Global Equivalence for Heat-Conducting Bodies with Fading Memory

Abstract

We study the notion of global physical equivalence between two heat-conducting deformable simple bodies with fading memory. For heat-conducting elastic bodies and heat-conducting rigid bodies this equivalence has been treated in previous works by the writer. It is understood in the sense that there exists a suitable bijective correspondence k between the bodies B and B′ such that any global thermokinetic process for B is admissible (in the sense of Coleman and Noll) if and only if its k-corresponding for B′ also is admissible. Here we extend the notion of global equivalence to all heat-conducting simple bodies with fading memory; we also give a definition of global equivalence which is properly weaker in all the cases considered above. Then we establish some conditions on the response functionals of B and B′ which are necessary and sufficient for these bodies to be globally equivalent in each one of the two senses. This theorem extends the ones mentioned above for heat-conducting elastic bodies and heat-conducting rigid bodies presented in previous works of the author. Lastly, we introduce the notion of entropy-equivalence for the response functionals of the heat flux in bodies that are globally equivalent. Then we characterize the class of all simple bodies that are globally equivalent to B in the cases where the heat flux response functionals of these bodies are entropy-equivalent or not, respectively, to the heat flux response functional of B. These characterizations put in evidence the existence of globally equivalent bodies whose heat fluxes are not entropy-equivalent.