Singular perturbation approach to an elastic dry friction problem with non-monotone coefficient

Renard, Yves

doi:10.1090/qam/1753401

Cited by 6 publications

(15 citation statements)

References 5 publications

(5 reference statements)

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“…An interesting open problem would be to study the evolution of the bifurcations (i.e., the evolution of the corresponding eigenvalue problems) when h and ∆t tend to zero. For the moment, no evidence of dynamic bifurcations has been exhibited for the continuous dynamical friction problem except when the friction coefficient is a decreasing function of the sliding velocity (see [18] and [26] for instance).…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

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

Hild¹,

Renard

2005

Quart. Appl. Math.

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“…However, following the analysis performed in Renard [26], this ill-posed problem may benefit from the well-posed property of the slider. To that end a thin rigid 'sole' is positioned on the friction surface {x = 0} of the slab (see Figure 4).…”

Section: The Surface Perturbation Modelmentioning

confidence: 99%

“…A mathematical analysis of the convergence as ε vanishes can be found in Renard [25,26]. It is shown that the unique solution v ε (t) to this Cauchy problem is close to the so-called 'maximum delay' solution v md (t) of the scalar equation (2.13)-(2.14).…”

Section: The Surface Perturbation Modelmentioning

confidence: 99%

“…The main motivation is to try to generalise theoretical and qualitative results established in Ionescu & Paumier [10] and Renard [26] for the one-dimensional problem to the multidimensional case. In the one-dimensional case, the analysis of the problem shows that the use of a non-monotone slip-dependent friction coefficient in the purely elastodynamic problem introduces a multiplicity of solutions and shocks in the velocity.…”

Section: Introductionmentioning

confidence: 99%

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Surface perturbation of an elastodynamic contact problem with friction

Paumier¹,

Renard

2003

Eur. J. Appl. Math

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International audienceWe consider the dynamical process of an elastic body in unilateral fricitonal contact with a rigid foundation. Friction is modelled with the Coulomb law with a coefficient that depends on the slip velocity. To allow for velocity discontinuities we use the elastodynamic (hyperbolic) framework. Nevertheless, this does not lead to a well-posed problem. To remedy this, we perturb the solution of the elastodynamical problem in a thin layer next to the contact boundary. This is a generalization of an approach previously studied in a one-dimensional case. We establish existence and uniqueness results for the perturbed and regularized problem and provide an interpretation of this perturbation

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“…The first method, due to Renard [17], [18], consists in adding a small mass concentrated on the friction surface. Using stability arguments, Renard has proved the convergence of the solution for a large class of loadings in the one-dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Viscosity solutions for dynamic problems with slip-rate dependent friction

Ionescu¹

2002

Quart. Appl. Math.

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Abstract.The dynamic evolution of an elastic medium undergoing frictional slip is considered. The Coulomb law modeling the contact uses a friction coefficient that is a non-monotone function of the slip-rate. This problem is ill-posed, the solution is nonunique and shocks may be created on the contact interface. In the particular case of the one-dimensional shearing of an elastic slab, the (perfect) delay convention can be used to select a unique solution. Different solutions in acceleration and deceleration processes are obtained. To transform the ill-posed problem into a well-posed one and to justify the choice of the perfect delay criterion, a visco-elastic constitutive law with a small viscosity is used here. An existence and uniqueness result is obtained in three dimensions. The assumptions on the functions implied in the contact model are weak enough to include both the normal compliance and the Tresca model. The following conjecture, based on results of numerical simulations, is stated: in the elastic case, the solution chosen by the perfect delay convention is the one obtained from the solutions of the problem with viscosity, when the viscosity tends to zero.

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Singular perturbation approach to an elastic dry friction problem with non-monotone coefficient

Cited by 6 publications

References 5 publications

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

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

Surface perturbation of an elastodynamic contact problem with friction

Viscosity solutions for dynamic problems with slip-rate dependent friction

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