Characterizing granular networks using topological metrics

We carry out a direct comparison of experimental and numerical realizations of the exact same granular system as it undergoes shear jamming. We adjust the numerical methods used to optimally represent the experimental settings and outcomes up to microscopic contact force dynamics. Measures presented here range from microscopic through mesoscopic to systemwide characteristics of the system. Topological properties of the mesoscopic force networks provide a key link between microscales and macroscales. We report two main findings: (1) The number of particles in the packing that have at least two contacts is a good predictor for the mechanical state of the system, regardless of strain history and packing density. All measures explored in both experiments and numerics, including stress-tensor-derived measures and contact numbers depend in a universal manner on the fraction of nonrattler particles, $f_{\mathrm{NR}}$. (2) The force network topology also tends to show this universality, yet the shape of the master curve depends much more on the details of the numerical simulations. In particular we show that adding force noise to the numerical data set can significantly alter the topological features in the data. We conclude that both $f_{\mathrm{NR}}$ and topological metrics are useful measures to consider when quantifying the state of a granular system.

Figure 1

Examples of an experimental (a) and a numerical (b) force network. Experimental and numerical examples are generated at the same strain amplitude $\gamma =14.9\%$ and $\varphi =0.8036$. The color scale represents the total force at each contact, normalized by the concurrent mean force. As described in the text, the simulation results in this and all the following figures include additional noise of amplitude 0.01 N, except if specified differently.

Figure 2

The evolution of the fraction of particles with contact numbers $Z_{2,3,4,5}$ as a function of the fraction of nonrattlers $f_{\mathrm{NR}}$, for both experiments ($\circ$) and simulations ($△$) for a range of $\varphi$'s (color scale).

Figure 3

(a) Pressure evolution versus $1-f_{\mathrm{NR}}$ for both experiments ($\circ$) and simulations ($▵$). (b) Normalized anisotropy $\tau /P$ evolution as a function of $f_{\mathrm{NR}}$. (c) Histogram of contact force magnitudes $\left|f\right|$ for all contacts in the five runs at $\varphi =0.7863$ for both numerics and experiments. The dot-dashed line indicates the force noise added after the numerical runs were completed. (d) The probability distribution function of the norm of the contact forces $P\left(\right|\mathbf{f}\left|\right)$ as a function of $f_{\mathrm{NR}}$ for both experiments ($\circ$) and simulations ($▵$) for the force bin centered around $1\langle \left|\mathbf{f}\right|\rangle$.

Figure 4

Betti numbers, ${\beta }_0$ (left) and ${\beta }_1$ (right), as a function of $f_{\mathrm{NR}}$ for both experiments (a,b: $\circ$) and simulations (c–f, $△$). Panels (c) and (d) show original numerical data. Panels (e) and (f) show the numerical data with 0.01 N noise added.

Figure 5

Integrated image difference $\mathrm{\Sigma }$ for the averaged experimental runs at different densities. There is a constant background difference due to pixel level noise and other artifacts. The dashed line at $f_{\mathrm{NR}}=0.95$ represents the cutoff we imposed.

Figure 6

Calibration curve from experiments (o) and curves using the nonlinear force model in Eqs. (B1) with different $\delta$.

Figure 7

Probability distribution function of contact forces for a force magnitude bin centered at $0.5\langle \left|\mathbf{f}\right|\rangle$ (a) and $1.5\langle \left|\mathbf{f}\right|\rangle$ (b).

Figure 8

Pressure, ${log}_{10}\left(P\right)$ and anisotropy, $\tau /P$ as a function of ${log}_{10}(1-f_{\mathrm{NR}})$ and $f_{\mathrm{NR}}$, respectively.