B. Audoly, S. Neukirch
Laboratoire de Modélisation en Mécanique,
The physical process of fragmentation is relevant to many areas of science and technology. Because different physical phenomena are at work, the fragmentation of solid bodies has mainly been studied from a statistical viewpoint. Nevertheless a growing number of works have included physical considerations: surface energy contributions, nucleation and growth properties of the fracture process, elastic buckling, and stress wave propagation.
Here we focus on the quasi-static fragmentation of brittle elastic rods. Our main purpose is to understand why brittle rods (like dry spaghetti) break into many fragments and not simply in half when bent beyond their rupture curvature.
Experiment. To study the dynamics of the rod following the first breaking event, we introduce a catapult experiment: a rod is bent quasi-statically and then suddenly released at one end. Surprisingly, instead of smoothly relaxing to its straight configuration, the rod often breaks at a intermediate point in between the free end and the clamped end shortly after its release.
Elastic model. We used the Kirchhoff equations for elastic rods to study the dynamics of the rod in this catapult geometry. When released, the rod follows three regimes successively: (1) the released end quickly straightens up at short times, giving birth to a burst of flexural waves that (2) travel along the rod to the clamped end and (3) are amplified by reflexions on the opposite (clamped) edge.
This behavior is described analytically by a self-similar solution with no adjustable parameter, whose relevance has been confirmed by numerical integration of the dynamic Kirchhoff equations.
Comparing the simulations and experiments. Our main result is that the flexural waves which travel along the rod locally increase the curvature. Since the rod was initially in a strongly bent state, this increase is sufficient to break it. We have performed 25 catapult experiments and recorded the time and location of breakings (shown as white circles, triangles and diamonds of the following figure, corresponding to various pasta radii). We have also computed the time and location of curvature records in the dynamics of the released rod, as predicted by a simulation of the nonlinear Kirchhoff equations: for each value of the arc-length, the shaded areas in the graph show the times where the curvature is higher than any time before. The agreement between computations and experimental data is excellent as the experimental breaking events fall perfectly onto the predicted maximum records when rescaled variables are used. No adjustable parameter is used in the comparison.
Conclusion We have shown that releasing an elastic brittle rod from a bent configuration is sufficient to make it break. This counter-intuitive result explains why brittle rods break in several pieces when bent beyond their limit curvature: a first breaking occurs when the curvature exceeds its limit value in some place, after which, as described above, flexural waves travel along the two newly formed halves of the rod, where they locally increase the curvature further. This increase leads to new breakings that give rise to new travelling waves, and a cascade mechanism can take place.