Eshelby’s tensor of a cubic piezoelectric crystal under plane strain condition and its application to elliptic cavity problems

Abstract

The general solution of two-dimensional problems in cubic piezoelectric crystals under plane strain condition is derived. The connection between the present solution and that for purely cubic elasticity is established. Accordingly, Eshelby’s tensor of an elliptic inclusion is obtained from the corresponding results in orthotropic elasticity. The tensor is expressed explicitly in terms of material constants and shape ratio of the inclusion. As an application, stress concentration problems of an elliptic hole under two different electric conditions (electrically free and electrically clamped) imposed on the end faces of the material are solved by the equivalent inclusion method. Explicit formulas of stress concentration factors are obtained for uniformly applied tension and shear. The link between the stress concentration factor of a slender cavity and the stress intensity factor of the associated slit crack is also developed and discussed.