Local and global measures of the shear modulus of jammed packings*

When the shape of the bounding box is controlled to generate jammed particle packings using isotropic compression, they can be stabilized with positive, as well as negative shear moduli.  We carry out computational studies of jammed disk packings using isotropic compression, while fixing the shape of the bounding box under periodic boundary conditions.  We first measure the ensemble-averaged shear modulus, <G_tot> = <G₊ − |G₋|> as a function of the pressure p, where G₊ and |G_–| are the contributions from configurations with positive and negative shear moduli, respectively.  We show that <G₊> and <|G₋|> obey different scaling relations with pressure above and below pN²∼1, where N is the system size. For pN² < 1, <|G₋|> ∼ p and <G₊> − G₀ ∼ p^0.7. In contrast, for pN² > 1, both <G₊> and <|G₋|> ∼ p^1/2.   We calculate the probability distribution of shear moduli P(G_tot), and show that it varies from a gamma distribution with shape parameter k = 1/2 for pN² « 1 to a left-skewed Gaussian distribution for pN² » 1. We also investigated the local shear moduli of the jammed packings, where the local strain is determined by the strain tensor of Delaunay triangles constructed from the disk centers. We show that the spatial correlations of the local shear moduli decrease with decreasing pressure. In addition, the local shear moduli can be negative even when G_tot > 0. We further show that it is unnecessary to remove jammed configurations with negative shear moduli from the ensemble to obtain a correspondence between an ensemble average and a spatial average in large systems.