13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Dispersion method from super atoms to actual atoms under the mechanical equivalence (Scaling from macro to micro under the conservative mechanical system) 2.2. Basic Equation In this analysis, the number of N of super atoms were placed between a crack and a super dislocation which represents dislocation groups. The equations of interaction forces exerted on super atoms were mechanically considered to be symmetry with those exerted on actual scale atoms [9]. The mechanical model used for this analysis was shown in Fig. 2. A crack tip exists at the site of x = 0 and shearing stress was exerted parallel to the crack that is mode II condition. The super dislocation exists at the site of x = d. The number of n of super atoms were placed the same scale interval. In this analysis, to avoid the jump out of atoms from this region during the process of analysis of obtaining the equilibrium positions of atoms between the crack and the super dislocation, fixed atoms were placed at the site of the crack tip and the super dislocation respectively. Local stress fields by the crack and super dislocation were given by boundary conditions in this analysis. The interaction forces between these local stress fields and fixed atoms were neglected. The interaction force exerted on each super atoms due to the crack, super dislocation and other super atoms in the array was given by Eq. (1), ( ) ∑ + ≠ = ∗ − − + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 1 0 7 13 * 2 6 12 4 N j i j i d i II j i j i i x x nA x K x x x x I π σ σ σ ε (1) Where i=1~(N-1), ε : constant with dimension of energy, σ : constant with dimension of length, KII：stress intensity factor of mode II which concerns local stress of the crack tip, xi：the position of the ith atom, n : intensity of the super dislocation, A*：intensity of an isolated actual dislocation. The first and second terms of right hand side of Eq. (1) were interactive forces due to other super atoms exerted on the ith atoms. The third and the forth terms of right hand side of Eq. (1) were interactive forces due to local stress fields of the crack and super dislocation exerted on the ith super atom. Figure 2. The mechanical equilibrium model of a crack, a super dislocation and super atoms Crack Super Atoms Super Dislocation 0 1 2 3 4 N-1 N N+1 ･･････ x=0 x=1 Self-similarity
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