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Intrusion Rheology

September 19, 2016

One advantage of living in most areas of the Northeastern United States is the proximity of ocean beaches. A common activity of beach-goers, both children and adults, is building sandcastles and other sand sculptures. These structures are temporary, since they are soon washed away by tides and waves. "Building castles in the sand" has become an expression of futile activity.


My sandcastles never looked like this!

Sand, as most other granular materials, will form a paste with the addition of a little water. The slight adhesion offered by the water leads to a glassy structure in which the particles are jammed into each other, becoming immobile, so the material acts as a solid.

(Via Wikimedia Commons.)

Building sandcastles isn't the only futile activity associated with sand. "Go pound sand" is an expression that seems to have originated in the Midwestern United States and is not common outside the US. Pounding sand is a futile task, since the sand is unchanged after all that work. An early use of this phrase is documented to 1857,[1] and it's been used in the present century.[2]

Dry sand will flow through a funnel, but damp sand will not, so this granular material has an unusual rheology. Rheology, from the Greek words, ρεω (rheo, flow) and λογια (logia, the study of), is the study of flow. The motto of The Society of Rheology is Παντα Ρει (Panta Rei, Everything Flows).

Logo of The Society of Rheology

Logo of The Society of Rheology.

(Society of Rheology image.)

The way that granular materials flow is important also when they must flow around an object, as when a shovel is pushed into sand. The force you need to exert to push into the granular material is called the resistive force, and that's the topic of a recent paper by MIT engineers in the journal, Nature Materials.[3-5] They found that the resistive force on mechanical "intruders" pushed into granular materials can be described by the equations of the simple resistive force hypothesis.[3]

Resistive Force Theory (RFT) was developed in the 1950s to describe the motion of such intruders through viscous fluids, such a honey. It was found that this theory produced better answers for granular materials than for viscous fluids, and the scientific reason for this was unknown.[5] As Ken Kamrin, an associate professor of mechanical engineering at MIT and co-author of the study, remarks, the unreasonable effectiveness of RFT in describing granular materials was "somewhat like magic."[5]
"People observed this concept worked but didn't know why, and that’s really shaky ground for scientists - is it just a coincidence?... Now we can explain the backbone of the granular resistive force theory."[5]

As discovered by a continuum model developed in this investigation, the resistive force in granular materials comes from local frictional yielding.[3] This is the difference between resistive force in granular materials and viscous materials, and it also allows a generalization to pastes, gels and muds.[3] The model is based on Coulomb's yield criterion, a simple model that predicts when granular materials will flow.[5]

In the Coulomb model, flow will occur when the ratio of the shear stress to the pressure is greater than the friction coefficient.[5] Kamrin extended this equation to account for the voids that are formed when an intruder is pushed into the granular material. Kamrin and his co-author, Hesam Askari, did finite element simulations of an intruder plate moving through the material at different orientations, and these confirmed their extended Coulomb model.[5]

Sand intruder experiments at MIT

A square intruder plunging into coarse, salt-like grains (white), fine sand (blueish green), and more viscous (green) and pasty (blue) materials.

(MIT image by Felice Frankel.)

Their model's validity was checked with odd shaped intruders, such as a circle, and a diamond shape with excellent results.[5] The MIT researchers also did experiments on pastes, gels, and mud, and these showed the predictive power of their model.[5] Looking towards applications, Kamrin says that
"... Scaling relations can be developed to understand pros and cons of different vehicle running gear and animal appendages, like, how do large and small tires compare? How do flipper-like feet versus long skinny feet compare? How do different body shapes affect sand-swimming performance?"[5]

Such knowledge would be useful for Martian rovers, and it could illuminate how lizards and worms burrow through earth.[5] Programmers might be interested in reviewing the Fortran source code in ref. 6.[6] This research was supported by the Army Research Office.[5]


  1. "Pound sand," The Word Detective.
  2. Citations for "pound sand," Wiktionary.
  3. Hesam Askari and Ken Kamrin, "Intrusion rheology in grains and other flowable materials," Nature Materials, Advanced Online Publication, August 29 2016, doi:10.1038/nmat4727.
  4. Supplementary information for ref. 3.
  5. Jennifer Chu, "Pushing through sand," MIT Press Release, August 29, 2016.
  6. Fortran source code associated with ref. 3.

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