A mixed finite element method and solution multiplicity for Coulomb frictional contact

Hassani, Riad; Hild, Patrick; Ionescu, Ioan R.; Sakki, Nour-Dine

doi:10.1016/s0045-7825(03)00419-5

Cited by 38 publications

(23 citation statements)

References 10 publications

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“…These solutions have been obtained for a large friction coefficient (F > 1) and for a tangential displacement having a constant sign. For the moment, it seems that no multi-solution has been exhibited for an arbitrary small friction coefficient in the continuous case, although such a result exists for finite element approximation in [6], but for a variable geometry. As far as we know, no uniqueness result has been proved even for a sufficiently small friction coefficient.…”

Section: A Uniqueness Criterionmentioning

confidence: 99%

A uniqueness criterion for the Signorini problem with Coulomb friction

Renard

2006

Analysis and Simulation of Contact Problems

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The 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 proven, here, that if a solutions 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 multi-solutions to the Coulomb problem, because it eliminates a lot of uniqueness situations.

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Section: A Uniqueness Criterionmentioning

confidence: 99%

A uniqueness criterion for the Signorini problem with Coulomb friction

Renard

2006

Analysis and Simulation of Contact Problems

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“…Several examples of nonunique solutions exist for the static case involving a finite or infinite number of solutions (see, e.g., [13]). Moreover it is possible to find (using finite element computations) for an arbitrary small friction coefficient a geometry with a nonuniqueness example (see [12]). …”

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Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics

Hild¹,

Renard

2005

Quart. Appl. Math.

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We dedicate this article to the memory of Jean-Claude Paumier. Abstract. This work is concerned with the frictional contact problem governed by the Signorini contact model and the Coulomb friction law in static linear elasticity. We consider a general finite-dimensional setting and we study local uniqueness and smooth or nonsmooth continuation of solutions by using a generalized version of the implicit function theorem involving Clarke's gradient. We show that for any contact status there exists an eigenvalue problem and that the solutions are locally unique if the friction coefficient is not an eigenvalue. Finally we illustrate our general results with a simple example in which the bifurcation diagrams are exhibited and discussed.Introduction. Friction problems are of current interest both from the theoretical and practical point of view in structural mechanics. Numerous studies deal with the widespread Coulomb friction law [6] introduced in the eighteenth century which takes into account the possibility of slip and stick on the friction area. Generally the friction model is coupled with a contact law, and very often one considers the unilateral contact allowing separation and contact and excluding interpenetration. Although quite simple in its formulation, the Coulomb friction law shows great mathematical difficulties which have not allowed a complete understanding of the model. In the simple case of continuum elastostatics (i.e., the so-called static friction law) only existence results for small friction [24,19,9]) as well as some examples of nonuniqueness of solutions for large friction coefficients [14,15]. As far as we know there does not exist any uniqueness result and/or nonexistence example for the continuous model.In the finite-dimensional context, the finite element problem admits always a solution which is unique provided that the friction coefficient is lower than a critical value vanishing when the discretization parameter h tends to zero (see e.g., [10]). In fact the critical value behaves like h 1/2 . Actually, it is not established if this mesh-size-dependent bound ensuring uniqueness represents a real loss of uniqueness or if it comes from a "nonoptimal" mathematical analysis. In particular we don't know if for a given geometry there may exist a mesh and a nonuniqueness example for an arbitrary small friction coefficient. Several examples of nonunique solutions exist for the static case involving a finite or infinite number of solutions (see, e.g., [13]). Moreover it is possible to find (using finite element computations) for an arbitrary small friction coefficient a geometry with a nonuniqueness example (see [12]).Our aim in this paper is to propose and to study a framework for the finite-dimensional problem in order to obtain results ensuring local uniqueness and smooth or nonsmooth continuation of solutions. As far as we know the only existing results concerned with uniqueness in the finite-dimensional case are global and assume that the friction coefficient is small. As a consequence t...

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“…µ > (1 − P )/P ) and tangential displacements with a constant sign on Γ C . Actually, it seems that no multi-solution has been detected for an arbitrary small friction coefficient in the continuous case, although such a result exists for finite element approximations in [28], but for a variable geometry. The forthcoming partial uniqueness result is obtained in [46]: it defines some cases where it is possible to affirm that a solution to the Coulomb friction problem is in fact the unique solution.…”

Section: The Frictional Contact Problem In Elasticitymentioning

confidence: 99%