Lattice stability for atomistic chains modeled by local approximations of the embedded atom method

The nucleation and motion of lattice defects such as dislocations and cracks can occur when the configuration loses stability at a critical strain. Thus, the accurate approximation of critical strains for lattice instability is a key criterion for predictive computational modeling of material deformation. Coarse-grained continuum approximations of atomistic models are needed to compute the long-range elastic interaction of defects with surfaces. In this paper, we present a comparison of the lattice stability for atomistic chains modeled by the embedded atom method (EAM) with their approximation by local Cauchy-Born models. The volume-based Cauchy-Born strain-energy density is given by the energy-density for a homogeneously strained lattice and is the typical strain energy density used by continuum models to coarse-grain the atomistic energy of a lattice [13]. The reconstruction-based Cauchy-Born model uses linear (and bilinear) extrapolation of local atoms (usually nearest-neighbor atoms in the reference lattice) to approximate the positions of nonlocal atoms and thus approximates a nonlocal atomistic site energy by a local atomistic site energy [18,5]. The reconstruction-based Cauchy-Born site energy has been proposed to accurately transition between an atomistic model used in the neighborhood of a defect and the coarse-grained volume-based local model. We find that both the volume-based local model and the reconstruction-based local model can give O (1) errors for the critical strain when the embedding energy density is nonlinear. In the physical case of a strictly convex embedding energy density, the critical strain predicted by the volume-based model is always equal to or larger than that predicted by the atomistic model (Theorem 5.1), but the critical strain for reconstruction-based models can be either larger, equal, or smaller than that predicted by the atomistic model (Theorem 5.2). If we further restrict our model to nearest-neighbor interactions, then the critical strain for the atomistic and reconstruction-based Cauchy-Born models are equal (Corollarys 4.1 and 4.2), but the critical strain for the volume-based Cauchy-Born model is O (1) larger (Theorem 4.3). We thus expect that reconstruction-based models are more accurate than volume-based models near lattice instabilities if nearest-neighbor interactions dominate, but we note that it is not known how to coarse-grain reconstruction-based models in three space dimensions.