Size-dependent Eshelby tensor fields and effective conductivity of composites made of anisotropic phases with highly conducting imperfect interfaces

In this work, Eshelby's results and formalism for an elastic inhomogeneity embedded in an elastic infinite matrix are extended to the thermal-conduction phenomenon in composites consisting of anisotropic phases with highly conducting imperfect interfaces. The generalized Eshelby's interior and exterior conduction tensor fields and localization tensor fields in the important cases of circular and spherical inhomogeneities are obtained in an explicit analytical way. Quite different from the relevant results of elasticity, the generalized Eshelby's conduction tensor field and localization tensor field inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of highly conducting imperfect interface. With the help of the obtained expressions for Eshelby's tensor fields and localization tensor fields, the size-dependent overall thermal-conduction properties of composites are estimated by using the dilute, Mori-Tanaka, self-consistent, and generalized self-consistent models. The analytical results are finally compared with numerical results delivered by the finite element method. The approach elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion, and flow in porous media and to the mathematically identical phenomenon of antiplane elasticity.