A dissipation inequality involving only the dynamic part of entropy (Q1339443)

This extensive study constitutes a very interesting thermodynamic investigation of bodies of differential type and arbitrary complexity. For this class of materials, two thermodynamic theories \(\tau^*\) and \(\tau\) are considered: \(\tau^*\) is a thermodynamic theory in which only the dynamic part \(\eta^ d\) of entropy is assumed as primitive concept, and the dissipative inequality involves only \(\eta^ d\). \(\tau\) is the classical thermodynamic theory based on Clausius-Duhem inequality in which the entropy is a primitive concept. The so-called rate of change of entropic equilibrium \(\upsilon^*\) is introduced in \(\tau^*\) by means of equilibrium stress power and equilibrium internal energy. The definition of \(\upsilon^*\) agrees with the rate of change of equilibrium entropy in \(\tau\). In \(\tau^*\) the Clausius-Duhem inequality, which is written in terms of \(\upsilon^*\), holds as a theorem. It is shown that the theory \(\tau^*\) is more general than the theory \(\tau\) because the class of constitutive functions defining a material in \(\tau^*\) is strictly larger than the analogous class in \(\tau\), and each differential type material in \(\tau\) is a material of the same kind in \(\tau^*\), too. It is shown that the existence of a response function of the entropic equilibrium is equivalent to certain integrability conditions \((C)\) for a system of partial differential equations entering into the equilibrium functionals of the stress and internal energy. By postulating the conditions \((C)\), the author obtains a thermodynamic theory \(\tau^*_ c\) in which there exists a response function for the entropic equilibrium and such that any theorem in \(\tau\) holds. The notions of entropy in \(\tau\) and \(\tau^*_ c\) agree, so that in this case entropic equilibrium can be called entropy as in \(\tau\).