Nonlinear Force Propagation During Granular Impact

We experimentally study nonlinear force propagation into granular material during impact from an intruder, and we explain our observations in terms of the nonlinear grain-scale force relation. Using high-speed video and photoelastic particles, we determine the speed and spatial structure of the force response just after impact. We show that these quantities depend on a dimensionless parameter, $M^{\prime }=t_c{v}_0/d$, where $v_0$ is the intruder speed at impact, $d$ is the particle diameter, and $t_c$ is the collision time for a pair of grains impacting at relative speed $v_0$. The experiments access a large range of $M^{\prime }$ by using particles of three different materials. When $M^{\prime }\ll 1$, force propagation is chainlike with a speed, $v_f$, satisfying $v_f\propto d/t_c$. For larger $M^{\prime }$, the force response becomes spatially dense and the force propagation speed departs from $v_f\propto d/t_c$, corresponding to collective stiffening of a strongly compressed packing of grains.

Figure 1

Force propagation after impacts with $v_0\approx 5\mathrm{m}/\mathrm{s}$. (a) Hard particles ($M^{\prime }\approx 0.1$) show sparse, chainlike force propagation. (b) Forces for intermediate particles ($M^{\prime }\approx 0.3$) are more dense spatially, but still relatively chainlike. (c) Soft particles ($M^{\prime }\approx 0.6$) show a dense force structure that propagates with a well-defined front.

Figure 2

(a)–(c) Space-time plots of propagating forces shown in Fig. 1 (see text for details). Dashed white lines indicate $v_f$. (d) These are plotted versus $v_0$. Symbol shape denotes particle stiffness (squares/triangles/circles are hard/medium/soft, respectively), and color denotes intruder diameter (red/blue/black for $6.35/12.7/20.32\mathrm{cm}$, respectively). (e) Plot of data in (d), but with $v_f$ and $v_0$ normalized by $v_b$. The fit lines correspond to Eq. (1) with $\alpha \approx 1.4$ (solid line) and $\alpha \approx 2.2$ (dashed line); see text for discussion.

Figure 3

(a) Plot of ratio of measured $v_f$ to $d/t_c$ versus $M^{\prime }$. We find $v_f\approx 5.3(d/t_c)$ to hold for small $M^{\prime }$, but for large $M^{\prime }$, $v_f$ is faster than this relation. (b) Participation ratio, $P$, versus $M^{\prime }$, which approximates the spatial density of propagating forces (see text for details).