A dissipation inequality involving only the dynamic part of entropy

For a general body B of the differential type and arbitrary complexity, we set up a thermodynamic theory T ∗ in which only the dynamic part of entropy is assumed as primitive. Indeed, in T ∗ the existence of the equilibrium entropy is not assumed and, furthermore, the dissipative inequality involves only the dynamic part of entropy. By a certain Gibbs relation proved here, a magnitude v ∗ , to be called ‘rate of change of the equilibrium entropic’, is defined by means of equilibrium stress power and equilibrium internal energy, without having at our disposal a response function for the equilibrium entropic. This definition agrees with the equality which yields the rate of change of the equilibrium entropy in the corresponding classical theory T, based on the Clausius-Duhem inequality and in which entropy is a primitive. The well-posedness of the definition given for v ∗ , from the physical point of view, is assured both by the uniqueness theorem for the response function of the stress and by the uniqueness theorem for the response function of the equilibrium internal energy, proved here for any complexity of the material. In T ∗ the Clausius-Duhem inequality is stated in terms of v ∗ and holds as a theorem. We show that the class of constitutive functions which represent a material in T ∗ is strictly larger than the analogous class in T. Indeed, in the former class the equilibrium entropic may have no response function, whereas in the latter class obviously the equilibrium entropy always has a response function. Furthermore, each material in T is a material in T ∗ too. Hence, theory T ∗ is more general than theory T. The existence of a response function for the equilibrium entropic is equivalent to a certain integrability condition, regarding a system of PDEs involving the equilibrium response functions of the stress and of the internal energy. By postulating this condition, we obtain a theory for which there exists a response function for the equilibrium entropic and such that any theorem of the classical theory T holds. In this case entropic can be called entropy, as in T