Planar granular shear flow under external vibration

We present results from a planar shear experiment in which a two-dimensional horizontal granular assembly of pentagonal particles sheared between two parallel walls is subjected to external vibration. Particle tracking and photoelastic measurements are used to quantify both grain scale motion and interparticle stresses with and without imposed vibrations. We characterize the particle motion in planar shear and find that flow of these strongly interlocking particles consists of transient vortex motion with a mean flow given by the sum of exponential profiles imposed by the shearing walls. Vibration is applied either through the shearing surface or as bulk vertical vibration of the entire shearing region with dimensionless accelerations $\mathrm{\Gamma }=A{\left(2\pi f\right)}^2/g\approx 0–2$. In both cases, increasing amplitude of vibration $A$ at fixed frequency $f$ leads to failure of the force network, reduction in mean stress, and a corresponding reduction in imposed strain. Vibration of the shearing surface is shown to induce the preferential slipping of large-angle force chains. These effects are insensitive to changes in frequency in the range studied ($f=30–120$ Hz), as sufficiently large displacements are required to relieve the geometrical frustration of the jammed states.

Figure 1

Experimental schematic (top view). A single layer of photoelastic grains (A) lies horizontally, confined above and below by acrylic plates. Shearing belts (B) held in position by fixed guide rails (C) move in opposite directions to apply shear. The belts move in equal and opposite directions and are driven by a single stepper motor (D). Circular end caps (E) contain the grains at the end of the channel. Vibration can be imposed through one of the shearing walls (sidewall) or vertically by shaking the entire experiment. Grains within the shaded region are imaged for analysis.

Figure 2

Experimental images under shear (indicated by arrows) corresponding approximately to the shaded region in Fig. 1. Top: Pentagonal grains (unmarked). Grains can also be marked with rectangular tag for automated particle tracking. Bottom: Photoelastic image of force network.

Figure 3

A commonly observed transient vortex in the velocity field in which a cluster of grains rotates in a cooperative motion of constituent grains. The total time is 7 s corresponding to a displacement of the shearing surface of $\approx 3.5d$.

Figure 4

(a) Mean velocity versus distance across the shearing channel. The overall velocity scale decreases with vibration amplitude with a corresponding decrease in stress. The solid line is a fit to the $\mathrm{\Gamma }=0$ data set using a sum of two exponential velocity profiles at the shearing surface [Eq. (1), $v_0=0.45d/\mathrm{s}$, $\ell =2.5d$, and $W=14.5]$. The inset shows a log-linear plot of $v_x\left(y\right)$, which is observed to be linear for each data set, confirming the exponential behavior for $y<0$. $\mathrm{\Gamma }=0.46$ is omitted as $v_x\approx 0$. (b) Shear rate for the four smallest vibrations from (a). The solid line corresponds to $d{v}_x/dy$ using Eq. (1) and the same parameters from part (a).

Figure 5

Mean stress versus time during shear. Strong fluctuations are observed compared to the mean stress (horizontal line).

Figure 6

Stress versus time. Individual runs of mean stress versus time are plotted for varying vibration amplitude at $f=30$ Hz. Each data set, corresponding to $A\approx 0$, 0.03, 0.1, 0.2, 0.3, and 0.4 mm, is shifted vertically by integer values after the background ${\sigma }_{bg}$ is subtracted.

Figure 7

(a) Stress versus amplitude at fixed frequency ($f=30\mathrm{Hz}$). (b) Stress versus frequency at fixed amplitude ($A=0.07\mathrm{mm}$). Both data sets correspond to a a similar range of $\mathrm{\Gamma }=0–1.6$. Error bars indicate the standard deviation for the stress time series which increases with mean stress and is around 20% for most parameter values.

Figure 8

Discrete probability distribution function for stress at various vibration amplitudes corresponding to the data shown in Fig. 7 (top). Each consists of 2036 data points or 17 min of data.

Figure 9

Sequential images at an interval of 0.2 s from an image sequence at 30 Hz showing a typical slip event at the shearing boundary (left edge of images). The boundary grain supporting the stress (circle) slips backwards relative to a fixed position (line) and relieves the stress in the force chain.

Figure 10

Distribution of angles for force chains in the absence of vibration ($•$) and large vibration ($\blacksquare$, $A\approx 0.2$ mm and $f=30$ Hz).

Figure 11

Average stress versus dimensionless acceleration for bulk vibration at fixed frequency and increasing amplitude. Inset shows stress versus dimensionless acceleration at fixed amplitude and increasing frequency.