Shear response of jammed disk and sphere packings

The response of purely repulsive disk and sphere packings to athermal, quasistatic simple shear near jamming onset is highly nonlinear. Previous studies have shown that the ensemble-averaged static shear modulus G is nearly constant at small pressure p, and at a characteristic pressure p*, G begins to increase as a power-law: G ~ p^α, where α=0.5. Also, p* decreases with increasing system size N, such that p* ~ N^-β, where β=1. Although scaling arguments have rationalized the scaling behavior of p* and G, there is currently no quantitative theoretical framework that can predict the values of α and β. Here, we carry out numerical simulations of 2D bidisperse disk packings near jamming onset undergoing athermal, quasistatic simple shear at fixed pressure to explain these exponents. We show that α and β can be understood by examining the "geometrical families" of jammed packings, which are intervals of shear or pressure where the packings maintain the same network of interparticle contacts without rearrangements. We present a statistical model based on random switching of the packings from one geometrical family to another to predict the values of the exponents α and β.