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

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.

Figures (2)

• Figure 1: Top row: the decay of $|D\bar{u}_i(m)|$ for $i=0,1,2$ (left to right). Middle row: the decay of forces $|{\rm Div}\nabla V(Du_{\rm pred}^{(i)})|$ for $i=0,1,2$ (left to right). Bottom row: the decay of the linear residual $|{\rm Div} \nabla^2V(0)D \bar{u}_i(m)|$ for $i=0,1,2$ (left to right). We can clearly see that while prescribing $u_{\rm pred}^{(1)} = \bar{u}_0 + \bar{u}_1$ as a boundary condition improves the decay of forces (middle row) and the linear residual (bottom row), it fails to improve the decay of $|D \bar{u}_1(m)|$ and additionally $\hat{u}_2$ is needed.
• Figure 2: The default interpolation (left) and the mirror interpolation (right). Three example bonds $b(m,\rho)$ are highlighted together with regions $R_{m,\rho}$ associated with it.

Theorems & Definitions (48)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6: adapted from 2018-antiplanecrack
• Definition 2.7: Discrete geometry corrector $\hat{G}_1$
• Proposition 2.8
• Theorem 2.9
• Theorem 3.1