13th International Conference on Fracture June 16-21, 2013, Beijing, China interfaces, internal boundaries or crack surfaces, it is beneficial to evaluate the integrals along large contours containing all N tips of a multiple crack system, see Fig. 6(a). For comparison, the crack paths are simulated by calculating the integral along small contours in the vicinity of the crack tip, as shown in Fig. 6(b). The resulting value of the Jk-integral along a large contour Γ0 equals the sum of −φ(1) φ(2) J(1) k J(2) k Γ0 ¯x1 ¯x2 P0 (a) −φ(1) φ(2) J(1) k J(2) k Γ1 Γ2 ¯x1 ¯x2 P0 (b) Figure 6. Integration contours Γ0, Γ1, Γ2 for path-independent Jk andMk-integral andJk-integral vectors J(i) k . the loading quantities of all Ncrack tips. J1 = N Σi=1 cos φ(i) J(i) k (19a) J2 = N Σi=1 sin φ(i) J(i) k (19b) The angle φ(i) is related to the global coordinate system ¯xi. The applied crack deflection criterion is that of the maximum energy release rate, i.e. the Jk-vector points into the direction of the crack propagationzk, see Eq. (4). This and the principle of minimum potential energy lead to the auxiliary condition, that the sum of the absolute values of the Jk-integrals related to every single crack tip, reaches a global maximum. Further, each absolute value J(i) k must be smaller or equal to the value of the critical energy release rate Gc. If another condition is required, in order to reduce the solution space it can be postulated that the crack deflection angle must be smaller or equal to a maximum value dφmax. The latter criterion is motivated by the fact that cracks usually show a smooth curvature whereas sharp kinks are only observed if the loading regime is subjected to a sudden and fundamental change. N Σi=1 J(i) k ! =max (20a) -8-
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