Faber series method for two-dimensional problems of an arbitrarily shaped inclusion in piezoelectric materials

Abstract

This paper presents a simple method for the two-dimensional electroelastic problem of an arbitrarily shaped inclusion in an infinite piezoelectric solid, which is subjected to out-of-plane shear stress and in-plane electric field at infinity. The most remarkable feature of the method is to express the complex potentials in the inclusion region in the form of Faber series with unknown coefficients. By use of continuity conditions on the interface, all unknown coefficients can be determined from a set of linear equations. At first, we derive a general solution for an arbitrarily shaped inclusion in form of infinite series. Then, we give exact solutions for an elliptical inclusion, and approximate solutions for a square inclusion and an equilateral triangle inclusion, respectively. More importantly, this method can be extended to solve the 2-D interface crack problems of any isotropic inclusion-matrix system, provided that one knows the mapping function which maps the infinite region outside the inclusion into the outside of a unit circle. In fact, the function has been well studied in Muskhelishvili’s masterpiece.