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

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 solicitation is smaller than a characteristic length of approximately 30 molecular sizes. Due to this surprisingly large effect ensembles containing up to $N=40\mathrm{}000$ 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 nonaffine part of the displacement fields under imposed elongation and shear. Similar correlations (vortices) are indeed observed on distances up to $\xi \approx 30$ particle sizes.