Continuum limit of amorphous elastic bodies: A finite-size study of low frequency harmonic vibrations

TL;DR

This study investigates the finite-size effects on the elastic continuum limit in amorphous solids, revealing a characteristic length scale of about 30 particles where classical elasticity fails, requiring large systems for accurate modeling.

Contribution

It demonstrates that classical continuum elasticity breaks down at wavelengths smaller than 30 particles, and confirms the existence of an effective length scale through vibrational mode analysis.

Findings

Classical elasticity fails below a wavelength of ~30 particles.

Large systems with up to 40,000 particles are needed for accurate continuum predictions.

An effective length scale  is identified and related to non-affine displacements.

Abstract

The approach of the elastic continuum limit in small amorphous bodies formed by weakly polydisperse Lennard-Jones beads is investigated in a systematic finite-size study. We show that classical continuum elasticity breaks down when the wavelength of the sollicitation is smaller than a characteristic length of approximately 30 molecular sizes. Due to this surprisingly large effect ensembles containing up to N=40,000 particles have been required in two dimensions to yield a convincing match with the classical continuum predictions for the eigenfrequency spectrum of disk-shaped aggregates and periodic bulk systems. The existence of an effective length scale \xi is confirmed by the analysis of the (non-gaussian) noisy part of the low frequency vibrational eigenmodes. Moreover, we relate it to the {\em non-affine} part of the displacement fields under imposed elongation and shear. Similar correlations (vortices) are indeed observed on distances up to \xi~30 particle sizes.