PSI - Issue 13

M.R. Ayatollahi et al. / Procedia Structural Integrity 13 (2018) 735–740

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Ayatollahi et al. / Structural Integrity Procedia 00 (2018) 000–000

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By employing Eq. (8), the variations of fracture initiation angle θ 0 with respect to B α for ν = 0.35 is plotted in Fig. 2. It can be seen from the figure that when B α is greater than 0.22 (for ν = 0.35), the minimum strain energy density does not occur in the initial plane of the crack. The critical value of B α above which crack kinking is expected to occur depends on the Poisson’s ratio. Consequently, it can be deduced that for higher values of T -stress or for large process zone sizes r c , it is expected that the crack kinks out of its initial direction. Similar results were reported by Smith et al. (2001). They proposed the generalized maximum tangential stress (GMTS) criterion and studied the effect of T -stress in mixed mode brittle fracture. It was shown that mixed mode brittle fracture is significantly influenced by T -stress values. The variation of crack initiation angle as a function of B α according to GMTS criterion is shown in Fig. 2. According to the GMTS criterion, when B α > 0.375 the crack deviates from its initial plane under mode I loading condition. According to Fig. 2, for the specimens with 0.22 < B α < 0.375, GMTS criterion doesn’t predict crack kinking, while GSED criterion predicts the deviation of the crack in this range of B α .

20 40 60 80 100

GMTS GSED

Fracture initiation angle, θ 0

‐100 ‐80 ‐60 ‐40 ‐20 0

0.1

0.2

0.3

0.4

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Bα

Fig. 2. Fracture initiation angle versus B α under mode-I loading for ν = 0.35.

To numerically obtain the crack growth path a crack tracking iterative model was used in finite element software. The values of SIFs and T -stress obtained from finite element models were considered as inputs for the GSED and GMTS models to obtain the theoretical prediction of crack growth path. In each increment of simulation, crack length was extended by a constant pre-specified incremental length of 1 mm. By substituting the numerical values of K I , K II and T -stress obtained in each simulation step into Eq. (4), the crack growth angle θ 0 can be determined. The crack geometry is then redefined by extending the incremental crack segment and the previous computational steps are repeated until the crack length reaches the boundaries of specimen. The elastic material properties of PMMA as mentioned before have been also considered in the 2D plane strain finite element models. As said above, all the fracture tests were conducted under pure mode I loading condition, so the first step of crack growth was modelled using the mode I relations. For the specimens with large values of T -stress, the crack growth path was deviated and, in these cases, the crack propagation was simulated using the mixed mode relations. 4. Results and Discussion Details of each test including the dimensions of specimens, the biaxiality ratio B , the fracture load and the measured fracture initiation angles are listed in Table 1. According to the GSED criterion, for specimens with large values of T -stress crack curving is expected to occur. In Table 1, the fracture initiation angle θ 0 for the five tested specimens were predicted using the GSED criterion by solving Eq. (4). For 0.22   B (e.g. for DCB 1 , DCB 2 and TDCB 2 specimens), the angle θ 0 does not coincide with the plane of the crack. By comparing the experimental values of the fracture initiation angle θ 0 with the theoretical ones based on GSED (see Table 1), it can be observed that the agreement is very good. It should be mentioned that all the tested specimens had B α < 0.375, therefore, the GMTS criterion failed to predict the crack deviation in the tested specimens giving a straight crack growth path. Fig. 3 shows a comparison between the theoretical curves predicted for the crack trajectory by employing the GSED and GMTS criteria and the experimentally observed curves. Good agreement can be observed between the experimental results and theoretical predictions based on GSED criterion. As shown in Fig. 3, for the CT and TDCB 1 specimens where B α is less than the critical value of 0.22, the crack propagated along its initial direction. However, for the other specimens with 0.22   B , the crack deviated from its initial plane generating a curvilinear crack trajectory. After the initial extensions of crack, unstable crack growth due to sudden fracture of the cracked specimens leads to a discrepancy between the numerical and experimental results particularly when the crack becomes larger. According to Fig. 3, cracks in the specimens with higher T -stress levels exhibit larger deviations from the initial crack plane.