13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- where ij and Ei are the strain tensor and electric field vector, and sijkl, dkij and T ik are the elastic compliance, direct piezoelectric constant and dielectric permittivity at constant stress, which satisfy the following symmetry relations: klij ijlk jikl ijkl s s s s , kji kij d d , T T ji ij (5) The strain and electric field may be expressed in terms of the displacement vector ui and the electric potential by ) ( 2 1 , , i j j i ij u u (6) i iE , (7) The constitutive equations (3) and (4) for PZT poled in the x3-direction are found in Appendix A. The enhanced electromechanical field level results in localized polarization switching. When the electromechanical work exceeds a critical value, polarization switching occurs [4]. Thus the switching criterion is defined as c s i i ij ij E P P E 2 (8) where Ps is spontaneous polarization, Ec is coercive electric field, and ij and Pi are the changes in the spontaneous strain s and spontaneous polarization during switching, respectively, and are given in Appendix B. The constitutive equations (3) and (4) after polarization switching are given by r ij kij k kl ijkl ij d E s ' (9) r i k ik kl ikl i E P D d T ' (10) where 31 15 1 ' { ( ) ( 2 )} 2 ikl i k l i kl i k l ik l i k l il k d dnnn dn nnn d n nnn n 33 (11) In Eq. (11), ni is the unit vector in the poling direction and ij is the Kroneker delta. In order to evaluate the energy release rate G of PZT, plane strain finite element analysis (FEA) was carried out for the cracked piezoelectric specimens under concentrated load P. The crack is assumed with faces normal (Case 1) or parallel (Case 2) to the polarization axis as shown in Fig.2. Let the coordinate axes x = x1 and z = x3 be chosen such that the y = x2 axis coincides with the thickness direction. The z axis is the poling direction. The three-point flexure specimen with span S = 13 mm is the beam of width W = 5 mm and length L = 15 mm containing a crack of length a. The length between two electrodes is L0 = 5 mm for Case 1 and W = 5 mm for Case 2. Because of symmetry, only the half of the model was used in the FEA. We first consider the Case 1 as shown in Fig. 2(a). The permeable crack model is treated, and the boundary conditions at z = 0 can be expressed in the form ( ,0) 0 (0 ) ( ,0) 0 ( ) z zz u x x W a x W a x W (12) ) (0 ( ,0) 0 x W x zx (13)
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