A dynamic thermoviscoelastic contact problem with the second sound effect

作者：Berti Alessia; Copetti Maria I M; Fernandez Jose R^*; Naso Maria Grazia

来源：Journal of Mathematical Analysis and Applications, 2015, 421(2): 1163-1195.

DOI：10.1016/j.jmaa.2014.07.049

• Dynamic thermoviscoelastic contact
• Second sound
• Energy decay
• Finite element approximation
• A priori error estimates
• Numerical simulations

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.

• 出版日期2015-1-15