In this work our main goal is to provide numerical simulations of a quasistatic frictionless contact problem arising in viscoplasticity taking into account the damage of the material and the adhesion to an obstacle. The mechanical damage, caused by excessive stress or strain, is modelled by an inclusion of parabolic type, and the adhesion by an ordinary differential equation. The contact is assumed with a deformable obstacle and then, a normal compliance contact condition is used. The variational formulation is provided for this mechanical problem and the existence of a unique solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the algorithm is deduced. Finally, some numerical examples are presented to show the performance of the method.

Key Words: Viscoplasticity, normal compliance, adhesion, damage, error estimates, numerical simulations.

2000 Mathematics Subject Classification: Primary: Primary: 74M15,
Secondary: 74S05, 65M15, 74D10.