Buckling of a rod penetrating into granular media

A. Seguin and P. Gondret

Phys. Rev. E 98, 012906 – Published 23 July 2018

Abstract

We investigate experimentally the possible buckling of a thin rod when penetrating downwards into a granular packing. When its bottom end reaches a specific depth, the rod may start buckling provided that the embrace is not enough to stop that phenomenon. The critical rod depth $z_{c}$ at buckling is observed to scale with the rod length $L$ either as $1 / L$ or $1 / L^{2}$ . These two scalings are shown to arise from the two resistant force terms that come into play during the rod penetration: a pressure force at the bottom of the rod that increases linearly with depth and a frictional force on the rod side that increases quadratically with depth. At the buckling point, the destabilizing force corresponds to the expected value given from conventional Euler's critical load for a rod bottom end considered as fixed in the granular clutch. Finally, we draw a buckling-nonbuckling phase diagram in a parameter space given by the rod aspect ratio and a rod-to-grain stress ratio.

Received 27 March 2018

DOI:https://doi.org/10.1103/PhysRevE.98.012906

Physics Subject Headings (PhySH)

Granular flows Granular materials

Polymers & Soft Matter

Authors & Affiliations

A. Seguin and P. Gondret

Laboratoire FAST, Univ. Paris-Sud, CNRS, Université Paris-Saclay, F-91405, Orsay, France

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Issue

Vol. 98, Iss. 1 — July 2018

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Images

Figure 1
(a)–(c) Images of the rod when penetrating into a granular packing (a) before and (c) after its buckling. (b) Experimental penetration force $F$ as a function of the penetration depth $z$ for a thin PVC rod of diameter $D = 4$ mm and length $L = 600$ mm into glass beads of diameter $d = 1$ mm. The rod buckling arises at $z_{c} = 0.13$ m for $F_{c} = 3.4$ N.
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Figure 2
(a) Penetration depth $z_{c}$ at buckling as a function of the penetration velocity $V$ . ( $\circ$ ) Data for a PVC rod of diameter $D = 4$ mm and length $L = 1000$ mm into glass beads of diameter $d = 1$ mm and (- - -) line of constant $z_{c} = 97$ mm. (b) Penetration depth $z_{c}$ as a function of the rod length $L$ at $V = 20$ mm/s for ( $\circ$ ) a $D = 4$ mm PVC rod, ( $▿$ ) a $D = 3$ mm PMMA rod, and ( $▹$ ) a $D = 3$ mm wood rod. (- - -) Power law fits $z_{c} = A / L$ and $z_{c} = B / L^{2}$ of the data with (o) $A = 0.14 \pm 0.01 m^{2}$ and $B = 0.09 \pm 0.01 m^{3}$ and ( $▿$ ) $A = 0.047 \pm 0.001 m^{2}$ and ( $▹$ ) $A = 0.060 \pm 0.001 m^{2}$ . The shaded region corresponding to $z_{c} > L$ is not allowed.
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Figure 3
(a) Critical buckling force $F_{c}$ as a function of the rod length $L$ for a $D = 4$ mm PVC rod into $d = 1$ mm glass beads. (- - -) Power law fit $F_{c} = A / L^{2}$ with $A = 1.5 \pm 0.1$ $N m^{2}$ . (b) Boundary condition parameter $ε$ as a function of the relative penetration depth at buckling $z_{c} / L$ for the same data set. (- - -) Line of constant $ε = 0.52$ .
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Figure 4
(a) Dimensionless vertical force $\tilde{F} = 4 F / π ϕ ρ g D^{3}$ on the rod as a function of the dimensionless rod penetration $z / D$ . Same experimental data as in Fig. 1 together with (—) the best fit by Eq. (3) with $C_{1} = 17$ and $C_{2} K μ = 0.85$ and (- - -) the two expected asymptotic scalings $\tilde{F} \sim z / D$ and $\tilde{F} \sim {(z / D)}^{2}$ at small and large $z / D$ , respectively. (b) Dimensionless rod penetration at buckling $z_{c} / D$ as a function of the stiffness parameter $E D / ρ g L^{2}$ for rods of different elastic modulus $E$ and diameter $D$ into $d = 1$ mm glass beads: (o) $D = 4$ mm in PVC, ( $▿$ ) $D = 3$ mm in PMMA, ( $▹$ ) $D = 3$ mm in wood, ( $⋄$ ) $D = 2$ mm in aluminum, ( $□$ ) $D = 5$ mm in PVC, ( $▵$ ) $D = 6$ mm in PMMA, and ( $⬠$ ) $D = 2$ mm in copper.
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Figure 5
Coefficients (a) $C_{1}$ and (b) $C_{2} K μ$ of the two force terms of Eqs. (1) and (2) as a function of the rod-to-grain size ratio $D / d$ . Same data symbols as in Fig. 4 with different colors for different grain sizes: $d = 0.3$ mm (green), 1 mm (black), 2 mm (blue), and 5 mm (red).
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Figure 6
Experimental observation of buckling (open symbols) or nonbuckling (filled symbols) in the parameter space given by the rod size aspect ratio $L / D$ and the rod-to-grain stress ratio $E / ρ g D$ for all the experiments in $d = 1$ mm grains [same data symbols as in Fig. 4]. (- - -) $L / D = α {(E / ρ g D)}^{1 / 4}$ inferred from the modeling Eqs. (1, 2, 3) with $z_{c} = L$ and $α = 1$ .
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