13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- - Type 2: bonding energy between the Al atom and the vacancy - Type 3: bonding energy between the vacancies (2) On the grain boundary: - Type 4: bonding energy between the Al atoms on the grain boundary - Type 5: bonding energy between the Al atom and the vacancy on the grain boundary - Type 6: bonding energy between the vacancies on the grain boundary (3) On the surface: - Type 7: surface energy of the Al atoms with different lattice orientation - Type 8: surface energy of the vacancies (4) In contact with the impurity (if any): - Type 9: adhesion energy between the Al atom and the impurity - Type 10: bonding energy between the vacancy and the impurity The bonding energy between the Al atoms inside the grain is 1.50 eV [21]. The bonding energy between the Al atom and the vacancy is 0.29 eV [22], and that between the vacancies is assumed to be zero. The surface energies of Al for lattice orientation (100), (110), and (111) are calculated as 0.95, 1.02 and 0.79 J/m2 respectively using the density function theory (DFT) and the VASP software [23]. The surface energy of the vacancy is assumed to be zero. The bonding energy for the atoms on the grain boundary is approximated to be 40% of that inside the grain [24, 25]. The adhesion energy between the Al atom and the impurity (e.g. Si) is 1.09 J/m2 [26], and 0.20 eV is used as the bonding energy between the vacancy and the Si impurity [27]. The Young’s Modulus of Al for lattice orientation (100), (110), and (111) are calculated as 63.05, 71.88 and 76.05 GPa respectively [28]. The probability of atom to fill in its neighboring vacant lattice site is calculated using Equation (3) [16], Probability ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ × Δ +Δ +Δ = − N k T E E E B pair str th exp (3) where th EΔ , str EΔ and pair EΔ are the change in thermal, strain and pairing energy between the target position and the original position of the atom respectively. To begin the void nucleation simulation, 0.01% to 0.1% vacancies are randomly generated inside the sub-model depending on whether it is inside the grain or on the grain boundary [29, 30]. A few Si impurities are generated on the grain boundary as well. As the voids are only visible at micrometer scale [7], we assume the initial size of the vacancy to be 1 µm3, which is also the size of one element in the sub-model. The thermal and strain energy of each element are calculated using Equation (1) and (2), and the pairing energy is calculated based on the position of the element and the number and type of its neighboring elements. In the Monte Carlo subroutine, a vacant site is randomly selected and total change in energy between the selected vacancy and its neighboring elements are computed. The neighboring element exchange position with the vacancy (i.e. fill in the position of the vacant site) if the probability of movement is larger than 0.5 [25]. The continuous position swapping process results in the movement of the vacancies and the nucleation of the voids. 3. Results and discussions The position of the vacancies at the beginning, during and the end of the simulation after 1500 Monte Carlo cycles are shown in Figure 6(a) (b), and (c) respectively.
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