The rules of wrinkling are surprisingly hard to pin down. Using experiments, simulations and mathematical proofs, a team has uncovered how curved thin sheets of plastic will wrinkle when flattened

We now know the rules for wrinkling, at least in certain circumstances. Using experiments, simulations and mathematical proofs, researchers have worked out how very thin curved shapes wrinkle when flattened.

Pinning down the mathematics of wrinkling is surprisingly hard. As something wrinkles, its properties change and so mathematical models of the situation must constantly change too.

To try to better understand how the process works, Eleni Katifori at the University of Pennsylvania and her colleagues investigated how curved pieces of a plastic material thousands of times thinner than a human hair wrinkle. The team chose this scenario because thin materials wrinkle particularly well and they had the right instruments to study the situation precisely.

##### Advertisement

The researchers made the curved pieces of plastic by depositing a flat piece on a curved glass surface like a sphere or a saddle and then spinning it, which thinned it out and cause it to take the shape. They then placed the plastic patches on top of water, allowing water tension to provide a consistent flattening force which collapsed them into wrinkly sheets.

Originally, the team couldn’t predict any wrinkling patterns. Sometimes, there would be rows or ripples on some parts of the shape while others stayed smooth, and some wrinkle patterns repeated throughout experiments while others didn’t. Simulating the situation on a computer led to similar results, but not much more understanding, says Katifori.

Ian Tobasco at the University of Illinois at Chicago had already independently developed a mathematical theory based on the energy cost of certain patterns of wrinkles and he realised the team’s work was putting it to the test. “It’s amazing, as a mathematician, to find experimentalists that are testing the oftentimes wildly theoretical predictions that you’re making,” he says.

Through a back-and-forth between the experiments, simulations and mathematics, they turned the theory into wrinkling rules for the different shapes.

They found that whether wrinkles form in rows of ripples or only on the edges of the patch depends on the shape of its curve before being put on water. For instance, saddle-like shapes stay smoother at their centre than sphere-like ones.

The researchers also found they could predict where wrinkles would form by splitting the shape into smaller sub-shapes. Tobasco’s mathematics then provided rules for what kind of ripples would show up in each of those smaller sections.

Robert Kohn at New York University says that predicting wrinkling patterns on a thin sheet is mathematically so complex that it is surprising to see the researchers determine any rules at all.

Dominic Vella at the University of Oxford says that the next challenge is to consider a broader range of patch shapes, including those that have curvature different from spheres or saddles or even those that get torn or pierced as they wrinkle.

Journal reference: *Nature Physics*, DOI: 10.1038/s41567-022-01672-2

More on these topics: