Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites

In a great number of situations of practical interest, the interfaces between the constituent phases of a composite turn out to be imperfect. In the context of thermal conduction, an interface is said to be imperfect if the requirement that both the temperature and the normal heat flux be continuous across the interface is not satisfied. A powerful method based on mathematical asymptotic analysis has been proposed and developed in the literature by several authors for the derivation of linear imperfect interface models of thermal conduction. This method consists in replacing an interphase of small uniform thickness between two-phases by an imperfect interface of null thickness characterized by the temperature and normal heat flux jump relations deduced by carrying out an appropriate asymptotic analysis. The objective of the present work is threefold. Firstly, it aims to explicitly show and emphasize the key role played by Hadamard’s relation in the method. Secondly, it has the purpose of using a coordinate-free differential geometry theory and Hadamard’s relation to render the method coordinate-free. Thirdly and most importantly, the present work gives a weak formulation for the problem concerning the steady thermal conduction in a composite with the interfaces described by the general temperature and normal heat flux jump relations derived. This weak formulation is a key step toward solving the problem by the extended finite element method (XFEM) presented in a companion paper.