Strong and weak solutions of history-dependent constrained evolutionary variational–hemivariational inequalities and application

Abstract

In this paper we study the well-posedness of evolutionary variational–hemivariational inequalities involving constraint and history-dependent operators. The strong and weak formulations of such inequalities are analysed. First, the existence and uniqueness of solutions to both formulations are proved, and results on solution dependence on functional parameters are delivered. Then, exploiting a fixed point argument, the well-posedness is established for a general form of history-dependent variational–hemivariational inequalities with constraints. As an illustrative example, we apply the theory to a dynamic frictional contact problem in viscoelasticity in which the contact is described by a frictionless Signorini-type unilateral boundary condition with a nonmonotone Clarke’s relation and the friction is modelled by a generalization of the evolutionary version of Coulomb’s law of dry friction with the friction bound depending on the normal and tangential components of the displacement.