A viscoplastic frictional contact problem

Abstract We consider a mathematical model, which describes the quasistatic contact process of an elastic‐viscoplastic body with a foundation. The process is quasistatic, the contact is bilateral and the friction is modeled with Tresca's friction law. We provide a new variational formulation of the model which is in the form of a system coupling a time‐dependent equation for the stress field, a nonlinear integral equation for the transformation strain field, and an evolutionary variational inequality for the displacement field. We use arguments of variational inequalities, fixed point, and various estimates in order to prove unique weak solvability of the model, together with a continuous dependence result of the weak solution with respect to the yield limit. Next, we use a finite element scheme to approximate the problem, based on the primal‐dual active set (PDAS) method. We implement it on the computer and provide numerical simulations which validate the theoretical continuous dependence result.