Numerical Methods for Strong Nonlinearities in Mechanics
Description
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.
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Persons
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.
Content
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 ¿ 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
1
Lagrangian and Nitsche Methods for Frictional Contact
Franz CHOULY1,2, Patrick HILD3 and Yves RENARD4,5
1 Université Bourgogne-Franche-Comté, Dijon, France
2 University of Chile, Santiago, Chile
3 CNRS, Université de Toulouse, France
4 Institut Camille Jordan, Université de Lyon, France
5 CNRS, INSA Lyon, France
1.1. Introduction
Augmented Lagrangians and Lagrangians are constrained optimization tools that very early have naturally been applied to contact problems with deformable solids (see, for example, Rockafellar 1974, 1976). The augmented Lagrangian has since quite widely become established for the approximation and resolution of contact problems in small and large strains, mainly following the research of Curnier and Alart (1988); Alart and Curnier (1991); Simo and Laursen (1992). The method by Nitsche (1971) was originally proposed to allow a Dirichlet-type boundary condition to be weakly taken into account, precisely avoiding the use of Lagrange multipliers. Only recently has it been extended to contact conditions with or without friction in Chouly and Hild (2013a); Annavarapu et al. (2014); Chouly (2014); Chouly et al. (2015). The close connection between Nitsche and Lagrangian methods is however quite clear and it is the objective of this chapter to shed some light on this relationship. This is achieved namely by looking into the mechanisms underlying these methods, and also by way of presenting some recent developments within the framework of small and large elastic strains.
Section 1.2 first presents the continuous problem of frictional contact between two elastic solids, within the framework of small strains. Section 1.3 is dedicated to finite element approximation within the framework of small strains, where mathematical analysis of numerical methods is possible. Section 1.4 finally presents the extension of the methods described in previous sections to the regime of large elastic transformations, as well as numerical results related to this context.
1.2. Small-strains frictional contact between two elastic bodies
The problem of frictional contact between two elastic solids is first described in section 1.2.1, and then in section 1.2.2, this problem is reformulated as a quasi variational inequality. Then, section 1.2.3 introduces the weak multiplier form, and section 1.2.4 introduces the proximal augmented Lagrangian formulation. These reformulations are the basis of the numerical approximations presented in sections 1.3 and 1.4.
1.2.1. Contact between two elastic bodies
We consider two elastic solids whose respective reference configurations are denoted by O1 and O2 corresponding to two domains of Rd (d = 2 or 3) of regular boundaries (piecewise of class ), as shown in Figure 1.1. At the boundaries ?O1 and ?O2 of O1 and O2, we can identify the boundaries and (with non-empty interiors) on which the elastic bodies are clamped, the boundaries and with an imposed force density and and which are the potential contact boundaries, slave and master, respectively. We assume that these boundaries form a partition without boundaries overlapping of ?O1 and ?O2.
The two elastic bodies are subjected to force densities (volumic forces if d = 3) denoted as f1 and f2 and on and to force densities (surface forces if d = 3) denoted as l1 and l2. The focus is now on expressing the contact condition with Coulomb friction. To this end, we consider the slave surface . For a point the point that potentially comes into contact therewith must be determined. This is called contact pairing. In the contact condition of small-strain approximation, this correspondence is determined on the reference configuration and is not questioned during deformation. In general, a projection is used, but it is not the only possible choice. Let us consider this correspondence:
Figure 1.1. Two bodies with their respective potential contact boundaries.
There are then two outward vectors of interest at point x (see Figure 1.1): the outward unit normal vector of O1, which we will denote by n1, and the outward unit vector in the direction of y = ?(x), which we will denote by n and which can be defined by:
since the last two cases (i.e. for (y - x) · n1 = 0) are expected, either when contact is established in the reference configuration, or if both domains are overlapping, which is a priori not prohibited. There is no reason that these two vectors n1 and n should be equal, in general, except by using the "ray tracing" strategy exposed in section 1.4.1. The vector n is usually called the contact normal. To express the contact condition, we need to determine what is the normal component of the stress. Let u1 : O1 Rd be the displacement of the first body and s(u1)its Cauchy stress tensor. So, we will denote:
the normal and tangential component decomposition of the stress on the slave contact boundary. We will also denote:
the initial gap between the two potential contact surfaces as well as:
the jumps of the displacements and of the normal displacements. Therefrom, the non-interpenetration condition, or Signorini condition, can be written on as the following complementarity relation:
[1.1]To write the friction condition, a coefficient of friction is of course needed, which will be denoted by F = 0 and rigorously a notion of sliding velocity. Here, in a supposedly quasi-static evolution, we will not use a sliding velocity but a tangential displacement increment that we will denote by dt. Duvaut and Lions (1972) and Kikuchi and Oden (1988) use the expression , leading to a problem which, although artificial, exhibits the same characteristics as that obtained for an expression of dt that would derive from a time discretization that can be written as:
where is the displacement jump at the previous time step. The friction condition is then written as:
[1.2]The second Newton law, or action-reaction principle, imposes that:
[1.3]where n2 is the outward unit normal to at point y = ?(x)and J? is the Jacobian of the transformation ? between the two surfaces and .
The description of the linearized elasticity law is carried out by the intermediate of the small strain tensor e(u) = (?u + ?uT)/2. The Cauchy stress tensor is then connected to the strain tensor by the fourth-order elasticity tensor A with the usual symmetry and coerciveness properties. This relationship is written as s(u) = A e(u). The displacements u1,u2 of the two elastic bodies are then subjected to the following equations on Oi, i = 1, 2 in addition to the contact and friction equations [1.1], [1.2] and [1.3]:
[1.4]1.2.2. The classical weak inequality form
The weak formulation in the form of inequality that can be found in Duvaut and Lions (1972) and Kikuchi and Oden (1988) can be constructed by introducing the following spaces:
and the set of admissible displacements:
The spaces of normal and tangential traces on are also introduced as:
as well as their respective topological duals and . Let (f1,f2) be in , (l1,l2) in , the following bilinear and linear forms on V are defined:
as well as the functional corresponding to the virtual work of the frictional force:
where s = -Fsn(u) is the friction threshold. The notation denotes the product of duality between the spaces and XN. When s is regular, this product is reduced to the integral . Therefore, the classical weak form associated with [1.1]-[1.4] is written as:
[1.5]When there is no friction (F = 0), the weak problem [1.5] is a variational inequality of the first kind. Then, the Stampacchia theorem allows us to conclude that it admits a unique solution, which moreover is the only minimizer of the functional on the convex set K. In the presence of friction, results of existence could be shown, for example, in Eck et al. (2005), given that the coefficient of friction F is small. A recent result in Ballard and Iurlano (2023) claims that existence holds for any friction coefficient. Regarding the uniqueness of the solution, counterexamples were presented for large friction coefficients in Hild (2003, 2004) and a criterion for characterizing the uniqueness of the solution was presented in Renard (2006). The uniqueness of the solution for a sufficiently small coefficient is still an open problem.
It should be noted that when the friction threshold is known...
