Variational inequality with almost history-dependent operator for frictionless contact problems

Abstract

We study two quasistatic contact problems which describe the frictionless contact between a body and deformable foundation on an infinite time interval. The contact is modelled by the normal compliance condition with limited penetration and memory. The first problem deals with evolution of a body made of a viscoplastic material and in the second problem the material is viscoelastic with long memory. The constitutive functions of these materials have a non-polynomial growth. For each problem we derive a variational formulation that has the form of an almost history-dependent variational inequality for the displacement field. We demonstrate existence and uniqueness results of abstract almost history-dependent inclusion and variational inequality in the reflexive Orlicz–Sobolev space. Finally, we apply the abstract results to prove existence of the unique weak solution to the contact problems.