A continuum model for brittle nanowires derived from an atomistic description by $\Gamma$-convergence

TL;DR

This paper develops a continuum model for brittle nanowires based on atomistic particle systems, capturing bending, torsion, and fracture behaviors, with a focus on applications to ceramic nanowires.

Contribution

It introduces a novel derivation of a continuum model from atomistic interactions using $ ext{Γ}$-convergence, encompassing various fracture modes in nanowires.

Findings

The fracture energy is characterized by an implicit cell formula.

The model applies to Lennard-Jones-type potentials.

Special cases allow simplification of the fracture energy formula.

Abstract

Starting from a particle system with short-range interactions, we derive a continuum model for the bending, torsion, and brittle fracture of inextensible rods moving in three-dimensional space. As the number of particles tends to infinity, it is assumed that the rod's thickness is of the same order as the interatomic distance. Fracture energy in the $\Gamma$-limit is expressed by an implicit cell formula, which covers different modes of fracture, including (complete) cracks, folds and torsional cracks. In special cases, the cell formula can be significantly simplified. Our approach applies e.g. to atomistic systems with Lennard-Jones-type potentials and is motivated by the research of ceramic nanowires.