Damage Mechanics

The main goal is to summarize thecurrent most effective methods for modeling, simulating, and optimizing metalforming processes and to present the main features of new, innovative method

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Preface xiii

Principle of Mathematical Notations xix

Chapter 1 Elements of Continuum Mechanics and Thermodynamics 1

1.1 Elements of kinematics and dynamics of materially simple continua 2

1.1.1 Homogeneous transformation and gradient of transformation 2

1.1.1.1 Homogeneous transformation 2

1.1.1.2 Gradient of transformation and its inverse 4

1.1.1.3 Polar decomposition of the transformation gradient 5

1.1.2 Transformation of elementary vectors, surfaces and volumes 5

1.1.2.1 Transformation of an elementary vector 6

1.1.2.2 Transformation of an elementary volume: the volume dilatation 6

1.1.2.3 Transformation of an oriented elementary surface 7

1.1.3 Various definitions of stretch, strain and strain rates 8

1.1.3.1 On some definitions of stretches 8

1.1.3.2 On some definitions of the strain tensors 10

1.1.3.3 Strain rates and rotation rates (spin) tensors 15

1.1.3.4 Volumic dilatation rate, relative extension rate and angular sliding rate 17

1.1.4 Various stress measures 19

1.1.5 Conjugate strain and stress measures 23

1.1.6 Change of referential or configuration and the concept of objectivity 23

1.1.6.1 Impact on strain and strain rates 24

1.1.6.2 Impact on stress and stress rates 26

1.1.6.3 Impact on the constitutive equations 29

1.1.7 Strain decomposition into reversible and irreversible parts 30

1.2 On the conservation laws for the materially simple continua 33

1.2.1 Conservation of mass: continuity equation 33

1.2.2 Principle of virtual power: balance equations 34

1.2.3 Energy conservation First law of thermodynamics 36

1.2.4 Inequality of the entropy Second law of thermodynamics 37

1.2.5 Fundamental inequalities of thermodynamics 38

1.2.6 Heat equation deducted from energy balance 39

1.3 Materially simple continuum thermodynamics and the necessity of constitutive equations 39

1.3.1 Necessity of constitutive equations 40

1.3.2 Some fundamental properties of constitutive equations 41

1.3.2.1 Principle of determinism or causality axiom 42

1.3.2.2 Principle of local action 42

1.3.2.3 Principle of objectivity or material indifference 42

1.3.2.4 Principle of material symmetry 43

1.3.2.5 Principle of consistency 43

1.3.2.6 Thermodynamic admissibility 44

1.3.3 Thermodynamics of irreversible processes The local state method 44

1.3.3.1 A presentation of the local state method 44

1.3.3.2 Internal constraints 52

1.4 Mechanics of generalized continua Micromorphic theory 55

1.4.1 Principle of virtual power for micromorphic continua 58

1.4.2 Thermodynamics of micromorphic continua 59

Chapter 2 Thermomechanically-Consistent Modeling of the Metals Behavior with Ductile Damage 63

2.1 On the main schemes for modeling the behavior of materially simple continuous media 64

2.2 Behavior and fracture of metals and alloys: some physical and phenomenological aspects 69

2.2.1 On the microstructure of metals and alloys 69

2.2.2 Phenomenology of the thermomechanical behavior of polycrystals 70

2.2.2.1 Linear elastic behavior 71

2.2.2.2 Inelastic behavior 72

2.2.2.3 Inelastic behavior sensitive to the loading rate 74

2.2.2.4 Initial and induced anisotropies 76

2.2.2.5 Other phenomena linked to the shape of the loading paths 77

2.2.3 Phenomenology of the inelastic fracture of metals and alloys 82

2.2.3.1 Micro-defects nucleation 84

2.2.3.2 Micro-defects growth 85

2.2.3.3 Micro-defects coalescence and final fracture of the RVE 85

2.2.3.4 A first definition of the damage variable 86

2.2.3.5 From ductile damage at a material point to the total fracture of a structure by propagation of macroscopic cracks 89

2.2.4 Summary of the principal phenomena to be modeled 90

2.3 Theoretical framework of modeling and main hypotheses 91

2.3.1 The main kinematic hypotheses 91

2.3.1.1 Choice of kinematics and compliance with the principle of objectivity 92

2.3.1.2 Decomposition of strain rates 94

2.3.1.3 On some rotating frame choices 99

2.3.2 Implementation of the local state method and main mechanical hypotheses 102

2.3.2.1 Choice of state variables associated with phenomena being modeled 103

2.3.2.2 Definition of effective variables: damage effect functions 108

2.4 State potential: state relations 113

2.4.1 State potential in case of damage anisotropy 114

2.4.1.1 Formulation in strain space: Helmholtz free energy 114

2.4.1.2 Formulation in stress space: Gibbs free enthalpy 121

2.4.2 State potential in the case of damage isotropy 124

2.4.2.1 Formulation in strain space: Helmholtz free energy 124

2.4.2.2 Formulation in stress space: Gibbs free enthalpy 128

2.4.3 Microcracks closure: quasi-unilateral effect 129

2.4.3.1 Concept of micro-defect closure: deactivation of damage effects 129

2.4.3.2 State potential with quasi-unilateral effect 132

2.5 Dissipation analysis: evolution equations 139

2.5.1 Thermal dissipation analysis: generalized heat equation 140

2.5.1.1 Heat flux vector: Fourier linear conduction model 141

2.5.1.2 Generalized heat equation 141

2.5.2 Intrinsic dissipation analysis: case of time-independent plasticity 143

2.5.2.1 Damageable plastic dissipation: anisotropic damage with two yield surfaces 144

2.5.2.2 Damageable plastic dissipation: anisotropic damage with a single yield surface 157

2.5.2.3 Incompressible and damageable plastic dissipation: isotropic damage with two yield surfaces 162

2.5.2.4 Incompressible and damageable plastic dissipation: single yield surface 169

2.5.3 Intrinsic dissipation analysis: time-dependent

plasticity or viscoplasticity 174

2.5.3.1 Damageable viscoplastic dissipation without restoration: anisotropic damage with two viscoplastic potentials 176

2.5.3.2 Viscoplastic dissipation with damage: isotropic damage with a single viscoplastic potential and restoration 182

2.5.4 Some remarks on the choice of rotating frames 186

2.5.5 Modeling some specific effects linked to metallic material behavior 189

2.5.5.1 Effects of non-proportional loading paths on strain hardening evolution 190

2.5.5.2 Strain hardening memory effects 191

2.5.5.3 Cumulative strains or ratchet effect 191

2.5.5.4 Yield surface and/or inelastic potential distortion 192

2.5.5.5 Viscosity-hardening coupling: the Piobert–Lüders peak 192

2.5.5.6 Accounting for the material microstructure 193

2.5.5.7 Some specific effects on ductile fracture 193

2.6 Modeling of the damage-induced volume variation 194

2.6.1 On the compressibility induced by isotropic ductile damage 195

2.6.1.1 Concept of volume damage 195

2.6.1.2 State coupling and state relations 196

2.6.1.3 Dissipation coupling and evolution equations 197

2.7 Modeling of the contact and friction between deformable solids 200

2.7.1 Kinematics and contact conditions between solids 201

2.7.1.1 Impenetrability condition 203

2.7.1.2 Equilibrium condition of contact interface 204

2.7.1.3 Contact surface non-adhesion condition 205

2.7.1.4 Contact unilaterality condition 205

2.7.2 On the modeling of friction between solids in contact 206

2.7.2.1 Time-independent friction model 206

2.8 Nonlocal modeling of damageable behavior of micromorphic continua 215

2.8.1 Principle of virtual power for a micromorphic medium: balance equations 217

2.8.2 State potential and state relations for a micromorphic solid 218

2.8.3 Dissipation analysis: evolution equations for a micromorphic solid 221

2.8.4 Continuous tangent operators and thermodynamic admissibility for a micromorphic solid 223

2.8.5 Transformation of micromorphic balance equations 224

2.9 On the micro–macro modeling of inelastic flow with ductile damage 226

2.9.1 Principle of the proposed meso–macro modeling scheme 227

2.9.2 Definition of the initial RVE 230

2.9.3 Localization stages 230

2.9.4 Constitutive equations at different scales 233

2.9.4.1 State potential and state relations 233

2.9.4.2 Intrinsic dissipation analysis: evolution equations 235

2.9.5 Homogenization and the mean values of fields at the aggregate scale 239

2.9.6 Summary of the meso–macro polycrystalline model 240

Chapter 3 Numerical Methods for Solving Metal Forming Problems 243

3.1 Initial and boundary value problem associated with virtual metal forming processes 244

3.1.1 Strong forms of the initial and boundary value problem 245

3.1.1.1 Posting a fully coupled problem 245

3.1.1.2 Some remarks on thermal conditions at contact interfaces 250

3.1.2 Weak forms of the initial and boundary value problem 252

3.1.2.1 On the various weak forms of the IBVP 252

3.1.2.2 Weak form associated with equilibrium equations 254

3.1.2.3 Weak form associated with heat equation 257

3.1.2.4 Weak form associated with micromorphic damage balance equation 258

3.1.2.5 Summary of the fully coupled evolution problem 258

3.2 Temporal and spatial discretization of the IBVP 259

3.2.1 Time discretization of the IBVP 259

3.2.2 Spatial discretization of the IBVP by finite elements 260

3.2.2.1 Spatial semi-discretization of the weak forms of the IBVP 260

3.2.2.2 Examples of isoparametric finite elements 266

3.3 On some global resolution scheme of the IBVP 270

3.3.1 Implicit static global resolution scheme 272

3.3.1.1 Newton–Raphson scheme for the solution of the fully coupled IBVP 273

3.3.1.2 On some convergence criteria 275

3.3.1.3 Calculation of the various terms of the tangent matrix 276

3.3.1.4 The purely mechanical consistent Jacobian matrix 280

3.3.1.5 Implicit global resolution scheme of the coupled IBVP 282

3.3.2 Dynamic explicit global resolution scheme 284

3.3.2.1 Solution of the mechanical problem 284

3.3.2.2 Solution of thermal (parabolic) problem 286

3.3.2.3 Solution of micromorphic damage problem 288

3.3.2.4 Sequential scheme of explicit global resolution of the IBVP 288

3.3.3 Numerical handling of contact-friction conditions 291

3.3.3.1 Lagrange multiplier method 293

3.3.3.2 Penalty method 295

3.3.3.3 On the search for contact nodes 296

3.3.3.4 On the numerical handling of the incompressibility condition 300

3.4 Local integration scheme: state variables computation 304

3.4.1 On numerical integration using the Gauss method 304

3.4.2 Local integration of constitutive equations: computation of the stress tensor and the state variables 305

3.4.2.1 On the numerical integration of first-order ODEs 306

3.4.2.2 Choice of constitutive equations to integrate 308

3.4.2.3 Integration of time-independent plastic constitutive equations: the case of a von Mises isotropic yield criterion 313

3.4.2.4 Integration of time-independent plastic constitutive equations: the case of a Hill quadratic anisotropic yield criterion 326

3.4.2.5 Integration of the constitutive equation in the case of viscoplastic flow 328

3.4.2.6 Calculation of the rotation tensor: incremental objectivity 333

3.4.2.7 Remarks on the integration of the micromorphic damage equation 335

3.4.3 On the local integration of friction equations 335

3.5 Adaptive analysis of damageable elasto-inelastic structures 337

3.5.1 Adaptation of time steps 339

3.5.2 Adaptation of spatial discretization or mesh adaptation 341

3.6 On other spatial discretization methods 347

3.6.1 An outline of non-mesh methods 348

3.6.2 On the FEM–meshless methods coupling 353

Chapter 4 Application to Virtual Metal Forming 355

4.1 Why use virtual metal forming? 356

4.2 Model identification methodology 359

4.2.1 Parametrical study of specific models 360

4.2.1.1 Choosing typical constitutive equations 360

4.2.1.2 Isothermal uniaxial tension (compression) load without damage 364

4.2.1.3 Accounting for ductile damage effect 383

4.2.1.4 Accounting for initial anisotropy in inelastic flow 396

4.2.2 Identification methodologies 413

4.2.2.1 Some general remarks on the issue of identification 414

4.2.2.2 Recommended identification methodology 416

4.2.2.3 Illustration of the identification methodology 422

4.2.2.4 Using a nonlocal model 429

4.3 Some applications 431

4.3.1 Sheet metal forming 431

4.3.1.1 Some deep drawing processes of thin sheets 432

4.3.1.2 Some hydro-bulging test of thin sheets and tubes 441

4.3.1.3 Cutting processes of thin sheets 447

4.3.2 Bulk metal forming processes 463

4.3.2.1 Classical bulk metal forming processes 463

4.3.2.2 Bulk metal forming processes under severe conditions 476

4.4 Toward the optimization of forming and machining processes 484

Appendix: Legendre–Fenchel Transformation 493

Bibliography 499

Index 515

As with other scientific fields where numerical simulation is essential, predictivecapabilities of virtual metal forming methods rely on: (i) advanced thermomechanicalconstitutive equations representing the mechanisms of the main thermomechanicalphenomena involved and their various couplings; (ii) high-performing numericalmethods adapted to the problem’s various nonlinearities; (iii) adaptive and user-friendly geometric tools for the spatial representation and spatial discretization ofsolids undergoing large transformations.

For scientists, numerical simulations allow researchers to explore phenomenaand to check out the credibility of assumptions of the scope of the experimentation

provided that these simulations are based on reliable solutions for relevant and

representative problems For engineers, simulating complex physical phenomena is

a trademark of requirement that guarantees optimum reliability and cost

management for maximum economic efficiency

Several books are devoted to the modeling of metal forming processes to obtainvarious optimum metallic components due to large inelastic deformations Carefulexamination of the literature on this subject allows their classification into threefamilies that differ by the modeling methodologies and recommended objectives.The first family relates to works where modeling methods and calculationprocedures are mainly analytical in nature, such as [THO 65], [AVI 68], [BAQ 73],[JOH 83], [MIE 91], and [MAR 02] The second is composed of essentiallynumerical methods, mainly based on the finite element method (FEM), such as[KOB 89], [ROW 91], [WAG 01], and [DIX 08], or even on more recent numericalmethods as meshfree or meshless methods [CHI 09] The third and last family which

is more technological aims to provide engineers with an insight into advanced metalforming technologies in relation to recent technological advances [SCH 98], and[COL 10] Each of these books has given the state-of-the-art including the most

recent advances to be used in improving metal forming and manufacturingprocesses As for all other engineering disciplines, these works are somehow the

“memory” of their time of major scientific and technical developments that supportthe present generation to deal with current problems and prepare for the newmethods of tomorrow

This book is intended to provide graduate students and researchers from both theacademic and industrial worlds, with a clear and thorough presentation of the recentadvances in continuum damage mechanics and its practical use in improvingnumerical simulations in virtual metal forming The main goal is to summarize thecurrent most effective methods for modeling, simulating, and optimizing metalforming processes and to present the main features of new, innovative methodscurrently being developed, which will no doubt be the industrial tools of tomorrow.Compared to recent books devoted to virtual metal forming, the main contribution ofthis book is found in Chapter 2 where the development of highly predictivemultiphysical and fully coupled constitutive equations is presented These can beused in computer codes to simulate and optimize all kinds of sheet or bulk metalforming processes by large inelastic strains, regarding the occurrence of ductiledamage

This book is organized into four main chapters The first aims to provide thereader with the basic theoretical “tools” needed to understand the models which willsubsequently be explored in this book This is essentially a brief introduction whichaims to combine scientific rigor with simple definitions in order to present: (i) themain measures of strains and stresses as well as their respective rates, (ii) the mainconservation laws for the materially simple continua, (iii) the thermodynamics ofirreversible processes with state variables firstly in the framework of materiallysimple continua (or Cauchy continua), followed by (iv) the generalization of theseconcepts to the materially non-simple continua, particularly the micromorphiccontinua in the framework of the generalized higher order continua For reasons ofbrevity, the mathematical aspects related to algebra and tensor analysis as well asthe convex functions’ analysis – concepts that are required for carrying out a number

of calculations examined – will not be discussed in this chapter nor in theappendices Rather, various academic books are referenced, which provide anoverview of these features However, the definition of Legendre–Fenchel transform,that is often used throughout the first two chapters of the book, is given inAppendix 1

The second chapter, the key part of this book, focuses solely on “advanced”modeling of the main physical phenomena characteristic of various behaviors andductile damages of metals under large strains by focusing on their various strong

couplings After a brief descriptive summary of the physical phenomena beingmodeled and their main physical mechanisms, the reader will find: (i) the mainassumptions adopted for accurate modeling in the context of thermodynamics ofirreversible processes with state variables; (ii) the construction of various statepotentials and the definition of state relations derived from them; (iii) analyzingdifferent sources of dissipations and deducing the evolutions equations fromappropriate load functions and adequate dissipation potentials; (iv) modeling thevolume variation induced by ductile damage; (v) modeling contact between solidsand friction along the contact interfaces; (vi) extending this to generalized continua

in the framework of a micromorphic theory in order to propose a rigorous non-localmodel that enables adequate prediction of the damage-induced localization zones;and finally (vii) giving a micro-macro modeling of polycrystalline plasticity withductile damage based on the mean fields approach

The third chapter introduces the numerical aspects, which allow us to obtain acredible “unique” solution of initial and boundary value problems (IBVP) in order tocompletely simulate various metal forming processes in the framework of what we call

virtual metal forming This chapter contains: (i) the pose of the main equations which

define the strong and weak forms of the IBVP; (ii) associated time discretization usingfinite difference method (FDM) as well as the space discretization using the finiteelement method (FEM) of the IBVP focusing on the most common currently usedelements for 2D and 3D problems; (iii) an overview of the main global resolutionschemes of the IBVP including the assessment of contact conditions; (iv) a detailedpresentation of the numerical aspects related to the iterative integration scheme of thefully coupled and highly nonlinear constitutive equations in each quadrature point

of each finite element in order to compute the overall state variables over each time

or load increment; (v) a summary of the adaptive methodology for virtual metalforming; and finally (vi) an insight into new meshless spatial discretization methods,and their possible link (or coupling) with the FEM Again for brevity, many aspectswere simply mentioned without going deeply into technical details but referring to acomplete list of references where the reader can obtain further information on thetopics under concern

Finally, the fourth chapter focuses on using the virtual adaptive virtual metalforming proposed in order to numerically simulate various metal forming andmachining processes The following aspects are examined: (i) the presentation ofthe methodology to follow in order to determine the material parameters enteringthe constitutive equations under concern A detailed parametric study is performed

in order to analyze the role of each material’s parameters; (ii) the application

of numerical simulation to sheet metal forming processes as deep drawing,hydroforming, or cutting of thin structures Bulk metal forming processes under

normal conditions (forging, stamping, extrusion, etc.) or under severe conditions(high-impact or high-velocity machining) are also presented All the examples used

in this chapter have been exclusively taken from research works in virtual metalforming performed at the University of Technology of Troyes (UTT) since 1995.For the sake of brevity, and with few exceptions, we only refer to academic orcollective books related to a given concept Therefore, we have deliberately left asideany reference to articles in scientific journals, except for a few review articles onaspects which are not treated by specific books The interested reader will have notrouble finding numerous articles through bibliographic searches on specialized sites.This book, which focuses on “advanced” modeling and numerical simulation inmetal forming and machining by large inelastic strains, is in fact an overview of theteaching and research activities of the author during his career Beginning atUTC in Compiègne since 1979, these activities have continued to evolve mainly atUTT since 1995 and also partially at ENSAM/CER in Chalons-en-Champagne,ENIM (Monastir, Tunisia), and ESSTT (Tunis, Tunisia) as invited professor formany years On the other hand, as a member of the French school of mechanics

of materials, the author has participated directly in the GRECO: Grandes

Déformations et Endommagement [Large deformations and damage] (1980–1988)

and in the MECAMAT association and indirectly in the CSMA (ComputationalStructural Mechanics Association) All this has greatly influenced the nature andcontent of this book

The author would, therefore, like to address his most sincere appreciation to allthose many people, who have directly or indirectly influenced the material of thisbook: the engineers who have attended his lectures; the PhD students who haveactively participated in the research from which a number of results have been takenfor the four chapters of this book, particularly in Chapter 4, and for which anexhaustive list of the PhD theses prepared at UTT over the past decade is provided

in the bibliographic list Finally, many colleagues and friends in both the French andinternational communities of solid and computational mechanics have, to a greater

or lesser extent, brought much to the author By the way, special thanks are due to

my friends and colleagues, Houssem Badreddine, Carl Labergère, and Pascal Lafonfrom UTT/LASMIS for their direct involvement in finalizing some of the results inChapter 4

Writing a book in combination with an increasing workload, strongly impacts

on the balance of family life The decision to start writing this book was takenwith and encouraged by my marvelous and adorable wife, Fathia, who acceptedthe obvious risk of spending many weekends and holidays without her other half

Her understanding, unwavering support, and ability to close her eyes each time Ispent an inordinately long time in front of the computer, have been instrumental in

me finishing this book To my dear Fathia and our three children Ilyes, Sarah andSlim, I dedicate this book as a token of the love I bear for them and that gave me,many times, the breath to continue during moments of doubt

Khemais SaanouniTroyes, March 2012

The symbols and notations used in this book are defined in the text upon theirfirst occurrence However, it should be stated that the principle of the main notationsused, by giving some non-exhaustive examples that allow readers to understand thecalculations carried out.

– x scalar variable.

– x x, i vector in 3.

– ,x x second-rank tensor in ij 3.

– ,x x ijk third-rank tensor in 3.

– x x, ijkl fourth-rank tensor in 3.

– tensorial product (external) of two tensors

– internal tensorial product: contraction on one indice

– : internal tensorial product: contraction on two indices.

– internal tensorial product: contraction on three indices

–:: internal tensorial product: contraction on four indices.

– T transpose of a matrix

– column matrix or vector

– line matrix or transpose of a column matrix

Elements of Continuum Mechanics and

Thermodynamics

This first chapter gives the main basic elements of mechanics and thermodynamics

of the materially simple continua A continuum is considered materially simple if theknowledge of the first transformation gradient is sufficient to define all the kinematicand state variables necessary for the characterization of the behavior of this medium.The main objective is to provide readers with the basic elements that will allow them

to follow and understand without difficulty the theoretical formulations of theconstitutive equations under large inelastic deformations used in virtual metal forming

In this chapter, readers will find the basic ideas of the kinematics and dynamics

of materially simple continua (section 1.1); the conservation laws or field equations(section 1.2); the thermodynamics of materially simple continua and specifically theso-called “local state method” in the framework of which the constitutive equationswill be formulated (section 1.3); finally, we will conclude by giving an introduction

to generalized continuum mechanics (GCM) by extending all kinematic andthermodynamic ideas to the context of generalized or materially non-simplecontinua (section 1.4) This extension allows the formulation of nonlocalconstitutive equations provided at the end of Chapter 2

For the sake of brevity, we will not recapitulate all of the mathematical detailsand rigorous demonstrations of all the ideas introduced In particular, we will neitherreview tensor algebra and tensor analysis nor convex analysis, ideas that areindispensable for the manipulation of all mechanical quantities For more details onthese subjects, we refer the reader to the excellent book by Truesdell and Noll, firstpublished in 1965 [TRU 65] and then republished by the same authors in a secondrevised and corrected edition in 1992 [TRU 92] A third edition appeared in 2004

[TRU 04] under the aegis of the publisher Springer-Verlag and with the support of

W Noll Directly or indirectly inspired by this work, at the origin of moderncontinuum mechanics, many other books have been published in which readers willfind the mathematical basics and physical justifications of all basic concepts ofmaterially simple continuum mechanics (MSCM): [CAL 60], [ERI 62], [FUN 65],[TRU 66], [ERI 67], [JAU 67], [PRI 68], [MAL 69], [KES 70], [GLA 71],[DAY 72], [SWA 72], [GER 73], [MAN 74], [SED 75], [BOW 76], [LEI 78],[KES 79], [MCL 80], [GUR 81], [HUN 83], [ZIE 83], [OGD 84], [TRU 84],[MÜL 85], [GER 86], [ABR 88], [SAL 88], [BOW 89], [DUV 90], [ERI 91],[DEH 93], [LAI 93], [SMI 93], [GON 94], [RAG 95], [BOU 96], [CHU 96],[COI 97], [ROU 97], [DUB 98], [CHA 99], [BAS 00], [SOU 01], [LIU 02],[GAS 03], [NEM 04], [ASA 06], and [WAT 07], among many others In the vastmajority of these books, the reader will find chapters or indices dedicated tomathematical reminders on vectors and tensor analysis as well as convex analysis.However, other specialized books may be of great help to readers who wish toimprove their understanding of tensor algebra and tensor analysis [LEL 63],[SOK 64], [LEG 71], [FLÜ 72], [SCH 75], [WIN 79], [ABR 88], [HLA 95],[ITS 07], or of convex analysis [MOR 66], [ROC 70], [EKE 74], [DAU 84],[SEW 87] In this book, a simple reminder of the definition and principal properties

of the Legendre–Fenchel transformation are provided in Appendix 1

1.1 Elements of kinematics and dynamics of materially simple continua

1.1.1 Homogeneous transformation and gradient of transformation

Let us consider a deformable solid occupying at time t a volume t, withboundary u F and u F u is the portion of the boundary wheredisplacements are imposed and F is the additional part of the boundary whereforces are imposed

1.1.1.1 Homogeneous transformation

Let us consider the description of the motion of the subdomain a part of solid Suppose that occupies at initial time t0 the initial non-deformedconfiguration C0 At any instant t t0, the subdomain occupies the currentdeformed configuration C t Using a direct orthonormal Euclidian space of base

Figure 1.1 Initial and deformed configurations of a deformable subdomain

and vectors transport

The components X X X X i: ,1 2, 3 of vector X X e X e1 1 2 2 X e3 3 are theLagrangian or material coordinates of point P0 in the reference configuration C0.The components x x x x i: , ,1 2 3 of vector x x e x e1 1 2 2 x e3 3 are the Eulerian orspatial coordinates of point P t in the current configuration C t corresponding topoint P0 ofC0

The vectorial field ( , )X t that allows the determination at any time t of the

position of point P t is a bijection of C0 on C t Thus, it allows a reciprocal function

1( , )x t , which at any point P t of C t is used to define in a unique manner itscorrespondent P0 in C0 The two vectorial functions ( , )X t and 1( , )x t arecontinuous and continuously differentiable (except possibly on certain surfaces ofdiscontinuity) with respect to the overall space and time variables

If the field ( , )X t is expressed at any time t in the form of an affine function

between the material coordinates X and the spatial coordinates x of the form:

1.1.1.2 Gradient of transformation and its inverse

The gradient of the transformation ( , )X t defined by [1.2] is given by:

( , )( ) x X t

F Grad

This is a “bipoint tensor” of the second-rank F (or F ij) called the gradient ofthe homogeneous transformation between C0 and C t According to Figure 1.1, thehomogeneous transformation is defined by:

in the shape, size, and orientation of the continuum as we will see later in thischapter

1.1.1.3 Polar decomposition of the transformation gradient

According to the well-known polar decomposition theorem, any homogeneoustransformation of a subdomain can be seen as the product of a pure rotation and

of a pure strain or stretch This means that any non-singular gradient of ahomogeneous transformation F defined by [1.5] can be multiplicativelydecomposed, in a unique manner, in the form:

where the symmetric and positive definite second-rank tensorsU and V are called

left and right pure strain or stretch tensors, and R is the rigid body orthogonal

rotation tensor (R R R R T T 1) U is a Lagrangian tensor defined with respect to

0

C , while V is purely Eulerian tensor, defined with respect to C t (see Figure 1.2)

Figure 1.2 2D schematic illustration of the polar decomposition of the

transformation gradient

1.1.2 Transformation of elementary vectors, surfaces and volumes

The affine nature of the relation [1.4] implies that any linear variety in thereference configuration C0 is transformed, in its transport by this homogeneousmotion, into a linear variety of the same order in the current configuration C t.This is particularly applicable to the transformation of elementary vectors, volumes,

or surfaces

1.1.2.1 Transformation of an elementary vector

We consider the set of particles occupying in C0 the segment P Q0 0 as definingthe Lagrangian elementary vector dX P Q0 0 (Figure 1.1) Due to the affine

character of the transformation [1.4], these particles occupy at time t in C t thesegment PQ t t defining the Eulerian vector dx PQ1 1 Thus, and according to [1.5],

the elementary vector dx is obtained by the transformation of the elementary vector

dX due to the homogeneous transformation between configurations C0andC t:

1.1.2.2 Transformation of an elementary volume: the volume dilatation

Given in the configuration C0 an elementary parallelepiped constructed withthe three non-coplanar vectors dX dX dX1, 2, 3 (Figure 1.3) Its volume in C0 isdefined by:

where (M) is the matrix, the columns of which are the three elementary vectors.

Moreover, in the current configuration C t, the parallelepiped formed by thevectors dx dx dx1, 2, 3, which are the transformation, respectively, of the vectors

Figure 1.3 Elementary volume transformation between C0 and C t

Finally, we note that it is possible to define the gradient of an isochoric orvolume preserving transformation by:

1/3

Thus, any homogeneous transformation can be decomposed into the product of

an isochoric or volume preserving transformation of gradient ˆF J F and of a1/3pure dilatation of gradient F J1 3/ 1, so as to have:

1.1.2.3 Transformation of an oriented elementary surface

Consider, in configuration C0 (see Figure 1.4), a plane elementary surfaceoriented by the normal vector n0 (surrounding, for example, the point P0) of area0

dA represented by the parallelogram formed by the two coplanar vectors dX dX, '.The “vector area” of this parallelogram is defined in C0 by dA0 dA n0 0 Thisoriented plane surface, transported by the motion into the configuration C t, istransformed into a plane surface with the normal n t surrounding point P t

represented by the parallelogram formed by the two vectors dx dx, (respectively,transformation of the vectors dX dX, by the gradient F ) of “vector area”

dA dA n By using the transformation relationships of elementary vectors as well

as [1.12], the following relationship between dA t and dA0 is obtained:

1

0 0( )T

Called Nanson’s relation, [1.16] will subsequently be used for the definition ofvarious forms of the stress tensor (see section 1.1.4)

Figure 1.4 Transport of elementary surface between C0 and C t

1.1.3 Various definitions of stretch, strain and strain rates

We will now give the main definitions of the strain undergone by the geometry

of area in the homogeneous transformation, between the reference configuration0

C and the current configurationC t , characterized by the gradient F

1.1.3.1 On some definitions of stretches

Let us consider two non-collinear vectors in configuration C0 named dX dX,with the common origin point P0; and let dx dx, be their respective transformedvectors to point P t in the current configuration C t The scalar product of these twovectors is given by:

'

Thus, we define in C0 the right CauchyíGreen stretch tensor C , Lagrangian,

symmetric and positive definite, by:

T

It is a matter of course that det( ) det( )C F F T det( )F 2 J2 and, due to

[1.8] the symmetry of U and the orthogonality of R , we have:

thereby allowing the definition in C t of the left Cauchy–Green stretch tensor B ,

Eulerian, symmetric, and positive definite, by:

1 ( 1) T 1

with det( ) det( )B F F T det( )F 2 J2 It is easy to verify, by using [1.8] and

given the properties of V and R , that:

2 T T

Due to the decomposition [1.14], we easily obtain the following decomposition

of the Cauchy–Green stretch tensors C and B:

These lengths are easily calculated by inserting dx' dx and dX' dX into[1.17] and [1.20] to obtain:

1

Taking the vectors dX and dX as equal and merging them with the unit base

vectors of the selected orthonormal triad, equation [1.27] permits an easy interpretation

of the different diagonal components of the right Cauchy–Green tensorC

The sliding of two initially orthogonal vectors can also be defined by calculatingthe angle of rotation of this pair of vectors in the current configurationC t by:

In order to define the strain of area under the effect of a homogeneoustransformation of the gradient F, it is appropriate to use symmetric second-ranktensors that have no physical dimension with zero value at the origin (i.e when1

F ) as well as for any rigid or non-deformable body motion A simple way to

assess the material deformation of an area in the homogeneous transformationthat causes it to change from the reference configuration C0 to the currentconfiguration C t consists of calculating the difference between the scalar products

of the elementary vectors dx dx dX dX previously calculated By using [1.17],this calculation leads to:

F The Eulerian Euler–Almansi strain tensor A is defined in a point P t ofC t by:

For different non-zero values of the natural integer m, we find various definitions

of strain tensors suggested in the literature Table 1.1 summarizes these differenttensors, which fulfill all the properties given at the beginning of this section.Particularly, it is easy to check that all of these strain measures shrinks to zero at theorigin of the motion (i.e when F 1) as well as for any rigid body motion

Reference configurationC0 Current configuration C t

Table 1.1 Various definitions of strain tensors in the two configurations C0 and C t

To illustrate these different strain measures, let us consider the one-dimensional

case of a bar occupying at the initial time t0the reference configuration defined by

initial section A0and initial length l0 Under the effect of an applied axial load, the

bar deforms (or elongates) to occupy at time t the current configuration defined by the current section A t and the current length l t We call ( ) ( / )t l l t 0 the ratio oflengths measuring the rate of elongation of the bar, and we calculate the variousstrain measurements in the reference configuration of the bar We obtain theexpressions given by:

The graphic representation of five different strain measures versus the elongationratio ( )t is given in Figure 1.5, thus illustrating the difference between these largestrain measures In particular, all of these strain measures are zero for ( )t 0 1and they are indistinguishable in proximity to 0, thus resulting in what iscommonly called a small strain hypothesis (SSH), as we will see later We also notethat all of these strain measures are bounded by the Green–Lagrange strain measurefor the upper bound and by the Karni strain measure for the lower bound.

Figure 1.5 Comparison between various Lagrangian strain measures

Let us now express, for example, the Lagrangian strain tensor C in terms of the

first displacement gradient (see [1.5]):

T T

[1.38]

The small strain theory mentioned above thus consists of assuming that all of thecomponents of the tensor Grad( u ) are very small compared to the unity, so we canreasonably disregard the term containing the double product of the displacementgradient in [1.38], and thus, obtain the definition of the small or infinitesimalstrain tensor:

This assumption can be easily illustrated in the case of the one-dimensional bardiscussed above In fact, we define the first component of the small strain tensor at

time t according to the elongation rate ȁ( ) t by:

2 2

Finally, we note that the tensor of infinitesimal rotations (or small rotations) isgiven by the antisymmetric part of the displacement gradient tensor:

T

1 Grad(u ) Grad(u )

We confirm that Grad( u ) is like any second-rank tensor

1.1.3.3 Strain rates and rotation rates (spin) tensors

The time derivative of [1.9] leads to:

Thus, the derivative with respect to time F of gradient F is nothing but the

Lagrangian gradient of the velocity vector of the material point

Moreover, the Eulerian velocity gradient is written considering [1.44]:

1

The second-rank tensor L is thus simply the Eulerian velocity gradient, which,

according to the inverse of the transformation gradient [1.9], allows us to express[1.43] versus dx:

Thus, E ( / 2)C is the symmetric Lagrangian strain rate and D is the Eulerian

strain rate defined as being the symmetric part of the Eulerian velocity gradient tensor

L defined by [1.45] Thus, as with any second-rank non-symmetric tensor, L can be broken down into a symmetric part D measuring the strain rates and an antisymmetric

part ȍ measuring the rotation rates (or spin) in the current configuration :C t

We note that, according to [1.53] and [1.54], the rotation of the rigid body affectsthe strain rate and that pure dilatation affects the material rotation rate ȍ, which isdistinct from the proper rotation rateW

1.1.3.4 Volumic dilatation rate, relative extension rate and angular sliding rate Now we will calculate the derivative of J with respect to time by using [1.12]:

Additionally, the derivatives with respect to time of the elementary volume dV t

are given by the mixed product of the three elementary vectors dx , dx , dx1 2 3 asshown below (see [1.11]):

The relationship [1.47] can be used to determine the elongation of a materialfiber in a particular directiondx dx , with:

called the rate of instantaneous relative elongation in the material direction carried

by m If m is collinear, for example to e1, the first vector of the orthonormal basis

of the Euclidian frame, we get:

1

11 1

The rate of angular sliding of two material directions dx dx m and

dx dx m in the current configurationC t can also be calculated:

If, for example, we take m collinear to e1 and m collinear to e2 (unit basevectors of the orthonormal Euclidian triad), then the rate of sliding is exactly twicethe value of the shearing component D12 of the Eulerian strain rate tensor:

12

Applied to the three principal directions of the strain rate tensor D takentwo-by-two, this relationship leads to 0, thus allowing the following definition:

“the principal directions of the strain rate tensor are the orthogonal directions for

which the rate of sliding is identically null”.

It remains to examine the evolution of an elementary surface by using therelationship of transformation of an oriented elementary surface [1.16], which werewrite in the following equivalent form:

1 0( )T

1.1.4 Various stress measures

Let us consider the area , initially occupying the configuration C0 and

currently the configuration C, and let us examine the elementary section dA0oriented by the normal n0 in C0 so that dA0 n dA0 0, which is transformed intoelementary section dA oriented by the normal n t in C t so that dA n dA t t t

(Figure 1.6) The elementary resultant force exerted at point P t of configuration C t

on the section oriented by the normal n t is written as ( )n ( )n

dF t dA where ( )n

t

t isthe elementary tension vector in this point

The most widely used measure of the stress in a point of a continuum is theCauchy stress (or true stress), which is defined using the measure of the elementaryinternal force in a point P t of current configuration C t The Euler–Cauchy principlepostulates that at point P t of configuration C t there is a symmetric second-rank

tensor called the Cauchy stress tensor linked to the elementary tension vector( )n

Figure 1.6 Representation of internal forces and definition of stress tensors

The resultant elementary force exerted in P t is thus written in configuration

The second-rank operator is called the Boussinesq or Piola–Lagrange stresstensor, defined by:

1 T

This tensor is clearly non-symmetric and, like F, it is neither purely Euleriannor purely Lagrangian It can serve perfectly to express the equilibrium of a solid,since it can be associated with the appropriate boundary conditions on the currentdeformed configuration C t.

Considering [1.69] and [1.16], let us now perform the convective inversetransformation of the elementary resultant force vector ( n )

Finally, we introduce the Eulerian Kirchhoff stress tensor as being the

“correction” by J det( F ) of the Cauchy stress tensor:

Note that the Boussinesq stress tensor that defines current stresses in thereference configuration is often called nominal stress tensor In fact, if themeasurement of the current force in the current configuration is very easy, this is not

the case for the measurement of the current deformed area, which is not a trivialtask Hence, defining a nominal stress tensor by relating current forces to thereference area holds obvious practical interest for engineers.

T

F.S.F J

Table 1.2 Relationships between various stress measures

As in section 1.1.3.2, for the strain measures, let us illustrate the differentrelationships between stress measures in the simple one-dimensional case of a bar

occupying at initial time t0 the reference configuration defined by the reference

section A0and the reference length l0 Subject to the effect of an axial tension force

F( t ) , the bar is deformed to occupy at time t the current configuration defined by current section A t and current length l t Let us consider, as in section 1.1.3.2, therelative elongation ( ) ( / )t l l t 0 and note ( ) ( / )t A A t 0 the ratio of the areas of

the bar during its deformation The Cauchy stress tensor in the bar at a given time t

has the following form:

( t ) J ( t ) ( t ) ( t ) ( t )

F( t ) ( t ) A

[1.75]

1.1.5 Conjugate strain and stress measures

We have seen several definitions of strain tensors and several definitions ofstress tensors Constitutive models (see section 1.3) are simply adequaterelationships between strain tensors and stress tensors In order to be able to saywhich strain tensor can be in relation with which stress tensor when constructing theconstitutive equations, we must express the density of massic power of the internalforces on the various configurations:

1.1.6 Change of referential or configuration and the concept of objectivity

All of the mechanical quantities introduced above are expressed in a directorthonormal Euclidian triad called . We examine now how these quantities areaffected when we proceed to a change of referential from the triad to the triadvia Euclidian transformation of type:

where c( t ) is the Euclidian vector representing the translation of the triad, Q( t ) is

an orthogonal tensor (Q.Q T Q Q T 1,Q 1 Q T, and det( Q ) 1) representingthe (rigid body) rotation of the triad, and t0 is the reference time We also suppose,

to simplify the matter, that the two triads and overlap at the origin of time

0 0

t , which givesQ( )0 1 and c( )0 0

In this change of referential, the transformation gradient given by [1.3] changesinto:

F is not strictly a tensor, but is often called a bipoint tensor.

If the two triads do not overlap at the origin of the time, then the ratio abovebecomes:

1.1.6.1 Impact on strain and strain rates

All Lagrangian strain measures and their rates measured with respect to theLagrangian triad are objective To prove this, we apply [1.79] to the rightCauchy–Green strain tensor to obtain, considering [1.80]: