A model considering both unilateral contact, Coulomb friction, and adhesion is presented. In the framework of continuum thermodynamics, the contact zone is considered as a material boundary and the local constitutive laws are derived by choosing two specific surface potentials : the free energy and the dissipation potential. Because of the non regular properties of these potentials, convex analysis is used to derive the local behavior laws from the state and the complementary laws. The adhesion is characterized by an internal variable β, introduced by Frémond, which represents the intensity of adhesion. The continuous transition from a total adhesive condition to a possible pure frictional one is enforced by using elasticity coupled with damage for the interface. Non penetration conditions and Coulomb law are strictly imposed without using any penalty. The variational formulation for quasistatic problems is written as the coupling between an implicit variational inequality, a variational inequality, and a differential equation. An incremental formulation is proposed. An existence result under a condition on the friction coefficient is given. A numerical method is derived from the incremental formulation and various algorithms are implemented : they solve a sequence of minimization problems under constraints. The model is used to simulate a micro-indentation experiment conducted to characterize the behavior of fiber/matrix interface in a ceramic composite. Identification of the constitutive parameters is discussed.

International audienceA quasistatic unilateral contact problem with a non-local friction law is considered We propose a new variational formulation of this problem consisting of two inequalities. By applying an implicit time discretization scheme, we obtain an incremental formulation which, if the friction coefficient is sufficiently small, has a unique solution for which appropriate estimations are obtained. This incremental solution enables us to construct a solution to the quasistatic problem by establishing the weak convergence of a subsequence of mappings interpolating the incremental solution. An algorithm is derived and a simple numerical example is presented

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Abstract. We consider a two dimensional elastic body submitted to unilateral contact conditions, local friction and adhesion on a part of his boundary. After discretizing the variational formulation with respect to time we use a smoothing technique to approximate the friction term by an auxiliary problem. A shifting technique enables us to obtain the existence of incremental solutions with bounds independent of the regularization parameter. We finally obtain the existence of a quasistatic solution by passing to the limit with respect to time.Résumé. Nous considérons un corpsélastique bidimensionnel soumisà des conditions de contact unilatéral avec frottement et adhésion sur une partie de sa frontière. Après avoir discrétisé la formulation variationnelle par rapport au temps, nous régularisons le terme de frottement dans un problème auxiliaire. Une technique de translation nous permet d'obtenir l'existence de solutions incrémentales bornées indépendamment du paramètre de régularisation. Nous obtenons finalement l'existence d'une solution quasi-statique en passantà la limite par rapport au temps.Mathematics Subject Classification. 35K85, 49J40, 73T05.

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