Tag: sand

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The trajectories of pointy intruders in sand

Original article: Collisional model of energy dissipation in three-dimensional granular impact

An alien spaceship commander was preparing to drop a cone-shaped spy shuttle into the sand of a Florida beach near Cape Canaveral. The shuttle needed to burrow deep enough that any passing humans wouldn’t see it while the aliens used it to spy on Earth’s space program. “From how high should I drop the shuttle so that it is hidden?” the commander asked their science advisor. The science advisor pulled out their alien high school mechanics book, hoping to calculate this based on the laws of motion and Earth’s gravitational force.

Not so fast, alien science advisor! While the mechanics of a falling shuttle are relatively simple, the forces the shuttle would experience while penetrating the sand are much more complicated. Sand and other granular materials are composed of millions of individual solid particles that, together, may be stiff like a solid or flow like a fluid. This interstellar scientist first needed to know how sand particles interact with one another and how the uneven distribution of forces between them before dropping the cone-shaped probe.

In “Collisional model of energy dissipation in three-dimensional granular impact”, C. S. Bester and R. P. Behringer asked a similar question. In their study, they looked at the forces that a conical object (such as the alien commander’s spy shuttle) experienced as it penetrated a granular material and investigated how these forces affect the depth a conical object will burrow. 

Bester and Behringer dropped conical intruders into a container filled with sand with a thin rod attached to the top of the intruder for tracking.  They filmed each falling intruder from the side with a high speed camera, from which they determined the depth z, velocity, and acceleration as the intruder penetrated the sand. For a video of the experiment, see here. They used the seven intruders shown in Figure 1 . The intruders all had the same mass m but different shapes. The sharpness of the intruder’s cone-shaped tip was characterized by a parameter $latex s= \frac{2 L_{tip} }{w}$, the ratio of its length to half its width. A higher value of s corresponded to a sharper cone. They were dropped from a range of heights between 6 cm and 2 m, which resulted in intruders reaching different speeds upon impact with the sand.

An image of seven intruders used in the experiment, ranging from a blunt intruder to an intruder with a conical top.
Figure 1: intruders of equal mass used in the experiment, from the bluntest (s = 0) to the sharpest (s = 2.1). Image adapted from original article.

Bester and Behringer measured the stopping depth zstop and the time to stop tstop as a function of the initial kinetic energy each intruder had upon hitting the sand, $latex K_i = \frac{1} {2}m z_i$. They found that sharp intruders penetrated deeper into the sand than blunt intruders with the same kinetic energy Ki, as shown in Figure 2a. Figure 2b shows that, above an initial kinetic energy of 1 J, the time the intruders took to stop was the same regardless of shape or initial energy. 

Graphs of stopping depth as a function of kinetic energy and stopping time as a function of kinetic energy.
Figure 2: (a) Stopping depth as a function of kinetic energy. (b) Stopping time as a function of kinetic energy. Blue represents blunt intruders while red represents sharp intruders. Figure adapted from original article.

To understand what forces the intruder experiences as it comes to a stop, the authors focused on the inertial drag, or the drag caused by the pressure of the sand on the intruder.  Previous studies hypothesized that the inertial drag depended on the penetration depth and was proportional to the velocity squared of the intruder as it enters the granular material.  Bester and Behringer found that this was not the whole story. They calculated the inertial drag coefficient h(z) from the intruder trajectories, as shown in Figure 3a. Surprisingly, they found that the drag coefficient oscillated as the intruder penetrated the material. This suggested that the inertial drag was caused by collisions of the intruder with particles that are part of “force chains”. Force chains in a granular material are made up of connected particles that bear the majority of the forces in the material (see this earlier Softbites post for a detailed description). When the intruder hit a force chain, the drag increased due to the added resistance. The drag then decreased again when the chain was broken.

To investigate how the drag force was affected by the shape of the intruder, Bester and Behringer used the sum of the drag coefficient as the intruder penetrated the sand $latex \int {h(z) dz}$ [1]. Blunt intruders had a drag that increased nearly linearly with depth, while the dependence of drag on depth was much more curved for sharp intruders, as seen in Figure 3b. The authors suggested that the nonlinear drag for sharper cones was caused by the changing surface area interacting with the grains. Upon impact, a sharp cone only interacted with the sand through the tip. As it sunk, the area that was in contact with the sand increased nonlinearly, which resulted in larger drag.  

Plots of the drag coefficient vs. height, showing a bumpy curve, and the drag dependence on the depth of different intruders, showing that the drag experienced by sharper intruders has a nonlinear relationship with depth.
Figure 3: (a) Drag coefficient h as a function of height z for an intruder shows fluctuations. (b) Drag dependence on depth for different intruders. The drag of blunt intruders has a roughly linear relationship with depth (blue curves) while that of sharp intruders has a nonlinear relationship with depth (yellow and red curves). Figure adapted from original article.

Bester and Behringer’s investigation into how the shape of a conical intruder falling into sand affects the forces it experiences is a beautiful example of how complex the interactions of everyday materials can be. According to their work, the aliens in our introduction should drop a pointy probe from very high up to make sure it gets buried — and also put some sensors on their probe to measure how its descent is interrupted by the force chains in the sand.  The aliens may have imaginary science fiction technology that allows them to traverse light years, but even they may marvel at the countless collisions that affect the path of something they drop on the beach once they reach the Earth.

[1] The sum of the drag coefficient  ($latex \int {h(z) dz}$) was calculated from the measured kinetic energy, and then the derivative of it was taken to obtain the drag coefficient h. Taking the derivative amplified the noise in the measurement.  Bester and Behringer compared the sum $latex \int {h(z) dz}$. for different intruders to avoid this amplified noise.

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Seeing Inside Sand: Visualizing Force Chains with Photoelastic Disks

Original Article: Contact force measurements and stress-induced anisotropy in granular materials

As their name suggests, so-called “granular materials” are made up of “grains” — small (but macroscopic) pieces of sand, glass beads, coffee grounds, or almost any other solid you can think of. Granular materials can flow like a liquid (like sand in an hourglass), resist deformation like a solid (like the sand under your feet at the beach), or quickly transition between these states (like pebbles in a rockslide).

Granular materials have properties that have no equivalent in regular materials like wood, metal, or rubber. In solids like these — the kind we learn about in materials science class — a force applied to the surface propagates through the material smoothly and predictably. If a uniform force is applied to the surface of a material, every equally sized cross-section of that material bears the same amount of load. In granular materials, however, the situation is very different: in a sand pile under stress (that is, when a force is applied to its surface), the force is distributed unevenly — some individual sand grains bear far more load than others. Surprisingly, this remains true even when the sand grains themselves are identical. What’s more, the load-bearing grains connect to one another to make a fractal, lightning-like pattern inside the material, like that shown in Figure 1. These string-like arrangements of load-bearing grains are called force chains.

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Figure 1 – Force chains in a computer simulation of a sand pile. The thickness of a black line indicates the magnitude of the force at that point inside the sand pile. (From Nadukuru & Michalowski (2012).)

As Figure 1 shows, force chains are easy to identify in a computer simulation. But can you “see” forces inside a real material? Today’s paper — which is from 2005 but has already proven to be a classic in the field — shows us how it can be done. The secret lies in a clever choice of “grain”: in their experiments, Majmudar and Behringer use about 2,500 transparent plastic disks, each about a centimeter in diameter and half a centimeter tall. These disks are placed in a thin container that confines them to a single plane — this experiment is similar to the board game Connect Four, but without the vertical rails.

Crucially, these plastic disks have a property called photoelasticity: when they are stretched or squeezed, they deform, and when they deform they alter the polarization state of light passing through them. For instance, linearly polarized light might be converted into circularly polarized light, or light that’s still linearly polarized, but along a different axis than before. Thus, placed between crossed (perpendicularly oriented) polarizers, an unstressed disk will appear dark, but a squeezed or stretched disk will appear bright, since any alteration of the polarization state of the incoming light will allow some of it to pass through the second polarizer. What’s more, the pattern of light — like that shown in Figure 2 — can be used to infer the normal and tangential forces acting on each disk.

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Figure 2 – Two plastic disks placed between crossed polarizers [1]. For an unstressed disk (top), the polarization state of the light is unaltered, and no light gets through the second polarizer. For a disk under load (bottom), the polarization state of transmitted light is altered — in this case, the polarization axis is rotated — allowing some light to pass through the second polarizer. The forces on the disk, indicated by thick black arrows, can be inferred from images such as the one on the bottom right
By imaging lots of disks at the same time, photoelasticity can be used to infer the overall stress pattern inside a granular material. Majmudar and Behringer are especially interested in two particularly simple situations: isotropic compression and shear. Under isotropic compression, the collection of disks is squeezed equally from all sides, while under shear, the collection of disks is squeezed on top and bottom, but allowed to expand by an exactly equal amount at the sides.

Interestingly, the system responds very differently to these two types of load: for isotropic compression, the force pattern, shown in the left panel of Figure 3, resembles a random network — short chains of highly stressed disks connect over distances of a few diameters. For shear (Figure 3, right panel), the situation is very different: long force chains, tens of disk diameters in length, extend in the direction along which the system is being squeezed. This phenomenon, where an applied stress causes the material itself to change in a direction-dependent manner, is called stress-induced anisotropy; it is not captured by the linear elasticity theory that students typically learn, even in advanced material science classes.

In the decades since this paper was published, the techniques pioneered by Majmudar and Behringer have allowed scientists to better understand properties of granular materials: under what circumstances force chains form, how they depend on properties of the disk such as shape and friction coefficient, and how they influence behaviors such as jamming – the rapid transition from a flowing state to a rigid one.

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Figure 3 – A granular material under isotropic compression (left), and shear (right). In the sheared system, long, oriented force chains are clearly visible.

 

Postscript: On the day of publication, we learned of the recent death of the PI of this paper, Bob Behringer, at the age of 69. This post highlights just one of the many contributions of this widely respected scientist to the field of soft matter physics. For a more detailed overview of Behringer’s life and work, see here.

 

Notes:

[1] The experiment described in the paper used crossed (oppositely oriented) circular polarizers rather than the linear ones shown here, but the principle is the same.