PSI - Issue 13

Igor Golovnev et al. / Procedia Structural Integrity 13 (2018) 1632–1637 Author name / Structural Integrity Procedia 00 (2018) 000–000

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2. Description of the physico-mathematical model

The perfect copper crystal shaped as a rectangular parallelepiped was used as a physical system; the number of crystalline cells in it is n x = 50, n y = n z = 5 along respective axes. The chosen crystal orientation [100] is in the X axis direction. To simulate the interatomic interaction, the potential based on the embedded atom method is chosen, Voter (1993). The initial data are assigned as follows. The atom coordinates and momenta correlating with the minimum of the potential structure energy are determined on the base of the perfect crystal with the FCC-structure by the artificial viscosity method Golovneva et al. (2003). These values of coordinates and momenta are used as the initial data for the system heating up to the needed temperature. Equations of motion were integrated using the second order velocity Verlet scheme, Allen et al. (1987). The time step in all numerical simulations is 10 -16 s. It should only be pointed that in the case of the isolated system, the energy error does not exceed 10 -5 % within the time interval of 50 ps. The leftmost immovable clamp is modeled as follows. The atoms of the leftmost atomic plane perpendicular to the X axis are fixed in the 3D biharmonic potential: ( ) ( ) ( ) ( ) 4 4 4 0 0 0 4 i i i i i i U k x x y y z z = − + − + − (1) Here, i r  is the radius-vector of the i-atom, 0 i r  is the radius-vector of the i-atom at the beginning of the process. The factor k is chosen in such a way that even under the intensive loading the atomic energy remains approximately in the minimum of the potential simulating the clamp. In these calculations, o 4 20000 J A k = . The moving clamp is simulated in the papers as follows. All atoms of the rightmost atomic plane perpendicular to the X axis moved as an integrity in accordance with the law ( ) 0 sin V t ω ⋅ . The forces acting on the remaining atoms of the nanostructure from the side of the rightmost plane, are calculated by the model of the used interatomic interaction potential. At the same time, the rightmost plane atoms themselves were excluded from the motion equations. Finally, the positions of these atoms are calculated in accordance with the simple law: ( ) 0 0 0 , , i i i i i i x x x t y y z z = + ∆ = = , and the interatomic force calculation procedure does not require any extra data aside for the information on the coordinates of all atoms. To find the shift ( ) x t ∆ of the utmost plane atoms along the X axis, the velocity expression ( ) 0 sin V t ω ⋅ is integrated over the time. As a result, the dependence of the shift of the nano-sized rod rightmost end on time looks like: ( ) 0 1 cos V x t ω ω   ∆ = ⋅ −     (2) Taking in to account that the oscillation period is found as the product of the steps number N ω by the calculation step τ : T N ω ω τ = ⋅ , the frequency is 2 N ω π ω τ = . Hence, ( ) ( ) 0 0 1 cos 1 cos 2 V N x t x t ω τ ω ω π   ∆ = ⋅ − ≡ ∆ ⋅ −     (3) Evident that the shift amplitude is similar for a whole set of parameters 0 V and N ω for which this product is constant. Thus, the parameters 0 x ∆ and N ω are convenient as characteristic ones. Free boundary conditions are used along the Y and Z axes. 2.1. Clamp simulation