Linearized thermoviscoelasticity with high temperature variations and related periodic problems

Abstract

In the first part of this paper we develop a linearization of the equations of the thermoviscoelastic field in the case of great temperature variations. The possibility of uncoupling the heat equation from the motion equation is discussed in Section 3. After recalling some results on duality and virtual work principle we then study the motion equation with temperature as data, i.e. a given function of time and space variables. More precisely we study existence, uniqueness, regularity and asymptotic stability of a T-periodic (stress) solution of the motion equation in the dynamical case (Section 7) and in the quasi-static case (Section 8), when the temperature field is T-periodic in time and with a constitutive equation of Maxwell type where the stiffness and viscosity matrix are temperature dependent and thus are T-periodic functions of time. In the proof of the theorems we use frequently an inequality of monotony which means that the material is dissipative on a period. This inequality hold if the stiffness is a slowly varying function of time (the temperature has a little effect on the stiffness), on the other hand, fortunately, there is no condition on the viscosity.