Basile Audoly and Sebastien Neukirch at the Université Pierre et Marie Curie in Paris, France have written a wonderful little paper on the fracture mechanics of thin rods. It’s beautiful science – we’ve all seen that spaghetti will break into multiple pieces when you bend it enough, and tends to not break nicely in half like you’d expect. What most of us then don’t do is wonder precisely why this happens – and it turns out that if delve a bit deeper, we gain some very interesting insights.
First let’s go over simple fracture that we’re all familar with. A rod is bent more and more, and soon the curvature becomes too much. The rod then breaks along a weak point, the free end falls off, and the fixed end returns to where it was before. This is what we typically expect.
The critical thing to realize here is the curvature of the rod around the time of break. Immediately before the break, it has a defined, positive curvature. After the beam breaks, the end is free, and structural mechanics tells us that a free end must have zero curvature. We can then deduce that there’s a small period of time where the end of the rod must transition from some defined curvature to zero curvature.
The clever thing that Audoly and Neukirch did was ignore the first break in the spaghetti. Basically, they decided to model spaghetti not from the first break, but from the instant that break happens, and see how the spaghetti then behaves. How can you do this? Well, you can bend the spaghetti very close to the point where it would break, hold it there (with curvature defined at the end) and then let it go (where curvature must then become zero, like the above example). From the paper, “the release of a rod mimics the initial breaking event“.
Well, this all seems very well and good in theory, but what actually happens? Well, it turns out that if you bend spaghetti to a point where it doesn’t break, then let it go, it breaks into a bunch of pieces! This seems totally counterintuitive – if we didn’t curve it enough initially to break it, how can the curvature increase afterwards enough to cause it to break into multiple pieces?
It turns out that the transition from a certain defined curvature to zero curvature causes “waves” of curvature to ring through the spaghetti like waves in a waterbed when you sit on it. This doesn’t typically happen in large beams like we’re familiar with, only thin brittle rods like spaghetti. One would assume that it would be impossible for these curvature waves to have any value beyond the initial curvature – however, this is not the case! This is the critical finding.
We can see the numerical analysis above – demonstrating that for the case of initial semicircular curvature, the peak curvature wave amplitude generated is 1.428 times as great as the initial curvature.
As the spaghetti breaks along a weak point, it send out these waves of higher curvature – which then cause the spaghetti to break along another weak point, and the cycle continues. This breaking process is eventually slowed (or else we’d end up with spaghetti dust) by forces which reduce these waves over time such as energy dissipation in transverse cracks and viscoelastic effects.
So there you have it! Every time you make dinner, you can witness a highly nonintuitive example of structural dynamics. You can see some movies the authors have made here.