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 Patrick Hild1, Yves Renard2 Abstract This work is concerned with the frictional contact problem governed by the Sig- norini contact model and the Coulomb friction law in static linear elasticity. We con- sider 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. Keywords: Coulomb friction, unilateral contact, local uniqueness, bifurcation, Clarke's gradient. 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.

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