A Quasistatic Electro-Elastic Contact Problem with Long Memory and Slip Dependent Coefficient of Friction

In this paper we consider a mathematical model which describes a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modelled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coefficient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a smallness assumption on the coefficient of friction, we prove an existence result of the weak solution of the model. We can show the uniqueness of the solution by adding another condition. The proofs are based on arguments of time -dependent variational inequalities, differential equations and Banach fixed point theorem.