Solution for a cracked piezoelectric plate subjected to variable load on plastic zones under mode-I deformation

A model for crack arrest is proposed for a poled piezoelectric plate weakened by a finite, hairline and straight crack. Remotely applied uniform, constant tension and electric field acting in the direction perpendicular to the rims of the crack opens the crack rims in mode-I type deformations. The plate is assumed to be mechanically brittle and electrically ductile. Under this assumption the plate will encounter mechanical singularity first. In this situation a plastic zone protrudes ahead of each tip of the crack. To arrest the crack from further opening, the rims of the developed plastic zones are subjected to normal cohesive quadratically varying yield point stress. Two cases investigated are: (i) when crack lie perpendicular to poling axis and (ii) crack lie parallel to poling axis. Each case is solved employing complex variable technique. Closed form analytic expressions are obtained for complex potentials, determining elastic and electric fields in the plate. A non-linear transcendental equation is obtained to determine plastic zone length, using Dugdale hypothesis.