Well-Posedness of an Elastic-Viscoplastic Contact Problem

We consider a mathematical model which describes the contact of an elastic-viscoplastic body with a foundation. The contact is frictionless and is modeled with normal compliance and unilateral condition. The weak formulation of the model is in a form of a system which couples a variational inequality for the displacement field with an implicit nonlinear equation for the stress field. We prove the unique solvability of the problem by using a fixed point argument for history-dependent operators. Next, we provide necessary and sufficient conditions which guarantee the convergence to the solution of this elastic-viscoplastic contact model. We use these results in order to obtain a continuous dependence result and to introduce two well-posedness concepts for the problem. Finally, we consider a one-dimensional example which allows us to compare the two well-posedness concepts employed.