Structure of force networks in tapped particulate systems of disks and pentagons. I. Clusters and loops

The force network of a granular assembly, defined by the contact network and the corresponding contact forces, carries valuable information about the state of the packing. Simple analysis of these networks based on the distribution of force strengths is rather insensitive to the changes in preparation protocols or to the types of particles. In this and the companion paper [Kondic et al., Phys. Rev. E 93, 062903 (2016)], we consider two-dimensional simulations of tapped systems built from frictional disks and pentagons, and study the structure of the force networks of granular packings by considering network's topology as force thresholds are varied. We show that the number of clusters and loops observed in the force networks as a function of the force threshold are markedly different for disks and pentagons if the tangential contact forces are considered, whereas they are surprisingly similar for the network defined by the normal forces. In particular, the results indicate that, overall, the force network is more heterogeneous for disks than for pentagons. Such differences in network properties are expected to lead to different macroscale response of the considered systems, despite the fact that averaged measures (such as force probability density function) do not show any obvious differences. Additionally, we show that the states obtained by tapping with different intensities that display similar packing fraction are difficult to distinguish based on simple topological invariants.

Figure 1

Sample snapshots of the disks (a) and (b) and pentagons (c) and (d) prepared by tapping at low tapping intensity, $\mathrm{\Gamma }=3.83\sqrt{dg}$ (see Sec. 2). The contact forces are indicated by segments connecting the touching particles colored according to the strength of the normal (a) and (c) or tangential (b) and (d) components of the force. The color scale indicates the force in units of $mg$, with $m$ the mass of a particle and $g$ the acceleration of gravity.

Figure 2

Steady state packing fraction, $\varphi$, as a function of $\mathrm{\Gamma }$ for disks and pentagons. The error bars indicate the standard error on the mean $\varphi$, which is averaged over 30 taps after reaching the steady state.

Figure 3

Probability density functions (PDF) (disks, low tapping).

Figure 4

${\beta }_0$ (disks, low tapping). Error bars correspond to the standard error estimated as $std/\sqrt{N}$ with $std$ the standard deviation and $N=500$, the number of realizations considered.

Figure 5

PDFs (disks, bottom slice).

Figure 6

Pair correlation function, $g\left(r\right)$ (disks, bottom slice).

Figure 7

Force correlation function, $f\left(r\right)$ (disks, low tapping). The $g\left(r\right)$ is also shown.

Figure 8

$f\left(r\right)/g\left(r\right)$ (disks, bottom slice).

Figure 9

(a) and (b) ${\beta }_0$, and (c) and (d) ${\beta }_1$ (disks, bottom slice). Error bars are as in Fig. 4.

Figure 10

PDFs (bottom slice, low tapping).

Figure 11

${\beta }_0$ (bottom slice, low tapping).

Figure 12

Average number of components (clusters) per particle for disks and pentagons in the tangential force network as a function of cluster size for various force thresholds, $F$. (a) $F=3$, (b) $F=2.0$, (c) $F=1.0,$ and (d) $F=0.5$ (bottom slice, low tapping).

Figure 13

${\beta }_1$ as a function of the force threshold $F$ for disks and pentagons. The trivial loops are included in (a) and (b) and omitted in (c) and (d) (bottom slice, low tapping).