13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Analyses were conducted for the following cases. (1) The case of local stress by the crack and the super dislocation being equivalent at the center position of (0, 1). (Condition 1) (2) The case of local stress by the crack being two times larger than that of super dislocation at the center position of (0, 1). (Condition 2) (3) The case of local stress by the crack being half of that of super dislocation at the center position of (0, 1). (Condition 3) Since the stress singularity of the crack against the distance from the crack tip (r-1/2) is smaller than that of super dislocation (r-1), the high stress region of the former is wider than that of the latter. 2.5. Fractal analysis based on Box counting method In this paper, fractal analysis was adopted to estimate the self-similarity of the distributed characteristics of D based on Box counting method. 3. Results of Analysis 3.1. The comparison of two methods and the characteristics of changing rate of atomic density, D The effects of atomic scale and number on the characteristics of D defined by Eq. (14) were shown in Figs. 4(a), (b) and (c) under three conditions. Results obtained by both of the direct and Verlet methods showed the same characteristics of atomic distributions that atomic density increase at the center position of analysis region, (0, 1) and decrease at positions of both ends, that is the crack tip and the super dislocation. The direct method can be applicable for any case of the moving distance of atom, however the latter is based on the Taylor expansion and the small moving distance is required. Furthermore, the calculating time of the former is much longer than that of the latter as shown in Fig. 5. In the region of D having a negative value as shown in Figs. 4 (a), (b) and (c), the breaking stress of atomic bond was applied. Therefore, this region is considered to be a dominating region of fracture. As shown in Figs. 4(a), (b) and (c), with increase in the effect of the crack or the super dislocation, correspondingly, the occurrence of the region of D having negative value near the crack or dislocation becomes typical. Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 Normalized distance, X The changing rate of atom density, Di N=250(Direct) N=250(Verlet) N=500(Direct) N=500(Verlet) N=1000(Direct) N=1000(Verlet) 0 0.2 0.4 0.6 0.8 1 -0.08 -0.06 -0.04 -0.02 0 (a) Condition 1 (b) Condition 2 (c) Condition 3 Figure 4. Plots of changing rate of atomic density D vs. normalized distance on each condition
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