13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- where 1 22 1 11 5 1 22 1 44 4 1 22 1 14 3 1 22 1 12 2 1 22 1 24 1 ˆ ˆ , ˆ ˆ , ˆ ˆ , ˆ ˆ , ˆ ˆ − − − − − − − − − − = = = = = L L a L L a L L a L L a L L a (7) and ˆ ˆ . 2 1 1 2 22 Ic Ic L K − = G (8) To obtain Ic G , values of I G given by 1 2 22 ˆ ˆ 2 1 I I L K − = G (9) are obtained at failure for each test and averaged. In Eq. (7), the parameters 1 ˆ− ij L ( , 1,...,4 = i j ) are found from Eq. (5). The phase angles are defined as . ˆ ˆ tan , ˆ ˆ tan 1 1 I II I IV K K K K − − φ= ψ= (10) Eq. (6) represents a three-dimensional failure surface for the case in which the crack faces are at an angle to the poling direction (which is in the plane) and in which the critical energy release rate cG is a function of the phase angles ψ and φ. There is an assumption that the crack propagates in a self-similar manner. Tests were carried out in [10] on four-point bend specimens fabricated from the piezoelectric ceramic PIC-151. This material is similar to PZT-5H. In these experiments, the crack faces were perpendicular to the poling direction and both mechanical loads and electric fields were applied. The electric field was perpendicular to the crack faces. Those experimental results were analyzed in [11]. In the analyses, II Kˆ was found to be zero implying that φ in Eq. (10) 2 is zero. Hence, Eq. (6) may be rewritten as ( ). tan 1 2 tan 2 4 1 = + ψ+ ψ a a Ic cG G (11) The experimental results and the failure curve of Eq. (10), shown as the solid line, are presented in Fig. 1. Note that a is crack length. The value of Ic G was found as 8.6 N/m. There is good agreement between the experimental data and the failure curve. To this end, the root mean square error (RMSE) given by ( ) N t c c ∑ − = N 2 heoretical experiment RMSE G G (12)
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