Exact solutions for piezoelectric materials with an elliptic hole or a crack under uniform internal pressure

The existing investigations on piezoelectric materials containing an elliptic hole or a crack mainly focus on remote uniform tensile loads. In order to have a better understanding for the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole or a crack in transversely isotropic piezoelectric materials subjected to uniform internal pressure and remote electro-mechanical loads. On the basis of the complex variable approach, analytical solutions of the elastic and electric fields inside and outside the defect are derived by satisfying permeable electric boundary condition at the surface of the elliptical hole. As an example of PZT-4 ceramics, numerical results of electro-elastic fields inside and outside the crack under various electric boundary conditions and electro-mechanical loads are given, and graphs of the electro-elastic fields in the vicinity of the crack tip are presented. The non-singular term is compared to the asymptotic one in the figures. It is shown that the dielectric constant of the air in the crack has no effect on the electric displacement component perpendicular to the crack, and the stresses in the piezoelectric material depend on the material properties and the mechanical loads on the crack surface and at infinity, but not on the electric loads at infinity. The figures obtained are strikingly similar to the available results. Unlike the existing work, the existence of electric fields inside an elliptic hole or a crack is considered, and the piezoelectric solid is subjected to complicated electro-mechanical loads.