13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- We are interested in the maximum values of different FIPs listed in Table 2. Scale factors ( na and nb ) and extreme index ξ are determined using the maximum likelihood method with a confidence interval of 99%. Figure 5 shows a comparison between the determined distributions and the samples for the Crossland FIP. The identified GEV density function and distribution function showed a good correlation with the probability density and the cumulative probability determined from the extreme values of FIPs database. 6. Results and discussions Figure 6 represents the mesoscopic thresholds (medians and also the values of 0.1 and 0.9 quantiles) for each loading condition. The mesoscopic thresholds are statistically determined from the FIPs at the grain scale considering the local stress state. This local stress state is computed by a finite element (FE) simulation. A comparison between the mesoscopic thresholds obtained by considering a 3D FE model (described above) and a 2D FE model [14] is presented in this figure. The mesoscopic thresholds are normalized by the macroscopic threshold to analyze the effect of microstructure variability. Referring to figure 6, for all studied loading conditions and studied FIPs, the normalized mesoscopic thresholds were always larger than 1 in the case of the local stress state computed by 2D FE model [14]. When the local stress state is determined by a 3D model, the mesoscopic threshold determined from all grains decreases for Matake and Dang Van criteria, and increases slightly in the case of Crossland criterion. When considering all the grains (bulk + surface), this threshold is higher for Crossland criterion and is close to 1 for Matake criterion and especially for Dang Van criterion. The mesoscopic threshold determined by considering only the surface grains becomes lower than the macroscopic threshold for Dang Van criterion. This is not verified for the other two criteria. This comparison highlights the ability of the Dang Van criterion to reflect the microstructural heterogeneities compared to the two other criteria. On the other hand, the mesoscopic thresholds, defined as the medians of the extreme value distribution of the studied FIPs depend on the loading case. This gap depends on the studied FIP: it is low in the case of the Crossland and Dang Van FIPs (Figure 6-(a) and (c)) and significant in the case of the Matake FIP (Figure 6-(b)). For this last FIP, the change in mesoscopic thresholds was observed especially for the biaxial loading with a phase shift of 90°. Figure 6. Evolution of the median (a probability of 0.50) of the extreme value distributions of (a) Crossland, (b) Matake and (c) Dang Van FIPs, as a function of loading conditions determined by 2D (in black), and 3D (in blue and red) FE model. The two limits of the interval correspond to a probability of 0.10 and 0.90. (a) Crossland FIP (b) Matake FIP (c) Dang Van FIP
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