Incompleteness of Sinclair-type continuum flexible boundary conditions for atomistic fracture simulations

TL;DR

This paper develops a higher-order asymptotic expansion for atomistic fracture simulations, revealing the incompleteness of Sinclair-type boundary conditions and proposing a more accurate boundary condition based on the expansion.

Contribution

It introduces a second-order asymptotic expansion for the elastic field around a crack and demonstrates the limitations of Sinclair's boundary conditions in atomistic fracture modeling.

Findings

Sinclair boundary conditions are incomplete for atomistic fracture simulations.

Higher-order expansion improves boundary condition accuracy.

New discrete geometry predictor accounts for crack tip geometry.

Abstract

The elastic field around a crack opening is known to be described by continuum linearised elasticity in leading order. In this work, we explicitly develop the next term in the atomistic asymptotic expansion in the case of a Mode III crack in anti-plane geometry. The aim of such an expansion is twofold. First, we show that the well-known flexible boundary condition ansatz due to Sinclair is incomplete, meaning that, in principle, employing it in atomistic fracture simulations is no better than using boundary conditions from continuum linearised elasticity. And secondly, the higher order far-field expansion can be employed as a boundary condition for high-accuracy atomistic simulations. To obtain our results, we develop an asymptotic expansion of the associated lattice Green's function. In an interesting departure from the recently developed theory for spatially homogeneous cases, this includes a novel notion of a discrete geometry predictor, which accounts for the peculiar discrete geometry near the crack tip.