Validity and size-dependency of Cauchy–Born hypothesis with Tersoff potential in silicon nano-structures

One of the most popular constitutive rules that correlate the continuum and atomic properties in multi-scale models is the Cauchy–Born (CB) hypothesis. Based on this constitutive law of continuum media, it assumes that all atoms follow the deformation subjected to the boundary of crystal. In this paper, the validity and failure of CB hypothesis are investigated for the silicon nano-structure by comparison of the continuum and atomic properties. In the atomistic level, the stresses and position of atoms are calculated using the molecular dynamics (MD) simulation based on the Tersoff inter-atomic potential. The stresses and strains are compared between the atomistic and continuous media to obtain the validity of CB hypothesis. The 3D validity surfaces are derived for various numbers of atoms to display the size-dependency of cubic specimens. It is shown that the CB is valid inside the surface, the deformations are homogeneous, and the material is elastic, while the CB is invalid outside of the surface, the deformations are inhomogeneous, and the material may remain elastic, or plastic.