Non-uniform Eshelby's tensor inside a spherical inclusion in a functionally graded space in transport phenomena

Within the framework of thermal conduction, we consider a functionally graded isotropic infinite medium containing a spherical inclusion which undergoes prescribed uniform heat flux-free temperature gradient. In this research the thermal conductivity is assumed to be exponentially varied in space. Analytical expressions in series form for the temperature and the second-order Eshelby's conduction tensor inside and outside the spherical inclusion are obtained. Our analytical results indicate that the second-order Eshelby's conduction tensor is non-uniform within the spherical inclusion and that it is in general not symmetric. Furthermore our numerical results quantitatively demonstrate how the Eshelby's tensor within the spherical inclusion is non-uniformly distributed due to the spatially varying thermal conductivity.