Eur. Phys. J. Plus (2020) 135: 480
https://doi.org/10.1140/epjp/s13360-020-00475-9

Regular Article

Analysis of a thermoelastic problem of type III

¹ Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI de Telecomunicación, Buzón 104, Campus As Lagoas Marcosende s/n, 36310, Vigo, Spain
² Departamento de Matemáticas, E.S.E.I.A.A.T.-U.P.C., Colom 11, 08222, Terrassa, Barcelona, Spain

^b jose.fernandez@uvigo.es

Received: 14 February 2020
Accepted: 19 May 2020
Published online: 8 June 2020

Abstract

This paper investigates several aspects of the linear type III thermoelastic theory. First, we consider the most general system of equations for this theory in the case that the conductivity rate is not definite and we prove an existence theorem by means of the semigroups theory. In fact, we show that the solutions of the problem generate a quasi-contractive semigroup. Then, assuming that the internal energy is positive definite, the numerical analysis of this problem is performed, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A discrete stability property and a priori error estimates are shown, from which the linear convergence of the algorithm is deduced. Finally, some one- and two-dimensional numerical simulations are presented, for the homogeneous and isotropic case, to demonstrate the accuracy of the approximation and the behaviour of the solution.