From Crushed Sugar Cubes to Exploded Ceramics, This Universal Law Predicts How Most Objects Will Shatter
A new equation calculates how many fragments of each size will be produced when an object breaks. The principle could help people prepare for rockfalls or other real-world scenarios
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When a delicate object crashes onto the floor, most people expect it to shatter into several pieces. What they might not know, however, is that the sizes of those fragments—whether from a broken plate, mirror or sugar cube—seem to follow a universal rule.
Researchers have long known that a dropped or smashed object will break into a consistent ratio of fragment sizes. In other words, if you grouped the splintered pieces by size and graphed their distribution, most broken objects would yield the same graph shape, per New Scientist’s Karmela Padavic-Callaghan.
Now, Emmanuel Villermaux, a physicist at Aix-Marseille University in France, has come up with an equation to describe this consistent pattern of fragmentation, which works across many materials. His study was published on November 26 in the journal Physical Review Letters.
“The simplicity and success of this approach are striking,” writes Ferenc Kun, a physicist at the University of Debrecen in Hungary who was not involved in the study, in a Physics Magazine commentary about the new work.
When studying fragmentation, researchers usually look at either fine-scale mechanisms, such as how cracks take shape, or a broader, general principle. Villermaux took the latter approach. He examined the most likely fragmenting outcomes in situations with few physical constraints, or a principle of “maximal randomness.”
Simply put, the most likely shattering pattern that will take place when an object breaks is the one that maximizes disorder, resulting in the messiest, most irregular-size pieces, Villermaux suggests in the study. But even maximal randomness has some limits. It obeys the previously mentioned rule about the ratio of shattered pieces always following the same pattern, which Villermaux and his colleagues reported in a 2015 study in the journal Proceedings of the Royal Society A.
Combining these two elements allowed Villermaux to figure out an equation to predict how many fragments of each size will come from an object cracking open. He then tested how the equation held up against data from previous shattering studies, including those with glass rods, dry spaghetti, exploding ceramic tubes, liquid drops and waves breaking in turbulent seas. He also sized up the equation with data he previously collected on how individual sugar cubes splintered when dropping objects on them from different heights.
“That was a summer project with my daughters. I did this a long time ago when my children were still young and then came back to the data, because they were illustrating my point well,” Villermaux tells New Scientist.
The objects fragmented in accordance with the new equation, he found.
Fun fact: Dry spaghetti conundrum
In 2005, researchers reported why dry spaghetti noodles always break into three or more pieces—never cleanly in two. The work earned them an Ig Nobel Prize, an award for imaginative, silly-sounding research that could ignite people’s interest in science.
The new work “shows that the statistical regularities of fragmentation can emerge from a combination of maximum randomness and kinematic constraints, without reference to any specific microscopic mechanism,” Kun writes.
Still, the math doesn’t work in all circumstances. In materials that are both springy and flowy, like silly putty, the cracks may “heal” and close, preventing some pieces from forming, according to the study. And in a jet of liquid breaking up into droplets, the fragmentation process is too regular for the equation to be applied.
This universal law of shattering might be relevant for more than just predicting messes from human clumsiness. For instance, Kun tells New Scientist, better understanding how objects break apart could help determine energy costs of mining for ore, which involves smashing rocks, or aid in preparing for rockfalls.
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