Atomic to continuum passage for nanotubes. Part I: a discrete Saint-Venant principle

Abstract

We consider general nanotubes of atoms in $\R^3$ where each atom interacts with all others through a two-body potential. When there are no exterior forces, a particular family of nanotubes is the set of perfect nanotubes at the equilibrium. When exterior forces are applied on the nanotube, we compare the nanotube to nanotubes of the previous family. This quantitative comparison is formulated in our main result as a Saint-Venant principle. This estimate can be derived for a large class of potentials (including Lennard-Jones potential), when the perfect nanotubes at the equilibrium are stable. The approach is designed to be applicable to general nanotubes that can be for instance carbon nanotubes or DNA. In a second paper (part II), we derive from our Saint-Venant principle, a macroscopic mechanical model for general nanotubes