Coupled tomography and distinct-element-method approach to exploring the granular media microstructure in a jamming hourglass

We describe an approach for exploring microscopic properties of granular media that couples x-ray microtomography and distinct-element-method (DEM) simulations through image analysis. We illustrate it via the study of the intriguing phenomenon of instant arching in an hourglass (in our case a cylinder filled with a polydisperse mixture of glass beads that has a small circular shutter in the bottom). X-ray tomography provides three-dimensional snapshots of the microscopic conditions of the system both prior to opening the shutter, and thereafter, once jamming is completed. The process time in between is bridged using DEM simulation, which settles to positions in remarkably good agreement with the x-ray images. Specifically designed image analysis procedures accurately extract the geometrical information, i.e., the positions and sizes of the beads, from the raw x-ray tomographs, and compress the data representation from initially 5 gigabytes to a few tens of kilobytes per tomograph. The scope of the approach is explored through a sensitivity analysis to input data perturbations in both bead sizes and positions. We establish that accuracy of size—much more than position—estimates is critical, thus explaining the difficulty in considering a mixture of beads of different sizes. We further point to limits in the replication ability of granular flows away from equilibrium; i.e., the difficulty of numerically reproducing chaotic motion.

Figure 1

(Color online) Combining Synchrotron x-ray tomography and DEM: an overview.

Figure 2

Tuning the sample-detector distance for optimal contrast enhancement in 2D projection: the line intensity profile (white mark) through the glass-air interface shows a peak of higher intensity corresponding to the refraction-based effects.

Figure 3

(Color online) Scheme of the synchrotron microtomography imaging setup.

Figure 4

The container (diameter $A=6.0\text{mm}$) was drilled from one block of boron nitride and has interchangeable openings (IO) (diameter $B=1.5\text{mm}$ and 2.0 mm). A paper shutter was positioned under the container at offset $C=1.0\text{mm}$ from its bottom. The container was mounted on the top of a metal sample holder CS.

Figure 5

Raw tomograph (and artifacts) used as input for image analysis: (a) Half of a horizontal cut ($z$ layer); (b) Newton ring; (c) splashes; (d) asphericity; (e) air bubbles.

Figure 6

Half of (a) a preprocessed $z$ layer and (b) a watershed $z$ layer.

Figure 7

Sphere fittings (white circles) superimposed on the original tomographs after (a) detection and (b) optimization.

Figure 8

Results of the detection: $n$, number of beads; tomographs are named after the corresponding simulation, the absence of the letter o indicates images taken before hole opening, a letter o indicates images taken after hole opening; white, correct detection; gray, automatically corrected for oversegmentation; black, corrected manually for undersegmentation and bad cuts.

Figure 9

Error estimation: symmetric difference of fitted spheres and preprocessed image (white shapes, particles in the preprocessed image absent in the sphere fittings; black shapes, opposite case).

Figure 10

(Color online) Average volume symmetric difference between a real bead and its sphere fitting.

Figure 11

(Color online) Average fitting error on positions and radii.

Figure 12

(Color) Three snapshots of simulation $S4\_5$ at times (a) 0.0 s (shutter closed); (b) 0.08 s (intermediate stage); (c) 0.3 s (arched configuration).

Figure 13

(Color online) Histogram of bead displacements between 0.05 and 0.4 s for simulation $S4\_5$ (outliers not shown).

Figure 14

Vertical cut highlighting the deviation between physical experiment 4_5 and simulation $S4\_5$: white shapes, bead traces in the simulation not coinciding with their physical experiment counterparts; black shapes, opposite case.

Figure 15

(Color online) 3D rendered volume of the deviation between physical experiment 4_5 and simulation $S4\_5$: bead traces in the physical experiment not coinciding with their simulation counterparts

Figure 16

Sensitivity analysis results showing the dominating effect of size vs position perturbations; ${\eta }_{0.3\text{s}}\left(ϵ\right)$ is given for experiments (a) 3_1, (b) $4b\_2$, and (c) $4b\_3$. Type II perturbations were carried out only for experiment 3_1. Lines represent the average value of the corresponding replications. Note that plots of types I and II coincide and dominate the plot of type III.

Figure 17

Plots of $E\left(ϵ\right)$, energy furnished to the system as a function of perturbation size, for experiment 3_1. Lines represent the average value of the corresponding replications. Note that the plot of type I dominates the coinciding plots of types II and III.