Once-baffling success of granular resistive force theory explained

How is a Mars rover like a sandfish skink? Whether you’re a $2.5 billion robot carefully rolling across Martian soil or a 10-cm-long North African lizard (see figure 1) that burrows in the desert sand to evade predators, your locomotion obeys the same set of rules that govern the deformation of granular media.

For close to a decade, researchers have successfully applied an empirical scheme called resistive force theory (RFT) to describe locomotion in dry granular environments. First proposed in the 1950s to calculate speeds of sea-urchin spermatozoa swimming in seawater, ¹ RFT approximates locomotion in viscous fluids relatively well, but it’s far from perfect. For granular media, though, it works bafflingly well. ²

To figure out the secret behind granular RFT’s success, Ken Kamrin of MIT and his postdoc Hesam Askari (now at the University of Rochester) devised the simplest continuum-mechanics equations that could describe granular flow around an object that intrudes into a medium. They then fed those equations into numerical simulations. ³ Their investigation uncovered the scaling laws that determine RFT’s validity. Ultimately, their insight could save RFT practitioners time and extend the theory’s range to other media.

Swimming in sand

Granular RFT got its start in 2008 when Daniel Goldman of Georgia Tech in Atlanta noticed that sandfish wriggle through sand like nematode worms swim through fluids. Goldman’s PhD student Yang Ding (now at the Beijing Computational Science Research Center) suggested trying out RFT to model the lizard’s motion. They knew that RFT when applied to viscous fluids had limitations, but, says Goldman, “We were naive and young and just said, well, why not.”

In essence, RFT states that the forces on any small element of a moving body are independent of the motions and positions of other body elements. (See the Quick Study by Yang Ding, Chen Li, and Daniel Goldman, Physics Today, November 2013, page 68 .) Thus the net force on the body is simply a superposition of forces on infinitesimal elements. To apply RFT to a wriggling sandfish swimming at constant speed, one sets the net force to zero and calculates the speed consistent with that requirement.

In viscous fluids, RFT is an approximation of the well-established Stokes equations that determine stress and flow fields in fluids. So the forces on infinitesimal segments of a swimming nematode can, in principle, be calculated from scratch.

No such constitutive equations existed for sand and other granular media. So Goldman, Ding, and their colleagues fabricated stainless steel cylindrical rods with roughly the same frictional properties as sandfish skin to serve as proxies for sandfish body segments. They measured the resistive forces on the cylinders as they moved them through sand at different orientations and in different directions. They could then use the measurements as inputs for RFT calculations.

In 2010 Kamrin saw Goldman give a talk on the sandfish study and was intrigued. “I couldn’t believe how well it worked,” he says. By 2013 Goldman’s group had shown that RFT also works for insects running on sand, and Kamrin wanted to know why. “It seemed clear to me that this was a problem worth solving at the fundamental level.”

Back at MIT, Kamrin enlisted Askari to help develop a theory that can explain RFT’s success. A natural approach to take—and one that’s commonly applied to granular material—might have been the discrete element method (DEM), which can be used to simulate the motions of millions of individual grains. But the method is computationally intensive and, Kamrin and Askari explain, the variables are so numerous that it can be difficult to derive a general theory from the results. They instead opted to see if RFT emerges from a simple continuum theory. The only inputs to their continuum model were the medium’s density and internal friction coefficient.

Tests and analysis

Kamrin and Askari did check their model’s predictions against published DEM results. ⁴ For the case of a thin, flat plate intruding into granular material, their model reassuringly reproduced the flow patterns in the material that DEM simulations did.

They then used their model to put RFT through a battery of tests. For the flat-plate scenario, they used their model to create resistive force plots—maps of predicted force profiles as functions of the plate’s orientation and direction of motion. They compared their plots against those produced experimentally and found good agreement.

The two theorists also looked at intruders of different shapes. They tested a cylinder, a rectangular prism, and a V-shaped object. The match between RFT predictions and their continuum model was so good, Askari says, “it was getting a little scary.”

All that raises the question, Why does RFT work so well? Because of the simplicity of a square plate’s geometry, Kamrin and Askari could gain insight from dimensional analysis. From the equations built into their model, the analysis implied that the granular resistive force on a square plate of width L, illustrated in figure 2, must scale as L³.

Kamrin and Askari further reasoned that if RFT works, they could shrink the plate width from L to a much smaller value λ, and then calculate the resistive force on the L-sized plate by summing the forces on a collection of λ-sized plates. And indeed, for granular media, RFT produces the correct scaling. In fact, for a square plate, it predicts the force exactly.

However, if one does the same analysis for a plate moving through a viscous fluid, RFT doesn’t scale correctly. The Stokes equations establish that the force must be proportional to L, but the sum delivers a net force that scales as L².

Kamrin and Askari call their analysis the garden hoe test, after the square plate’s resemblance to the familiar tool. They envision applying it to predict how well RFT will perform for other types of media. For instance, the test shows that for certain types of gels, pastes, and muds—media that produce drag forces proportional to L²—RFT correctly predicts the forces.

In addition, the pair explains that the agreement between granular RFT and their continuum model means that the model can be used to generate inputs for RFT calculations. That could spare researchers the labor-intensive force measurements currently required. And RFT could be used for off-Earth or other environments that can’t easily be reproduced in the lab.