Analysis of a dynamic contact problem involving a nonlinear thermoviscoelastic beam with second sound

In this paper we consider a dynamic contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law, relying the heat flux to the gradient of the temperature. The beam is rigidly clamped at its left end whereas the right end of the beam is supposed to move vertically between two stops reactive and behaving as a nonlinear spring. Existence and uniqueness of the solution are stated, as well as the exponential decay of the related energy. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme to approximate the spatial variable and to discretize the time derivatives, respectively. An a priori error estimates result is obtained, from which the linear convergence of the algorithm is deduced. Finally, a numerical simulation is presented to demonstrate the behavior of the solution.