13th International Conference on Fracture June 16-21, 2013, Beijing, China F1 F2 Figure 5. FE-mesh of the grown curved crack shown in a deformed configuration. acting at the front end leading to a curved crack path, see Fig. 5. The corresponding Jk-integral calculated within the range [0, ˆxc] is shown in Fig. 4(b). The function Jk (ˆxc) has got nothing to do with path-dependence but enables to localize the incorrect contribution to the crack face integral. The value that Jk reaches approaching the crack tip ˆx1 =1 is the relevant loading quantity according to Eqs. (12) and (1). For the sake of comparison, results from the Crack Tip Element (CTE) method [9, 1] have been included in the figure. The choice of the parameter δhas an influence on the results. For δ<δc, not all inaccurate values are excluded from the numerical integration and therefore the resulting values of Jk are inaccurate. For δ>δc, the values of Jk obtained by the extrapolation method (Method 2) show little influence. On the other hand, the values of Jk calculated by the analytic extension method (Method 1) show increasing deviation for increasing δ. The research revealed that the best choice of δc includes the region of the first three element rings around the crack tip. Both methods are suitable for the accurate calculation of the coordinate MII 2 aswell. Table 1. Comparison of stress intensity factors calculated from the CTE-method and the Mk-integral considering the actual crack face integral (ACFI) and the fictitious crack face integral (FCFI), neglecting FCFI only or neglecting both FCFI and ACFI. CTE Mk Mk without FCFI Mk without FCFI and ACFI KI/(MPa√mm) 240.39 240.31 276.77 261.72 KII/(MPa√mm) -2.86 -2.95 -38.38 7.26 Results of SIF calculated from the CTE-method [9, 1] and theMk-integral are presented in Tab. 1. It is obvious, that neglecting the integration along the fictitious crack faces of the auxiliary fields produces considerable errors. The same holds, when neglecting both the fictitious crack face integral (FCFI) and the actual crack face integral (ACFI), as shown in the last column of Tab. 1. 4. Systems of multiple interacting cracks The loading quantities of multiple crack systems, related to every single crack tip, are calculated by path independent integrals. With regard to crack tips approaching interfaces such as material -7-
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