Anti-plane problems of a piezoelectric inclusion with an elliptic hole or a crack in an infinite piezoelectric matrix

Based on the complex potential method and linear-elastic piezoelectric constitutive equation, the anti-plane problems of a piezoelectric inclusion with an elliptic hole or crack in an infinite piezoelectric matrix are studied. Firstly, by using the conformal transformation and Taylor series, the complex potential functions in the piezoelectric matrix and inclusion are given, respectively, in form of series. Secondly, the unknown coefficients are obtained in terms of the boundary conditions. Finally, the electric and stress fields of the piezoelectric matrix and inclusion are solved. The numerical results show that the field intensity factors changes along with the material constants of the matrix and inclusion. It is also found that for the “soft inclusion”, the field intensity factors decrease with the increase of the size ratio between the crack and inclusion, and for the “hard inclusion”, the field intensity factors increase with the increase of the size ratio between the crack and inclusion.