Electro-elastic fields of piezoelectric materials with an elliptic hole under uniform internal shearing forces

The existing investigations on piezoelectric materials containing an elliptic hole mainly focus on remote uniform tensile loads. In order to have a better understanding of the fracture behavior of piezoelectric materials under different loading conditions, theoretical and numerical solutions are presented for an elliptic hole in transversely isotropic piezoelectric materials subjected to uniform internal shearing forces based on the complex potential approach. By solving ten variable linear equations, the analytical solutions inside and outside the hole satisfying the permeable electric boundary conditions are obtained. Taking PZT-4 ceramic into consideration, numerical results of electro-elastic fields along the edge of the hole and axes, and the electric displacements in the hole are presented. Comparison with stresses in transverse isotropic elastic materials shows that the hoop stress at the ends of major axis in two kinds of material equals zero for the various ratios of major to minor axis lengths; If the ratio is greater than 1, the hoop stress in piezoelectric materials is smaller than that in elastic materials, and if the ratio is smaller than 1, the hoop stress in piezoelectric materials is greater than that in elastic materials; When it is a circle hole, the shearing stress in two materials along axes is the same. The distribution of electric displacement components shows that the vertical electric displacement in the hole and along axes in the material is always zero though under the permeable electric boundary condition; The horizontal and vertical electric displacement components along the edge of the hole are symmetrical and antisymmetrical about horizontal axis, respectively. The stress and electric displacement distribution tends to zero at distances far from the elliptical hole, which conforms to the conclusion usually made on the basis of Saint-Venant’s principle. Unlike the existing work, uniform shearing forces acting on the edge of the hole, and the distribution of electro-elastic fields inside and outside the elliptic hole are considered.