Analysis of a dynamic frictional contact problem with damage

This work studies variationally and numerically a dynamic problem which models the bilateral frictional contact between a viscoelastic body and a foundation. A regularized version of Tresca's law is employed to model the friction. Material damage, which results from tension or compression, is taken into account in the constitutive law and its evolution is described by a nonlinear parabolic partial differential equation. The problem is formulated as a coupled system of a nonlinear variational equation for the velocity field and an evolutionary nonlinear variational equation for the damage field. The existence of a unique solution is established. The proof uses a priori estimates and the theory of evolution equations for pseudomonotone operators. A fully discrete numerical scheme is introduced, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme obtained. Finally, the results of simulations of three two-dimensional examples are presented.