Critical Scaling in Linear Response of Frictionless Granular Packings near Jamming

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Figure 1

Force response networks for a point loading with pressure as indicated. Blue (red) lines indicate positive (negative) changes in contact force, the thickness indicating the amount. The particles themselves are not drawn.

Figure 2

(a) Scaling of bulk or shear modulus with pressure, obtained from point response (squares), and global compression or shear (diamonds). The fitted exponents are $0.38\pm 0.03$ for the bulk and $0.70\pm 0.08$ for the shear modulus. (b) Distribution function of the scaled transverse response for shear (see text). (c),(d) Distribution of the relative displacement angle $\alpha$ for (c) shear and (d) compression of packings for a range of pressures. Insets: scaling of the width of the peak $\sim p^{1/3}$.

Figure 3

(a) Shear stress ${\sigma }_{xy}$ for point loading in the center and $p={10}^{-3}$. The response is coarse grained over a grain diameter [18] and averaged over 216 realizations (solid) and fitted to linear elasticity (dashed). (b)–(d) Spatial decay of the force fluctuations $h$. Squares, diamonds, crosses, and plus signs represent increasing pressures, ranging from $p=4.6\times {10}^{-5}$ to $p=3.2\times {10}^{-2}$, respectively. (b) $h$ as function of $r$. (c) Scaling collapse when $h$ is plotted as function of $r\Delta z$. (d) Scaling collapse of relative fluctuations.