A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory

Abstract

In this work, a model for the quasistatic frictionless contact between a viscoelastic body with long memory and a foundation is studied. The material constitutive relation is assumed to be nonlinear and the contact is modelled with the normal compliance condition, i.e., the obstacle is assumed deformable. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, which is modelled by a nonlinear partial differential equation. We derive a variational formulation for the problem and prove the existence of its unique weak solution. Then, we introduce a fully discrete scheme for the numerical solutions of the problem, based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, and we obtain error estimates on the approximate solutions. Finally, some numerical results are presented in the simulation of two-dimensional test problems.