Generalized Hill-Mendel lemma and equivalent inclusion method for determining the effective thermal conductivity of composites with imperfect interfaces

Abstract

The present work aims at determining the effective thermal conductivity of two- or three-dimensional composites with imperfect interfaces between their constituent phases. These imperfect interfaces are described by the highly conducting, lowly conducting or general thermal imperfect model. To achieve the objective, the classical Hill-Mendel lemma is first extended to include the effects of imperfect interfaces and an equivalent inclusion method (EIM) is proposed. The basic idea of EIM is to replace an inclusion embedded in a matrix via an imperfect interface by an equivalent inclusion inserted in the same matrix via a perfect interface. Using EIM and applying the dilute distribution, Mori-Tanaka, self-consistent, generalized self-consistent and differential schemes, the effective thermal conductivities of layered composites and some particle-reinforced composites with imperfect interfaces are analytically and explicitly determined. These results are compared with the Voigt, Reuss and Hashin-Shtrikman bounds and checked against the numerical results provided by the fast Fourier transform (FFT) method. These comparisons and checks show that the methods proposed in this work are particularly efficient. The methods and results of the present work are directly transposable to other transport phenomena and anti-plane elasticity by their strict mathematical analogy with thermal conduction.