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 perpendicu
