Torque Hyperuniformity in Frictional Granular Matter - Theory and Experiments

Abstract

A question of some fundamental importance is whether a given assembly of frictional granules (say sand or powder) will exhibit stress autocorrelations with long-range anisotropic decay as determined by the elastic Green's function. In Hamiltonian systems with central forces, mechanical balance and material isotropy demand the stress auto-correlation matrix to be fully determined by the pressure auto-correlation only. If the local pressure fluctuations are normal, it follows that stress autocorrelations decay at long distance like the elastic Green's function. With friction, Hamiltonian symmetry is lost, and one may expect more constraints. Indeed, it was shown recently that for frictional amorphous solids mechanical balance and material isotropy demand the stress auto-correlation matrix to be fully determined by two spatially isotropic functions: the pressure and torque auto-correlations. Elastic-like decay of the stress autocorrelations follows from normal fluctuations of the pressure and from the torque fluctuations being hyperuniform. The theoretical discovery of these conditions required experimental confirmation, to test whether these conditions are generically obeyed in actual frictional amorphous solids. Recently the confirmation was announced for 2-dimensional amorphous assemblies of frictional disks under isotropic load, in which torque is caused by tangential forces only. In this paper we review that case and report confirmation of the theoretical predictions in 2-dimensional systems of disks under shear and in isotropically loaded frictional ellipses, where contributions to torque come also from normal forces. The paper ends with physical explanations of the hyperuniformity of the torque fluctuations and predictions for how the results are expected to extend to d-dimensions.

Figures (8)

• Figure 1: Schematic of the experimental setup. Inside we show a typical image in which half the cell exhibits disks and the other half shows force chains.
• Figure 2: Definitions of $\mathbf{r}_i$, $\mathbf{r}_{ij}$ and $\mathbf{f}_{ij}$.
• Figure 3: (a) The $R$ dependence of the variance of of torque fluctuations. The linearity in R is a direct evidence of hyperuniformity. Below we explain that the result is exact, i.e. the growth of variance of the torque is like $R^\alpha$ with $\alpha=1$. (b) The $R$ dependence of the variance of of pressure fluctuations. (c) $S(k)$ of torque. (d) $S(k)$ of pressure.
• Figure 4: $\sigma^2(R)$ and $S(k)$ for systems at similar pressure $p\approx 25\ \mathrm{N\,mm^{-1}}$ with different packing fractions $\phi$ and strains $\epsilon$.
• Figure 5: The evolution of the $\sigma^2(R)$ and $S(k)$ curves with strain $\epsilon$ in systems of packing fraction $\phi = 0.859$.