I’m going to follow in John and Mary’s footsteps and give a brief description of some work of mine that was published a little while ago by PRE. At 18 months after my defense, this allows me to squeak past John’s recent publication, which came a mere 17 months post-defense. I won’t get too comfortable, though, as John already has another one in the hopper (pun intended). Plus there’s the mysterious Wambaugh scaling paper, on which I’m rumored to be a co-author. I believe it’s something like the “true” director’s cut of Blade Runner: no one is really sure if it exists.
My work looked at response to a boundary force in granular systems, but let me start with the back story. One of the curious properties of granular systems is the so called “stress dip” at the bottom of a sandpile. If you pour grains onto a table, they form a pile. We (by which I mean other people) can measure the amount of force being applied on the table by the pile at each point under the pile. Intuitively, we would expect this force to be greatest at the middle of the pile, where there’s the most material overhead. This intuition is correct in, say, rubber, but in sandpiles the force exerted below the peak is often a local minimum. I say ‘often’ because it turns out to depend on how you made the sandpile; for other preparation procedures, the force actually is maximal under the peak. This is the sort of weird thing that keeps certain physicists up at night.
Eventually a few theorists came up with a model that explained the experimental measurements rather well. The only problem was that their model defied conventional elasticity theory, the 19th century theory of the reversible deformations of solids. If you ignore a material’s atomistic structure and treat it as a continuum, the equations describing the transmission of force through a static body are of a class called elliptical partial differential equations — think Laplace equation. The new model yielded hyperbolic pde’s — think wave equation. In other words, you can describe sandpiles if you postulate that they behave in a way that’s qualitatively different from conventional materials; indeed, in a way that’s seemingly incompatible with the tried-and-true theory of elasticity.
On some level, this should not be shocking. I mentioned that elasticity treats materials as a continuum. Granular systems are discrete on the scale of a grain diameter, many orders of magnitude bigger than atomistic solids. Perhaps the departure from elasticity has something to do with the discreteness. In fact, although the model I described above was postulated more or less ad hoc, later researchers were able to derive hyperbolic equations by starting from a microscopic picture and then coarse-graining. There’s a growing consensus that the discreteness of a granular material matters most when it’s closest to the “jamming transition” — the transition from solid to liquid-like behavior as density (not temperature) is tuned. Very much an open question is whether everyday granular materials, like a sandpile, are, practically speaking, near the jamming transition.
To see if the new equations were correct, experimentalists at Duke tested one of their simplest predictions. Think of a box filled with grains. What happens when you poke on one grain on the boundary of the material? Specifically, if you measure force at each point along the opposite wall, how is it distributed? This is called the “response”. The new hyperbolic equations say that there will be two peaks, while old fashioned elasticity says there will be just one. Sure enough, the physicists found two-peaked response, but only sometimes. And instead of happening when the material was closest to the jamming transition, it happened when the material was most ordered.
So, was this evidence that the new theory is right? I don’t think it’s particularly convincing, but it doesn’t rule it out. After all, elasticity doesn’t seem to explain the two-peaked result, either. Or can it? The point of our paper was to show that two-peaked response consistent with that seen in the experiment can happen in elastic materials.
It’s known that you can have two-peaked response when a material is anisotropic, like when it’s got crystalline order. And the Duke group found two-peaked response when they carefully arranged their granular packing in a triangular lattice. The thing is that triangular lattices are “not very” anisotropic; more specifically, conventional elasticity theory predicts that they show single-peaked response. Conventional elasticity theory is linear, which should describe the response accurately if, when you push on the boundary, you don’t push too hard. If you push harder, you need nonlinear elasticity theory. And within nonlinear elasticity theory, the difference between a triangular lattice and a disordered packing shows up — the two have different response. Specifically, the triangular lattice’s response becomes two peaked.
So one way to understand the Duke experiment is to imagine there is some sort of “enhanced” anisotropy, so that you see the effects of nonlinear elasticity in a situation where you wouldn’t normally expect it. Why? We suggested that it’s due to the fact that the material is discrete on the grain scale, and the measurements were made in a system that was only a couple dozen grain diameters deep. But, to be honest, that’s just a guess.
So now you have one set of experiments with two possible explanations. One says that you should get the observed response when you softly poke a system prepared in a fragile state. But the two-peaked response shows up when the packing is prepared in an ordered state, which isn’t necessarily fragile. Another theory says that the observed response should show up when the system is ordered, which sounds good. Only then you have to assume the system responds as if you’re poking it rather hard, which isn’t the case. Neither explanation is totally satisfactory. On the other hand, if you’re going to say that the experiments aren’t necessarily evidence of hyperbolic response, it’s nice to have an alternative explanation to point to.