The problem of sand

2009 November 22
by Daniel Lakeland

I'm working with a model for the dynamics of sand that is a little like Renormalization Group theory, a little like Statistical Mechanics, and a little like Lagrangian mechanics. Imagine the following:

You have a sample of sand, and it's got variable sized grains. Perhaps like this picture:

Sand grains at geology.com

Sand grains at geology.com

These grains of sand are tightly packed, irregularly shaped, and made of a variety of different materials. Seemingly nothing like the ideal spheres used in Discrete Element Models.

We have no hope of describing the mechanics of any particular sample of sand in detail. Doing so would require a model for the shape, stiffness, and surface properties of every grain of sand. We don't have this information available even if we were willing to put it all into the computer and run the simulation for a couple of years.

The traditional method of dealing with missing information in Physics is called Statistical Mechanics. Traditional Statistical Mechanics is interested in atomic level motion. So called "heat". For example, molecules of water are essentially identical, there is no surface roughness or material differences between them. Furthermore the interactions between molecules are more smooth. Sand grains have essentially no interaction until they come into contact, and then they have a very stiff interaction that depends on the shape in the local region of contact. Molecules have medium range interactions that can extend over several molecular diameters via Coulomb or Pauli interactions. Molecules do not "suddenly" come into contact.

So what can we do? My intuition is telling me to take a sample of sand, and create a model for it. This model will be what geometers call a smooth manifold. At every point in time, space is filled with not grains of sand, but statistical quantities that describe these grains of sand. We can think of these statistical quantities as the averages over all the grain-grain interactions in a certain region around the point of interest. But we don't know those interactions, so calculating the statistical quantities goes backwards. If I go to a High School, I don't know the SAT score of every student, but based on my knowledge of the statistics of SATs, I can still predict the average SAT score of a given class. Statistical averages are much easier to predict than individual scores, and the change in the averages due to things happening are easier to predict than the change in individual scores.

So now that I have space filled with statistics, what do I do? I need a way to describe how those statistics change with time so that I can figure out whether some of those statistics imply engineering failure???

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