On the Microscopic Foundations of Elasticity

TL;DR

This paper develops a microscopic foundation for elasticity theory, deriving exact expressions for displacement, strain, and stress fields, and explores the scale-dependent validity of continuum elasticity in disordered and nanoscale solids.

Contribution

It introduces a microscopic approach to elasticity, providing exact expressions and analyzing the conditions for the applicability of continuum elasticity at small scales.

Findings

Standard continuum elasticity applies only above certain length scales.

Microscopic expressions for displacement, strain, and stress are derived.

Conditions for linear elastic behavior are identified and analyzed.

Abstract

The modeling of the elastic properties of disordered or nanoscale solids requires the foundations of the theory of elasticity to be revisited, as one explores scales at which this theory may no longer hold. The only cases for which microscopically based derivations of elasticity are documented are (nearly) uniformly strained lattices. A microscopic approach to elasticity is proposed. As a first step, microscopically exact expressions for the displacement, strain and stress fields are derived. Conditions under which linear elastic constitutive relations hold are studied theoretically and numerically. It turns out that standard continuum elasticity is not self-evident, and applies only above certain spatial scales, which depend on details of the considered system and boundary conditions. Possible relevance to granular materials is briefly discussed.