Collision-based understanding of the force law in granular impact dynamics

We study the stopping force felt by an intruder impacting onto a granular medium. Variations in the shape of the intruder can influence the penetration depth by changing the inertial drag. We find this observed correlation can be explained by associating the velocity-dependent inertial drag to the energy dissipation that occurs through intermittent collisions of force-chain-like clusters, the mean behavior of which can be statistically described. In consequence, the stopping force can be captured through a proposed collisional model with good accuracy, and the observed impact dynamics data can be reproduced quantitatively.

Figure 1

(a) Visualization of the force networks excited by intruders of different shapes. All the five shaped intruders have the same diameter $D_p=4.15$ cm ($\sim 21r_g$) and mass $M=101$ g. Colors indicate the elastic energy, $E$, stored in each particle relative to the average elastic energy, $E_0$. Note that only those particles with $E>E_0$ are considered part of force chains [26]. The solid black curves mark out the free surface of the target grains during impact. (b, c) Kinematics and the total drag acceleration of a sphere dropped onto a granular bed. The scattered points are color coded according to the initial impact velocity $v_0$. By simultaneously adjusting $k$ and $b$, the prediction of Eq. (1) is reasonably fitted to the data. (d) Sketch of a sphere with velocity $v$, impacting inelastically with a chain of particles that behaves as a quasiparticle [18]. These intruder-cluster collisions gradually dissipate the kinetic energy of the intruder and eventually stop the penetration.

Figure 2

(a) The total drag force versus square of velocity at five chosen specific depths. The dotted curves represent linear fits, where the proportionality coefficients are determined to be around 32.3. (b) The depth-dependent drag, $\left|F_{\mathrm{drag}}\right|-b{v}^2$, versus depth. The solid curves represent impact data and are colored based on initial impact velocity as in Fig. 1. The dashed line represents fit to $k\left|z\right|$.

Figure 3

Scaling behavior of the normalized penetration depth $d/r_g$ versus the normalized total drop distance $H/r_g$ for sphere ($●$), disk ($▬$), ${120}^{\circ }$ cone ($★$), ${90}^{\circ }$ cone ($\times$), and ${60}^{\circ }$ cone ($▾$), colored based on initial impact velocity $v_0$ as in Fig. 1. The shaded region corresponding to $H<d$ is by definition not allowed. The solid curves ($—$) show the depth predicted by Eq. (1) with parameters fitted in Fig. 2. The dashed gray lines ($‐‐$) show power-law scaling rules, and the fitting exponent changes greatly with intruder shape, indicating that the intruder shape has a strong effect on the interactions with ambient granular grains.

Figure 4

(a) Probability distribution of elastic energy stored in each particle for a spherical intruder with $v_0=5.0$ m/s at the following times: $t_1=0.002$ s ($◆$), $t_2=0.005$ s ($+$), $t_3=0.025$ s ($▸$), and $t_4=0.05$ s ($◂$). The distribution density $\varphi$ of force chains can then be measured from energy distributions considering the stress concentration within strong networks [26], which is almost time invariant during impact. Inset: The distribution density $\varphi$ as a function of impact velocity. $\varphi$ only changes slightly with the increase of the impact velocity and with different intruder shapes, and gives a universal value of 0.24 $\pm$ 0.02. The symbol code is the same as in Fig. 3. (b) Scaled energy transmission versus initial kinetic energy in impact tests on collections of several particles. The particles are arranged in quasilinear alignment but with random position disturbance to mimic the force-chain-like clusters during granular impact. An intruder of mass $M$ is released to impact this cluster at velocity $v$. The cluster length $n$ is varied in $\left\{6,7,8,9\right\}$, the friction coefficient $\mu$ is varied in $\left\{0.4,0.5,0.6,0.7,0.8\right\}$, and the relative angle $\theta$ is varied in $\left\{0^{\circ },{30}^{\circ },{60}^{\circ }\right\}$. Note that each color in the figure corresponds to a unique friction coefficient $\mu$, each symbol corresponds to a unique cluster length $n$, and each point corresponds to 100 runs with different initial configurations. We conduct about 60 impact tests with different parameter sets ($\mu$, $n$, $\theta$). The test data collapse to one line when scaled by ${\left(n\mu \right)}^{-1}$ as expected in Eq. (3), yielding a damping dissipation coefficient $e=3.2\pm 0.3$. Also in (b) are representative snapshots in an impact test with the parameter set $n=8$, $\mu =0.6$, and $\theta ={30}^{\circ }$, showing the breakage of the force-chain-like cluster.

Figure 5

The dependence of inertial term $b$ on scaled intruder area $I\mu {\rho }_{g}A$. The symbol shapes are coded as in Fig. 3. The solid line indicates the fitting to Eq. (4) with slope equal to $6.65\pm 0.21$, close to the semi-analyzed value (dashed line), $6.14\pm 0.58$, expected by the collisional model, showing that the proposed energy dissipation mechanism effectively captures the physical origin of inertial term.