Numerical analysis of a dynamic viscoelastic contact problem with damage

Abstract

In this work, a dynamic frictionless viscoelastic contact problem is considered. The contact with the foundation is modelled by a normal compliance contact condition. The mechanical damage of the material, caused by excessive stress or strain, is included into the model through a differential inclusion. The weak formulation leads to a nonlinear system including a parabolic variational inequality for the damage field coupled with a variational equation for the displacement field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and a finite difference to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the scheme, under suitable regularity conditions, can be derived. Finally, some numerical results on a two-dimensional problem are presented to show the performance of the scheme.