Two models of three-dimensional thin interphases with variable conductivity and their fulfillment of the reciprocal theorem

Interphases appear in heterogeneous media in a variety of forms. Often the treatment of a thin interphase as a separate phase in a multiphase solid is not convenient in analytical or numerical solutions of those systems. Thus, approximate models of a thin interphase that make possible to obtain a solution for the fields in the media adjacent to it, without the need of determining the fields within the interphase itself, become a necessity in many cases. The question then arises whether a global property which was present in the original heterogeneous medium will continue to prevail after an approximate representation of the thin interphase has been introduced in the system. A global property, known to have important consequences on the behavior of the heterogeneous solid, is the “reciprocity” relation between a pair of two different solutions, as stated by the reciprocal theorem. Since the formulation of an approximate model for the thin interphase involves several assumptions, the fulfillment of the reciprocal theorem in the original system does not necessarily imply its fulfillment in the transformed system in which an approximate model of the thin interphase has been intoduced. The preservation of the reciprocity relation by the approximate model, if proved, would be considered to be an important consistency quality of the model. In this paper we consider steady thermal conduction phenomena, and generalize the two approximate models of a thin interphase by Bövik (1994), Benveniste (2006), Benveniste and Berdichevsky (2010) to the case of thin interphases with a variable conductivity. The fulfillment of the reciprocity property in the presence of those models, which was not discussed in the above papers, is investigated here in the context of their presently developed generalized version, and it is proved that both models fulfill the reciprocal theorem.