Local and global measures of the shear moduli of jammed disk packings

TL;DR

This study investigates the distribution and spatial correlations of local and global shear moduli in jammed disk packings, revealing how negative shear moduli contribute to mechanical response and how their statistical properties evolve with pressure and system size.

Contribution

The paper introduces a detailed decomposition of the global shear modulus into positive and negative components, deriving their scaling laws and analytical distribution, and examines local shear moduli and their spatial correlations.

Findings

Global shear modulus distribution transitions from gamma to skew-normal with increasing pressure.

Negative local shear moduli can occur even when the global modulus is positive.

Spatial correlations of local shear moduli develop long-range order with four-fold symmetry at higher subsystem sizes.

Abstract

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as $\langle G\rangle = (1-{\cal F}_-) \langle G_+ \rangle + {\cal F}_- \langle G_-\rangle$, where ${\cal F}_-$ is the fraction of jammed packings with negative shear moduli and $\langle G_+\rangle$ and $\langle G_-\rangle$ are the average values from packings with positive and negative moduli, respectively. We show that $\langle G_+\rangle$ and $\langle|G_-|\rangle$ obey different power-law scaling relations above and below $pN^2 \sim 1$. We then calculate analytically that ${\cal P}(G)$ is a Gamma distribution in the $pN^2 \ll 1$ limit. As $pN^2$ increases, the skewness of ${\cal P}(G)$ decreases and ${\cal P}(G)$ becomes a skew-normal distribution with negative skewness in the $pN^2 \gg 1$ limit. We also partition jammed disk packings into subsystems using Delanunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when $G > 0$. The spatial correlation function of local shear moduli $C({\vec r})$ displays weak correlations for $pn_{\rm sub}^2 < 10^{-2}$, where $n_{\rm sub}$ is the number of particles within each subsystem. However, $C({\vec r})$ begins to develop long-ranged spatial correlations with four-fold angular symmetry for $pn_{\rm sub}^2 \gtrsim 10^{-2}$.