Damage Mechanics

Damage Mechanics in Metal Forming Damage Mechanics in Metal Forming Advanced Modeling and Numerical Simulation Khemais Saanouni Series Editor Pierre Devalan First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2012 The rights of Khemais Saanouni to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988 Library of Congress Cataloging-in-Publication Data Saanouni, Khemais, 1955Damage mechanics in metal forming : advanced modeling and numerical simulation / Khemais Saanouni p cm Includes bibliographical references and index ISBN 978-1-84821-348-7 Metals Plastic properties Metal-work Mathematical models Metal-work Quality control Deformations (Mechanics) Mathematical models Boundary value problems I Title TA460.S12 2012 620.1'6 dc23 2011051811 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-348-7 Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY Image: created by UTT/LASMIS Table of Contents Preface xiii Principle of Mathematical Notations xix Chapter Elements of Continuum Mechanics and Thermodynamics 1.1 Elements of kinematics and dynamics of materially simple continua 1.1.1 Homogeneous transformation and gradient of transformation 1.1.1.1 Homogeneous transformation 1.1.1.2 Gradient of transformation and its inverse 1.1.1.3 Polar decomposition of the transformation gradient 1.1.2 Transformation of elementary vectors, surfaces and volumes 1.1.2.1 Transformation of an elementary vector 1.1.2.2 Transformation of an elementary volume: the volume dilatation 1.1.2.3 Transformation of an oriented elementary surface 1.1.3 Various definitions of stretch, strain and strain rates 1.1.3.1 On some definitions of stretches 1.1.3.2 On some definitions of the strain tensors 1.1.3.3 Strain rates and rotation rates (spin) tensors 1.1.3.4 Volumic dilatation rate, relative extension rate and angular sliding rate 1.1.4 Various stress measures 1.1.5 Conjugate strain and stress measures 1.1.6 Change of referential or configuration and the concept of objectivity 1.1.6.1 Impact on strain and strain rates 1.1.6.2 Impact on stress and stress rates 2 5 8 10 15 17 19 23 23 24 26 vi Damage Mechanics in Metal Forming 1.1.6.3 Impact on the constitutive equations 1.1.7 Strain decomposition into reversible and irreversible parts 1.2 On the conservation laws for the materially simple continua 1.2.1 Conservation of mass: continuity equation 1.2.2 Principle of virtual power: balance equations 1.2.3 Energy conservation First law of thermodynamics 1.2.4 Inequality of the entropy Second law of thermodynamics 1.2.5 Fundamental inequalities of thermodynamics 1.2.6 Heat equation deducted from energy balance 1.3 Materially simple continuum thermodynamics and the necessity of constitutive equations 1.3.1 Necessity of constitutive equations 1.3.2 Some fundamental properties of constitutive equations 1.3.2.1 Principle of determinism or causality axiom 1.3.2.2 Principle of local action 1.3.2.3 Principle of objectivity or material indifference 1.3.2.4 Principle of material symmetry 1.3.2.5 Principle of consistency 1.3.2.6 Thermodynamic admissibility 1.3.3 Thermodynamics of irreversible processes The local state method 1.3.3.1 A presentation of the local state method 1.3.3.2 Internal constraints 1.4 Mechanics of generalized continua Micromorphic theory 1.4.1 Principle of virtual power for micromorphic continua 1.4.2 Thermodynamics of micromorphic continua 29 30 33 33 34 36 37 38 39 39 40 41 42 42 42 43 43 44 44 44 52 55 58 59 Chapter 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 2.2 Behavior and fracture of metals and alloys: some physical and phenomenological aspects 2.2.1 On the microstructure of metals and alloys 2.2.2 Phenomenology of the thermomechanical behavior of polycrystals 2.2.2.1 Linear elastic behavior 2.2.2.2 Inelastic behavior 2.2.2.3 Inelastic behavior sensitive to the loading rate 2.2.2.4 Initial and induced anisotropies 2.2.2.5 Other phenomena linked to the shape of the loading paths 2.2.3 Phenomenology of the inelastic fracture of metals and alloys 64 69 69 70 71 72 74 76 77 82 Table of Contents 2.2.3.1 Micro-defects nucleation 2.2.3.2 Micro-defects growth 2.2.3.3 Micro-defects coalescence and final fracture of the RVE 2.2.3.4 A first definition of the damage variable 2.2.3.5 From ductile damage at a material point to the total fracture of a structure by propagation of macroscopic cracks 2.2.4 Summary of the principal phenomena to be modeled 2.3 Theoretical framework of modeling and main hypotheses 2.3.1 The main kinematic hypotheses 2.3.1.1 Choice of kinematics and compliance with the principle of objectivity 2.3.1.2 Decomposition of strain rates 2.3.1.3 On some rotating frame choices 2.3.2 Implementation of the local state method and main mechanical hypotheses 2.3.2.1 Choice of state variables associated with phenomena being modeled 2.3.2.2 Definition of effective variables: damage effect functions 2.4 State potential: state relations 2.4.1 State potential in case of damage anisotropy 2.4.1.1 Formulation in strain space: Helmholtz free energy 2.4.1.2 Formulation in stress space: Gibbs free enthalpy 2.4.2 State potential in the case of damage isotropy 2.4.2.1 Formulation in strain space: Helmholtz free energy 2.4.2.2 Formulation in stress space: Gibbs free enthalpy 2.4.3 Microcracks closure: quasi-unilateral effect 2.4.3.1 Concept of micro-defect closure: deactivation of damage effects 2.4.3.2 State potential with quasi-unilateral effect 2.5 Dissipation analysis: evolution equations 2.5.1 Thermal dissipation analysis: generalized heat equation 2.5.1.1 Heat flux vector: Fourier linear conduction model 2.5.1.2 Generalized heat equation 2.5.2 Intrinsic dissipation analysis: case of time-independent plasticity 2.5.2.1 Damageable plastic dissipation: anisotropic damage with two yield surfaces 2.5.2.2 Damageable plastic dissipation: anisotropic damage with a single yield surface 2.5.2.3 Incompressible and damageable plastic dissipation: isotropic damage with two yield surfaces 2.5.2.4 Incompressible and damageable plastic dissipation: single yield surface vii 84 85 85 86 89 90 91 91 92 94 99 102 103 108 113 114 114 121 124 124 128 129 129 132 139 140 141 141 143 144 157 162 169 viii Damage Mechanics in Metal Forming 2.5.3 Intrinsic dissipation analysis: time-dependent plasticity or viscoplasticity 2.5.3.1 Damageable viscoplastic dissipation without restoration: anisotropic damage with two viscoplastic potentials 2.5.3.2 Viscoplastic dissipation with damage: isotropic damage with a single viscoplastic potential and restoration 2.5.4 Some remarks on the choice of rotating frames 2.5.5 Modeling some specific effects linked to metallic material behavior 2.5.5.1 Effects of non-proportional loading paths on strain hardening evolution 2.5.5.2 Strain hardening memory effects 2.5.5.3 Cumulative strains or ratchet effect 2.5.5.4 Yield surface and/or inelastic potential distortion 2.5.5.5 Viscosity-hardening coupling: the Piobert–Lüders peak 2.5.5.6 Accounting for the material microstructure 2.5.5.7 Some specific effects on ductile fracture 2.6 Modeling of the damage-induced volume variation 2.6.1 On the compressibility induced by isotropic ductile damage 2.6.1.1 Concept of volume damage 2.6.1.2 State coupling and state relations 2.6.1.3 Dissipation coupling and evolution equations 2.7 Modeling of the contact and friction between deformable solids 2.7.1 Kinematics and contact conditions between solids 2.7.1.1 Impenetrability condition 2.7.1.2 Equilibrium condition of contact interface 2.7.1.3 Contact surface non-adhesion condition 2.7.1.4 Contact unilaterality condition 2.7.2 On the modeling of friction between solids in contact 2.7.2.1 Time-independent friction model 2.8 Nonlocal modeling of damageable behavior of micromorphic continua 2.8.1 Principle of virtual power for a micromorphic medium: balance equations 2.8.2 State potential and state relations for a micromorphic solid 2.8.3 Dissipation analysis: evolution equations for a micromorphic solid 2.8.4 Continuous tangent operators and thermodynamic admissibility for a micromorphic solid 2.8.5 Transformation of micromorphic balance equations 2.9 On the micro–macro modeling of inelastic flow with ductile damage 2.9.1 Principle of the proposed meso–macro modeling scheme 174 176 182 186 189 190 191 191 192 192 193 193 194 195 195 196 197 200 201 203 204 205 205 206 206 215 217 218 221 223 224 226 227 Table of Contents 2.9.2 Definition of the initial RVE 2.9.3 Localization stages 2.9.4 Constitutive equations at different scales 2.9.4.1 State potential and state relations 2.9.4.2 Intrinsic dissipation analysis: evolution equations 2.9.5 Homogenization and the mean values of fields at the aggregate scale 2.9.6 Summary of the meso–macro polycrystalline model 230 230 233 233 235 239 240 Chapter Numerical Methods for Solving Metal Forming Problems 243 3.1 Initial and boundary value problem associated with virtual metal forming processes 3.1.1 Strong forms of the initial and boundary value problem 3.1.1.1 Posting a fully coupled problem 3.1.1.2 Some remarks on thermal conditions at contact interfaces 3.1.2 Weak forms of the initial and boundary value problem 3.1.2.1 On the various weak forms of the IBVP 3.1.2.2 Weak form associated with equilibrium equations 3.1.2.3 Weak form associated with heat equation 3.1.2.4 Weak form associated with micromorphic damage balance equation 3.1.2.5 Summary of the fully coupled evolution problem 3.2 Temporal and spatial discretization of the IBVP 3.2.1 Time discretization of the IBVP 3.2.2 Spatial discretization of the IBVP by finite elements 3.2.2.1 Spatial semi-discretization of the weak forms of the IBVP 3.2.2.2 Examples of isoparametric finite elements 3.3 On some global resolution scheme of the IBVP 3.3.1 Implicit static global resolution scheme 3.3.1.1 Newton–Raphson scheme for the solution of the fully coupled IBVP 3.3.1.2 On some convergence criteria 3.3.1.3 Calculation of the various terms of the tangent matrix 3.3.1.4 The purely mechanical consistent Jacobian matrix 3.3.1.5 Implicit global resolution scheme of the coupled IBVP 3.3.2 Dynamic explicit global resolution scheme 3.3.2.1 Solution of the mechanical problem 3.3.2.2 Solution of thermal (parabolic) problem 3.3.2.3 Solution of micromorphic damage problem 3.3.2.4 Sequential scheme of explicit global resolution of the IBVP 3.3.3 Numerical handling of contact-friction conditions ix 244 245 245 250 252 252 254 257 258 258 259 259 260 260 266 270 272 273 275 276 280 282 284 284 286 288 288 291 x Damage Mechanics in Metal Forming 3.3.3.1 Lagrange multiplier method 3.3.3.2 Penalty method 3.3.3.3 On the search for contact nodes 3.3.3.4 On the numerical handling of the incompressibility condition 3.4 Local integration scheme: state variables computation 3.4.1 On numerical integration using the Gauss method 3.4.2 Local integration of constitutive equations: computation of the stress tensor and the state variables 3.4.2.1 On the numerical integration of first-order ODEs 3.4.2.2 Choice of constitutive equations to integrate 3.4.2.3 Integration of time-independent plastic constitutive equations: the case of a von Mises isotropic yield criterion 3.4.2.4 Integration of time-independent plastic constitutive equations: the case of a Hill quadratic anisotropic yield criterion 3.4.2.5 Integration of the constitutive equation in the case of viscoplastic flow 3.4.2.6 Calculation of the rotation tensor: incremental objectivity 3.4.2.7 Remarks on the integration of the micromorphic damage equation 3.4.3 On the local integration of friction equations 3.5 Adaptive analysis of damageable elasto-inelastic structures 3.5.1 Adaptation of time steps 3.5.2 Adaptation of spatial discretization or mesh adaptation 3.6 On other spatial discretization methods 3.6.1 An outline of non-mesh methods 3.6.2 On the FEM–meshless methods coupling 293 295 296 300 304 304 305 306 308 313 326 328 333 335 335 337 339 341 347 348 353 Chapter Application to Virtual Metal Forming 4.1 Why use virtual metal forming? 4.2 Model identification methodology 4.2.1 Parametrical study of specific models 4.2.1.1 Choosing typical constitutive equations 4.2.1.2 Isothermal uniaxial tension (compression) load without damage 4.2.1.3 Accounting for ductile damage effect 4.2.1.4 Accounting for initial anisotropy in inelastic flow 4.2.2 Identification methodologies 4.2.2.1 Some general remarks on the issue of identification 4.2.2.2 Recommended identification methodology 4.2.2.3 Illustration of the identification methodology 4.2.2.4 Using a nonlocal model 355 356 359 360 360 364 383 396 413 414 416 422 429 Table of Contents 4.3 Some applications 4.3.1 Sheet metal forming 4.3.1.1 Some deep drawing processes of thin sheets 4.3.1.2 Some hydro-bulging test of thin sheets and tubes 4.3.1.3 Cutting processes of thin sheets 4.3.2 Bulk metal forming processes 4.3.2.1 Classical bulk metal forming processes 4.3.2.2 Bulk metal forming processes under severe conditions 4.4 Toward the optimization of forming and machining processes xi 431 431 432 441 447 463 463 476 484 Appendix: Legendre–Fenchel Transformation 493 Bibliography 499 Index 515 508 Damage Mechanics in Metal Forming [LOW 05] LOWEN R., VERSCHOREN A (eds), Foundation of Generic Optimization Vol 1: A Combinatorial Approach to Epistasis, Springer Verlag, Dordrecht, 2005 [LOW 07] LOWEN R., VERSCHOREN A (eds), Foundation of Generic Optmization Vol 2: Application of Fuzzy Control, Genetic Algorithms and Neural Networks, Springer Verlag, New York, 2007 [LUB 02] LUBARDA V.A., Elastoplasticity Theory, CRC Press, Boca Raton, 2002 [LUB 90] LUBLINER J., Plasticity Theory, McMillan, New York, 1990 [MAA 02] MAARCINIAK Z., DUNCAN J.L., HU S.J., Mechanics of Sheet Metal Forming, 2nd edition, Butterworth-Heinemann, London, 2002 [MAL 69] MALVERN L.E., Introduction to the Mechanics of Continuous Medium, Prentice Hall, Englewood Cliffs, 1969 [MAN 66] MANDEL J., Mécanique des milieux continus Vols I and II, Gauthier-Villars, Paris, 1966 [MAN 73] MANDEL J., “Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques”, International Journal of Solids and Structures, vol 9, no 6, pp 725–740, 1973 [MAN 74] MANDEL J., Introduction la mécanique des milieux continus déformables, Editions Scientifiques de Pologne, Varsovie, 1974 [MAN 78] MANDEL J., Propriétés mécaniques des matériaux, Eyrolles, Paris, 1978 [MAR 75] MARTIN J.B., Plasticity: Fundamentals and General Results, MIT Press, Cambridge, 1975 [MAU 92] MAUGIN G.A., The Thermomechanics of Plasticity and Fracture, Cambridge University Press, Cambridge, 1992 [MCC 66] MCCLINTOCK F.A., ARGON A.S., Mechanical Behaviour of Materials, Addison Wesley, Reading, MA, 1966 [MCL 80] MCLELLAN A.G., The Classical Thermodynamics of Deformable Materials, Cambridge University Press, Cambridge, 1980 [MEN 12] MENG F., Optimisation multi-objectif d’une gamme de forge partir de métamodèles, PhD thesis, University of Technology of Troyes, Troyes, 2012 [MIE 91] MIELNIK E.M., Metalworking Science and Engineering, McGraw-Hill, New York, 1991 [MIE 99] MIETTINEN K., Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Dordrecht, 1999 [MON 86] MONTHEILLET F., MOUSSY F., Physique et mécanique de l’endommagement, Les Editions de Physique, Les Ulis, 1986 [MOR 66] MOREAU J.J., Fonctionnelles convexe, Cours Collège de France, Paris, 1966 Bibliography 509 [MÜL 95] MÜLHAUS H., Continuum Models for Materials with Microstructure, John Wiley & Sons, New York, 1995 [MÜL 85] MÜLLER I., Thermodynamics, Pitman, Boston, 1985 [MUN 04] MUNJIZA A., The Combined Finite-discrete Element Method, John Wiley & Sons, Chichester, 2004 [MUR 87] MURA T., Micromechanics of Defects in Solids, Martinus Nijhoff Publishers, Dordrecht, 1987 [MUR 12] MURAKAMI S., Continuum Damage Mechanics A Continuum Mechanics Approach to the Analysis of Damage and Fracture, Springer Verlag, Dordrecht, 2012 (forthcoming) [NAY 92] NAYROLES B., TOUZOT G., VILLON P., “Generalizing the FEM: Diffuse approximation and diffuse elements”, Comput Mech., vol 10, pp 307–318, 1992 [NEM 93] NEMAT-NASSER S., HORI M., Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam, 1993 [NEM 04] NEMAT-NASSER S., Plasticity A Treatise on Finite Deformation of Heterogeneous Inelastic Materials, Cambridge University Press, Cambridge, 2004 [NGU 12] NGUEN D T, Anisotropie de l’endommagement et simulation numérique en mise en forme par grandes déformations plastiques, PhD thesis, University of Technology of Troyes, Troyes 2012 [NIC 03] NICHOLSON D.W., Finite Element Analysis: Thermomechanics of Solids, CRC Press, Boca Raton, 2003 [ODE 72] ODEN J.T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972 [OGD 84] OGDEN R.W., Non-linear Elastic Deformations, Ellis Horwood/John Wiley & Sons, Chichester, 1984 [OHN 87] OHNAMI M., Plasticity and High Temperature Strength of Materials, Combined Micro and Macro Mechanical Approaches, Elsevier, Norwich, 1987 [OWE 80] OWEN D.R.J., HINTON E., Finite Elements in Plasticity Theories and Practice, Pineridge Press Limited, Swansea, 1980 [PAL 10] PALLASCHKE D., ROLEWICZ S., Foundation of Mathematical Optimization: Convex Analysis Without Linearity, Kluwer Academic Publishers, Dordrecht, 2010 [PAR 08] PARIS T., Modélisation du comportement mécanique des liaisons soudées hétérogènes Ta/TA6V: comportement et critère de rupture, PhD thesis, University of Technology of Troyes, no 2008TROY0026, Troyes, 2008 [PRA 58] PRAGER W., Problèmes de plasticité théorique, Dunod, Paris, 1958 [PRI 68] PRIGOGINE I., Introduction la thermodynamique des processus irréversibles, Dunod, Paris, 1968 [RAA 98] RAABE D., Computational Material Science: The Simulation of Materials Microstructures and Properties, Wiley-VCH, Weinheim, 1998 510 Damage Mechanics in Metal Forming [RAB 65] RABINOWICZ E., Friction and Wear of Materials, John Wiley & Sons, New York, 1965 [RAD 74] RADENKOVIC D., SALENÇON J., Plasticité et viscoplasticité, McGraw-Hill, New York, 1974 [RAG 95] RAGONE D.V., Thermodynamics of Materials, John Wiley & Sons, New York, 1995 [RAO 78] RAO S.S., Optimization: Theory and Application, John Wiley & Sons, New York, 1978 [RAO 82] RAO S.S., The Finite Element Method in Engineering, 2nd edition, Pergamon Press, Oxford, 1982 [RAO 95] RAOUS M., MOREAU J.J., Contact Mechanics, Plenum Press, New York, 1995 [RED 93] REDDY J.N., An Introduction to the Finite Element Method, McGraw-Hill, New York, 1993 [RED 04] REDDY J.N., An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Oxford, 2004 [REE 82] REES D.W., “Yield functions that account for the effects of initial and subsequent plastic anisotropy”, Acta Mechanica, vol 43, pp 223–241, 1982 [ROC 70] ROCKAFELLAR R.T., Convex Analysis, Princeton University Press, Princeton, 1970 [ROU 97] ROUGEE P., Mécanique des grandes transformations, Springer Verlag, Heidelberg, 1997 [ROU 85] ROUSSET M., Surface seuil de plasticité: détermination automatique et modélisation, PhD thesis, University Pierre and Marie Curie Paris VI, Paris, 1985 [ROW 91] ROWE G.H., STURGUESS C.E.N., HARTLEY P., PILLINGER I., Finite Element Plasticity and Metal Forming Analysis, Cambridge University Press, Cambridge, 1991 [SAA 03] SAANOUNI K (ed.), Numerical Modelling in Damage Mechanics, Kogan Page Science, Guildford, 2003 [SAL 77] SALENÇON J., Application of the Theory of Plasticity in Soil Mechanics, John Wiley & Sons, Chichester, 1977 [SAL 83] SALENÇON J., Viscoélasticité, Presses de l’école des Ponts et Chaussées, Paris, 1983 [SAL 88] SALENÇON J., Mécanique des milieux continus, tome 1: Concepts généraux, tome 2: Elasticité-milieux curviligne, Ellipses, Paris, 1988 [SCH 68] SCHMID E., BOAS W., Plasticity of Crystals, Chapman & Hall, London, 1968 [SCH 98] SCHULLER GMBH, Metal Forming Handbook, Springer Verlag, Heidelberg, 1998 [SCH 75] SCHWARTZ L., Les tenseurs, Hermann, Paris, 1975 [SÉD 75] SEDOV L., Mécanique des milieux continus, tomes I and II, Editions Mir, Moscow, 1975 Bibliography 511 [SEW 87] SEWELL M.J., Maximum and Minimum Principles, Cambridge University Press, Cambridge, 1987 [SHI 04] SHILLOR M., SOFONEA M., TELEGA J.J., Models and Analysis of Quasistatic Contact: Variational Methods, Springer Verlag, Berlin, 2004 [SHP 95] SHPENKO G.P., Friction Surface Phenomena, Elsevier, Amsterdam, 1995 [SID 01] SIDOROFF F., DOGUI A., “Some issues about anisotropic elastic-plastic models at finite strain”, International Journal of Solids and Structures, 38, pp 9569–9578, 2001 [SIM 98] SIMO J.C., HUGHES T.J.R., Computational Inelasticity, Springer Verlag, New York, 1998 [SKR 99] SKRZYPECK J., GANCZARSKI A., Modeling of Material Damage and Failure of Structures, Springer Verlag, Berlin, 1999 [SMI 93] SMITH D.R., An Introduction to Continuum Mechanics, Kluwer Academic Publisher, Dordrecht, 1993 [SOF 06] SOFONEA M., HUN W., SHILLOR M., Analysis and Approximation of Contact Problems with Adhesion and Damage, Chapman & Hall/CRC, 2006 [SOK 64] SOKOLNIKOFF I.S., Manifolds, Tensor Analysis and Applications, 2nd edition, John Wiley & Sons, New York, 1964 [SOR 07] SORNIN D., Sur les formulations non locales en gradient d’endommagement, PhD thesis, University of Technology of Troyes, no 2007TROY0011, Troyes, 2007 [SOU 01] SOUCHET R., Leçons sur les grandes déformations, Cépaduès, Toulouse, 2001 [STE 03] STEIN E (ed.), Error-controlled Adaptive Finite Elements in Solid Mechanics, John Wiley & Sons, Chichester, 2003 [STO 70] STOJANOVIC R., Recent Developments in the Theory of Polar Continua (CISM Lecture Notes), Springer Verlag, Vienna, 1970 [STO 87] STOLZ C., Anélasticité et stabilité, PhD thesis, University Pierre and Marie Curie Paris VI, Paris, 1987 [STO 09] STOLZ C., Milieux continus en transformations finies: hyperélasticité, rupture, élastoplasticité, Editions de l’Ecole polytechnique, Ellipse, Paris, 2009 [STR 76] STRANG G., FIX G.J., An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, 1976 [SUQ 97] SUQUET P (ed.), Continuum Micromechanics, CISM Courses and Lectures, Springer Verlag, Udine, Italy, 1997 [SWA 72] SWALIN R.A., Thermodynamics of Solids, John Wiley & Sons, New York, 1972 [SZA 91] SZABO B., BABUSKA I., Finite Element Analysis, John Wiley & Sons, New York, 1991 512 Damage Mechanics in Metal Forming [TAN 94] TANAKA M., BUI H.D (eds), Inverse Problems in Engineering Mechanics I, Springer Verlag, New York, 1994 [TAN 00] TANAKA M., DULIKRAVICH G.S (eds), Inverse Problems in Engineering Mechanics II, Elsevier, Paris, 2000 [TAN 03] TANAKA M (ed.), Inverse Problems in Engineering Mechanics IV, Elsevier, Paris, 2003 [THO 65] THOMSEN E.G., YANG C.T., KOBAYASHI S., Mechanics of Plastic Deformation in Metal Processing, The Macmillan Company, New York, 1965 [TOL 98] TOLLENAERE M (ed.), Conception de Produits Mécaniques: Méthodes, Modèles et Outils, Hermès, Paris, 1998 [TRU 65] TRUESDELL C., NOLL W., The Non-linear Filed Theories of Mechanics, 1st edition, Springer Verlag, New York, 1965 [TRU 66] TRUESDELL C., The Elements of Continuum Mechanics, Springer Verlag, New York, 1966 [TRU 84] TRUESDELL C., Rational Thermodynamics, Springer Verlag, New York, 1984 [TRU 92] TRUESDELL C., NOLL W., The Non-linear Filed Theories of Mechanics, 2nd edition, Springer Verlag, New York, 1992 [TRU 04] TRUESDELL C., NOLL W., The Non-linear Filed Theories of Mechanics, 3rd edition, Springer Verlag, New York, 2004 [VER 96] VERFÜRTH R., A Review of A Posteriori Error Estimation and Adaptive MeshRefinement Techniques, John Wiley & Sons, Chichester, 1996 [VIL 02] VILLON P., BOROUCHAKI H., SAANOUNI K., “Transfert de champs plastiquement admissibles”, Comptes rendus de l’Académie des sciences, mécanique, vol 330, pp 313–318, 2002 [VOY 99] VOYIADJIS G.Z., KATTAN P.I., Advances in Damage Mechanics: Metals and Metal Matrix Composite, Elsevier, Amsterdam, 1999 [WAG 01] WAGONER R.H., CHENOT J.L., Metal Forming Analysis, Cambridge University Press, Cambridge, 2001 [WAT 07] WATZKY A., Thermodynamique Macroscopique, De Boeck Université, Brussels, 2007 [WIN 79] WINOGRADZKI J., Les méthodes tensorielles de la physique Calcul tensoriel dans un continuum amorphe, Dunod, Paris, 1979 [WRI 99] WRIGGERS P., PANAGIOTOPOULOS P., New Developments in Contact Problems, Springer Verlag, Vienna, 1999 [WRI 02] WRIGGERS P., Computational Contact Mechanics, John Wiley & Sons, New York, 2002 [WRI 08] WRIGGERS P., Nonlinear Finite Element Methods, Springer Verlag, Berlin, 2008 Bibliography 513 [YAN 93] YANG W., LEE W.B., Mesoplasticity and Its Applications, Springer Verlag, Berlin, 1993 [ZHO 93] ZHONG Z.H., Finite Element Procedures for Contact-Impact Problems, Oxford University Press, Oxford, 1993 [ZIE 83] ZIEGLER H., An Introduction to Thermomechanics, Springer Verlag, Berlin, 1983 [ZIE 95] ZIEGLER H., Mechanics of Solids and Fluids, 2nd edition, North-Holland Publications, Amsterdam, 1995 [ZIE 05] ZIEKIEWICZ O.C., TAYLOR L.R., The Finite Element Method for Solids and Structural Mechanics, 6th edition (1st edition in 1967), Elsevier, Burlington, 2005 [ZYC 81] ZYCZKOWSKI M., Combined Loadings in the Theory of Plasticity, Polish Scientific Publishers, Warsaw, 1981 Index A adaptation of spatial discretization, 341-347 of time steps, 339-341 adiabatic, 143, 479 algorithm(s), 292, 295, 297-300, 308, 313, 316, 325, 329, 337, 341-343, 351, 416, 484, 485, 491, 492 anisotropy damage, 91, 113, 114, 138, 175, 412, 413 induced, 31, 412 assembly, 69, 265, 290, 294, 350, 351, 356, 446 B, C behavior elastic, 71-72, 76, 125, 228, 233, 301 inelastic, 64, 72-74, 77, 92, 240, 301 thermoelastic, 104 changing reference configuration, 23, 24, 26, 43 referential, 23, 24, 26-27, 29, 42 chips, 356, 457, 477, 479, 480 closing microcracks, 129, 130, 131, 138, 162, 244, 308, 384, 392, 394-396, 419 coalescence, 65, 83, 85, 86, 89-91, 105, 140, 194, 200, 215, 227, 228 coefficient friction, 212, 433, 435, 439, 449, 452, 459, 464, 466, 473, 478, 487 Lamé, 125, 219 Lankford, 402-404, 423, 424f, 425, 427 Poisson, 125 compressibility, 64, 90, 91, 114, 125, 149, 194, 195, 200, 236, 301, 495 concept of objectivity, 23 condition of coherence, 124, 129, 486 consistency, 50, 51, 150, 151, 159, 164-166, 171, 175, 179, 199, 214, 222, 310, 354 impenetrability, 203-205, 209 incompressibility, 271, 272, 293, 300, 301, 333 non-adhesion, 203, 205 conditions initial, 27, 28, 99, 248, 259, 282, 288 516 Damage Mechanics in Metal Forming Kuhn–Tucker, 49, 50, 150, 175, 211, 249, 310, 317, 328, 336 limit, 24, 44, 56, 57, 72, 73, 101, 103, 106-108, 114 rupture, 82, 297, 385, 390, 464, 484 configuration current, 3, 5, 6, 8, 10-12, 16-19, 21, 22, 27, 29, 30, 34, 35, 37, 38, 47, 58-61, 93, 204, 246, 346 intermediate, 30, 31, 93, 97 reference, 3, 5, 8, 11, 12, 21, 22, 24, 26, 27, 29, 35, 37, 38, 43, 45-47, 93, 252, 253, 259, 260, 458 conservation laws, 1, 33, 34, 40, 41, 43, 58, 203, 204, 206, 216 conserving material, 43 constraint-deformation conjugation, 53 contact interfaces, 63, 64, 201, 203, 204, 205, 209, 212, 213, 246, 250 contact nodes, 292, 294-297, 300 convergence, 226, 266, 272, 275, 276, 281, 283, 284, 304, 317, 322, 328, 332, 340, 405, 430 Coulomb, 206, 209-212, 214, 244, 249, 336, 433, 435, 439, 446, 449, 452, 457, 459, 461, 464, 466, 473, 479, 481, 487 coupling behavior-damage, 109, 112, 113, 155, 168, 195, 297, 383-385, 398, 406, 408-410, 413, 418, 419, 427 dissipation, 197 multiphysical, 41, 63, 272, 355 state, 196 thermoelastic, 104, 115, 119, 419 thermomechanical, 39, 90, 113, 417, 464 viscocity-hardening, 192 cracks in chevrons, 471, 472 criteria damage, 147, 150, 151, 157, 163-165, 179 flow, 64, 145, 174, 306, 339 friction, 209, 210, 213, 214, 215f, 336, 337 Hill quadratic, 310, 397 non-quadratic, 145 plasticity, 150, 155, 157, 158, 165 crystal aggregates, 70, 72, 73 crystallographic orientations, 66-68, 99, 227, 230 cutting, a sheet, 452 cyclic hardening, 78, 79f, 80, 81f, 374-376, 377f, 379 cyclical loadings, 82, 417 D damage, 30, 45, 50, 61, 63, 64, 68, 73, 82, 83, 84f, 85-87 anisotropic, 105, 108, 114, 118, 126, 128, 132, 138, 139, 142, 144, 149, 157, 158, 161, 162, 164, 167, 168, 176, 194, 412 ductile, 63-64, 82-91, 102-108, 113, 124, 130-132, 140-153, 157-165, 169, 173, 182-184, 192-195, 200, 215-217, 226-229, 233-245, 258, 284, 291, 301, 308, 355-356, 383-397, 410-413, 419, 427, 431, 438, 447, 463, 465, 473-479, 483-486 isotropic, 103, 132, 184, 195, 216, 217, 258, 308, 385, 397, 486 micromorphic, 217, 220, 223-226, 244-247, 258-267, 271, 274, 279, 283-284, 288-290, 306, 335, 421, 430 Index deburring, 465, 470, 471 decomposition additive, 32, 95, 98, 103, 114, 186, 233, 311, 313 deformation, 30 multiplier, 186 polar, 5, 7, 15, 32, 53, 94, 101, 238 deformation forced, 250 inelastic, plastic, 393, 411, 427, 431 derivative convective, 27 covariant, 28 Jaumann, 97, 98, 101, 311, 312 Lie contravariant, 27 rotational, 28, 29, 94, 413 Truesdell, 28 dilations left Cauchy-Green, 102, 121 pure right, 104 thermal, 116, 118, 125, 135, 196, 495 volume, 6, 31 direction principal of anisotropy, 397 Dirichlet, 34, 250, 256, 353 discretization spatial, 216, 252, 253, 259, 260, 262, 266, 267, 339, 341, 342, 345, 347, 420 temporal, 243, 259, 338 dissipation damageable plastic, 144-157, 162, 169, 200, 449, 464 intrinsic, 140, 142, 143, 161, 168, 173, 174, 180, 186, 221, 224, 236, 278, 305, 345 instrinsic volume, 247 thermal, 47, 103, 140-143, 174, 244 517 dissipative phenomena, 39, 41, 44-50, 52, 57, 64, 86, 92, 104, 105, 121, 128, 139, 140, 143, 148, 162, 164, 170, 175, 177, 182, 201, 220, 233, 235, 236, 244, 306, 362 instantaneous, 49, 140 E easy opening, 460 elasticity domain, 72, 73, 103104 elasticity limit, 72, 73 elastic prediction, 312, 316, 317f, 324-326, 329, 330, 335 elastic rigidity, 115 elements diffuse, 348 finite, 65, 66, 68, 241, 243, 244, 260, 266-268, 335, 338, 339, 347, 350f, 357, 431, 435 energy free, 37, 38, 40, 45, 47, 134 Helmholtz, 39, 58, 114-121, 124-128, 134, 218, 233, 234, 495 specific, 40 internal, 36-38, 44 environments continuous of superior degree, 56 of superior order, 56 generalized, 64, 68, 105, 193, 224 materially simple, 64, 108, 120 equation balance, 34-36, 39-41, 54, 56, 57, 59, 64, 217, 224, 244, 247, 250, 252, 258, 271 complementary, 64 evolution, 44, 47-49, 51, 52, 57, 66, 91, 109, 121, 128, 139, 143, 148 518 Damage Mechanics in Metal Forming heat, 39, 41, 140-143, 187, 247, 257, 283, 291, 419, 464 result, 31, 43, 65, 95, 96, 127, 151, 166, 197, 219, 275, 375 error estimators, 338, 339 error indicator, 338, 339, 341-342, 343f, 344, 345, 349 Eulerian deformation, 30, 36, 41, 142, 301 expansion test, 458 extrusion, 355, 463, 471-473 F first principle of thermodynamics, 37, 141 force micromorphic, 57, 59, 61, 218, 225, 247 micromorphic volume, 59 thermodynamic linked to damage, 120, 123, 124, 129, 134, 136, 137, 169, 220, 322, 385, 386f, 391f, 392f forging, 295, 355, 356, 360, 463, 465, 468-470, 473, 474f, 486, 488 form strong, 245, 249, 252, 257, 258, 351, 352 weak, 244, 252, 254-263, 265, 266, 276, 291, 293, 294, 305, 312, 338, 347, 351 formulation mixed, 253, 269, 294, 302, 352 non-associated, 153, 209, 214, 215, 405 on a single surface, 157 friction Coulomb, 211, 212, 244, 435, 439, 459, 461, 464, 466, 473, 478, 481, 487 friction model, 206, 209, 211-213, 247, 256, 263, 291, 355, 417, 433, 446, 449, 452, 459, 481 function convex, 39, 44, 47, 49, 99, 113, 141, 218, 493-496 effect of damage, 112, 389 interpolation, 260-263, 266, 277, 352, 354 G Galerkin, 261, 262, 294, 340 generalized continuum mechanics, 1, 61, 64 germination, 89 Gibbs free enthalpy, 47, 114, 121, 128, 136, 495 gradient Eulerian velocities, 15, 16, 25, 32 Lagrangian velocity vector, 15 temperature, 104, 140, 141 transformation, 1, 5, 15, 23, 24, 30f, 31, 32, 40, 43, 53-55, 93, 94, 100, 101, 238, 282, 288, 292, 303, 312, 334 velocity, 15, 16, 25, 28, 32, 41, 95-98, 101, 189, 239, 333, 334 Green–Naghdi, 28, 32, 101, 102, 152, 154, 187-189, 312, 334, 409 Green–Lagrange,11, 13, 32, 39 guillotining, 350, 354, 356, 455f, 456f H, I hardening kinematic, 73, 77, 78f, 103-107, 108f, 110, 112, 113, 117-120, 123-125 isotropic, 103-105, 110, 112, 119, 124, 143, 144, 153, 159, 178, 182-184, 191 micromorphic, 219, 421 Index hardening memory, 191 heat flux vector, 36, 40, 104, 141 Helmholtz, 37, 39, 45, 58, 110, 111, 114, 124, 134, 136, 218, 224, 225, 233, 234, 495 heterogeneities, 65, 66, 86, 89, 91 Hill parameters, 398, 404, 410 plastic orthotropy, 146 hydraulic presses, 486 hydroforming, 355, 444, 445f, 446, 447, 448f hypothesis of equivalence in total energy, 110, 111f hysteresis, 80, 374, 376f, 394, 419 identification, 157, 338, 356, 359, 360, 413-420, 422, 425-427, 430 models, 359 identification methodology, 359, 360, 413, 416, 417, 420, 425 impact, 24-26, 29, 32, 187, 188, 222, 292, 297, 299, 418, 481, 482f, 483f, 484, 488, 490 implicit prior Euler, 287, 295 incompressibility, 64, 74, 103, 114, 121, 125, 194, 254, 269, 271, 272, 293, 300-303, 401, 402 inequality Clausius–Duhem, 38, 45, 54, 57, 59, 60, 120, 121, 139, 140, 156 entropy, 56, 57, 59 inextensibility in certain directions, 54 internal connections, 347, 348 internal length, 56, 68, 193, 216, 225, 226, 421, 430, 431 isoparametric, 261, 263, 266-269, 301 isotropy, damage, 124 519 J, K, L Jaumann, 28, 97, 98, 101, 102, 144, 152, 154, 159, 160, 166, 167, 171, 173, 186-189, 199, 223, 311, 312, 334, 408, 409, 413 Kuhn–Tucker, 49, 50, 150, 175, 205, 211, 249, 294, 310, 317, 328, 336 Lagrangian deformations, 40, 41 Lagrange multipliers Lagrangian increased, 295 restricted, 54 trapeze, 338 landmark principal of anisotropy, 397 large deformations, 250, 253, 341 left Cauchy-Green, 9, 25, 121 linear thermo-elasticity, 103, 495 local state, 1, 40, 44, 52, 57, 58, 68, 91, 102, 105, 108, 113, 139 localization and homogenization, 66 localization flow, 215 temperature, 464 M manufacture, 356, 357, 358, 463, 473 material rotations, 17, 28, 96, 99, 100, 238, 312, 335, 408 matrix capacity, 264, 265, 287, 290 coherent material Jacobean, 325 concentrated, 84, 263 conduction, 264, 266 mass, 263-265, 284, 285, 289, 291 micromorphic damage, 264, 290 tangent, 272, 274, 276, 277, 279, 282, 284, 291, 292 meshing, 433, 435, 443f, 444f, 452, 461, 483, 486, 487 520 Damage Mechanics in Metal Forming metallic materials, 32, 33, 48, 49, 52, 63, 64, 69, 70, 73, 82, 86, 90, 91, 99, 100 metamodel, 485, 491 method arc length, 273 complete fields, 67 error estimation a posteriori, 338 Gaussian, 490 Gaussian quadrature, 280, 304, 305 mean fields, 66 Newtonian, 273, 282 quasi-Newtonian, 273 selective reduced integration, 303 self-consistent, 66, 229 microcracks, 83, 86, 109, 129-131, 138, 162, 195, 228, 232, 236, 244, 308, 384, 392, 394, 395, 396f, 419, 461 micromorphic, 55, 57-61, 64, 68, 105, 193, 215-226, 244-247, 249, 258, 260-262, 264-267, 271, 274, 279, 283, 284, 288-290, 306, 335, 421, 429-431 stress, 60 inertia, 59, 225 microstructure micromorphic, 68, 225 modulus elasticity, 71, 85, 108, 110, 125, 126, 131 multiplier “Friction”, 211, 214, 337 Lagrange, 50, 51, 54, 150, 159, 164, 171, 179, 180, 211, 256, 263 plastic, 50, 150, 153, 158, 159, 186-189, 199, 222, 223, 306, 318, 362, 365, 370, 400 N, O necking geometric, 425, 429 Neumann, 35, 250, 251, 255, 258, 352 Newton–Raphson, 273, 274, 281-284, 288, 289, 297, 306, 312, 317, 321, 322, 325, 327, 332 nodal approximation, 260, 261, 294, 302, 347 non-associative normality, 51, 406, 411-413, 435 nonlinear elasticity, 72 non-associative theory, 51, 52, 143, 170, 208, 236 Norton–Hoff, 176, 179, 180, 182, 183, 185, 192, 193, 237, 310, 363, 364, 366, 380-383, 419, 464, 466, 486 normality rule, 47, 50-52, 99, 139, 148, 153, 155, 159, 161, 164, 170, 175, 177, 183, 191, 209, 211, 214, 221, 236, 404-406, 409-413, 433, 435, 436, 439, 459 nucleation, 65, 83, 84, 89, 140, 194, 200, 215, 227, 228 numerical experiment plan, 489 numerical simulation, 63, 65, 82, 243, 244, 267, 272, 291, 296, 301, 303, 347, 353, 355, 358, 415, 416, 420, 421, 427, 447, 454, 464, 465, 473, 485-487, 490 objectivity constitutive equations, 32, 33, 94, 97, 282 incremental, 333, 335, operators continuous elastoplastic tangents, 154-155 continuous tangents, 160, 167, 173, 180, 186, 188, 189, 200, 215, 223, 281, 326, 333 Index damage effect, 112, 113, 116, 117, 149, 153, 194 optimization procedure, 486, 489 P Pareto diagram, 489-491 Pareto front, 484, 486, 490-492 piercing, 461, 462 plasticity time-independent, 74, 143, 174-180, 189, 211, 227, 244, 248, 308, 310, 314, 316, 328-332, 362, 365, 367-380, 384-386, 432 plastic correction, 312, 316, 317, 322, 324-326, 329, 335 polycrystal, 70, 226, 231, potential damage, 114, 115, 118, 147, 148, 163, 170, 181 damageable thermo-inelastic, 114 dissipation, 41, 56, 58, 64, 66, 91, 109, 113, 114, 139, 143, 144, 148, 156, 162, 170, 176, 215, 233, 236, 249, 339, 360 dual, 114, 121, 122, 128, 136, 137, 141, 142, 493-497 flow, 51, 52 friction, 208, 210 plastic, 51, 143-145, 148, 149, 157-159, 162, 169, 170, 192, 195, 197, 198, 221, 397, 399, 400, 404-406, 410 state, 39, 44, 45, 48, 57, 58, 60, 64, 109, 113-114, 116, 119, 120, 124, 126, 132-134, 138, 139, 143, 175, 196, 208, 218, 233, 234, 248, 249, 493 thermo-elastic damageable, 114 viscoplastic, 175, 176, 182, 185, 194, 379 521 principal of consistency, 43 of determinism, 42, 44, 53, 55 of energy conservation, 36, 37, 57 of local action, 35, 42, 55, 56 of material symmetry, 43 of objectivity, 42-43, 92 of material indifference, 92 of virtual power, 34-36, 39, 56-59, 217, 218, 255, procedure cutting, 451 expansion, 460 pseudopotential dissipations, 47, 99 dual, 49 R rate adhesion, 209 deformation process, 68, 253 elongation, 14 sliding, 17, 204, 207, 209, 211, 212, 249, 293 angular, 17, 18 volume dilation, 6, 31 referential corotational, 28, 101, 152, 171, 186, 223, 312, 409 Euclidian, 24-26, 42, 53 relations complementary, 168 state, 39, 44-48, 53, 57, 60, 64, 109, 113, 119, 124, 126, 139, 148, 156, 157, 161, 162, 169, 175, 196, 208, 209, 218, 219, 221, 222, 224, 233, 234, 240, 244, 249 evolution, 249, 422 micromorphic, 224 522 Damage Mechanics in Metal Forming restoration, 75, 175, 176, 182, 183, 191, 308, 360 rotation material, 17, 28, 96, 99, 100, 238, 312, 335, 408 plastic, 96, 97, 189, 230 total, 96-98, 238 rupture, 82, 297, 385, 390, 464, 484 ductile, 297 S scheme backward Euler, 286, 287, 307 for local integration, 304, 335 resolution explicit dynamic, 284, 288 global, 270, 272, 282, 296, 304, 312 implicit, 272, 282 implicit static, 272, 312 sequential, 284 second principle of thermodynamics, 38, 57 segment, 6, 298-300 shear bands, 85, 86, 89, 192, 215, 216, 354, 464, 479 adiabatic, 479 sidepressing a cylinder, 463, 467 sliding, 10, 17, 18, 19, 69, 74, 204, 206, 207, 209, 211, 212, 215, 249, 251, 252, 293, 297, 336, 337, 433, 478 slitting sheets, 356 small elastic strains, 98, 101, 104, 121, 126, 137, 142, 187, 228, 238, 311-313, 495 softening, 61, 64, 70, 78, 79, 82, 83, 85, 117, 190, 215, 216, 219, 266, 342, 376, 377, 384, 410, 413, 419, 420, 425, 429 space strain, 44-48, 58, 78, 111, 114, 124, 137, 145, 153, 164, 169, 170, 174, 191, 192, 213, 218, 224, 366, 370, 496 real, 260 reference, 260, 261, 266, 270, 277, 280, 294, 305, 351 spatial semi-discretization, 260 specific heat at constant strain, 142, 495 stamping, 356, 463 stability, 86, 192, 266, 271, 272, 282, 286, 287, 290, 304, 350, 352 stress Cauchy, contact, 335, 336 drag, internal, 153, 232 micromorphic, 219-222, 224 plane, viscous, structures massive, 463 thin, 397, 447, 460 surface damage, 105, 108, 147, 164, 180 equipotential, 175 load, 306 master, 205, 297, 299, 300 response, 352 slave, 202, 297, 299 stretch right Cauchy-Green, 9, 10, 53, 54 left Cauchy-Green, 9, 25 T tensor Boussinesq or Piola–Lagrange, 20, 35 Index Cauchy, 1923, 26, 27, 35, 55, 59, 119, 121, 154, 167, 228, 235, 246, 255, 278, 281 constraint Eulerian strain rate, 18, 19, 23, 32, 36, 95 Euler-Almansi, 11 Lagrangian Green–Lagrange, 11 strains, 11, 13, 32, 39 Kirchhoff, 22, 23, 26, 104, 121 Mandel, 99, 121 Piola-Kirchhoff, 21, 23, 26, 36-38, 46, 53-55 thermodynamics admissibility, 44, 104, 141, 156, 157, 161, 168, 173, 180, 185, 223, 224 irreversible processes, 38, 44, 56, 65, 66, 68, 91, 99, 102, 175, 201, 206, 208, 213, 233, tension uniaxial, 70, 75, 83, 130, 364, 368, 398 transformation homogenous, 2, 4-8, 10, 11, 31, 53, 55 isochorous, 6, 7, 54, 55 Legendre-Fenchel, 2, 47,110, 114, 121, 128, 136, 493-497 523 U, V unilateral contact, 201, 202 variables flux, 46-49, 139, 140, 209, 221, 30575-77, 90 micromorphic, 57, 59, 61, 218, 221, 222, 335 optimization, 414, 484-486, 489 state, 1, 29, 39, 41, 44, 45, 47, 55, 57, 58, 60, 64-66, 91, 92, 102 effective, 108, 128, 134 internal, 106, 191, 417 micromorphic, 57, 60, 105, 217 observable, 105, 106 variation in volume, 17, 64, 74, 90, 103, 149, 194, 195, 200, 229, 237, 308 virtual forming, 357, 358, 431 virtual power, 34, 35, 39, 56-59, 217, 218, 224, 255 viscoplasticity, 50, 75, 76, 174, 175, 179-182, 189, 192, 206, 214, 216, 227 viscoplastic flow, 139, 175-178, 180-182, 228, 328-330, 332, 360, 379, 383, 396, 397, 418, 419 volume cavity fraction, 88, 195, 200 Von Mises, 106, 163, 172, 183, 195, 200, 221, 310, 313, 314, 316, 319, 329, 331, 361, 363, 364, 396, 403, 441, 487, 488 [...]... in 2004 2 Damage Mechanics in Metal Forming [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 modern continuum mechanics, many other books have been published in which readers will find the mathematical basics and physical justifications of all basic concepts of materially simple continuum mechanics. .. professor for many 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 (Computational Structural Mechanics Association) All this has greatly influenced the nature and content of... ductile damage; (v) modeling contact between solids and 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-local model that enables adequate prediction of the damage- induced localization zones; and finally (vii) giving a micro-macro modeling of polycrystalline plasticity with ductile damage based... 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 xvi Damage Mechanics in Metal Forming 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... tomorrow This book is intended to provide graduate students and researchers from both the academic and industrial worlds, with a clear and thorough presentation of the recent advances in continuum damage mechanics and its practical use in improving numerical simulations in virtual metal forming The main goal is to summarize the current most effective methods for modeling, simulating, and optimizing... contraction on four indices – – matrix T transpose of a matrix – column matrix or vector – line matrix or transpose of a column matrix Chapter 1 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 the knowledge of the first transformation gradient... with an insight into advanced metal forming 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 xiv Damage Mechanics in Metal Forming recent advances to be used in improving metal forming and manufacturing processes As for all other engineering disciplines, these works are somehow the “memory” of their... affine function between the material coordinates X and the spatial coordinates x of the form: xj j xi (t ) X i Xj c j (t ) then the transformation between C0 and Ct is considered homogeneous [1.2] 4 Damage Mechanics in Metal Forming 1.1.1.2 Gradient of transformation and its inverse The gradient of the transformation x( X , t) X Grad ( ) F ( X , t ) defined by [1.2] is given by: [1.3] This is a “bipoint... this homogeneous motion, into a linear variety of the same order in the current configuration Ct This is particularly applicable to the transformation of elementary vectors, volumes, or surfaces 6 Damage Mechanics in Metal Forming 1.1.2.1 Transformation of an elementary vector We consider the set of particles occupying in C0 the segment P0Q0 as defining the Lagrangian elementary vector dX P0Q0 (Figure... parallelogram is defined in C0 by dA0 dA0 n0 This oriented plane surface, transported by the motion into the configuration Ct , is transformed into a plane surface with the normal nt surrounding point Pt 8 Damage Mechanics in Metal Forming represented by the parallelogram formed by the two vectors dx , dx (respectively, transformation of the vectors dX , dX dAt by the gradient F ) of “vector area” dAt nt By ... 2.5.2.1 Damageable plastic dissipation: anisotropic damage with two yield surfaces 2.5.2.2 Damageable plastic dissipation: anisotropic damage with a single... 2.5.2.3 Incompressible and damageable plastic dissipation: isotropic damage with two yield surfaces 2.5.2.4 Incompressible and damageable plastic dissipation: single... 157 162 169 viii Damage Mechanics in Metal Forming 2.5.3 Intrinsic dissipation analysis: time-dependent plasticity or viscoplasticity 2.5.3.1 Damageable viscoplastic