On the self-consistent, energetically consistent, and electrostatic traction approaches in piezoelectric fracture mechanics

In terms of crack opening, the present work studies the self-consistent, energetically consistent, and electrostatic traction approaches. In the self-consistent approach, crack will not open if no mechanical load is applied. The energetically consistent approach under a given electric field yields a threshold stress for crack opening and a bifurcation stress, which is higher than the threshold stress. Between the bifurcation and threshold points there are two solutions for crack opening. The electrostatic traction approach considers electrostatic tractions along crack faces and infinite boundaries. The electrostatic traction along infinite boundaries is equivalent to an additional mechanical load for a given electric field, which is tensile and promotes crack opening, if the dielectric constant of the infinite medium is smaller than that of the material. Energy release rate is also comprehensively analyzed in each of the three approaches.