An anisotropic piezoelectric half-plane containing an elliptical hole or crack subjected to uniform in-plane electromechanical loading

We derive a series solution for the electro-elastic field inside an anisotropic piezoelectric half-plane containing an elliptical hole or a crack when the half-plane is subjected to in-plane mechanical and electric loadings.Our solution is based on a specific type of conformal map which allows for the mapping of a complete half-plane (without a hole) onto the interior of the unit circle in the imaginary plane.We illustrate our solution with several examples.We show that with decreasing distance between the hole and the edge of the half-plane, the maximum hoop stress around the hole increases rapidly under mechanical loading but slowly in the presence of electric loading.In particular, for a crack with particular orientation in a piezoelectric half-plane subjected to pure shear, we find that the mode-II stress intensity factor at the crack tip farthest from the edge of the half-plane may decrease as the crack approaches the edge.Moreover, if the distance between the crack or the elliptical hole and the edge of the halfplane exceeds four times the size of the hole or semi-length of the crack, the half-plane can be treated essentially as a whole plane without inducing significant errors in the stress concentration around the hole or in the stress and electric displacement intensity factors at the crack tips.