Numerical Methods for Strong Nonlinearities in Mechanics: Contact and Fracture

Numerical Methods for Strong Nonlinearities in Mechanics deals with recent advances in the numerical treatment of contact/friction and damage phenomena. Although physically distinct, these phenomena both lead to a strong nonlinearity in the mechanical problem, therefore limiting the regularity of the problem, which is now non-differentiable. This has two direct consequences: on the one hand, the mathematical characteristics of the problem deviate from wellestablished forms, requiring innovative discretization schemes; on the other hand, the low regularity makes it particularly difficult to solve the corresponding large-scale algebraic systems robustly and efficiently. In addition, neither the uniqueness, nor the existence of solutions, remain assured, resulting in bifurcation points, limit loads and structural instabilities, which are always tricky to overcome numerically.

Table of Contents:
Preface Jacques Besson, Frédéric Lebon And Éric Lorentz Part 1 Contact and Friction 1 Chapter 1 Lagrangian and Nitsche Methods for Frictional Contact 3 Franz Chouly, Patrick Hild And Yves Renard 1.1 Introduction 3 1.2 Small-strains frictional contact between two elastic bodies 4 1.2.1 Contact between two elastic bodies 4 1.2.2 The classical weak inequality form 7 1.2.3 The principle of duality and the weak form with multipliers 8 1.2.4 Proximal augmented Lagrangian: principle and use 9 1.3 Finite element approximation in small deformations 12 1.3.1 State of the art, methods with multipliers 13 1.3.2 Absence of inf-sup condition and stabilized methods 15 1.3.3 Nitsche’s method seen as a limit stabilized method model 16 1.3.4 Relationship between Nitsche and proximal augmented Lagrangian 19 1.3.5 The connection between Nitsche and penalty 20 1.4 Large strain finite element approximation 21 1.4.1 About contact pairing and gap function 23 1.4.2 Formulation of contact and friction conditions 26 1.4.3 Augmented Lagrangian and penalization 28 1.4.4 Nitsche’s method 33 1.4.5 About the value of the parameter γ 36 1.4.6 Numerical tests 36 1.5 Acknowledgments 41 1.6 References 41 Chapter 2 High-performance Computing in Multicontact Mechanics: From Elastostatics to Granular Dynamics 47 Pierre Alart 2.1 Introduction 47 2.2 Multicontact in elastostatics 49 2.2.1 Development framework 49 2.2.2 Parallel solver preconditioning 51 2.2.3 Domain decomposition: Newton–Schur solver 53 2.3 Diffuse non-smoothness in discrete structures: tensegrity 57 2.3.1 Motivation 57 2.3.2 Domain decomposition: micro-macro LATIN solver 58 2.4 Granular dynamics 61 2.4.1 Velocity-impulse formulation 61 2.4.2 Parallelized and parallelizable solvers 63 2.4.3 Conjugate projected gradient solver 65 2.4.4 Domain decomposition: FETI-NLGS solver 66 2.5 Conclusion 73 2.6 References 75 Chapter 3 Numerical Methods in Micromechanical Contact 79 Vladislav A. Yastrebov 3.1 Introduction 79 3.1.1 Plan 80 3.2 Contact micromechanical problem 80 3.2.1 Surface geometry: mathematical description 80 3.2.2 Surface geometry: examples and discussions 83 3.2.3 Roughness models 85 3.2.4 Contact formalization 86 3.2.5 Laws of friction 88 3.3 Finite element method 90 3.3.1 Convergence, parameters and loading step 91 3.3.2 Convergence of friction problems 92 3.3.3 Quadratic convergence 94 3.3.4 Mesh and computation time 95 3.3.5 Contact constraint 95 3.3.6 Surface regularity 97 3.4 Application I: study of an isolated asperity 98 3.4.1 Elastic asperity 98 3.4.2 Elastoplastic asperity 102 3.5 Application II: rough surface contact 109 3.6 Conclusion 113 3.7 References 114 Part 2 Damage and Cracking 135 Chapter 4 Numerical Methods for Ductile Fracture 137 Jacques Besson 4.1 Introduction 137 4.2 Physical mechanisms of ductile fracture 138 4.3 Some ductile fracture models 139 4.3.1 Rice and Tracey model and fracture criteria 139 4.3.2 The Gurson–Tvergaard–Needleman model 140 4.3.3 Other models 143 4.4 Performing ductile fracture simulations with a finite elements code 143 4.4.1 Calculation parameters 143 4.4.2 Pressure control 145 4.4.3 Application of the Rice and Tracey criterion 146 4.4.4 GTN model application 148 4.4.5 Pragmatic solution 149 4.5 Localization origin 150 4.6 Regularization methods 152 4.6.1 Integral methods 152 4.6.2 Explicit or implicit gradient methods 153 4.6.3 Micromorphic models 157 4.6.4 Enhanced energy models 160 4.6.5 Example 161 4.7 Conclusion 164 4.8 References 167 Chapter 5 Quasi-brittle Fracture Modeling 175 Éric Lorentz 5.1 What are the approaches for predicting quasi-brittle fracture? 175 5.2 Materials with internal lengths 178 5.2.1 Localization and non-locality 178 5.2.2 Risks of ignoring the non-local nature inherent to damage 181 5.2.3 Limitations of a localization: regularization by viscosity 183 5.2.4 Characterization of the internal length: toward a fracture model 185 5.3 Non-local formulations 189 5.3.1 Formulation of the mechanical problem at the structural scale 189 5.3.2 Some non-local model classes 194 5.3.3 Qualitative analysis of non-local formulations 204 5.3.4 Phase field models and damage gradient models 207 5.3.5 Approximating a cohesive model with a gradient model 212 5.4 Phenomenological aspects of quasi-brittle behavior 214 5.4.1 Isotropy or anisotropy? 215 5.4.2 Unilateral nature 218 5.4.3 Asymptotic fracture behavior 222 5.5 Numerical solving methods 227 5.5.1 Impact of non-locality 228 5.5.2 Difficult to perform computations 238 5.6 Conclusion 249 5.7 References 251 Chapter 6 Extended Finite Element (XFEM) and Thick Level Set (TLS) Methods 261 Nicolas Moës 6.1 Introduction 261 6.2 Categorization of approaches to cracking 262 6.3 The XFEM method for cracking in non-softening media 264 6.4 XFEM-TLS for cracking in softening media 271 6.4.1 TLS V1 and V2 models 274 6.4.2 Relation to the Griffith model and the cohesive model 278 6.4.3 TLS: implementation aspects 278 6.5 XFEM-TLS simulation examples 282 6.5.1 Torsional chalk fracture 283 6.5.2 Multiple cracking in a block with holes 284 6.5.3 Three-point bending for a beam and cohesive crack 285 6.6 Conclusion 286 6.7 References 287 Chapter 7 Damage-to-Crack Transition 293 Sylvia Feld-Payet 7.1 Introduction 293 7.1.1 Continuous damage models and their limitations 293 7.1.2 Modeling a discontinuity 295 7.1.3 Definition of a damage-to-crack transition strategy 303 7.1.4 Study objective and framework 304 7.2 Localizing discontinuity 305 7.2.1 Formulation of an orientation criterion 305 7.2.2 From orientation criterion to crack surface 311 7.2.3 Basic evaluation methods for more regularity 315 7.2.4 Advanced evaluation methods ensuring more regularity 317 7.2.5 Constructing a continuous discretized surface in 3D 329 7.3 Inserting a discontinuity 336 7.3.1 Objectives and connection with the orientation criterion 336 7.3.2 The different insertion criteria 338 7.3.3 Challenges associated with front determination 341 7.3.4 Outlook: strengthening the link with physics 343 7.4 Resuming computations after inserting a discontinuity 344 7.4.1 Issues 344 7.4.2 Field transfer 345 7.4.3 Reequilibrium 351 7.5 Conclusion 353 7.6 References 353 List of Authors 363 Index 365