Constitutive Models in Thermomechanics

Abstract

The macroscopic behaviour of material bodies is not determined by mechanical processes alone. Among the numerous non-mechanical effects thermomechanical energy transformations play a vital role. They change the temperature field and influence the mechanical behaviour of the material. Elasticity parameters, viscosities, relaxation spectra, yield limits, rate-dependent and rate-independent hysteresis effects, every material property can depend to a greater or lesser degree on temperature. In order to take the temperature-dependence of the material behaviour into account it is not sufficient to simply insert the temperature into the material description as an additional parameter. The modelling of thermomechanical material behaviour has to observe the fundamental laws of mechanics. It also has to adhere to the natural laws of thermodynamics, i.e. the energy balance (first law) (3.18),

$$\dot e\left( {X,t} \right) = - \frac{1}{{{\rho _R}}}Div{q_R} + r + \frac{1}{{{\rho _R}}}\tilde T\cdot \dot E$$

(13.1)

and the principle of irreversibility (second law). The formulation of the principle of irreversibility usually employed in continuum thermomechanics has its roots in the entropy production (3.39),

$$\gamma = \dot s\left( {X,t} \right) + \frac{1}{{{\rho _R}}}Div\left( {\frac{{{q_R}}}{\Theta }} \right) - \frac{r}{\Theta },$$

(13.2)

and demands that this must be non-negative:

$$\gamma \geqslant 0.$$

(13.3)