The New Math of Wrinkling Patterns - Quanta Magazine

A few minutes into a 2018 talk at the University of Michigan, Ian Tobasco picked up a large piece of paper and crumpled it into a seemingly disordered ball of chaos.

Tobasco has been working on this for years, studying equations that describe thin elastic materials — stuff that responds to a deformation by trying to spring back to its original shape.

In January 2021, Tobasco published a paper that answered that question in the affirmative — at least in the case of a smooth, curved, elastic sheet pressed into flatness (a situation that offers a clear way to explore the question).

And he co-wrote a paper, published last month, that shows a new physical theory, grounded in rigorous mathematics, that could predict patterns in realistic scenarios.

“The role of the math here was not to prove a conjecture that physicists had already made,” said Robert Kohn, a mathematician at New York University’s Courant Institute, and Tobasco’s graduate school adviser, “but rather to provide a theory where there was previously no systematic understanding.”.

The goal of developing a theory of wrinkles and elastic patterns is an old one.

“This has been going on for about the last 10 years, where we’re doing experiments first and then trying to find the theory to understand them,” said the mathematician Dominic Vella of the University of Oxford.

Ian Tobasco has developed a theory that mathematically describes the kinds of wrinkles that emerge when curved surfaces are pressed flat.

Then in 2017, Vella led the analysis of the wrinkling instabilities of a thin elastic film under pressure, characterizing how the number of wrinkles changed according to the depth of the initial poke and other specific details.

“They showed that you could get these random disordered wrinkle patterns — when you did the experiment twice, you got two different patterns,” said Tobasco, who’s now an assistant professor at the University of Illinois, Chicago.

That is, as the thin film settles on a flat surface, it morphs until it finds the arrangement of wrinkles, disordered or not, that takes the least amount of energy to maintain.

“You can organize patterns by the amount of energy that’s stored when [the pattern] manifests,” Tobasco said.

They had suspected that Gaussian curvature was important to wrinkling, but Vella said it was a surprise that the domains depended so heavily on the sign.

What was more, Tobasco’s theory also applies to a broad spectrum of elastic materials, not just Paulsen’s forms.

“It’s a nice geometric construction that shows where wrinkles will appear,” Vella said.

In early 2018, Tobasco had his theory mostly settled — but even though it worked on paper, he couldn’t be sure it would be accurate in the real world.

“He’s a mathematician and making all these things non-dimensional,” Paulsen said, referring to how Tobasco’s ideas about curvature could be extended far beyond two-dimensional flat materials.

For the next two years, the three researchers hashed out the details, showing that Tobasco’s theory really did predict — exactly — the arrangement of wrinkles that Paulsen saw in his experiments and Katifori found in her computer models

On August 25, they published a paper in Nature Physics showing how the three approaches all converge on the same, straightforward geometric arrangement of wrinkles

“It’s very focused on situations where [Gaussian curvature] is either positive or negative, but there are a lot of situations with some regions that are positive and some negative,” Vella said

But Paulsen said the theory, even as it currently stands, is beautiful and surprising