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

The elastic field around a crack opening is known to be described by continuum linearized elasticity in leading order. In this work, we rigorously derive the next term in the atomistic asymptotic expansion in the case of a mode III crack in antiplane 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 linearized 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 prove 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.