Issue 45

G. Gomes et alii, Frattura ed Integrità Strutturale, 45 (2018) 67-85; DOI: 10.3221/IGF-ESIS.45.06 70 The decomposition of Eqn. (7) leads to, 2 2 , ' ' I II I II K K J J E E   (8) The integration process takes place along a circular path around the crack tip, as illustrated in Fig. 1. A simple trapezoidal rule can be adopted along the circular path. Figure 1: Circular contour path for calculation of the J-integral [12]. Prediction of Fatigue Crack Growth According to [14], there are several criteria to compute the crack-extension direction in the linear elastic regime, as the Maximum Principal Stress (MPS) criterion. This criterion postulates that the crack growth will occur in a perpendicular direction to the maximum principal stress, that is, the crack propagates in the direction that maximizes the circumferential stress in a closed area at the crack tip. Here, this criterion will be used since it describes the local direction for the crack propagation considering mixed mode fracture mechanics, with the SIFs calculated by the J-integral technique. According to the MPS criterion, the local direction of crack-extension  t is determined by the expression, sin (3cos 1) 0 I t II t K K      (9) where  t is an angular coordinate with centre at the crack tip and measured from the crack axis ahead of the tip. In an incremental analysis, the procedure used to set the direction of the nth crack extension increment requires a correction angle β in the tangential direction θ t(n) of the crack path due to the continuity criterion of Eqn. (9), as shown in Fig. 2. Accordingly, where the length increment of the crack propagation,  a, tends to zero, the angle  t (n + 1) tends to zero as well, and therefore the angle  is fixed meaning that, in the limit, the crack-extension direction tends to the tangential direction of the continuous crack path. This correction angle  is given by  =  t(n+1) /2 , where  t(n+1) corresponds to the direction of the next crack-extension increment, which is calculated by the MPS criterion. The fatigue crack propagation analysis presented here stems from the results extracted from the incremental analysis of the crack growth, considering the stress intensity factors calculated for each increment. Thus, based on the fundamental postulate of LEFM, a fracture criterion involving the SIFs is given by, Ieq Ic K K  (10) where K Ieq is the stress intensity factor (mode I), defined in a mixed-mode analysis, and K Ic is the stiffness fracture plane strain, or a critical SIF value defined as a material property. Thus, we consider a load cycle with constant amplitude, for simplicity, described by a static load level and stress amplitude ratio R, given by, Ieq min min max Ieq max K R K     (11)