Classical results for a non-classical theory: Remarks on thermodynamic relations in Green-Naghdi thermo-hyperelasticity

In this contribution we first investigate the thermodynamic’s relation between the entropy flux H and the heat flux Q in the non-classical theory of Green and Naghdi imposed by the entropy equality used in their approach. We show that the classical assumption entropy flux = heat flux/absolute temperature holds for isotropic materials by exploiting the entropy principle of Müller and Liu. As opposed to the usual procedure, we formulate the balance laws with respect to the material description. Subsequently, further restrictions are obtained by exploiting the entropy principle as suggested by Green and Naghdi (in Proc. R. Soc. Lond. 432, 171–194, 1991). This yields the classical potential relations between the entropy and the free energy and between the second Piola-Kirchhoff stress tensor and the free energy