This research was concerned with a linear theory of thermoelasticity with microtemperatures where the second thermal displacement gradient and the second gradient of microtemperatures are included in the classical set of independent constitutive variables. The master balance laws of micromorphic continua, the theory of the strain gradient of elasticity, and Green-Naghdi thermomechanics were used to derive a second gradient theory. The semigroup theory of linear operators allowed us to prove that the problem of the second gradient thermoelasticity with microtemperatures is well-posed. For the equations of isotropic rigids, we presented a natural extension of the Cauchy-Kovalevski-Somigliana solution of isothermal theory. In the case of stationary vibrations, the fundamental solutions of the basic equations were obtained. Uniqueness and instability of the solutions were obtained in the case of antiplane shear deformations.

In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain gradient theory. An existence and uniqueness result is proved by using semigroup arguments, and the exponential energy decay is obtained. The lack of differentiability for the semigroup of contractions is also shown. Then, fully discrete approximations are introduced by using the finite element method and the backward time scheme, for which a discrete stability property and a priori error estimates are proved. The linear convergence is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy decay.