Evolution of force networks during stick-slip motion of an intruder in a granular material: Topological measures extracted from experimental data

In quasi-two-dimensional experiments with photoelastic particles confined to an annular region, an intruder constrained to move in a circular path halfway between the annular walls experiences stick-slip dynamics. We discuss the response of the granular medium to the driven intruder, focusing on the evolution of the force network during sticking periods. Because the available experimental data do not include precise information about individual contact forces, we use an approach developed in our previous work [Basak et al., J. Eng. Mech. 147, 04021100 (2021)] based on networks constructed from measurements of the integrated strain magnitude on each particle. These networks are analyzed using topological measures based on persistence diagrams, revealing that force networks evolve smoothly but in a nontrivial manner throughout each sticking period, even though the intruder and granular particles are stationary. Characteristic features of persistence diagrams show identifiable slip precursors. In particular, the number of generators describing the structure and complexity of force networks increases consistently before slips. Key features of the dynamics are similar for granular materials composed of disks or pentagons, but some details are consistently different. In particular, we find significantly larger fluctuations of the measures computed based on persistence diagrams and, therefore, of the underlying networks, for systems of pentagonal particles.

Figure 1

Schematic of the experimental setup. An intruder is attached to a torsion spring (shown in blue), connected to a stepper motor rotating at a fixed rate $\omega$, that provides the driving force. Granular particles are shown in green.

Figure 2

(a) Photoelastic image of the annulus packed with disks. (b) Enlargement of the outlined region of (a) showing the details of the photoelastic signal. (c) Processed image of the configuration from (a) showing force magnitude per particle. (d) Resulting force network. Color bars in (c) and (d) show the forces normalized by the current value of the intruder force. (Note that some particles, in particular the ones close to the intruder, exceed the range shown on the color bar; we choose a maximum of 0.5 for visualization purposes.)

Figure 3

$G_s^2$ data from a single disk experiment while pushing a disk against the intruder, which is held in place. The horizontal axis shows the total force applied on a particle, which is twice the applied force in these experiments. Error bars show the standard deviation, and the calibration curve (in red) is the best cubic fit to the data.

Figure 4

Example of (a) the intruder velocity and (b) the force on the intruder showing a small part of the analyzed data set. The full set involves hundreds of slip events. The dashed red line denotes the threshold value used to define the beginning and the end of slip events, as described in the text. Red dots show the beginning and end of the particular stick period, which is more detailed in the Figs. 8 and 9.

Figure 5

$G^2$ signal for the data shown in Fig. 4, without (a) and with (b) normalization by the intruder force shown in Fig. 4.

Figure 6

Persistence diagrams (PDs) corresponding to one experimental image at $\mathrm{\Delta }\theta =0$, from the stick period denoted in Fig. 4: (a) components, B0, (b) loops, B1. Note that the forces used are normalized by the intruder force; blue points are inside of the “band” that is not considered due to experimental noise (see the text for a full description of how this band is defined). Animations showing all PDs for the considered stick period are available [42].

Figure 7

Generators for the B0 persistence diagram (red dots) and the corresponding force network at $\mathrm{\Delta }\theta =0$ for the period shown in Fig. 4. Color bar shows the (nondimensional) force magnitude. Animations are available [42].

Figure 8

Time series of the considered measures during the stick period marked by red dots in Fig. 4: intruder's speed (a) and force (b) in the azimuthal direction, $N_{G}0$ (c), TP0 (d), $N_{G}1$ (e), TP1 (f), $W_2\beta 0$ (g), $W_2\beta 1$ (h). See the text for a description of plotted quantities. The value $\mathrm{\Delta }\theta =0$ indicates the time when a slip event starts.

Figure 9

Time series of the considered measures for the part of the data set shown in Fig. 4. (a) $N_{G}0$, (b)TP0, ($\mathrm{c})N_{G}1$, (d) TP1, (e) $W_2\beta 0$, and (f) $W_2\beta 1$.

Figure 10

Cumulative number of stick periods, $N_E$, with a duration larger than the angular distance, $\mathrm{\Delta }\theta$, to the upcoming slip event (occurring at $\mathrm{\Delta }\theta =0$).

Figure 11

Distribution of the force on the intruder at which slips occur for three available volume fractions.

Figure 12

Persistence-derived measures averaged over all stick periods (a), (b) TP and the number of generators for components; (c), (d) TP and generators for loops; (e), (f) ratio of $B_k$ to $W_2$ for components and loops. Red lines show the averages, and blue lines their standard error.

Figure 13

Number of broken contacts and average broken force (averaged over all stick periods).

Figure 14

Ratio of the standard deviation ($\sigma$) to the average ($\mu$) of (a) $N_{G}0$, (b) $N_{G}1$ for disks. $N$ count stick events; each point in the plot corresponds to a separate stick event.

Figure 15

Example of (a) intruder's velocity and (b) force on the intruder showing a small part of the analyzed data set for pentagons, $\varphi =0.65$. Red dots show the beginning and the end of a particular stick period that we consider in more detail in Figs. 16 and 17.

Figure 16

Persistence diagrams (PDs) corresponding to the experimental image at $\mathrm{\Delta }\theta =0$ from the stick period denoted in Fig. 15 (animations of all PDs corresponding to this stick period are available [42]. (a) Components, B0; (b) loops, B1.

Figure 17

Generators for a PD and corresponding network at $\mathrm{\Delta }\theta =0$ for the period shown in Fig. 15. Color bar shows the (nondimensional) force magnitude. Animations are available [42].

Figure 18

Pentagons: Cumulative number of stick periods, $N_E$, with an angular duration larger than $\mathrm{\Delta }\theta$, where $\mathrm{\Delta }\theta =0$ corresponds to the initiation of the upcoming slip event. Note that the larger values of $N_E$ for pentagons compared to disks (see Fig. 10) is due to the set of data analyzed for pentagons being larger: the number of sticks per unit time is comparable for the considered systems.

Figure 19

Distribution of the force on the intruder at which slip occurs for different volume fractions.

Figure 20

Persistence-derived measures averaged over all stick periods: (a), (b) TP and number of generators for components; (c), (d) TP and number of generators for loops; (e), (f) ratio of $B_k$ to $W_2$ for components and loops.

Figure 21

Ratio of the standard $\mathrm{deviation},\sigma$, to the average, $\mu$, for pentagons. (a) $N_{G}0$. ($\mathrm{b})N_{G}1$. (See Fig. 14 for disks and the caption of Fig. 18 regarding the range of $N$.)