Ploughing friction on wet and dry sand

R. W. Liefferink, B. Weber, and D. Bonn

Phys. Rev. E 98, 052903 – Published 6 November 2018

Abstract

The friction for sliding objects over partially water-saturated granular materials is investigated as a function of the water volume fraction. We find that ploughing friction is the main sliding mechanism: The slider leaves a deep trace in the sand after its passage. In line with previous research and everyday experience, we find that the friction force varies nonmonotonically with the water volume fraction. The addition of a small amount of water makes the friction force sharply drop, whereas too much added water causes the friction force to increase again. We present a ploughing model that quantitatively reproduces the nonmonotonic variation of the friction force as a function of water volume fraction without adjustable parameters. In this model, the yield stress of the water-sand mixture controls the depth to which the hemisphere sinks into the sand and the force that is required to plough through the water-sand mixture. We show that the model can also be used for other ploughing friction experiments, such as an ice skate that leaves a ploughing track on ice.

Received 27 August 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052903

Physics Subject Headings (PhySH)

Friction Hardness Wear

Granular materials Wet granular materials

Fluid DynamicsInterdisciplinary Physics

Authors & Affiliations

R. W. Liefferink¹, B. Weber^1,2, and D. Bonn¹

¹Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands
²Advanced Research Center for Nanolithography, Science Park 110, 1098 XG Amsterdam, Netherlands

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Issue

Vol. 98, Iss. 5 — November 2018

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Images

Figure 1
Schematic representation of the ploughing experiment. (a) Side view: Normal ( $N$ ) and frictional ( $F$ ) forces act on the sliding hemisphere. All experiments were performed using polydisperse ( $100 - 1000 μ m$ grains) ISO 679 standard sand, which is first dried in an oven and cooled down to room temperature. The sphere is pulled forward by a Zwick/Roell Z2.5 tensile tester, which imposes a constant displacement speed (4 mm/s) and measures the force $F$ required with a load cell (Z6FD1, precision of 5 mN). Subsequently, the final radius $r$ of the track drawn in the water-sand mixture perpendicular to the movement is measured. (b) Front view: the hemisphere with radius $R$ , where the ploughing cross section $A_{P}$ (red) can be calculated with use of the track radius $r$ . The cross section can be written as $A_{P} \approx \frac{4}{3} r d$ , with the depth of penetration $d \approx \frac{r^{2}}{2 R}$ if $d ≪ R$ . Consequently, the ploughing area is $A_{P} = \frac{2 r^{3}}{3 R}$ . Note that the assumption $d ≪ R$ results in a relative error in $A_{P}$ and $μ$ [with Eq. (2)] of less than $4.5 %$ for $r \leq 20 mm$ ( $\frac{r}{R} \leq 0.38$ ). However, for $r = 40 mm$ , the relative error increases to $21 %$ , which results in an underestimation of the friction coefficient for $ϕ_{w} = 0 %$ in Fig. 2. (c) Top view: the hemisphere with the projected area of contact $A_{c}$ (blue).
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Figure 2
Evolution of the friction coefficient $μ$ for the water volume fraction $ϕ_{w}$ . The friction coefficient obtained from the measured friction force and fixed normal force ( $N = 7.9 N$ ) in black circles display a nonmonotonic behavior for increasing water fraction. With use of the ploughing model, the friction coefficient can be modeled either based on the ploughing track radius $r$ [Eq. (2), red triangles] or based on the penetration hardness $P_{h}$ of the water-sand mixture [Eq. (3), dashed line]. The inset shows the friction force as a function of the normal force for various water volume fractions experimentally (squares) and predicted by the ploughing model based on the penetration hardness $P_{h}$ (dashed line).
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Figure 3
Ploughing radius $r$ as a function of water volume fraction $ϕ_{w}$ for a fixed normal force of $N = 7.9 N$ . Like the friction coefficient, the ploughing track width (black circles) evolves nonmonotonically with the water content, also shown based on the ploughing model (dashed line, $r = \sqrt{\frac{N}{\frac{1}{2} π P_{h} (ϕ_{w})}}$ ). The inset shows images of the experiments for $ϕ_{w} = 0 %$ , 8%, and $25 %$ with red lines highlighting the ploughing tracks.
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Figure 4
Penetration hardness $P_{h}$ as a function of water volume fraction $ϕ_{w}$ . The contact pressure $P_{h} = N / A_{c}$ , where the projected contact area $A_{c} = \frac{1}{2} π r^{2}$ is obtained from the ploughing track radius $r$ . With use of varying dead weight, the normal force is varied from 2.5 up to $16 N$ for each water-sand mixture and subsequently the track radius is measured, which enables one to calculate an average penetration hardness. The error bars represent the standard deviation. The solid line describes the data based on a first-order increase due to liquid bridge formation and an exponential decrease based on the coalescence of these liquid bridges.
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Figure 5
Ploughing radius $r$ as a function of normal force $N$ for various water volume fractions. The squares represents the experimental data and the dashed lines reflects the model ( $r = \sqrt{{\frac{N}{\frac{1}{2} π P_{h} (ϕ_{w})}}^{}}$ ) based on the penetration hardness. The radius increases with the normal force and has a minimum for $ϕ_{w} = 8 %$ . Note that the nonmonotonic behavior of the track radius for increasing water content holds over the full domain of normal forces.
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Figure 6
Master curve which shows the collapse of ploughing through sand and ice (data from [27]) when the ploughing friction is plotted versus the relative penetration hardness with use of rescaling based on the ploughing model (3). The friction coefficient for ploughing through sand is given for $P_{h max} = 76.7 kPa$ and $N = 7.9 N$ . Weber et al. [27] obtained the steel-on-ice friction coefficient as a function of temperature, where the penetration hardness decreases linearly with increasing temperature. The high-temperature regime ( $T = [- 21, 0]^{\circ} C$ ), where the plastic ploughing dominates, is replotted with use of the sphere radius $R = 2.38 mm$ , the normal force $N = 1 N$ , the maximum penetration hardness $P_{h max} = 233.1 MPa$ , and a surface friction contribution of $μ_{0} = 0.01$ .
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