Entropy Flux Relation for Viscoelastic Bodies

Thermodynamic restrictions of elastic materials in general are well-known based on the Clausius–Duhem inequality by employing the simple Coleman–Noll procedure. One of the basic assumptions in this entropy inequality is that the entropy flux is defined as the heat flux divided by the absolute temperature. To avoid this unnecessary and possibly too restrictive assumption, the general entropy inequality has been proposed and its thermodynamic consequences exploited following the Müller–Liu procedure in which supply-free bodies are considered and Lagrange Multipliers are introduced. In this new thermodynamic theory, the entropy flux and heat flux relation identical to the above assumption has not been proved for elastic bodies in general. For isotropic elastic bodies, it was proved by Müller in 1971, using explicit isotropic representations for constitutive functions. Unfortunately, the procedure contains a flaw which was later pointed out, but can not be easily resolved. Although it was shown later that it can be proved by Müller–Liu procedure, it has not been available in the literature. In this paper, we shall establish this result, providing the missing details in the previous proof. The analysis will be carried out for isotropic viscoelastic materials and the case of elastic materials follows as a special case.