Euler elastica as a $\Gamma$-Limit of discrete bending energies of one-dimensional chains of atoms

TL;DR

This paper rigorously derives the continuum limit of a discrete atomistic model of a graphene sheet's bending energy, showing it converges to Euler's elastica as the bond length approaches zero.

Contribution

It provides a rigorous mathematical justification that discrete atomic bending energies converge to Euler's elastica in the continuum limit.

Findings

Discrete bending energies $ ext{Γ}$-converge to Euler's elastica

Validates continuum modeling of graphene's bending behavior

Establishes a link between atomistic and continuum mechanics

Abstract

This work is motivated by discrete-to-continuum modeling of the mechanics of a graphene sheet, which is a single-atom thick macromolecule of carbon atoms covalently bonded to form a hexagonal lattice. The strong covalent bonding makes the sheet essentially inextensible and gives the sheet a resistance to bending. We study a one-dimensional atomistic model that describes the cross-section of a graphene sheet as a collection of rigid links connected by torsional springs. $\Gamma$-convergence is used to rigorously justify an upscaling procedure for the discrete bending energy of the atomistic model. Our result establishes that as the bond length in the atomistic model goes to 0, the bending energies $\Gamma$-converge to Euler's elastica.