New physics equations explain universal laws of how things break, from glass to pasta

A fallen vase, a crushed sugar cube, and exploding bubbles all have something in common. A new mathematical equation reveals that they break down in a similar way.

French scientists recently discovered a mathematical formula that describes the size distribution of the pieces that form when something is crushed. According to a new study published Nov. 26 in the journal Physical Review Letters, this equation applies to a variety of materials, including solids, liquids, and gas bubbles.

Although cracks often propagate in objects in unpredictable ways, research has shown that the size distribution of the resulting debris appears to be consistent regardless of its material. You can always expect a certain ratio of large and small pieces. Scientists suspected that this consistency indicated something universal about the fragmentation process.

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Rather than focusing on how the debris forms, Emmanuel Villemaux, a physicist at the University of Aix-Marseille in France, studied the debris itself. In the new study, Villermaux argued that the fragmentation of objects follows the principle of “maximum randomness.” This principle suggests that the most likely fragmentation pattern is the one that is the most disordered, that is, the one that maximizes entropy or disorder.

However, that randomness must follow certain restrictions. To explain this, Villermaux introduced a conservation law he discovered with colleagues in 2015. This law adds physical constraints to the density of debris in space when an object is shattered.

By combining two principles, Villermaux derived a mathematical formula that describes the pattern of fragment sizes from crushed objects. He then tested the equation’s predictions by comparing it to years of debris data collected from a variety of objects, including glass, spaghetti, droplets, air bubbles, plastic debris in the ocean, and even fragments of early stone tools. All matched the predicted size distribution.

Villermaux also tested his equation by dropping heavy objects onto sugar cubes and observing how they fragmented. “It was a summer project with my daughters,” Villermaux told New Scientist. “I did this a long time ago when my kids were young, and then I came back to the data because it explained my point well.”

However, the newly discovered law does not always apply. It does not apply to situations without randomness, such as a smooth stream of liquid breaking into droplets of the same size. Also, conditions where debris interacts with each other, such as certain plastics, are not covered.

Ferenc Kun, a physicist at the University of Debrecen in Hungary, told New Scientist that understanding fragmentation could help scientists determine how energy is spent crushing ore in the mining industry, or how to prepare for rockfall.

Future research could include determining the smallest possible size of the fragments, Villermaux told New Scientist.

It’s also possible that the shapes of the various fragments have similar relationships, Kuhn writes in an accompanying viewpoint article.