Pressure-dependent shear response of jammed packings of frictionless, spherical particles

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 <G-G₀> scales with p^α, where α ≈ 1, but above a characteristic pressure p**, <G-G₀> scales with p^β, where β ≈ 0.5. However, we find that the shear modulus Gⁱ 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, <G> also depends on a contribution from discontinuous jumps in Gⁱ that occur at the transitions between geometrical families. For p > p**, geometrical-family and rearrangement contributions to <G> are of opposite signs and remain comparable for all system sizes. <G> can be described by a scaling function that smoothly transitions between two power-law exponents α and β. We also demonstrate the phenomenon of compression unjamming, where a jammed packing unjams via isotropic compression.