Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D Elasticity

TL;DR

This paper develops two atomistic models for 1D elastic crystals, deriving their continuum limits and identifying energy contributions for smooth and sharp interfaces, including a novel repulsion term.

Contribution

It introduces two atomistic models with Taylor expansions revealing elastic, sharp-interface, smooth-interface energies, and a new interface repulsion term, advancing understanding of atomistic-to-continuum limits.

Findings

Taylor expansion coefficients correspond to elastic and interface energies

Second model predicts a repulsion force between sharp interfaces

Models unify atomistic and continuum descriptions in 1D elasticity

Abstract

In this paper we present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the first model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the first terms of the Taylor expansion (with respect to the parameter representing the interatomic distance) of the atomistic energy, and obtain that the coefficients of that Taylor expansion represent, respectively, an elastic energy, a sharp-interface energy, and a smooth-interface energy. The second atomistic model is a variant of the first one, and its Taylor expansion predicts, in addition, a new term that accounts for the repulsion force between two sharp interfaces.