Intrusion into Granular Media Beyond the Quasistatic Regime

Leah K. Roth, Endao Han, and Heinrich M. Jaeger

Phys. Rev. Lett. 126, 218001 – Published 25 May 2021

Abstract

The drag force exerted on an object intruding into granular media is typically assumed to arise from additive velocity and depth dependent contributions. We test this with intrusion experiments and molecular dynamics simulations at constant speed over four orders of magnitude, well beyond the quasistatic regime. For a vertical cylindrical rod we find velocity dependence only right after impact, followed by a crossover to a common, purely depth-dependent behavior for all intrusion speeds. The crossover is set by the timescale for material, forced to well up at impact, to subsequently settle under gravity. These results challenge current models of granular drag.

Received 23 October 2019
Revised 9 February 2021
Accepted 13 April 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.218001

Physics Subject Headings (PhySH)

Friction Granular compaction Granular flows Granular materials Granular packing Jamming Non-Newtonian fluids

Polymers & Soft Matter

Authors & Affiliations

Leah K. Roth ^*, Endao Han ^†, and Heinrich M. Jaeger^‡

James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA

^*lk.roth.20@gmail.com
^†Present address: Center for the Physics of Biological Function, Princeton University, Princeton, New Jersey 08544, USA.
^‡h-jaeger@uchicago.edu

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Issue

Vol. 126, Iss. 21 — 28 May 2021

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Images

Figure 1
(a) Experimental setup. (b) Rendered image from simulation. Grains are colored by velocity magnitude ( $v_{0} = 100 cm / s$ ). (c) Expected results, based on Eq. (1), for the drag force on an intruder moving at a constant speed $v_{0}$ . (d) Experimental and (e) simulation results for the drag force on a cylindrical intruder as a function of $z$ for different $v_{0}$ .
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Figure 2
(a) Force on intruder right after impacting the free surface, as function of intrusion depth $z$ scaled by grain diameter $d_{g}$ . (b) Peak force as function of $v_{0}$ . (c), (d) Normalized drag force (c) and change in packing fraction (d), both from simulations, as a function of $z / d_{g}$ .
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Figure 3
Force on intruder in fluidized region following the initial peak, as a function of depth $z$ scaled by rod diameter $D$ . Data shown are from simulations, smoothed (via moving average, window size $\sim 0.8 d_{g}$ ) and with the initial force peak removed. For given $v_{0}$ this region ends at depth $z^{*}$ or, equivalently, dimensionless time $\tilde{t} = 1$ (see text) and the force merges onto a common, linear depth dependence.
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Figure 4
Simulation results showing flow fields and vertical velocity component $v_{z}$ at times $\tilde{t} = 0.7$ , 1.0, and 1.3 (top to bottom rows). Left column (a),(b),(c): Flow field in the $z − r$ plane for $v_{0} = 160 cm / s$ . Streamlines are shown in black, $v_{z}$ is indicated by color. The white rectangle in the upper left of each panel gives the rod position. Right column (d),(e),(f): Vertical velocity profiles, averaged over the strip shown shaded in (a), for five values of $v_{0}$ . This strip extends from $r / D = 0.75$ to 1.25. Vertical lines delineate the rod face. Near the surface of the medium ( $z \sim 0$ ), $v_{z}$ transitions from positive to negative once $\tilde{t} > 1$ .
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Figure 5
Simulation results. (a) Summed upward momentum $\sum p$ of all grains at $r > D / 2$ , binned by vertical velocity magnitude. The momentum carried by the fastest grains ( $v_{z} > v_{0} / 10$ , dashed line) decreases sharply at $\tilde{t} = 1$ , while increasing for the slowest grains ( $v_{0} / 80 \geq v_{z} > v_{0} / 320$ , solid line). (b) The number of grains $N$ in the fastest bin also decreases at $\tilde{t} = 1$ , while the number of slow grains grows. (c) The average upward momentum carried per grain decreases throughout the intrusion and collapses onto a $v_{0}$ -independent trend (the line indicates $1 / \tilde{t}$ ) for $\tilde{t} > 1$ . (d) Drag force on the rod plotted against the mass of all grains moving with an upward velocity of at least $v_{z} > v_{0} / 320$ multiplied by the friction coefficient. A $1 ∶ 1$ line is plotted in black.
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