Physicists have worked out a universal law for how objects shatter

A dropped plate, a broken sugar cube, and a broken glass all seem to follow the same law of physics in terms of how many fragments of a given size they will break.

For several decades, researchers have known that there is something universal about the process of fragmentation, when an object breaks into several parts when it falls or crashes. If you counted the number of existing fragments for each possible size and made a graph of that distribution, it would have the same shape regardless of which object broke. Emmanuel Villermaux of Aix-Marseille University in France has now derived an equation that explains this shape, thereby formulating a universal law for how objects break.

Instead of focusing on the details of how cracks appear in an object before it fragments, he took a more zoomed-in approach. Villermaux considered all possible sets of fragments into which an object could break. Some sets would include very specific outcomes, such as a vase breaking into four equal pieces. He selected the most likely set, the one with the highest entropy, which captured the disordered and irregular breaks. This is similar to how many laws regarding large ensembles of particles were derived in the 19^th century, he said. Additionally, Villermaux used a law of physics that describes changes in the total density of fragments when the object breaks, something he and his colleagues had already discovered.

Together, these two ingredients allowed him to derive a simple equation predicting how many fragments of each size a breakable object should produce. To see how well it worked, Villermaux compared it to a whole series of past experiments involving breaking glass bars, dry spaghetti, plates, ceramic tubes and even plastic fragments in the ocean and waves crashing on rough seas. Overall, the way fragmentation appears in each of these scenarios follows its new law, capturing the ubiquitous graphical form that researchers had seen before.

He also carried out a series of experiments in which he broke a sugar cube by dropping an object on it from different heights. “It was a summer project with my daughters. I did it a long time ago, when my children were still young, then I came back to the data, because it illustrated my point well,” says Villermaux. The equation doesn’t work in cases where there is no randomness and the fragmentation process is too regular, such as when a jet of liquid breaks into several uniformly sized droplets following the deterministic laws of fluid physics, and in some cases where the fragments interact with each other as they burst, he says.

Ferenc Kun, of the University of Debrecen in Hungary, says that because the graphic form explained by Villermaux’s analysis is so ubiquitous, it is not surprising that it arises from a larger principle. At the same time, it’s amazing how well this principle works to a large extent and how it can be modified in some cases where there are additional stresses, such as in the case of plastic where cracks can sometimes “heal”, he says.

Fragmentation is not just an interesting physics problem. Understanding this better could have real implications for how energy is spent breaking ore in industrial mines, for example, or how we prepare for rockfalls that increasingly occur in mountainous regions as global temperatures rise, Kun says.

In the future, Kun says it might be interesting to consider the distribution of not only the size of the fragments, but also their shapes. Additionally, it remains an open question as to what the smallest possible size of a fragment could be, Villermaux explains.