A dynamic thermoviscoelastic contact problem with the second sound effect

This paper deals with a contact problem describing the mechanical and thermal
evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law,
relating 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 moves vertically between
reactive stops like a nonlinear spring. Existence and uniqueness of the solution is
proved, 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 proved, from which the
linear convergence of the algorithm is deduced. The case where the two stops are
rigid is also studied from the point of view of the existence and longtime behavior
of the solutions. Finally, some numerical simulations are presented to demonstrate
the accuracy of the approximation and the behavior of the solution