Granular response to impact: Topology of the force networks

The impact of an intruder on granular matter leads to the formation of mesoscopic force networks, which were seen particularly clearly in the recent experiments carried out with photoelastic particles [Clark et al., Phys. Rev. Lett. 114, 144502 (2015)]. These force networks are characterized by complex structure and evolve on fast time scales. While it is known that total photoelastic activity in the granular system is correlated with the acceleration of the intruder, it is not known how the structure of the force network evolves during impact, and if there are dominant features in the networks that can be used to describe the intruder's dynamics. Here, we use topological tools, in particular persistent homology, to describe these features. Persistent homology allows quantification of both structure and time evolution of the resulting force networks. We find that there is a clear correlation of the intruder's dynamics and some of the topological measures implemented. This finding allows us to discuss which properties of the force networks are most important when attempting to describe the intruder's dynamics. In particular, we find that the presence of loops in the force network, quantified by persistent homology, is strongly correlated to the deceleration of the intruder. In some cases, particularly for the impact on soft particles, the measures derived from the persistence analysis describe the deceleration of the intruder even better than the total photoelastic activity. We are also able to define an upper bound on the relevant time scale over which the force networks evolve.

Figure 1

Example of image processing for (a), (b) medium (6.35–4.5) and (c), (d) soft (6.35–1.3) particles. Here, (a), (c) are raw images, and (b), (d) show the postprocessed images (using the matlab jet color scheme). See the text for details on image processing. Note in particular that our image-processing approach removes the preexisting force chains that are clearly visible at the bottom part in (c) but not in (d). The naming convention is such that the first number specifies the radius of the intruder in cm, and the second one specifies the impact speed in m/s. The images are shown at the times 0.276 ms (medium) and 16 ms (soft) after impact. We will refer to the medium 6.35–4.5 experiment as the reference one for the remainder of the paper.

Figure 2

Persistence diagrams (PDs) for the reference experiment (medium 6.25–4.5) (same time as shown in Fig. 1). Parts (a) and (b) show the PDs for components and loops, ${\mathrm{PD}}_0$ and ${\mathrm{PD}}_1$, respectively. Parts (c) and (d) show the corresponding lifespans. The $x$-axis range corresponds to brightness, covering the full range $[0:255]$. Note the large number of generators with very small lifespans, showing that there are many points in the diagrams that are near the diagonals and represent essentially noise. The Supplemental Material [47] shows the PDs and lifespans for all considered images of the reference experiment.

Figure 3

Total persistence for the components, ${\mathrm{TP}}_0$, for the loops, ${\mathrm{TP}}_1$, the photoelastic intensity, and the intruder's acceleration for the reference case (medium 6.35–4.9). The Supplemental Material [47] shows the above curves jointly with the (processed) photoelastic images.

Figure 4

Total persistence for the components, ${\mathrm{TP}}_0$, for the loops, ${\mathrm{TP}}_1$, the photoelastic intensity, and the intruder's acceleration for the impact on soft particles (soft 6.35–1.3).

Figure 5

Betti numbers, ${\beta }_0$ (a) and ${\beta }_1$ (b), for the reference case, respectively. No smoothing is used here since the amount of data, particularly for ${\beta }_1$, is limited. The results are shown for three different brightness threshold levels, as discussed in the text.

Figure 6

Wasserstein distance between the consecutive images for the reference case as a different amount of noise (shown by the values of $N$ in the legends) is assumed for components (a) and loops (b). The distances are essentially insensitive to noise.

Figure 7

Wasserstein distance between the images [(a) components; (b) loops] for the reference case assuming that the generators closer than 30 pixels to the diagonal can be ignored. The given number of “skips” specifies the number of images that were skipped in the calculation of the Wasserstein distance. The insets show how the maxima of the distances depend on the time interval between the images considered. For the insets, $\mathrm{\Delta }t$ on the horizontal axes correspond to the number of skipped images multiplied by the time between two consecutive images, which is given by the inverse frame rate. Since the data for the maxima are noisy, we smoothed the distances data from the main panels for the purpose of calculating the maxima shown in the insets.