Analysis of electro-viscoelastic antiplane contact problem with total slip rate dependent friction

We consider a mathematical model which describes the antiplane shear deformation of a cylinder in frictional contact with a rigid foundation. The contact is bilateral and is modelled with a total slip rate dependent friction law. The material is assumed to be electro-viscoelastic and the foundation is assumed to be electrically conductive. First, we describe the classical formulation for the antiplane problem and we give the corresponding variational formulation which is given by a system coupling an evolutionary variational equality for the displacement field and a time-dependent variational equation for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of variational inequalities and by using the Banach fixed-point Theorem.