A bending-torsion theory for thin and ultrathin rods as a $\Gamma$-limit of atomistic models

TL;DR

This paper derives two continuum models for the bending and torsion of thin rods as limits of 3D atomistic models, capturing effects at nanoscales and connecting to classical rod theories.

Contribution

It introduces a novel ultrathin rod theory with surface energy effects and discrete terms, and links atomistic models to classical Kirchhoff's rod theory through rigorous limits.

Findings

Ultrathin rod model includes surface energy and discrete effects.

Connection established between atomistic models and classical rod theories.

Derived models applicable to nanowire mechanical analysis.

Abstract

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as $\Gamma$-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness $h$ and interatomic distance $\varepsilon$. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres ($\varepsilon\sim h$), which incorporates surface energy and new discrete terms in the limiting functional. This can be thought of as a contribution to the mechanical modelling of nanowires. Second, we treat the case where $\varepsilon\ll h$ and recover a nonlinear rod model $-$ the modern version of Kirchhoff's rod theory.