  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.
