A Generalized Quasi-Nonlocal Atomistic-to-Continuum Coupling Method with Finite Range Interaction

TL;DR

This paper introduces a generalized quasi-nonlocal atomistic-to-continuum coupling method that extends to finite range interactions, providing a stable and accurate approach for modeling crystalline solids with lattice defects.

Contribution

It develops a new formulation of the quasi-nonlocal method for arbitrary finite range interactions and analyzes its stability and accuracy for strains up to lattice instabilities.

Findings

The generalized method extends quasi-nonlocal coupling to finite range interactions.

Stability and accuracy are established for strains up to lattice instabilities.

The method is applicable in one-dimensional models of crystalline solids.

Abstract

The accurate and efficient computation of the deformation of crystalline solids requires the coupling of atomistic models near lattice defects such as cracks and dislocations with coarse-grained models away from the defects. Quasicontinuum methods utilize a strain energy density derived from the Cauchy-Born rule for the coarse-grained model. Several quasicontinuum methods have been proposed to couple the atomistic model with the Cauchy-Born strain energy density. The quasi-nonlocal coupling method is easy to implement and achieves a reasonably accurate coupling for short range interactions. In this paper, we give a new formulation of the quasi-nonlocal method in one space dimension that allows its extension to arbitrary finite range interactions. We also give an analysis of the stability and accuracy of a linearization of our generalized quasi-nonlocal method that holds for strains up to lattice instabilities.