Intruder friction effects on granular impact dynamics

There is considerable recent interest in intruders impacting into granular materials. Many studies focus on a collisional model where the drag force acting on an intruder varies as the square of the intruder speed. However, it is unclear how intruder friction affects granular impact dynamics. Here, we experimentally study impacts into quasi-two-dimensional beds of photoelastic granular beds of three circular intruders of similar size and mass, but with varying friction coefficients associated with the intruder edges (smooth, “sandy,” and gear). We compare typical measures of the dynamics for the three intruders, including impact depth and speed vs time. We show that the smooth and sandy intruders share similar impact dynamics, while the gear intruder displays smaller impact depth, speed, and impact time. We attribute the differences between the gear intruder's dynamics and those of the other two to differences in the collision-generated force networks associated with the grain-scale roughness of the gear intruder. For the smooth and sandy intruders, the force networks align close to the normal direction of the intruder boundaries. For the gear intruder, the grain-scale geometric roughness leads to force chains that are closer to vertical, rather than in the coarse-grained normal direction to the intruder edge. This leads to a stronger drag force for the gear intruder. Hence, in the range that we have explored, the granular impact dynamics are highly sensitive to grain-scale roughness of the intruder and relatively insensitive to microscale roughness that is associated with the conventional friction coefficient.

Figure 1

Snapshots of force propagation in the granular bed after impact with $v_0\approx 2.2\mathrm{m}/\mathrm{s}$. (a) Bronze intruder with a relatively smooth boundary. (b) “Sandy” intruder coated with 1-mm-diameter sand, which has a large conventional friction coefficient. (c) Gear intruder, showing the nature of the grain-scale geometric friction.

Figure 2

(a) Intruder trajectories for for three different friction coefficient intruders with similar initial impact speeds, where $v_0\approx 2.2\mathrm{m}/\mathrm{s}$ ($\circ$, smooth intruder; $*$, sandy intruder; and $\square$, gear intruder). Inset: the differences of impact depth during impacting ($·$, the depth difference between smooth and sandy intruders; $\nabla$, the depth difference between sandy and gear intruders; and $▵$, the depth difference between smooth and gear intruders). (b) Impact speeds vs time. Here, $t=0$ of (a) and (b) is the first contact with the granular surface, measured from the photoelastic response. (c) Final impact depth $z_{\mathrm{stop}}$ vs initial impact energy $K_0$ (blue line is the fitting curve for smooth intruder, with $a=52.7$, $b=55.4$, and $c=10.0$; orange line is the fitting curve for sandy intruder, with $a=54.4$, $b=52.4$, and $c=10.1$; black line is the fitting curve for gear intruder, with $a=59.6$, $b=29.8$, $c=10.2$). (d) Total impact times with the different $K_0$.

Figure 3

(a) Schematic of the definition of $\theta$, which is the angle between force chain segment contacting the surface of the intruder and the extension of the radial direction. $r$ is the intruder radius, $R$ corresponds to two particle diameters plus the intruder radius, which is used to characterize forces chain around intruder, $v$ is the direction of gravity. (b) The mean value of $\theta$ generated by three different intruders. Inset: the probability distribution function of the force chain relative to the vertical direction ${\theta }_{\mathrm{ver}}$ for gear intruder.

Figure 4

The cumulative distribution function $P(\mathrm{\Theta }>\theta )$ for the three different intruders vs the angle between interface force chains and the reference radial direction $\theta$, with the initial kinetic energy $K_0=0.4J$ for all of three intruders. Inset: The probability distribution function $P\left(\theta \right)$ for the three different intruders vs $\theta$.