Jacques Besson is Research Director at the CNRS, France, where he conducts research into damage and fracture modeling of metallic materials.Frédéric Lebon is Professor of Solid Mechanics at Aix-Marseille University and the Mechanics and Acoustics Laboratory (LMA), France.Eric Lorentz is a senior expert at EDF R&D, France, where he conducts studies on damage modeling, applied to the performance of power generation structures.

• PrefaceJacques Besson, Frédéric Lebon And Éric LorentzPart 1 Contact and Friction 1Chapter 1 Lagrangian and Nitsche Methods for Frictional Contact 3Franz Chouly, Patrick Hild And Yves Renard1.1 Introduction 31.2 Small-strains frictional contact between two elastic bodies 41.2.1 Contact between two elastic bodies 41.2.2 The classical weak inequality form 71.2.3 The principle of duality and the weak form with multipliers 81.2.4 Proximal augmented Lagrangian: principle and use 91.3 Finite element approximation in small deformations 121.3.1 State of the art, methods with multipliers 131.3.2 Absence of inf-sup condition and stabilized methods 151.3.3 Nitsche’s method seen as a limit stabilized method model 161.3.4 Relationship between Nitsche and proximal augmented Lagrangian 191.3.5 The connection between Nitsche and penalty 201.4 Large strain finite element approximation 211.4.1 About contact pairing and gap function 231.4.2 Formulation of contact and friction conditions 261.4.3 Augmented Lagrangian and penalization 281.4.4 Nitsche’s method 331.4.5 About the value of the parameter γ 361.4.6 Numerical tests 361.5 Acknowledgments 411.6 References 41Chapter 2 High-performance Computing in Multicontact Mechanics: From Elastostatics to Granular Dynamics 47Pierre Alart2.1 Introduction 472.2 Multicontact in elastostatics 492.2.1 Development framework 492.2.2 Parallel solver preconditioning 512.2.3 Domain decomposition: Newton–Schur solver 532.3 Diffuse non-smoothness in discrete structures: tensegrity 572.3.1 Motivation 572.3.2 Domain decomposition: micro-macro LATIN solver 582.4 Granular dynamics 612.4.1 Velocity-impulse formulation 612.4.2 Parallelized and parallelizable solvers 632.4.3 Conjugate projected gradient solver 652.4.4 Domain decomposition: FETI-NLGS solver 662.5 Conclusion 732.6 References 75Chapter 3 Numerical Methods in Micromechanical Contact 79Vladislav A. Yastrebov3.1 Introduction 793.1.1 Plan 803.2 Contact micromechanical problem 803.2.1 Surface geometry: mathematical description 803.2.2 Surface geometry: examples and discussions 833.2.3 Roughness models 853.2.4 Contact formalization 863.2.5 Laws of friction 883.3 Finite element method 903.3.1 Convergence, parameters and loading step 913.3.2 Convergence of friction problems 923.3.3 Quadratic convergence 943.3.4 Mesh and computation time 953.3.5 Contact constraint 953.3.6 Surface regularity 973.4 Application I: study of an isolated asperity 983.4.1 Elastic asperity 983.4.2 Elastoplastic asperity 1023.5 Application II: rough surface contact 1093.6 Conclusion 1133.7 References 114Part 2 Damage and Cracking 135Chapter 4 Numerical Methods for Ductile Fracture 137Jacques Besson4.1 Introduction 1374.2 Physical mechanisms of ductile fracture 1384.3 Some ductile fracture models 1394.3.1 Rice and Tracey model and fracture criteria 1394.3.2 The Gurson–Tvergaard–Needleman model 1404.3.3 Other models 1434.4 Performing ductile fracture simulations with a finite elements code 1434.4.1 Calculation parameters 1434.4.2 Pressure control 1454.4.3 Application of the Rice and Tracey criterion 1464.4.4 GTN model application 1484.4.5 Pragmatic solution 1494.5 Localization origin 1504.6 Regularization methods 1524.6.1 Integral methods 1524.6.2 Explicit or implicit gradient methods 1534.6.3 Micromorphic models 1574.6.4 Enhanced energy models 1604.6.5 Example 1614.7 Conclusion 1644.8 References 167Chapter 5 Quasi-brittle Fracture Modeling 175Éric Lorentz5.1 What are the approaches for predicting quasi-brittle fracture? 1755.2 Materials with internal lengths 1785.2.1 Localization and non-locality 1785.2.2 Risks of ignoring the non-local nature inherent to damage 1815.2.3 Limitations of a localization: regularization by viscosity 1835.2.4 Characterization of the internal length: toward a fracture model 1855.3 Non-local formulations 1895.3.1 Formulation of the mechanical problem at the structural scale 1895.3.2 Some non-local model classes 1945.3.3 Qualitative analysis of non-local formulations 2045.3.4 Phase field models and damage gradient models 2075.3.5 Approximating a cohesive model with a gradient model 2125.4 Phenomenological aspects of quasi-brittle behavior 2145.4.1 Isotropy or anisotropy? 2155.4.2 Unilateral nature 2185.4.3 Asymptotic fracture behavior 2225.5 Numerical solving methods 2275.5.1 Impact of non-locality 2285.5.2 Difficult to perform computations 2385.6 Conclusion 2495.7 References 251Chapter 6 Extended Finite Element (XFEM) and Thick Level Set (TLS) Methods 261Nicolas Moës6.1 Introduction 2616.2 Categorization of approaches to cracking 2626.3 The XFEM method for cracking in non-softening media 2646.4 XFEM-TLS for cracking in softening media 2716.4.1 TLS V1 and V2 models 2746.4.2 Relation to the Griffith model and the cohesive model 2786.4.3 TLS: implementation aspects 2786.5 XFEM-TLS simulation examples 2826.5.1 Torsional chalk fracture 2836.5.2 Multiple cracking in a block with holes 2846.5.3 Three-point bending for a beam and cohesive crack 2856.6 Conclusion 2866.7 References 287Chapter 7 Damage-to-Crack Transition 293Sylvia Feld-Payet7.1 Introduction 2937.1.1 Continuous damage models and their limitations 2937.1.2 Modeling a discontinuity 2957.1.3 Definition of a damage-to-crack transition strategy 3037.1.4 Study objective and framework 3047.2 Localizing discontinuity 3057.2.1 Formulation of an orientation criterion 3057.2.2 From orientation criterion to crack surface 3117.2.3 Basic evaluation methods for more regularity 3157.2.4 Advanced evaluation methods ensuring more regularity 3177.2.5 Constructing a continuous discretized surface in 3D 3297.3 Inserting a discontinuity 3367.3.1 Objectives and connection with the orientation criterion 3367.3.2 The different insertion criteria 3387.3.3 Challenges associated with front determination 3417.3.4 Outlook: strengthening the link with physics 3437.4 Resuming computations after inserting a discontinuity 3447.4.1 Issues 3447.4.2 Field transfer 3457.4.3 Reequilibrium 3517.5 Conclusion 3537.6 References 353List of Authors 363Index 365