Variational Analysis of a Quasistatic Contact Problem

We start by proving an existence and uniqueness result for a new class of variational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities which involve history-dependent operators. The proof is based on arguments of monotonicity, convexity and fixed point. Then, we consider a mathematical model which describes the frictional contact between an elastic-viscoplastic body and a moving foundation. The mechanical process is assumed to be quasistatic, and the contact is modeled with a multivalued normal compliance condition with unilateral constraint and memory term, associated to a sliding version of Coulomb's law of dry friction. We prove that the model casts in the abstract setting of variational inequalities, with a convenient choice of spaces and operators. Further, we apply our abstract result to prove the unique weak solvability of the contact model.