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

The purpose of this note is to establish two continuum theories for the bending and torsion of inextensible rods as $\mathrm{\Gamma }$-limits of 3D atomistic models. In our derivation we study simultaneous limits of vanishing rod thickness $h$ and interatomic distance $\epsilon$. First, we set up a novel theory for ultrathin rods composed of finitely many atomic fibres ($\epsilon \sim h$), 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 $\epsilon \ll h$ and recover a nonlinear rod model $-$ the modern version of Kirchhoff’s rod theory.