A contact problem in thermoviscoelastic diffusion theory with second sound^∗,∗∗

¹ École Nationale d’Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire, 7035 Menzel Abderrahman, Tunisia .
moncefaouadi@yahoo.fr
² Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900 Santa Maria, RS Brasil.
mimcopetti@ufsm.br
³ Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain
jose.fernandez@uvigo.es

Received: 22 June 2015
Revised: 6 May 2016
Accepted: 16 May 2016

Abstract

We consider a contact problem in thermoviscoelastic diffusion theory in one space dimension with second sound. The contact is modeled by the Signorini’s condition and the stress-strain constitutive equation is of Kelvin−Voigt type. The thermal and diffusion disturbances are modeled by Cattaneo’s law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions and the lack of regularity. We prove that the viscoelastic term provides additional regularity leading to the existence of weak solutions. Then, fully discrete approximations to a penalized problem are considered by using the finite element method. A stability property is shown, which leads to a discrete version of the energy decay property. A priori error analysis is then provided, from which the linear convergence of the algorithm is derived. Finally, we give some computational results.