Modified Eshelby tensor for an ellipsoidal inclusion imperfectly embedded in an infinite piezoelectric medium

The modified Eshelby tensor for predicting the effective moduli of particle-reinforced piezoelectric composites is derived for the problem of an ellipsoidal inclusion which is imperfectly bonded to the matrix. A linear interface relation is adopted, which involves discontinuities of the mechanical displacements and electric potential across the interface, and assumes that the corresponding jumps are proportional to the continuous stresses and electric displacements at the interface. The piezoelectric field induced by a uniform eigenstrain given only in the inclusion is deduced analytically. As the induced piezoelectric field is no longer uniform, the average strains and electric displacements are calculated, and the modified piezoelectric Eshelby tensor is evaluated by both an iterative method and a direct method. By comparison, it is shown that the iterative method yields rapidly convergent results.