Variational and numerical analysis of a frictionless contact problem for elastic-viscoplastic materials with internal state variables

We consider a quasistatic frictionless contact problem between a deformable body and a foundation. The material is assumed to have a nonlinear behaviour that we model with a rate-type viscoplastic constitutive law involving internal state variables. The contact is modelled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution. We then derive error estimates for a fully discrete scheme to solve the problem. Under appropriate regularity assumptions on the exact solution, we establish optimal-order error estimates. Finally, we present numerical examples which show a very good performance of the fully discrete scheme.