Depth-Dependent Resistance of Granular Media to Vertical Penetration

We measure the quasistatic friction force acting on intruders moving downwards into a granular medium. By utilizing different intruder geometries, we demonstrate that the force acts locally normal to the intruder surface. By altering the hydrostatic loading of grain contacts by a sub-fluidizing airflow through the bed, we demonstrate that the relevant frictional contacts are loaded by gravity rather than by the motion of the intruder itself. Lastly, by measuring the final penetration depth versus airspeed and using an earlier result for inertial drag, we demonstrate that the same quasistatic friction force acts during impact. Altogether this force is set by a friction coefficient, hydrostatic pressure, projectile size and shape, and a dimensionless proportionality constant. The latter is the same in nearly all experiments, and is surprisingly greater than one.

Figure 1

Friction force versus depth for (a) cylindrical, (b) conical, and (c) spherical intruders of different geometry; $R$ is radius and $\varphi$ is apex half-angle. The scaling of the $y$ axes, and the curves through the data, are the expectations of Eq. (3) for $d\mathbf{F}$ normal to the intruder surface area elements. The dashed (dotted) line in (c) represents expectation for $z>R$ for $d\mathbf{F}$ normal (tangential) to the intruder.

Figure 2

Friction force versus depth for (a) cylindrical, (b) conical, and (c) spherical probes, normalized as in Fig. 1 and also by $1-U/U_c$, where $U$ is airspeed and $U_c$ is airspeed at fluidization. The solid curves are simultaneous fits to Eq. (3) with $\alpha =35\pm 5$. All data from Fig. 1 are replotted as small grey points.

Figure 3

Penetration depth versus normalized airspeed for (a) a sphere of mass 0.07 kg and (b) a cylinder of mass 0.2 kg, from various drop heights $h$. The grains are $300\mu \mathrm{m}$ and the packings are 19.4 cm deep, except for two special cases in (a) where the depth and the grain size are decreased. The curves represent solutions of Eq. (2) and (a) in Eq. (3) with $\alpha =35$, taking $b$ to match the data at $U=0$.