13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- from the work of Lukas and Kunz [12]. The parameters iβ are identical for the three criteria ( 1− = = =t dv c m β β β ). The expression of iα are given in Table 2. A comparison between the mesoscopic FIPs predictions and the macroscopic (original) criteria is shown in Figure 2. This comparison shows the existence of grains at the surface (highlighted in light blue) or in the bulk of the volume element from which the FIP exceeded the macroscopic threshold. The macroscopic threshold is not applicable at the grain scale. This is especially true for the Crossland criterion where most of the grains are above the threshold. For other criteria, only a small number of grains exceeded the macroscopic threshold. A statistical analysis of the microstructure-sensitivity of the different FIPs will be presented in the next section. This statistical analysis will determine a new mesoscopic threshold to take into account the microstructure heterogeneities. Figure 2. Locus of the 200 FIPs at the grain length scale (gray dots), and macroscopic criterion (black dot) in the case of tension loading ( 1 =− ∑R ). The FIPs corresponding to the surface grains are highlighted (in light blue) and the black straight line is the experimental macroscopic threshold 4. Microstructure sensitivity The HCF strength is related to the critical grain whose response leads to the maximum value of the FIP. These extreme values are located at the tails of the probability density functions of mesoscopic responses (see Figure 3) and are highly sensitive to the microstructure attributes. To study these critical grains, several statistical approaches are possible. The method selected for this work was based on the extreme value probability. This method involves the statistical analyze of the maximum values of the mesoscopic FIP, corresponding to the various studied volume element, by considering a single value for each aggregate. Figure 3. Localization of the extreme values in the tails of the Dang Van FIP distribution (a) Crossland FIP (b) Matake FIP (c) Dang Van FIP
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