The developments in the preceding chapter assumed nothing about the constitutive response of the continuum; we now restrict attention to a special class of constitutive laws – the so-called thermoelastic materials introduced previously in Chapter 5. Our discussion in Chapter 5 was focused entirely on the energy wells of the characterizing energy potential. Here we discuss thermoelastic materials and nonlinear thermoelasticity in more detail.
In Section 7.2 we state the constitutive law of nonlinear thermoelasticity, in which stress and specific entropy are specified as functions of deformation gradient and absolute temperature through the Helmholtz free energy potential. An equivalent alternate form of the constitutive law, in which stress and temperature are given in terms of deformation gradient and specific entropy by means of the internal energy potential, is also discussed. The expression for the driving force is then specialized to this setting. Next we state the heat conduction law, and in Section 7.2.3, we write out the full theory in the form of four scalar partial differential equations involving the three components of displacement and temperature. The accompanying jump conditions are also laid out. In the final subsection we specialize the results to a state of thermomechanical equilibrium.