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,
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© 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
1. Metals--Plastic properties. 2. Metal-work--Mathematical models. 3. Metal-work--Quality control. 4.
Deformations (Mechanics)--Mathematical models. 5. 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 1. Elements of Continuum Mechanics and Thermodynamics . . .

1

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 . . . . . . . . . . . . . . . .

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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. . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . .
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. .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . .

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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 .

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Chapter 3. 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 . . . . . . . .

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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 . . . . . . . . . . . .

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. 353

Chapter 4. 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

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356
359
360
360

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xi

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.

431
431
432
441
447
463
463
476

. . . . .

484

Appendix: Legendre–Fenchel Transformation . . . . . . . . . . . . . . . . . .

493

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

499

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

515

Preface

As with other scientific fields where numerical simulation is essential, predictive
capabilities of virtual metal forming methods rely on: (i) advanced thermomechanical
constitutive equations representing the mechanisms of the main thermomechanical
phenomena involved and their various couplings; (ii) high-performing numerical
methods adapted to the problem’s various nonlinearities; (iii) adaptive and userfriendly geometric tools for the spatial representation and spatial discretization of
solids undergoing large transformations.
For scientists, numerical simulations allow researchers to explore phenomena
and 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 obtain
various optimum metallic components due to large inelastic deformations. Careful
examination of the literature on this subject allows their classification into three
families that differ by the modeling methodologies and recommended objectives.
The first family relates to works where modeling methods and calculation
procedures 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 essentially
numerical 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 numerical
methods 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 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 time of major scientific and technical developments that support
the present generation to deal with current problems and prepare for the new
methods of 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 metal
forming processes and to present the main features of new, innovative methods
currently being developed, which will no doubt be the industrial tools of tomorrow.
Compared to recent books devoted to virtual metal forming, the main contribution of
this book is found in Chapter 2 where the development of highly predictive
multiphysical and fully coupled constitutive equations is presented. These can be
used in computer codes to simulate and optimize all kinds of sheet or bulk metal
forming processes by large inelastic strains, regarding the occurrence of ductile
damage.
This book is organized into four main chapters. The first aims to provide the
reader with the basic theoretical “tools” needed to understand the models which will
subsequently be explored in this book. This is essentially a brief introduction which
aims to combine scientific rigor with simple definitions in order to present: (i) the
main measures of strains and stresses as well as their respective rates, (ii) the main
conservation laws for the materially simple continua, (iii) the thermodynamics of
irreversible processes with state variables firstly in the framework of materially
simple continua (or Cauchy continua), followed by (iv) the generalization of these

concepts to the materially non-simple continua, particularly the micromorphic
continua in the framework of the generalized higher order continua. For reasons of
brevity, the mathematical aspects related to algebra and tensor analysis as well as
the convex functions’ analysis – concepts that are required for carrying out a number
of calculations examined – will not be discussed in this chapter nor in the
appendices. Rather, various academic books are referenced, which provide an
overview of these features. However, the definition of Legendre–Fenchel transform,
that is often used throughout the first two chapters of the book, is given in
Appendix 1.
The second chapter, the key part of this book, focuses solely on “advanced”
modeling of the main physical phenomena characteristic of various behaviors and
ductile damages of metals under large strains by focusing on their various strong


Preface

xv

couplings. After a brief descriptive summary of the physical phenomena being
modeled and their main physical mechanisms, the reader will find: (i) the main
assumptions adopted for accurate modeling in the context of thermodynamics of
irreversible processes with state variables; (ii) the construction of various state
potentials and the definition of state relations derived from them; (iii) analyzing
different sources of dissipations and deducing the evolutions equations from
appropriate load functions and adequate dissipation potentials; (iv) modeling the
volume variation induced by 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 on the mean fields approach.
The third chapter introduces the numerical aspects, which allow us to obtain a
credible “unique” solution of initial and boundary value problems (IBVP) in order to
completely 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 using
finite difference method (FDM) as well as the space discretization using the finite
element method (FEM) of the IBVP focusing on the most common currently used
elements for 2D and 3D problems; (iii) an overview of the main global resolution
schemes of the IBVP including the assessment of contact conditions; (iv) a detailed
presentation of the numerical aspects related to the iterative integration scheme of the
fully 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 metal
forming; and finally (vi) an insight into new meshless spatial discretization methods,
and their possible link (or coupling) with the FEM. Again for brevity, many aspects
were simply mentioned without going deeply into technical details but referring to a
complete list of references where the reader can obtain further information on the
topics under concern.
Finally, the fourth chapter focuses on using the virtual adaptive virtual metal
forming proposed in order to numerically simulate various metal forming and
machining processes. The following aspects are examined: (i) the presentation of
the methodology to follow in order to determine the material parameters entering
the 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


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 chapter have been exclusively taken from research works in virtual metal
forming 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 or
collective books related to a given concept. Therefore, we have deliberately left aside
any reference to articles in scientific journals, except for a few review articles on
aspects which are not treated by specific books. The interested reader will have no
trouble finding numerous articles through bibliographic searches on specialized sites.
This book, which focuses on “advanced” modeling and numerical simulation in
metal forming and machining by large inelastic strains, is in fact an overview of the
teaching and research activities of the author during his career. Beginning at
UTC in Compiègne since 1979, these activities have continued to evolve mainly at
UTT since 1995 and also partially at ENSAM/CER in Chalons-en-Champagne,
ENIM (Monastir, Tunisia), and ESSTT (Tunis, Tunisia) as invited 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 this book.
The author would, therefore, like to address his most sincere appreciation to all
those many people, who have directly or indirectly influenced the material of this
book: the engineers who have attended his lectures; the PhD students who have
actively participated in the research from which a number of results have been taken
for the four chapters of this book, particularly in Chapter 4, and for which an
exhaustive 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 and
international 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 Lafon
from UTT/LASMIS for their direct involvement in finalizing some of the results in
Chapter 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 taken
with and encouraged by my marvelous and adorable wife, Fathia, who accepted
the obvious risk of spending many weekends and holidays without her other half.


Preface

xvii

Her understanding, unwavering support, and ability to close her eyes each time I
spent 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 and
Slim, 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 Saanouni
Troyes, March 2012


Principle of Mathematical Notations

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

calculations carried out.
– x scalar variable.
– x , xi vector in

3

.
3

– x, xij second-rank tensor in
– x , xijk third-rank tensor in

3

– x , xijkl 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.
–
–

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 is sufficient to define all the kinematic
and 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 the
constitutive 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 the
so-called “local state method” in the framework of which the constitutive equations
will be formulated (section 1.3); finally, we will conclude by giving an introduction
to generalized continuum mechanics (GCM) by extending all kinematic and
thermodynamic ideas to the context of generalized or materially non-simple
continua (section 1.4). This extension allows the formulation of nonlocal
constitutive equations provided at the end of Chapter 2.
For the sake of brevity, we will not recapitulate all of the mathematical details
and rigorous demonstrations of all the ideas introduced. In particular, we will neither
review tensor algebra and tensor analysis nor convex analysis, ideas that are
indispensable for the manipulation of all mechanical quantities. For more details on
these subjects, we refer the reader to the excellent book by Truesdell and Noll, first
published in 1965 [TRU 65] and then republished by the same authors in a second
revised and corrected edition in 1992 [TRU 92]. A third edition appeared 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 (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 vast
majority of these books, the reader will find chapters or indices dedicated to
mathematical reminders on vectors and tensor analysis as well as convex analysis.
However, other specialized books may be of great help to readers who wish to
improve 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 , with
boundary
. u is the portion of the boundary where
u
F and
u
F
displacements are imposed and F is the additional part of the boundary where
forces are imposed.
1.1.1.1. Homogeneous transformation
a part of solid
Let us consider the description of the motion of the subdomain
. Suppose that

occupies at initial time t0 the initial non-deformed
configuration C0 . At any instant t t0 , the subdomain
occupies the current
deformed configuration Ct . Using a direct orthonormal Euclidian space of base
(O , e1 , e2 , e3 ) , in any homogeneous transformation moving
from C0 to Ct , a
point P0 of coordinates X in C0 transforms into Pt of coordinates x ( X , t ) in Ct
by (see Figure 1.1):
x

( X , t)

[1.1]


Elements of Continuum Mechanics and Thermodynamics

3

Figure 1.1. Initial and deformed configurations of a deformable subdomain
and vectors transport

The components X i: X 1 , X 2 , X 3 of vector X X 1e1 X 2 e2 X 3e3 are the
Lagrangian or material coordinates of point P0 in the reference configuration C0 .
The components xi: x1 , x2 , x3 of vector x x1e1 x2 e2 x3e3 are the Eulerian or
spatial coordinates of point Pt in the current configuration Ct corresponding to
point P0 of C0 .
The vectorial field

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

position of point Pt is a bijection of C0 on Ct . Thus, it allows a reciprocal function
1

( x , t ) , which at any point Pt of Ct is used to define in a unique manner its

1
correspondent P0 in C0 . The two vectorial functions ( X , t ) and
( x , t ) are
continuous and continuously differentiable (except possibly on certain surfaces of
discontinuity) 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:
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 tensor” of the second-rank F (or Fij ) called the gradient of
the homogeneous transformation between C0 and Ct . According to Figure 1.1, the
homogeneous transformation is defined by:
x

( X , t)

X

[1.4]

u( X , t)

where u ( X , t ) designates the displacement vector expressed in the same basis. The
gradient of this homogeneous transformation is thus given by:

F

x( X , t)
X

Grad ( )

where Grad (u )

1 Grad (u ) or Fij

xi
Xj

ij

ui
Xj

[1.5]

ui / X j is a non-symmetric second-rank tensor that can be

broken down into symmetric and antisymmetric parts, as we will see later on. Note
that in order for [1.2] and [1.4] to define correctly the motion of a continuum, we

must have:
J

0

[1.6]

det( F )

Since J is not zero in any point of
inverse gradient called F
F

1

grad (

1

)

1

, the second-rank operator F allows an

defined by:
X
so that F . F
x

1

1

[1.7]

Finally, note that for this theory of materially simple continua, the knowledge of
the gradient F is amply sufficient for the complete definition of the transformation
kinematic of the continuum
in that it allows the complete description of changes
in the shape, size, and orientation of the continuum as we will see later in this
chapter.


Elements of Continuum Mechanics and Thermodynamics

5

1.1.1.3. Polar decomposition of the transformation gradient
According to the well-known polar decomposition theorem, any homogeneous
transformation 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 a
homogeneous transformation F defined by [1.5] can be multiplicatively
decomposed, in a unique manner, in the form:

F

R.U

V .R

[1.8]

where the symmetric and positive definite second-rank tensors U and V are called
left and right pure strain or stretch tensors, and R is the rigid body orthogonal
rotation tensor ( R T .R R.RT 1 ). U is a Lagrangian tensor defined with respect to
C0 , while V is purely Eulerian tensor, defined with respect to Ct (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 the
reference configuration C0 is transformed, in its transport by 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 1.1). Due to the affine
character of the transformation [1.4], these particles occupy at time t in Ct the
segment PQ

PQ
t t defining the Eulerian vector dx
1 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 C0
and Ct :
dx

[1.9]

F .dX

1.1.2.2. Transformation of an elementary volume: the volume dilatation
Given in the configuration C0 an elementary parallelepiped constructed with
the three non-coplanar vectors dX 1 , dX 2 , dX 3 (Figure 1.3). Its volume in C0 is
defined by:
dV0

dX 1.( dX 2

dX 3 )

det M

det dX 1 dX 2

dX 3

[1.10]

where (M) is the matrix, the columns of which are the three elementary vectors.
Moreover, in the current configuration Ct , the parallelepiped formed by the
vectors dx1 , dx2 , dx3 , which are the transformation, respectively, of the vectors
dX 1 , dX 2 , dX 3 , has a volume dVt defined by:
dVt

det dx1 dx2

[1.11]

dx3

Due to [1.9], the following relationship between the two volumes can be easily
obtained:
dVt

det(F )dV0

JdV0 or J

det( F )

dVt
dV0

[1.12]

Thus, J defines the volume dilatation in the homogeneous transformation between
C0 and Ct . If J det( F ) 1 , then the volume is preserved and the homogeneous
transformation is called isochoric or incompressible (see section 1.3.3.2). It should be

noted that according to [1.8], we have det(U ) det(V ) det( F ) J .


Elements of Continuum Mechanics and Thermodynamics

7

Figure 1.3. Elementary volume transformation between C0 and Ct

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

J

1/ 3

F with det(Fˆ )

det(J

1/ 3

F)

J 1J

1

[1.13]

Thus, any homogeneous transformation can be decomposed into the product of
an isochoric or volume preserving transformation of gradient Fˆ J 1/ 3 F and of a
pure dilatation of gradient F J 1 / 3 1 , so as to have:
F

F . Fˆ

[1.14]

It then results, by using the polar decomposition theorem [1.8], that:
Fˆ

R.( J

1/ 3

U)

R.Uˆ and Fˆ

(J

V ). R Vˆ .R

1/ 3

[1.15]

where Uˆ and Vˆ are the left and right pure stretch tensors of a purely isochoric or

volume preserving transformation.
1.1.2.3. Transformation of an oriented elementary surface
Consider, in configuration C0 (see Figure 1.4), a plane elementary surface
oriented by the normal vector n0 (surrounding, for example, the point P0 ) of area
dA0 represented by the parallelogram formed by the two coplanar vectors dX , dX ' .

The “vector area” of this 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 using the transformation relationships of elementary vectors as well

as [1.12], the following relationship between dAt and dA0 is obtained:
nt dAt

J ( F 1 )T n0 dA0

[1.16]

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

Figure 1.4. Transport of elementary surface between C0 and Ct

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
in the homogeneous transformation, between the reference configuration
of area
C0 and the current configuration Ct , 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 transformed
vectors to point Pt in the current configuration Ct . The scalar product of these two
vectors is given by:
dx.dx '

( F .dX ).( F .dX )

( F .dX )T ( F .dX )

dX .( F T . F ).dX

dX .C.dX

[1.17]


Elements of Continuum Mechanics and Thermodynamics

9

Thus, we define in C0 the right CauchyíGreen stretch tensor C , Lagrangian,
symmetric and positive definite, by:
F T .F

C

[1.18]
det( F T .F )

It is a matter of course that det(C )

det( F )

2

J 2 and, due to

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

C

U .R T .R.U

U2

[1.19]

Moreover, the scalar product of the two Lagrangian vectors dX , dX ' leads to:
dx. ( F 1 )T .( F 1 ) .dx

( F 1.dx ).( F 1.dx )

dX .dX

dx.B 1.dx

[1.20]

thereby allowing the definition in Ct of the left Cauchy–Green stretch tensor B ,
Eulerian, symmetric, and positive definite, by:
B

( F 1 )T . F

1

with det( B )

1

det( F . F T )

or B

F .F T

det( F )

2

[1.21]
J 2 . It is easy to verify, by using [1.8] and

given the properties of V and R , that:
B

F .F T

V .R.R T .V

V2

[1.22]

Due to the decomposition [1.14], we easily obtain the following decomposition
of the Cauchy–Green stretch tensors C and B :
C

C.Cˆ with C

J 2 / 3 1 and Cˆ

J

2/3

C

[1.23]

B

B. Bˆ with B

J 2 / 3 1 and Bˆ

J

2/3

B

[1.24]

Let us finally note that extension in a given direction, for example direction dX ,
can be defined as being the ratio

( dX ) of the length of the transformed vector

(here dx ) to that of the corresponding vector in C0 (here dX ):
( dX )

dx
dX

[1.25]