K2 = 400 MPa1/n2
n1 = 5 and n2 = 5 (the other parameters being the same as in Sect. 5.6.3). For a value of D similar to the one found in Sect. 5.6.4, we simulate again a tensile test at imposed strain rate E˙ 22 = 10−3 s−1 . The proposed method can be expanded along several lines, by identifying the concentration law not on a self-consistent localization rule, but on other kinds of models, for example the three-phase model, as in [FOR95b], or more complex morphological patterns [SUQ97]. With increasing computer speed, it is also possible to consider building a “numerical tensile test machine” from a realistic mesh of microstructures [BAR01a, BAR01b] (see Sect. 5.7).
5.6.5. Polycrystal behavior A polycrystal is a random heterogeneous material whose constituents are grains or crystallites that differ one from another generally only by their crystallographic orientation. The most simple, and now classical, homogenization models of the polycrystal resort to a notion of phase distinct from the notion of grain: we call the set of all grains of a RVE having a given crystallographic orientation up to a certain solid angle, a crystallographic phase. This concept is sufficient to incorporate the information about the crystallographic texture and a simplified vision of the intergranular heterogeneities in the modeling. The stress and strain in each phase correspond then to values averaged over the set of grains of the RVE belonging to the considered phase. These are the quantities that must be estimated by means of the available homogenization techniques. For reasons determined in the statistical analysis of Sect. 5.3.4, the self-consistent model is a good candidate to predict the effective properties of the polycrystal, at least in the linear case (thermoelasticity and electromagnetic properties [KRE89]). Many developments in the nonlinear case (elastoplasticity [BER79], viscoplasticity [MOL87]) follow or extend Hill’s method, presented in Sect. 5.6.1. Remarkable success has been obtained with these models notably for the prediction of texture evolution during metal forming processes such as rolling, drawing, extrusion, etc [KOC98]. The advantage of the self-consistent method over so-called Taylor’s model (Voigt method applied to polycrystal), also widely used, lies in the prediction
Elements of microstructural mechanics
255
of more realistic macroscopic stress states during the forming processes. A simplified polycrystal model is presented here that makes use of the methods of Sect. 5.6.4, in order to illustrate the wide range of mechanical properties that this kind of polycrystal models is able to reproduce [CAI92].
A model of polycrystal The ingredients of the model have already been introduced: the behavior of the components and the scale transition rule. The mechanical behavior attributed to each phase corresponds to the one of a single crystal deforming by crystallographic slip. A set of constitutive laws of the elastoviscoplastic single crystal has been provided in Sect. 3.10.3 and is adopted here. The average stress and strain in each phase g are noted σ∼ g and ε∼ g . The volume fraction fg of phase g corresponds to the fraction of grains contributing to the component g of the discretized texture. The number of phases is then the number of orientations in the chosen discretization. For computational efficiency, this number is limited in general from 40–100 (for identification procedures) to 1000 or 2000 for more precise results. Moreover, we restrict ourselves to isotropic homogeneous elasticity. Macroscopic stress and strain rate are then defined by: fg σ g fg ε˙ pg E˙ = E˙ e + E˙ p E˙ p = = ∼
∼
∼
∼
∼
g
∼
∼
g
The local stress is given explicitly by a concentration rule of the form (5.168) to (5.169) introducing intergranular accommodation variables. The well-known Schmid criterion is the local plasticity criterion and the increment of crystallographic slip is deduced from it according to the normality rule for the system s of grain g:
γ˙
gs
|τ gs − x gs | − r gs = k
n sign(τ gs − r gs )
The local isotropic and kinematic hardening variables and their evolution laws have been introduced in Sect. 3.10.3. We will insist in particular on the presence of an interaction matrix between slip systems of a given grain allowing us to model selfhardening and latent hardening. The viscoplastic framework chosen here has the advantage of avoiding the possible indetermination on active slip systems that arises in the rate-independent case. Local plastic strain is then due to the contribution of all slip systems: γ˙ gs { mgs ⊗ ngs } ε∼˙ pg = s
An extension of the previous model to finite transformations by using the concepts presented in Chap. 6 is possible and is given in [FOR99]. The self consistent model
256
Non-Linear Mechanics of Materials
in elastoplasticity has been extended to finite strain formalism in [LIP90]. These extensions are necessary in order to predict the crystallographic texture evolution during deformation.
Distortion of yield surface The yield surface of a metallic polycrystal is generally subjected to a translation (kinematic hardening), an expansion (isotropic hardening) but also a distortion during deformation. The complexity of the convex shape of the yield surface after deformation has been observed many times on many materials thanks to multiaxial test machines [BUI69, ROU85b]. The explicit phenomenological descriptions of the yield surfaces are in general not able to rationalize such distortions and deal only with translation and expansion mechanisms. To the contrary, the polycrystal model uses a large number of internal variables that define implicitly an effective yield surface, a sort of envelope of the polyhedral Schmid yield surfaces. One should also note that using explicitly the crystallographic texture of the material allows one to introduce naturally the anisotropy of the plastic behavior and its evolution, which remains out of reach of classical purely phenomenological approaches. Figure 5.23 shows the evolution of the yield surface during a non-proportional loading at imposed stress in the plane σ11 –σ12 . The shape of the surface depends on
Figure 5.23. Evolution of the yield surface during a non-proportional path under tensionshear loading (polycrystal with 1000 grains, initially isotropic and presenting local isotropic and kinematic hardening): initial state and four successive states
Elements of microstructural mechanics
257
the offset of irreversible deformation used to detect plastic loading. One can imagine plotting the curve representing the limit of the yield surface, or a line corresponding to a (very small) given increment of strain by following a star-shape path (as in the actual experimental tests), or an equipotential line in the viscoplastic case. The case illustrated here was calculated with a microstructure of 1000 randomly oriented grains. The material is then initially isotropic and, when the chosen offset is very small, the obtained yield surface is of Tresca type (Fig. 5.23). In contrast, for a higher detection threshold the surface is more similar to von Mises criterion. The kinematic intragranular part is significant, as shown by the translation of the surface, that follows the loading direction with a low expansion. The distortion is high, the surface sharpening on one side in the direction of loading and flattening in the opposite direction. The successive points for which the current yield surfaces have been explored are indicated on the loading curve.
Overhardening during non-proportional loading The polycrystalline model allows us to model with high accuracy low cycle fatigue tests of polycrystalline metallic materials, under uniaxial as well as under multiaxial conditions provided that local isotropic and kinematic hardening variables are introduced when necessary [CAI92]. The objective is then to account for cyclic hardening or softening of materials. Some materials (316L steel, Waspaloy. . .) exhibit additional hardening for complex multiaxial cyclic paths [LAM78, CAI84, BEN87]. In the case of a material presenting intragranular hardening, the maximum hardening obtained during a biaxial test with a 90◦ phase difference between the components E11 and E22 of the prescribed strain is then much larger than the maximum hardening obtained by loading uniaxially the material with respect to each of these components for a given strain amplitude. Whether there is overhardening or not for a given material can be correlated with the value of the stacking fault energy of the material [DOQ90]. The polycrystal model is able to reproduce such effects through the interaction matrix hrs introduced in Sect. 3.10.3. Indeed, the complex loading paths lead in general to the activation of a greater number of systems during loading. The so-called latent hardening out of diagonal terms of the interaction matrix lead then to additional hardening, actually observed experimentally. Thus, Fig. 5.24 shows a test that starts in tensioncompression, and continues by non-proportional loading at second strain amplitude level. After a few cycles of the first level, the maximum hardening in proportional loading is reached. The following circular deformation path in the (E11 , E22 ) space shows the additional hardening capacity of the material, induced by activating new slip systems. The modeling of overhardening effects reveals the limitations of most models corresponding to purely phenomenological approaches, that must incorporate special variables to work accurately [CAL97], as indicated in Sect. 3.8.6. Even greater overhardening can be obtained for deformation paths more complex than circles [PIL90a].
258
Non-Linear Mechanics of Materials
Figure 5.24. Overhardening obtained for a deformation path corresponding to a circle (equivalent strain of 0.005%) in the plane (E11 , E22 ), compared to cyclic tension-compression of same amplitude, (a) response in the stresses plane (b) evolution of von Mises equivalent stress (same material as in Fig. 5.25)
Memory effects Like the overhardening effect, the memory effect associated with maximum strain has been thoroughly studied since its first description [CHA79b]. It is related to the fact that dislocation structures that form when the material is at maximum strain are not the same as the ones that develop at low strain levels (dislocations walls/cells). The crystallographic model reproduces, with no additional ingredient, the effect simulated by the specific model of Sect. 3.8.4. Figure 5.25 represents thus a tension-compression test consisting of three groups of cycles with a symmetric imposed loading, the first and the third with an amplitude of 0.5% and the second, 1.5%. One observes, actually, that the maximum stress reached during the third group is higher than the one of the first group.
5.7. Computation of RVE A recent trend consists in supporting the development of approximate methods such as the self-consistent model, by large scale finite element simulations of realizations of heterogeneous materials, and, if possible, of Representative Volume Elements. This becomes possible only thanks to the growing power of computers. It generally requires parallel computing. It is possible, for instance, to use the finite element method coupled to a realistic microstructure model. The first question to be solved is the proper size of the considered volume elements. This essential question of the RVE size is addressed below in the case of elastic polycrystals.
Elements of microstructural mechanics
259
Figure 5.25. Maximum strain memory effect during a cyclic test: first level ±0.005, second level: ±0.015, third level ±0.005. The maximum stress at the third level is higher than at level 1 (polycrystal with 1000 isotropic grains, having local isotropic and kinematic hardening)
Figure 5.26. Distribution of 238 grains in one representative volume element of a polycrystal; mesh by corresponding finite elements (20 nodes and 27 Gauss points elements)
260
Non-Linear Mechanics of Materials
5.7.1. Representative volume element size Computational homogenization methods are efficient tools to estimate effective properties of heterogeneous materials. They can take realistic distributions of phases and sophisticated constitutive equations of the constituents into account [CAI03]. A keypoint in such models is the determination of the appropriate size of volume elements of heterogeneous materials to be computed in order to get a precise enough estimation of effective properties. This is related to the long-standing problem of the determination of the size of the Representative Volume Element (RVE) in homogenization theory. The present section also raises the question of the existence and determination of the RVE in the case of thin polycrystalline structures (metal sheets, plates, layers, films. . .). For that purpose, the apparent mechanical properties of volumes of increasing sizes are compared in the case of polycrystalline thin sheets but also bulk polycrystals taken as a reference. The statistical and numerical methodology proposed in [KAN03] is used to estimate the size of a RVE in isotropic linear elastic copper polycrystals. The method follows three main steps: the choice of a random model for polycrystalline microstructures containing a finite number of grains; the resolution of boundary value problems on such polycrystalline aggregates of increasing sizes; the analysis of the convergence of the calculated apparent properties towards an asymptotic value as a function of the number of grains and of the boundary conditions. The asymptotic value is regarded as the effective property [SAB92]. In other words, the objective is to find the minimum number of grains required in a volume element to estimate the effective elastic property with a given accuracy. The size of the RVE for several cubic elastic bulk polycrystals was investigated in [NYG03] using three-dimensional FE simulations and periodic boundary conditions. A relationship between the RVE size and the anisotropy coefficient of each material was identified. The author links the notion of representativity of considered material volumes with the decay of the scatter in the calculated apparent properties for increasing grain numbers, as done in [KAN03]. The present contribution lays the stress on the dependence of the result on the choice of boundary conditions and the determination of a statistical parameter quantifying the decrease in scatter with increasing grain number. Such a parameter makes it possible to compare RVE sizes for other microstructures and properties. Thin polycrystalline materials have only a limited number of grains through the thickness, in contrast to bulk polycrystals. Such microstructures are increasingly encountered in micro-electromechanical systems (MEMS) but also in coatings and layered microstructures [HOM01]. The existence of effective properties and their determination can be useful for subsequent structural computations, since one usually cannot afford considering all grains in the simulations. The proposed methodology for the determination of effective properties is applied to copper thin plates with a fixed number of grains through the thickness. The lateral dimensions of the plate are
Elements of microstructural mechanics
261
then increased in order to determine the wanted RVE size, if it exists. The possible bias induced by the boundary conditions in the computation must be carefully investigated. The determination of effective properties in textured copper thin films on a substrate was tackled in [WIL02] but the question of the existence of a RVE and the effect of boundary conditions were not addressed. All simulations are carried out for linear elastic copper polycrystals with a uniform distribution of crystal orientations, leading to an isotropic texture in both bulk and thin polycrystals. The cubic elasticity constants of pure copper at 300 K are taken from [GAI81]: C11 = 168400 MPa, C12 = 121400 MPa, C44 = 75390 MPa. The corresponding value of the anisotropy coefficient a = 2C44 /(C11 − C12 ) is 3.2.
Generation of microstructures Voronoï mosaics are used here as a random model to represent the polycrystalline morphology, as explained in [KAN03]. For each realization, one given volume V (a cube or a plate with fixed thickness) that contains a given number Ng of Voronoï cells is simulated. In the following, n realizations of volume V are considered. The number of cells for each realization of the microstructure obeys a Poisson distribution with given mean value N¯g = N . The average volume of one Voronoï cell is equal to 1. No unit length is introduced because the models involved in this section cannot account for absolute size effects. As a result, one has N = V . A crystal orientation is attributed to each Voronoï cell which is then regarded as an individual grain of the polycrystal. The crystallographic texture is assumed to be uniformly random. It is possible to impose a geometrical periodicity constraint at the boundary of the polycrystalline cube or thin structures, as shown in Fig. 5.27 (see also [KAN03]). This condition is enforced in the subsequent FE simulations involving periodicity conditions, if not otherwise stated. It results in a slight decrease of the dispersion of the apparent properties when compared to simulations relying on the initial Voronoï model.
FE meshing of microstructures and parallel computing The so-called multi-phase element technique is used in order to superimpose a regular 3D FE mesh on the Voronoï tessellation. The crystal orientation of the closest voxel is attributed to every integration point of each element of the mesh [KAN03]. The elements are 20-node quadratic bricks with 27 Gauss points. Figures 5.27(a) and (b) show such meshes made of 16 × 16 × 16 and 7 × 36 × 36 elements respectively. The main drawback of the technique is that one element may contain integration points belonging to several grains. The bias introduced by this meshing technique was investigated in [KAN03, BAR01a]. In particular, it does not affect the mean stress and strain values computed over the whole volume. The effect of mesh density, i.e., of
262
Non-Linear Mechanics of Materials
Figure 5.27. Regular FE mesh superimposed on Voronoï mosaics using the multiphase element technique: (a) cube containing 50 grains with periodicity constraint at the boundaries, (b) layer with 3 grains within the thickness containing 657 grains, with in-plane periodicity constraint. Arbitrary colors are attributed to the grains
the number of elements per grain, on apparent shear modulus μapp was investigated for an elastic polycrystalline volume, containing 50 grains, when the number of finite elements is increased. For each simulation, the geometry of the microstructure is unchanged but the number of degrees of freedom, namely, the unknown displacement components, was changed from 5568 to 56355. From these results, a resolution of 16 elements per grain was chosen for the following calculations. The largest volume computed in this section is a cube with 423 = 74088 elements, i.e., 937443 degrees of freedom. Such computations are made possible in a reasonable time by using parallel computing. The FE program used in this work implements the subdomain decomposition method FETI [Z-01]. The mesh is split into 32 subdomains and the tasks are distributed on a platform of 32 processors (768 MB RAM, 800 MHz). Compatibility and equilibrium at interfaces between subdomains are restored by an iterative procedure. The whole resolution requires 21 GB of memory.
Elements of microstructural mechanics
263
Boundary conditions and definition of apparent moduli Three types of boundary conditions to be prescribed on an individual volume element V are recalled: • Kinematic uniform boundary conditions (KUBC): The displacement vector u is imposed at all points x belonging to the boundary ∂V according to: 1 u=E .x ∀x ∈ ∂V =⇒ ε := εdV = E (5.175) ∼ ∼ ∼ V V∼ where E is a given constant symmetrical second-rank tensor. The macroscopic ∼ is then defined as the spatial average of the local stress tensor σ∼ . stress tensor ∼ • Static uniform boundary conditions (SUBC): The traction vector is prescribed at the boundary ∂V according to: 1 σ∼ .n = .n ∀x ∈ ∂V =⇒ σ := σ dV = (5.176) ∼ ∼ ∼ V V ∼ where is a given constant symmetrical second-rank tensor. The outer normal ∼ is then defined as to ∂V at x is denoted by n. The macroscopic strain tensor E ∼ the spatial average of the local strain ε∼ . • Periodicity conditions (PERIODIC): The displacement field over the entire volume V takes the form u=E .x + v ∀x ∈ V (5.177) ∼ where the fluctuation v is periodic. v (resp. σ∼ .n) takes the same value (resp. opposite values) at two homologous points on opposite sides of V . The local behavior at every integration point inside each grain in the simulation is described by the fourth-rank linear elasticity tensor ∼c: ∼
σ∼ (x) = ∼c(x) : ε∼ (x)
(5.178)
∼
For a given volume V, and owing to the linearity of the considered boundary value app problems, fourth-rank tensors of apparent moduli C and apparent compliances ∼E ∼
app
S∼ can be defined by the following macroscopic relations: ∼
= σ∼ = ∼
1 V
V
app
σ∼ dV = C ∼E ∼
:E ∼
E = ε∼ = ∼
1 V
V
app
ε∼ dV = S∼
∼
: (5.179) ∼
The first relation is used for KUBC and PERIODIC problems, the second one for app SUBC problems. Note that in general, the tensor S∼ cannot be expected to coinapp
∼
cide with the inverse of C . However, for sufficiently large volumes V (along all ∼E ∼
264
Non-Linear Mechanics of Materials
three directions of space), the apparent moduli do not depend on the type of boundary conditions any longer and coincide with the effective properties of the medium [SAB92]: app−1 app eff = S∼ eff −1 = C =C (5.180) S∼
∼ ∼E ∼
∼
∼
∼
For intermediate volumes V , the following inequalities, written in the sense of quadratic forms, hold [HUE90]: app−1
S∼
∼
app
eff ≤C ≤C ∼ ∼E ∼
(5.181)
∼
eff and the periodic estimations are checked to remain In the next sections, both C ∼ ∼
between the bounds defined by (5.181). The following two shear loading conditions E and are used in this section: ∼μ ∼μ ⎡
=⎣ E ∼μ
0 1 2
0
1 2
0 0
⎤ 0 0 ⎦ 0
⎡
0 ⎣ a = μ ∼ 0
a 0 0
⎤ 0 0 ⎦ 0
with a = 1 MPa
(5.182)
in the particular Cartesian coordinate frame attached to the cubic volume element. In the case of KUBC and PERIODIC conditions prescribed for a given volume V , one app defines the apparent modulus μE by the work of internal forces in the volume V : subjected to the loading E ∼μ 1 app = σ12 dV (5.183) μE (V ) := σ∼ : ε∼ = σ∼ : E μ ∼ V V app
In the case of SUBC boundary conditions, an apparent shear modulus μ is defined : as the work of internal forces generated in V by the application of the loading ∼μ 2a a2 := σ∼ : ε∼ = : ε∼ = app ∼μ V μ (V )
ε12 dV
(5.184)
V
These definitions remain formal insofar as the apparent elasticity properties of a given material volume element V are not necessarily isotropic.
5.7.2. A definition of the RVE size It is known that the RVE size is property and morphology dependent, but a well-suited parameter is necessary for quantitative comparisons. The choice of the boundary conditions applied to volumes of heterogeneous materials introduce a bias in the estimation of the apparent mechanical properties. A deterministic definition of the RVE is related to the volume size at which the estimated properties no longer depend, within a
Elements of microstructural mechanics
265
given statistical precision, on the choice of boundary conditions [SAB92]. This leads however to large volume sizes that are sometimes hardly tractable numerically. In contrast, a series of simulations for various microstructures have shown that the use of periodicity conditions provide, with relatively small volumes, estimations that are close to the wanted effective properties [KAN03, NYG03]. Such estimations based on rather small volumes require however a sufficient number of realizations of the microstructures to get accurate enough estimates. This suggests a pragmatic and statistical definition of the RVE size, as the minimum size for which the effective properties are estimated with a wanted precision [KAN03] for a well-chosen set of boundary conditions. This is the definition of RVE size adopted in this book. A quantitative definition of this RVE size is given in Sect. 5.7.3.
5.7.3. RVE size for bulk copper polycrystals Apparent shear moduli for bulk polycrystalline copper Due to the uniform distribution of crystal orientations, the effective medium exhibits an isotropic linear elastic behavior, described by effective bulk and shear moduli k eff and μeff . For cubic symmetry, the apparent bulk modulus is not a random variable [GAI81]. It is uniquely determined from the single crystal elasticity constants according to the formula k app = k eff = (C11 + 2C12 )/3 = 137067 MPa. As a result, the homogenization problem reduces to the estimation of apparent shear properties μapp and in fine of the effective shear modulus μeff . It is shown in [KAN03] that the fourthapp (V ) obtained for a finite domain V containing rank tensor of apparent moduli C ∼E ∼
¯ app (V ), i.e., its Ng grains is generally not isotropic. However, its ensemble average C ∼E ∼
mean value over a sufficiently large number of realizations, turns out to be isotropic. This has been checked here for polycrystalline copper aggregates. The shear modu¯ app (V ) coincides with μ¯ app (V ), lus associated with the isotropic elasticity tensor C E ∼E ∼
app
the ensemble average of the apparent shear moduli μE (V ) defined by (5.183) and computed for a domain V of given size (or equivalently containing N = V grains in app average). Accordingly, the estimation of μ¯ E (V ) only requires the determination of app μE (V ) for each realization. This is the computation strategy adopted in this section. app Similarly, using SUBC conditions, it is sufficient to compute μ for each realization according to equation (5.184). The apparent shear moduli μapp (V ) were estimated using cubic volume elements V of increasing size, ranging from V = 25 to V = 5000 grains, with n(V ) realizations for every volume. The value of n is such that the estimation of the mean μ¯ app (V ) is obtained with a precision better than 1%. Simulation results for bulk copper polycrystals are shown in Table 5.1.
266
Non-Linear Mechanics of Materials
V KUBC KUBC KUBC KUBC PERIODIC PERIODIC PERIODIC PERIODIC SUBC SUBC SUBC SUBC
25 400 1000 5000 25 123 400 500 25 400 1000 5000
n 100 50 25 10 100 50 50 50 100 50 25 10
μ¯ app (MPa) 52543 50088 49787 49336 49669 48886 48784 48764 43397 47308 47566 48390
Dμ (V )(MPa) 3186 836 533 222 3162 1400 811 778 3185 823 538 178
rel 1.2% 0.4% 0.4% 0.2% 1.2% 0.8% 0.4% 0.4% 1.4% 0.4% 0.4% 0.2%
Table 5.1. Mean apparent shear modulus, associated scatter and relative error on the mean as a function of the domain size and of the number of realizations for three different boundary conditions (bulk copper polycrystals)
Figure 5.28. Mean values and intervals of variation for the shear modulus μapp as a function of domain size, for three different boundary conditions (bulk copper polycrystals)
Mean values and intervals of variation for the apparent shear modulus [μ¯ app (V ) − 2Dμ (V ), μ¯ app (V )+2Dμ (V )], are plotted in Fig. 5.28, as a function of volume size V . The mean apparent shear moduli strongly depend on the domain size and on the boundary conditions. However, the values converge towards an asymptotic constant μeff as the volume size increases, as expected. A striking feature of these results is the very fast convergence of the periodic solution and, in contrast, the very slow con-
Elements of microstructural mechanics
267
vergence associated with homogeneous boundary conditions. The periodic estimate is bounded by the KUBC and SUBC estimates: app
μReuss ≤ μ¯
app
app
≤ μ¯ PERIODIC ≤ μ¯ E
≤ μV oigt
(5.185)
where μReuss and μV oigt denote the first order lower and upper bounds for the effective shear modulus of the polycrystal. For decreasing values of V , the apparent app app moduli μ¯ E (V ) (resp. μ¯ (V )) get closer to the upper (resp. lower) limit μV oigt (resp. μReuss ). The bias observed on the mean value for all loading conditions for small volumes is clearly due to the specific boundary layer effect induced by each type of boundary condition. Another important result is the rate of decrease in the variance Dμ2 (V ) of μapp with increasing V for all three types of boundary conditions. Finally, the estimated effective shear modulus is compared to the self-consistent estimate according to [GAI81] in Fig. 5.28. The self-consistent method predicts a shear modulus of 48167 MPa, which is 1.2% lower than the periodic solution found with 500 grains. This difference lies within the numerical precision associated with the mesh density chosen in Sect. 5.7.1. Note that this result a priori depends on the anisotropy coefficient of the considered cubic material.
Size of the RVE for bulk polycrystalline copper The notion of RVE is necessarily related to the choice of a statistical precision in the estimation of the investigated effective property. First, we set a tolerance error α on the bias and find a corresponding volume V0 such that: |μ¯ app (V0 ) − μeff | ≤ α
(5.186)
This condition sets a lower bound for the size of the RVE. Then, the relative precision of the estimation of the mean μ¯ app (V ) of apparent shear moduli for a given volume V ≥ V0 and a given number of realizations n, can be defined according to the sampling theory by: 2Dμ (V ) (5.187) rel = app √ μ¯ (V ) n This definition includes explicitly the number of realizations n. In turn, the number of realizations required to correctly estimate μ¯ app (V ) is deduced from (5.187) provided that the variance Dμ2 (V ) is known. According to homogenization conditions (5.182), (5.183) and (5.184), the apparent shear modulus is obtained by averaging an additive scalar over the volume V . As a result, for asymptotically large volumes, the variance Dμ2 (V ) of μapp (V ) is given by: Dμ2 (V ) = Dμ2
A3 V
(5.188)
268
Non-Linear Mechanics of Materials
where A3 is the integral range, a well-established quantity for additive geometrical properties such as volume fraction. It has the dimension of a volume. Dμ2 is the point variance of c1212 (x), which depends on the crystal orientation at x. For uniform orientation distributions, it can be expressed in terms of the single crystal cubic elasticity constants as follows: Dμ2 = (c∼ : c∼)1212 − c1212 2 ∼
∼
with c1212 =
1 (C11 − C12 + 3C44 ) 5
(5.189)
1 2 2 2 +2C11 +6C11 C44 +15C44 ) (5.190) (−6C44 C12 −4C12 C11 +2C12 35 where . denotes here averaging over uniformly distributed crystal orientations. For pure copper, one gets Dμ = 13588 MPa. We choose to identify the integral range A3 from the results obtained with periodicity conditions because they introduce the smallest bias in the estimated effective shear modulus. We find A3 = 1.43, to be compared with the mean grain size set to 1. It can also be compared to the integral range for the volume fraction of a given orientation A3 = 1.17 given in [KAN03]. The integral range A3 is a well-suited parameter to compare RVE sizes for different properties and morphologies. It characterizes the rate of decrease in the dispersion of apparent properties for increasing volume sizes, according to (5.188). (c∼ : ∼c)1212 = ∼
∼
Equations (5.186), (5.187) and (5.188) can now be used quantitatively to determine a minimal size of RVE for a given precision rel and a given number of realizations n: V =
A3 4 2 Dμ 2 n rel (μeff )2
(5.191)
In the case of periodic boundary conditions, the choice (rel , n) = (1%, 10) gives a minimal volume corresponding to V = 445. For n = 100 computations, this volume reduces to 45. This low number of grains still remains in the domain range for which periodic boundary conditions introduce only a slight bias in the estimation of the effective property. The obtained results compare quite well with the prediction of the formula identified in [NYG03] that relates the size of the RVE for a given precision to the anisotropy coefficient a. Taking a relative error rel = 1% and 20 realizations, Nygårds’ formula predicts V = 265, vs 220 according to our formula (5.191). Interestingly, the simulations of [REN02] for 2D elastic copper polycrystals lead to a value V = 484, but in the case of homogeneous boundary conditions and without any information about the variance of the results.
5.7.4. RVE size for thin polycrystalline copper sheets In this section, the question of the existence of homogeneous equivalent properties and of the size of the corresponding representative volume element is investigated in the
Elements of microstructural mechanics
269
case of linear elastic polycrystalline copper sheets exhibiting a small and fixed number of grains through the thickness. The crystallographic orientations are distributed randomly among the grains in each simulation, which results in an isotropic crystallographic texture. The grain morphology and distribution in space still correspond to the Voronoï mosaic random model. The polycrystalline copper layer of Fig. 5.27(b) is obtained by superimposing a regular mesh of a plate with a given thickness on a bulk Voronoï mosaic. As in the previous section, a crystallographic orientation is attributed to each integration point in each element according to the color of the underlying voxel. The same mesh density as in Sect. 5.7.3 is used for the plate, namely 16 elements per grain. Two thicknesses are considered corresponding respectively to an average of 1 grain and 3 grains through the thickness. Square sheets are considered, the normal to the sheet being the direction 3. Two orthogonal in-plane directions parallel to edges of the square plate are labeled 1 and 2. In all simulations presented in this section, the thickness of the plate and grain size are kept constant whereas the width is increased gradually leading to larger volumes V and therefore larger numbers of grains. The boundary conditions KUBC and PERIODIC defined in Sect. 5.7.1 are applied to the entire outer surface of the plates, namely both faces normal to direction 3, and the four lateral faces. In the case of PERIODIC boundary conditions, a periodicity constraint is imposed in the morphology of the grains, as depicted in Sect. 5.7.1 but only with respect to lateral faces. This accelerates slightly the convergence of the apparent moduli. No periodicity constraint is prescribed to the morphology of the grains on the faces normal to direction 3. Special boundary conditions called mixed PERIODIC–SUBC are introduced that are especially relevant for free-standing films. According to these conditions, periodicity conditions for all displacement components are imposed on the lateral faces of the plate, whereas the stress vector is prescribed on both surfaces normal to direction 3. The volume sizes, or equivalently the numbers of grains in the sample plates, considered in the simulations of the elastic deformation of copper polycrystalline plates are listed in Table 5.2 in the case of one grain through the thickness and in Table 5.3 in the case of three grains, for all types of boundary conditions. They range from 50 to 400 grains per polycrystalline plate in average. The numerical analysis is restricted to the computation of the apparent shear moduli of a large number of plates corresponding to different realizations of the microstructure. Whatever the type of boundary conditions, the apparent shear moduli are defined as proper ratios of mean shear stress and shear strain components over the whole volume of the plate, according to the formula: ⎞ ⎛ app ⎞⎛ ⎞ ⎛ C44 × × 2E23
23 app ⎝ 31 ⎠ = ⎝ × (5.192) × ⎠ ⎝ 2E31 ⎠ C55 app
12 2E12 × × C66 In the case of KUBC and PERIODIC boundary value problems, the components
270
Non-Linear Mechanics of Materials
KUBC C66 KUBC C44 KUBC C66 KUBC C44 KUBC C66 KUBC C44 KUBC C66 KUBC C44 PERIODIC C66 PERIODIC C44 PERIODIC C66 PERIODIC C44 PERIODIC C66 PERIODIC C44 PERIODIC C66 PERIODIC C44 mixed PERIODIC–SUBC C66
V
n
50 50 120 120 260 260 400 400 50 50 120 120 260 260 400 400
50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
app app C¯ 66 or C¯ 44 (MPa) 52498 53234 52143 52842 52350 52608 52211 52428 48794 49884 48467 49609 48788 49405 48604 49431
400
50
46350
Dμ (V ) (MPa) 2039 2771 1372 1310 1101 945 797 675 2066 2985 1437 1375 1149 1024 785 905
rel 1.1% 1.5% 0.7% 0.7% 0.6% 0.5% 0.4% 0.4% 1.2% 1.7% 0.8% 0.8% 0.7% 0.6% 0.5% 0.5%
791
0.5%
Table 5.2. Mean apparent shear moduli, associated scatter and relative error on the mean as a function of the domain size and of the number of realizations for three different boundary conditions (1-grain-thick polycrystalline copper sheets)
app
app
E12 = 1/2 and E23 = 1/2 are imposed successively to estimate C66 and C44 respectively, the remaining components of the average strain tensor E being set to ∼ zero. In the case of mixed PERIODIC–SUBC problems, the components 12 = a are prescribed successively to and 23 = a(= 1 MPa) of the average stress tensor ∼ the volumes, the remaining components being set to zero. app app It cannot be expected a priori that the mean apparent properties C¯ 66 and C¯ 44 will coincide, even for a sufficiently high number of realizations or number of grains, due to the special morphology of the plates with a small number of grains through the thickness. That is why both properties are evaluated. For reasons of statistical homogeneity of grain distribution and isotropy of the given crystallographic texture, app app the mean apparent property C¯ 55 coincides with C¯ 44 . It is therefore not reported here. For the same reasons, the components labeled with a cross × in (5.192) vanish in average (but, of course, not in general for an individual plate). A number of 50 realizations for each investigated volume size corresponding to distinct grain and orientation distributions was found to be sufficient to ensure a relative error always smaller than 1.7% in the estimation of the apparent properties. A relative precision of
Elements of microstructural mechanics
271
Figure 5.29. Finite element computations of a plate containing 500 grains with one grain within the thickness: (a) finite element mesh and morphology of the grains, (b) stress heterogeneities induced by in-plane shear under mixed PERIODIC–SUBC conditions. The deformation state of the grains is magnified for the illustration. The represented variable is the min normalized equivalent von Mises stress σeq /σeq
0.4–0.5% was even reached for larger volumes. An example of such computations is given in Fig. 5.29. The finite element mesh of a thin sheet with one grain through the thickness is shown in Fig. 5.29(a). The Fig. 5.29(b) gives the fields of stress concentration in the case of in-plane shear loading according to mixed PERIODIC–SUBC boundary conditions. The stress concentration is defined here as the ratio of the local von Mises equivalent stress σeq divided by the minimum value reached in the aggregate for a given mean shear strain E12 (resp. shear stress 12 ). The roughness of the deformed plate and the stress gradient that develops along the normal direction are visible. app
Mean values and intervals of variation for the apparent shear moduli C44 and are plotted, as a function of volume size V , in Fig. 5.30 in the case of one grain
app C66
272
Non-Linear Mechanics of Materials
Figure 5.30. Mean values and intervals of variation for the shear modulus μapp as a function of domain size, for three different boundary conditions (one-grain thick copper polycrystalline sheets)
through the thickness. Each mean apparent shear modulus for each type of boundary conditions is found to slightly depend on the volume size and to converge towards an asymptotic value. These limit values can be defined unambiguously since the mean apparent properties do not significantly vary for the considered largest volume sizes. They are called C¯ 44 and C¯ 66 respectively. In the meantime, the scatter of the individual apparent properties decreases slowly with increasing volume size. Two main results can be pointed out. First, the asymptotic mean apparent shear modulus C¯ 66 (resp. C¯ 44 ) depends on the type of boundary conditions: KUBC PERIODIC mixed C¯ 66 = C¯ 66 = C¯ 66
KUBC PERIODIC mixed C¯ 44 = C¯ 44 = C¯ 44
(5.193)
The same ranking of asymptotic shear moduli according to the type of boundary conditions was found for bulk polycrystals: mixed PERIODIC KUBC C¯ 66 < C¯ 66 < C¯ 66
mixed PERIODIC KUBC < C¯ 44 < C¯ 44 C¯ 44
(5.194)
PERIODIC and C ¯ KUBC is significant compared The found difference of 7% between C¯ 66 66 to the relative precision of the estimations. It can be noticed that the asymptotic inPERIODIC = 48604 MPa is close to the effective shear modulus plane shear modulus C¯ 66 eff μ = 48764 MPa found for bulk copper polycrystals. Second, an anisotropy of asymptotic shear properties pertains for large plates: KUBC KUBC C¯ 66 = C44
PERIODIC PERIODIC = C¯ 44 C¯ 66
(5.195)
unlike what happens to bulk polycrystals. The found differences of about 1.7% bePERIODIC and C ¯ PERIODIC is significant, compared to the relative precision of tween C¯ 66 44
Elements of microstructural mechanics
273
the estimations. The effective shear properties based on KUBC boundary conditions are close to the values that one gets by considering 2D crystal aggregates under plane strain conditions [REN02].
Discussion on the existence of a RVE for thin structures The main feature of the analysis of thin layers is that the boundary conditions induce a bias in their asymptotic elastic behavior, even for very large plates. This is in contrast to the bulk behavior of elastic polycrystals for which the apparent shear modulus μ¯ app was found to converge towards a single asymptotic value μeff irrespective of the type of boundary conditions. Even though asymptotic shear properties can be associated to the polycrystalline plate for a given type of boundary conditions, we reserve the term of effective properties to the case of asymptotic values that eventually do not depend on the type of boundary conditions. Accordingly, unambiguous effective properties do not exist for copper polycrystalline layers with one grain through the thickness. As a result, a RVE cannot be defined unambiguously in this case. In practice, the asymptotic values found for each type of boundary conditions can be used to model the elastic behavior of thin copper layers provided that the in situ material layers undergo similar boundary or interface conditions. For instance, in the case of thin copper sheets with free surfaces, the asymptotic shear properties obtained from mixed PERIODIC–SUBC conditions (or equivalently SUBC conditions) will be the relevant ones for further structural analyses. The found slight anisotropy of the asymptotic tensor of elastic moduli should also be taken into account. The authors in [WIL02] also study the evolution of scatter when increasing the number of grains in textured copper one-grain thick coatings on a substrate in the case of mixed KUBC– SUBC–PERIODIC boundary conditions. They also found asymptotic values of the apparent elastic properties, which are specific to the chosen boundary conditions. When the number of grains within the thickness increases, the bias introduced by the boundary conditions is reduced. The case of thin copper sheets having 3 grains through the thickness was investigated. Table 5.3 shows that 50 simulations over 400 grains with PERIODIC boundary conditions provide mean apparent shear properties close to the shear modulus of bulk polycrystalline copper (the difference is less than 0.8%). There is still a difference of 2% between the asymptotic values of the apparent PERIODIC and C ¯ KUBC . The bias introduced by the boundary conditions properties C¯ 66 66 does still exist but is significantly smaller than for one-grain-thick sheets. FurtherPERIODIC and more, the anisotropy of the shear moduli has almost disappeared since C¯ 66 PERIODIC C¯ 44 do not by more than 0.05% with a relative precision of 0.35% on the mean. Accordingly, the concepts of effective properties and RVE are found to be meaningful in the case of 3-grain thick copper polycrystalline sheets with a tolerance of 2%. The layer that concept of RVE is also characterized by the value of the integral range A3 layer can be defined as in Sect. 5.7.3, (5.188). The value A3 = 1.40 is deduced from
274
Non-Linear Mechanics of Materials
KUBC C66 KUBC C44 PERIODIC C66 PERIODIC C44 PERIODIC C66 PERIODIC C44 KUBC C66 KUBC C44 KUBC C66 KUBC C44 KUBC C66 KUBC C44
V 120 120 123 123 400 400 1000 1000 2000 2000 5000 5000
n 50 50 50 50 50 50 50 50 10 10 1 1
app app C¯ 66 or C¯ 44 (MPa) 51212 50814 48805 48914 49123 49148 50418 50993 50464 50891 50200 51000
Dμ (V )(MPa) 1483 1516 1400 1400 821 614 530 446 243 274
rel 0.8% 0.8% 0.8% 0.8% 0.5% 0.35% 0.3% 0.25% 0.3% 0.3%
Table 5.3. Mean apparent shear modulus, associated scatter and relative error on the mean as a function of the domain size and of the number of realizations for PERIODIC conditions (3-grain-thick polycrystalline copper sheets)
the results of Table 5.3, which is almost equal to the integral range found for bulk polycrystals. Finally, for a given precision of rel = 1% in the estimated mean and for n = 10 realizations and periodic boundary conditions, the RVE size is found to be equal to 435 grains, which is close to the result of 445 grains found for bulk polycrystals. The main difference is that, in the case of copper polycrystalline layers, the RVE does not have the shape of a cube. It is a plate with 3 grains within the thickness and 12 grains along the edges, instead of a cube of about 8×8×8 grains.
5.7.5. Elastoplastic behavior of polycrystalline aggregates The global behavior of polycrystals can be deduced from a calculation on a finelymeshed polycrystalline aggregate incorporating hundreds of grains [EBE98, BAR01a, BAR01b, FOR00]. Thanks to this approach, one has also access to the heterogeneity of strain in the phases, which is ignored in classical approaches. This technique extends, of course, to any situation for which it is possible to simulate a realization of the microstructure with a maximum of geometrical details. It is recommended for periodic but also for random conditions, on condition that one considers a large enough RVE, or a sufficient number of realizations. The complexity of local behaviors requires, in general, three-dimensional modeling. Figures 5.26 and 5.31 show for example the realization of a polycrystalline aggregate from Voronoï polyhedron as well as the associated plastic deformation state during an imposed kinematic-boundary test.
Elements of microstructural mechanics
275
Figure 5.31. Accumulated plastic strain field in the polycrystalline aggregate for a deformation value of E33 = 0.2% in tension (parallel calculation on 8 processors, imposed kinematic boundary E33 = −2E11 = −2E11 )
5.8. Homogenization of “coarse grain structures” Previous methods are valid only when macroscopic strain is quasi-constant on distances of the RVE size. This hypothesis breaks down in the case of many industrial components, because the unit cell of a sandwich structure, for instance, does not always have a size that can be neglected with respect to the component size. We call “coarse grains structures”, such components that are made of heterogeneous materials with constituents of non-negligible size. In particular, the complex geometries of these components lead to large thermomechanical loading gradients. The previous methods cannot be used in this case, because the condition l Lω is not fulfilled. A way of extending these methods consists in introducing non-homogeneous boundary conditions on the RVE, of a realistic type, if possible, with respect to the final application. It follows that the resulting HEM is not anymore a classical continuum but a generalized
276
Non-Linear Mechanics of Materials
continuum (for example second gradient continuum, or Cosserat continuum or micromorphic continuum). Kinematics and balance equations of a Cosserat continuum will be presented at the end of Chapt. 6 and constitute a simple example of a generalized continuum.
5.8.1. An example of inhomogeneous average loading of a unit cell We consider a material whose microstructure is periodic and whose unit cell is provided in Fig. 5.32a. It is made of a succession of steel sheets incorporating rods and by elastomeric layers. We are interested in the effective linear properties of the material in the two-dimensional case under plane strain conditions. The values of the classical moduli are obtained by considering tensile and simple shear tests (Figs. 5.32b and 5.33), in accordance to the periodic homogenization theory, already presented. Moreover, we try to associate a bending modulus to this cell. For that purpose, a nonlinear contribution is added to the classical expression of the imposed displacement on the cell boundary in kinematic boundary condition: ui = Eij xj + ij k Kj l xl xk
(5.196)
If K vanishes, we recognize the classical conditions. If K33 or K31 are not zero, we ∼ obtain respectively: u1 = −K31 x1 x2 u1 = −K33 x2 x3 or (5.197) u2 = K33 x1 x3 u2 = K31 x12 which are torsion or bending-like prescribed conditions. In the periodic case considered here, the previous expression is enhanced by introducing periodicity conditions
Figure 5.32. Unit cell of a periodic heterogeneous material (left); simple extension test to the left in direction 1 (right)
Elements of microstructural mechanics
277
Figure 5.33. Simple extension test in direction 2 (left) and simple shear (right); deformations have been magnified significantly
Figure 5.34. Imposed bending boundary conditions (left) and in a periodic way (right)
on the perturbation v: ui = Eij xj + ij k Kj l xl xk + vi
(5.198)
We exemplify the non-homogeneous boundary conditions in Fig. 5.34a and the periodic bending conditions in Fig. 5.34b.
278
Non-Linear Mechanics of Materials
5.8.2. Generalized Hill–Mandel condition Applying the previous non-homogeneous boundary conditions leads to the following expression of the work of internal forces: σ∼ : ε∼ = :E +M :K ∼ ∼ ∼ ∼
(5.199)
with = σ∼ and Mij = ikl xk σlj ∼ and K as macroscopic strain and macroscopic As a consequence, if one considers E ∼ ∼ effective curvature tensors and if one defines the effective force-stresses and torquestresses tensors by the relations above, (5.199) represents the work of internal forces of an effective Cosserat continuum. Thus, we have a generalized Hill–Mandel condition to varying mean fields provided that the effective medium is interpreted as a Cosserat continuum and not anymore as a classical Cauchy continuum. This is then the transition from a local heterogeneous classical continuum to a generalized effective continuum. The choice of the type of strain heterogeneities and hence of effective generalized medium is not unique and depends on the type of main loading seen by the final structure in the desired application. Here, only the bending aspect has been introduced. The macroscopic generalized continuum introduced here is actually a Cosserat are symmetric. It is then continuum called “constrained”, for which the stresses ∼ possible to determine all the effective moduli linking generalized stresses and strains: ⎧ ⎤ ⎡ ⎤⎡ ⎤ ⎡ 0 0 0 0
11 Y1111 Y1122 ε11 ⎪ ⎪ ⎨ ⎥ ⎢ ⎢ ⎢
22 ⎥ 0 0 0 0 ⎥ ⎥ ⎢ Y1122 Y2222 ⎥ ⎢ ε22 ⎥ ⎢
⎥ ⎥ ⎢ ⎢ ⎢ 0 0 ⎥ ⎢ ε12 ⎥ 0 Y1212 Y1221 ⎪ 12 ⎥ ⎢ 0 ⎥ ⎢ ⎪ =⎢ ⎩ ⎥ ⎥ ⎢ ε21 ⎥ ⎢
Y 0 0 0 0 Y 21 1221 1212 ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ ⎣ M31 ⎦ ⎣ 0 0 ⎦ ⎣ K31 ⎦ 0 0 0 C3131 M M32 K32 0 0 0 0 0 C3232 One will find application to structural analysis showing the gain linked to these methods, as well as extensions to other generalized continuum theories and to nonlinear local behavior in [TRI96, SUQ97, FOR98a, FOR97, FOR98b, GEE01, BOU01].
Chapter 6
Inelastic constitutive laws at finite deformation
Introducing the finite deformation formalism requires us to revisit elementary notions of geometry, kinematics and statics of continua. Note: in agreement with some traditions, the “.” will be omitted in multiplications between second-rank tensors in this chapter.
6.1. Geometry and kinematics of continuum 6.1.1. Observer and change of observer Any rigid body, or more generally a triad of rigid vectors, generates an observer and a space frame modeled by a three-dimensional affine euclidean space E, with its associated vector space E. Hereafter, we shall denote by (E, E) the current frame of observation. The absolute time variable is denoted by t. A change of frame/observer from (E, E) to (E , E ) is possible at any time using an orthogonal second-rank tensor Q(t) and a translation V (t) according to: ∼
x = Q(t).x + V (t) ∼
(6.1)
where x and x are the positions of a point M of space, respectively in E and E. At each instant t, an identification of spaces E and E is possible so that Q will be treated ∼ as an endomorphism. Equation (6.1) is also called Euclidean transformation. For the J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_6,
280
Non-Linear Mechanics of Materials
problems of continuum mechanics handled in this book, it is sufficient to consider rotation Q(t) only (i.e., with det Q = 1). ∼
∼
6.1.2. Objective tensors Let x 1 and x 2 be two fixed points in E and u = x 2 − x 1 ∈ E a rigid line. Let then x 1 and x 2 be the positions of these points in the frame E , u = x 2 − x 1 ∈ E . We have then: (6.2) u = Q(t).u. ∼
u
We see that depends on time. Any vector field u whose frame transformation is given by (6.2) is called objective. The meaning of such a notion is clear from the previous example. More generally, we shall say that an n-rank tensor of the form:
is objective if:
T(n) = u1 ⊗ · · · ⊗ un
(6.3)
= u1 ⊗ · · · ⊗ un T(n)
(6.4)
where the ui follow the transformation rule (6.2). The second-rank tensor T∼ = u ⊗ v, for example, is objective if: T∼ = u ⊗ v = QT∼ QT ∼
∼
This notion is immediately generalized to a n-rank tensor. An n-rank tensor is called invariant by Euclidean transformation if: T(n) = T(n)
(6.5)
When n = 0 (scalar), the notions of invariance and objectivity coincide. The mass m is considered objective (i.e., invariant).
6.1.3. Position of the material body A material body can be thought of as independent of any observer and modeled through a differentiable manifold B. It is possible to follow its position t at any time in E (i.e., with respect to the frame E) by means of the position function: a ∈ B → x = pt (a) = p(a, t) ∈ t ⊂ E
(6.6)
The current position of material point a in E is x. We can record, for instance, the position 0 of B in E at t = t0 . 0 will be called a reference position. The reference
Inelastic constitutive laws at finite deformation
281
position of material point a is written X. According to another terminology, t is called current configuration and 0 , reference configuration. In what follows, we will consider only observers that share the same reference configuration: Q(t0 ) = 1∼ (6.7) ∼
x and X are respectively Eulerian and Lagrangian representations of a ∈ B. There exists then the correspondence , assumed bijective, between these representations: x = t (X) = (X, t) = pt op0−1 (X)
(6.8)
In the sequel, we will no longer distinguish the material point a from reference position X.
6.1.4. Local placement and metrics Deformation gradient To describe motion in the vicinity of X, we consider the gradient of the transformation : ∂ F∼ = grad x = ⊗ ei (6.9) ∂Xi where the ei form a Cartesian orthonormal basis of E. If E0 is the tangent space to E at X and Et the tangent space to E at x, E0 and Et can be identified with E since E is Euclidean. However, the deformation gradient (6.9) should always be seen as a linear transformation from E0 in E (with which Et is identified): dx = F∼ .dX
(6.10)
The deformation gradient makes it possible to consider the convective frame b from the basis b0 : (6.11) bi = F∼ .bi0 The grid or micro-grid experimental technique exactly illustrates the use of such a convective basis. To the element of matter dV0 = det[dX1 , dX2 , dX3 ] is associated its current volume: dV = det[dx 1 , dx 2 , dx 3 ] which allows us to define the volume dilatation: J = det F∼ =
dV ρ0 = dV0 ρ
282
Non-Linear Mechanics of Materials
where ρ is the local density. The deformation gradient is assumed invertible and its determinant is then positive by convention. The deformation gradient is not an objective physical quantity since it works on the two spaces E0 , E. It transforms like F∼ = QF∼ ∼
(6.12)
in a change of frame.
Convective transport For the sake of clarity, we distinguish here the tangent spaces E0 and E. The deformation gradient can be used to transport a vector u0 ∈ E0 to a vector u ∈ E by the rule u = F∼ .u0 . An inverse transport can be defined based on F∼ −1 . The vectors u0 and F∼ .u0 have the same components in the basis b0 and the convective frame respectively. Other vector transport rules are obtained by using F∼ −T and F∼ T . For linear transformations, one can then define four convective transports. Indeed, if T∼ is considered as a linear transformation on E, four convected tensors T∼ on E0 can be defined, namely: (6.13) τF −1 F (T∼ ) = F∼ −1 T∼ F∼ τF −1 F −T (T∼ ) = F∼ −1 T∼ F∼ −T
(6.14)
τF T F −T (T∼ ) = F∼ T T∼ F∼ −T
(6.15)
τF T F (T∼ ) = F∼ T T∼ F∼
(6.16)
In general, the scalar product of two tensors is not conserved in such transports, except for the following combinations: A :B = τF −1 F (A ) : τF T F −T (B ) = τF T F (A ) : τF −1 F −T (B ) ∼ ∼ ∼ ∼ ∼ ∼
(6.17)
Metrics The components of the metric tensor 1∼ with respect to the convective bases are: 1∼ = b0i ⊗ b0i = Ckl bk ⊗ bl
(6.18)
Ckl = FkiT Fil
(6.19)
with The components Ckl can be interpreted as the Cartesian components of the tensor: = F∼ T F∼ = Ckl b0k ⊗ b0l C ∼
(6.20)
Inelastic constitutive laws at finite deformation
283
The tensor C is called the dilatation tensor or the right Cauchy–Green tensor. We find ∼ and the metric of E by noting that: the link that exists between C ∼ .dX1 ).dX2 = dX1 .(C .dX2 ) dx 1 .dx 2 = (C ∼ ∼
(6.21)
.dx 1 ).dx 2 = dx 1 .(B dX2 ) = dx 1 .B .dx 2 dX1 .dX2 = (B ∼ ∼ ∼
(6.22)
Similarly,
where B = F∼ F∼ T is the left Cauchy–Green tensor. We note that C is an invariant ∼ ∼ is an objective tensor. A given positivetensor upon change of observer whereas B ∼ (X) is called compatible if there exists a transfordefinite symmetrical tensor field C ∼ T (X) = (grad x).(grad x) . One can then obtain the mation x = (X) such that C ∼ compatibility equations by writing that the curvature tensor associated to the metric must be zero because E is Euclidean. induced by C ∼
6.1.5. Rates; strain-rate The velocity field is defined by: V =
∂x (X, t) ∂t
(6.23)
One can then define a Lagrangian (V (X, t)) and an Eulerian (v(x, t)) velocity field. The derivative with respect to time of the vector dx = F∼ .dX is (dx). = F∼˙ .dX = F∼˙ F∼ −1 .dx
(6.24)
We check that the tensor introduced in that way is nothing but the gradient of the Eulerian velocity field: = F∼˙ F∼ −1 = gradc v (6.25) L ∼ where gradc is the gradient operator with respect to Eulerian coordinates. Decomin its symmetric and skew–symmetric parts, leads to the definition of the posing L ∼ and the spin tensor : so-called strain rate tensor D ∼ ∼ } = {L D ∼ ∼
{ = }L ∼ ∼
(6.26)
where the brackets and the inverted brackets denote the symmetric and skew-symmetrdeic parts of the considered second-rank tensor, respectively. One can say that D ∼ scribes the strain rate of the continuum in the following sense: d T (dx 1 .dx 2 ) = (L .dx 1 ).x 2 + x 1 .(L .dx 2 ) = (L .dx 1 ).dx 2 + (L .dx 1 ).dx 2 ∼ ∼ ∼ ∼ dt ˙ .dX2 = 2dx 1 .D .dx 2 = dX1 .C (6.27) ∼ ∼
284
Non-Linear Mechanics of Materials
Here is the transformation rule upon change of observer for the velocity gradient: ˙ QT = QL QT + Q L ∼ ∼ ∼
D = QD QT ∼ ∼ ∼
∼
∼
(6.28)
∼ ∼
˙ QT = Q QT + Q ∼ ∼ ∼
∼
∼ ∼
(6.29)
As a consequence, L and are not objective tensors, whereas the strain rate tensor is ∼ ∼ objective.
6.1.6. Objective derivatives We consider again the vector u = x 2 − x 1 from Sect. 6.1.2, that links two points of an unloaded solid at rest in E. Its derivative with respect to time u˙ of course vanishes in the frame E. Let us consider now the same vector from the point of view of the “mobile” frame E : u = Q.u, and calculate its derivative with respect to time: ∼
˙ QT .u u˙ = Q ∼ ∼
It is not zero! Moreover, it appears that the derivative with respect to time, called the particle derivative, of an objective vector is not an objective vector. Yet, in the previous example we would rather expect from a derivative that it provides the stretching rate of the vector u. We will see that it is possible to define objective derivatives by adding appropriate correcting terms in their definitions. The choice of well-suited objective derivatives in a constitutive model at finite deformation is not an easy task and strongly depends on the studied material or structure. Very often, it will be more convenient to consider derivatives of scalar or invariant tensors. In this latter case, objective derivatives will arise after applying transport rules to the evolution equations.
Jaumann derivative To eliminate the rotation rate of the observer in the previous example, one is tempted to introduce the following “derivative”: D J u = u˙ − .u ∼
(6.30)
is the rotation rate of the continuum. Note that one should actually write: where ∼ J Dx u = u˙ − (x).u ∼
(6.31)
Inelastic constitutive laws at finite deformation
285
i.e., u is considered as an element of the tangent space in x and the derivative depends on this position. J u = 0, ∀x ∈ . In the introductory example, indeed, Dx
We extend the definition to second-rank tensors of the form T∼ = u ⊗ v by imposing that: T∼ J = D J u ⊗ v + u ⊗ D J v (6.32) hence
T∼ J = T∼˙ + T∼ − T ∼ ∼ ∼
(6.33)
We will come back to another interpretation of Jaumann’s derivative at the end of this section.
Convective derivatives Another idea consists in taking advantage of the Lagrangian approach (also called material approach). According to this procedure, a convective transport rule is used before applying time derivation and pulling back the quantity to the considered configuration:
·
T + T∼ L T∼ = F∼ F∼ −1 T∼ F∼ F∼ −1 = T∼˙ − L ∼ ∼ ∼
(6.34)
T T −L T T∼ = F∼ F∼ −1 T∼ F∼ −T F∼ T = T∼˙ − L ∼ ∼ ∼ ∼
(6.35)
·
T∼ = F∼ −T
T∼ = F∼
·
T T T − T∼ L F∼ T T∼ F∼ −T F∼ T = T∼˙ + L ∼ ∼ ∼
(6.36)
T T + T∼ L F∼ T T∼ F∼ F∼ −1 = T∼˙ + L ∼ ∼ ∼
(6.37)
−T
·
Note that the already defined Jaumann derivative can be expressed as a combination of the previous convective derivatives: T∼ J =
1 (T + T∼ ) 2 ∼
(6.38)
The definition in part 2 of this chapter of the second Piola–Kirchhoff stress tensor will lead us to the definition of the following transport rule: τJ F −1 F −T (T∼ ) = J F∼ −1 T∼ F∼ −T
(6.39)
286
Non-Linear Mechanics of Materials
whose associated derivative is:
·
1 T T − T∼ L + T∼ Tr L T∼ = F∼ τJ F −1 F −T (T∼ ) F∼ T = T∼˙ − L ∼ ∼ ∼ ∼ J ◦
(6.40)
also called the Truesdell derivative. The derivatives thus defined are independent of the choice of the reference configuration 0 since they involved the Eulerian velocity gradient only.
Derivative in a local objective frame In some cases, the constitutive equations display a simpler form when they are written with respect to some privileged frame attached to each material point. Such frames can follow the evolution of the material point in some sense to be specified for each kind of material. In the case of a deformable medium, we need a family of frames Ex , where Ex is the local privileged frame at x and Qx the associated rotation. However, ∼ we restrict ourselves to objective local frames for which: Qx = Qx QT ∼
∼
(6.41)
∼
when changing from one current frame to another. The derivative of T∼ with respect to Ex is then defined as:
·
DE x T∼ = QTx Qx T∼ QTx Qx ∼ ∼ ∼ ∼
(6.42)
Corotational frame There exists a single family of local objective frames Exc such that at any point and at any time, the instantaneous rotation rate of the medium with respect to that frame is zero. Indeed: ˙ QT + QQT ∀x ∈ =Q (6.43) ∼ ∼ ∼ ∼
∼
∼
= 0, we must have So that ∼
˙c = Q ˙ Tc Qc = −QTc Q ∼ ∼
∼
∼
(6.44)
∼
This relation, added to the initial condition (6.7), defines entirely Exc up to a translation. If x = P∼ (t)x, ˙ QT + QQT = Q ˙ QT + QP˙ T P QT + QP T P QT =Q ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼ ∼
∼
∼
∼
∼
Inelastic constitutive laws at finite deformation
287
which shows that the Exc are objective. With respect to that frame, matter in the vicinity of x deforms without instantaneous rotation. In a way, this is the frame with respect to which the continuum rotates the least. This frame is called corotational. The associated derivative is
·
− T DE c T∼ = QTc (Qc T∼ QTc ) Qc = T∼˙ + T∼ ∼ ∼ ∼ ∼
∼
∼
∼
(6.45)
We recognize the Jaumann derivative. More generally, the derivative with respect to the local objective frame El , whose , is: rotation rate is ∼ El − T (6.46) DEl T∼ = T∼˙ + T∼ ∼ El ∼ El ∼ One can verify that DEl T∼ is an objective derivative as soon as the local frame El is objective. The convective derivatives can be interpreted as derivatives with respect to a deformable frame (change of frame given by F∼ , which is not a Euclidean transformation).
Eigenrotation frame of the polar decomposition of F∼ (cf. next section) can also be The rotation part R ∼ used to build an objective local frame ER : T Q ER = R ∼ ∼
(6.47)
The associated derivative depends on the choice of the reference configuration.
6.1.7. Strain tensors Strain means change of shape with respect to a reference shape. The notion of strain is linked to the more precise notion of metric. When one has to characterize the deformation of the continuum, an infinity of choices appear. The final choice is governed by the strain domain considered and. . . personal preferences.
Polar decomposition is positive-definite. One can build the The deformation gradient being invertible, C ∼ 2 . Taking det F > 0, such that C =U unique symmetric positive-definite tensor U ∼ ∼ ∼ ∼
288
Non-Linear Mechanics of Materials
−1 is a rotation. The unique polar decomposition of the it turns out that R = F∼ U ∼ ∼ deformation gradient then is U = V∼ R (6.48) F∼ = R ∼ ∼ ∼
is called the stretch tensor. U (Lagrangian) is invariant whereas V∼ (Eulerian) is U ∼ ∼ objective.
Strain measurements To characterize the shape change from one state to another, we actually have a wealth of strain indicators that deserve the name of strain measures. A “strain tensor” should fulfill the following conditions: • it must be either invariant or objective; • it must cancel out when F∼ = 1∼; • a series expansion at F∼ = 1∼ at first order must give the tensor ε∼ of the small strain theory (i.e., a constraint over the derivative in 1∼). The following strain measures are based on the dilatation tensor: E = ∼ A = ∼
1 (C − 1) 2 ∼ ∼
Green–Lagrange tensor
1 ) (1 − B −1 ) = τF−1 T F (E ∼ 2 ∼ ∼ =U − 1∼ E ∼1 ∼
Euler–Almansi tensor
A = 1∼ − V∼ −1 ∼ −1
(6.49) (6.50) (6.51)
With index notation, Eij = + uj,i + uk,i uk,j ) where u is the displacement , purely Lagrangian, is invariant with respect to a change of observer, whereas field. E ∼ , purely Eulerian, is objective. More generally, one can define the following two A ∼ families of strain measures: 1 2 (ui,j
E = ∼n
1 n (U − 1∼) n ∼
A = ∼n
1 n (V − 1∼) n ∼
(6.52)
=E and A =A . for any non–zero integer n. In particular, E ∼ ∼2 ∼ ∼ −2 To extend these definitions to n = 0, one defines the logarithmic strain: ⎡ ⎤ log(1 + λ1 ) 0 0 ⎦ P −1 0 0 log(1 + λ2 ) E = log U = P∼ ⎣ ∼0 ∼ ∼ 0 0 log(1 + λ3 ) = logV∼ A ∼0
(6.53) (6.54)
Inelastic constitutive laws at finite deformation
289
where the λi are the eigenvalues of E . These strain measures have actually the same ∼1 principal directions but distinct principal strains. In the one dimension case, if the length of the sample is changed from l0 to l:
2
2 l l B= C= l0 l0 l − l0 l0
2 l 1 −1 E2 = 2 l0 E1 =
E0 = log
l l0
l − l0 l
2 1 l0 = 1− 2 l
A−1 = A−2
A0 = log
l l0
One verifies that the measures are zero at 1 and have a common tangent of slope 1. The measures A−|n| have a better resolution in compression. Composition of deformations If the continuum undergoes a transformation F∼ 1 from a reference configuration, then a transformation F∼ 2 from configuration 1 , the overall transformation is F∼ = F∼ 2 F∼ 1
(6.55)
The composition of the gradients of the transformation is then multiplicative. However, the consequences on the strain tensors are more complicated. For instance, C = F∼ T2 C F = C C ∼ ∼1 ∼2 ∼2 ∼1
(6.56)
Likewise, E = E +E . In the uniaxial case only, the composition of logarithmic ∼ ∼1 ∼2 measures is additive.
290
Non-Linear Mechanics of Materials
6.2. Sthenics and statics of the continuum Sthenics concerns the representation of stresses and statics the equilibrium conditions of the continuum.
6.2.1. The method of virtual power methods; principle of objectivity of stress The first step consists in choosing a space of virtual motions V: the velocity fields v(x, t), continuous and continuously differentiable over . Within the framework of is assumed to contribute to exchanges the first gradient theory, the velocity gradient L ∼ of energy. Modeling the stresses consists in building dual quantities of the velocity and gradient of velocity fields, dual in the sense that their definition involves a linear form acting on the latter fields. This linear form represents the power of the forces in a given virtual motion. In mechanics, one usually divides forces into internal and external forces. One postulates then the principle of virtual power: “In a Galilean frame, at any time and for any system, the virtual power of all—internal as well as external— applied forces is zero whatever the virtual motion”. It is also called D’Alembert’s principle as noted in [SZA87].
Virtual power of internal forces The starting point is the expression of the power of internal forces, and this for two reasons. Firstly, because it is the actual new concept in continuum mechanics. Secondly, the application of the principle of objectivity of stress that is going to follow, gives insight on the form that should be proposed to model external forces. We postulate the principle of objectivity of stress: “The virtual power of internal forces in a sub-domain D ⊂ is objective; it is the same at a given time whatever the frame of observation”. In the principle of objectivity, the issue is that all the possible Euclidean frames are involved and not only Galilean transformations. Let D be any subsystem of . We assume that power can be written as a volume integral thanks to a volume density of internal forces: pd (6.57) P (i) = − D
In that section, we work with the Eulerian description of fields because balance of forces is fundamentally written with respect to the current configuration. The most : simple hypothesis consists in assuming that the power is linear with respect to v and L ∼ p = k.v + σ∼ : L ∼
(6.58)
Inelastic constitutive laws at finite deformation
291
To ensure that p is invariant upon change of frame, we must have: k=0
σ∼ = σ∼ T
(6.59)
To ensure the fulfillment of the principle of stress objectivity, it is sufficient to consider a linear form involving only objective or only invariant tensors. In the first case, one must use D instead of L . The velocity field is clearly not objective and cannot appear ∼ ∼ in the power of internal forces. σ∼ : D d (6.60) P (i) = − ∼ D
Applying the divergence theorem leads to: P (i) = (divc σ∼ ).vd − D
∂D
(σ∼ .n).vdS
(6.61)
Power of external forces One introduces volume forces, dual quantities to v, D and : ∼ ∼ P (e) = (f .v + :D +C : )d ∼ ∼ ∼ ∼ D
(6.62)
T = −C ) is a density of volume torques (with C f is the density of volume forces, C ∼ ∼ ∼ a field of symmetric double forces as they may exist in a first gradient continand ∼ uum according to [GER73a, MAU80]. Finally, P (e) = (f − divc C − div ).vd + (C .n + .n).vdS. (6.63) c∼ ∼ ∼ ∼
D
∂D
Virtual power of contact forces Finally, contact/surface forces are considered. We can introduce a priori surface densities of forces, couples and double forces. But, according to (6.61), surface torques and double forces are not counterbalanced, and must therefore be zero. (c) T .vd (6.64) P = D
Applying the principle of virtual power The field equations are deduced from the application of the variational principle of virtual power: (6.65) ∀v ∈ V P (i) + P (c) + P (ext) = 0
292
Non-Linear Mechanics of Materials
Hence:
D
(f + divc τ∼ ).vd +
∂D
(T − τ∼ .n).vdS = 0
(6.66)
where: τ∼ = σ∼ − C − ∼ ∼
(6.67)
τ∼ is the stress tensor. In general C = = 0 so that τ∼ = σ∼ is symmetric. By choosing ∼ ∼ fields v canceling on ∂V, we deduce the balance of momentum equation: divc τ∼ + f = 0 ∀x ∈ D.
(6.68)
By choosing fields v canceling out on a compact set having a common part with ∂D, arbitrarily chosen, we obtain the boundary conditions T = σ∼ .n
∀x ∈ ∂D
(6.69)
Then, by taking D = itself, we obtain the necessary and sufficient conditions of the equilibrium of the continuum and the boundary conditions. In what follows = = 0. C ∼ ∼
6.2.2. Lagrangian formulation of equilibrium The natural formulation of equilibrium is attached to the Eulerian representation, because it is in the current configuration that the notion of stress takes all its meaning. The equivalent Lagrangian formulation is derived in this section.
Stress tensors One considers the mass density of internal forces: pm =
σ∼ :D ρ ∼
(6.70)
where σ∼ is the Cauchy stress tensor defined on the current configuration. In accordance with transport rules established in Sect. 6.1.4, it appears that: pm =
S 1 ˙ = ∼ : F˙ τF −1 F −T (σ∼ ) : τF T F (D )= ∼ :E ∼ ∼ ρ ρ0 ρ0 ∼
where we noticed that ˙ = F T DF E ∼ ∼ ∼ ∼ = J σ∼ F∼ −T ∼
(6.71)
∂v and F∼˙ = ∂X
(6.72)
S∼ = J F∼ −1 σ∼ F∼ −T
(6.73)
Inelastic constitutive laws at finite deformation
293
(∈ L(E0 , E)) is the first Piola–Kirchhoff tensor or Boussinesq tensor; S∼ (La∼ grangian) is the second Piola–Kirchhoff tensor or Piola–Lagrange tensor. The latter is a symmetric second-rank tensor. Thus, it is the stress tensor dual to the Green– is not symmetric Lagrange strain tensor in the power of internal forces. In contrast, ∼ and F∼ is not a strain tensor. More generally, to each strain tensor, one can associate a dual stress tensor from the corresponding power.
Transport of a surface element A surface element dS = dSn is a linear form that computes the volume contained in a cylinder obtained by translating the surface element with a vector u. The mass contained in this cylinder is: dm = ρdS.u = ρ0 dS 0 .u0
(6.74)
where u = F∼ u0 and dS 0 = NdS0 . Hence: dS = J F∼ −T dS 0 .
(6.75)
The pulling back of the traction vector dT = σ∼ .ndS allows a better understanding of the previous stress tensors: .dS 0 dT = σ∼ .dS = ∼
(6.76)
dT 0 = F∼ −1 .dT = S∼ .dS 0
(6.77)
Lagrangian formulation of the principle of virtual power By taking into account the conservation of mass, we rewrite the equilibrium in the Lagrangian description: ˙ d0 + − : F f .V d + T 0 .V dS0 = 0 (6.78) 0 0 ∼ ∼ 0
0
0
dS T = J F∼ −T N T . The method of virtual power where f 0 = J f and T 0 = dS 0 provides then the Lagrangian form of the balance of momentum equation:
+ f0 = 0 div ∼ .N = T 0 ∼
on 0
on ∂0
(6.79) (6.80)
294
Non-Linear Mechanics of Materials
6.2.3. Thermodynamics We present here some aspects of continuum thermodynamics that belong to the field of irreversible thermodynamics [GER83].
Conservation of energy The first principle of thermodynamics is written (in a Galilean frame): E˙ + K˙ = P (e) + P (c) + Q
(6.81)
where E is the total internal energy of the body, K, the kinetic energy and Q, the remaining supplied energy. A corollary of the principle of virtual power is the kinetic energy theorem: P (i) + P (c) + P (e) = K˙ (6.82) One assumes that there exists a mass density of internal energy: ρed = ρ0 ed0 E= D
D0
(6.83)
The heat rate supplied to the system results from a volume supply r per mass unit and a surface supply through the surface ∂D, characterized by the heat flux q: ρrd − q.dS (6.84) Q= D
∂D
Then, the Eulerian form of energy balance is written d ρed = (σ∼ : D + ρr)d − q.dS ∼ dt D D ∂D
(6.85)
The local form of this balance equation can be derived in the same way as for the balance of momentum: + ρr − divc q (6.86) ρ e˙ = σ∼ : D ∼ The Lagrangian version is obtained by introducing the heat-flux Lagrangian vector q 0 = J F∼ −1 .q: ˙ + ρ0 r − div q 0 ρ0 e˙ = S∼ : E (6.87) ∼ Thermodynamic state of the continuum The material is described at each point M by state variables and, in the case of dissipative processes, by a set of additional internal variables governed by evolution laws.
Inelastic constitutive laws at finite deformation
295
Clausius–Duhem inequality The second principle of thermodynamics of irreversible processes states that:
S˙ ≥ S (ext)
(6.88)
where S = D0 ρ0 sd0 = D ρsd is the total entropy of the system D. If T is the absolute temperature, the quantity: q.n ρr d − dS (6.89) S (ext) = D T ∂D T is the form postulated for the external supply in entropy. The local Eulerian form of this inequality is then: q r ρ s˙ ≥ − div (6.90) T T Combining (6.86) and (6.90) provides the inequality: q −ρ(e˙ − T s˙ ) − .grad T + σ∼ : D ≥0 (6.91) ∼ T Introducing the Helmholtz free energy function = e − T s, this inequality becomes: q ˙ + s T˙ ) + σ : D − .grad T ≥ 0 −ρ( (6.92) ∼ ∼ T called Clausius–Duhem inequality. Its Lagrangian formulation is: q ˙ − 0 .grad T ≥ 0 (6.93) −ρ0 (e˙ − T s˙ ) + S∼ : E ∼ T The quantity: = i + th (6.94) q th is called the rate of dissipation, where = − T . grad T . The quantity ˙ + s T˙ ) + σ : D i = −ρ( ∼ ∼
(6.95)
is called the intrinsic dissipation rate.
6.3. Constitutive laws 6.3.1. Formulation of constitutive laws Functional approach vs. internal variable approach Whereas the balance equations described in the previous section have a character of universality, there exists a wealth of constitutive laws that reflect the variety of materials. However, some fundamental rules govern the formulation of these constitutive
296
Non-Linear Mechanics of Materials
laws. These rules were formalized in the 1960s and are criticized, for example in [RIV69]. Generally, a constitutive law must allow us to calculate the current stress as a function of the whole history of deformation of the continuum. That is why the functional approach of the behavior was first preferred by the authors in [TRU65]. The linear viscoelasticity theory at small strain represents a great success of the functional approach, introducing in the constitutive law an integral over all the past response of the material. But because of the complexity linked to the required functional analysis, one often prefers the internal variables approach, meant to characterize the state and history of the continuum by their value at any time and any point. The stress state and its evolution must be deducible at any time from the state of the material (strain and internal variables).
Principle of material frame indifference We require the constitutive law of the material to be frame independent. In other words, the form of the constitutive law should not change upon change of observer. It should be invariant under Euclidean transformation. This principle is called the form invariance principle or material frame indifference. This principle applies to any Euclidean frame and not only to Galilean frames. It is quite an obvious requirement in the sense that we do not expect stresses to develop in a body simply by running around it! However, the principle has raised much criticism mainly due to the confusion of the material frame indifference principle with the principle of invariance of the constitutive law with respect to rigid body motion which is sometimes claimed instead of the form invariance principle. The kinetic theory of gases provides examples of constitutive laws that do not fulfill the condition of invariance under superimposed rigid body motions, but that satisfy the principle of material frame indifference [MUR83b]. The principle of material frame indifference still generates confusing debates [BER01, MUR03] but its rational use leaves room for an extremely large variety of behaviors observed in nature and industry. An efficient way to automatically fulfill this principle consists in writing only relations between objective quantities. From this point of view, the Lagrangian approach has the merit of providing invariant quantities. However, it is unable to account for the simple notion of principal direction of stress or strain, as they vary with the choice of the convective transport. The Lagrangian formulations may display then more complex forms and they are more difficult to invent. It is nevertheless widely used in hyperelasticity. The Eulerian tensors σ∼ (or rather, the Kirchhoff stress σ∼ /ρ) and D ∼ are, on the other hand, unquestioned tools to develop new constitutive laws. The Eulerian approach is not ideal, though, as it is necessary to work with the objective derivative of stress or internal variables, a derivative whose choice is contingent. We present another major drawback of the purely Eulerian approach at the end of the next
Inelastic constitutive laws at finite deformation
297
section. We are left with the material approach, presented for example in [ROU97], but it is still not widely used.
Respect of material symmetries The continuum is said to admit a material symmetry QS if ∼
σ∼ = F (F∼ )
=⇒
F (F∼ QS ) = σ∼
(6.96)
∼
The group of symmetry G of the continuum, a subgroup of the orthogonal group, is the set of orthogonal transformations QS that lead to the same response from the material. ∼ A material is said to be isotropic if its group of symmetry is equal to the orthogonal group itself. Writing constitutive laws for anisotropic materials is not an easy task. In particular, the purely Eulerian approach does not allow us to write anisotropic laws! Let us imagine a constitutive law of the form: ) T∼ = f (D ∼
(6.97)
where T∼ = σ∼ or T∼ = σ∼ is an objective derivative of stress. The principle of material frame indifference implies: ), T∼ = f (D ∼
i.e.,
QT∼ QT = f (QD QT ) ∼ ∼
∼
∼
∼
Qf (D )QT =f (QT QT ), ∀D .
f must then be an isotropic function of its hence ∼ ∼ ∼ ∼ ∼ ∼ ∼ arguments in the mathematical sense of the word. Thus, stresses could not be an anisotropic function of the principal strains. Then, writing anisotropic constitutive laws involves using invariant quantities (the Lagrangian approach or in local objective frames, as we shall see).
6.3.2. Elasticity Elasticity laws By elasticity law we mean here a relationship of the form stress = f (strain). Previous remarks show that any relationship of that type is valid if both stress and strain tensors are invariant upon change of observer. Particularly, the following laws are valid: ) S∼ = f (E ∼n
(6.98)
σ∼ = λ Tr(log V∼ )1∼ + 2μlog V∼
(6.99)
One shall choose among the multitude of possibilities the one that best fits the experimental results.
298
Non-Linear Mechanics of Materials
Hypoelasticity This concerns constitutive laws of the form:
σ∼ = f (D ) ∼
(6.100)
where ∇ is any objective derivative. The drawback of that type of law is that, integrated along a closed strain path, final stress may be non-zero, while starting from a natural state (zero initial stress)! That is not the idea we have of reversibility in elasticity.
Approach based on local objective frames We consider a law of the form:
∗ σ∼˙ ∗ = f (D ) ∼
(6.101)
∗ = Q∗ D Q∗T are the Cauchy stress and strain rate seen where σ∼ ∗ = Q∗ σ∼ Q∗T and D ∼ ∼ ∼ ∼ ∼ ∼ ∗ being invariant, the choice of f is free, from a local objective frame ∗. σ∼ ∗ and D ∼ as long as it respects the material symmetries. Let us write, for example, using the corotational frame: c c )1∼ + 2μD (6.102) σ∼˙ c = λ(Tr D ∼ ∼
Then pulling back the constitutive law in the current observation frame, we find: )1 + 2μD QcT σ∼˙ c Qc = σ∼ J = λ(Tr D ∼ ∼ ∼ ∼
∼
(6.103)
where we recognize the Jaumann derivative. If f is isotropic, a law written in a local objective frame is equivalent to a hypoelastic law.
Hyperelasticity Hyperelasticity is the most satisfying formulation of elasticity as it is derived from continuum thermodynamics arguments. Free energy (T , E ) can be considered, for ∼ example, as a function of temperature and Green–Lagrange strain. Then, intrinsic dissipation is written
∂ ˙ ∂ ˙ ˙ + S : E˙ :E + i = −ρ0 T + s T ∼ ∼ ∼ ∂E ∂T ∼
∂ ˙ − ρ s + ∂ T˙ ≥ 0 :E = S∼ − ρ (6.104) ∼ ∂E ∂T ∼
Inelastic constitutive laws at finite deformation
299
Variables E and T lead to independent evolutions and are called, because of this, ∼ normal, and, following an argument due to Coleman [COL67], this implies: ∂ ∂E ∼
(6.105)
∂ ∂T
(6.106)
S∼ = ρ0 s=−
By using other strain measures and working with Eulerian representation, the first hyperelastic relation is replaced by: σ∼ = ρ
∂ ∂H ∼
(6.107)
= log V∼ is the logarithmic measure, also called Hencky’s measure. One can where H ∼ also suggest: ∂ B (6.108) σ∼ = 2ρ ∼ ∂B ∼ But, again, this purely Eulerian treatment applies only to isotropic materials. Examples of hyperelastic laws will be given in Sect. 6.4. Fundamental applications to elastomers and rubber materials can be found in [OGD97, HEU95].
6.3.3. Viscoelasticity On that matter, see the book by Truesdell and Noll [TRU65] and [SID96]. We will note also the current trend to substitute the internal variables approach to the classical hereditary approach of viscoelasticity by using elastoviscoplastic models, presented in what follows, and by letting the introduced threshold go to 0. The basic tool is then the introduction of kinematic hardening type variables.
6.3.4. Plastic and viscoplastic fluids Here, we consider materials for which the Eulerian approach is natural: ) σ∼ = f (D ∼
(6.109)
where f is isotropic. For this kind of materials, all the energy i = σ∼ : D is dissi∼ pated.
300
Non-Linear Mechanics of Materials
For a viscoplastic incompressible rigid material (det F∼ = 1∼, Tr D = 0), we intro∼ ): duce the dual potentials (σ∼ ) and ∗ (D ∼ σ∼ = −p1∼ +
∂∗ ∂D ∼
(6.110)
or D = ∼
∂ ∂σ∼
(6.111)
The Norton–Hoff law is obtained for: ∗ = =
n+1 k (D : D ) 2 n+1 ∼ ∼
σ∼ = −p1∼ + k(D :D ) ∼ ∼
n+1 n (σ∼ dev : σ∼ dev ) 2n k(n + 1)
D = ∼
n−1 2
D ∼
1−n 1 dev : σ∼ dev ) 2n σ∼ dev . (σ∼ k
(6.112)
(6.113)
It is used for the description of the pure creep response of materials. The Newtonian fluid is obtained for n = 1.
6.3.5. Elastoviscoplasticity Decomposition of the deformation gradient The problem of decomposing the deformation gradient F∼ in an elastic F∼ e and an inelastic F∼ p part is a fundamental step when modeling elastoplastic finite deformations. As often in large strain mechanics, the response is not unique, contrary to the small strain case where the decomposition is additive. For nonlinear behavior at finite deformation, the interested reader is referred to [MAU92, BER05].
Possible decompositions Then, let us look for a function F∼ = f (F∼ e , F∼ p ). It must satisfy in particular: • f isotropic: f (QF∼ e QT , QF∼ p QT ) = Qf (F∼ e , F∼ p )QT ; ∼
• f (F∼ e , 1∼) = F∼ e ; • f (1∼, F∼ p ) = F∼ p .
∼
∼
∼
∼
∼
Inelastic constitutive laws at finite deformation
301
This fits the idea of elastic and inelastic deformations. Let us consider the Taylor expansion of f about 1∼: f (F∼ e , F∼ p ) = f (1∼, 1∼) + (F∼ e − 1∼) :
∂f ∂f + (F∼ p − 1∼) : ∂F∼ e ∂F∼ p
1 ∂ 2f 1 ∂ 2f + (F∼ e − 1∼)(F∼ e − 1∼) e2 + (F∼ p − 1∼)(F∼ p − 1∼) p2 2 2 ∂F∼ ∂F∼ + (F∼ e − 1∼)(F∼ p − 1∼)
∂ 2f + ··· ∂F∼ e ∂F∼ p
(6.114)
The condition f (F∼ e , 1∼) = F∼ e implies that: ∂if =0 ∂F∼ ei
∀i > 1
(6.115)
∀i > 1
(6.116)
The condition f (1∼, F∼ p ) = F∼ p implies that: ∂if =0 ∂F∼ pi
We deduce the general form that such a decomposition can take: (F p − 1∼) F∼ = F∼ e + F∼ p − 1∼ + (F∼ e − 1∼)H ∼ ∼ ∼ ∼
(6.117)
where H must be isotropic (15 invariants in the general case of the 6th-rank tensor, ∼ ∼ ∼
[SIE92, TRO93]). The terms that can then intervene are Tr(F∼ e − 1∼) Tr(F∼ p − 1∼)1∼ (F∼ e − 1∼) : (F∼ p − 1∼)1∼ −F∼ e − F∼ p + 1∼ + F∼ p F∼ e −F∼ e − F∼ p + 1∼ + F∼ e F∼ p (Tr(F∼ e − 1∼))(F∼ p − 1∼) (Tr(F∼ p − 1∼))(F∼ e − 1∼) If the decomposition involves several of these terms, the coefficients that balance them are material parameters.
Hyperelasticity condition However, we are going to see that it is possible to narrow even more the possibilities by imposing hyperelasticity conditions. The result is due to [SIE92]. First, we note that it is inappropriate to try to decompose F∼ itself as it is not an objective quantity so that its decomposition cannot be unique. We then prefer the form: F∼ F∼ = R ∼
where
Ue ∼
(symmetric) and
Fp ∼
e with F∼ = f (U , F∼ p ) ∼
(6.118)
are assumed to be invariant tensors. We impose again:
302
Non-Linear Mechanics of Materials e , 1) = U e ; • f (U ∼ ∼ ∼
• f (1∼, F∼ p ) = F∼ p . The first condition implies that R is the rotation part of the polar decomposition of ∼ F∼ e . We define the rotated stress tensors σ
∼
T =R ∼
ρ
σ∼ ρ
R = F∼ ∼
S∼
ρ0
F∼ T
(6.119)
The Clausius–Duhem inequality (excluding, for sake of simplicity, the terms due to the hardening and considering the isothermal case) is written ˙ − ρ0 ˙ ≥0 S∼ : E ∼
(6.120)
2 , in the pure elasFor objectivity reasons, free energy can only be a function of U ∼ e2 ). When F˙ p = 0, the previous tic case. In the elastoplastic case, we keep (U ∼ ∼ inequality is an equality from which we deduce:
S∼ :
∂E d ∼ = ρ0 e2 e2 ∂U dU ∼ ∼
(6.121)
e ), the second principle (see next section) leads =U In the pure hyperelastic case (U ∼ ∼ to: d σ∼ = 2ρU . .U (6.122) ∼ 2 ∼ dU ∼
Then, if we require, in the elastoplastic case, that there exists a hyperelasticity relae , the decomposition has a tionship having the previous form by substituting U by U ∼ ∼ single form and we have actually the following theorem:
e σ∼ = 2ρU . ∼
d .U e e2 ∼ dU ∼
⇐⇒
e p F∼ = U F ∼ ∼
(6.123)
The demonstration from the right to the left does not pose any problem using techniques explained in the next section. The reverse way is less classical: When F∼˙ p = 0, we have S∼ :
∂E ∂ ∼ = ρ0 e2 e2 ∂U ∂U ∼ ∼
(6.124)
which can be written as: ∂ S∼ = ρ0 : e2 ∂U ∼
∂E ∼ e2 ∂U ∼
−1 (6.125)
Inelastic constitutive laws at finite deformation
303
Hence: ρ0 −T e ∂ F = F∼ −1 2ρ0 U U e F∼ −T ∼ e2 ∼ ρ ∼ ∂U ∼
∂ e −T T e −T = 2ρ0 U F∼ U F∼ ∼ e2 ∼ ∂U ∼ ∂ e −T e −T = ρ0 : 2(U F∼ .1∼T .U F∼ ) ∼ ∼ e2 ∂U ∼ ∼
S∼ = F∼ −1
(6.126)
T . We get where 1∼T is the tensor such that 1∼T A =A ∼ ∼ ∼
∼
∂E 1 ∼ = G .1T .G e2 2 ∼ ∼∼ ∼ ∂U ∼
(6.127)
e−1 . = F∼ T U where G ∼ ∼ T − 1) so that: = 12 (G U e2 G Furthermore E ∼ ∼ ∼ ∼ ∼
= dE ∼
1 e2 T T} dU G ) + { dG U e2 G (G ∼ ∼ ∼ ∼ ∼ ∼ 2
(6.128)
hence, with (6.127) and (6.128), ∂G ∼ e2 ∂U ∼
=0
(6.129)
e−1 = G(F p ), i.e., a function of F p only, so = F∼ T U We deduce that G ∼ ∼ ∼ ∼ ∼
e T F∼ = U G (F∼ P ) ∼ ∼
(6.130)
e = 1, F p ) = F p entirely determines G and finally: The condition F∼ (U ∼ ∼ ∼ ∼ ∼
e p F∼ = U F ∼ ∼
(6.131)
Notion of intermediate isoclinic configuration The multiplicative decomposition F∼ = F∼ e F∼ p
(6.132)
thus brought to light, appeared after the end of the 1950s in the continuum theory of dislocations, i.e., in the study of the plasticity of the single crystal. The initial idea consisted in introducing in a point a local relaxation of stresses leading to a so-called relaxed intermediate configuration. This elastic unloading is, however, defined only
304
Non-Linear Mechanics of Materials
Figure 6.1. Kinematics of an elastoplastic continuum with an isoclinic released configuration
up to a rotation so that the proposed decomposition was satisfying only for isotropic materials, as for these materials undetermined rotation has no effect on the behavior (moreover, this implies that F∼ p must be symmetric). Mandel [MAN72, MAN66] gives then a clear definition of the same decomposition in the anisotropic case, whose archetype is the single crystal and for which F∼ p is generally non-symmetric. He proposed to introduce a triad of directors associated with the anisotropic material and to perform the unloading while bringing it back to an initially fixed orientation. This is called the isoclinic released intermediate configuration. This boils down, in fact, to working in a local objective frame, which is the one attached to the crystal. Note that the intermediate configuration does not generally correspond to a global position of the body. It is a purely local configuration. The reason is that, generally, F∼ e is not a compatible field and cannot be integrated to an “elastic” displacement field. This local released configuration may even not exist since viscoplastic flow may start again before total unloading of the material element. This is however not a problem since this unloading must be seen as a thought experiment. It simply represents an idealized construction for writing constitutive equations. This linear extrapolation can always be constructed, which makes this released intermediate configuration unambiguous. Note also that there is no imperious need for using a stress-free intermediate configuration. This is simply convenient for the further construction of the constitutive equations. Making use of the polar decomposition of elastic deformation: e F∼ e = V∼ e R ∼
(6.133)
e orthogonal. R e sets the current orientation of the triad where V∼ e is symmetric and R ∼ ∼ of directors with respect to the axes of the current frame. In the case of a single crystal e corresponds to lattice rotation. undergoing infinitesimal elastic strain, R ∼
Decomposition (6.132) is not, as we saw, the only possible decomposition, but it
Inelastic constitutive laws at finite deformation
305
is a well-suited one for materials with purely elastic domains and when one wants to favor the notion of hyperelasticity. An additive decomposition of Green–Lagrange tensor is sometimes used. We can see that it is a particular case contained in the general expression (6.117).
Generalized standard materials When adopting the multiplicative decomposition (6.132) and the isoclinic released e , α , T , where the meaintermediate configuration, free energy is only a function of E ∼ ∼ sure of the Lagrangian elastic deformation with respect to the released configuration has been chosen as: 1 e = (F∼ eT F∼ e − 1∼) (6.134) E ∼ 2 The advantage of the isoclinic configuration is that the tensors of physical properties (elasticity moduli, thermal expansion coefficients. . .) take their simplest form. Likewise, stresses attached to the isoclinic intermediate configuration are defined: S∼ e =
ρi e−1 F σ∼ F∼ e−T ρ ∼
(6.135)
where ρi is the mass density measured with respect to the released configuration. The . The scalar and/or tensorial internal variables are gathered in one symbolic notation α ∼ power of internal forces takes now the form: 1 1 ˙ e + (F eT F e S e ) : (F˙ p F p−1 )) σ∼ : F∼˙ F∼ −1 = (S∼ e : E ∼ ∼ ∼ ∼ ∼ ∼ ρ ρi
(6.136)
Clausius–Duhem inequality is then written
˙ e − ρ s + ∂ T˙ − ρ ∂ α˙ :E ∼ ∼ ρi ∂T ∂α ∼
1 Se : F∼˙ p F∼ p−1 − q. grad T ≥ 0 + ρ F∼ eT F∼ e ∼ ρi T
ρ
S∼ e
−
∂ e ∂E ∼
which must hold for all possible increments of state and internal variables. The lhs of ˙ e and T˙ . the inequality is a linear form with respect to the independent increments E ∼ Following Coleman and Noll’s argument, a linear function can be always positive only if it vanishes, so that: ∂ (6.137) s=− ∂T S∼ e ρi
=
∂ e ∂E ∼
(6.138)
306
Non-Linear Mechanics of Materials
The intrinsic dissipation rate is written : F∼˙ p F∼ p−1 + X :α ˙ D= ∼ ∼ ∼ where = F∼ eT F∼ e S∼ e /ρi and ∼ X = −ρi ∼
∂ ∂α ∼
(6.139)
(6.140)
is the thermodynamic force associated with internal variables. The material is then , X ) such that: called generalized standard if there exists a viscoplastic potential ( ∼ ∼ ∂ F∼˙ p F∼ p−1 = ∂ ∼ α ˙ = ∼
∂ ∂X ∼
(6.141)
(6.142)
arising in the dissipation and taken as argument of the viscoplasThe stress tensor ∼ tic potential is called the Mandel stress tensor [HAU00]. The elastoplastic (rateindependent) case will be considered, in most cases, as a limit case of viscoplasticity. , X) Otherwise, the elastoplastic generalized standard material admits a criterion F ( ∼ ∼ such that: ∂F (6.143) F∼˙ p F∼ p−1 = λ˙ ∂ ∼ ∂F α ˙ = λ˙ ∼ ∂X ∼
(6.144)
where the plastic multiplier λ˙ is determined by the consistency condition.
Elastoviscoplastic constitutive laws in local objective frames Using a local objective frame provides a systematic method to transpose constitutive laws developed in the small strain framework to the finite deformation case. Let us consider for instance a set of elastoviscoplastic constitutive laws of the form: ε∼˙ = ε∼˙ e + ε∼˙ p ε∼˙ p = f (σ∼ , α ) ∼ σ∼ = C : ε∼ e ∼ ∼
α ˙ = h(α , ε˙ p ) ∼ ∼ ∼ where α represents the internal variables. At finite deformation, D and σ∼ /ρ are the ∼ ∼ privileged variables in the current frame E. Let us consider then a local objective
Inelastic constitutive laws at finite deformation
307
frame E at each point whose evolution with respect to E is described by the rotation Q. ∼
The kinematic and static quantities are convected then in E : Q e∼˙ = QT D ∼ ∼
s = QT
∼
∼
∼
ρ0 TQ ρ ∼∼
(6.145) (6.146)
One can verify that e∼˙ and ∼s are invariant upon any other change of frame, since D ∼ and σ∼ are objective. As a consequence, any relationship between e∼ and ∼s fulfills automatically the principle of materials indifference. The only constraint governing the choice of this relationship is to observe the material symmetries. One extends then the additive decomposition of deformations to strain-rates and keeps all the evolution equations unchanged: e∼˙ = e∼˙ e + e∼˙ p e∼˙ p = f (s∼, α ) ∼ s=C : e∼e ∼
∼
∼
α ˙ = h(α , e˙ p ) ∼ ∼ ∼ Internal variables are then defined in the local frame and can be pulled back in E through the reverse transport. This systematic method has been proposed in [LAD80] and developed in [DOG86]. The choice of the local objective frame is the delicate step. We have seen several examples in Sect. 6.1.6. This method is used in most commercial finite element codes because of its systematic character. It has however the
308
Non-Linear Mechanics of Materials
drawback that the model does not admit a free energy potential, the reason being that the elasticity law is fundamentally hypoelastic when formulated in the current frame. It does not seem to be of importance as long as elastic strains remain significantly smaller than plastic strains. Constitutive laws and material coefficients determined through small strain characterizations cannot, actually, be simply extrapolated to finite deformations. One must perform experimental characterization in this domain and modify, if necessary, the parameters or evolution laws themselves accordingly. In particular, the evolution of the isotropic and kinematic hardening contributions at finite deformation is still a matter of debate, important especially for forming and spring-back.
6.4. Application: Simple glide Studying simple glide at finite deformation is a very rich example as it involves, so to speak, as much rotation as stretching. It allows us therefore to test the validity range of the proposed models. The experimental execution of a simple glide test, that allows one, in principle, to discriminate between models, is confronted with great difficulties as deformation becomes heterogeneous rather quickly, so that there remains a lot of mysteries to be elucidated concerning very large strains.
6.4.1. Rotation of material fibres We consider the following simple glide test:
Let dx be a material segment drawn on the sample, θ0 and θ being respectively the initial and current orientations of the segment with respect to a fixed axis. The relation between θ0 , θ and the amount of shear γ is: tan θ = tan θ0 + γ .
(6.147)
Inelastic constitutive laws at finite deformation
309
The rotation rate of the material segment is: θ˙ =
γ˙ 1 + (tan θ0 + γ )2
(6.148)
It is important to note that, even if the deformation gradient F∼ = 1∼ + γ e1 ⊗ e2
(6.149)
is homogeneous, the rotation rates of the material fibres depend on their orientation: dx |θ0 =π/2 does not rotate whereas dx |θ0 =0 undergoes the maximum rotation rate. One can calculate the two following rotation rate averages: γ˙ θ˙L = π
π/2 −π/2
1 γ˙ dθ0 = 2 1 + (tan θ0 + γ ) 2(1 +
γ2 4 )
(6.150)
if C0 is taken as reference configuration and γ˙ θ˙e = π
π/2 −π/2
γ˙ 1 dθ = 2 2 1 + tan θ
(6.151)
if the current configuration is taken as reference configuration (updated Lagrangian, θ0 → θ when γ → 0). These two averages are called respectively Lagrangian and Eulerian averages of the rotation rates. At constant shear rate, the Lagrangian rotation rate tends towards 0 when the amount of gliding tends towards infinity, whereas the Eulerian rate remains constant. One can show that θ˙L is also the rotation rate of the eigenrotation frame defined in Sect. 6.1.6. The definition of the eigenrotation frame is then linked to the choice of a reference configuration. In contrast, θ˙e is the rotation rate of the corotational frame Ec . θL reaches the limit π/2, whereas θe tends towards infinity (Fig. 6.2). During simple glide, the corotational frame rotates without end. Thus, it can be said that the eigenrotating frame follows at best a material fibre in a Lagrangian sense, i.e., with respect to a given reference configuration. In contrast, the corotational frame follows at best the material lines instantaneously as its rotation rate is exactly the average of all the instantaneous rotation rates of the fibres in the material element. We can see that neither of these two frame is entirely satisfactory.
6.4.2. Analysis in elasticity and elastoplasticity Very often, the objection is made to using Jaumann derivative or the corotational frame that, in some cases, they generate stress oscillations during simple glide which may
310
Non-Linear Mechanics of Materials
Figure 6.2. Rotation of the eigenrotation and corotational frames during a simple glide test
Figure 6.3. Comparison of various isotropic elasticity formulations at finite deformation for simple glide
be physically irrelevant. Indeed, infinite rotation of Ec makes possible a periodic mechanical response. We are going to show here that the model indeed predicts such oscillations in elasticity and in elastoviscoplasticity but that, in the latter case, the oscillations disappear for the classical values of the hardening parameters. In Fig. 6.3 the stress-strain curves of simple glide are compared for the following elastic and hypoelastic formulations:
Inelastic constitutive laws at finite deformation
311
Lagrangian formulation S∼ = 2μE + λ Tr E 1 ∼ ∼∼
(6.152)
S∼ and E are respectively the second Piola–Kirchhoff and the Green–Lagrange tensors. ∼ In this case, we give the component 12 of the Cauchy tensor: σ12 = μγ + (λ + 2μ)
γ3 2
Even so, the components σ11 , σ22 , σ33 are not zero but have a zero tangent at the origin, as expected at small strain.
Eulerian formulation σ∼ = 2μ log V∼ + λ Tr(log V∼ )1∼
(6.153)
V∼ intervenes in the polar decomposition F∼ = V∼ R . In this case: ∼ ⎛ σ12 =
2μ 1+
γ2 4
log ⎝
⎞
γ + 2
1+
γ2 4
⎠
Local objective frame formulation s = 2μe∼ + λ Tr e∼1∼
∼
(6.154)
where e∼ and ∼s are given by (6.145) and (6.146); we take successively Q=Q ∼
∼
Q=R ∼ ∼
c
(corotational)
(6.155)
(eigenrotation)
(6.156)
Solutions respectively are:
σ12 =
μ 1+
γ2 4
σ12 = μ sin(γ )
γ
γ2 γ2 4atan −γ 1− + 2γ log 1 + 2 4 4
Using the corotational frame results in shear stress oscillations of period γ = 2π. All these results are gathered in Fig. 6.3.
312
Non-Linear Mechanics of Materials
Plasticity In the rigid-plastic case, we introduce isotropic and kinematic hardening-variables, defined in the corotational frame.
Criterion ) = J (s∼ − X )−R f (s∼, R, X ∼ ∼
(6.157)
where p˙ =
2 e˙ : e˙ 3∼ ∼
J (s∼ − X )= ∼
3 dev (s −X ) : (s∼dev − X ) ∼ ∼ 2 ∼
(6.158)
Flow rule e∼˙ = p˙
∂f ∂s∼
(6.159)
Evolution laws ˙ = 2 C e˙ − D pX ˙∼ X ∼ 3 ∼
(6.160)
The calculation for simple glide gives: σ11
σ12 =
D C D 1 − exp − √ γ = 2 cos γ + √ sin γ = −σ22 D +3 3 3
D C exp − √ γ sin γ 3 3
D CD D R 1 − exp − √ γ +√ cos γ + √ sin γ +√ 3(D 2 + 3) 3 3 3
This proves that pure isotropic hardening induces no oscillation. In contrast, linear kinematic hardening (D = 0) gives rise to oscillations of period γ = 2π, which are damped or suppressed (aperiodic response) as soon as a nonlinear component is introduced (Fig. 6.4). Adding elasticity promotes the appearance of oscillations, even in the linear isotropic case, the period of which decreases when H /μ increases (H hardening modulus, μ shear modulus). For usual values of H /μ, the oscillations are elusive (Fig. 6.5).
Inelastic constitutive laws at finite deformation
313
Figure 6.4. Disappearance of the oscillations of shear stress σ12 during simple glide when a nonlinear term is introduced in the kinematic hardening evolution law
Figure 6.5. Appearance of oscillations in simple glide when the hardening modulus becomes close to the shear modulus (elastoplasticity in corotational frame, linear isotropic hardening, μ = 80000 MPa)
314
Non-Linear Mechanics of Materials
6.4.3. Single crystal plasticity The deformation of the single crystal can be accompanied by rotation of the crystal lattice with respect to the current frame, that differs from the rotation of material lines. This is due to the fact that lattice direction (directions of rows of atoms but not the atom rows themselves) are no material directions. We are going to see however that the situation in simple glide is even more complicated because some specific orientations are stable whereas others lead to an endless rotation of the lattice. We use the following model of the elastoviscoplastic single crystal. Following Mandel’s notation, we define the multiplicative decomposition of the deformation gradient: P (6.161) F∼ = E ∼ ∼ Plastic strain is due to slip, according to slip systems linked to the crystallographic structure of the single crystal: γ˙ s ms ⊗ ns (6.162) P∼˙ P∼ −1 = s
where ms and ns are respectively the slip direction, and the normal to the slip plane for the system s in the isoclinic released intermediate configuration. The slip rate is given by a viscoplastic law: s |τ − x s | − r s n sign (τ s − x s ) (6.163) γ˙ s = K r s = r0 + Q
N
hsr (1 − exp(−bv r ))
(6.164)
r=1
rs
v˙ s = |γ˙ s |
(6.165)
x s = cα s
(6.166)
α˙ s = γ˙ s − d v˙ s α s
(6.167)
xs
and are the isotropic and kinematic hardening variables attached to each system. The resolved shear stress τ s is the component in the glide direction of the traction vector acting on the slip plane: τs = : ms ⊗ ns ∼
(6.168)
and the criterion used is the Schmid criterion (cf. [FRA91]). The material parameters are k, n, r0 , q, b, hrs , c and d. This model can be studied based on numerical simulations. We consider the so-called octahedral slip systems of a face-centered cubic crystal (slip directions: 110, slip planes: {111}, cf. [FRA91]). We are going to
Inelastic constitutive laws at finite deformation
315
Figure 6.6. Simple glide of a single crystal of initial orientation [100], [001]
consider successively several orientations of the single crystal with respect to loading axes of simple glide. e1 = [100]/e2 = [001] The vectors e1 and e2 refer to the basis vectors of the figure in Sect. 6.4.1. The glide direction is e1 whereas e3 is normal to the plane of shear. The vectors e1 and e2 coincide initially respectively with the crystallographic directions [100] and [001]. Figure 6.6 shows the response of the single crystal when no hardening is introduced (material parameters: E = 200000 MPa, ν = 0.33, r0 = 50 MPa, k = 5 MPa s1/n , n = 10, hij = δij ). A cyclic behavior is obtained. One can show, actually, that the crystal exactly follows the corotational frame. During the endless rotation of the crystal, some slip systems are successively activated and de-activated. Eight systems are always simultaneously activated (Fig. 6.7). Introducing linear isotropic hardening breaks the periodicity and leads to an increase of the maximum stress at each cycle (Fig. 6.8). The case of linear kinematic hardening is even more complex (Fig. 6.9). e1 = [110]/e2 = [111] The imposed slip direction coincides with a crystallographic slip direction of the crystal < 110 > and slip occurs on a plane of {111} type. Only one slip system is then activated ([110](111)). The stress σ12 is constant and the crystal lattice does not rotate with respect to the loading axes. It is a stable orientation. The crystal deforms like a deck of cards.
316
Non-Linear Mechanics of Materials
Figure 6.7. [100], [001]
Activated slip systems during the simple glide simulation for a single crystal
Figure 6.8. Simple glide of a single crystal with initial orientation [100], [001] in the case of a linear isotropic hardening (H = 100 MPa)
e1 = [210]/e2 = [123]
For this orientation, four slip systems are activated (Fig. 6.10). A representation of the rotation of axis e2 in the standard triangle (stereographic projection of this direction in the crystal frame) shows that the crystal rotates with respect to the loading axis (Figs. 6.11 and 6.12). We can see that e2 tends then towards a stable orientation [111].
Inelastic constitutive laws at finite deformation
317
Figure 6.9. Simple glide of a single crystal with initial orientation [100], [001] in the case of a linear kinematic hardening (C = 100 MPa)
Figure 6.10. Simple slip of a single crystal with initial orientation [210], [123]: Activated slip systems
318
Non-Linear Mechanics of Materials
Figure 6.11. Simple slip of a single crystal with initial orientation [210], [123]: Stress evolution
Figure 6.12. Simple glide of a single crystal with initial orientation [210], [123]: Rotation of axis 2 in the standard triangle; the initial orientation is indicated by a triangle
Inelastic constitutive laws at finite deformation
319
6.5. Finite deformations of generalized continua Generalized continuum mechanics (GCM) introduces material characteristic lengths in continuum mechanics and enables us to account for size effects observed in mechanics of materials. There is a growing interest for GCM and two applications are proposed in the chapters dedicated to homogenization and localization. Its use is relevant in the nonlinear case and in particular at finite deformation. Nonlinear GCM is still a widely open field of research. We propose here to treat the case of the Cosserat continuum, one of the most simple cases of GCM [ERI99, ERI02, FOR03b, FOR06b, FOR06a].
6.5.1. Kinematics of Cosserat continuum A material point M ∈ B at time t0 is described by its position X and its internal (X, t0 ), for some initial placement, chosen as reference state described by a rotation R ∼ (X, t), in a given configuration. At time t, its position is x(X, t) and its internal state R ∼ frame E. If (d i )i=1,3 are three privileged orthogonal vectors (called “directors” of the continuum and attached to the microstructure) and (d 0i )i=1,3 their initial position in is defined through E, then the rotation R ∼ .d 0i di = R ∼
(6.169)
with R T = 1∼, R ∼ ∼
R (X, t0 ) = 1∼ and ∼
det R =1 ∼
(6.170)
is attached to the microstructure at each point M ∈ B A local objective frame by using the rigid triad of directors. Each tensorial variable y considered with respect to E will be denoted by y. E (M)
(X, t) can be replaced by the vector field (X, t) given by The rotation field R ∼ R = exp 1∼ × = exp − . (6.171) ∼ ∼
and = θn
(6.172)
where θ is the rotation angle with respect to rotation axis n. Accordingly: R = cos θ 1∼ + ∼
1 − cos θ sin θ ⊗+ 1× θ ∼ θ2
(6.173)
The three components of are three degrees of freedom of the continuum in addition to the three components of the displacement field: u(X, t) = x(X, t) − X
(6.174)
320
Non-Linear Mechanics of Materials
Thus, u and are considered as independent kinematic variables that are linked at the level of the constitutive laws, of the balance equations, or by an additional internal constraint. The deformation gradient classically links a current material segment dx with its initial position dX dx = F∼ .dX (6.175) so that F∼ − 1∼ = u ⊗ ∇ = ui,j ei ⊗ ej
(6.176)
(without additional indication, partial derivatives are taken with respect to Xj ). Simiof microrotation along a material segment dX: larly, we calculate the variation dR ∼ )R T = 1∼ × δ = .δ (dR ∼ ∼
(6.177)
1 δ = − ∼ : (dR RT ) ∼ ∼ 2 1 T = − ij k dRj m Rmk ei 2 1 = ikj Rkm Rj m,n dXn ei 2 =
.dX ∼
(6.178)
∼
with
1 : (R (R T ⊗ ∇)) (6.179) 2∼ ∼ ∼ is generally not invertible. With respect to the local Contrary to F∼ , the new tensor
∼ frame E , these quantities become =
∼
where
dx
=
R T .dx ∼
and
d
=
dx = F∼ .dX
(6.180)
δ =
.dX ∼
(6.181)
R T .d ∼
T F∼ = R F∼ ∼
and
T
=R
. ∼ ∼ ∼
It can be seen that F∼ and
are invariant under Euclidean transformation. A Euclid∼ ean transformation, i.e., a coordinate transformation, in the case of a Cosserat continuum takes the form: x = Q(t)x(X, t) + c(t) ∼
= Q(t)R (X, t) R ∼ ∼ ∼
(6.182)
Inelastic constitutive laws at finite deformation
321
where Q belongs to the orthogonal group and Q(t0 ) = 1∼. As a consequence, they ∼ ∼ are natural Cosserat strain measures for the development of constitutive equations. They are called respectively Cosserat deformation tensor and wryness (or bend-twist, or curvature) tensor. Expression (6.179) is also written
1
dX = − ∼ : (dR RT ) ∼ ∼ ∼ 2 1 = − ∼ : (R (R T dR )R T ) ∼ ∼ ∼ ∼ 2 1 T = − ∼ : (R dR ) ∼ ∼ 2
so that
1 T
= − ∼ : (R (R ⊗ ∇)) ∼ ∼ ∼ 2
(6.183)
We define next the velocity v = u˙ = u˙ i ei and the gyration tensor: ˙ RT ϑ∼ = R ∼ ∼
(6.184)
× 1 ϑ = − ∼ : ϑ∼ 2
(6.185)
which can be replaced by the vector
as it is skew-symmetric. The time derivative of the strain tensors can be linked to the gradient of the previous quantities:
T ˙ −1 ˙ RT ) R (F∼ F∼ − R F∼˙ F∼ −1 = R ∼ ∼ ∼ ∼ T =R (v ⊗ ∇ c − ϑ∼ )R ∼ ∼ ×
T =R (v ⊗ ∇ c − 1∼ × ϑ)R ∼ ∼
(6.186)
∂. where ∇ c = ∂x ei = F∼ −T .∇ (Eulerian representation, c stands for current). i The relative velocity gradient is v ⊗ ∇ c − ϑ∼ . It describes the local motion of the particle with respect to the microstructure. Taking the time derivative of (6.181), we
322
Non-Linear Mechanics of Materials
get: ˙ .dX ( d)· =
∼ 1 T = − ∼ : (R dR )· ∼ ∼ 2 1 ˙ T dR + R T d R˙ ) = − ∼ : (R ∼ ∼ ∼ ∼ 2 1 T ˙ R T )R ) = − ∼ : (R (R RT + d R R˙ T dR ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼ 2 1 T ˙ RT ) =− R . : (R R˙ dR R T + d R ∼ ∼ 2∼ ∼ ∼∼ ∼∼ 1 T ˙ RT ) =− R . : (R˙ R T R dR T + d R ∼ ∼ 2∼ ∼ ∼∼ ∼ ∼ 1 T =− R .( : dϑ∼ ) 2∼ ∼ ×
T .d ϑ =R ∼
× T .dX =R ϑ ⊗∇ ∼
Hence
˙ = RT
∼ ∼
× ϑ ⊗∇
(6.187)
(6.188)
or, using Eulerian representation
˙ F −1 = R T
∼ ∼ ∼
× c ϑ ⊗∇ R ∼
(6.189)
6.5.2. Sthenics In order to introduce forces and stresses and to deduce the equilibrium equations, we resort to the method of virtual power. The virtual motions are the velocity v and the × gyration ϑ (or microrotation rate vector). The next step is to choose the form of the virtual power of a system of forces. Within the framework of a first gradient theory, the virtual power of the internal forces is a linear form of the virtual motions and their gradients. The principle of material frame indifference requires that this linear form be invariant under any Euclidean transformation. That is why we will work with × the objective quantities v ⊗ ∇ c − ϑ∼ and ϑ ⊗∇ c . The dual quantities involved in the linear form of the virtual power are denoted by σ∼ and μ respectively and must then be ∼ objective. For objectivity reasons, the dual variable associated with v is zero. For any
Inelastic constitutive laws at finite deformation
323
subdomain D ⊂ B
× c c σ∼ : (v ⊗ ∇ − ϑ∼ ) + μ : (ϑ ⊗∇ ) dV P(i) = − ∼ D
× σij vi,j + μij ϑ i,j − σij ϑij dV = − D
× × σij vi + μij ϑ i σij,j vi + (μij,j − ikl σkl )ϑ i dV = − dV + D
D
,j
× × σij vi + μij ϑ i nj dS + σij,j vi + (μij,j − ikl σkl )ϑ i dV = − ∂D D
× × × v.σ∼ + ϑ .μ .ndS + v.div σ∼ + ϑ .(div μ + 2 σ ) dV = −
∼
∂D
∼
D
(in this subsection the partial derivatives are taken with respect to the current configuration). The virtual power of external forces is
× f .v + c. ϑ dV (6.190) P(e) = D
The virtual power of contact forces can then be defined:
× P(c) = t.v + m. ϑ dS ∂D
(6.191)
The dual quantities of velocity and microrotation rate in P(e) and P(i) have the dimensions of force and moment respectively. The principle of virtual power then states that: ×
∀D ⊂ B, ∀(v, ϑ)
P(i) + P(e) P(c) = 0
(6.192)
In particular ×
×
∀D ⊂ B, ∀(v, ϑ)/v = ϑ = 0 on ∂D
× × v.(div σ∼ + f ) + ϑ .(div μ + 2 σ +c) dV = 0 ∼
D
(6.193)
Assuming that the quantities are continuous on B, the local equilibrium equations follow from (6.193): div σ∼ + f = 0 (6.194) div μ + 2 σ × +c = 0 ∼
As a result, the principle of virtual power becomes:
× × (σ∼ .n − t).v + (μ.n − m). ϑ dS = 0 ∀D ⊂ B, ∀(v, ϑ) ∂D
∼
(6.195)
324
Non-Linear Mechanics of Materials
from which the boundary conditions are deduced σ∼ .n = t μ.n = m
(6.196)
∼
σ∼ is called the force stress tensor and μ the couple stress tensor. They are generally ∼ not symmetric.
6.5.3. Hyperelasticity Energy balance Let ε be the internal energy per unit mass, q the heat flux vector, ρ the current density. The energy balance equation reads then ×
ρ ε˙ = σ∼ : (v ⊗ ∇ c − ϑ∼ ) + μ : (ϑ ⊗∇ c ) − divq ∼
(6.197)
(any other heat supply is excluded for brevity). According to the thermodynamics of irreversible processes, the entropy principle is written ρ η˙ + div
q T
≥0
(6.198)
where T denotes the temperature and η the entropy per unit mass. Introducing the free energy ψ = ε − ηT and combining the energy and entropy equations, the Clausius– Duhem inequality is derived ×
−ρ(ψ˙ + ηT ) + σ∼ : (v ⊗ ∇ c − ϑ∼ ) + μ : (ϑ ⊗∇ c ) − ∼
1 q.T ∇ c ≥ 0 T
(6.199)
or, equivalently, ˙ F −1 ) − 1 q.T ∇ c ≥ 0 −ρ(ψ˙ + ηT ) + σ∼ : ( F∼˙ F∼ −1 ) + μ : (
∼ ∼ ∼ T where
σ
∼
μ ∼
T σR =R ∼ ∼ ∼ T μR =R ∼ ∼
(6.200)
(6.201)
∼
are rotated stress tensors with respect to the space frame E attached to the microstructure.
Inelastic constitutive laws at finite deformation
325
A material is said to be hyperelastic if its free energy and entropy are functions of and
only. Clausius–Duhem inequality (6.200) becomes: ∼ ∼
∂ψ ∂ψ ˙ − ρ − σ∼ F∼ −T : F∼˙ − ρ − μ F∼ −T :
∼ ∼ ∂ F∼ ∂
∼
∂ψ ˙ 1 − ρη + ρ T − q.T ∇ c ≥ 0 ∂T T
F
˙ and T˙ , the last inequality implies Since this expression is linear in F∼ −T ,
∼ η=−
∂ψ ∂T
(6.202)
⎧ F T ⎨ σ∼ = ρ ∂ψ ∂ F ∼
and
⎩ μ = ρ ∼
∼
(6.203)
∂ψ T F∼ ∂
∼
⎧ ∂ψ ⎨σ∼ = ρR FT ∼ ∂ F ∼
or
∼
(6.204)
∂ψ ⎩μ = ρR FT ∼ ∂ ∼ ∼
∼
Linear case; isotropic elasticity Deformation and curvature are small if F∼ − 1∼ 1 and
l 1, where l is ∼ a characteristic length. If, in addition, microrotations remain small, i.e., if 1, then 1∼ + 1∼ × = 1∼ − . (6.205) R ∼ ∼
u ⊗ ∇ 1 and ⊗ ∇l 1
(6.206)
F∼ 1∼ + u ⊗ ∇ + . = 1∼ + e∼
(6.207)
⊗ ∇ = κ∼ ∼
(6.208)
∼
μ. Accordingly, for linear elasticity, two fourth-rank Furthermore, ∼ σ∼ and ∼ ∼ elasticity tensors are introduced σ
μ
: e∼ σ∼ = E ∼ ∼
and μ = C : κ∼ ∼ ∼
∼
(6.209)
(no coupling between deformation and curvature is possible as soon as point symmetry is assumed even for the least symmetric solid). Some symmetry properties of these tensors are derived from the hyperelastic conditions (6.203) Eij kl = Eklij
and Cij kl = Cklij
(6.210)
326
Non-Linear Mechanics of Materials
Further symmetry conditions can be gained if materials symmetries are taken into account. The form of the Cosserat elasticity tensors for all symmetry classes are available in [ILC86]. For a triclinic solid, 90 independent constants are necessary instead of the 21 in the classical case (for a solid without point symmetry, Kessel found 171 constants!). For cubic symmetry, 10 constants instead of 3. Let us now consider the isotropic case. The two classical Lamé constants λ, μ are complemented by 4 additional moduli according to σ∼ = λ1∼ Tr e∼ + 2μ{ e∼} + 2μc } e∼{
(6.211)
μ = α1∼ Tr κ∼ + 2β { κ∼ } + 2γ } κ∼ {
(6.212)
∼
Relation (6.211) can be inverted in e∼ =
1 { } λ 1 } { σ − σ 1(Tr σ∼ ) + 2μ ∼ 2μ(3λ + 2μ) ∼ 2μc ∼
(6.213)
The expression of the free energy is 1 1 e:E :e+ κ :C :κ 2 ∼ ∼∼ ∼ 2 ∼ ∼∼ ∼ 1 1 = eij Eij kl ekl + κij Cij kl κkl 2 2
ψ(e∼, κ∼ ) =
Stability conditions require the non-negativeness of the elasticity tensors: ⎧ ⎨3λ + 2μ ≥ 0 μ≥0 ⎩ μc ≥ 0 and similar relations hold for α, β and γ .
Example 1: simple bending in the linear case
(6.214)
(6.215)
Inelastic constitutive laws at finite deformation
327
We consider the simple bending of an isotropic elastic material under plane strain conditions (see the figure for the definition of axes). We postulate the form of the stress components: (6.216) σ∼ = Ax2 e1 ⊗ e1 + σ3 e3 ⊗ e3 We first give the solution in the classical case for which no microstructure is considered M (6.217) σ3 = νAx2 and A = − I where M is the applied bending moment. The displacement field is ⎧ A 2 ⎨u1 = E (1 − ν )x1 x2 A(1−ν 2 ) 2 νA 2 ⎩u2 = − 2E (1 + ν)x2 − 2E x1 u3 = 0 We can compute the material curvature tensor χ (curvature of the material fibres) ∼
×
A (6.218) χ = F ⊗∇ = − (1 − ν 2 )e3 ⊗ e1 ∼ E For the Cosserat continuum, we investigate the existence of a solution with the same form of stress tensor as (6.216) and the following form of rotation field = Bx1 e3
(6.219)
From elastic relations (6.213) it follows that σ3 = νAx2 and A νA x2 (1 − ν 2 )e1 ⊗ e1 − x2 (1 + ν)e2 ⊗ e2 E E We must then solve the partial differential equations A νA x2 (1 + ν) u1,1 = x2 (1 − ν 2 ) u2,2 = − E E u2,1 − Bx1 = 0 u1,2 + Bx1 = 0 e∼ = u ⊗ ∇ + . = ∼
(6.220)
The proposed form is a solution if and only if A B = − (1 − ν 2 ) (6.221) E In that case, the displacement field is also given by (6.5.3). Then we compute the curvature tensor and associated couple stress κ∼ = Be3 ⊗ e1
(6.222)
μ = B((β + γ )e3 ⊗ e1 + (β − γ )e1 ⊗ e3 )
(6.223)
∼
Condition (6.221) implies that κ∼ = χ , which means that the microstructure moves ∼ with the material lines. If this constraint is abandoned, a more complicated form of the solution must be worked out.
328
Non-Linear Mechanics of Materials
Example 2: Simple glide in the linear case We study the problem of an infinite plate, of height H , fixed at the bottom and with prescribed microrotation or couple-stress vector at the top, as shown on Fig. 6.13.
Figure 6.13. Simple glide problem for the Cosserat continuum
We assume plane strain conditions: u3 = 0 and that 1 = 2 = 0, and a displacement field and rotation field of the form: ⎛ ⎞ ⎛ ⎞ u(y) 0 u=⎝ 0 ⎠ and = ⎝ 0 ⎠ 0 (y) The deformations associated with the kinematic fields defined above write: ⎛ ⎞ 0 u + 0 0 0 ⎠ e∼ = ∇ u + . = ⎝ − ∼ ∼ 0 0 0 and
⎛
0 κ∼ = ⊗ ∇ = ⎝ 0 0
0 0
⎞ 0 0 ⎠ 0
The elastic constitutive equations allow us to obtain the expression of the fields σ∼ and μ: ∼ ⎡ ⎤ 0 μu + μc (u + 2) 0 0 0 ⎦ σ∼ = ⎣ μu − μc (u + 2) 0 0 0 and
⎡
0 0 0 μ=⎣ 0 ∼ 0 (β + γ )
⎤ 0 (β − γ ) ⎦ 0
In general, we assume that the coefficients β and γ are equal so that μ23 = 0 and μ32 = 2β . This hypothesis allows us to fulfill the plane strain conditions. On the other hand, we notice on these expressions the non-symmetry of the strain and stress
Inelastic constitutive laws at finite deformation
329
tensors. The equilibrium equations write:
⎡
⎤ 0 ⎦ 0 with σ = ⎣ −μc (u + 2)
div σ∼ + f = 0 div μ + 2 σ × +c = 0
×
∼
with: σ12,2 = 0 μ32,2 + 2 σ3 × = 0 σ12 = μu + μc (u + 2) = Cst = σ0 2β − 2μc (u + 2) = 0
σ0 −2μc μ+μc β(μ + μc )
u =
− 2μμc = μc σ0
σ0 The particular solution of the differential equation in is 1 = − 2μ .
The solution of the homogeneous system is: 2 = Aeλy +Be−λy with λ =
2μμc β(μ+μc ) .
We notice that λ1 has the dimension of a length. The general solution for the kinematic fields u and is: σ0 + Aeλy + Be−λy = 1 + 2 = − 2μ 2μc u = σμ0 y − λ(μ+μ (Aeλy − Be−λy ) + C c) The constants obtained after integrating the equilibrium equations are determined from the boundary conditions. We note that, whatever the boundary conditions are, the solution has a hyperbolic form, which differs completely from the classical case of single slip where the deformations are homogeneous and the displacements are linear in y. We consider the case where we impose a microrotation 0 at the upper surface, which is free from force stresses. The boundary conditions are then: ⎧ •lower surface:u = = 0 ⎪ ⎪ ⎨ $ $ = 0 ⎪ ⎪ ⎩•upper surface: $ $e1 .σ .e2 = σ12 = 0 ∼ Hence
⎧ σ12 (H ) = 0 ⎪ ⎪ ⎨ (0) = 0 (H ) = 0 ⎪ ⎪ ⎩ u(0) = 0
⇒ ⇒ ⇒ ⇒
σ0 = 0 A+B =0 AeλH + Be−λH = 0 2μc − λ(μ+μ (A − B) + C = 0 c)
330
Non-Linear Mechanics of Materials
The analytical solution for the elastic problem with imposed microrotation is written (y) = 0 u(y) =
sinh(λy) sinh(λH)
2μc 0 (1 − cosh(λy)) λ(μ + μc ) sinh(λH)
The solution shows that a boundary layer of size characterized by λ exists at the top of the strip. This solution can be used to test numerical implementation of the Cosserat model. The solution of the corresponding problem at finite transformations is not available.
6.5.4. Elastoplasticity Decomposition of deformation and curvature We introduce the elastic and plastic contributions to deformation and curvature: F∼ e , F p , e and p . It is then necessary to propose a partition of total deformation ∼ ∼ ∼ and total curvature-torsion. We follow here [SIE92] and [SIE98b]. We can keep using the multiplicative partition of total deformation, by generalizing the classical case: F∼ = F∼ e F∼ p
(6.224)
F∼˙ F∼ −1 = F∼˙ e F∼ e−1 + F∼ e F∼˙ p F∼ p−1 F∼ e−1
(6.225)
The expression
must then be substituted in Clausius–Duhem inequality (6.200). The most natural hypothesis consists in assuming that the hyperelastic relations still have the form: ⎧ eT ⎨ σ∼ = ρ ∂ψ e F ∼ ∂ F ∼ (6.226) F eT ⎩ μ = ρ ∂ψ ∂ e ∼ ∼
∼
We can actually show as in Sect. 6.3.5 that the multiplicative partition (6.224) results from hyperelastic relations (6.226a). Similarly, for these relations (6.226) and to be fulfilled, a specific form of the partition of the total torsion-curvature is needed. We find: e p p
=
F∼ +
(6.227) ∼ ∼ ∼ This curvature decomposition introduces the notion of released configuration for curvature, for which couples stresses are unloaded. This notion is pertinent in particular
Inelastic constitutive laws at finite deformation
331
in the case of crystalline plasticity. A slightly different formulation of hyperelasticity conditions leads to a purely additive decomposition:
e p
=
+
∼ ∼ ∼
(6.228)
On the other hand, it is important to note that a decomposition such as: e p R =R R ∼ ∼ ∼
(6.229)
has no sense, even if we are tempted to write it, inspired from (6.224)! Indeed, the rotation R has the same status as the displacement u, non-objective quantity for which ∼ it is not legitimate to introduce a decomposition of the type u = ue + up for example!
Cosserat single crystal plasticity The curvature of the crystal lattice affects hardening of crystalline materials. In this case, lattice rotation is no longer a mere hidden variable as in the classical case already studied, but rather a real additional degree of freedom whose gradient generates couples-stresses. The previous elastoplastic formulation is well adapted to the case of the Cosserat single crystal and we can enrich the scheme of Fig. 6.1 in the following way:
The chosen intermediate isoclinic configuration represents a released configuration for forces and couples-stresses simultaneously. In this state of the material element, there remains only the permanent deformation and curvature. By coming back to the current frame of observation, the decomposition (6.227) is written =R
=
F +
∼ ∼ ∼ ∼e ∼ p ∼p
with
=R
, = R
and F∼ p = F∼ p . ∼e ∼ ∼e ∼p ∼ ∼p
Let dx be a material segment attached to the current configuration, and dX the corresponding segment on the initial configuration. We have: dx = F∼ dX = F∼ e dx i
with dx i = F∼ p dX.
332
Non-Linear Mechanics of Materials
Similarly, the variation of microrotation along a material segment is given by dX =
dx i +
dX
∼ ∼e ∼p We see that one can consider simultaneously the elastic contributions to the deformation and to the total curvature with respect to the material segment dx i on the intermediate configuration [SIE98b, FOR00]. This definition of the simultaneously released configuration is not possible with the purely additive decomposition (6.228). Applications of Cosserat crystal plasticity to size effects can be found in [FOR00, FOR01, FOR03a].
Chapter 7
Nonlinear structural analysis
This chapter is mainly devoted to the application of the various nonlinear behaviors presented above to structural computations. This application is achieved by implementing the nonlinear constitutive equations in a finite element code. This chapter deals with the numerical description of a behavior (Sect. 7.1), the implementation of some particular models (Sect. 7.2) and finally some special treatments linked to the finite element method (Sect. 7.3). Finite deformations (Sect. 7.3.5) are treated in this last part: it is, after the behavior itself, a second source of nonlinearity.
7.1. The material object This section introduces the numerical description of materials behavior [BES97, BES98a, BES98b, FOE97]. The behaviors we wish to implement make use of various quantities, which can be identified as follows: External parameters (EP). External parameters are the parameter fields imposed by the user. They are not calculated while solving the finite element problem. It can be, for instance, temperature imposed during a mechanical calculation. Let us note that this temperature can be the result of another finite element calculation. Of course, this example is not limitative and one can wish to impose any field: humidity, grain size, etc. Integrated variables (V int ). The state of the material is described by variables whose evolution must be time integrated. In the case of standard materials, they are internal variables. Generally, they will be referred to as integrated variables. They can be, for example, elastic strain, accumulated plastic strain, damage, porosity, etc. J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_7,
334
Non-Linear Mechanics of Materials
Problem small strain mechanics finite deformation mechanics Cosserat thermal problem electrostatics magnetostatics
primal ε∼ F∼ (ε∼ , κ∼ ) (T , gradT ) gradV rot A
dual σ∼ ∼ (σ∼ , μ) ∼ (H, q) E H
Table 7.1. Examples of primal/dual couples
ε∼ strain tensor, F∼ deformation gradient, T temperature, V electric potential, A vector potential, σ∼ Cauchy stress tensor, S∼ second Piola– Kirchhoff tensor, κ∼ curvature tensor, μ volume torques tensor, H en∼ thalpy, q thermal flux, E electric field, H magnetic field.
Auxiliary variables (Vaux ). Auxiliary variables are variables that are used during calculations, and that are available in post-processing, for example Hill’s equivalent stress, internal stress, stress triaxiality, etc.
Coefficients (CO). Coefficients allow us to describe the constitutive law. They can be, for example, Young modulus, hardening modulus, etc. Coefficients can depend simultaneously on external parameters, integrated variables and auxiliary variables. CO = CO(EP, Vint , Vaux )
(7.1)
Let us note that the calculation of auxiliary variables can use some coefficients. It is then possible that some coefficients depend on other coefficients.
Primal and dual variables (primal/dual). A behavior is driven externally by imposing a variable that will be called “primal” (primal) in the following. In mechanics, it is the strain tensor, but this notion is general and can be applied to other types of behaviors (Table 7.1). To each primal variable is associated a “dual” variable (dual) that represents the response of the behavior to the primal variable (which is then driven). In mechanics, it is of course the stress tensor. Table 7.1 gives some examples of primal/dual couples for various types of physical problems.
Nonlinear structural analysis
335
Figure 7.1. Generic interface to use a constitutive law
General formulation of constitutive laws Constitutive laws are formulated as differential equations on integrated variables. We will then have, in a very general way: dVint dVint = dt dt
dEP dprimal Vint , Vaux , CO, EP, , primal, dt dt
(7.2)
The variation rate of integrated variables depends then on quantities defined above as well as on the derivative of imposed quantities (i.e., primal and EP). The dual variables are calculated after integration of Vint .
Interface of a behavior An interface between behavior and finite elements has already been presented in Sect. 2.8.9 in the framework of small strain mechanics. This interface can be generalized. We consider the case of an incremental evaluation of the behavior. We try to achieve a time step t (t → t + t) by imposing an increment primal of the t , driven variables. At the beginning of the increment at t, all quantities are known: Vint t , EPt , primalt , dualt . The user of the behavior sets the values at the end of the Vaux increment of the driven quantities: primalt+t and EPt+t . In return, the behavior t+t t+t , and dualt+t at the end of the increment. , Vaux must provide the values of Vint The scheme of the interface is presented in Fig. 7.1. One will notice in particular that the interface is general and can be used in a context different from the finite element method.
336
Non-Linear Mechanics of Materials
Tangent matrix The behavior must sometimes provide the tangent matrix, consistent with the integration scheme that one can write formally: ∂dual ∂primal
(7.3)
This matrix can be calculated analytically, by perturbation (see Sect. 2.7.4) or by integrating the constitutive law with a θ -method (see Sect. 2.7.4 in the case of mechanics).
7.2. Examples of implementations of particular models 7.2.1. Introduction Our objective, here, is to illustrate with some examples the implementation of numerical integration methods of constitutive laws. Explicit methods do not pose, generally, any difficulties save for the calculation of the plastic multiplier. For implicit methods, the main difficulty is to find at the end of an increment an admissible stress state. To fulfill this condition, many methods have been proposed. They all consist in projecting stress over the yield surface (plasticity) or on the viscoplastic equipotential (viscoplasticity) in order to bring stress to an admissible state while iterating. Thus we distinguish the “elastic predictor/radial return” method [WIL64, MEN68], the “secant stiffness” method [RIC73], and the “tangent stiffness/radial return” [KRI77, SCH79, YOD84]. The “single-step generalized midpoint” method [SIM85a, ORT85b] or θ method, is equivalent to the “elastic predictor/radial return” method for θ = 1. It can, in addition, be interpreted as a projection method over a yield surface or an equipotential; it can be formulated without using explicitly the notion of “projection”. The “single-step generalized midpoint” method is nowadays the most efficient and the most general one, so it is the one that will be exemplified hereafter, when considering the implicit case. Let us recall that this method allows the direct calculation of the consistent tangent matrix (see Sect. 2.7.4).
7.2.2. Prandtl–Reuss law Here we implement explicit and implicit integration schemes for simple laws.
Nonlinear structural analysis
337
Law formulation The Prandtl–Reuss law is an isotropic plasticity law with isotropic hardening. The yield function is given by: f (σ∼ ) = J (σ∼ ) − R(p) (7.4) The behavior is characterized by two internal variables: ε∼ e , elastic strain tensor, and p, accumulated plastic strain. If f < 0 or f = 0 and f˙ < 0, the behavior is elastic and we have: : ε∼ e σ∼ = ∼
(7.5)
p˙ = 0
(7.6)
∼
If f = 0 and f˙ = 0, the behavior is plastic. Following normality rule, the plastic strain-rate tensor is given by: (7.7) ε∼˙ p = p˙ n ∼ is the flow direction given by: where n ∼ n = ∼
∂f 3 ∼s = ∂σ∼ 2 J (σ∼ )
(7.8)
Runge–Kutta integration In the case of explicit integration, it is suitable to give the evolution of ε∼ e and p at time t for a given total strain-rate and for fixed internal variables. One can also take into account thermal strain. From ε∼ e one can calculate σ∼ = : ε∼ e and test whether the yield condition is ∼ reached or not. If f < 0 we have:
∼
ε∼˙ e = ε∼˙ − ε∼˙ th p˙ = 0
(7.9) (7.10)
In the case where the yield condition is reached, one calculates p˙ from consistency equation f˙ = 0. Then: n : : (˙ε∼ − ε∼˙ th ) ∼ ∼ ∼ p˙ = (7.11) H +n : :n ∼ ∼ ∼ ∼
where
dR = R the hardening modulus (7.12) dp The plastic strain-rate tensor is then easily calculated: ε∼˙ p = p˙ n . The elastic strain ∼ rate tensor is given by: (7.13) ε∼˙ e = ε∼˙ − ε∼˙ th − ε∼˙ p H =
338
Non-Linear Mechanics of Materials
Integration by the θ -method In the case of the implicit method, it is suitable to calculate the increments ε∼ e and p for a given increment ε∼ − ε∼ th . We test first whether this increment is purely elastic assuming a purely elastic strain increment given by ε∼ e∗ = ε∼ − ε∼ th and checking if: f : ε∼ e∗ t+t < 0 (7.14) ∼ ∼
with ε∼ e∗ = ε∼ et + ε∼ e∗ . In this case, resolution is immediate. Otherwise, the system to solve is: = ε∼ − ε∼ th r∼e = ε∼ e + p n ∼θ
(7.15)
rp = f (σ∼ t+t ) = 0
(7.16)
This solution is the one proposed in [ORT85b], which does not fulfill the quadratic convergence conditions, because all the equations are not considered simultaneously [SIM91]. We would find the same convergence by replacing (7.16) with: rp = f (σ∼ t+θt ) = 0
(7.17)
This point is discussed later. To implement Newton’s resolution method of this system of equations, it is necessary to calculate the Jacobian. We denote by Xθ the value of the variable X at time t + θ t. We also write X0 = Xt and X1 = Xt . We notice that: ∂(X0 + θ X) ∂Xθ = =θI ∂X ∂X With N = ∼ ∼
∂n ∂ 2f 1 ∼ = = ∂σ∼ J (σ∼ ) ∂σ∼ 2
3 ⊗ n J∼ − n 2∼ ∼ ∼
(7.18)
(7.19)
where J∼ is the fourth-rank tensor that associates a second-rank tensor with its deviator: ∼
s = J∼ : σ∼ . We obtain:
∼
∼
∂σ∼ ∂ε∼ e ∂n ∂r∼e ∼θ = I + p : : = I∼ + θ pN : ∼ ∼ ∼ ∂ε∼ e ∂σ∼ ∂ε∼ e ∂ε∼ e ∼ ∼ ∼θ ∼θ ∂r∼e =n ∼θ ∂p ∂rp = :n ∼ ∼1 ∂ε∼ e ∼1 ∂rp ∂R = −H1 =− ∂p ∂p 1
(7.20) (7.21) (7.22) (7.23)
Nonlinear structural analysis
339
The Jacobian matrix can then be built from the previous derivatives: [J ] =
: I∼ + θ pN ∼ ∼
n ∼θ
:n ∼ ∼1
−H1
∼
∼
∼θ
θ
∼1
(7.24)
Tangent and consistent matrices By using the expression of the plastic multiplier p˙ (7.11) one can show that the tangent operator ∂ σ∼˙ /∂ ε∼˙ is: L = ∼ ∼
:n ) ⊗ ( :n ) ( ∂ σ∼˙ ∂σ∼ ∼ ∼ ∼ ∼ ∼ ∼ = − = =L ∼ ∼ ∂ ε∼˙ H + n : : n ∂ε∼ ∼ ∼c ∼ ∼ ∼
(7.25)
∼
is, generally, not equal to the consistent tangent The instantaneous tangent operator L ∼ ∼
operator L . The difference can simply be illustrated in the isotropic elastic case, for ∼ ∼c
which = λI∼ ⊗ I∼ + μI∼ . In this case, the spherical part disappears in dyadic products ∼ ∼
∼
with n (which is deviatoric), and L is simply written ∼ ∼ ∼
L = λI∼ ⊗ I∼ + μI∼ − ∼ ∼
∼
4μ2 n ⊗n ∼ ∼ H + 3μ
(7.26)
The calculation of the consistent tangent operator, which is explicit in this case, involves the variation of the normal during the step, which does not appear in the previous expression, indeed: : d(ε) − d(p)n − p d(n ) (7.27) d(σ∼ ) = ∼ ∼ ∼ ∼
We get then: =L − : pN : =L − 4μ2 pN L ∼ ∼ ∼ ∼ ∼ ∼ ∼ ∼c
∼
∼
∼
∼
∼
∼
(7.28)
Remarks • When using the θ-method, it is not necessary to write the consistency condition. One writes more directly the condition f (σ∼ t+t ) = 0 (7.16). • In the case of more complex constitutive models, as, for instance, the one introducing nonlinear kinematic hardening, the consistent tangent matrix becomes non-symmetric [CHA96b].
340
Non-Linear Mechanics of Materials
• Implementing Newton’s method to solve the system formed by (7.15) and (7.16) provides almost without additional computational cost the term necessary to get the consistent tangent operator. The inverse of matrix [J ] of (7.24) is, indeed, made of blocks:
J∼ ε∼ e ε ∼e (7.29) = . 0 p so that: L = : J∼ ∼ ∼ ∼c
∼
(7.30)
∼e
This method applies regardless of the constitutive model used; in practice, it is not useful to look for the explicit form of the consistent tangent operator (see also Sect. 2.7.4). • By writing the condition f (σ∼ ) = 0 at the end of the increment and not at t + θ t, one verifies the yield condition at mechanical equilibrium. Rigorous application of the θ -method leads rather to f (σ∼ t+θt ) = 0, but it is however necessary to compute the stresses at the end of the increment as: σ∼ t+t = : ε∼ et+t . This stress tensor does not ensure the yield condition at the end ∼ ∼ t+t
of the increment, f (σ∼ t+t ) = 0, which can lead to oscillations of the solution. In this case, one would write instead of (7.16), (7.22) and (7.23): = f (σt+θt )
(7.31)
= :n ∼ ∼θ
(7.32)
= −Hθ
(7.33)
= ε∼ et+θt + (1 − θ )ε∼ e
(7.34)
pt+t = pt+θt + (1 − θ )p
(7.35)
rp ∂rp ∂ε∼ e ∂rp ∂p then ε∼ et+t and
∼θ
Figure 7.2 exemplifies the θ -method integration of the Prandtl–Reuss law with θ = 12 , writing the yield condition at θ = 12 or θ = 1. We notice, indeed, oscillations. The hardening law is: R(p) = 300 + 100(1 − exp(−200p)) (MPa)
(7.36)
Varying temperature We assume now that the yield stress R depends also on temperature T (which is here an external parameter). The consistency condition is now written : σ∼˙ − R,p p˙ − R,T T˙ = 0 f˙ = n ∼
(7.37)
Nonlinear structural analysis
341
Figure 7.2. θ -method integration of the Prandtl–Reuss law with θ = 12 , writing yield condition at θ = 12 or θ = 1. Tension until ε = 0.02. E = 200 GPa. Equation (7.15) is written here at θ = 12
The plastic multiplier is then given by: p˙ =
n : : (˙ε∼ − ε∼˙ th ) − R,T T˙ ∼ ∼ ∼
(7.38)
R,p + n : :n ∼ ∼ ∼ ∼
Figure 7.3 presents the previous example with R(p, T ) = [300 + 100(1 − exp(−200p))][1 − T /200]
(7.39)
We show in particular that omitting the term R,T T˙ in the consistency formulation leads to grossly wrong results. The same kind of problem is encountered in the case of kinematic hardening, when one introduces in the differential system the variable X ∼ instead of α (see Sect. 7.2.3). ∼ In the case where the coefficients of elasticity matrix depend also on tempera∼ ∼
ture, one must take into account this dependency in the formulation of the consistency. The plastic multiplier is then written p˙ =
n : : (˙ε∼ − ε∼˙ th ) − T˙ n : : ε∼ e − R,T T˙ ∼ ∼ ,T ∼ ∼ ∼
∼
R,p + n : :n ∼ ∼ ∼
(7.40)
∼
Taking into account all dependencies can then be delicate when several external fields are imposed.
342
Non-Linear Mechanics of Materials
Figure 7.3. Integrating the Prandtl–Reuss law with varying temperature. We compare Runge– Kutta integration conducted with a wrong plastic multiplier in varying temperature condition (7.11) to exact Runge–Kutta integration (7.38) and to the θ-method with θ = 12
Let us note that these problems are not encountered with the θ -method as we then write directly the condition ft+t = 0 and not f˙ = 0.
Viscoplasticity When the material is viscoplastic, p˙ is directly obtained using a specific creep law: p˙ = φ(f, . . . )
(7.41)
with f = J (σ∼ ) − R. This equation replaces (7.11), (7.38) or (7.40) in the case of Runge–Kutta integration. For the θ -method, (7.16) is replaced by: rp = p − t φ(J − R, . . . )θ = 0
(7.42)
Nonlinear structural analysis
343
Figure 7.4. Viscoplastic Prandtl–Reuss law integration with varying temperature for different values of K with n = 10 and comparison with the plastic model
The partial derivatives involved in the Jacobian calculation are: ∂rp = −θ t φ,f :n ∼ ∼ ∂ε∼ e ∼ ∂rp = 1 + θ t R,p φ,f ∂p
(7.43) (7.44)
There is no more problem linked to the formulation of consistency. One can use creep law to “imitate” plasticity. For example, a Norton type law is often used:
f φ(f ) = K
n (7.45)
where K and n are material coefficients. For n large enough, we will have J −R K. Figure 7.4 uses the example of Fig. 7.3 with n = 10 and K = 1, 10, 50. When K becomes small enough, the viscoplastic solution tends towards the plastic solution.
7.2.3. Multikinematic law We present here the implementation of the multikinematic hardening model in a finite element code.
344
Non-Linear Mechanics of Materials
Formulation of the law The law is formulated with the following internal variables: ε∼ e α ∼i r
elastic strain tensor tensor of the kinematic-hardening variables isotropic hardening variable
is free. We will use in addition the following The number of kinematic variables α ∼i auxiliary variables: ε∼ p plastic strain tensor p accumulated plastic strain
(7.46) (7.47)
The forces associated with the internal variables are: σ∼ = : ε∼ e ∼
(7.48)
=C :α X ∼i ∼i ∼i
(7.49)
∼
∼
R = Qr
(7.50)
X is the internal stress associated with the variable α . Internal stress X is defined as: ∼i ∼ ∼i
= X (7.51) X ∼ ∼i i
is the center of the domain of elasticity. R is the size increase of the domain of X ∼ elasticity due to isotropic hardening. Evolution laws are given by: ε∼˙ e + ε∼˙ p = ε∼˙ α ˙ = ε∼˙ p − pD ˙ ∼i : α ∼i ∼i
(7.52) (7.53)
r˙ = p˙ − pbr ˙
(7.54)
∼
(7.53) and −pbr ˙ (7.54) represent a dynamic recovery The recovery terms −pD ˙∼ :α ∼i ∼
i
mechanism. The yield criterion is written f = σ − X ∼
∼
M
− σy − R ≥ 0
(7.55)
σy is the initial size of the domain of elasticity. The norm . M is a norm in the space of deviatoric tensors defined by: a ∼
M
1 2 = a∼ : M : a ∼ ∼ ∼
(7.56)
is a fourth-rank tensor whose kernel is the space of tensors of the form where M ∼ ∼
{aI∼ , a real}. The formulation of consistency (plasticity) in the most general case is
Nonlinear structural analysis
345
complicated. To simplify the problem, we consider viscoplasticity, and plasticity is assumed to be a limit case. We have then: p˙ = φ(f, . . . )
It can be shown that: p˙ =
(7.57)
−1 : ε ε∼˙ p : M ˙p ∼ ∼
(7.58)
∼
−1 is defined for any deviatoric tensors. The normal to the criterion is defined by: M ∼ ∼
n = ∼
∂f 1 = M : (σ∼ − X ) ∼ ∂σ∼ σ∼ − X ∼∼ ∼ M
(7.59)
We define also: N = ∼ ∼
∂n ∂ 2f ∼ = ∂σ∼ ∂σ∼ 2
1 = σ∼ − X ∼ M
− M ∼
: (σ∼ − X )) ⊗ (M : (σ∼ − X )) (M ∼ ∼ ∼ ∼ ∼
∼
σ∼ − X 2 ∼ M
∼
(7.60)
,D and M allow us to account for an anisotropic behavior. The isotropic Tensors C ∼ ∼ ∼ ∼i
∼
i
∼
case is recovered for C = Ci 1∼, D = Di 1∼ and M = J∼ . ∼ ∼ ∼ ∼
∼i
∼
i
∼
∼
∼
Runge–Kutta integration In the framework of viscoplasticity, integrating the constitutive law with the Runge– Kutta method consists in integrating (7.52), (7.53), (7.54) and (7.57).
θ -method integration Discretizing (7.52), (7.53), (7.54) and (7.57) gives: r∼e = ε∼ e + pn − ε∼ = 0∼ ∼ r∼αi = α − p n + pD :α = 0∼ ∼i ∼ ∼i ∼i ∼
rr = r − p(1 − br) = 0 rp = p − φ(f, . . . )t = 0
(7.61) (7.62) (7.63) (7.64)
, r are considered at time t + θ t, and are respectively approached Variables ε∼ e , α ∼i (t) + θ α and r + θ r. The associated Jacobian matrix is then by ε∼ e (t) + θ ε∼ e , α ∼i ∼i
346
Non-Linear Mechanics of Materials
calculated as follows: ∂r∼e ∂ε∼ e ∂r∼e ∂α ∼i ∂r∼e ∂r ∂r∼e ∂p ∂r∼αi ∂ε∼ e ∂r∼αi ∂α ∼i ∂r∼αi ∂r ∂r∼αi ∂p ∂rr ∂ε∼ e ∂rr ∂α ∼i ∂rr ∂r ∂rr ∂p ∂rp ∂ε∼ e ∂rp ∂α ∼i ∂rp ∂r ∂rp ∂p
= 1∼ + θ pN : ∼ ∼ ∼
∼
∼
= p
∂α i ∂n ∂X i ∼ : ∼ : ∼ = −θ pN :C ∼ ∼i ∂X ∂α ∂α ∼ ∼ ∼i ∼i ∼i
(7.65) (7.66)
=0
(7.67)
=n ∼
(7.68)
= −p
∂σ ∂ε e ∂n ∼ : ∼e : ∼ e = −θ pN : ∼ ∼ ∂σ∼ ∂ε∼ ∂ε∼ ∼ ∼
(7.69)
= 1∼ + θ pD ∼i
(7.70)
=0
(7.71)
∼
∼
= −n +D :α ∼i ∼ ∼i
(7.72)
= 0∼
(7.73)
= 0∼
(7.74)
= 1 + θ p d
(7.75)
= br
(7.76)
∼
=−
∂ε e ∂φ ∂f ∂σ∼ : e : ∼ e = −θ tφ,f n : ∼ ∼ ∂f ∂σ∼ ∂ε∼ ∂ε∼ ∼
= θ tφ,f n :C ∼ ∼i ∼
=− =1
∂φ ∂R ∂r t = θ tcφ,f ∂R ∂r ∂r
(7.77) (7.78) (7.79) (7.80)
In the framework of the θ -method, it is possible to treat plasticity by replacing (7.64) with: − R − σy = 0 rp = σ∼ − X (7.81) ∼ M To avoid oscillations (see Sect. 7.2.2), it is then possible to write this equation at t +t.
Nonlinear structural analysis
347
Reduced integration We note that the system (7.61)–(7.64) can be reformulated, keeping only the direction n and the plastic strain increment as unknowns. We obtain then: ∼ n=0 =M : (σ∼ − X ) − σ∼ − X W ∼ ∼ M ∼ ∼ ∼ ∼ ∼ − R − k t = 0 w = p − φ σ∼ − X ∼ M
(7.82) (7.83)
This system can be solved using Newton’s method. The stress is then calculated as follows: ε∼ e = ε∼ − pn ⇒ σ∼ θ = : (ε∼ e0 + θ ε∼ e ) (7.84) ∼θ ∼ ∼
We obtain then α by solving (7.62): ∼i −1 α = p 1 + θ pD : n − D α i i i i0 ∼ ∼ ∼ ∼ ∼ ∼ ∼
and r by solving (7.63): r =
∼
∼
1 − b r0 p 1 + θ p d
(7.85)
(7.86)
To avoid oscillations, one could write (7.83) at t + t and (7.82) at t + θ t.
Static recovery We can easily add a static recovery mechanism by modifying (7.53) and (7.54), which become: α ˙ = ε∼˙ p − pD ˙ ∼i : α − S∼ i : α ∼i ∼i ∼i
(7.87)
r˙ = p˙ − pbr ˙ − sr
(7.88)
∼
∼
In the calculation of the Jacobian, it is necessary to add −θ tS∼ i to ∂r∼αi /∂α and ∼i
−θ ts to ∂r∼αi /∂r.
∼
Varying temperature The coefficients intervening in the constitutive law can depend on external parameters and internal variables. It is then suitable to evaluate them accurately at the corresponding time (t, t + θ t or t + t, accordingly). Besides, using a creep law in the
348
Non-Linear Mechanics of Materials
case of the Runge–Kutta method enables us to circumvent the problem of consistency formulation. ) in the law, instead A mistake is however possible if one uses internal stresses (X ∼i of the α kinematic variables. For the demonstration, we will limit to the isotropic i ∼ 2 case with one kinematic hardening. Using relation X = Cα one must write: 3 ∼ ∼ 2 ∂C ˙ ˙ = 2 C ε˙ p − D pX ˙∼ + Tα X ∼ ∼ 3 3 ∂T ∼
(7.89)
Omitting the term in T˙ in this equation leads to a model different from the thermodynamic one exposed until now, and which is not physically realistic [CHA93a]. Figure 7.5 compares the solutions obtained using the standard model (7.53) and the modified model. Calculation is realized with: C = 30000(1 − T /200) K = 20
n = 10
D = 200 R = 300
E = 200000
The loading is indicated in Fig. 7.5. In both cases, we observe the ratchet phenomenon but solutions differ.
Remarks • The methods exposed above have the advantage of being generic, and can easily be used when the number of internal variables that the user will consider is unknown from the programmer. In addition, any type of elasticity can be introduced. • The formalism of the θ-method with θ = 1 boils down to the classical radial return method [BUS77, KRI77]. This is exemplified for a material with isotropic hardening and with isotropic elasticity by the decomposition of Fig. 7.6a, in which the deviatoric part of each term of (7.61) is multiplied by the elasticity tensor, which causes the appearance of the increment s∼e = 2με of the de, carried by the normal to viatoric trial stress ∼s e , and the radial return pn ∼1 the final surface. This scheme is also simple to establish in the case of linear kinematic hardening [NGU77] to the extent that the evolution of the direction of the kinematic variable is then defined by the final normal. The geometrical construction prevails (Fig. 7.6b) with nonlinear kinematic hardening to the extent that it is possible to establish the direction of the final normal from values at the beginning of the increment of the kinematic state variable αt and of the evolution of equivalent plastic strain [AUR94, CHA96b]: proportional to ∼s e − n ∼
2C/3 αt 1 + Dp ∼
(7.90)
Nonlinear structural analysis
349
Figure 7.5. Comparing the two evolution laws for the kinematic hardening variable
Figure 7.6. Illustration of the equivalence between the θ-method and the radial return (a) isotropic hardening, (b) nonlinear kinematic hardening
350
Non-Linear Mechanics of Materials
7.2.4. Porous materials We detail here the implementation of constitutive laws of porous materials presented in Sect. 3.11.
Formulation of the model The state of the material is characterized by: ε∼ e p η
elastic strain tensor accumulated plastic strain of the matrix porosity
To simplify the presentation, we will not take into account thermal strain. We use strain decomposition and we obtain: ε∼˙ e + ε∼˙ p = ε∼˙
(7.91)
The yield function used in plasticity or viscoplasticity is written f = σ − R ≥ 0
(7.92)
σ is the effective stress of the porous material defined by equation ψ(σ∼ , η, σ ) = 0 (3.11.1). σ is then the effective stress acting on the material constituting the matrix of the porous material. Plastic flow direction is then given by the normality rule and we get:
Let us recall that: = n ∼ We obtain also:
n ∼
=
ε∼˙ p
=
∂f ∂σ = ∂σ∼ ∂σ∼
(1 − η)pn ˙∼
∂σ ∂ψ = −h ∂σ∼ ∂σ∼
with
1 ∂ψ = h ∂σ
∂n ∂ 2 σ ∂ 2ψ ∂ψ ∂ 2ψ ∼ N = = =h h ⊗ − ∼ ∂σ∼ ∂σ∼ ∂σ ∂σ∼ ∂σ∼ 2 ∂σ∼ 2 ∼
Likewise = h2 n ∼ ,η
∂ 2 ψ ∂ψ ∂ 2ψ − ∂σ ∂η ∂σ∼ ∂η∂σ∼
(7.93) (7.94) (7.95)
(7.96)
(7.97)
The evolution of porosity results from conservation of mass, i.e., ) η˙ = (1 − η) Tr(˙ε∼ p ) = (1 − η)2 p˙ Tr(n ∼
(7.98)
Nonlinear structural analysis
351
The evolution of the accumulated plastic strain of the matrix is given by: p˙ = φ(σ − R) σ − R = 0
in viscoplasticity in plasticity
(7.99)
Implementation After time discretization of the previous equations, we obtain the following system, to solve by the θ -method: r∼e = ε∼ e + (1 − η)pn − ε∼ = 0∼ ∼ rp = p − φ(σ − R)t = 0 or rp = σ − R = 0
(7.100) (7.101)
)=0 rη = η − (1 − η)2 p Tr(n ∼
(7.102)
To simplify, the implementation of the θ -method will be exposed in the case of viscoplasticity. Using the following notation: ⎧ ⎫ ⎧ e ⎫ ⎨ ε∼ e ⎬ ⎨ ε∼ ⎬ {U } = p p {U } = ⎩ ⎭ ⎩ ⎭ η η ⎫ ⎧ ⎫ ⎧ ⎨ σ∼ ⎬ ⎨ r∼e ⎬ rp p {r} = {U } + {r2 } {V } = {r} = ⎭ ⎩ ⎭ ⎩ rη η
(7.103)
(7.104)
the system to solve is written: {r}({U }) = 0. The Jacobian matrix is then: ∂{r2 } ∂{V } ∂{U } ∂{r} = [1] + ∂{U } ∂{V } ∂{U } ∂{U }
(7.105)
where [1] is the unit matrix. We have, of course: ⎡ ∂{V } ⎣ = ∂{U }
⎤
∼ ∼
⎦
1 1
and
∂{U } = θ [1] ∂{U }
(7.106)
352
Non-Linear Mechanics of Materials
∂{r2 }/∂{V } remains to be calculated. Thus (the zero derivatives are omitted): ∂r∼e2 = (1 − η)pN ∼ ∂σ∼ ∼
(7.107)
∂r∼e2 = −pn + (1 − η)pn ∼ ∼ ,η ∂η ∂r∼e2 = (1 − η)n ∼ ∂p p ∂r2 = −φ,η tn ∼ ∂σ∼
(7.108) (7.109) (7.110)
p
∂r2 = φ,η tR,p ∂p p ∂r2 ∂φ = φ,η ht ∂η ∂η η ∂r2 = −(1 − η)2 p1∼ : N ∼ ∂σ∼ ∼
(7.111) (7.112) (7.113)
η ∂r2 = (1 − η)p1∼ : 2n − (1 − η)n ∼ ∼ ,η ∂η η ∂r2 = −(1 − η)2 1∼ : n ∼ ∂p
(7.114) (7.115)
It is then necessary to calculate the following partial derivatives: ∂ψ ∂σ∼
∂ 2ψ ∂σ∼ 2
∂ψ ∂η
∂ψ ∂σ
∂ 2ψ ∂σ∼ ∂η
∂ 2ψ ∂σ∼ ∂σ
∂ 2ψ ∂σ ∂η
(7.116)
Nucleation Plastic strain can trigger the creation of new cavities, for example by decohesion between matrix and inclusions. We limit here to the case of strain controlled nucleation. Porosity then obeys the following evolution law: η˙ = (1 − η) Tr(˙ε∼ p ) + Ap˙
(7.117)
where A is a coefficient that can depend on η, p or on stress triaxiality. One must then η η η add the term −Ap to r2 , −A/θ + A,p p to ∂r2 /∂p and −A,η p to ∂r2 /∂η.
Nonlinear structural analysis
353
Adiabatic heating During strain localization that accompanies fracture, materials can locally undergo adiabatic heating. Temperature evolves then according to: CpP T˙ = β ε∼˙ p : σ∼
(7.118)
where CpP is the heat capacity of the porous material. The coefficient β, which is lower than but close to 1, is present in “engineering” approaches to dissipation. For an evaluation of the dissipation consistent with the constitutive law, one has to refer to (2.39) of Sect. 2.3.5. Dissipation is linked to the heat capacity of the dense material (CpD ) through the relation: CpP = (1 − η)CpD
(7.119)
To implement adiabatic heating, it is necessary to add temperature to integrated variables. This adds an equation to solve: T −
β β (1 − η)pn : σ∼ = T − D pn : σ∼ = 0 ∼ ∼ CpP Cp
(7.120)
The following terms must be added to the Jacobian: ∂r2T β = − D p N : σ + n ∼ ∼ ∼ ∂σ∼ Cp ∼
(7.121)
∂r2T β : σ∼ = D pn ∼ ,f ∂η Cp
(7.122)
∂r2T β :σ = − Dn ∂p Cp ∼ ∼
(7.123)
In general, the only material parameters depending strongly on temperature are the yield stress R and the creep law. We have then: p
∂r2 = (−φ,T + φ,f R,T )t ∂T
(7.124)
Managing fracture Numerically, fracture is accounted for by the fact that for any strain, stress must be zero. Fracture will then occur at Gauss point after Gauss point. The problem is then that the tangent matrix of the behavior is zero so that the global matrix can become non invertible. To prevent these issues, it is possible to use several numerical techniques.
354
Non-Linear Mechanics of Materials
• After fracture, the behavior is replaced by a very soft elastic behavior so that σ∼ 0∼. The tangent matrix is equal to the elasticity matrix. • After fracture, we impose σ∼ = 0∼; we give a fictive non-zero stiffness matrix to avoid that the global system becomes singular. • When an element has a number of Gauss points broken large enough to make its stiffness matrix non invertible, this element is removed (simple remeshing).
Behavior of a cax8r element Because of their softening, constitutive laws of porous materials result in damage and plastic strain localization. Results of the calculations are then strongly meshsize dependent. A common practice, in particular with the Gurson model, consists in setting mesh size as a constant of the material [ROU87, XIA95]. The following example treats the behavior of an 8-nodes axisymmetrical element containing 4 Gauss points (cax8r) in tension along direction 2. Two cases are considered: (a) the right side of the element is free, (b) displacement along direction 1 on the right side is uniform. Calculations are conducted in finite deformation (updated Lagrangian Sect. 7.3.5). The constitutive law is the same as the one used in Sect. 3.11.9). = ∂σ∼ /∂ε∼ is not positiveThe softening behavior allows that tangent matrix L ∼ ∼
definite anymore. There is no more uniqueness of solution (see Chap. 8). Figure 7.7 shows the evolution of porosity at each Gauss point as a function of the elongation L/L0 . We observe clearly the appearance of the bifurcation [DOG95]. The latter depends also on boundary conditions: bifurcation appears earlier for (a) than for (b).
Tension behavior of an axisymmetric rod Tensile fracture of a specimen cannot be simulated using a single element. Indeed, fracture is generally preceded by necking that must be accounted for by finely meshing the specimen. Figure 7.8 presents the simulation of a tensile test specimen. Ductility is measured using the relative diameter variation R/R0 where R is the variation of the minimum radius in the necking zone and R0 the initial value of the specimen radius. The force F applied to the specimen is normalized by the initial cross section S0 = πR02 . The curve F /S0 –R/R0 presents a large change of slope that corresponds to the initiation of a central crack leading to failure of the structure. For the sake of comparison, the result obtained with a von Mises material is also shown in Fig. 7.8. In this case, we get a slightly larger force than with the Gurson model before slope change. Necking is also present in this case.
Nonlinear structural analysis
355
Figure 7.7. Evolution of porosity at four Gauss point and boundary conditions; case (a): solid line, case (b): dashed line
Figure 7.8 presents also the value at Gauss points of porosity η, plastic strain p and stress in tension direction σ22 for the point situated at the change of slope. We notice that damage is essentially localized in the center of the necking zone; i.e., where stress triaxiality is maximum because of the notch created by necking. Plastic strain is more uniformly distributed. Moreover, we observe that the center of the specimen is unloaded (low σ22 ) because of damage. It is important to note that, in spite of the stress decrease, this part keeps deforming plastically. In addition, we observe high stresses beyond this area.
Behavior of a cracked structure Modeling ductile crack propagation in a structure remains a very active field of research. Various methods can be used: (i) nodal relaxation according to a global criterion such as J integral [RIC68] or crack tip opening displacement [MCG80, SHA93], (ii) nodal relaxation based on a local criterion such as critical cavity growth ratio [BER81a], (iii) using interface elements allowing decohesion [NEE90, CHA97,
356
Non-Linear Mechanics of Materials
Figure 7.8. Simulation of a tensile test specimen. Values at Gauss points of η, p and σ22 just before the appearance of the central crack ( point on the F /S0 –R/R0 curve)
SIE98a], (iv) using a coupled damage model [XIA95] as it has been done previously for creep fracture [LIU94, HAY75, HAL96, SAA89]. This latter approach is illustrated here. The main problem in this case is that the results depend on the mesh size [LIU94]. This problem is linked to the softening character of the material. The solution used by many authors is to set the mesh size by considering it as an adjustable
Nonlinear structural analysis
357
Figure 7.9. Simulation of crack extension: load vs. crack-extension curve. The maps represent the value at Gauss points of plastic strain p, of the stress along the normal to the crack plane σ22 and the stress triaxiality ratio for a crack extension a = 3 mm; broken Gauss points appear in white in the various maps
parameter of the model [ROU87, XIA95]. This solution, although numerically efficient, is not perfect because the mesh size is used simultaneously to discretize the geometry and to set a characteristic length of the material. A solution could consist in using non-local constitutive laws [BAZ88, BOR91, PEE96] which are under development in the domain of ductile fracture [GOL97, MÜH98] but that have already been used to model propagation by creep [SAA89]. In any case, it is important to set a characteristic length of the material. Let us note that the same type of problem is encountered when using non-coupled models [BER81b, DEC98]. It is however possible to use a fine mesh and to consider the average of stress and strain in the vicinity of the crack tip [KIM98]. In the example of Fig. 7.9 we use the same constitutive equation as above with a mesh size of 200 × 200 μm2 at the crack tip, in plane strain. It is possible to plot the extension of the crack a as a function of the applied load or the macroscopically dissipated energy.
7.3. Specificities related to finite elements 7.3.1. The “volume element” element In order to test new constitutive equations or to use constitutive equations with a large number of internal variables, one often has to do calculations for which the stress and
358
Non-Linear Mechanics of Materials
strain states are homogeneous. One uses then only one (2D or 3D) finite element. In order to minimize the calculation time, it is however possible to use a special element whose unknowns are the deformations. There are no nodes or positions. The element contains only one Gauss point, to which is associated a volume Ve . One can then use the principle of finite element discretization (cf. Sect. 2.8.6) to define degrees of freedom as well as associated internal forces. The degrees of freedom are the deformations of the volume element and the associated forces are the product of stresses with the volume Ve . To calculate internal forces and the elementary stiffness matrix one can always apply (2.271) and (2.263) with a matrix [B] equal to identity. Deformations being degrees of freedom, it is possible to apply boundary conditions of any type: imposed value, imposed nodal reaction, multipoint constraint. . .
7.3.2. Treating incompressibility Some materials have an incompressible or quasi-incompressible behavior (rubber, metals during shaping, etc.). Incompressibility is written as the rate condition: div u˙ = 0 (7.125) The displacement formulation of a finite element does not allow us to obtain directly this condition as it is impossible to build directly a displacement field verifying incompressibility. The stress tensor can be decomposed into its deviatoric ∼s and hydrostatic −p parts so that: (7.126) σ∼ = ∼s − p1∼ Pressure-displacement formulation This technique consists in adding to each node some pressure degrees of freedom [ODE72, MAL80, ODE78]. The pressure field is then defined by:
p = N k pk
(7.127)
Displacement and position are given by: u = N k uk
x = N k xk
(7.128)
We use a higher interpolation order for displacement than for pressure so that the derivatives of u (deformation) have the same order of interpolation as pressure. Figure 7.10 illustrates these elements. In general, positions x are interpolated in the same way as displacements (i.e., N k = N k ).
Nonlinear structural analysis
359
Figure 7.10. Examples of elements used for mixed pressure–displacement formulations
In that case, mechanical equilibrium (2.9) and incompressibility condition (7.125) must be solved simultaneously. The unknowns of the problem are nodal pressures and displacements. To write equilibrium we can use Greenberg’s formulation of Sect. 2.8.4 ˙ ∼, where ∼s˙ is given by the behavior of the material (˙∼s = L : ε∼˙ ) and p˙ with σ∼˙ = ∼s˙ − p1 ∼ by interpolating nodal pressures. One can write: p = {Hp }.{p}
∼
(7.129)
where {Hp } is the vector of the shape functions associated with nodes for which pressure is an unknown and {p} the nodal pressures vector. Thus: ! σ∼˙ = L .[B].{u} ˙ − ({Hp }.{p}){m} ˙ (7.130) ∼ ∼
with: {m} = {1 1 1 0 0 0} {Hp } = {N1 . . . Nnp }
(7.131) (7.132)
To discretize the incompressibility equation, we apply Galerkin method for which we take pressure as a test function. We have then: " δ p˙ u˙ i,i dV = 0 (7.133) −
"
so −
{δ p}.{H ˙ ˙ dV = 0 p } ({m}.[B].{u})
(7.134)
For each element, we then obtain internal forces associated with the unknowns referred to as {D}: % & ' # $ T {σ } dV [B] {u} V ∼ &e {D} = {Fi } = (7.135) {p} − Ve {Hp }divu dV
360
Non-Linear Mechanics of Materials
The stiffness matrix of the element is given by: & & T ].[B]dV − Ve [B]T .{m} ⊗ {Hp }dV Ve [B] .[L ∼ ∼ & [Ke ] = − Ve {Hp } ⊗ {m}.[B]dV [0]
(7.136)
We notice that if [L ] is symmetric, so is [Ke ]. ∼ ∼
Approximated formulation: penalization This solution is simpler to implement than the previous one. Pressure is directly calculated by the element (behavior gives always only ∼s ) by using p = −λui,i
(7.137)
where λ is a penalization factor that can be assimilated to a compressibility. For λ large enough, we obtain divu ≈ 0. The unknowns are only the displacements. Internal forces are then still: " [B]T .{σ∼ } dV (7.138) Ve
The elementary stiffness matrix is given by: " T ! [B] . L .[B] + λ[B]T .{m} ⊗ {m}.[B] dV [Ke ] = ∼ Ve
∼
(7.139)
Approximated formulation: selective integration In the case of calculations where plastic strain is important, we often observe strong variations of pressure inside the elements. This is due to plastic strain incompressibility. In order to easily treat these problems (i.e., without modifying material behaviors), one can use selective integration of the volume variation [NAG74, HUG80, SIM85b]. In small strain we have: (7.140) {ε∼ } = [B].{u} We can then decompose [B] in a deviatoric part [Bdev ] and a dilation part [Bdil ]: [B] = [Bdev ] + [Bdil ]. We have then: {e∼} = [Bdev ] .{u} deviatoric part εkk {m} = [Bdil ] .{u} dilation part 3 We define the average of [Bdil ] as: ! 1 B¯ dil = |Ve |
(7.141) (7.142)
" [Bdil ] dV Ve
(7.143)
Nonlinear structural analysis
We use then a modified [B] matrix: ! ! B ∗ = [Bdev ] + B¯ dil
361
(7.144)
We verify that the average of [B ∗ ] is equal to the average of [B]. Deformation is then calculated with [B ∗ ]: ! (7.145) {ε∼ } = B ∗ .{u} The volume variation is then constant in the element. This particular treatment will be more particularly adapted to linear elements. The case of quadratic elements is treated in [ODE83]. Figure 7.11 provides a comparison between a classical calculation and a calculation using selective integration. It is the calculation of an elementary cell containing an elastic inclusion surrounded by a perfectly plastic matrix, undergoing a uniaxial tensile test. The calculation results compare the values of σkk /3 for both simulations. Maps indicate values at Gauss points. We notice for classical calculation strong variations of pressure inside the elements. These variations disappear for selective integration. On the other hand, von Mises stress (independent of the pressure) is almost identical in both cases Remark: We almost obtain an analogous result to selective integration by postprocessing the classical calculation as follows: 1. Carrying out classical calculation. 2. Averaging σkk (σ¯ kk ) per element. 3. Calculating a corrected stress σij∗ so that: σ∼ ∗ = ∼s +
σ¯ kk 1 3 ∼
7.3.3. Plane stress We assume here 2D elements. The plane stress state is along direction 3, so that the stress tensor has the form: ⎡ ⎤ σ11 σ12 0 σ∼ = ⎣ σ12 σ22 0 ⎦ (7.146) 0 0 0 A 2D finite element allows us to calculate the components of the strain tensor in the plane 1–2. Component ε33 must be calculated from the condition σ33 = 0. Conditions
362
Non-Linear Mechanics of Materials
Figure 7.11. Comparing classical calculation and calculation with selective integration in the case of the calculation of the pressure field surrounding an elastic inclusion. Inclusion: Young’s modulus 400 GPa, Poisson’s ratio 0.2, Matrix: Young’s modulus 70 GPa, Poisson’s ratio 0.3, yield stress 200 MPa; the calculation is conducted in small strain, applied macroscopic strain is 10%
σ13 = σ23 = 0 are immediately fulfilled as ε13 = ε23 = 0. In the case where applied forces in the 1–2 plane can generate ε13 or ε23 deformations, the behavior cannot be used in plane stress (anisotropic material loaded “out of axis”). Plane stress conditions can be treated either by modifying the behavior or by modifying the formulation of the elements. Generally, one modifies the behavior. For example, in the case of isotropic elasticity, one uses the following elasticity matrix to link σ∼ and ε∼ : ⎧ ⎫ ⎡ 1 ⎨ σ11 ⎬ E ⎣ ν σ22 = ⎩ ⎭ 1 − ν2 0 σ12
ν 1 0
⎫ ⎤⎧ 0 ⎨ ε11 ⎬ 0 ⎦ ε22 ⎩ ⎭ 1−ν ε12
(7.147)
Nonlinear structural analysis
363
In the case of nonlinear behaviors, the solution consists in imposing, during integration of constitutive equations, the condition σ33 = 0 as a supplementary equation to solve when integrating using the θ -method (the associated unknown is ε33 ). In the case of Runge–Kutta integration, we have, when elastic strain is an integrated quantity: e e e e σ33 = [33][11] ε11 + [33][22] ε22 + [33][33] ε33 + [33][12] ε12 =0
(7.148)
after derivation we get: e e e e ˙ [33][11] ε11 ˙ [33][22] ε22 ˙ [33][33] ε33 ˙ [33][12] ε12 0= + + + e e e e + [33][11] ε˙ 11 + [33][22] ε˙ 22 + [33][33] ε˙ 33 + [33][12] ε˙ 12
(7.149)
e . It is important not to forget the derivatives of the stiffness maWe obtain then ε˙ 33 trix n cases where the elastic stiffness matrix depends on an external parameter. ∼ ∼
Plane stress condition can also be treated specifically in the framework of the θ methods [SIM86]. These methods work well but introduce a link between element and behavior that implies treating the “plane stress” case for each new behavior. In order to suppress this dependency, it can be useful to develop a “plane stress” element that will avoid the necessity to modify the behavior [BES97, BES98a]. The solution consists in adding new degrees of freedom in order to apply a plane stress condition. We associate to each Gauss point, a degree of freedom corresponding to strain in direction 3. Strain at Gauss point g is then given by: ⎧ ⎫ 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ! g 0 g ε∼ = B g .{u} + ε33 1 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0
(7.150)
g
One shows that the reaction associated to the degree of freedom ε33 is: g
{Fi }εg = σ33 v g
(7.151)
33
g
In the absence of boundary conditions imposed on ε33 , once the problem is solved g (see (2.276)), we obtain {Fi }εg = 0 hence σ33 = 0. 33
This method has the advantage of making the finite element independent of the behavior, to be able to treat plane stress states without modifying the implementation of the behavior. However, it implies treating more degrees of freedom and it is then slower.
364
Non-Linear Mechanics of Materials
7.3.4. Periodic structures To homogenize periodic structures, it is necessary to be able to impose any macroscopic strains to a structure. In the case where the average strains of the structure are all imposed, it is possible to treat the problem in the framework of classical elements, by applying periodicity conditions on the displacement of the borders of the structure. This is not possible if one wishes to impose average stress. The solution consists in adding to the problem new degrees of freedom corresponding to macroscopic deformations [BES88]. Displacements u are then perturbations. Global displacement U is given by: U (x) = u(x) + E .x (7.152) ∼ is the macroscopic deformation. After finite element discretization of the where E ∼ field u, deformation is: ε∼ = [B].{u} + E (7.153) ∼ ; they correspond to The added degrees of freedom correspond to the components of E ∼ degrees of freedom associated to a set of elements. One can then show that reactions associated with variables Eij are: "
FEi ij = σij dV = σij v g = ||σij (7.154)
elements Gauss pts
where σij is the average of σij over the volume . Using Voigt notation, the term E12 (ibid for E23 and E31 ) works at the same time with σ12 and σ21 ; the reaction associated with E12 is then: FEi 12 = 2||σ12 (7.155) Figure 7.12 gives an example of the use of periodic conditions on a 2D mesh containing two inclusions.
7.3.5. Large deformations In the case of finite transformations, displacements are not negligible with respect to the dimensions of structures, and deformations become important. We consider three instants: t beginning of the increment, t end of the increment, and t0 instant of the reference configuration used for the calculation of the mechanical equilibrium. We look then for the displacement increment u corresponding to the time interval [t, t ]. With t = t − t and S∼ t ,t0 the second Piola–Kirchhoff tensor at t expressed in the reference configuration at t0 , the deformation gradient is given by: F∼ t1 →t2 =
∂X t2 ∂X t1
(7.156)
Nonlinear structural analysis
365
Figure 7.12. Calculation of a composite with periodic conditions: macroscopic strain imposed: E22 = 0.1, macroscopic stress imposed: σ¯ 11 = σ¯ 12 = 0.0. Displacement map (u1 and u2 ) on the mesh deformed by the perturbation field
We have then:
−1 −T S∼ t ,t0 = det(F∼ t0 →t ) Ft0 →t .σ∼ t . F∼ t0 →t
(7.157)
σ∼ t is a Cauchy tensor at time t. We have, of course: S∼ t,t = σ∼ t
(7.158)
We can then consider three possibilities for the reference configuration: (i) configuration corresponding to the beginning of the problem t0 = 0 (total Lagrangian), (ii) configuration corresponding to the beginning of the increment t0 = t (updated Lagrangian I), (iii) configuration corresponding to the end of the increment t0 = t (updated Lagrangian II). Let {D}t be the values of the displacements of the element at time t. We will denote by {D}t1 →t2 the increment of the discretized values corresponding to the displacement field increment U t1 →t2 = U t2 − U t1 . We have then: F∼ t0 →t = I∼ + [BF ]t0 .{D}t0 →t (7.159)
366
Non-Linear Mechanics of Materials
[BF ]t0 is a matrix depending on {D}t0 . We define also the tensor: e∼t0 →t = We have then:
We will note also:
e∼t0 →t = [BF ]t0 .{D}t0 →t
ε∼ t0 →t =
∂U t0 →t ∂Xt0
∂U t0 →t ∂X t0
= sym.
1 T e∼t0 →t + e∼t0 →t 2
(7.160)
(7.161)
(7.162)
This tensor is assimilated to the strain tensor in the small strain case. ε∼ is linked to displacements by: (7.163) ε∼ t0 →t = [Bε ]t0 .{D}t0 →t where [Bε ]t0 depends only on the configuration at t0 . The Green-Lagrange tensor is then given by: 1 1 1 T 1 F E = e∼T + e∼ + e∼T .e∼ = ε∼ + e∼T .e∼ = .F − 1 ∼ ∼ ∼ ∼ 2 2 2 2
(7.164)
We introduce matrices [ML ] and [MR ] (left and right multiplications) so that: ! a∼ .b∼ = ML (a∼ ) . b∼ (7.165) ! b∼ .a∼ = MR (a∼ ) . b∼ (7.166) Matrices [ML ] and [MR ] are linear with respect to tensor a∼ . Matrix [ML ] (resp. [MR ]) is the one that, multiplied by tensor b∼ in Voigt notation, gives the tensor a∼ .b∼ (resp. b∼.a∼ ) in Voigt notation. Transposition matrix [T ] is such that: T a∼ = [T ] . a∼ (7.167) We have the following relations:
(2)
!T ! ML (a∼ T ) = ML (a∼ ) !T ! MR (a∼ T ) = MR (a∼ )
(7.169)
(3)
[T ]T = [T ]
(7.170)
(1)
!T ! (4) si a∼ = a∼ then ML (a∼ ) = ML (a∼ ) !T ! (5) si a∼ = a∼ T then MR (a∼ ) = MR (a∼ ) T
(7.168)
(7.171) (7.172)
At time t , the power of inner forces for a virtual infinitesimal displacement δu (associated strain field δε∼ ) is written " Wi = σ∼ t : δε∼ dV (7.173) t
Nonlinear structural analysis
It can also be expressed in the configuration at t0 by the transformation: " S∼ t ,t0 : δE dV Wi = ∼
367
(7.174)
t0
where δE is the infinitesimal variation of the Green–Lagrange tensor corresponding ∼ to the infinitesimal variation δε∼ . Differentiating 7.164, we get: 1 1 δE = δε∼ t0 →t + δe∼Tt0 →t .e∼t0 →t + e∼Tt0 →t .δe∼t0 →t (7.175) ∼ 2 2 This variation can be linked to the infinitesimal variation of displacements ({δD}). δε∼ t0 →t is calculated by (using notation: {δD} = δ{D}t0 →t ): (7.176) δε∼ t0 →t = [Bε ]t0 .{δD} We have:
! e∼Tt0 →t .δe∼t0 →t = ML (e∼Tt0 →t ) . [BF ]t0 .{δD}
Finally, δE can be calculated: ∼ ! 1 T = [Bε ]t0 + ([1] + [T ]) . ML (e∼t0 →t ) . [BF ]t0 .{δD} δE ∼ 2 ≡ [BE ]t0 →t .{δD}
(7.177)
(7.178)
is written Matrix [BE ]t0 →t depends on configurations at t0 and t . Product S∼ t ,t0 : δE ∼ (7.179) = St ,t0 j BE t0 →t i,j δD|j = [BE ]Tt0 →t . S∼ t ,t0 .{δD} S∼ t ,t0 : δE ∼ The forces associated with the degrees of freedom of the element are then: " {Fi }t = [BE ]Tt0 →t . S∼ t ,t0 dV
(7.180)
Vt0
The elementary stiffness matrix is calculated as: [K] =
∂{Fi }t ∂{Fi }t = ∂{D}t ∂{D}t→t
(7.181)
To calculate this derivative, the product [BE ]Tt0 →t .{S∼ t ,t0 } is then rewritten: 1 !T [Bε ]Tt0 . S∼ t0 →t + [BF ]Tt0 . ML (e∼Tt0 →t ) .([1] + [T ]T ). S∼ t ,t0 2 This formula can be simplified using the following properties: (1) (2) (3) (4)
[T ]T = [T ] 1 ([1] + [T ]). S∼ t ,t0 = S∼ t ,t0 as S∼ is symmetric 2 ! ! T ML (e∼Tt0 →t ) = ML (e∼t0 →t ) ! ! ML (e∼t0 →t ) . St ,t0 = MR (S∼ t ,t0 ) . e∼t0 →t
(7.182)
368
Non-Linear Mechanics of Materials
We have then: ! [BE ]Tt0 →t . S∼ t ,t0 = [Bε ]Tt0 . S∼ t ,t0 + [BF ]Tt0 . MR (S∼ t ,t0 ) . e∼t0 →t
(7.183)
which can be differentiated with respect to {D}t→t : [BE ]Tt0 →t .
! ∂{et →t } ∂{S∼ t ,t0 } ∂{E t →t } . ∼0 + [BF ]Tt0 . MR (S∼ t ,t0 ) . ∼ 0 ∂{E } ∂{D} ∂{D}t→t t→t ∼ t0 →t ! ∂{S∼ t ,t0 } L = ∼ } ∂{E ∼ ∼ t0 →t
is the stiffness matrix of the behavior. The elementary stiffness is then " ! ! . [BE ]t0 →t + [BF ]Tt0 . MR (S∼ t ,t0 ) . [BF ]t0 dV [K] = [BE ]Tt0 →t . L ∼ Vt0
∼
(7.184)
(7.185)
(7.186)
Remarks • Behaviors in finite deformation receive then, as input, deformation gradient F∼ . As output, they must provide a measure of the stress. It will then be sometimes necessary to transport stress (for instance Cauchy towards Piola–Kirchhoff) before calculating internal forces. In this case, it will also be necessary to transport the stiffness matrix using the following transition formula between configuration at t2 and at t1 : −1 −1 −T −T F : L (7.187) = J ⊗ F : F ⊗ F L t1 →t2 ∼ t1 →t2 ∼ ∼ t1 →t2 ∼ ∼ t1 →t2 ∼ t1 →t2 ∼ t1
∼ t2
• Three matrices of interpolation of the degree of freedom must be calculated: [Bε ]t0 , [BF ]t0 and [BE ]t0 →t . [Bε ]t0 and [BF ]t0 depend only on configuration at t0 whereas [BE ]t0 →t depends also on the state at t . [Bε ] is calculated as in small strain, paying attention to using node positions at time t0 , likewise for [BF ]t0 . [BE ]t0 →t can be calculated from [Bε ]t0 , [BF ]t0 and the increment of the degrees of freedom {D}t0 →t (quantity modified from one iteration to another). • When t0 = t , the tensor e∼t0 →t is zero so that [BE ]t →t = [Bε ]t . • One generally uses the “total lagrangian” formulation in the case of behavior whose formulation uses the pair Piola–Kirchhoff tensor/Green–Lagrange tensor. It is the case, for instance, of an elastomer. • In plasticity and viscoplasticity we usually treat equilibrium in the configuration at final time t .
Nonlinear structural analysis
369
• In the case of the configuration at t , some boundary conditions involving the borders of the domain (for example when applying a pressure) depend on the degrees of freedom and involve additional terms in the global stiffness matrix. We could then prefer to look at the configuration at time t. • As in the small strain case, it is possible to apply the same approach to treat periodic conditions and plane stress. In the former, one adds degrees of freedom representing the average deformation gradient; node displacements represent perturbation. In the plane stress case, one adds for each Gauss point a degree of freedom representing the deformation gradient in the third direction: ∂Uz /∂z. One can use both modifications to obtain periodic elements in plane stress. Vector {D} used above represents then all the degrees of freedom attached to the element.
7.3.6. Cosserat elements To exemplify the generality and versatility of the object-oriented programming scheme applied to structural analysis, we show that with the base tools introduced until now, implementing a Cosserat element and the associated behavior does not pose any additional problem. Equilibrium equations and nonlinear behavior of Cosserat continua in small strain are given in Sects. 8.2.2 and 8.2.3. Cosserat continua are part of the generalized continua introducing additional degrees of freedom. The equivalent for a 1D or 2D continuum would be Timoshenko beam and Mindlin shell, for which a finite element formulation is proposed in [BAT91]. Each node is associated with the following vector of degrees of freedom: {ddl} = {U1 , U2 , 3 }
(7.188)
in the 2D case where Ui stands for displacement components and 3 the microrotation with respect to third axis. Reactions at the nodes correspond then to forces and microtorques. We define then the matrix providing the Cosserat measure of deformation: {e} = [B].{ddl} (7.189) with {e} = {e11 e22 e33 e12 e21 κ31 κ32 }T ⎡ ∂ ⎢ ∂x [B] = ⎢ ⎣ 0 0
0 ∂ ∂y 0
0
∂ ∂y 0
0
1
0
0 ∂ ∂x −1
0 0 ∂ ln ∂x
(7.190) 0
⎤T
⎥ 0 ⎥ ⎦ ∂ ln ∂y
(7.191)
A normalizing length ln is introduced for a better conditioning of the matrices. Dual quantities associated with strain measures are the force-stress and couple-stress stored
370
Non-Linear Mechanics of Materials
in the vector: {σ } = {σ11 σ22 σ33 σ12 σ21 μ31 μ32 }T
(7.192)
The finite element problem is based on the variational formulation associated to the principle of virtual power for the Cosserat continuum: " " " ˙ ˙ (σ∼ : e∼˙ + μ : κ∼˙ )dV = (f .u˙ + c.)dV + (t.u˙ + m.)dS (7.193) V
∼
V
∂V
It can be written in a general way: " " " ˙ {σ }.{e} ˙ = {F }.{ddl}dV + V
V
˙ {T }.{ddl}dV
(7.194)
∂V
which differs from the formulation of classical elements seen above only by the larger size of the vectors. Boundary conditions extend then automatically without additional programming. Treating constitutive equations calls upon all the tools already mentioned. It is necessary, of course, to add the class of non-symmetric tensors with associated operations. Fourth-rank tensors that do not possess minor symmetries must be introduced. In the case of a nonlinear behavior such that Cosserat elastoplasticity presented in Sect. 8.2.3, one uses explicit or implicit integration methods, described previously. The variable to be integrated is then the accumulated plastic curvature-deformation p, possibly accompanied by additional hardening variables.
Chapter 8
Strain localization phenomena
Strain localization phenomena represent extreme limits of material’s behavior and often are precursors of its fracture. This is an essential aspect of nonlinear materials mechanics. It can sometimes be interpreted as an unstable response associated, for example, with the occurrence of a quasi-surface discontinuity inside the material. Such bifurcation modes are presented here within the framework of elastoplasticity. The difficulties associated with the simulation of strain localization phenomena are illustrated in order to show the need for regularization methods. One such regularization method, based on the Cosserat continuum, is described in detail.
8.1. Bifurcation modes in elastoplasticity 8.1.1. Formulation of the boundary value problem Linear incremental formulation Under small strain conditions, total strain rate can be decomposed into elastic and plastic parts: (8.1) ε∼˙ = ε∼˙ e + ε∼˙ p The constitutive behavior of an elastoplastic solid is described by • elastic law σ∼ = : ε∼ e ∼ ∼
where is the fourth rank elasticity tensor. ∼ ∼
J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7_8,
(8.2)
372
Non-Linear Mechanics of Materials
• yield function f (σ∼ , α ) where α represents the set of hardening variables ∼ ∼ • plastic flow rule
˙ ε∼˙ p = λP ∼
(8.3)
• evolution equations for the hardening variables that are assumed to have the form ˙ (8.4) α ˙ = λh ∼ ∼ The plastic loading condition is
λ˙ > 0
(8.5)
f = 0 and f˙ = 0
(8.6)
f˙ = n : σ∼˙ − λ˙ H ∼
(8.7)
The consistency conditions are
It follows that where n = ∼
∂f ∂σ∼
H =−
∂f :h ∼ ∂α ∼
H is the hardening modulus. Non-associated plasticity means that P∼ = n . As a result ∼ λ˙ =
n : σ∼˙ ∼ H
If the material has a hardening behavior (H > 0), the plastic loading condition can be written n : σ∼˙ > 0. ∼ This does not hold for strain softening materials (H < 0), for which we derive from (8.1), (8.2) and (8.7), n : : ε∼˙ ∼ ∼ ∼ λ˙ = H +n : : P∼ ∼ ∼ ∼
Provided that H + n : : P∼ > 0 (no snap-back behavior) the plastic loading ∼ ∼ condition (8.5) becomes
∼
n : : ε∼˙ > 0 ∼ ∼ ∼
The constitutive equations take the incremental form : ε∼˙ σ∼˙ = L ∼ ∼
(8.8)
Strain localization phenomena
373
where L = ∼ ∼
if f < 0 or (f = 0 and n : : ε∼˙ ≤ 0) ∼ ∼
=D L ∼ ∼
if f = 0 and n : : ε∼˙ > 0 ∼ ∼
∼ ∼
∼
∼
∼
∼
1 with = − D ∼ ∼ ∼
∼
( : P∼ ) ⊗ (n : ) ∼ ∼ ∼ ∼
∼
H +n : : P∼ ∼ ∼
(8.9) (8.10)
(8.11)
∼
Formulation of the rate problem For given fields of the rates of body forces f˙, surface forces F˙ on the part Sf of the boundary of the domain D, and displacement V over Sd , the rate problem consists in finding the velocity field v in B satisfying: 1 T v) (∇ v + ∇ ∼ 2 ∼ div σ∼˙ + f˙ = 0 ε∼˙ =
σ∼˙ = L : ε∼˙ ∼ ∼
(8.12) (8.13) (8.14)
σ∼˙ n = F˙
on Sf
(8.15)
v=V
on Sd
(8.16)
The previous problem is nonlinear because of relations (8.9) and (8.10).
8.1.2. Loss of uniqueness, general bifurcation modes Hill’s condition Let (σ∼˙ A , ε∼˙ A ) and (σ∼˙ B , ε∼˙ B ) be two distinct solutions of the boundary value problem and let us consider the differences σ∼˙ = σ∼˙ A − σ∼˙ B and ˙ε∼ = ε∼˙ A − ε∼˙ B . By applying the principle of virtual power successively to solutions A and B on body B, we get a necessary condition for loss of uniqueness σ∼˙ : ˙ε∼ dV = 0 B
A local sufficient condition for uniqueness is then σ∼˙ : ε∼˙ > 0
374
Non-Linear Mechanics of Materials
for any kinematically admissible velocity fields. This condition is called positiveness of the local second-order work. When a linear incremental formulation of the constitutive equations like (8.8) exists, the positiveness of the local second-order work is s , where s denotes equivalent to the definite positiveness of the fourth-rank tensor D ∼ ∼
symmetrization. A necessary condition for loss of uniqueness is then that at some material point s det D =0 ∼ ∼
When uniqueness is lost, general bifurcation modes become possible. They can be diffuse or localized deformation modes. Bifurcation modes associated with strain rate jumps at the boundary of the localization zone can occur later on. The corresponding bifurcation criteria will be derived in Sect. 8.1.4. It is important to note that the analysis can be conducted for discrete systems, such as the one associated with finite element discretization. In this case, a linear system involving a global stiffness matrix K must be solved. The uniqueness is lost when the determinant of K becomes zero. Stability of the discrete problem may be lost as soon as (K + KT ) vanishes. These results are especially important for finite element structural analyses. However, we insist on the difference between the conditions of loss of uniqueness for discrete and continuous systems.
General bifurcation modes in non-associated elastoplasticity It is necessary to perform a spectral analysis (determining eigenvalues and eigentens in order to obtain explicit criteria sors) of the elastoplastic tangent tensors D and D ∼ ∼ ∼
∼
has six symfor possible loss of uniqueness and the associated bifurcation modes. D ∼ ∼
metric eigentensors and three skew-symmetric ones. Let x∼ i be the six orthogonal symmetric eigentensors and ωi the respective eigenvalues ranged in increasing order of the eigenvalues. Any bifurcation mode can then be written ε∼˙ = αi x∼ i . Uniqueness may be lost as soon as ε∼˙ : D : ε∼˙ = αi2 ωi = 0 ∼ ∼
This occurs for the first time when ω1 = 0. The associated bifurcation mode is proportional to x 1 and is called the fundamental mode. When ω1 < 0, other bifurcation modes become possible. As a basis of the symmetric tensors, we can take the eigenfor instance. In the isotropic case, 3k (bulk modulus) tensors of the elasticity tensor ∼ ∼
is an eigenvalue of first order, 1∼ being the associated eigentensor and 2μ (shear modulus) is an eigenvalue of order 5. One can choose five basis tensors of the deviatoric
Strain localization phenomena
375
space (with 1 = −1) ⎡ 1 ⎣ [e∼1 ] = √ 6 ⎡ 1 [e∼3 ] = √ ⎣ 2 ⎡ 1 ⎣ [e∼5 ] = √ 2
⎡ 1 ⎣ [e∼2 ] = √ 2 ⎡ 1 [e∼4 ] = √ ⎣ 2
⎤ 1 0 0 0 2 0 ⎦ 0 0 1 ⎤ 0 1 0 1 0 0 ⎦ 0 0 0 ⎤ 0 0 0 0 0 1 ⎦ 0 1 0
⎤ 1 0 0 0 0 0 ⎦ 0 0 1 ⎤ 0 0 1 0 0 0 ⎦ 1 0 0
In a special case we try now to find the critical hardening modulus H u for which are in the space uniqueness may be lost. Let us assume for simplicity that P∼ and n ∼ generated by two base vectors P∼ = p1 e∼1 + p2 e∼2 s respec(λi , e∼i ) and (ωi , x∼ i ) still being the eigenvalues and eigentensors of and D ∼ ∼ s under the form tively, we look for an eigenvector of D ∼
∼
x∼ = ξ1 e∼1 + ξ2 e∼2 The equation s .x∼ = ω x∼ D ∼ ∼
leads to a homogeneous system with two unknowns (ξ1 , ξ2 ). The determinant of this system must be zero if non-trivial solutions exist. This condition gives a relationship between the hardening modulus and ω. H u is then obtained for ω = 0. A more systematic analysis [RUN89] leads to: H > Hu =
1 : :n P∼ : : P∼ − n : : P∼ ) ( n ∼ ∼ ∼ ∼ ∼ ∼ 2 ∼ ∼ ∼
The fundamental bifurcation mode is x∼ =
P∼ P∼ : : P∼ ∼
+
∼
n ∼ n : :n ∼ ∼ ∼ ∼
For associated plasticity P∼ = n and the critical hardening modulus is then ∼ Hu = 0
376
Non-Linear Mechanics of Materials
8.1.3. Well-posedness of the rate boundary value problem for the linear comparison solid The linear comparison solid introduced by Hill is obtained when elastic unloading is excluded, so that =D L ∼ ∼ ∼
∼
In the incrementally linear case, necessary and sufficient conditions can be worked out for the well-posedness of the rate problem [HIL58]. The rate boundary value problem is said to be well-posed if it admits a finite number of linearly independent solutions which depend continuously on the data (especially boundary conditions), and which constitute diffuse modes of deformation. It can be shown that the linear rate problem for a solid with boundary and possible interfaces is then well-posed if and only if the following conditions are met: (i) the ellipticity condition, (ii) the boundary complementing condition, (iii) the interfacial complementing condition. Condition (i) states that the rate equilibrium equations must remain elliptic in the body B. It is equivalent to the following conditions (take care of the distinction be(flow direction)): tween the vector n (normal to a surface) and the second-rank tensor n ∼ (i)
det n.D .n = 0 ∀n = 0 et ∀M ∈ B, ∼
(i)
bifurcation modes involving jumps of the velocity gradient are precluded,
∼
(i) stationary acceleration waves are precluded. The meaning of conditions (i) , (i) and (i) will be given in the next section. The link between the existence of discontinuous bifurcation modes and of stationary acceleration waves has been seen by Mandel [MAN66]. The rate problem written for finite bodies can become ill-posed while the incremental field equations remain elliptic, as a consequence of the failure of the complementary condition at the boundaries or interfaces. When condition (ii) is not fulfilled anymore, so called Rayleigh surface waves can develop. This can be investigated by looking for wave solutions: v = w exp ik.x Body waves correspond to solutions with real components of k. If the solid extends over the half-space x1 ≥ 0, the boundary being free of traction, solutions with k1
Strain localization phenomena
377
complex lead to Rayleigh free surface waves, which decay exponentially into the body if: Im k1 ≥ 0 Similar conditions exist for the existence of Stoneley stationary interfacial waves, which are deformation modes localized at each side of an interface. The interface can be perfect or described by a constitutive equation. If no length scale is introduced in the constitutive equations, the wave length of body, surface or interfacial waves remains arbitrary. The mismatch in mechanical properties at an interface can induce stress and strain concentration that may act as initiation sites for localization. Conversely, grain boundaries can act as barriers to localization that originates in the bulk of the material.
8.1.4. Existence of velocity gradient discontinuities Acceleration waves; acoustic tensor; discontinuous bifurcation modes in the static case We investigate the possibility of emergence of deformation modes involving jumps of the acceleration, of the velocity gradient and of the stress rate across a moving surface S with normal n [[u]] ¨ = 0 [[u˙ ⊗ ∇]] = 0 and [[σ∼˙ ]] = 0 (8.17) But it is assumed that [[u]] = [[u]] ˙ = 0 and [[σ∼ ]] = 0
(8.18)
The discontinuity surface S propagates with the speed U = x.n where x ∈ S. According to Hadamard’s compatibility conditions [HAD03], the jump of the velocity
378
Non-Linear Mechanics of Materials
gradient at the interface must have the form ∃g/[[u˙ ⊗ ∇]] = g ⊗ n
(8.19)
A detailed description of singular surfaces can be found in [TRU60]. Similarly, ∃g /[[σ∼ ⊗ ∇]] = g ⊗ n. ∼
∼
Furthermore conditions (8.18) must hold at any time, so that [[u]] ¨ − Ug = 0 [[σ∼˙ ]] − U g = 0 ∼
Applying the previous relation to n, one gets [[σ∼˙ ]].n − U [[div σ∼ ]] = 0 On the other hand, the equilibrium equation implies ¨ [[div σ∼ ]] = ρ[[u]] If the constitutive equations have the linear incremental form (8.8), and assuming that ]] = 0, one gets: [[L ∼ ∼
Q.g = ρU 2 g
(8.20)
Qij = nk Diklj nl
(8.21)
∼
where under plastic loading conditions on each side of S. Q is called the acoustic tensor. ∼ The previous relation will be written Q = n.D .n ∼ ∼
∼
in the sequel. As a result, the existence of discontinuity surfaces of the type (8.17) depends on the solutions of the eigenvalue problem (8.20).
Static case We consider now the following jump conditions across a surface S which is not moving ˙ = Cst (8.22) [[u˙ ⊗ ∇]] = 0 and [[u]]
Strain localization phenomena
379
Condition (8.19) must still hold and we make use of the linear incremental form (8.8). ]] = 0. Elastic/plastic bifurWe investigate plastic/plastic bifurcations for which [[L ∼ ∼
cation modes will be considered in the next section. The equilibrium condition at the interface S becomes in the static case [[σ∼˙ ]].n = 0
(8.23)
Combining (8.19) and (8.23) leads to Q.g = 0 ∼
where the acoustic tensor is still defined by (8.21). Non-trivial solutions exist if the acoustic tensor becomes singular. Discontinuous bifurcation modes can then be regarded as stationary plastic waves in the dynamic case. It can be shown that there is an equivalence between the loss of ellipticity of the governing equations and the existence of discontinuous bifurcation modes.
Conditions for plastic/plastic and elastic/plastic bifurcations If plastic loading occurs on each side of S, the bifurcation modes are called plastic/plastic localization modes and necessary and sufficient conditions for such modes to become possible inside the body are: .n = 0 and n. : P∼ = 0 det n.D ∼ ∼ ∼
∼
(8.24)
If elastic unloading occurs on one side of S, the associated bifurcation modes are called elastic/plastic localization modes. Necessary and sufficient conditions for their existence inside the body are: det n.D .n < 0 and n. : P∼ = 0 ∼ ∼ ∼
∼
This result will be proved in Sect. 8.1.5. It means that in the case of usual elastoplastic materials for which the work-hardening rate decreases with strain, an elastic/plastic localization mode involving elastic unloading cannot occur before the condition for plastic/plastic bifurcation modes is fulfilled. However it does not exclude that elastic unloading occurs on one side just after plastic/plastic bifurcation.
Bifurcation at the boundary or at an interface We seek necessary and sufficient conditions for a discontinuity surface S to appear at or to reach the boundary of a solid [BEN89b]. To conditions (8.22) and (8.23), the following boundary conditions must be added: σ∼˙ 2 .m = σ∼˙ 1 .m = F˙
380
Non-Linear Mechanics of Materials
where unit vector m is normal to the boundary. For a plastic/plastic bifurcation mode to appear, this implies: .n .g = 0 m.D ∼ ∼
Combining (8.11) and (8.27) (see next section) with this last condition, we obtain: −1 m. .n .n : P : P∼ (8.25) n. n. = m. ∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
Thus necessary and sufficient conditions for plastic/plastic localization to become possible on a boundary are: : ε∼˙ 0 = F˙ (i) ∃˙ε∼ 0 such that m.D ∼ ∼
(ii) det n.D .n = 0 ∼ ∼
(iii) (m. .n).(n. .n)−1 .(n. : P∼ ) = m. : P∼ ∼ ∼ ∼ ∼ ∼
∼
∼
∼
(iv) n. : P∼ = 0 ∼ ∼
The case of elastic/plastic localization modes is obtained by replacing (i) and (ii) by: (i) ∃˙ε∼ 0 such that m.D : ε∼˙ 0 = F˙ and n. : ε∼˙ 0 > 0 ∼ ∼ ∼
(ii)
∼
det n.D .n < 0 ∼ ∼
These results imply that bands in the bulk and bands at the boundary are usually misaligned. Bands of localized deformation may display a kink when approaching a free surface (see examples in [SUO92, FOR95a]).
8.1.5. Bifurcation analysis in elastoplasticity Critical hardening modulus for a given normal n Rice [RIC76] has determined the critical hardening modulus for which a plastic/plastic bifurcation becomes possible across a surface S of given normal n. One starts from the equilibrium equation on S : [[˙ε∼ ]].n = 0 D ∼ ∼
where the linear incremental form (8.8) has been used. Taking (8.19) and (8.11) into account, one obtains
1 .n − : P ) ⊗ (n : .n) .g = 0 n. (n. ∼ ∼ ∼ ∼ ∼ A ∼ ∼ ∼
Strain localization phenomena
381
A=H +n : : P∼ ∼ ∼ ∼
It can be checked that H (n) = −n : : P∼ + (n : .n).(n. .n)−1 .(n. : P∼ ) ∼ ∼ ∼ ∼ ∼ ∼
(8.26)
.n)−1 .(n. : P∼ ) g ∝ (n. ∼ ∼
(8.27)
∼
and
∼
∼
∼
∼
∼
are solutions of the previous equations. The uniqueness of this solution remains to be proved. For that purpose we investigate both plastic/plastic and elastic/plastic bifurcation modes.
Plastic/plastic bifurcation modes We consider the following eigenvalue problem for the acoustic tensor: Q.y = λQe .y ∼
∼
where .n Qe = n. ∼ ∼
is positive definite. If
Pe ∼
=
Qe−1 , ∼
∼
the equation to be solved can be written .y = λ y B ∼
with
1 e P (b ⊗ a) a=n : .n b = n. : P∼ ∼ ∼ ∼ A∼ ∼ ∼ λ = 1 is an eigenvalue of order 2. The associated eigenvectors are orthogonal to a. The remaining eigenvalue is obtained by applying the trace operator = 1∼ − B ∼
= 2 + λ3 = 3 − Tr B ∼
1 1 a.P∼ e .b ⇒ λ3 = 1 − a.P∼ e .b A A
One checks that y 3 = P∼ e b is the associated eigenvector. The solution of the initial problem, i.e., the eigenvalues of the acoustic tensor, is obtained only for λ3 = 0 which gives relation (8.26). The expression of y 3 is then identical to (8.27).
Elastic/plastic bifurcation modes If elastic behavior is assumed on one side of S, the equilibrium conditions become : ε∼˙ 2 − : ε∼˙ 1 ).n = 0 (D ∼ ∼ ∼
∼
382
Non-Linear Mechanics of Materials
where ε∼˙ 2 − ε∼˙ 1 = { g ⊗ n} Hence, the equation to be solved now is Q.g =
1 ( .n : P ) ⊗ (n : ) : ε ˙ 1 ∼ ∼ ∼ A ∼∼ ∼ ∼
Q.g =
α b A
∼
Eliminating ε∼˙ 2 gives ∼
with α = n :E : ε∼˙ 1 ≤ 0 ∼ ∼ ∼
(elastic unloading). Eliminating ε∼˙ 1 gives Qe .g = ∼
β b A
with β = n :E : ε∼˙ 2 > 0 ∼ ∼ ∼
(plastic loading). The solution is then g=
n : : ε∼˙ 2 ∼ ∼ ∼
A
P∼ e .b
In this case the amplitude of the discontinuity is not arbitrary any more. Furthermore it can be noticed that α Q.g = Qe g ∼ β ∼ It means that an elastic/plastic bifurcation is possible only if α/β is an eigenvalue of . Yet α/β < 0, so that the only possibility is λ3 = α/β. This is the the previous B ∼ proof of the result given in Sect. 8.1.4: An elastic/plastic bifurcation cannot occur before the condition for plastic/plastic bifurcation is fulfilled.
Critical hardening modulus and orientation of the first possible shear band For materials with decreasing hardening modulus, the critical hardening modulus for which the first discontinuous bifurcation mode becomes possible is H cr = sup H (n) n=1
where H (n) is given by (8.26). The orientation of the first possible band of localized deformation is given by the values of n for which H cr is reached.
Strain localization phenomena
383
Three-dimensional case The problem reduces to maximizing the Lagrangian function: L(n, λ) = H (n) − λ(n2 − 1)
(8.28)
For isotropic elasticity,
.n) − (n.P∼ .n)(n.n .n) − P∼ : n H (n) = 2μ 2(n.P∼ ).(n ∼ ∼ ∼ ν − .n − Tr n ) (n.P∼ .n − Tr P∼ )(n.n ∼ ∼ 1−ν The associated vector g is g = 2P∼ .n −
1 ν .n)n + (n.P∼ .n)(n.n (Tr P∼ )n ∼ 1−ν 1−ν
We refer to [BIG91] to solve this maximization problem for general non-associated elastoplasticity. The present analysis is restricted to associated and incompressible plasticity P∼ = n and Tr P∼ = 0 ∼ Equation (8.28) combined with (8.26) and written in the principal axes of P∼ becomes then:
3 Pi2 − L(n, λ) = 2μ 2n2i Pi2 − i=1
Writing
∂L ∂λ
=
nk
Pk2
∂L ∂nk
3 1 n2i − 1 (n2i Pi )2 − λ 1−ν i=1
= 0, makes the problem equivalent to the resolution of the system:
1 λ − (n2i Pi )Pk − 1−ν 4μ
=0
k ∈ {1, 2, 3} (no sum over k)
If n1 n2 n3 = 0, the system is indeterminate or impossible depending on P1 , P2 , P3 . If nk = 0 and ni nj = 0 (i, j , k distinct), – if Pi = Pj , the system is indeterminate; – if Pi = Pj , n2i =
Pi + νPk Pi − Pj
and
n2j = 1 − n2i
(8.29)
384
Non-Linear Mechanics of Materials
The jump of the strain rate across the surface of normal n is also described by g: gi = (Pi − Pj )ni gj = (Pj − Pi )nj gk = 0 [[˙εi ]] = Pi + νPk [[˙εj ]] = Pj + νPk [[˙εk ]] = 0 the corresponding critical hardening modulus being H cr = −EPk2
(8.30)
where E is Young’s modulus. If n2i = 1 and nj = nk = 0,
H
cr
(Pj + νPk )2 + (1 + ν)Pk2 = −2μ 1−ν
The first possible band is given by the combination i, j, k for which H cr has the highest value.
Two-dimensional case The plane strain case can be solved using the three-dimensional analysis. If the deformation plane is normal to direction 3 p
e + ε˙ 33 ε˙ 33 = ε˙ 33
If plastic flow is large enough, and assuming that elastic strain remains small enough, the contribution ε˙ 33 can be neglected. Furthermore the direction 3 is supposed to be a principal direction of P∼ . As a result, we have almost P3 = 0 so that (8.30) becomes H dp = 0 We have also P1 + P2 = 0, so that (8.29) gives n21 = n22 =
1 2
The orientation of shear bands under plane strain conditions is 45◦ .
Strain localization phenomena
385
The plane stress case must be reexamined. The solid is considered two-dimensional. This means in particular that compatibility conditions in the third direction will not be ensured, which will promote in general earlier loss of ellipticity than for threedimensional solids. The condition ε33,12 = 0 holds only if ε33 is linear in x1 and x2 . are supposed to lie in plane 1–2. Lastly, the elasticity matrix in The vectors g and n ∼ ∼ the isotropic case takes a special form ⎡ ⎤ ⎡ ⎤⎡ e ⎤ ε11 σ11 1 ν 0 ⎢ e ⎥ ⎣ σ22 ⎦ = 2μ ⎣ ν 1 0 ⎦ ⎣ ε22 (8.31) ⎦ 1−ν e 0 0 1−ν σ12 ε12 In these conditions, the critical hardening modulus for a given orientation n is H cp (n) = 2μ(2(n.P∼ ).(n .n) − (n.P∼ .n)(n.n .n) ∼ ∼ − P∼ : n − ν(n.P∼ .n − Tr P∼ )(n.n .n − Tr n )) ∼ ∼ ∼ where n = [n1 n2
]T
(8.32)
and P∼ and Q are also two-dimensional, and g is given by ∼
g = 2P∼ .n − (1 + ν)(n.P∼ .n)n + ν(Tr P∼ )n We consider now only the case of incompressible associated plasticity, 2
2 2 2 2 2 2 2 2 H (n) = 2μ 2Pi ni − (Pi ni ) − Pi − ν ni Pi − Pi i=1
i=1
∂H = −4μ(1 + ν)(P1 − P2 )(n21 P1 + n22 P2 − P1 − P2 ) ∂n21 If P1 = P2 , H cr = −2μ(1 + ν)P12 and n is arbitrary. If P1 = P2 , n21 =
P1 P1 − P2
n22 = 1 − n21
(8.33)
g1 = (P1 − P2 )n1 g2 = −(P1 − P2 )n2 and H cp = 0 The jump is then proportional to [[˙ε1 ]] = P1 and [[˙ε2 ]] = P2 . The calculation proves that for plane stress also the critical hardening modulus for discontinuous bifurcations is generally zero. The two-dimensional solid is therefore more prone to localization than the real solid. The reason is that some bifurcation modes though incompatible (in the sense of Hadamard) in the direction 3 are compatible in the plane 1–2 and are then full discontinuous bifurcation modes for the 2D solid.
386
Non-Linear Mechanics of Materials
8.1.6. Stability Concerning stability we present Mandel’s approach [MAN66]. Let us consider a solid in a state of equilibrium. Perturbations of the equilibrium may be represented by plane waves u = f (n.x ± U t)c (8.34) One assumes that the initial equilibrium stress state is homogeneous. σ∼ denotes then the stress variation due to the perturbation. It follows that div σ∼ = ρ u¨ Furthermore it is assumed that the constitutive equations take the linear incremental is initially homogeneous and that it remains constant around the form (8.8), that D ∼ ∼
equilibrium state. In these conditions and taking (8.34) into account, the last equation becomes Dij kl uk,lj = u¨ i (8.35) ni Dij kl nl ck = ρU 2 ci i.e., Qc = ρU 2 c
(8.36)
∼
As a result plane waves and acceleration waves are governed by the same equations (compare (8.20) and (8.36)), even though they represent two distinct phenomena. The assumptions in the previous calculation correspond to the so-called “acoustic approximation” in fluid mechanics, which explains the denomination of tensor Q. When the ∼ eigenvalues of the acoustic tensor are real and positive, equilibrium is said to be stable because plane waves can propagate without growing. If they are real but negative, equilibrium is unstable with divergence of the perturbations. Complex eigenvalues lead to a phenomenon called flutter instability in aerodynamics.
8.1.7. Localization criteria Summary The various bifurcation criteria that have been established in the previous sections are summed up in the following figure. Two critical hardening moduli have been defined: H u corresponds to the possible loss of uniqueness and H cr is associated with the possible existence of discontinuous bifurcation modes. An additional critical hardening modulus can be determined for which general bifurcation modes become compatible, i.e., are of the form [[˙ε∼ ]] = { g ⊗ n} , and ε∼˙ : D : ε∼˙ = 0 ⇒ g.Q.g = g.{ Q} .g = 0 ∼ ∼
∼
∼
Strain localization phenomena
387
which corresponds to the loss of positive definiteness of the symmetrized acoustic tensor. This condition is called loss of strong ellipticity.
Accounting for thermal effects and thermomechanical couplings on localization phenomena is treated in [BEN91, BEN89b]. This leads in particular to a description of adiabatic shear bands [DUS93]. In the general case, the sign of the three critical hardening moduli can be positive or negative. In particular, bifurcation may occur while the material is hardening. In contrast, for associated plasticity at small strains, H u = 0 so that bifurcations are possible only for ideal plastic or strain-softening materials. Furthermore D and Q ∼ ∼
∼
are then symmetric tensors so that H se = H cr . In general bifurcation may occur before the extremum σ∼˙ = 0 of the loading curve. The associated hardening modulus H l for which det D = 0 corresponds to a limit point. General bifurcation modes ∼ ∼
may induce diffuse or localized deformation. It is difficult in practice to distinguish localized general bifurcations from discontinuous ones (shear bands strictly speaking). However one can assume that a general bifurcation which is also compatible in the sense of Hadamard will induce localized rather than diffuse deformation. That is why it is sometimes said that strong ellipticity loss criterion is a good criterion for localization of deformation.
Example: tensile test Bifurcation in the bulk We consider the example of an elastoplastic model with the von Mises criterion
388
Non-Linear Mechanics of Materials
f = J (σ∼ ) − R
with J (σ∼ ) =
3 s:s 2∼ ∼
s denoting the deviatoric part of σ∼ . In that case,
∼
D = − ∼ ∼ ∼
∼
2μ2 ∼s ⊗ ∼s H /3 + μ ∼s : ∼s
For isotropic elasticity, 1∼ is an eigentensor of D and the associated eigenvalue is 3k; ∼ ∼
s d∼ 1 = √ ∼ s∼ : ∼s
is an eigentensor for the eigenvalue ω1 = 2μH /(H + 3μ). The remaining eigentensors are in {1∼, ∼s }⊥ and the eigenvalue is 2μ. Accordingly, only one eigenvalue depends on the hardening modulus. Uniqueness is lost for ω1 = 0 = H . For tensile straining ⎡ ⎤ −1 0 0 1 ⎣ 0 2 0 ⎦ [d∼ 1 ] = √ 6 0 0 −1
(8.37)
This is an incompatible diffuse mode that gives rise to the necking phenomenon. We restrict ourselves to the tension of a plate to study the bifurcation modes in the plane 1–2. The mode (8.37) is assumed to occur within a band of normal n. The components of d 1 in the basis (m, n) (see next figure) are ⎡
2 sin2 θ − cos2 θ 1 ⎣ [d∼ 1 ] = √ 3 cos θ sin θ 6 0
3 cos θ sin θ 2 cos2 θ − sin2 θ 0
⎤ 0 0 ⎦ 1
Strain localization phenomena
389
The bifurcation mode becomes compatible in the sense of Hadamard in the 1–2 plane if 2 sin2 θ − cos2 θ = 0, i.e., tan2 θ =
1 2
θ = ±35.26◦
It means that deformation can localize within a band with the previous orientation. However, compatibility is not ensured in the third direction until the critical hardening modulus for discontinuous bifurcations is reached H cr = −E/4 after (8.30). But the orientation of the band is then, according to (8.29) n21 =
1+ν 3
which gives θ = 42◦ for ν = 0.33.
Bifurcation at the boundary We provide now an example of application of condition (iii) of Sect. 8.1.4 in the case of isotropic elasticity. The following expressions are useful: : P∼ .n = λ(Tr P∼ )n + 2μP∼ .n ∼ ∼
390
Non-Linear Mechanics of Materials
m. .n = μm ⊗ n + μn ⊗ m + λm.n1∼ ∼ ∼
(n. .n)−1 = − ∼ ∼
λ+μ n⊗n 1 + 1 μ(λ + 2μ) n.n μ n.n ∼
Without assuming right now that n and m are unit vectors, condition (iii) is written (n.n)2 (λ(Tr P∼ )m + 2μP∼ .m) = 2μ(m.n)(n.n)P∼ .n
λ 2μ 2 Tr P∼ (n.n) + (n.P∼ .n)(n.n) m +λ λ + 2μ λ + 2μ
λ λ+μ Tr P∼ (m.n)(n.n) − 2 (n.P∼ .n)(m.n) + 2μ λ + 2μ λ + 2μ + (m.P∼ .n)(n.n) n For an incompressible plastic flow Tr P∼ = 0, and for a tensile test ⎡
⎤ 0 0 ⎦ −a/2
−a/2 0 a P =⎣ 0 0 0 the plane bifurcation modes are given by sin θ = 0 or
tan2 θ =
2−ν 3λ + 4μ = 3λ + 2μ 1+ν
where θ denotes the angle between the normal n to the discontinuity surface and the normal m to the boundary. In the case of plane strain and neglecting elastic strain, we have ⎡
−a P =⎣ 0 0 Taking m =
1 0 0
0 a 0
⎤ 0 0 ⎦ 0
as free boundary, condition (iii) of (8.1.4) yields θ = ±45◦
This means that for plane strain, shear bands, regarded as bifurcation modes, undergo no deviation when approaching a free boundary.
Strain localization phenomena
391
8.1.8. Numerical simulation of some localization modes in elastoplasticity Introductory example: slope localization We consider a section of a soil with a slope and a rigid footing at the top. The vertical displacement of a node of the footing is prescribed. Finite elements calculations are performed at plane strain with a strain-softening elastoplastic behavior. Slightly after the load reaches a maximum, strain localizes within a thin curved layer (Fig. 8.2). This coincides with an abrupt load drop. In the following we will focus on shear bands formation triggered by the presence of geometrical or material defects.
Presentation of the calculations We consider the tension of a plate with a geometrical imperfection. The mesh corresponds to one quarter of the entire plate and loading and boundary conditions are described in Fig. 8.1. A thickness heterogeneity is introduced according to
m (x2 − y0 )2 −1 x1 = l 1 + a tanh L2
with L = 2, l = 1, a = 0.002, m = 25 and y0 = 2 or 1.5.
Figure 8.1. 2D plate in tension with an initial flaw
392
Non-Linear Mechanics of Materials
Figure 8.2. Strain localization in a softening plate
A classical phenomenological elastoplastic model has been chosen involving two hardening variables: Isotropic hardening and kinematic hardening. The constitutive equations are f = J (σ∼ − X ) − R(p) − σy ∼ λ˙ = p˙ =
2 p p ε˙ : ε˙ 3∼ ∼
R(p) = Q(1 − e−bp ) = X ∼
2 Cα 3 ∼
α ˙ = ε∼˙ p − D pα ˙∼ ∼
Strain localization phenomena
393
Figure 8.3. Tensile behavior of the sane material (on the left); Load drop due to strain localization in a plate with initial defect (on the right)
The following parameters have been chosen so that the material first hardens and then softens. The maximum is reached after small deformation (Fig. 8.3). E = 200000 MPa, σy = 102 MPa
ν = 0.3
Q = −100 MPa
C = 91800 MPa
b = 100
D = 800
The FE calculations are performed with the code ZéBuLoN, developed at Mines ParisTech. We use a local implicit integration method (θ -method). For global resolution we use also a Newton type algorithm. In the two-dimensional case, we use quadratic elements with eight nodes and nine integration points. In the three-dimensional case, we use 20-node bricks with 27 or 8 Gauss points.
2D plane stress In Fig. 8.3b, the behavior of the flawed plate in tension under plane stress conditions in elastoplasticity is compared to the homogeneous deformation of a plate without initial imperfection. Slightly after the maximum, the load-displacement curve displays an abrupt load drop and strain localizes within two narrow shear bands. Their orientation is that predicted in Sect. 8.1.5, namely 54.7◦ with respect to the tensile axis.
394
Non-Linear Mechanics of Materials
2D plane strain; dissipative structures We consider the problem of the tension of a plate under plane strain conditions. The geometrical imperfection is now located at three quarters of the height. Shortly after the maximum load but before the load drop, we observe several reflecting and intersecting bands of intense deformation (Fig. 8.4). This may be interpreted as a resonance between shear band orientation (45◦ as expected) and specimen geometry. This deformation mode can be regarded as a dissipative structure in the sense of Prigogine. After the load drop only three shear bands remain, constituting half of the previous pattern. Thus the initial imperfection has first induced a symmetric non-homogeneous deformation pattern and then triggered a symmetry-breaking localization mode.
Figure 8.4. Strain localization in a plate under plane strain conditions
3D case The plate is now a three-dimensional solid and has a one-element thickness. Elements are 20-node bricks with 8 Gauss points. Once again shear bands appear in a softening material with the same orientation but the bands are much wider and strain in them
Strain localization phenomena
395
Figure 8.5. Strain localization in a 3D plate
much lower than for the 2D plate (Fig. 8.5). The reason is that the observed localization mode is only compatible in the plane 1–3, and the incompatibility in direction 2 results in necking. That is why we obtain a more diffuse strain field.
8.2. Regularization methods 8.2.1. Mesh dependence The previous calculations have been carried out successively for 200, 800 and 1800 elements. The phenomenon exhibits a strong mesh dependence. The load-displacement curves of Fig. 8.7 prove that the finer the mesh, the later localization occurs, the size of the defect being unchanged. The following two features depend on mesh size: • localization time; • band width, always equal to one element. On the other hand, band orientation is independent of the mesh as long as one uses quadratic elements (linear elements usually lead to “hour-glass” phenomenon exemplified in Fig. 8.8). These difficulties are linked to ellipticity loss of the partial
396
Non-Linear Mechanics of Materials
Figure 8.6. Influence of mesh size on the band width
Figure 8.7. Strain localization in tension: Influence of mesh size on the global response
Strain localization phenomena
397
Figure 8.8. Localization and hourglass effect for linear elements
differential equations of the problem. Finite element calculation sorts out one possible solution of the problem! One can also show that the results of the calculation depend not only on the discretization but also on the algorithms used for local and global integration, on their precision, on the tolerance of the global residual. . . The influence of viscoplasticity on shear banding has been investigated. Two viscosity parameters are introduced in the previous model to provide an elastoviscoplastic model s−X 3 ε∼˙ p = p˙ ∼ ∼ 2 J (σ∼ − X ) ∼ with
p˙ =
J (σ∼ − X ) − R − σy ∼
n
K
The problem of the plate in tension is reconsidered with this model (strain-rate ε˙ = 10−3 s−1 ). The obtained shear bands are broader than for the rate-independent material and the strain reached inside the band is lower. Figure 8.9 shows the initiation and growth of a wide non-homogeneous deformation pattern. In contrast the profile of a shear band in a plastic material looks like a Dirac function. Figure 8.10 shows that
398
Non-Linear Mechanics of Materials
Figure 8.9. Initiation and growth of a shear band in the elastoplastic case (up) and viscoplastic (down), (increasing level numbers indicate an increasing level of global imposed displacement)
for a slightly viscous material (K = 10 MPa s1/n , n = 5), localization is still possible (corresponding to a sharp load drop) but higher viscosity delays and damps the load
Strain localization phenomena
399
Figure 8.10. Influence of viscosity on load drop in localization
Figure 8.11. Influence of mesh size in the viscoplastic case
drop (K = 50 MPa s1/n ). Viscoplasticity may even preclude shear band initiation. Moreover viscoplasticity significantly reduces mesh-dependence even for slightly viscous materials (Fig. 8.11, K = 10 MPa s1/n ). It subsists nevertheless, so that the problem of simulation of localization phenomena is not really solved by introducing spurious viscosity. The spurious mesh-dependence is due to the loss of ellipticity of the governing equilibrium equations in the static case and of hyperbolicity in the dynamic case.
400
Non-Linear Mechanics of Materials
Harirêche and Loret [HAR92] have shown that hyperbolicity is preserved if viscoplasticity is used in the dynamic case. It is not the case in statics. Any complement to a model that allows the preservation of well posedness is called a regularization method. It is imperative to use one of these methods to quantitatively analyse localization phenomena and avoid useless calculations. . . These methods have in common the introduction of one or more characteristic length(s) in the model, which is not the case in classical continuum mechanics (at least in the static and non-viscous case). Generalized continuum mechanics provides a systematic regularization framework with many variants: • Non-local models (as used in damage mechanics of concrete for example); • Introduction of higher gradients (non-simple materials); • Continuum theories with additional degrees of freedom. The last approach will be exemplified in the next section, through Cosserat mechanics.
8.2.2. Cosserat continuum at small deformation The general theory of Cosserat continua is presented at the end of Chap. 6 on finite strain. Here some definitions are recalled in the small deformation framework.
Kinematics A Cosserat continuum is a continuum where each material point is characterized by a displacement u(x) and an independent rotation (x). We define the deformation tensors: Cosserat deformation tensor: e∼ = grad u + . ∼ Wryness tensor: κ∼ = grad
Statics The balance of momentum and moment of momentum equations concerning force and couple-stress tensors are: divσ∼ + f = 0 ×
divμ + 2 σ +c = 0 ∼
Strain localization phenomena
and the boundary conditions:
401
σ∼ .n = T μ.n = M ∼
×
σ represents the vector associated with the skew-symmetric part of σ∼ . Elasticity We introduce an elastic potential (e∼, κ∼ ). In the linear case: (e∼, κ∼ ) =
1 1 : e∼ + κ∼ : C : κ∼ e∼ : ∼ ∼ 2 2 ∼ ∼
Thus, by deriving , we obtain the linear elastic constitutive equations: σ∼ = : e∼ ∼ ∼
μ=C : κ∼ ∼ ∼
∼
The expression of and C can be simplified in the isotropic case and we get: ∼ ∼ ∼
∼
σ∼ = λ Tr e∼1∼ + 2μ{ e∼} + 2μc } e∼{ μ = α Tr κ∼ 1∼ + 2β { κ∼ } + 2γ } κ∼ { ∼
From two constants in the case of classical continuum mechanics, we pass to six for isotropic Cosserat continua. We define a characteristic length associated with elasticity. β le = μ
Plasticity criterion We extend the von Mises criterion to Cosserat continua by taking into account the non-symmetric force stress tensor σ∼ and the existence of the couple stress tensor μ: ∼
f (σ∼ , μ) = J − R − σy ∼
with
(8.38)
3 a1 ∼s : ∼s + a2 ∼s : ∼s T + b1 μ : μ + b2 μ : μT ∼ ∼ ∼ ∼ 2 where R represents isotropic hardening. ∼s is the deviatoric part of the force stress tensor. Such a formulation has been proposed in [BOR91]. When the couple stresses J22 =
402
Non-Linear Mechanics of Materials
vanish and the force stresses are symmetric, the criterion coincide with the classical von Mises criterion provided that a1 + a2 = 1. We define a characteristic length lp attached to plastic curvature: a1 lp = (8.39) b1
Plastic curvature flow Total curvature has an elastic and a plastic contribution: κ∼ = κ∼ e + κ∼ p The flow rule is written κ∼˙ p = λ˙
∂f ∂μ ∼
Some analytical solutions of Cosserat plasticity problems based on this extension can be found in [FOR03b].
8.2.3. Elastoplastic Cosserat continuum and strain localization phenomena The problem consists in studying the behavior of a plate of a Cosserat continuum undergoing simple tension, as in Fig. 8.1. We have chosen the plate height H = 2, 4L to avoid the initiation of several shear bands that can appear after reflexion on the boundaries. The geometrical defect has been replaced by a material defect, namely a yield stress slightly lower than the one of other elements. The material has an isotropic linear elastic behavior and follows the von Mises criterion extended to Cosserat continuum (8.38) with a1 = a2 = 0.5, b1 = b2 and R(p) = σy + Q 1 − e−bp + Q 1 − e−b p For the whole study, we take σy = 102 MPa, Q = −100 MPa, Q = 114 MPa, b = 100, b = 800, to model the strain softening behavior. Classical elastic constants are E = 200000 MPa and ν = 0, 3. Other Cosserat parameters will be set during the study.
Study of the loading curve To analyse the influence of Cosserat parameters, we study first the global response (force-displacement).
Strain localization phenomena
403
Figure 8.12. Loading curves for various values of lp = le
Case where le = lp In a first step, we assume that the elastic le and plastic lp lengths are equal. These values vary from 0.015 to 0.035 mm. Figure 8.12 represents the behavior of the plate for these different values, compared to the classical and homogeneous cases. We note that response of the Cosserat continuum lies between classical and homogeneous responses. The load drop is less abrupt and takes place later than in the classical case. One could say as in [BOR91] that the shorter the characteristic length, the more brittle the material is (narrow bands) and the larger the characteristic length, the more ductile the material becomes (diffuse localization). Case where lp varies We study the problem by decoupling the intrinsic lengths. We consider that elastic length le is set at 0.025 and the plasticity characteristic length lp is varied from 0.01 to 0.05. Figure 8.13 represents the behavior for various values of lp . Conclusions are the same as in the previous case. By varying plastic length, we observe that the behavior tends to homogenize when lp increases. At last, we find that for a fixed value of lp , the behavior evolves marginally with respect to le . This suggests that the intrinsic length characteristic of plastic strain localization is the internal length lp .
Band width analysis After analyzing Cosserat parameters influence on the behavior of the structure, we look more precisely at the localization zone. The next figure shows that the shear band width does not depend on the mesh size (Fig. 8.14). We have analyzed the accumulated plastic strain curve on a sample section during loading. The curve has
404
Non-Linear Mechanics of Materials
Figure 8.13. Loading curves for various values of lp
Figure 8.14. Finite band width in a plate in tension
normally a peak when the section crosses the shear band and has a homogeneous threshold far from this band. To study the evolution of the band, we have then chosen to analyse the width corresponding to the half-height between the peak and the deformation minimum. We observe that this band width goes through a minimum at the onset of localization. This is the one plotted in the presented curves. Case le = lp We consider the influence of the Cosserat lengths on band width. Figure 8.15, represents band width vs. The Cosserat length lp . We observe that the curve is almost a straight line of slope 8. Variable lp In Fig. 8.16, we observe that when le is set to 0.025, band width is also a linear function of lp of approximate slope 7. Case where le varies Figure 8.17 represents band width evolution as a function of le for two values of lp . We notice that, for a low value of lp , band width is identical, whatever the elastic length is. In contrast, for a higher value, band width becomes a linear function of le of slope approximately 3.
Strain localization phenomena
405
Figure 8.15. Localization-band width for different values of lp = le
Figure 8.16. Band width for various values of lp
We can then conclude from this analysis that plastic internal length lp governs localization phenomena. The intrinsic plastic length lp allows us to determine the main features of the behavior and the order of magnitude of the shear band width. The Cosserat continuum represents an efficient regularization especially for granular media for localization modes involving significant rotations. The more general micromorphic continuum invented by Mindlin and Eringen [ERI99, ENG03, PEE04, FOR06b] considers deformable triads of directors instead of the rigid one of the
406
Non-Linear Mechanics of Materials
Figure 8.17. Band width for different values of le
Cosserat medium. Its application to localization phenomena is presented in [BES04]. In particular the formation of crushing or tearing bands is simulated by means of the micromorphic model in [DIL06].
Appendix
Notation used
A.1. Tensors scalar (0th rank) vector (1st rank) 2nd-rank tensor 3rd-rank tensor 4th-rank tensor
a a a∼ a∼ or a ∼ A ∼
nth-rank tensor (n > 4)
A(n)
∼
Contracted product: ., :, ::, etc. x = a.b x = a∼ . b x∼ = a∼ . b∼ x = a∼ : b∼ x∼ = A : b∼ ∼
x = ai bi xi = aij bj xij = aik bkj x = aij bij xij = Aij kl bkl
x=A :: B ∼ ∼
x = Aij kl Bij kl
∼
∼
∼
Dyadic product: ⊗, ⊗ , etc. x∼ = a ⊗ b X = a∼ ⊗ b∼ ∼
xij = aj bj Xij kl = aij bkl
X = a∼ ⊗ b∼ ∼
Xij kl = aik bj l
X = a∼ ⊗ b∼ ∼
Xij kl = ail bkj
∼ ∼ ∼
J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7,
408
Non-Linear Mechanics of Materials
Special tensors 1
unit tensor: 1∼, 1∼ or sometimes I∼ , I∼
J∼
tensor such that: J∼ : a∼ = deviator(a∼ )
K ∼
tensor such that: K : a∼ = ∼
∼
∼
∼
∼
∼
∼
1 3
Tr(a∼ )1∼
A.2. Vectors, matrices {v} [M]
vector (any dimension n) matrix (any dimension n × m)
Contracted products x = {v} . {w} {x} = [M] . {v} x = [M] : [N ]
x = vi vj xj = Mij vj x = Mij Nij
Dyadic product [X] = {v} ⊗ {w}
Xij = vi wj
A.3. Voigt notation To denote second and fourth-rank tensors, as well as to represent them numerically, we use Voigt’s notation that consists in noting second-rank tensors as vectors and fourthrank tensors as matrices. To note explicitly the use of such a notation, we will use the following conventions: Second-rank tensor {a∼ } Fourth-rank tensor
] [A ∼ ∼
Usual Voigt notation distinguish the stress and strain tensors (symmetric case): ⎧ ⎧ ⎫ ⎫ ε11 σ11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε σ22 ⎪ ⎪ ⎪ ⎪ ⎪ 22 ⎪ ⎪ ⎪ ⎨ ⎨ ⎪ ⎬ ⎬ ε33 σ33 and σ∼ → (A.1) ε∼ → γ12 = 2ε12 ⎪ σ12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ23 = 2ε23 ⎪ σ23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎪ ⎭ ⎭ γ31 = 2ε31 σ31
Notation used
409
We can also use another notation that allows us to work on the algebra of symmetric second-rank tensors: ⎧ ⎫ a11 ⎪ ⎪ ⎪ ⎪ a ⎪ ⎪ ⎪ 22 ⎪ ⎪ ⎪ ⎪ ⎨ a ⎪ ⎬ 33 (A.2) a∼ → √ 2a12 ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ 2a ⎪ ⎪ ⎪ ⎪ ⎩√ 23 ⎪ ⎭ 2a31 This solution allows a uniform notation. Moreover, we check that: a∼ : b∼ = {a∼ }.{b∼}
(A.3)
Remarks Using an object-oriented language that authorizes operator overloading allows us to manipulate tensors directly [BES98a]. The second notation fits best this methodology. Fourth-rank tensors must be written differently according to the kind of notation used for second-rank tensors.
Bibliography [ABO88] M. A BOUAF, J.L. C HENOT, G. R AISSON, and P. BAUDUIN. Finite element simulation of hot isostatic pressing of metal powders. Int. J. Numer. Methods Eng., 25(1):191–212, 1988. [ACH93] P. ACHON. Comportement et tenacité d’alliages d’aluminium. PhD thesis, École Nationale Supérieure des Mines de Paris, 1993. [ALL90] O. A LLIX, P. L ADEVÈZE, E. L E DANTEC, and E. V ITTECOQ. Damage Mechanics for composite laminates under complex loading. In J. B OEHLER, editor, Yielding, Damage and Failure of Anisotropic Solids, pages 551–569. Mechanical Engineering Publications, London, 1990. [ALL94] L. A LLAIS, M. B ORNERT, T. B RETHEAU, and D. C ALDEMAISON. Experimental characterization of the local strain field in a heterogeneous elastoplastic material. Acta Mater., 42:3865–3880, 1994. [ANA96] L. A NAND and M. KOTHARI. A computational procedure for rate-independent crystal plasticity. J. Mech. Phys. Solids, 44:525–558, 1996. [AND86] S. A NDRIEUX, Y. BAMBERGER, and J.-J. M ARIGO. Un modèle de matériau microfissuré pour les bétons et les roches. J. Méc. Théor. Appl., 5(3):471–513, 1986. [ARG75] A.S. A RGON. Constitutive Equations in Plasticity. MIT Press, Cambridge, 1975. [ARM66] P.J. A RMSTRONG and C.O. F REDERICK. A mathematical representation of the multiaxial Bauschinger effect. Technical Report RD/B/N731, CEGB, 1966. [ARN97] S. A RNDT, B. S VENDSEN, and D. K LINGBEIL. Modellierung der Eigenspannungen and der Rißspitze mit einem Schägigungsmodell. Tech. Mech., 17(4):323–332, 1997. [ASA77] R.J. A SARO and J.R. R ICE. Strain localization in ductile single crystals. J. Mech. Phys. Solids, 25:309–338, 1977. [ASA83a] R.J. A SARO. Crystal plasticity. J. Appl. Mech., 50:921–934, 1983. [ASA83b] R.J. A SARO. Micromechanics of crystals and polycrystals. Adv. Appl. Mech., 23:1– 115, 1983. [ASA85] R.J. A SARO and A. N EEDLEMAN. Texture development and strain hardening in rate dependent polycrystals. Acta Metall., 33:923–953, 1985. [ASH80] M.F. A SHBY and D.R.H. J ONES. Engineering Materials, volume 1: An Introduction to Their Properties and Applications. Pergamon, Elmsford, 1980. [AUB99] M. AUBERTIN, O.M.L. YAHYA, and M. J ULIEN. Modeling mixed hardening of alkali halides with a modified version of an internal state variables model. Int. J. Plast., 15:1067–1088, 1999. J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7,
412
Non-Linear Mechanics of Materials
[AUR94] [BAL65] [BAR01a]
[BAR01b]
[BAS89]
[BAT82] [BAT91] [BAX94] [BAZ88] [BEC86] [BEC88] [BEN87] [BEN89a] [BEN89b]
[BEN91] [BER79] [BER81a] [BER81b]
[BER97] [BER01]
F. AURICCHIO and R.L. TAYLOR. A generalized elastoplastic plate theory and its algorithmic implementation. Int. J. Numer. Methods Eng., 37:2583–2608, 1994. A. BALTOV and A. S AWCZUK. A rule of anisotropic hardening. Acta Mech., 1:81– 92, 1965. F. BARBE, L. D ECKER, D. J EULIN, and G. C AILLETAUD. Intergranular and intragranular behavior of polycrystalline aggregates, part 1: FE model. Int. J. Plast., 17(4):513–536, 2001. F. BARBE, S. F OREST, and G. C AILLETAUD. Intergranular and intragranular behavior of polycrystalline aggregates, part II: results. Int. J. Plast., 17(4):537–566, 2001. S. BASTE, R. E L G UERJOUMA, and A. G ÉRARD. Mesure de l’endommagement anisotrope d’un composite céramique-céramique par une méthode ultrasonore. Rev. Phys. Appl., 24:721–731, 1989. K.J. BATHE. Finite Element Procedures in Engineering Analysis. Prentice Hall, New York, 1982. J.L. BATOZ and G. D HATT. Modélisation des Structures par Éléments Finis, I–III. Hermès, Paris, 1991. C. BAXEVANAKIS. Comportement statistique à rupture des composites stratifiés. PhD thesis, École Nationale Supérieure des Mines de Paris, 1993. Z.P. BAZANT and G. P IJAUDIER -C ABOT. Nonlocal continuum damage, localization, instability and convergence. J. Appl. Mech., 55:287–294, 1988. R. B ECKER and A. N EEDLEMAN. Effect of yield surface curvature on necking and failure in porous plastic solids. J. Appl. Mech., 53:491–499, 1986. R. B ECKER, A. N EEDLEMAN, O. R ICHMOND, and V. T VERGAARD. Void growth and failure in notched bars. J. Mech. Phys. Solids, 36:317–351, 1988. A. B ENALLAL and D. M ARQUIS. Constitutive equations for nonproportional cyclic elasto-visco-plasticity. J. Eng. Mater. Technol., 109:326, 1987. A. B ENALLAL. Thermoviscoplasticité et endommagement des structures. Doctorat d’Etat, Université Pierre et Marie Curie, Paris 6, 1989. A. B ENALLAL, R. B ILLARDON, and G. G EYMONAT. Conditions de bifurcation à l’intérieur et aux frontières pour une classe de matériaux non-standards. C. R. Acad. Sci. Paris, Série II, 308:893–898, 1989. A. B ENALLAL. Quelques remarques sur le rôle du couplage thermomécanique dans les phénomènes de localisation. C. R. Acad. Sci. Paris, Sér. II, 312:117–122, 1991. M. B ERVEILLER and A. Z AOUI. An extension of the self-consistent scheme to plastically flowing polycrystal. J. Mech. Phys. Solids, 26:325–344, 1979. F.M. B EREMIN. Cavity formation from inclusions in ductile fracture. Metall. Trans., 12A:723–731, 1981. F.M. B EREMIN. Study of fracture criteria for ductile rupture of a 508 steel. In D. F RANÇOIS et al., editors, International Conference of Fracture, Advances in Fracture Researches, Cannes. Pergamon, Elmsford, 1981. M. B ERVEILLER and F.D. F ISCHER, editors. Mechanics of Solids with Phase Change. CISM Courses and Lectures, volume 368. Springer, Berlin, 1997. A. B ERTRAM and B. S VENDSEN. On material objectivity and reduced constitutive equations. Arch. Mech., 53:653–675, 2001.
Bibliography
[BER05] [BES88]
[BES92] [BES97] [BES98a] [BES98b]
[BES04] [BIG91] [BIS51] [BLA87]
[BOE78] [BOR91] [BOR01] [BOU96]
[BOU01]
[BUI69]
[BUR87]
[BUS77]
413
A. B ERTRAM. Elasticity and Plasticity of Large Deformations. Springer, Berlin, 2005. J.-M. B ESSON, M. JAEGER, and O. D ÉBORDES. Homogénéisation de composites à base de tissus de fibres. In D. G AY and C. B ORD, editors, Composite Structures, pages 77–90, 1988. J. B ESSON and M. A BOUAF. Rheology of porous alumina and simulation of hot isostatic pressing. J. Am. Ceram. Soc., 75:2165–2172, 1992. J. B ESSON and R. F OERCH. Large scale object-oriented finite element code design. Comput. Methods Appl. Mech. Eng., 142:165–187, 1997. J. B ESSON and R. F OERCH. Object-oriented programming applied to the finite element method, part I: general concepts. Rev. Eur. Élém. Finis, 7(5):535–566, 1998. J. B ESSON, R. L E R ICHE, R. F OERCH, and G. C AILLETAUD. Object-oriented programming applied to the finite element method, part II: application to material behaviors. Rev. Eur. Élém. Finis, 7(5):567–588, 1998. J. B ESSON. Local Approach to Fracture. Les Presses de l’Ecole des Mines de Paris, Paris, 2004. D. B IGONI and T. H UECKEL. Uniqueness and localization, I: associative and nonassociative elastoplasticity. Int. J. Solids Struct., 312:117–122, 1991. J.F.W. B ISHOP and R. H ILL. A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philos. Mag., 42:414–427, 1951. D. B LANC and J.L. S TRUDEL. The Portevin-Le Chatelier instability: on a heterogeneous flow with a microscopic origin. In Proceedings of the International Symposium on Metallurgy and Materials Sciences: Constitutive Relations and Their Physical Basis, pages 227–238, Risø, Holland, 1987. J.P. B OEHLER. Lois de comportement anisotropes des milieux continus. J. Méc., 17(2):153–190, 1978. R. DE B ORST. Simulation of strain localization: a reappraisal of the Cosserat continuum. Eng. Comput., 8:317–332, 1991. M. B ORNERT, T. B RETHEAU, and P. G ILORMINI. Homogénéisation en Mécanique des Matériaux. Hermès, Paris, 2001. A. B OURSIN, J.-L. C HABOCHE, and F. ROUDOLFF. Mécanique de l’endommagement avec conditions unilatérales et stockage d’énergie élastique. C. R. Acad. Sci. Paris, Sér. II, 323:369–376, 1996. F. B OUYGE, I. JASIUK, and M. O STOJA -S TARZEWSKI. A micromechanically based couple-stress model of an elastic two-phase composite. Int. J. Solids Struct., 38:1721–1735, 2001. H.D. B UI. Etude de l’évolution de la frontière du domaine élastique avec l’écrouissage et relation de comportement élastoplastique des métaux cubiques. PhD thesis, Université Pierre et Marie Curie, Paris 6, 1969. H. B URLET and G. C AILLETAUD. Modeling of cyclic plasticity in finite element codes. In C.S. D ESAI, E. K REMPL, P.D. K IOUSIS, ET AL ., editors, International Conference on Constitutive Laws for Engineering Materials: Theory and Applications, pages 1157–1164, Tucson, AZ, 1987. D. B USHNEL. A strategy for the solution of problems involving large deflections, plasticity and creep. Int. J. Numer. Methods Eng., 11:683–708, 1977.
414
Non-Linear Mechanics of Materials
[BUS96] [CAI79]
[CAI80] [CAI84]
[CAI87]
[CAI92] [CAI94] [CAI95] [CAI00]
[CAI03]
[CAL97]
[CHA74] [CHA77]
[CHA78]
[CHA79a]
E.P. B USSO and F.A. M C C LINTOCK. A dislocation mechanics-based crystallographic model of a B2-type intermetallic alloy. Int. J. Plast., 12:1–28, 1996. G. C AILLETAUD and J.-L. C HABOCHE. Macroscopic description of the microstructural changes induced by varying temperature: example of IN100 cyclic behaviour. In K.J. M ILLER and R.F. S MITH, editors, 3rd International Conference on Mechanical Behaviour of Metals, pages 23–32, Cambridge, UK, 1979. G. C AILLETAUD, H. P OLICELLA, and G. BAUDIN. Mesure de déformation et d’endommagement par mesure électrique. Rech. Aérospatiale, 1:69–75, 1980. G. C AILLETAUD, H. K ACZMAREK, and H. P OLICELLA. Some elements on multiaxial behaviour of 316L stainless steel at room temperature. Mech. Mater., 3:333– 347, 1984. G. C AILLETAUD. Une approche micromécanique phénoménologique du comportement inélastique des métaux. PhD thesis, Université Pierre et Marie Curie, Paris 6, 1987. G. C AILLETAUD. A micromechanical approach to inelastic behaviour of metals. Int. J. Plast., 8:55–73, 1992. G. C AILLETAUD and P. P ILVIN. Utilisation de modèles polycristallins pour le calcul par éléments finis. Rev. Eur. Élém. Finis, 3(4):515–541, 1994. G. C AILLETAUD and K. S AÏ. Study of plastic/viscoplastic models with various inelastic mechanisms. Int. J. Plast., 11:991–1005, 1995. G. C AILLETAUD, C. D EPOID, D. M ASSINON, and E. N ICOULEAU -B OURLES. Elastoviscoplasticity with aging in aluminium alloys. In G.A. M AUGIN, R. D ROUOT and F. S IDOROFF, editors, Continuum Thermomechanics: The Art and Science of Modelling Material Behaviour (Paul Germain’s Anniversary Volume), pages 75–86. Solid Mechanics and Its Applications. Kluwer Academic, Dordrecht, 2000. G. C AILLETAUD, S. F OREST, D. J EULIN, F. F EYEL, I. G ALLIET, V. M OUNOURY, and S. Q UILICI. Some elements of microstructural mechanics. Comput. Mater. Sci., 27:351–374, 2003. S. C ALLOCH and D. M ARQUIS. Additional hardening due to tension–torsion nonproportional loadings: influence of the loading path shape. In S. K ALLURI and P.J. B ONNACUSE, editors, Multiaxial Fatigue and Deformation Testing Techniques, pages 113–130, 1997. J.-L. C HABOCHE. Une loi différentielle d’endommagement de fatigue avec cumulation non linéaire. Revue Fr. Méc., 50–51:71–82, 1974. J.-L. C HABOCHE. Sur l’utilisation des variables d’état interne pour la description de la viscoplasticité cyclique avec endommagement. In Problèmes Non Linéaires de Mécanique, Symposium Franco-Polonais de Rhéologie et Mécanique, pages 137– 159, 1977. J.-L. C HABOCHE. Description phénoménologique de la viscoplasticité cyclique avec endommagement. Doctorat d’Etat, Université Pierre et Marie Curie, Paris 6, Paris, Juin 1978. J.-L. C HABOCHE. Le concept de contrainte effective appliqué à l’élasticité et à la viscoplasticité en présence d’un endommagement anisotrope. In J.-P. B OEHLER, editor, Colloque EUROMECH 115, pages 737–760. Editions du CNRS, Grenoble, 1979.
Bibliography
415
[CHA79b] J.-L. C HABOCHE, K. DANG VAN, and G. C ORDIER. Modelisation of the strain memory effect on the cyclic hardening of 316 stainless steel. In Proceedings SMIRT, volume 5, page L11/3. Berlin, 1979. [CHA81] J.-L. C HABOCHE. Continuous damage mechanics: a tool to describe phenomena before crack initiation. Nucl. Eng. Des., 64:233–247, 1981. [CHA86] A. C HAREWICZ and I.M. DANIEL. Fatigue damage mechanisms and residual properties of graphite/epoxy laminates. Eng. Fract. Mech., 25:793–808, 1986. [CHA88] J.-L. C HABOCHE. Continuum damage mechanics, parts I and II. J. Appl. Mech., 55:59–72, 1988. [CHA89] J.-L. C HABOCHE. Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int. J. Plast., 5:247–302, 1989. [CHA91] J.-L. C HABOCHE. On some modifications of kinematic hardening to improve the description of ratcheting effect. Int. J. Plast., 7:661–678, 1991. [CHA92a] J.-L. C HABOCHE. Damage induced anisotropy: on the difficulties associated with the active/passive unilateral condition. Int. J. Damage Mech., 1(2):148–171, 1992. [CHA92b] J.-L. C HABOCHE. A new unilateral condition for the description of material behaviour with anisotropic damage. C. R. Acad. Sci. Paris, Sér. II, 314:1395–1401, 1992. [CHA93a] J.-L. C HABOCHE. Cyclic viscoplastic constitutive equations, part I: a thermodynamically consistent formulation. J. Appl. Mech., 60:813–821, 1993. [CHA93b] J.-L. C HABOCHE. Development of continuum damage mechanics for elastic solids sustaining anisotropic and unilateral damage. Int. J. Damage Mech., 2:311–329, 1993. [CHA93c] J.-L. C HABOCHE. Cyclic viscoplastic constitutive equations, parts I & II. J. Appl. Mech., 60:813–831, 1993. [CHA94] J.-L. C HABOCHE. Modeling of ratchetting: evaluation of various approaches. Eur. J. Mech. A, Solids, 13:501–518, 1994. [CHA95] J.-L. C HABOCHE, N. E L -M AYAS, and P. PAULMIER. Modélisation thermodynamique des phénomènes de viscoplasticité, restauration et vieillissement. C. R. Acad. Sci. Paris, Sér. II, 320:9–16, 1995. [CHA96a] J.-L. C HABOCHE. Unified Cyclic Viscoplastic Constitutive Equations: Development, Capabilities and Thermodynamic Framework, pages 1–68. Academic Press, New York, 1996. [CHA96b] J.-L. C HABOCHE and G. C AILLETAUD. Integration methods for complex plastic constitutive equations. Comput. Methods Appl. Mech. Eng., 133:125–155, 1996. [CHA97] J.-L. C HABOCHE, R. G IRARD, and P. L EVASSEUR. On the interface debonding models. Int. J. Damage Mech., 6:220–257, 1997. [CHA02] J.-L. C HABOCHE and J.F. M AIRE. A new micromechanics based CDM model and its application to CMC’s. Aerospace Sci. Technol., 6(2):131–145, 2002. [CHI69] G.Y. C HIN and W.L. M AMMEL. Generalization and equivalence of the minimum work (Taylor) and maximum work (Bishop–Hill) principles for crystal plasticity. Trans. Metall. Soc. AIME, 245:1211–1214, 1969. [CHO87] C.L. C HOW and J. WANG. An anisotropic theory of continuum damage mechanics for ductile fracture. Eng. Fract. Mech., 27(5):547–558, 1987.
416
Non-Linear Mechanics of Materials
[CHO89] C.L. C HOW and T.J. L U. On evolution laws of anisotropic damage. Eng. Fract. Mech., 34:679–701, 1989. [CHR76]
M. C HRZANOWSKI. Use of the damage concept in describing creep-fatigue interaction under prescribed stress. J. Mech. Sci., 18:69–73, 1976.
[CHR92]
A. C HRYSOCHOOS and F. B ELMAHJOUB. Thermographic analysis of thermomechanical couplings. Arch. Mech., 44:55–68, 1992.
[CHR98]
A. C HRYSOCHOOS and H. L OUCHE. Analyse thermographique des mécanismes de localisation dans des aciers doux. C. R. Acad. Sci. Paris, Sér. II, 326:345–352, 1998.
[COL67]
B.D. C OLEMAN and M.E. G URTIN. Thermodynamics with internal variables. J. Chem. Phys., 47:597–613, 1967.
[CON89] E. C ONTESTI and G. C AILLETAUD. Description of creep-plasticity interaction with non-unified constitutive equations. Nucl. Eng. Des., 116:265–280, 1989. [COR79]
J.-P. C ORDEBOIS and F. S IDOROFF. Anisotropie élastique induite par endommagement. In J.-P. B OEHLER, editor, Colloque EUROMECH 115. Editions du CNRS, Grenoble, 1979.
[COR82]
J.-P. C ORDEBOIS and F. S IDOROFF. Endommagement anisotrope en élasticité et plasticité. J. Méc. Théor. Appl., Numéro spécial:45–60, 1982.
[COU91] O. C OUSSY. Mécanique des Milieux Poreux. Technip, Paris, 1991. [CUI92]
A.M. C UITINO and M. O RTIZ. Computational modelling of single crystals. Model. Simul. Mater. Sci. Eng., 1:225–263, 1992.
[DAR87]
F. DARVE. Manuel de Rhéologie des Géomatériaux. Presses des Ponts et Chaussées, Paris, 1987.
[DEC98]
K. D ECAMP, L. BAUVINEAU, J. B ESSON, and A. P INEAU. Size and geometry effects on ductile rupture of notched bars in a C–Mn steel: experiments and modelling. Int. J. Fract., 88(1):1–18, 1998.
[DEL87]
P. D ELOBELLE and C. OYTANA. Modeling of 316 stainless steel mechanical properties using biaxial experiments, part I: experiments and basis of the model; part II: model and simulation. J. Press. Vessel Technol., 109:149, 1987.
[DEL95]
P. D ELOBELLE, P. ROBINET, and L. B OCHER. Experimental study and phenomenological modelization of ratchet under uniaxial and biaxial loading on an austenitic stainless steel. Int. J. Plast., 11:295–330, 1995.
[DEL96]
P. D ELOBELLE, P. ROBINET, P. G EYER, and P. B OUFFIOUX. A model to describe the anisotropic viscoplastic behavior of zircaloy-4 tubes. J. Nucl. Mater., 238:135– 162, 1996.
[DES89]
C.S. D ESAI and Q.S.E. H ASHMI. Analysis, evaluation and implementation of a nonassociative model for geologic materials. Int. J. Plast., 5(4):397–420, 1989.
[DEV97a] L. D EVILLERS -G UERVILLE. Rupture d’aciers inoxydables austéno-ferritiques moulés, fragilisés par vieillissement à 350–400◦ . PhD thesis, École Nationale Supérieure des Mines de Paris, 1997. [DEV97b] L. D EVILLERS -G UERVILLE, J. B ESSON, and A. P INEAU. Notch fracture toughness of a cast duplex stainless steel: modelling of experimental scatter and size effect. Nucl. Eng. Des., 168:211–225, 1997.
Bibliography
[DIL06]
[DOE95] [DOG86] [DOG95] [DOG00] [DOL00] [DOQ90] [DOU03] [DRA76] [DUS93] [DUV72] [EBE98] [ENG03]
[ERI99] [ERI02] [EST98]
[FAV95]
[FAV99]
[FOE97]
[FOR95a]
417
T. D ILLARD, S. F OREST, and P. I ENNY. Micromorphic continuum modelling of the deformation and fracture behaviour of nickel foams. Eur. J. Mech. A, Solids, 25:526–549, 2006. E. D OEGE, T. E L -D SOKI, and D. S EIBERT. Prediction of necking and wrinkling in sheet-metal forming. J. Mater. Process. Technol., 50:197–206, 1995. A. D OGUI and F. S IDOROFF. Rhéologie anisotrope en grandes déformations. In Rhéologie des Matériaux Anisotropes, pages 27–38. Cepadues, Toulouse, 1986. I. D OGHRI and R. B ILLARDON. Investigation of localization due to damage in elasto-plastic materials. Mech. Mater., 19:129–149, 1995. I. D OGHRI. Mechanics of Deformable Solids. Linear and Nonlinear, Analytical and Computational Aspects. Springer, Berlin, 2000. I. D OLTSIRIS, editor. Elements of Plasticity, Theory and Computation. WIT Press, Southampton, 2000. V. D OQUET and A. P INEAU. Extra-hardening due to non-proportional loading of an austenitic stainless steel. Scr. Metall. Mater., 24:433, 1990. P. D OUMALIN, M. B ORNERT, and J. C RÉPIN. Caractérisation de la répartition de la déformation dans les matériaux hétérogènes. Méc. Ind., 4:607–617, 2003. A. D RAGON. On phenomenological description of rock-like materials with account for kinetics of brittle fracture. Arch. Mech. Stosow., 28(1):13–30, 1976. M.K. D USZEK -P ERZYNA and P. P ERZYNA. Adiabatic shear band localization in elastic-plastic single crystals. Int. J. Solids Struct., 30(1):61–89, 1993. G. D UVAUT and J.-L. L IONS. Les Inéquations en Mécanique et en Physique. Dunod, Paris, 1972. F. E BERL, F. F EYEL, S. Q UILICI, and G. C AILLETAUD. Approches numériques de la plasticité cristalline. J. Phys. IV, 8:Pr4-15–Pr4-25, 1998. R.A.B. E NGELEN, M.G.D. G EERS, and F.P.T. BAAIJENS. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. International Journal of Plasticity, 19:403–433, 2003. A.C. E RINGEN. Microcontinuum Field Theories. Springer, New York, 1999. A.C. E RINGEN. Nonlocal Continuum Field Theories. Springer, New York, 2002. Y. E STRIN, L.S. T ÓTH, A. M OLINARI, and Y. B RÉCHET. A dislocationbased model for all hardening stages in large strain deformation. Acta Mater., 46(15):5509–5522, 1998. N. FAVROT. Etudes des différentes étapes de l’élaboration de structures graduelles en Wc-Co : compactage à froid, frittage, homogénéisation. PhD thesis, École Nationale Supérieure des Mines de Paris, 1995. N. FAVROT, J. B ESSON, C. C OLIN, and F. D ELANNAY. Cold compaction and solid state sintering of wc–co based structures: experiments and modeling. J. Am. Ceram. Soc., 82(5):1153–1161, 1999. R. F OERCH, J. B ESSON, G. C AILLETAUD, and P. P ILVIN. Polymorphic constitutive equations in finite element codes. Comput. Methods Appl. Mech. Eng., 141:355–372, 1997. S. F OREST and G. C AILLETAUD. Strain localization in single crystals: effect of boundaries and interfaces. Eur. J. Mech. A, Solids, 14(5):747–771, 1995.
418
Non-Linear Mechanics of Materials
[FOR95b] S. F OREST and P. P ILVIN. Modelling the cyclic behaviour of two-phase single crystal nickel-base superalloys. In A. P INEAU and A. Z AOUI, editors, Micromechanics of Plasticity and Damage of Multiphase Materials. Proceedings of the IUTAM Symposium, Sèvres, Paris, France, 29 August–1 September 1995, pages 51–58. Kluwer Academic, Dordrecht, 1995. [FOR97] S. F OREST. Homogenization methods and the mechanics of generalized continua. In G.A. M AUGIN, editor, International Seminar on Geometry, Continuum and Microstructure, Paris, France, 28–29 May 1997. Collection Travaux en Cours, volume 60, pages 35–48. Hermann, Paris, 1997. [FOR98a] S. F OREST. Mechanics of generalized continua: construction by homogenization. J. Phys. IV, 26:Pr4-39–Pr4-48, 1998. [FOR98b] S. F OREST and K. S AB. Cosserat overall modeling of heterogeneous materials. Mech. Res. Commun., 25:449–454, 1998. [FOR99] S. F OREST and P. P ILVIN. Modelling finite deformation of polycrystals using local objective frames. Z. Angew. Math. Mech., 79:199–202, 1999. [FOR00] S. F OREST, F. BARBE, and G. C AILLETAUD. Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multiphase materials. Int. J. Solids Struct., 37:7105–7126, 2000. [FOR01] S. F OREST, P. B OUBIDI, and R. S IEVERT. Strain localization patterns at a crack tip in generalized single crystal plasticity. Scr. Mater., 44:953–958, 2001. ˇ [FOR03a] S. F OREST and R. S EDLÁCEK . Plastic slip distribution in two-phase laminate microstructures: dislocation-based vs. generalized-continuum approaches. Philos. Mag. A, 83:245–276, 2003. [FOR03b] S. F OREST and R. S IEVERT. Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech., 160:71–111, 2003. [FOR05] S. F OREST. Generalized continua. In K.H.J. B USCHOW, R.W. C AHN, M.C. F LEMINGS, B. I LSCHNER, E.J. K RAMER and S. M AHAJAN, editors, Encyclopedia of Materials: Science and Technology updates, pages 1–7. Elsevier, Oxford, 2005. [FOR06a] S. F OREST. Milieux Continus Généralisés et Matériaux Hétérogènes. Les Presses de l’Ecole des Mines de Paris, Paris, 2006. [FOR06b] S. F OREST and R. S IEVERT. Nonlinear microstrain theories. Int. J. Solids Struct., 43:7224–7245, 2006. [FRA80] P. F RANCIOSI, M. B ERVEILLER, and A. Z AOUI. Latent hardening in copper and aluminium single crystals. Acta Metall., 28:273, 1980. [FRA83] G. F RANCFORT. Homogenization and linear thermoelasticity. SIAM J. Math. Anal., 14:696–708, 1983. [FRA85] P. F RANCIOSI. On flow and work hardening expression correlations in metallic single crystal plasticity. Rev. Phys. Appl., 33:1601–1612, 1985. [FRA91] D. F RANÇOIS, A. P INEAU, and A. Z AOUI. Comportement Mécanique des Matériaux. Volume 1 : Élasticité et Élastoplasticité. Hermès, Paris, 1991. [FRA93] D. F RANÇOIS, A. P INEAU, and A. Z AOUI. Comportement Mécanique des Matériaux. Volume 2 : Endommagement, Mécanique de la Rupture, Mécanique du Contact. Hermès, Paris, 1993.
Bibliography
419
[FRA98]
D. F RANÇOIS, A. P INEAU, and A. Z AOUI. Mechanical Behaviour of Materials, volumes I and II. Kluwer, Dordrecht, 1998.
[FRI64]
J. F RIEDEL. Dislocations. Pergamon, Elmsford, 1964.
[GAI81]
B.K.D. G AIROLA and E. K RÖNER. A simple formula for calculating the bounds and the self-consistent value of the shear modulus of polycrystalline aggregates of cubic crystals. Int. Eng.Sci., 19:865–869, 1981.
[GEE01]
M.G.D. G EERS, V. KOUZNETSOVA, and W.A.M. B REKELMANS. Gradientenhanced computational homogenization for the micro-macro scale transition. J. Phys. IV, 11:Pr5-145–Pr5-152, 2001.
[GER73a] P. G ERMAIN. La méthode des puissances virtuelles en mécanique des milieux continus, première partie : théorie du second gradient. J. Méc., 12:235–274, 1973. [GER73b] P. G ERMAIN. Mécanique des Milieux Continus. Masson, Paris, 1973. [GER83]
P. G ERMAIN, Q.S. N GUYEN, and P. S UQUET. Continuum thermodynamics. J. Appl. Mech., 50:1010–1020, 1983.
[GOL83]
G.H. G OLUB and C.F. VAN L OAN. Matrix Computations. The John Hopkins University Press, Baltimore, 1983.
[GOL94]
M. G OLOGANU, J.B. L EBLOND, and J. D EVAUX. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. Trans. ASME, J. Eng. Mater. Technol., 116:290–297, 1994.
[GOL97]
M. G OLOGANU, J.B. L EBLOND, and J. D EVAUX. Recent extensions of Gurson’s model for porous ductile metals. In P. S UQUET, editor, Continuum Micromechanics. CISM Courses and Lectures, volume 377. Springer, Berlin, 1997.
[GOO84] A.M. G OODMAN. Development of constitutive equations for computer analysis of stainless steel components. Nucl. Eng. Des., 83:349, 1984. [GRE72]
R.J. G REEN. A plasticity theory for porous solids. Int. J. Mech. Sci., 14:215–224, 1972.
[GUI92]
C. G UIONNET. Modeling of ratchetting in biaxial experiments. J. Eng. Mater. Technol., 114:56–62, 1992.
[GUI98]
C. G UILLEMER -N EEL, X. F EAUGAS, V. B OBET, and M. C LAVEL. Damage kinetics in ferritic spheroidal graphite cast iron. In ECF 12, Fracture from Defect, volume II, pages 721–726, 1998.
[GUR77] A.L. G URSON. Continuum theory of ductile rupture by void nucleation and growth, part I: yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol., 44:2–15, 1977. [HAD03] J. H ADAMARD. Leçons sur la Propagation des Ondes et les Équations de l’Hydrodynamique. Hermann, Paris, 1903. [HAL75]
B. H ALPHEN and Q.S. N GUYEN. Sur les matériaux standards généralisés. J. Méc., 14:39–63, 1975.
[HAL87]
B. H ALPHEN and J. S ALENÇON. Élasto-plasticité. Presses des Ponts et Chaussées, Paris, 1987.
[HAL96]
F.R. H ALL, D.R. H AYHURST, and P.R. B ROWN. Prediction of plane-strain creep crack growth using Continuum Damage Mechanics. Int. J. Damage Mech., 5(4):353–402, 1996.
420
Non-Linear Mechanics of Materials
[HAN92] N.R. H ANSEN and H.L. S CHREYER. Thermodynamically consistent theories for elastoplasticity coupled with damage. In Damage Mechanics and Localization. AMD, volume 142, 53–167, ASME, New York, 1992. [HAR92] O. H ARIRÊCHE and B. L ORET. 3D dynamic strain-localization: shear band pattern transition in solids. Eur. J. Mech. A, Solids, 11(6):735–751, 1992. [HAS94a] T. H ASSAN and S. K YRIAKIDES. Ratcheting of cyclically hardening and softening materials, part I: uniaxial behavior. Int. J. Plast., 10:149–184, 1994. [HAS94b] T. H ASSAN and S. K YRIAKIDES. Ratcheting of cyclically hardening and softening materials, part I: multiaxial behavior. Int. J. Plast., 10:185–212, 1994. [HAU00] P. H AUPT. Continuum Mechanics and Theory of Materials. Springer, Berlin, 2000. [HAV92] K.S. H AVNER. Finite Plastic Deformation of Crystalline Solids. Cambridge University Press, Cambridge, 1992. [HAY72] D.R. H AYHURST. Creep rupture under multiaxial state of stress. J. Mech. Phys. Solids, 20(6):381–390, 1972. [HAY75] D.R. H AYHURST, P.R. D IMMER, and M.W. C HERNUKA. Estimates of the creep rupture lifetime of structures using the finite element method. J. Mech. Phys. Solids, 23:335, 1975. [HER90] E. H ERVÉ and A. Z AOUI. Modelling the effective behaviour of non-linear matrixinclusion composites. Eur. J. Mech. A, Solids, 9:505–515, 1990. [HEU95] P. H EUILLET, P. ROUMAGNAC, and L. D UGAUTIER. Modélisation du comportement hyperélastique des élastomères. Mécamat-Info, 15:15–21, 1995. [HIL58] R. H ILL. A general theory of uniqueness and stability in elastic plastic solids. J. Mech. Phys. Solids, 6:236–249, 1958. [HIL65] R. H ILL. Continuum micro-mechanisms of elastoplastic polycrystals. J. Mech. Phys. Solids, 13:89–101, 1965. [HIL66] R. H ILL. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J. Mech. Phys. Solids, 14:95–102, 1966. [HIL72] R. H ILL and J.R. R ICE. Constitutive analysis of elastic–plastic crystals at arbitrary strains. J. Mech. Phys. Solids, 20:401–413, 1972. [HIL89] R. H ILL. The Mathematical Theory of Plasticity. Oxford Engineering Science Series, volume 11, 1989. [HOM01] M. H OMMEL and O. K RAFT. Deformation behavior of thin copper films on deformable substrates. Acta Mater., 49:3935–3947, 2001. [HUE85] C. H UET. Eléments de thermomécanique des matériaux hétérogènes. In Cours de DEA de Génie Mécanique, ENPC, 1985. [HUE90] C. H UET. Application of variational concepts to size effects in elastic heterogeneous bodies. J. Mech. Phys. Solids, 38:813–841, 1990. [HUG80] T.J.R. H UGHES. Generalization of selective integration procedures to anisotropic and nonlinear media. Int. J. Numer. Methods Eng., 15:1413–1418, 1980. [HUG87] T.J.R. H UGHES. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, New York, 1987. [HUL79] J. H ULT. Continuum damage mechanics. Capabilities limitations and promises. In Mechanisms of Deformation and Fracture, pages 233–347. Pergamon, Oxford, 1979.
Bibliography
[ILC86] [JAO85] [JIA94] [JON77] [JOU99] [JU89] [KAC58] [KAC80] [KAC93]
[KAL92]
[KAN82] [KAN03]
[KAN06]
[KIM98] [KOC66] [KOC98] [KOI60] [KOP88] [KRA81] [KRA84]
421
L.B. I LCEWICZ, M.N.L. NARASIMHAN, and J.B. W ILSON. Micro and macro material symmetries in generalized continua. Int. J. Eng. Sci., 24:97–109, 1986. B. JAOUL. Etude de la Plasticité et Application aux Métaux. Dunod, Paris, 1985. Y. J IANG and H. S EHITOGLU. Multiaxial cyclic ratchetting under multiple step loading. Int. J. Plast., 10:849–870, 1994. J.J. J ONAS and B.B. BAUDELET. Effect of crack and cavity generation on tensile stability. Acta Metall., 25:43–50, 1977. H. J OUSSET. Comportement en fatigue et relaxation à basse température d’un alliage base titane. PhD thesis, École Nationale Supérieure des Mines de Paris, 1999. J.W. J U. On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct., 25(7):803–833, 1989. L.M. K ACHANOV. Time of the rupture process under creep conditions. Isv. Akad. Nauk. SSR, Otd. Tekh. Nauk, 8:26–31, 1958. M. K ACHANOV. Continuum model of medium with cracks. J. Eng. Mech. Div., 106(EM5):1039–1051, 1980. M. K ACHANOV. Elastic solids with many cracks and related problems. In J. H UTCHINSON and T. W U, editors, Advances in Appl. Mech., volume 30, pages 259–445. Academic Press, New York, 1993. S.R. K ALIDINDI, C.A. B RONKHORST, and L. A NAND. Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solids, 40:536–569, 1992. K.I. K ANATANI. Plasticity theory for the kinematics of ideal granular materials. Int. J. Eng. Sci., 20(1):1–13, 1982. T. K ANIT, S. F OREST, I. G ALLIET, V. M OUNOURY, and D. J EULIN. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct., 40:3647–3679, 2003. T. K ANIT, F. N GUYEN, S. F OREST, D. J EULIN, M. R EED, and S. S INGLETON. Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng., 195:3960–3982, 2006. A.S. K IM, J. B ESSON, and A. P INEAU. Global and local approaches to fracture normal to interfaces. Int. J. Solids Struct., 36:1845–1864, 1998. U.F. KOCKS and T.J. B ROWN. Latent hardening in aluminium. Acta Metall., 14:87–98, 1966. U.F. KOCKS, C.N. T OMÉ, and H.R. W ENK. Texture and Anisotropy. Cambridge University Press, Cambridge, 1998. W.T. KOITER. In R.H.I.N. S NEDDON, editor, General Theorems for Elastic– Plastic Solids, volume 6, pages 167–221. North-Holland, Amsterdam, 1960. J. KOPLIK and A. N EEDLEMAN. Void growth and coalescence in porous plastic solids. Int. J. Solids Struct., 835–853, 1988. D. K RAJCINOVIC and G.U. F ONSEKA. The continuous damage theory of brittle materials, parts 1 and 2. J. Appl. Mech., 48:809–824, 1981. D. K RAJCINOVIC. Continuum damage mechanics. Appl. Mech. Rev., 37(1):1–6, 1984.
422
Non-Linear Mechanics of Materials
[KRE89] [KRI77] [KRÖ61] [KRÖ76] [KRÖ77] [KRÖ81]
[LAD80] [LAD83] [LAD84] [LAD94] [LAD95] [LAM78] [LEB95] [LEC74] [LEC80] [LEM78] [LEM84] [LEM85a] [LEM85b] [LEM96] [LIP90]
[LIU94]
W. K REHER and W. P OMPE. Internal Stresses in Heterogeneous Solids. AkademieVerlag, Berlin, 1989. R.D. K RIEG and D.B. K RIEG. Accuracies of numerical solution for the elasticperfectly plastic model. J. Press. Vessel Technol., 99:510–515, 1977. E. K RÖNER. Zur plastischen Verformung des Vielkristalls. Acta Metall., 9:155– 161, 1961. E. K RÖNER and H. KOCH. Effective properties of disordered materials. SM Arch., 1:183–238, 1976. E. K RÖNER. Bounds for effective moduli of disordered materials. J. Mech. Phys. Solids, 25:137–155, 1977. E. K RÖNER. Linear properties of random media—the systematic theory. In C. H UET and A. Z AOUI, editors, Comportements Rhéologiques et Structure des Matériaux, pages 15–40. Editions ENPC, Paris, 1981. P. L ADEVÈZE. Sur la théorie de la plasticité en grandes déformations. Rapport Interne LMT No. 9, 1980. P. L ADEVÈZE. Sur une théorie de l’endommagement anisotrope. Rapport Interne 34, Laboratoire de Mécanique et Technologie, Cachan, 1983. P. L ADEVÈZE and J. L EMAITRE. Damage effective stress in quasi-unilateral condition. IUTAM Congress, Lyngby, Danemark, 1984. P. L ADEVÈZE, A. G ASSER, and O. A LLIX. Damage mechanics modelling for ceramic composites. J. Eng. Mater. Technol., 16, 1994. P. L ADEVÈZE. A damage computational approach for composites: basic aspects and micromechanical relations. Comput. Mech., 17:142–150, 1995. H.S. L AMBA and O.M. S IDEBOTTOM. Cyclic plasticity for non-proportional paths, parts 1 and 2. J. Eng. Mater. Technol., 100:96–112, 1978. J.B. L EBLOND, G. P ERRIN, and J. D EVAUX. An improved Gurson-type model for hardenable ductile metals. Eur. J. Mech. A, Solids, 14(4):499–527, 1995. F.A. L ECKIE and D.R. H AYHURST. Creep rupture of structures. Proc. R. Soc. Lond., 340:323–347, 1974. F.A. L ECKIE and E.T. O NAT. Tensorial nature of damage measuring internal variables. In Physical Non-Linearities in Structural Analysis. Springer, Berlin, 1980. J. L EMAITRE and J.-L. C HABOCHE. Aspect phénoménologique de la rupture par endommagement. J. Méc. Appl., 2(3):317–365, 1978. J. L EMAITRE. How to use damage mechanics. Nucl. Eng. Des., 80:233–245, 1984. J. L EMAITRE. A continuum damage mechanics model for ductile fracture. J. Eng. Mater. Technol., 107:83–89, 1985. J. L EMAITRE and J.-L. C HABOCHE. Mécanique des Matériaux Solides. Dunod, Paris, 1985. J. L EMAITRE. A Course on Damage Mechanics. Springer, Berlin, 1996. P. L IPINSKI, J. K RIER, and M. B ERVEILLER. Elastoplasticité des métaux en grandes déformations : comportement global et évolution de la structure interne. Rev. Phys. Appl., 25:361–388, 1990. Y. L IU, S. M URAKAMI, and Y. K ANAGAWA. Mesh-dependence and stress singularity in finite element analysis of creep crack growth by continuum damage mechanics approach. Eur. J. Mech. A, Solids, 13(3):395–417, 1994.
Bibliography
[LUB90] [LUB93] [LUE84] [MAC95] [MAI96] [MAI97a] [MAI97b] [MAL80] [MAN65] [MAN66] [MAN72] [MAN78] [MAR89]
[MAT79] [MAT94]
[MAU80] [MAU92] [MAZ82] [MAZ86]
[MAZ89] [MCC66]
423
J. L UBLINER. Plasticity Theory. Mc Millan, New York, 1990. V.A. L UBARDA and D. K RAJCINOVIC. Damage tensors and the crack density distribution. Int. J. Solids Struct., 30:2859–2877, 1993. D.G. L UENBERGER. Linear and Nonlinear Programming. Addison-Wesley, Reading, 1984. D.L. M AC D OWELL. Stress state dependence of cyclic ratcheting behavior of two rail steels. Int. J. Plast., 11:397–421, 1995. J.-F. M AIRE and D.D. PACOU. Essais de Traction-Compression-Torsion sur Tubes Composites Céramique-Céramique, pages 1225–1234. AMAC, Paris, 1996. J.-F. M AIRE and J.-L. C HABOCHE. A new formulation of continuum damage mechanics for composite materials. Aerospace Sci. Technol., 4:247–257, 1997. J.-F. M AIRE and P.-M. L ESNE. A damage model for ceramic matrix composites. Aerospace Sci. Technol., 4:259–266, 1997. D.S. M ALKUS. Finite elements with penalties in non linear elasticity. Int. J. Numer. Methods Eng., 16:121–136, 1980. J. M ANDEL. Une généralisation de la théorie de la plasticité de W.T. Koiter. Int. J. Solids Struct., 1:273–295, 1965. J. M ANDEL. Cours de Mécanique des Milieux Continus, tomes I et II. GauthierVillars, Paris, 1966 (réédition 1994, J. Gabay). J. M ANDEL. Plasticité Classique et Viscoplasticité. CISM Courses and Lectures, volume 97. Springer, Berlin, 1972. J. M ANDEL. Propriétés Mécaniques des Matériaux. Eyrolles, Paris, 1978. D. M ARQUIS. Phénoménologie et thermodynamique : couplage entre thermoélasticité, plasticité, vieillissement et endommagement. PhD thesis, Université Pierre et Marie Curie, Paris 6, 1989. H. M ATTHIES and G. S TRANG. The solution of non-linear finite element equations. Int. J. Numer. Methods Eng., 14:1613–1626, 1979. K.K. M ATHUR, A. N EEDLEMAN, and V. T VERGAARD. Ductile failure analyses on massively parallel computers. Comput. Methods Appl. Mech. Engrg., 119:283–309, 1994. G.A. M AUGIN. The method of virtual power in continuum mechanics: application to coupled fields. Acta Mech., 35:1–70, 1980. G.A. M AUGIN. Thermomechanics of Plasticity and Fracture. Cambridge University Press, Cambridge, 1992. J. M AZARS. Mechanical damage and fracture of concrete structures. In Advances in Fracture Research, volume 4, pages 1499–1506. Pergamon, Elmsford, 1982. J. M AZARS. A model of a unilateral elastic damageable material and its application to concrete. In Fracture Toughness and Fracture Energy of Concrete. Elsevier, Lausanne, 1986. J. M AZARS and G. P IJAUDIER -C ABOT. Continuum damage theory: application to concrete. J. Eng. Mech., 115(2):345–365, 1989. F.A. M C C LINTOCK and A.S. A RGON. Mechanical Behavior of Materials. Addison-Wesley, Reading, 1966.
424
Non-Linear Mechanics of Materials
[MCG80] P.A. M C G UIRE, S.G. S AMPATH, C.H. P OPELAR, and M.F. K ANNINEN. A theoretical model for crack propagation and crack arrest in pressurized pipelines. In G.T. H AHN and M.F. K ANNINEN, editors, Crack Arrest Methodology and Applications, pages 341–358, 1980. [MEA85] M.E. M EAR and J.W. H UTCHINSON. Influence of yield surface curvature on flow localization in dilatant plasticity. Mech. Mater., 4:395–407, 1985. [MEN68] A. M ENDELSON. Plasticity: Theory and Application. Macmillan, New York, 1968. [MOL87] A. M OLINARI, G.R. C ANOVA, and S. A HZI. A self-consistent approach to the large deformation polycrystal viscoplasticity. Acta Metall., 35:2983–2994, 1987. [MOR83] J.-J. M OREAU. In P. G ERMAIN and B. NAYROLLES, editors, Application of Convex Analysis to the Treatment of Elastoplastic Systems. Springer, Swansea, 1983. [MRÓ67] Z. M RÓZ. On the description of anisotropic work hardening. J. Mech. Phys. Solids, 15:163, 1967. [MRÓ95] Z. M RÓZ, D. W EICHERT and S. D OROSZ, editors. Inelastic Behaviour of Structures Under Variable Loads. Kluwer Academic, Dordrecht, 1995. [MÜH98] U. M ÜHLICH and R. K IENZLER. Nonlocal modification of the Gurson-model. In A. B ERTRAM and F. S IDOROFF, editors, Euromech—Mecamat’97, 2nd European Mechanics of Materials Conference on Mechanics of Materials with Intrinsic Length Scale: Physics, Experiments, Modelling and Applications, pages 277–284, 1998. [MUN94] E. M UNIER. Identification expérimentale et modélisation statistique multi-échelle du comportement mécanique de composites SiC/SiC tissés. PhD thesis, École Nationale Supérieure des Mines de Paris, 1994. [MUR80] S. M URAKAMI and N. O HNO. A continuum theory of creep and creep damage. In A.R.S. P ONTER and D.R. H AYHURST, editors, 3rd Creep in Structures Symposium, Leicester, pages 422–443. Springer, Berlin, 1980. [MUR83a] S. M URAKAMI. Notion of continuum damage mechanics and its application to anisotropic creep damage theory. J. Eng. Mater. Technol., 105:99, 1983. [MUR83b] A.I. M URDOCH. On material frame-indifference, intrinsic spin and certain constitutive relations motivated by the kinetic theory of gases. Arch. Ration. Mech. Anal., 83:185–194, 1983. [MUR87] T. M URA. Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht, 1987. [MUR03] A.I. M URDOCH. Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Contin. Mech. Thermodyn., 15:309–320, 2003. [NAG74] J.C. NAGTEGALL, D.M. PARKS, and J.R. R ICE. On numerically accurate finite element solutions in the fully plastic range. Comput. Methods Appl. Mech. Eng., 4:153–177, 1974. [NEE90] A. N EEDLEMAN. An analysis of decohesion along an imperfect interface. Int. J. Fract., 42:21–40, 1990. [NEM80] S. N EMAT-NASSER and A. S HOKOOH. On finite plastic flow of compressible materials with internal friction. Int. J. Solids Struct., 16:495–514, 1980. [NEM98] S. N EMAT-NASSER and Y. L I. Flow stress of FCC polycrystals with application to OFHC Cu. Acta Mater., 46(2):565–577, 1998.
Bibliography
425
[NGU77] Q.S. N GUYEN. On the elastic–plastic initial boundary problem and its numerical integration. Int. J. Numer. Methods Eng., 11:817–832, 1977. [NOU90] D. N OUAILHAS. Un modèle de viscoplasticité cyclique pour matériaux anisotropes à symétrie cubique. C. R. Acad. Sci. Paris, Sér. II, 310:887–890, 1990. [NOU92] D. N OUAILHAS and G. C AILLETAUD. Comparaison de divers critères anisotropes pour monocristaux cubiques à faces centrées (CFC). C. R. Acad. Sci. Paris, Sér. II, 315:1573–1579, 1992. [NOU94] D. N OUAILHAS, J.-P. C ULIÉ, G. C AILLETAUD, and L. M ÉRIC. FE analysis of the stress-strain behaviour of single-crystal tubes. Eur. J. Mech. A, Solids, 14(1):137– 154, 1994. [NOU95] D. N OUAILHAS and G. C AILLETAUD. Tension-torsion behavior of single-crystal superalloys: experiment and finite element analysis. Int. J. Plast., 11(4):451–470, 1995. [NYG03] M. N YGÅRDS. Number of grains necessary to homogenize elastic materials with cubic symmetry. Mech. Mater., 35:1049–1057, 2003. [ODE72] J.T. O DEN. Finite Elements on Nonlinear Continua. McGraw-Hill, New York, 1972. [ODE78] J.T. O DEN. A theory of penalty methods for finite element approximations of highly nonlinear problems in continuum mechanics. Comput. Struct., 8:445–449, 1978. [ODE83] J.T. O DEN and G. C AREY. Finite Elements, volume IV: Mathematical Aspects. Prentice-Hall, Englewood Cliffs, 1983. [OGD97] R.W. O GDEN. Non-Linear Elastic Deformations. Dover, New York, 1984–1997. [OHA83] Y. O HASHI, M. K AWAI, and T. M OMOSE. Effect of prior plasticity on subsequent creep of type 316 stainless steel at elevated temperature. J. Eng. Mater. Technol., 105:257–263, 1983. [OHN93a] N. O HNO and J.D. WANG. Kinematic hardening rules with critical state for the activation of dynamic recovery, part I: formulation and basic features for ratchetting behaviour. Int. J. Plast., 9:375–390, 1993. [OHN93b] N. O HNO and J.D. WANG. Kinematic hardening rules with critical state for the activation of dynamic recovery, part II: application to experiments of ratchetting behaviour. Int. J. Plast., 9:390–403, 1993. [ORT85a] M. O RTIZ. A constitutive theory for the inelastic behavior of concrete. Mech. Mater., 4:67–93, 1985. [ORT85b] M. O RTIZ and E.P. P OPOV. Accuracy and stability of integration algorithms for elastoplastic constitutive equations. Int. J. Numer. Methods Eng., 21:1561–1576, 1985. [PAR04] R. PARISOT, S. F OREST, A. P INEAU, F. G RILLON, X. D ÉMONET, and J.-M. M ATAIGNE. Deformation and damage mechanisms of zinc coatings on galvanized steel sheets, part I: deformation modes. Metall. Mater. Trans., 35A:797–811, 2004. [PEE96] R.H.J. P EERLINGS, R. D E B ORST, W.A.M. B REKELMANS, J.H.P. D E V REE, and I. S PEE. Some observations on localisation in non-local and gradient damage models. Eur. J. Mech. A, Solids, 15(6), 1996. [PEE04] R.H.J. P EERLINGS, T.J. M ASSART, and M.G.D. G EERS. A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking. Comput. Methods Appl. Mech. Engrg., 193:3403–3417, 2004.
426
Non-Linear Mechanics of Materials
D. P IERCE, R.J. A SARO, and A. N EEDLEMAN. Material rate dependence and localized deformation in crystalline solids. Acta Metall., 31:1951, 1985. [PIL90a] P. P ILVIN. Approches multiéchelles pour la prévision du comportement inélastique des métaux. PhD thesis, Université Pierre et Marie Curie, Paris 6, 1990. [PIL90b] P. P ILVIN and G. C AILLETAUD. Intergranular and transgranular hardening in viscoplasticity. In M. Z YCZKOWSKI, editor, Creep in Structures, volume 4, pages 171– 178, 1990. [PIL94] P. P ILVIN. The contribution of micromechanical approaches to the modelling of inelastic behaviour. In A. P INEAU, G. C AILLETAUD and T. L INDLEY, editors, 4th International Conference on Biaxial/Multiaxial Fatigue, Saint-Germain, France, 31 May–3 June 1994, volume 1, pages 31–46. ESIS, Paris, 1994. [PIN76] A. P INEAU. Influence of uniaxial stress on the morphology of coherent precipitates during coarsening—elastic energy considerations. Acta Metall., 24:559–564, 1976. [PIN81] A. P INEAU. Review of fracture mechanisms and local approaches to predicting crack resistance in low strength steels. In D. F RANÇOIS et al., editor, International Conference of Fracture, Advances in Fracture Researches. Pergamon, Elmsford, 1981. [PLU86] A. P LUMTREE and J.O. N ILSSON. Damage mechanics applied to high temperature fatigue. In Journées Internationales de Printemps, Fatigue at High Temperature, Paris, 1986. [POI96] N. P OINT and E. S ACCO. A delamination model for laminated composites. Int. J. Solids Struct., 33(4):483–509, 1996. [PON98] P. P ONTE C ASTAÑEDA and P. S UQUET. Nonlinear composites. Adv. Appl. Mech., 34:171–302, 1998. [PRA98] F. P RAT, M. G RANGE, J. B ESSON, and E. A NDRIEU. Behavior and rupture of hydrided zircaloy-4 tubes and sheets. Metall. Trans., 29A:1643–1651, 1998. [PRE88] W.H. P RESS, R.O. F OSCHI, S.A. T EUKOLSKY, and W.T. V ETTERLING. Numerical Recipes in C. Cambridge University Press, Cambridge, 1988. [RAB69] Y.N. R ABOTNOV. Creep Problems in Structural Members. North-Holland, Amsterdam, 1969. [RAM90] S. R AMTANI. Contribution à la modélisation du comportement multiaxial du béton endommagé avec description du caractère unilatéral, Thèse de doctorat, Université Pierre et Marie Curie, Paris 6, 1990. [REN02] Z.-Y. R EN and Q.-S. Z HENG. A quantitative study of minimum sizes of representative volume elements of cubic polycrystals-numerical experiments. J. Mech. Phys. Solids, 50:881–893, 2002. [RIC68] J.R. R ICE. A path independent integral and the approximate analysis of strain concentration by notched and cracks. J. Appl. Mech., 35:379, 1968. [RIC69] J.R. R ICE and D.M. T RACEY. On the enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids, 17:201–217, 1969. [RIC70] J.R. R ICE. On the structure of stress-strain relations for time-dependent plastic deformation in metals. J. Appl. Mech., 37:728, 1970. [RIC71] J.R. R ICE. Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids, 19:433–455, 1971. [PIE85]
Bibliography
[RIC73]
[RIC76] [RIK79] [RIV69]
[ROU85a]
[ROU85b] [ROU87] [ROU97] [RUN89] [SAA88]
[SAA89]
[SAA92]
[SAB92] [SAI95] [SAL84] [SCH65] [SCH79] [SHA93]
[SHI76] [SID96]
427
J.R. R ICE and D.M. T RACEY. Computational fracture mechanics. In S.J. F ENVES, editor, Symposium on Numerical Methods in Structural Mechanics, Urbana, IL. Academic Press, New York, 1973. J.R. R ICE. The localization of deformation. In W.T. KOITER, editor, Theoretical and Applied Mechanics. North Publishing Company, Philadelphia, 1976. E. R IKS. An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct., 15:529–551, 1979. R.S. R IVLIN. Red herrings and sundry unidentified fish in non-linear mechanics. Technical Report No. CAM-100-9, Centre for the Application of Mathematics, Lehigh University, 1969. G. ROUSSELIER, J.-J. E NGEL, and J.C. M ASSON. Etude comparative de modèles de comportement pour la simulation d’essais en traction-pression sur tubes en acier inoxydable. Technical Report 8, EDF-DER/GIS Rupture à Chaud, 1985. M. ROUSSET. Surface seuil de plasticité : détermination automatique et modélisation. PhD thesis, Université Pierre et Marie Curie, Paris 6, 1985. G. ROUSSELIER. Ductile fracture models and their potential in local approach of fracture. Nucl. Eng. Des., 105:97–111, 1987. P. ROUGÉE. Mécanique des Grandes Transformations. Springer, Berlin, 1997. K. RUNESSON and Z. M RÓZ. A note on non-associated flow rules. Int. J. Plast., 5:639–658, 1989. K. S AANOUNI. Sur l’analyse de la fissuration des milieux élasto-viscoplastiques par la théorie de l’endommagement continu. Doctorat d’Etat, Université de Technologie de Compiègne, 1988. K. S AANOUNI, J.-L. C HABOCHE, and P.-M. L ESNE. On the creep crack growth prediction by a nonlocal damage formulation. Eur. J. Mech. A, Solids, 8(6):437–459, 1989. K. S AANOUNI and P.-M. L ESNE. Sur la description phénoménologique des déformations anélastiques dans les composites endommageables. C. R. Acad. Sci. Paris, Sér. II, 315:1165–1170, 1992. K. S AB. On the homogenization and the simulation of random materials. Eur. J. Mech. A, Solids, 11:505–515, 1992. K. S AI and G. C AILLETAUD. Study of plastic/viscoplastic models with various inelastic mechanisms. Int. J. Plast., 11(8):991–1005, 1995. J. S ALENÇON. Viscoélasticité. Presses des Ponts et Chaussées, Paris, 1984. A. S CHOFIELD and P. W ROTH. Critical States Soil Mechanics. McGraw-Hill, New York, 1965. H.L. S CHREYER, R.L. K ULAK, and J.M. K RAMER. Accurate numerical solutions for elastic–plastic models. J. Press. Vessel Technol., 101:226–234, 1979. G.X. S HAN, O. KOLEDNIK, F.D. F ISHER, and H.P. S TÜWE. A 2D model for numerical investigations of stable crack growth in thick smooth fracture mechanics specimens. Eng. Fract. Mech., 45(1), 1993. S. S HIMA and M. OYANE. Plasticity theory for porous metals. Int. J. Mech. Sci., 18:285–291, 1976. F. S IDOROFF. Formalisme grandes déformations en hyperélasticité et viscoélasticité. In G. C AILLETAUD, editor, Colloque National Mécamat Aussois, 1996.
428
Non-Linear Mechanics of Materials
R. S IEVERT. Eine Systematik für elastisch-plastische Stoffgleichungen. PhD thesis, Dissertation, Schriftenreihe Phys. Ing.-Wiss., Band 23, TU Berlin, 1992. [SIE98a] T. S IEGMUND and W. B ROCKS. Tensile decohesion by local failure criteria. Tech. Mech., 18(4):261–270, 1998. [SIE98b] R. S IEVERT, S. F OREST, and R. T ROSTEL. Finite deformation Cosserat-type modelling of dissipative solids and its application to crystal plasticity. J. Phys. IV, 8:Pr8357–Pr8-364, 1998. [SIM85a] J.C. S IMO and R.L. TAYLOR. Consistent tangent operators for rate independent elasto-plasticity. Comput. Methods Appl. Mech. Eng., 48:101–118, 1985. [SIM85b] J.C. S IMO, R.L. TAYLOR, and K.S. P ISTER. Variational and projection methods for the volume constraint in finite deformation elastoplasticity. Comput. Methods Appl. Mech. Eng., 51:177–208, 1985. [SIM86] J.C. S IMO and R.L. TAYLOR. Return mapping algorithm for plane stress elastoplasticity. Int. J. Numer. Methods Eng., 22:649–670, 1986. [SIM87] J.C. S IMO and J.W. J U. Strain- and stress-based continuum damage models, I: formulation. Int. J. Solids Struct., 23(7):821–840, 1987. [SIM91] J.C. S IMO and S. G OVINDJEE. Nonlinear B-stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity. Int. J. Numer. Methods Eng., 31:151–176, 1991. [SIM97] J.C. S IMO and T.R.J. H UGHES. Computational Inelasticity. Springer, Berlin, 1997. [SIQ93] C. S IQUIERA. Développement d’un essai biaxial sur plaques composites et utilisation pour la modélisation du comportement d’un matériau SMC. PhD, Université de Franche-Comté, 1993. [SKR98] J. S KRZYPEK and A. G ANCZARSKI. Modeling of Material Damage and Failure of Structures. Springer, Berlin, 1998. [STA98] A. S TAROSELSKI and L. A NAND. Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. J. Mech. Phys. Solids, 46(4):671–696, 1998. [STE97] A.C. S TEENBRINK, E. VAN DER G IESSEN, and P.D. W U. Void growth in glassy polymers. J. Mech. Phys. Solids, 45(3):405–437, 1997. [SUO92] Z. S UO, M. O RTIZ, and A. N EEDLEMAN. Stability of solids with interfaces. J. Mech. Phys. Solids, 40:613–640, 1992. [SUQ97] P. S UQUET, editor. Continuum Micromechanics. CISM Courses and Lectures, volume 377. Springer, Berlin, 1997. [SZA87] I. S ZABO. Geschichte der mechanischen Prinzipien. Birkhäuser, Basel, 1987. [TAB97] L. TABOUROT, M. F IVEL, and E. R AUCH. Generalised constitutive laws for FCC single crystals. Mater. Sci. Eng. A, 234–236:639–642, 1997. [TAH99] S. TAHERI and E. L ORENTZ. An elastic–plastic constitutive law for the description of uniaxial and multiaxial ratchetting. Int. J. Plast., 15:1159–1180, 1999. [TAL85] R. TALREJA. A continuum mechanics characterization of damaged composite materials. Proc. R. Soc. Lond., 195–216, 1985. [TAN70] K. TANAKA and T. M ORI. The hardening of crystals by non-deforming particles and fibres. Acta Metall., 18:931–941, 1970. [SIE92]
Bibliography
429
[TAY38a] G.I. TAYLOR. Analysis of Plastic Strain in a Cubic Crystal, pages 218–224. Mc Millan, New York, 1938. [TAY38b] G.I. TAYLOR. Plastic strain in metals. J. Inst. Met., 62:307–324, 1938. [TEO76] C. T EODOSIU and F. S IDOROFF. A theory of finite elastoviscoplasticity of single crystals. Int. J. Eng. Sci., 14:165–176, 1976. [TEO93] C. T EODOSIU, J.-L. R APHANEL, and L. TABOUROT. Finite element simulation of the large elastoplastic deformation of multicrystals. In C. T EODOSIU, J.-L. R APHANEL and F. S IDOROFF, editors, 5th MECAMAT International Seminar on Large Plastic Deformations: Fundamental Aspects and Applications to Metal Forming, Fontainebleau, France, pages 153–168. Balkema, Rotterdam, 1993. [TEO97] C. T EODOSIU, editor. Constitutive Modeling of Polycrystalline Metals at Large Strains. CISM Courses and Lectures, volume 376. Springer, Berlin, 1997. [TOU93] G. T OUZOT and J. DABOUNOU. Intégration numérique de lois de comportement élastoplastique. Rev. Eur. Élém. Finis, 2(4):465–494, 1993. [TRI96] N. T RIANTAFYLLIDIS and S. BARDENHAGEN. The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. J. Mech. Phys. Solids, 44:1891–1928, 1996. [TRO93] R. T ROSTEL. Vektor- und Tensor-Algebra. Mathematische Grundlagen der technischen Mechanik. I: Beiträge zur theoretischen Mechanik, Vieweg, Wiesbaden, 1993. [TRU60] C. T RUESDELL and R. T OUPIN. The classical field theories. In S. F LÜGGE, editor, Handbuch der Physik, volume 3.1. Springer, Berlin, 1960. [TRU65] C. T RUESDELL and W. N OLL. The non-linear field theories of mechanics. In S. F LÜGGE, editor, Handbuch der Physik. Springer, Berlin, 1965 (reedited in 2005). [TVE84] V. T VERGAARD and A. N EEDLEMAN. Analysis of cup-cone fracture in a round tensile bar. Acta Metall., 32:157–169, 1984. [TVE90a] V. T VERGAARD. Effect of fibre debonding in a whisker-reinforced metal. Mater. Sci. Eng., A125:203–213, 1990. [TVE90b] V. T VERGAARD. Material failure by void growth to coalescence. Adv. Appl. Mech., 27:83–151, 1990. [VAK71] A.A. VAKULENKO and M.L. K ACHANOV. Continuum theory of medium with cracks. Mech. Solids, 6(4):145–151, 1971 (English transl. of Mekhanika Tverdogo Tela (in Russian)). [VER83] P.A. V ERMEER. Frictional slip and non-associated plasticity. Scand. J. Metall., 12:268–276, 1983. [VOY93] G.Z. VOYIADJIS and P. K ATTAN. Damage of fiber reinforced composite materials with micromechanical characterization. Int. J. Solids Struct., 30(20):2757–2778, 1993. [WIL64] M.L. W ILKINS. Calculation of Elastic–Plastic Flow. Academic Press, New York, 1964. [WIL02] A. W ILKSTRÖM and M. N YGÅRDS. Anisotropy and texture in thin copper films— an elastoplastic analysis. Acta Mater., 50:857–870, 2002. [XIA95] L. X IA, C.F. S HIH, and J.W. H UTCHINSON. A computational approach to ductile crack growth under large scale yielding conditions. J. Mech. Phys. Solids, 43(3):389–413, 1995.
430
Non-Linear Mechanics of Materials
[YOD84] P.J. YODER and R.L. W HIRLEY. On the numerical implementation of elastoplastic models. J. Appl. Mech., 51(2):283–287, 1984. [Z-01] Z- SET PACKAGE. www.nwnumerics.com, www.mat.ensmp.fr, 2001. [ZAO85] A. Z AOUI. Approximate statistical modelling and applications. In E. S ANCHEZ PALENCIA and A. Z AOUI, editors, Homogenization Techniques for Composite Media. Lecture Notes in Physics, volume 272, pages 337–397. Springer, Berlin, 1985. [ZAO93a] A. Z AOUI. Matériaux Hétérogènes. Cours de l’Ecole Polytechnique, 1993. [ZAO93b] A. Z AOUI and J.L. R APHANEL. On the nature of the intergranular accommodation in the modelling of elastoviscoplastic behaviour of polycrystalline aggregates. In J.L. R APHANEL, C. T EODOSIU and F. S IDOROFF, editors, 5th MECAMAT International Seminar on Large Plastic Deformations, pages 27–38. Balkema, Rotterdam, 1993. [ZAR87] J. Z ARKA. Systematic structural modeling of metals. In J. G ITTUS and J. Z ARKA, editors, Modelling Small Deformations of Polycrystals, pages 374–410. Elsevier, Amsterdam, 1987. [ZHA95] Z.L. Z HANG and E. N IEMI. A new failure criterion for the Gurson–Tvergaard dilational constitutive model. Int. J. Fract., 70:321–334, 1995. [ZHU95] Y.Y. Z HU and S. C ESCOTTO. Fully coupled elasto-visco-plastic damage theory for anisotropic materials. Int. J. Solids Struct., 32(11):1607–1641, 1995. [ZIE96] J. Z IEBS, J. B RESSERS, H. F RENZ, D.R. H AYHURST, H. K LINGELHÖFFER, and S. F OREST. International Symposium on Local Strain and Temperature Measurements in Non-uniform Fields at Elevated Temperatures. Woodhead, Berlin, 1996. [ZIE00] O.C. Z IENKIEWICZ and R.L. TAYLOR. The Finite Element Method, volumes I–III. Butterworth-Heinemann, Stoneham, 2000.
Index θ -method, 51, 336 Accommodation elastoplastic, 250 Acoustic tensor, 377, 386 Adaptive step, 50 Adiabatic, 13 Aging, 100 Anisotropy, 36, 96, 164 Arc length, 46 Assembly, 62 Asymptotic methods, 213, 237 Average ensemble, 205 spatial, 205 BFGS method, 41 Bifurcation, 371 Bounds, 216 Hashin–Shtrikman, 233 Reuss, 218 Voigt, 218 Compatibility, 377 Configuration isoclinic, 305 Conjugate gradient, 43 Consistency, 21, 74, 81, 173, 306, 337, 341, 372 Consistent tangent matrix, 52, 65, 336 Constitutive equation, 65 linear elasticity, 14 Constitutive law, 67, 70, 295 Continua
generalized, 400 Continuum Cosserat, 278, 319 generalized bending, 276 curvature, 276 Continuum mechanics generalized, 319 Convective transport, 282 Convergence, 47 Convexity, 75 Corotational, 286 Cosserat, 278, 319, 369, 400 Coupling, 16, 147 Crack, 129, 157, 354 Creep, 22 Criterion, 68, 120 anisotropic, 36 Drucker–Prager, 33, 77 Hill, 36 Mohr–Coulomb, 34 Tresca, 32, 77 von Mises, 31, 76 Damage, 14, 127 deactivation, 162, 189 scalar, 138 tensorial, 138, 180 Decomposition multiplicative, 300, 314, 330 polar, 287 Deformation, 288 Derivative convective, 285
J. Besson et al., Non-Linear Mechanics of Materials, Solid Mechanics and Its Applications 167, c Springer Science+Business Media B.V. 2010 DOI 10.1007/978-90-481-3356-7,
432
Non-Linear Mechanics of Materials
Jaumann, 284 Differential equations, 48 Discrete integration, 55 Discretization, 61 Dissipation, 11, 70, 72, 84, 149, 295, 324 Dissipative structures, 394 Effective moduli, 214 Eigenrotation, 287 Ellipticity, 386 Energy free, 83 stored, 84 Entropy, 10, 295, 324 Erasing, 27 Ergodicity, 207 Error control, 50 Eshelby, 231 External forces, 291 Fading memory, 93, 95 Finite deformation, 279, 364 Finite element, 54 Flow, 68, 73, 76, 80, 120, 153 Flutter instability, 386 Fracture, 129, 157, 353 Frame corotational, 286 Free energy, 11 Gauss point, 55 Generalized standard materials, 69 Green operator, 221 Hadamard conditions, 377, 378 Hardening, 14, 69, 81, 97, 122 isotropic, 21, 82, 97 kinematic, 82, 92, 97 multiplicative, 100 nonlinear, 82, 92 overhardening, 257 product, 28 Hashin–Shtrikman bounds, 233
Hencky–Mises, 79 Heterogeneous elasticity, 214 Hill condition, 373 Hill–Mandel lemma, 212 Homogenization, 195 in thermoelasticity, 235 nonlinear, 241 Hyperelasticity, 298, 324 Hypoelasticity, 298 Inclusion Eshelby, 231 Incompressibility, 358 Integration θ -method, 51 explicit, 49 implicit, 51 Runge–Kutta, 49 Interaction matrix, 102, 103 Interface (of a behavior), 335 Internal forces, 8, 290 Internal variable, 11, 70, 333 Invariant, 30 Irreversible processes, 10 Isoclinic (configuration), 305 Isoparametric elements, 57 Isotropic hardening, 21 Iterative method, 40, 43 Jaumann derivative, 287 Kinematics, 279 Kuhn–Tucker, 75 Large deformation, 279, 364 Loading cyclic, 86, 92, 95 non-proportional, 95 Localization, 371, 386 Loss of uniqueness, 373 Mandel stress, 306 Material frame indifference, 296
Index
Material object, 333 Materials compressible, 115, 350 generalized standard, 305 heterogeneous, 195 porous, 115, 116, 350 Medium generalized continuous, 275 Memory, 93, 95, 258 Mesh, 395 Model 3 phase, 228 β, 250 Berveiller–Zaoui, 243 generalized standard, 15 Kröner, 243 Prager, 19 self-consistent, 227 Multikinematic, 92, 343 Multimechanism, 92, 101, 108, 110, 314 Newton method, 38 Newton–Raphson, 38 Normality, 72, 73, 76, 169 Norton–Hoff, 28 Nucleation, 123 Numerical solution, 48 Objectivity, 7, 280, 290, 306 Overhardening, 257 Periodicity, 364 Plastic work, 72, 74 Plasticity, 18, 67, 72, 110, 120, 300 non associated, 81 perfect, 74 Polycrystal, 254 Potential, 11 Prager, 19, 79 Prandtl–Reuss, 78, 336 Principle of virtual power, 8 Progressive deformation, 88, 95, 348 Propagation, 129, 157, 354
433
Ratchet, 88, 95, 348 Rayleigh waves, 376 Recovery, 27, 97, 347 Regularization, 395 Relaxation, 22 Rheology, 17 Riks method, 44 Runge–Kutta, 49, 336 Second-order work, 374 Single crystal, 101, 103, 314 Single slip, 308, 328 Spatial discretization, 54 Standard generalized, 305 Stoneley waves, 377 Strain, 288 Strain measurement, 288 Stress effective, 122, 131, 140, 155 internal, 20 plane, 361 Structure large grains, 275 Symmetry, 297 Systematic theory, 221 Temperature, 29 Thermodynamical force, 15 Thermodynamics, 10, 294 Thermoelasticity, 12, 235, 237 Unilateral effect, 162 Uniqueness, 373 Varying temperature, 340, 347 Virtual power, 61, 290 Virtual works, 8 Viscoelasticity, 22, 299 Viscoplasticity, 25, 67, 71, 97, 110, 120, 177, 306, 342 Yield, 29 Yield surface distortion, 256
