Collisional model of energy dissipation in three-dimensional granular impact

We study the dynamic process occurring when a granular assembly is displaced by a solid impactor. The momentum transfer from the impactor to the target is shown to occur through sporadic, normal collisions of high force carrying grains at the intruder surface. We therefore describe the stopping force of the impact through a collisional-based model. To verify the model in impact experiments, we determine the forces acting on an intruder decelerating through a dense granular medium by using high-speed imaging of its trajectory. By varying the intruder shape and granular target, intruder-grain interactions are inferred from the consequent path. As a result, we connect the drag to the effect of intruder shape and grain density based on a proposed collisional model.

Figure 1

(a) Schematic of the top view of experiment setup. A high-speed camera is used to view an intruder which is released from a height above and impacts into the granular bed. A mirror is placed at 45 degrees to the impact to measure any tilting during penetration. (b) Diagram of the impact of a conical intruder with a chain of particles of the granular target (represented by an orange ellipse). (c) Image of all intruders of $w=3.8$ cm, varying $s$ through the following values: 0, 0.2, 0.5, 1, 1.4, 1.7, 2.1.

Figure 2

(a) $z$ vs $t$, (b) $\dot{z}$ vs $t$, and (c) $\ddot{z}$ vs $t$ of a $s=1$ intruder impacting couscous. The gray region surrounding each curve indicates the standard deviation of five runs.

Figure 3

Comparison of $\ddot{z}\left(t\right)$ of $s=2.1$ intruder impacting couscous as determined simultaneously by high speed imaging (black line) and by direct force measurements via accelerometer (red line). The shape of the curves matches for these approaches.

Figure 4

(a) $z$ vs $t$, varying $s$, all impacting sand from the same release height. Penetration depth $z_{\mathrm{stop}}$ increases with increasing $s$. (b) $\dot{z}$ vs $t$. The decline of $\dot{z}\left(t\right)$ changes with increasing $s$, but all curves reach $\dot{z}=0$ at approximately the same time. (c) $z_{\mathrm{stop}}$ vs $K_i$. Higher $s$ shifts curve to higher values but does not change the trend. The fit of $z_{\mathrm{stop}}\propto aln(d{K}_i+1)$ where $a$ and $d$ are constants, is shown with the data of the $s=0$ and $s=2.1$ intruders. (d) $t_{\mathrm{stop}}$ vs $K_i$. Above 1 J, $t_{\mathrm{stop}}$ is constant for all $K_i$ and $s$. Solid line shows the fit of Eq. (4) to data $t_{\mathrm{stop}}$($K_i$) of $s=0$ intruder. Data sets correspond to $s$ values given by the colorbar.

Figure 5

(a) Inset shows $K$ versus $z$ of $s=0$ intruder impacting sand. Each curve represents a trajectory of a different impact velocity. Main panel shows $K_p$ versus $z$, as determined from all pairs of $K\left(z\right)$ trajectories of the inset plot. The gray region surrounding the curve indicates the standard deviation from the mean of $K_p$. (b) $\int h\left(z\right)dz$ versus $z$. (c) $h\left(z\right)$ versus $z$ for a $s=0$ intruder impacting sand, showing oscillations about a constant value.

Figure 6

(a) $\int h\left(z\right)dz$ versus $z$ for all intruders, ranging from $s=0$ to $s=2.1.$ Colors correspond with $s$ values of Fig. 4. Higher $s$ lowers the curve to smaller inertial drag values. The trend in data also changes from approximately linear to increasingly curved. (b) The data collapse to one curve when scaled as a function of $s$. (c) Collapse with consideration of added contribution from tip for $s>1$ intruders.

Figure 7

Inset shows $\int h\left(z\right)dz$ versus $z$ of $s=1$ intruder, varying width $w$: 2.2 cm (red), 3.0 cm (blue), and 3.8 cm (green). Lower $w$ decreases the curve but, in this case, the trend in data is not affected. Main panel: When scaled as a function of $w$, the data collapse to a single curve.

Figure 8

(a) $z_{\mathrm{stop}}$ versus $K_i$, varying target grain density ${\rho }_g$ to sand ($⧫$) or couscous ($•$). In each case, $z_{\mathrm{stop}}$ shows similar dependence on $K_i$, yet the intruder penetrates to higher $z_{\mathrm{stop}}$ with decreasing ${\rho }_g$. (b) When scaled as a function of ${\rho }_g$, the data collapse to a single curve. (c) $t_{\mathrm{stop}}$ versus $K_i$, varying ${\rho }_g$. Stopping time $t_{\mathrm{stop}}$ is constant for $K_0>0.5$ J. The behavior for sand and couscous are similar; however, $t_{\mathrm{stop}}$ increases for decreased ${\rho }_g$. (d) The data collapse when scaled by ${\rho }_g^{1/2}$. All data are determined by using an $s=0$ intruder.

Figure 9

Inset shows $\int h\left(z\right)dz$ versus $z$ for $s=0$ intruder, varying ${\rho }_g$: sand (red line) or couscous (black line). Higher ${\rho }_g$ (sand) increases inertial drag $h\left(z\right)$. Main panel shows scaled $h\left(z\right)$, $\int h\left(z\right)dz/{\rho }_g$ versus $z$ for the two granular targets, collapsing data.