We consider a mathematical model for the process of a frictionless contact between an elastic-viscoplastic body and a reactive foundation. The material is elastic-viscoplastic with internal state variable which may describe the damage of the system caused by plastic deformations. We establish a variational formulation for the model and prove the existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear equations with monotone operators, on parabolic type inequalities and fixed point.Mathematics Subject Classification (2010). Primary 74M15; Secondary 74D10.