Sand tapping experiments and MCMC

2014 September 26
by Daniel Lakeland

It’s a well known phenomenon in granular materials that if you fill up a tube full of sand and then you tap the tube repeatedly, the sand will settle down to a certain stable height in the tube. Typically the variability between the “least dense” and “most dense” states is a few percent of the height. So for example you might start with 10cm of sand, tap it for a while and wind up with 9cm of sand. Note that it’s also possible though difficult to get your sand into a state where it actually expands as you tap it, but generally doing so requires you to crush the sand into the tube initially, when poured into the tube the sand will generally be less than or about equal to equilibrium density.

During my PhD I spent a lot of time thinking about how to model this process. One of the key issues is that we have essentially no information about the sand. For example the position, orientation, shape, and material properties (elasticity, surface/friction properties, etc) of the individual grains. It’s tempting to say that this is similar to the situation in the ideal gas where we have no idea where, how fast, or in what direction any of the atoms are. That’s true, in so far as it goes. But whereas in the ideal gas we have no interactions between the gas molecules, in the static sand condition we have essentially nothing but interactions between the sand grains. At first glance it seems hopeless to predict what will happen when what will happen is caused by interactions, and we have virtually no information about those interactions.

However, it does also depend on what you want to predict, and for someone interested in say soil liquefaction, the main thing to predict is how some disturbance such as a shear wave will affect the density of the soil, and in particular when that soil is saturated with water.

So consider a sand tapping experiment. We have a short-ish column of sand at uniform porosity $$\phi$$ (the fraction of the volume taken up by voids), and we tap this tube of sand with a blow from a hammer having kinetic energy $$dE$$ which is small compared to the total gravitational potential of the deposit relative to the bottom of the tube (you won’t be lifting the whole tube off the table and putting it into near earth orbit), but large compared to the gravitational potential of a single grain sitting at the top of the tube (you may very well bounce the grains sitting at the surface up a few millimeters), and given this energy, the sand grains bounce around a bit. Most of the sand grains will move not-very-far, you won’t have a grain go from the bottom of the tube to the top for example. The average center-of-mass distance traveled is likely to be considerably less than a typical grain diameter. However, the orientations of the grains may change by larger fractions, it wouldn’t be completely unheard of for a grain to rotate 180 degrees around some axis.

This tapping process is in many ways like the process of a random “proposal” in MCMC. It moves the grains to a nearby state, one in which the total energy is within about $$dE$$ of the initial energy. It makes sense to ask the question: “Given that the final state is somewhere in a very high dimensional state space which has energy within about $$dE$$ of the initial energy, what is the $$d\phi$$ that we’re likely to observe?”

It is, in general, hopeless to try to compute this from first principles for realistic sands, you might get somewhere doing it for idealized spherical beads or something like that, but it isn’t hopeless to try to observe what actually happens for some sample of sand, and then describe some kind of predictive model. In particular it seems like what we’d want is a kind of transition kernel:

\[P(\phi_1 | \phi_0, dE)\]

at least for $$\phi_0,dE$$ in a certain range.

So, while I didn’t get around to doing it in my PhD dissertation, I may very well need to go out and buy a bag of sand, a clear plastic tube, some kind of small hammer, and a bit of other hardware and have a go at collecting some data and seeing what I get.


2 Responses
  1. September 29, 2014

    I have an intuition that together, dimensionless analysis and maximum entropy reasoning could completely or nearly completely solve the problem. Alas, I have a day job.

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