Rothe method for an adhesive contact of a nonlinear viscoelastic rod

We consider a dynamic process of an adhesive contact between a viscoelastic rod and a foundation. The problem in its mathematical formulation is a coupled system of two relations: a second order Clarke subdifferential inclusion describing the evolution of the rods displacement and an ordinary differential equation that governs a behavior of the bonding field. Both relations in the system are nonlinear, which introduces the main difficulty to its analysis. Our goal is to provide the existence of a weak solution for the problem under consideration. It is achieved by means of the Rothe method based on the temporal discretization strategy.