  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 $\langle G \rangle$ increases with
pressure $P$ as $\sim P^{(\alpha-3/2)/(\alpha-1)}$ at large pressures. However,
a deep theoretical understanding of this scaling behavior is lacking. We show
that the shear modulus of jammed packings of frictionless, spherical particles
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. We show that the shear modulus of the first
geometrical family for jammed packings can be collapsed onto a master curve:
$G^{(1)}/G_0 = (P/P_0)^{(\alpha-2)/(\alpha-1)} - P/P_0$, where $P_0 \sim
N^{-2(\alpha-1)}$ is a characteristic pressure that separates the two power-law
scaling regions and $G_0 \sim N^{-2(\alpha-3/2)}$. Deviations from this form
can occur when there is significant non-affine particle motion near changes in
the contact network. We further show that $\langle G (P)\rangle$ is not simply
a sum of two power-laws, but $\langle G \rangle \sim (P/P_c)^a$, where $a
\approx (\alpha -2)/(\alpha-1)$ in the $P \rightarrow 0$ limit and $\langle G
\rangle \sim (P/P_c)^b$, where $b \gtrsim (\alpha -3/2)/(\alpha-1)$ above a
characteristic pressure $P_c$. In addition, the magnitudes of both
contributions to $\langle G\rangle$ from geometrical families and changes in
the contact network remain comparable in the large-system limit for $P >P_c$.
