Electroelastic Eshelby tensors for an ellipsoidal piezoelectric inclusion

Abstract

The present paper extends Eshelby's tensors for elastic isotropic inclusions to the piezoelectric cases. By utilizing the eigenstrain formulation and Cauchy's residue theorem, a simple and unified explicit expression for electroelastic Eshelby tensors is presented. As in the uncoupled elastic cases, the resulting electroelastic Eshelby tensors are functions of the shape of the inclusion and the properties of the surrounding matrix. These tensors are applicable not only to the general ellipsoidal inclusion in a piezoelectric solid but also to the elastic anisotropic inclusion problems when piezoelectric coupling is absent. In particular, the electroelastic Eshelby tensors are obtained in the closed form when the piezoelectric medium is transversely isotropic and the shapes of the inclusions are elliptic, rod-shaped, penny-shaped, and ribbon-like. Finally, it is illustrated that the coupled electric field and elastic strains inside the inclusion are uniform when the eigenstrains are constant.