In this talk we present the numerical analysis of a dynamic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage, which measures the density of the microcracks in the material and results from tension or compression ([
]), is then involved in the constitutive law (see [
] for details), and modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary inequalities for which we state the existence of a unique weak solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme is deduced. Finally, numerical results are presented for some two-dimensional examples in order to show the accuracy of the algorithm.