Granular avalanches of entangled rigid particles

In granular mechanics, the shape of grains plays a critical role in the overall dynamics and significantly affects the macroscopic properties of the system. Using a dam break setup, granular collapses of nonconvex (cross-shaped) plastic particles assumed quasirigid are conducted experimentally and simulated numerically for a wide range of aspect ratios $a=H_0/L_0$, with $H_0$ the initial height of the column and $L_0$ its initial length. We report avalanche dynamics such as the top-driven collapse and the buckling collapse, as well as an intermittent flow behavior where reproducibility is lost and where the stability of the column is determined by the random initial configuration of the assembly of entangled particles. While counterintuitive and despite fundamentally different dynamics, we find that the runout distance $L_{\infty }$ and the final height $H_{\infty }$ of our granular collapses of crosses agree with those of spherical particles both experimentally and numerically. Our discrete element method simulations are able to reproduce all flow behaviors observed experimentally and they show excellent quantitative agreement with the experimental data. In the simulations, extra care is given to adopting a tangential friction force model based on the cumulative tangential displacement at the contact point, critical to represent stable cases, and to determining the contact model parameters. The analysis of (i) the force network via the average probability density function of contact force magnitude and (ii) the fabric anisotropy suggests that the stability of the column is a complex problem determined by mesoscale properties that we could not reliably identify at that point.

Figure 1

Top-level view of a soft-body DEM algorithm.

Figure 2

In soft-body DEM, a contact is modeled with an overlap (note that the overlap is intentionally magnified here for visual purposes).

Figure 3

Linear spring-dashpot contact model as a mechanical analog in (a) translation and (b) rotation. The dashed line represents the tangential contact plane.

Figure 4

Schematics of the experimental setup: (a) dam break apparatus and (b) shape of the particles.

Figure 5

(a) Example of a postprocessing output for the runout distance $L_{\infty }$ of a simulation, from a top view of the avalanche. The solid line corresponds to the intensity of a given pixel column (the maximum intensity corresponds to the top of the image and the minimum intensity to the bottom). The dashed line indicates the runout distance; in this case, $L_{\infty }=32$ cm. (b) Example of a postprocessing output for the final height $H_{\infty }$ of a simulation. The solid line indicates the average final height and the dotted lines show the first and third quartiles. For visual purposes, an example with a particularly wide interquartile range (IQR) is shown (a typical IQR is half the size of that shown on this figure).

Figure 6

Snapshots of a top-driven collapse with an initial aspect ratio $H_0/L_0=6$ from (a) experiments and (b) simulation. The time increment between each frame is $0.145\mathrm{s}$. Videos of this collapse are available in [47].

Figure 7

Snapshots of a buckling collapse with an initial aspect ratio of $H_0/L_0\approx$ 7 from (a) experiment and (b) simulation. The time increment between each frame is $0.12\mathrm{s}$. Videos of this collapse are available in [48].

Figure 8

Snapshots of (a) a simulated collapsing assembly and (b) a simulated stable assembly. The initial dimensions of both assemblies are identical; only the initial microstructure differs between these two intermittent cases. The frames are taken at the following times (from left to right): $t=0$, 0.30, 0.60, 0.90, 1.20, and $2.00\mathrm{s}$.

Figure 9

Flow maps from (a) experiments and (b) simulations. Each cross represents an experiment or a simulation. Two flow regimes are identified, repeatable and intermittent, and two collapse behaviors are reported, top driven (observed in all configurations) and buckling (observed only for narrow columns).

Figure 10

Nondimensional runout distance $(L_{\infty }-L_0)/L_0$ as a function of the initial aspect ratio $H_0/L_0$. Error bars are depicted with vertical segments.

Figure 11

Nondimensional height $H_0/H_{\infty }$ as a function of the initial aspect ratio $H_0/L_0$. Error bars are depicted with vertical segments.

Figure 12

(a) Comparison of the average nondimensional translational particle velocity versus nondimensional time for systems of spherical and cross-shaped particles in systems of initial aspect ratio $a=H_0/L_0=3.2$. Also shown are snapshots of the collapse of (b) a small and (c) a large system of crosses at $t=3{\tau }_c$, colored by the norm of their translational velocity from dark blue (smallest) to red (largest). The small system has 3200 spheres or 1000 crosses and the large system $15000$ spheres or 3000 crosses.

Figure 13

Comparison of the average nondimensional translational particle velocity versus nondimensional time for a staggered collapse (solid line) and a regular collapse (dashed line) in the case of an initial aspect ratio $a=H_0/L_0=1.75$. The only difference between the two collapses is the random initial packing.

Figure 14

Visualization of the contact network in a granular column composed of spheres and crosses. Each segment links the centers of mass of two contacting particles, and its color and width vary according to the magnitude of the total contact force magnitude $F_c$ divided by the average contact force magnitude $\langle F_c\rangle$. In (a) all contacts are shown, while in (b) only the contacts associated with a force magnitude greater than five times the average force magnitude are plotted.

Figure 15

Probability density function of the contact force magnitude $F_c$ divided by the average contact force magnitude $\langle F_c\rangle$ just before the gate is opened, in (a) linear scale and (b) logarithmic scale, in the cases of assemblies of crosses (black) and spheres (blue).

Figure 16

Definition of the branch vector projected onto the vertical $xz$ plane and of the angle $\theta$ it forms with the horizontal direction.

Figure 17

Average fabric anisotropy of various granular assemblies prior to the opening of the gate. (a) Comparison of the fabric anisotropy of spheres (averaged over 30 simulations) to crosses (averaged over 60 simulations) and coins (only one simulation shown); the curve for coins has been scaled down for readability. (b) Comparison of the fabric anisotropy of crosses about to collapse in a top-driven fashion (averaged over 50 simulations) to crosses about to collapse in a buckling fashion (averaged over 6 simulations). (c) Comparison of the fabric anisotropy of crosses in the intermittent regime about to collapse (averaged over 18 simulations) to remaining stable (averaged over 26 simulations).

Figure 18

Visualization of the force network of two intermittent simulations. Each segment links a contact point to the center of mass of a particle and is colored according to the magnitude of the normal component of the contact force. (a) The opening of the right gate leads to a granular collapse. (b) The opening of the right gate leads to a stable assembly.

Figure 19

Snapshots of simulations (a) with and (b) without memory in the contact model. The frames are taken at the following times (from left to right): $t=0$, 0.15, 0.30, 0.45, 0.60, 0.75, 0.90, 1.05, and $3.00\mathrm{s}$.

Figure 20

Variation of the flow behavior with respect to the tangential parameters. Each dot corresponds to a simulation with specific tangential parameters ($k_t$ and ${\gamma }_t$). Region 2 correctly reproduces experiments. Region 1 leads to avalanches that collapse like convex particles, i.e., the base of the pile drives the motion. Region 3 leads to fully entangled systems that do not collapse.

Figure 21

Surface plots of the tangential parameters (a) $k_t$ and (b) ${\gamma }_t=\frac{{\eta }_t}{2m}$ for $e_n$ and $e_t$ varying from 0 to 1.

Figure 22

Time evolution of the average (a) translational velocity and (b) angular velocity for a system of 750 spheres settling in a box. The dotted line represents no memory term in the contact model and the solid line represents a memory term in the contact model.

Figure 23

Variation of the repose angle $\theta$ with the static friction coefficient ${\mu }_c$. The solid line shows results from the present study and the dashed line shows results adapted from Yan et al. [45].

Figure 24

Nondimensional final $x$ displacement of the center of mass of the granular assembly (diamonds) and nondimensional runout distance (all other symbols) as a function of the initial aspect ratio.

Figure 25

Schematics of the avalanche profiles of the granular assembly for (a) low and (b) high aspect ratios, under the assumption that the slope of the avalanched profile is constant for a given aspect ratio.

Figure 26

Probability density functions of the contact forces $F_c$ just before the gate is opened, in (a) and (c) linear scale and (b) and (d) logarithmic scale, for the cases of (a) and (b) a top-driven regime (black) with a buckling regime (blue) and (c) and (d) an intermittent regime leading to stable (black) and collapsed (blue) behaviors.