13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 2.3. The algorithm of mechanical dispersion of atom groups The value of x+ i,n is taken as the non-dimensional variable given by Eq. (2). σ is defined as l. Concerning the dispersion up to the scale of atom, 1/(βN)13 and 1/(βN)7 was exerted to the first and the second terms of Eq. (1) respectively to decrease and disperse interactive forces between super atoms with increase in the number of atoms. By converting Eq. (1) using Eq. (2), the non-dimensional representation of Eq. (1) was given by Eq. (3). That is, 1/(βN) is an adjusting parameter of the reasonable intensity of potential field corresponding with each scale of atoms. l x x i n i n , , = + , l =σ (2) Where, +x is non-dimensional position of atom, β is a dispersion coefficient of super atoms. l is a representative length and it equals to value of σ. ( ) ( ) ( ) ( ) ( ) ∑ ≠ = + − + ∗ + − + − + − + − + − ∗ − − + ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − − − = N j i j i n d i n II j n i n j n i n i n x x nA x K l x N x x N x I 1 , 1 , 1 7 , 1 , 1 7 13 , 1 , 1 13 , 2 1 6 1 12 4 π β β ε (3) 2.4. Method of analysis On the basis of the basic equation, to obtain equilibrium positions of super atoms, following two methods were adopted. 2.4.1. Direct method Equation (2) represents forces exerted on each super atom. When value of Ii * is not zero, each super atom moves toward the corresponding equilibrium position dominated by Eq. (4). ∗ = i n i n I dt d x M , 2 , 2 (4) Where n is the step number when corresponding atom moves step by step toward the equilibrium position under the Eq. (3). By Eq. (4), velocity of each atom at the time incremental value, Δt is given by Εq. (5). Therefore, moving distance during the time incremental value, Δt is given by Εq. (6). Using Εq. (6), the position of each atom at the time of (t+Δt) is given by Εq. (7). Where M is an imaginary mass and for convenience from the view point of calculation, M was taken as unity. , 1 * , , − = Δ + i n i n i n V I t V (5) x V t i n i n Δ = Δ , , (6) i n i n i n x x x , , , 1 = +Δ + (7) This method [9] can be applied to any cases of moving distance of atoms, however the calculation time of numerical analysis becomes longer as compared with Verlet method. 2.4.2. Verlet method On the basis of Taylor expansion, Verlet method was proposed to obtain numerical solutions for the type of Eq. (4) with high accuracy and short calculation time. This method is mainly applied to molecular dynamics. This method is summarized as follows. On the basis of Taylor expansion, finite difference representation of two order derivative was given by Eq. (8).
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