13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- Finally, the mesoscopic threshold, common to all the loading cases was determined as the average of the thresholds associated to each loading conditions. This mesoscopic average threshold is shown in Figure 6 by the dashed horizontal lines passing through all intervals bounded by the values of 0.1 and 0.9 quantiles in the case of Crossland and Van Dang FIPs. For Matake FIP, this was also true except for the case of biaxial loadings with a phase shift of 90° due to the reasons mentioned above. (a) Crossland (b) Matake (c) Dang Van Figure 7. Predictions from (a) Crossland, (b) Matake and (c) Dang Van at the macroscopic length scale (black line) and the mesoscopic length scale considering all grains (gray dots) and surface grains (gray dots). The mesoscopic threshold of different FIPs is the average (over the different loading conditions) of the medians of the extreme value distributions. Keeping the same value for the iα parameter, the new criterion containing microstructural heterogeneities contribution at the mesoscopic length scale is plotted in Figure 7. This Figure illustrates for each criterion two mesoscopic thresholds : the first was determined from all the grains of the aggregate (red line) and the second was computed by considering only the surface grains. When the mesoscopic threshold is close to the macroscopic one (black line), the microstructure heterogeneities are taken into account by the original criterion. This is especially the case of Dang Van criterion, when the mesoscopic threshold determined for all the aggregate grains is equal to the macroscopic thresholds, and to a lesser extent the case of Matake criterion (Table 3). For the Crossland criterion, the distance between the two straight lines is important in the case of thresholds determined from all the grains and from the surface grains (table 3). This comparison shows that critical plane type approaches can capture the microstructure heterogeneity despite simplifying assumptions [11]. Table 3. Values of macroscopic thresholds and mesoscopic thresholds obtained by 2D and 3D FE modeling Criterion Macroscopic Thresholds Mesoscopic Thresholds 2D model (all grains) 3D model (all grains) 3D model (surface grains) Crossland 36.15 54.78 55.68 51.48 Matake 36.15 41.97 39.21 36.82 Dang Van 36.15 39.81 36.36 33.59
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