Decompaction wave propagation in a vibrated fine-powder bed

We experimentally study the crack formation and decompaction-wave propagating in a vibrated powder bed consisting of glass beads of 5 $\mu \mathrm{m}$ in diameter. The vibrated powder bed exhibits three distinct phases depending on the vibration conditions: consolidation (CS), static fracture (SF), and dynamic fracture (DF). Particularly, we found an upward wave propagation in the DF regime when the powder bed is strongly vibrated. As a remarkable feature, we found that in fine cohesive powders, the decompaction-wave propagation speed normalized to gravitational speed is independent of the shaking strength. This result implies that the wave propagation speed is governed by the balance between gravity and cohesion effect rather than vibration strength. We also explore the universality of wave propagation phenomenon in coarser and low-density granular powders.

Figure 1

The experimental setup of the vibrated granular powder bed. The inset shows a schematic of the vibrated fractured powdered bulk consisting of horizontal cracks. In DF phase, these cracks would open in an successive manner from bottom to the top giving an impression of vertically upward traveling wave. The parameters $w,L$, and $h$ are crack width, crack length, and a traveling height of the crack wave, respectively. We use ImageJ [22] to obtain the crack measurements.

Figure 2

(a) Phase diagram for $f=30\mathrm{Hz}$, RH $=40\%$, and the corresponding raw images of: (b) and (c) consolidated state (CS), (d) static fracture (SF), and (e) dynamic fracture (DF). The real-time videos of three phases can be found in the Supplemental Material (SM) [26]. The red line and hollow circles on the phase diagram indicate the boundary separating consolidated state and the fractured state for RH $=40\%$ and RH $=85\%$, respectively. The star pointers on the phase diagram indicate the data points for the corresponding raw images. The nondeforming stable voids in CS regime can be qualitatively differentiated with SF regime, where stable voids deform into fine cracks and ruptured top surface. DF regime is quantitatively differentiated using criterion based on crack area, which should be greater than 10 ${\mathrm{mm}}^2$. Details of phase classification is written in Appendix pp1-s2.

Figure 3

The Phase diagram for $f=30\mathrm{Hz}$ and RH $=85\%$.

Figure 4

The Phase diagrams at RH $=40\%$ for (a) $f=25\mathrm{Hz}$ and (b) $f=40\mathrm{Hz}$.

Figure 5

Decompaction-wave propagation in DF regime. ImageJ [22] was used to track cracks in the bulk. The red dashed lines represent mean locations of the cracks. The standard deviation about mean of the data is defined here as a crack width $w$.

Figure 6

The relation among $v/\sqrt{gw}$ and $S$ at RH $=40\%$. $K=0.16\pm 0.04$ is obtained by averaging across $m$ and $f$.

Figure 7

Comparing $v/\sqrt{gw}$ for different relative humidities and container types. The base is vibrated at $f=30\mathrm{Hz}$, and at $a=2.4$ mm.

Figure 8

Consolidated (CS) phase. ImageJ [22] is used to observe deformations in the bulk. During CS phase, we do not observe any deforming voids. The experimental conditions are $m=30\mathrm{g}$, $f=30\mathrm{Hz}$, and $a=1.2\mathrm{mm}$.

Figure 9

Static fracturing (SF) phase. ImageJ [22] is used to observe the deformations in the bulk. The cracks/voids present in the bulk keep deforming to form elongated horizontal cracks. The experimental conditions are $m=30\mathrm{g}$, $f=30\mathrm{Hz}$, and $a=2.1\mathrm{mm}$.

Figure 10

Transition from static (SF) to dynamic fracturing (DF). ImageJ [22] is used to observe the deformations of widening cracks. The cracks formed during the SF phase widen when the vibrational strength is increased. This is the onset of dynamic phase. This experimental conditions are $m=30\mathrm{g}$, $f=30\mathrm{Hz}$, and $a=2.7\mathrm{mm}$.

Figure 11

Agglomerates on the top free surfaces of the consolidated powder bed. This image corresponds to the top view of Fig. 2 in this paper.

Figure 12

The relation among $v/\sqrt{gw}$ and $\mathrm{\Gamma }$ at RH $=40\%$. $K=0.16\pm 0.04$ is obtained by averaging all the data.

Figure 13

The Phase diagram for a powder with mean diameter size $d=18\mu \mathrm{m}$ at $f=30$ Hz and RH $=45\%$. The regime CF represents convective fracture (shown by vertical green dashed lines).

Figure 14

The relation among $v/\sqrt{gw}$ and $S$ for powders: (a) $d=18\mu \mathrm{m}$ at 30 Hz and (b) $d=50\mu \mathrm{m}$ at two different frequencies.