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

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. For this reason, discrete terms and energy contributions from the ultrathin rod’s lateral surface appear in the limiting functional. 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—we illustrate this by the example of a full crack and also show that the energy of a mere fold is strictly lower for a class of models. Our approach applies e.g. to atomistic systems with Lennard–Jones-type potentials and is motivated by the research of ceramic nanowires.