Dynamic Buckling and Breaking of Thin Rods

Josh Gladden,  Nestor Handzy,   Andrew Belmonte,   Emmanuel Villermaux

Popular science and media reports: Feature article in Science News (November 12, 2005 issue) An interview with CBC Toronto Radio (broadcast November 19, 2005)

    How do things break   -   when you smash them?

Pushing hard enough on something solid usually results in bending it or denting it - depending on what it is, what it's made of. Push even harder, and it may break! A late night experiment breaking pasta in the sink - after years of bending spaghetti and seeing small pieces fly across the room - led us to study how thin rods break in the lab. And since pasta is easy to get in the quantities we needed, why not start with that? But what we eventually found out applies equally to glass, steel, teflon... paper, plastic. A summary of this story is given on this webpage. We also published some of what we found.

    Let's Begin With Pasta...

If you hold a piece of spaghetti in your hands, you can easily bend it - and break it. Try it yourself. Not so easy to break in half! More on breaking by hand can be found here. We decided instead to try dropping heavy weights on the spaghetti - and it does some interesting things!!

The image on the right shows a sequence from our high speed video of spaghetti (San Giorgio #8, diameter 1.9 mm) as it buckles and breaks under the impact of a brass cylinder moving at 3.5 m/s. The images are separated by 250 microseconds. The first thing to notice is the sinusoidal perturbation in the second frame, which grows until the pasta fractures at - or near - the points of highest curvature. Using a pneumatic cannon, we can produce projectile speeds of up to 30 m/s so that the dynamic buckling wavelength (lambda) can be explored over a range of impact velocities. These experiments we performed for materials including teflon, pasta, glass, and stainless steel. The measured wavelength for each material is then plotted log-log as a function of impact speed (see below), where the solid line represents a slope of -1/2 indicating that the wavelength scales with the inverse square root of the impact speed U_o. Here is some of our data:

    What's The Wavelength?

This scaling is in fact encoded in classical elasticity theory. We start with the dynamical equation for the lateral displacement (zeta) of a straight elastic beam under an arbitrary applied force along its axis F(x,t).

A stability analysis results in a range of modes which are growing exponentially in time and we pick the fastest growing mode to be the one experimentally observed. Also obtained from this analysis is the characteristic buckling time (the inverse of the growth rate of the fastest growing mode). To compare with our experiments, we need the spatial distribution of the force in the rod due to the projectile as a function of time. The form of F(x,t) was first solved by Saint-Venant for a rigid projectile striking an infinitely long elastic rod. Letting x=0 be the impact end, this leads to

where c is the speed of sound in the material. After reflecting off of the far end (for times t > L/c), the stress profile becomes more complicated, but can be reasonably approximated by multiplying the stress by a factor of two - the reflected stress pulse is superimposed on the initial pulse and is of roughly the same size. We account for this by including a multiplying factor (gamma) to the force - equal to 1 before the reflection, 2 after one reflection, etc. In our experiments, the buckling time was never larger than 2L/c, thus never more than one reflection. Combining these results we obtain a scaling law for the dynamic buckling wavelength and buckling time for a circular rod of diameter d:

   and   

Note that what we mean by "buckling time" is actually the inverse of the growth rate (discussed below). The scaling law allows us to replot the data, and the solid line is the law itself, with no adjustable parameters:

    How Fast Does it Buckle?

We also used our high speed video to directly measure the growth rate of the preferred buckling mode. The figure on the right shows the buckling amplitude as a function of time, for two different lengths of spaghetti (L = 22 cm and L = 6 cm). For the long pieces, the stress pulse has not had time to reach the fixed end and reflect, thus gamma=1. However the incident pulse for the short pieces has reflected and traveled more than half way back up the rod, so gamma=2. The measured exponential growth confirms the stability analysis approach.

    Picking Up The Pieces

Our experiments on breaking spaghetti made it clear that there is a connection between the buckling wavelength and the length of the fragments. The typical fragment size is about 1/2 of the wavelength, since breaks seem to occur at points of highest curvature. To investigate this, we had to break alot of pasta: 300 pieces of Barilla angel hair (d=1.1mm), resulting in 1200 fragments, and 200 pieces of San Giorgio #8 spaghetti (d=1.9mm). The plots below show the length distributions (normalized by the wavelength). In both cases, there are two peaks just under the lambda/2 and lambda/4 lengths (indicated by the arrows). We attribute this downshift to secondary fragmentation in which smaller pieces break off the ends of the larger fragments.

The full reference for our published work: J. R. Gladden, N. Z. Nandzy, A. Belmonte, and E. Villermaux, Dynamic buckling and fragmentation in brittle rods.  Physical Review Letters 94, 35503 (2005);   download the PDF.

Here is a video (animated GIF) of this sequence (file size ~ 3MB).

A link to the related work by B. Audoly and S. Neukirch, Université de Paris VI.

An Intro to Buckling - Dept of Mechanical & Materials Engineering, Univ Western Australia (Douglas Wright)

Video of Richard Feynman and spaghetti breaking here.

A YouTube demo of breaking it in half (in Scotland??)

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