The Eshelby problem of the confocal N-layer spheroid with imperfect interfaces and the notion of equivalent particle in thermal conduction

Abstract

The solution to the generalized conduction Eshelby problem of a confocal N-layer spheroid with low or highly conducting interfaces between isotropic layers is provided thanks to a decomposition in series of harmonics. A generic workflow is detailed for practical numerical implementation including a fine analysis to assess the influence of the level of truncation of the infinite series. The case of perfect interfaces presents a particular interest insofar as it is characterized by an equivalent conductivity tensor obtained from a recursive procedure. The notion of equivalent conductivity is then investigated and applied to a uniform spheroid surrounded by an imperfect interface. Some approximated models are developed, either based on a surface description of the interface or on a thin interphase, casting a new light on published models and proposing a unified framework for new ones. These approximated models are finally analyzed by comparison to the exact solution.