Second gradient thermoelasticity with microtemperatures

Abstract

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.

Keywords

• elastic solids with microtemperatures
• solids with microstructure
• second gradient theory
• constitutive equations
• well-posed problems