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 o f entropy is assumed as primitive. Indeed, in T the existence of the equilibrium entropy is not assumed and, furthermore, the dissipative i nequality involves only the dynamic part of entropy. By a certain Gibb s relation proved here, a magnitude upsilon, to be called rate of cha nge of the equilibrium entropic, is defined by means of equilibrium st ress power and equilibrium internal energy, without having at our disp osal a response function for the equilibrium entropic. This definition agrees with the equality which yields the rate of change of the equil ibrium entropy in the corresponding classical theory T, based on the C lausius-Duhem inequality and in which entropy is a primitive. The well -posedness of the definition given for upsilon, from the physical poi nt 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 comp lexity of the material. In T the Clausius-Duhem inequality is stated in terms of upsilon and holds as a theorem. We show that the class of constitutive functions which represent a material in T is strictly l arger than the analogous class in T. Indeed, in the former class the e quilibrium entropic may have no response function, whereas in the latt er class obviously the equilibrium entropy always has a response funct ion. Furthermore, each material in T is a material in T too. Hence, t heory T is more general than theory T. The existence of a response fu nction for the equilibrium entropic is equivalent to a certain integra bility condition, regarding a system of PDEs involving the equilibrium response functions of the stress and of the internal energy. By postu lating this condition, we obtain a theory for which there exists a res ponse function for the equilibrium entropic and such that any theorem of the classical theory T holds. In this case entropic can be called e ntropy, as in T.