A uniqueness criterion for the Signorini problem with Coulomb friction

International audienceThe purpose of this paper is to study the solutions to the Signorini problem with Coulomb friction (the so-called Coulomb problem). Some optimal a priori estimates are given, and a uniqueness criterion is exhibited. Recently, nonuniqueness examples have been presented in the continuous framework. It is proved, here, that if a solution satisfies a certain hypothesis on the tangential displacement and if the friction coefficient is small enough, it is the unique solution to the problem. In particular, this result can be useful for the search of multisolutions to the Coulomb problem because it eliminates a lot of uniqueness situations

Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers

International audienceThe purpose of this work is to present in a general framework the hybrid discretization of unilateral contact and friction conditions in elastostatics. A projection formulation is developed and used. An existence and uniqueness results for the solutions to the discretized problem is given in the general framework. Several numerical methods to solve the discretized problem are presented (Newton, SOR, fixed points, Uzawa) and compared in terms of the number of iterations and the robustness with respect to the value of the friction coefficient

An error estimate for the Signorini problem with Coulomb friction approximated by finite elements

International audienceThe present paper is concerned with the unilateral contact model and the Coulomb friction law in linear elastostatics. We consider a mixed formulation in which the unknowns are the displacement field and the normal and tangential constraints on the contact area. The chosen finite element method involves continuous elements of degree one and continuous piecewise affine multipliers on the contact zone. A convenient discrete contact and friction condition is introduced in order to perform a convergence study. We finally obtain a first a priori error estimate under the assumptions ensuring the uniqueness of the solution to the continuous problem

Fixed point strategies for elastostatic frictional contact problems

International audienceSeveral fixed point strategies and Uzawa algorithms (for classical and augmented Lagrangian formulations) are presented to solve the unilateral contact problem with Coulomb friction. These methods are analyzed, without introducing any regularization, and a theoretical comparison is performed. Thanks to a formalism coming from convex analysis, some new fixed point strategies are presented and compared to known methods. The analysis is first performed on continuous Tresca problem and then on the finite dimensional Coulomb problem derived from an arbitrary finite element method

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

International audienceThe 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas

The non-smooth contact dynamics method

International audienceThe main features of the Non-Smooth Contact Dynamics method are presented in this paper, the use of the dynamical equation, the non-smooth modelling of unilateral contact and Coulomb's law, fully implicit algorithms to solve the dynamical frictional contact problem for systems with numerous contacting points, in particular large collections of rigid or deformable bodies. Emphasis is put on contact between deformable bodies. Illustrating numerical simulation examples are given for granular materials, deep drawing and buildings made of stone blocks

Interface models coupling adhesion and friction

International audienceInterface models coupling friction and adhesion, where adhesion is regarded as interface damage, are briefly reviewed. The most widely used cohesive zone models are presented and discussed. A general framework for these laws, recently developed by Del Piero and Raous in the form of a unified model, is outlined. As an example, it is here established that the RCCM (Raous-Cangémi-Cocou-Monerie) model is a specific case in this general framework. The variational formulation and some associated solvers are briefly recalled in the context of non-smooth mechanics in the cases of both quasi-static and dynamic problems. A few examples in various fields of application are given. Lastly, some open problems and ongoing researches in this field are presented and discussed

Space-time adaptive finite elements for nonlocal parabolic variational inequalities

This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic variational inequalities associated to the fractional Laplacian we obtain a priori and a posteriori error estimates and study the resulting space-time adaptive mesh-refinement procedures. Particular emphasis is placed on mixed formulations, which include the contact forces as a Lagrange multiplier. Corresponding results are presented for elliptic problems. Our numerical experiments for $2$-dimensional model problems confirm the theoretical results: They indicate the efficiency of the a posteriori error estimates and illustrate the convergence properties of space-time adaptive, as well as uniform and graded discretizations.Comment: 47 pages, 20 figure

Existence results for a contact problem with varying friction coefficient and nonlinear forces

We consider the rate-independent problem of a particle moving in a three - dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed by Mielke and co-workers. We generalize the time-incremental minimization procedure of Mielke and Rossi for the present situation of a non-associative flow rule