Existence and uniqueness of the weak solution for a contact problem

Abstract

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, the friction is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field, a time-dependent variational equation for the potential field and a differential equation for the bounding field. Then we prove the existence of a unique weak solution for the model. The proof is based on arguments of evolution equations and the Banach fixed point theorem.