A quasistatic viscoelastic frictional contact problem with multivalued normal compliance, unilateral constraint and material damage

The goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoelastic body and a foundation. We are interested in the foundation which is made of a hard material covered by a thin layer composed of a rigid crust and a viscoplastic material with finite thickness. The mechanical damage, caused by excessive stress or strain, is emerged in the viscoelastic body during the contact process. We provide the variational formulation of the model which is a system consisting of a history-dependent variational–hemivariational inequality, a fully nonlinear integral equation and a parabolic variational inequality. Then, in order to solve this complex system, we deliver abstract results on a class of history-dependent variational–hemivariational inequalities. Finally, our abstract results are applied to derive the unique weak solvability of the frictional contact problem under consideration.