Shear response of granular packings compressed above jamming onset

We investigate the mechanical response of jammed packings repulsive, frictionless spherical particles undergoing isotropic compression. Prior simulations soft-particle model, where repulsive interactions scale as a power-law in interparticle overlap with exponent $\alpha$, have found that ensemble-averaged shear modulus $\langle G \rangle$ increases pressure $P$ $\sim P^{(\alpha-3/2)/(\alpha-1)}$ at large pressures. However, deep theoretical understanding this scaling behavior is lacking. show frictionless, has two key contributions: 1) continuous variations function along geometrical families, for which contact network does not change, and 2) discontinuous jumps during compression arise from changes network. first family can be collapsed onto master curve: $G^{(1)}/G_0 = (P/P_0)^{(\alpha-2)/(\alpha-1)} - P/P_0$, $P_0 \sim N^{-2(\alpha-1)}$ characteristic separates regions $G_0 N^{-2(\alpha-3/2)}$. Deviations form occur when there significant non-affine particle motion near further (P)\rangle$ simply sum power-laws, but \rangle (P/P_c)^a$, $a \approx (\alpha -2)/(\alpha-1)$ $P \rightarrow 0$ limit (P/P_c)^b$, $b \gtrsim -3/2)/(\alpha-1)$ above $P_c$. In addition, magnitudes both contributions to G\rangle$ families remain comparable large-system >P_c$.