13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- conditions, there were discrepancies between the measured and calculated curves. A recent review article may be found in [20]. 2. Fracture Criteria A generalized fracture criterion for piezoelectric ceramics requires the development of a unified theory, which consists of mixed mode behavior, together with application of the energetically consistent crack face boundary conditions. In [11], the derivation for the energy release rate begins with the expression [21] k L k1 2 1 − = T G (1) where k is the intensity factor vector given by [ ] IV III I II T K K K K, , , = k (2) and the superscript T represents transpose. In Eq. (1), the 4 4× matrix L is one of the Barnett-Lothe tensors whose components are related to material properties. For the numerical calculations, the intensity factors were normalized so that k V k1 ˆ − = (3) where = e L G L E L E L T A A 26 0 0 0 0 0 0 0 0 0 0 0 0 V (4) AE is Young's modulus in the poling direction, TG is the shear modulus perpendicular to the poling direction, 26 e is a contracted piezoelectric constant and L is a geometrical length parameter. The Barnett-Lothe tensor 1− L is normalized as . ˆ 1 1 L VLV− − = (5) In this way, the diagonal and off-diagonal elements of 1− L are the same order of magnitude contributing to the accuracy of the intensity factor calculation [22]. Note that the units of 1− L are N/m. The criterion presented in [11] with mode III deformation omitted is given by ( )φ φ+ ψ φ+ ψ+ = + ψ+ 2 5 2 4 3 2 1 tan tan 2 tan tan 2 tan 1 2 tan a a a a a Ic cG G (6)
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