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In this work, we study a dynamic problem arising in the MGT-thermoelasti-city. Under some assumptions on the constitutive tensors, the basic equations of the model are derived, leading to a coupled system written in terms of the displacements and the thermal displacements. A MGT dissipation mechanism is considered for the heat equation. An existence and uniqueness result is proved in the three-dimensional case, and the one-dimensional version is analyzed in the case that the coefficients of the constitutive functions are assumed constants. We prove that at least a polynomial decay is found and that, in general, an exponential decay cannot be expected. A couple of remarks concerning particular situations are also proposed. Then, a fully discrete approximation is introduced by using the finite element method and the implicit Euler scheme, proving an a priori error estimates result which, under some additional regularity, leads to the linear convergence of the approximation. Finally, we perform some one-dimensional numerical simulations which show the accuracy and the behavior of the solution.

Analysis of a strain-gradient problem arising in MGT thermoelasticity