Jammed frictionless discs: connecting local and global response

TL;DR

This study explores how soft, frictionless disc packings respond to small external forces near the jamming point, revealing a diverging length scale that connects local disorder to global elastic behavior.

Contribution

It identifies a diverging length scale $ ext{l}^*$ that links local packing disorder to the global elastic response in jammed systems, unifying vibrational and mechanical properties.

Findings

The length scale $ ext{l}^*$ diverges as $1/ ext{Δ}z$ near jamming.

Elastic properties are well described by continuum elasticity at large scales.

Non-affine displacements are characterized by the displacement angle distribution.

Abstract

By calculating the linear response of packings of soft frictionless discs to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and non-affine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale $\ell^*$ up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as $1/\Delta z$, where $\Delta z$ is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the non-affine displacements of the particles using the \emph{displacement angle distribution}, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.