Shear Response Of Granular Packings Compressed Above Jamming Onset

We investigate the mechanical response of jammed packings of repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations of the soft-particle model, where the repulsive interactions scale as a power law in the interparticle overlap with exponent alpha, have found that the ensemble-averaged shear modulus < G(P)> increases with pressure P as similar to P(alpha-3/2)/(alpha-1) at large pressures. < G > has two key contributions: (1) continuous variations as a function of pressure along geometrical families, for which the interparticle contact network does not change, and (2) discontinuous jumps during compression that arise from changes in the contact network. Using numerical simulations, we show that the form of the shear modulus G(f) for jammed packings within near-isostatic geometrical families is largely determined by the affine response G(f) similar to G(a)(f) where G(a)(f)/G(a0) = (P/P-0)((alpha-2)/(alpha-1)) -P/P-0, P-0 similar to N-2(alpha-1) is the characteristic pressure at which G(a)(f) = 0, G(a0) is a constant that sets the scale of the shear modulus, and N is the number of particles. For near-isostatic geometrical families that persist to large pressures, deviations from this form are caused by significant nonaffine particle motion. We further show that the ensemble-averaged shear modulus < G(P)> is not simply a sum of two power laws, but < G(P)> similar to (P/P-c)(a), where alpha approximate to (alpha - 2)/(alpha - 1) in the P -> 0 limit and < G(P)> similar to (P/P-c)(b), where b >= (alpha - 3/2)/(alpha - 1), above a characteristic pressure that scales as P-c similar to N-2(alpha-1).