Keywords: nanomechanics, atomistic model, molecular dynamics, finite element analysis, static equilibrium, multi-scale technique.

Recently, much attention has been devoted to the study of material behavior at the atomistic or molecular level. As a result, molecular dynamics models or atomistic models have been developed to simulate the material behavior at the nano- scale [1,2]. Most of the simulations were dynamic analysis using Newton's second law and different potential energies among neighboring atoms or molecules. Static equilibrium positions are time averages of the dynamic equilibrium motions. It is quite a time consuming process to compute the static equilibrium positions of atoms or molecules from dynamic analysis.

In this paper a new algorithm was developed to determine static equilibrium positions directly. The interactive forces among atoms or molecules are highly nonlinear. Therefore, interaction between any two atoms or molecules is considered as a nonlinear spring with internal forces. An iterative solution technique was applied to determine the static equilibriums of all nonlinear springs. The convergence was checked with the difference between the present and previous positions of atoms during iterations. In this study, Morse potential energy [3] was selected to calculate the interactive forces and the equivalent nonlinear spring constants. In addition, the present static nanomechanics model was coupled with the continuum-based finite element analysis model. As a result, the static atomistic model can be embedded into a static finite element model. This coupling provided flexibility of applying constraints to the atomic model.

Several example cases were solved using the present atomistic model for static equilibrium. One of the examples considered the atomic displacements around a nano-scale hole in an array of atoms subjected to a tensile load. Another example investigated the dislocation of atoms under a load. Different orientations of dislocations were studied. The interaction between a hole and dislocation was also investigated. Furthermore, coupled examples between the atomic and finite element models were studied. In addition, the atomic model was implemented into a multi- scale approach for woven fabric composite structures. An example showed prediction of the effective elastic moduli of plain weave and 2/2-twill composites made of carbon natotubes and a matrix material. For this example, the Abell- Tersoff-Brenner potential was used.

One of examples is shown here. This is a case of prismatic dislocation. Figure 2.1(a) shows the equilibrated atoms with dislocation. A line in the atom array denotes the dislocation position. Atoms on the four boundaries were considered to move only tangentially along the boundaries but not perpendicularly. As the top boundary was displaced uniformly in the vertical direction, the movement of atoms to the new equilibrium position is plotted in Figure 2.1(b). The direction and magnitude of movement of each atom is shown in the figure.

1

J. M. Haile, "Molecular Dynamics Simulation: Elementary Method", John Wiley & Sons, New York, 1997.

2

D. C. Rapaport, "The Art of Molecular Dynamics Simulation", Cambridge University Press, Cambridge, United Kingdom, 1995.

3

L. A. Girifalco and V. G. Weizer, "Application of the Morse Potential Function to Cubic Metals", Physical Review, 114 (3), 687-690, 1959. doi:10.1103/PhysRev.114.687

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