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 . 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.
Bibliographical noteFunding Information:
This work was supported in part by DMS-0757355, DMS-0811039, the Institute for Mathematics and Its Applications, and the University of Minnesota Supercomputing Institute. This work was also supported by the Department of Energy under Award Number DE-SC0002085.
- Atomistic to continuum
- Embedded atom model
- Error analysis