The mechanical response of packings of purely repulsive, spherical particles to athermal, quasistatic simple shear near jamming onset is highly nonlinear. Previous studies have shown that, at small pressure $p$, the ensemble-averaged static shear modulus $langle G-G_0 rangle$ scales with $p^alpha$, where $alpha approx 1$, but above a characteristic pressure $p^{**}$, $langle G-G_0 rangle sim p^beta$, where $beta approx 0.5$. However, we find that the shear modulus $G^i$ for an individual packing typically decreases linearly with $p$ along a geometrical family where the contact network does not change. We resolve this discrepancy by showing that, while the shear modulus does decrease linearly within geometrical families, $langle G rangle$ also depends on a contribution from discontinuous jumps in $langle G rangle$ that occur at the transitions between geometrical families. For $p > p^{**}$, geometrical-family and rearrangement contributions to $langle G rangle$ are of opposite signs and remain comparable for all system sizes. $langle G rangle$ can be described by a scaling function that smoothly transitions between the two power-law exponents $alpha$ and $beta$. We also demonstrate the phenomenon of {it compression unjamming}, where a jammed packing can unjam via isotropic compression.
Download