Lattice stability for atomistic chains modeled by local approximations of the embedded atom method
X. H. Li and M. Luskin
Computational Materials Science, 66, 96-103 (2013).
ABSTRACT
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.