Anisotropic ellipsoidal inclusion with an anisotropic shell in an isotropic medium subjected to a uniform electric field

An electrostatic problem has been solved for a dielectric inclusion that consists of an anisotropic core and an anisotropic shell. The inclusion is immersed in a uniform isotropic medium (matrix) subjected to a uniform electric field. It is assumed that the outer boundaries of the core and shell are ellipsoidal and become confocal after a linear nonorthogonal transformation that removes the anisotropy of the dielectric properties of the shell. Analytical expressions have been derived for the potential and strength of the electric field in the matrix and also in the shell and core of the inclusion, and an expression for the polarizability tensor of the inclusion has been deduced. It has been shown that the results agree with the well-known solutions in partial (limiting) cases.