Local and global measures of the shear moduli of jammed disk packings
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 $pn_{\rm sub}^2 \gtrsim 10^{-2}$.