Analysis of two frictional viscoplastic contact problems with damage

In this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material, caused by excessive stress or strain and modelled by an inclusion of parabolic type. The variational formulation is provided for both problems and the existence of a unique solution is proved for each of them. 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.