On molecular statics and surface-enhanced continuum modeling of nano-structures

Abstract Possible links between discrete and continuum formulations have been discussed in the scientific community since several decades. Different atomistic expressions for the continuum fields (most importantly stresses) were proposed. As an example, it is possible to link discrete models to a continuum formulation based on spatial averaging in the Eulerian configuration followed by probability (statistical) averaging. As an alternative to Eulerian averaging, Lagrangian averaging has been recently proposed. Both approaches allow calculation of the local continuum fields from atomistic simulations. From the continuum formulation perspective, the behavior of solids at the nanoscale can be captured based on a surface-enhanced continuum (SEC) theory whereby the surface is equipped with its own constitutive structure. A distinct advantage of continuum models over their atomistic counterparts is the increased computational efficiency. In this contribution we compare atomistic fields obtained from molecular statics (MS) simulations to their counterpart, obtained from numerical approximations to the SEC theory. Bulk elastic parameters for the continuum constitutive model are obtained directly from the atomistic model. A representative numerical simulation of face-centered-cubic (FCC) copper is used to compare the two approaches. The ability of the continuum formulation enhanced with a surface energy to model size effects, as observed in the atomistic simulations, is shown. The local fields evaluated using both the continuum and discrete approach are in a good agreement. The dependence of the results obtained from the atomistic-to-continuum procedure on both the averaging radius and the proximity of the free surface is studied. Eulerian and Lagrangian averaging approaches are shown to give comparable results for the here considered FCC crystal.

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