STUDIA UNIV. “BABES ¸–BOLYAI”, MATHEMATICA, Volume LIV, Number 1, March 2009 ANALYSIS OF A ELECTRO-ELASTIC CONTACT PROBLEM WITH FRICTION AND ADHESION SALAH DRABLA AND ZILOUKHA ZELLAGUI Abstract. We consider a mathematical model which describes the qua- sistatic frictional contact between a piezoelectric body and an obstacle, the so-called foundation. A nonlinear electro-elastic constitutive law is used to model the piezoelectric material. The contact is modelled with Signorini’s conditions and the associated with a regularized Coulomb’s law of dry fric- tion in witch the adhesion of contact surfaces is taken into account. The evolution of the bonding field is described by a first order differential equa- tion. We derive a variational formulation for the model, in the form of a coupled system for the displacements, the electric potential and the adhe- sion. Under a smallness assumption on the coefficient of friction, we prove the existence of a unique weak solution of the model. The proof is based on arguments of time-dependent quasi-variational inequalities, differential equations and Banach’s fixed point theorem. 1. Introduction The piezoelectric effect is characterized by the coupling between the mechani- cal and electrical properties of the materials. Indeed, the apparition of electric charges on some crystals submitted to the action of body forces and surface tractions was ob- served and their dependence on the deformation process was underlined. Conversely, it was proved experimentally that the action of electric field on the crystals may gen- erate strain and stress. A deformable material which presents such a behavior is called a piezoelectric material. Piezoelectric materials are used extensively as switches and Received by the editors: 01.03.2008. 2000 Mathematics Subject Classification. 74H10, 74H10, 74M15, 74F25, 49J40. Key words and phrases. piezoelectric material, electro-elastic, erictional contact, nonlocal Coulomb’s law, adhesion; quasi-variational inequality, weak solution, fixed point. 75