Experimental results for cracked piezoelectric materials demonstrate that applied electric fields can enhance or inhibit crack growth, which cannot be explained via stress intensity factor K and energy release rate G, but may be predicted according to the fracture criterion of mechanical strain energy release rate Gm. However, the Gm criterion fails to predict the experimental observation that purely electric fields can drive crack growth in a poled ferroelectric ceramic. In this paper, using electric boundary conditions dependent on the crack-opening displacement, the nonlinear relation between the electric displacement at the crack faces and applied loading is given explicitly. The Fourier transform technique is employed to reduce the problem to dual-integral equations. Solving the resulting equations, expressions for the electroelastic field in the entire plane are obtained for a piezoelectric material weakened by a Griffith crack perpendicular to the poling axis. The distribution of asymptotic field and the intensity factors of electroelastic field as well as the elastic T-stress are determined. In particular, the strain intensity factor Ks is suggested as a fracture criterion for piezoelectric materials, which can explain successfully relevant experimental results. Results reveal that an applied electric field tends to cause crack to open or close, depending on its positive or negative direction. A comparison of numerical results for PbðZr, TiÞO3-5H indicates that the Ks criterion is superior to other fracture criteria.