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 < G > = (1 - F-)< G(+)> + F-< G(-)>, where F- is the fraction of jammed packings with negative shear moduli and < G(+)> and < G(-)> are the average values from packings with positive and negative moduli, respectively. We show that < G(+)> and <|G(-)|> obey different power-law scaling relations above and below pN(2) similar to 1. For pN(2) > 1, both < G(+)> N and <|G(-)|> N similar to ( pN(2))(beta), where beta similar to 0.5 for repulsive linear spring interactions. Despite this, < G > N similar to ( pN(2))(beta)' with beta greater than or similar to 0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN(2) and different values of p and N. We calculate analytically that P(G) is a distribution in the pN(2) << 1 limit. As pN(2) increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN(2) >> 1 limit. We also partition jammed disk packings into subsystems using Delaunay 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((r) over right arrow) displays weak correlations for pn(sub)(2) < 10(-2), where n(sub) is the number of particles within each subsystem. However, C(<(r)over right arrow>) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn(sub)(2) greater than or similar to 10(-2).