Analysis of Two Dynamic Frictionless Contact Problems for Elastic-visco-plastic Materials

We consider two mathematical models which describe the contact between an elastic-visco-plastic body and an obstacle, the so-called foundation. In both models the contact is frictionless and the process is assumed to be dynamic. In the first model the contact is described with a normal compliance condition and, in the second one, is described with a normal damped response condition. We derive a variational formulation of the models which is in the form of a system coupling an integro-differential equation with a second order variational equation for the displacement and the stress fields. Then we prove the unique weak solvability of the models. The proofs are based on arguments on nonlinear evolution equations with monotone operators and fixed point. Finally, we study the dependence of the solution with respect to a perturbation of the contact conditions and prove a convergence result.