Asymptotic behavior in nonlocal Mindlin's strain gradient thermoelasticity with voids and microtemperatures

Abstract

A nonlocal theory for thermoelastic materials with voids and microtemperatures based on Mindlin's strain gradient theory was derived in this paper. The obtained system of equations is a coupling of a two second-order in time equations with higher gradients terms due to the strain gradient length scale parameter l and the elastic nonlocal parameter ϖ coupled with two parabolic equations. This poses some new mathematical difficulties due to the lack of regularity. Using the semigroup theory, we show the well-posedness of the one dimensional problem. By an approach based on the Gearhart-Herbst-Prüss-Huang theorem, we prove that the associated semigroup is exponentially stable; but not analytic.