On the derivation of linear elasticity from atomistic models

Abstract

We derive linear elastic energy functionals from atomistic models as a -limit when the number of atoms tends to infinity, respec tively, when the interatomic distances tend to zero. Our approach generalizes a recent result of Braides, Solci and Vitali (2). In particular, we study mass spring models with full nearest and next-to-nearest pair interactions. We also consider boundary value problems where a part of the boundary is free. 1. Introduction. The passage from discrete atomic models to continuum theories is an active area of current research in continuum mechanics. For elastic systems one usually refers to the Cauchy-Born rule to obtain macroscopic energy densities from atomistic interaction functionals. The Cauchy-Born rule states that - roughly speaking - each individual atom follows the macroscopic deformation gradient and in particular does not take into account fine scale oscillations on the microscopic scale. For a two-dimensional mass spring model, the validity of the Cauchy-Born rule for deformations close to a rigid motion has been proved by Friesecke and Theil in (8). Their result has been generalized to arbitrary dimensions by Conti, Dolzmann, Kirchheim and Muller in (3). If the deformation gradients are very close to SO(d), the set of orientation pre- serving rigid motions, then we expect linear elasticity theory to apply. This relation has been made rigorous by Dal Maso, Negri and Percivale who derive the energy functional of linear elasticity as a -limit of nonlinear el asticity for small displace- ments in (5). (See also the author's article (10) for a strong convergence result for the associated minimum problems.) Recently it has been noted that one can derive linear elasticity functionals directly from certain atomistic pair potentials: For a special class of pair interaction models Braides, Solci and Vitali prove -convergence of the discre te energy functionals to the energy functional of an associated continuum linear elasticity energy functional (see (2)). In this set-up one has to deal with two small parameters e and δ measuring the typical interatomic distance and the local distance of the deformations from the set of rigid motions, respectively. The aim of the present article is to extend these results in three directions. Firstly, we will drop the assumption that atoms are allowed to interact only along