Variational Analysis of a Dynamic Frictional Contact Problem with Adhesion and Long Memory in Viscoelastic Materials

Abstract

We consider a mathematical model that describes a dynamic frictional contact between a foundation and a viscoelastic body with long memory. The contact is modelled with a normal compliance condition in that the penetration is limited and restricted to a unilateral constraint and associated to the nonlocal friction law with adhesion, where the coefficient of friction is an independent solution. The adhesion of the contact surfaces is considered and modelled with a surface variable. We derive a variational formulation written as the coupling between a variational inequality and a differential equation. The existence and uniqueness result of the weak solution under a smallness assumption on the coefficient of friction is established. The proof is based on arguments of nonlinear evolution equation with monotone operators, a classical existence, differential equations, and the Banach fixed point theorem.