Issue34

Y. Sumi, Frattura ed Integrità Strutturale, 34 (2015) 43-59; DOI: 10.3221/IGF-ESIS.34.04 47 Eq. (17) with respect to  , we can obtain the equation which gives rise to the direction as   sin 3cos 1 0, I II k k      (25) Another criterion is the geometry-based one, the so-called local symmetry criterion proposed by Banichuk [1] and Goldstein and Salganik [2, 3], where they assume that the crack extends under pure Mode-I condition along the crack extension, i.e.       II 21 I 22 II 0. K F k F k       (26) In contrast to the condition (25), the stress intensity factor after crack extension is required for the crack path prediction. Sometimes the explicit simplicity of the criterion (25) is in favor over the criterion (26), which may be consistent but numerically implicit. In order to examine whether the above two criteria are equivalent to each other or not, we calculate determinant, D 1 of Eqs. (25) and (26) with respect to k I and k II ,       1 22 21 3 5 7 2 4 2 sin 3cos 1 4 1 4 53 1 ( ), 3 4 90 32 D F F O                                 (27) where F 2q ( q =1,2) are expanded in terms of  . The result does not vanish so that these two criteria are independent with each other, and the difference appears in the third order term with respect to  . A more fundamental question raised from this result is that since the crack path predicted by the maximum hoop-stress criterion certainly induces a finite Mode-II stress intensity factor at the crack tip after an infinitesimally small crack extension, we cannot expect a smooth trajectory of a crack path i.e., an infinitesimally small zig-zag crack path along the kink. The energy-based criterion is the one which maximizes the energy release rate so that the potential energy of the elastic body is minimized, Wu [9]. Using the energy release rate calculated by Eq. (24), this condition is attained by I II I II 1 0, 4 dK dK dG K K d d d                (28) and 2 2 0. d G d   (29) Suppose that the equivalence of the criteria (26) and (28) holds, then the second term of the right-hand side of Eq. (28) vanishes so that I I 0. dKK d   (30) Since K I can naturally be assumed to be positive during the fracturing process, we must have the stationary condition of K I , i.e.,     I 11 I 12 II ' ' 0, dK F k F k d       (31) in which prime denotes the differentiation with respect to   and   1 ' q F  ( q =1,2) are given by   3 5 7 11 2 4 2 3 4 5 2 11 119 ' ( ), 4 32 3 12 2560 F O                             (32)