Temperature-Rate Dependence Thermoelasticity Theory with Memory-Dependent Derivative: Stability and Uniqueness

Abstract

This article discusses the stability analysis of thermal signals of a thermodynamic consistent model including temperature-rate dependence thermoelasticity theory (Green-Lindsay) with a memory-dependent derivative (MDD). A unifying approach (an extension of Lyapunov’s original method to the stability theory developed by Zubov together with Korn’s inequality in elasticity under homogeneous boundary condition on displacement components) is employed to characterize the stability of the present thermoelastic system. In continuation, a uniqueness of the solutions of the present thermoelastic system is presented as a corollary of the stability theorem, and corresponding results in the absence of MDD are also mentioned as special cases. Finally, based on theoretical importance and understanding, with the help of an analogy between the homogeneous and nonhomogeneous boundary conditions on the displacement components, an open mathematical problem as alternate to the employed unifying approach is proposed.