Analysis of an atomistic model for anti-plane fracture

TL;DR

This paper develops and analyzes an atomistic model for anti-plane fractures on a square lattice, establishing existence, uniqueness, stability, and decay properties of solutions under small loads.

Contribution

It introduces a rigorous atomistic model for anti-plane fractures, proving key mathematical properties and decay estimates that advance understanding of lattice defect behavior.

Findings

Existence and uniqueness of solutions for small loads

Stability of the fracture solutions

Sharp decay estimates for the lattice Green's function

Abstract

We develop a model for an anti-plane crack defect posed on a square lattice under an interatomic pair-potential with nearest-neighbour interactions. In particular, we establish existence, local uniqueness and stability of solutions for small loading parameters and further prove qualitatively sharp far-field decay estimates. The latter requires establishing decay estimates for the corresponding lattice Green's function, which are of independent interest.