Dynamical Thermoelastic and Adiabatic Theories

Abstract

This part deals with aspects of behavior of thermoelastic bodies with heat conduction and viscosity in dynamical situations. The governing equations are the balance equations of Sects. 3.4–3.7 combined with the constitutive equations of Sect. 9. I; moreover, the constitutive equations must be compatible with the Clausius—Duhem inequality for smooth processes, see Sect. 9.2. In their full generality, the constitutive equations allow for a dissipation by the heat conduction and viscosity, but they also include as special cases the thermoelastic materials 9.1.3(1), where the heat conduction is present and the viscosity is absent, and the idealized dissipationless (‘adiabatic’) materials 9.1.3(2), where both the heat conduction and viscosity are absent. The behavior and interpretation of solutions strongly depend on whether the equations describe a material in which the viscosity and heat conduction are really present or whether they describe one of the two subcases mentioned above. Moreover, there are also situations when one can pass to isothermal or isentropic dynamic theories, thus omitting formally the thermal phenomena altogether. The latter two theories are often referred to by the common name ‘elastodynamics’. Their basic equations are formally identical, but for a given material, the isothermal elastodynamics uses the stress expressed as a function of the deformation gradient at a given fixed temperature, whereas the isentropic dynamics uses the stress expressed as a function of the deformation gradient at a fixed entropy.