Effect of volume fraction on granular avalanche dynamics

We study the evolution and failure of a granular slope as a function of prepared volume fraction, ${\varphi }_0$. We rotated an initially horizontal layer of granular material (0.3-mm-diam glass spheres) to a ${45}^{\circ }$ angle while we monitor the motion of grains from the side and top with high-speed video cameras. The dynamics of grain motion during the tilt process depended sensitively on ${\varphi }_0\in [0.58–0.63]$ and differed above or below the granular critical state, ${\varphi }_c$, defined as the onset of dilation as a function of increasing volume fraction. For ${\varphi }_0-{\varphi }_c<0$, slopes experienced short, rapid, precursor compaction events prior to the onset of a sustained avalanche. Precursor compaction events began at an initial angle ${\theta }_0=7.7\pm 1.4^{\circ }$ and occurred intermittently prior to the onset of an avalanche. Avalanches occurred at the maximal slope angle ${\theta }_m=28.5\pm 1.0^{\circ }$. Granular material at ${\varphi }_0-{\varphi }_c>0$ did not experience precursor compaction prior to avalanche flow, and instead experienced a single dilational motion at ${\theta }_0=32.1\pm 1.5^{\circ }$ prior to the onset of an avalanche at ${\theta }_m=35.9\pm 0.7^{\circ }$. Both ${\theta }_0$ and ${\theta }_m$ increased with ${\varphi }_0$ and approached the same value in the limit of random close packing. The angle at which avalanching grains came to rest, ${\theta }_R=22\pm 2^{\circ }$, was independent of ${\varphi }_0$. From side-view high-speed video, we measured the velocity field of intermittent and avalanching flow. We found that flow direction, depth, and duration were affected by ${\varphi }_0$, with ${\varphi }_0-{\varphi }_c<0$ precursor flow extending deeper into the granular bed and occurring more rapidly than precursor flow at ${\varphi }_0-{\varphi }_c>0$. Our study elucidates how initial conditions—including volume fraction—are important determinants of granular slope stability and the onset of avalanches.

Figure 1

Experiment setup and volume fraction evolution during tilting of the granular bed. (a) Air flow through a porous floor in the fluidized bed and mechanical vibration prepare granular material to the desired initial $\varphi$. The bed rotates about midpoint and is imaged from the top and side views. The coordinate system is in the frame of reference of the rotating bed. (b) The average instantaneous volume fraction of the granular bed $\varphi \left(t\right)$ vs tilt angle, $\theta$, for 14 experiments at different ${\varphi }_0$. As the layer is inclined, compaction or dilation precursor events precede an avalanche. The preavalanche and failure dynamics are sensitive to initial ${\varphi }_0$.

Figure 2

Avalanche dynamics. (a) Change in volume fraction $\Delta \varphi$ is dependent on initial ${\varphi }_0$ and changed sign at ${\varphi }_c=0.595\pm 0.003$. Each point in (a)–(c) represents a single experiment. Gray bar represents uncertainty in ${\varphi }_c$. (b) Initial angle of avalanche precursor, ${\theta }_0$, and maximum angle of stability, ${\theta }_m$, are sensitive to ${\varphi }_0$. (c) Angle of repose is insensitive to ${\varphi }_0$.

Figure 3

Profiles of granular displacement during precursor failure. Displacement profiles of a loose-packed (a), a critical (b), and a close-packed (c) granular material shown from left to right. For each initial condition, we show a vector displacement field of flow, $x$ displacement ($d_x$), and $y$ displacement ($d_y$). The red line indicates surface location. The vector field is in the rotating frame of reference of the granular bed, and the downslope direction is to the left.

Figure 4

Mean granular displacement during precursor avalaching. (a) and (b) Depth-averaged vertical ($\langle d_y\rangle$) and horizontal ($\langle d_x\rangle$) displacement of granular material at different prepared ${\varphi }_0-{\varphi }_c$. (c) Tangent of the dilation angle, $\psi$, vs ${\varphi }_0-{\varphi }_c$. Horizontal lines correspond to close- and loose-packed examples described in the text.

Figure 5

Spatiotemporal evolution of granular flow during slow tilting of the bed. (a) Difference images of granular flow at ${\theta }_0$ (left) and ${\theta }_m$ (right). The red line indicates the top surface. White spots correspond to regions that underwent a rearrangement of particle positions. The white vertical rectangle in the center illustrates the region plotted vs time below. (b) Space-rotation angle profiles of granular flow generated by evaluating image differences along a vertical strip in the image center over time [white rectangle in (a)]. Gray-scale signifies the intensity of pixel differences shown in (a), however gray-scale intensity does not map linearly to granular displacement (see Sec. I). The intensity plots of the flow profile are shown above the space-rotation angle plots (arbitrary units). Top plots are loose pack and show a precursor event occurring at ${\theta }_0=6.2^{\circ }$. Several precursors occur prior to the initiation of a surface avalanche at ${\theta }_m=28.1^{\circ }$ (gray regions signify avalanche regime).

Figure 6

Spatiotemporal evolution of granular flow during precursors and avalanches. (a) and (b) Space-time images of precursor events at ${\theta }_0$ in a loose- (a) and close- (b) packed granular media. In example (a), flow begins at the surface and propagates vertically into the granular layer (black arrow) at a speed of approximately $v=1.08$ m/s, to a maximum depth $d_a$. In example (b), flow begins simultaneously throughout the layer. The intensity of the image is arbitrary and chosen to highlight flow shape. (c) Time duration of precursor events as a function of ${\varphi }_0-{\varphi }_c$. (d) Depth dependence of precursor and avalanche flow as a function of initial ${\varphi }_0-{\varphi }_c$. White circles are precursor flow and black circles are avalanching flow. The black and green lines show the fit functions described in the text.