Upper bounds on decay rates in Mindlin’s Form II gradient thermoelasticity of type III

In this paper we derive a nonlinear theory of Mindlin’s Form II gradient for thermoelastic materials where the heat propagation model proposed by Green and Naghdi, taking into account micro-inertia effects as well. The elastic behavior is assumed to be consistent with Mindlin’s Form II gradient elasticity theory, while the thermal behavior is based on type III entropy balance, where the second gradient of the thermal displacement is included in the set of independent constitutive variables. This leads to a fourth-order equations for the temperature. The equations of the linear theory are also obtained. Semigroup theory is then used to prove the well-posedness of the obtained problem. We show that, in general, the one-dimensional problem is not exponentially stable but that there exists polynomial stability with rates that depend on the micro-inertia parameter h and the regularity of the initial data. Using a resolvent criterion developed by Borichev and Tomilov, we prove that the polynomial decay rate is t −1/3 when h > 0, while it is t −1 when h = 0. By following a result due to Arendt-Batty, we show that the considered problem is strongly stable whenever the value of h .