When curved materials flatten, simple geometry can predict wrinkle patterns that appear

Nature Communications (2017). DOI: 10.1038 / ncomms15809″ width=”800″ height=”530″/>

Scaling the domain size. Typical equilibrium wrinkle patterns in a hexagonal section of a thin spherical shell on a liquid substrate. All simulation parameters are kept constant, except for plate thickness. The system’s Föppl – von Kármán number is indicated above each pattern. While the upper left pattern is clearly divided into six domains, increasing γ appears to decrease the typical domain size. The scale bar to the right of each scaling pattern shows the typical field sizes predicted by equation (20), down to a constant factor (only the ratios between the bars are meaningful because equation (20) only suggests a scaling of the typical field and leaves the initial factor unknown). attributed to him: Nature Communications (2017). DOI: 10.1038 / ncomms15809

An intrinsically flat object, for example a piece of paper, can be shaped into a cylinder without stretching or tearing. However, the same is not true for something intrinsically curved like contact lenses. When compressed between two flat surfaces or placed on water, curved objects will flatten, but with wrinkles that form when they contract.

Now, research from the University of Pennsylvania, the University of Illinois Chicago (UIC), and Syracuse University has shown that with some simple geometry, it is possible to predict the patterns of those wrinkles, both where they will form and in some cases their direction. The results are published in Nature Physicshas a range of implications, from how materials react to moisture and reflect sunlight in nature to the way a flexible electronic device might bend.

“The beauty of this work is how simple it really is,” says Eleni Katifori, associate professor in the Department of Physics and Astronomy at the University of Pennsylvania. “Beyond that is very complex, the physics translated by these rules that we found, but the rules themselves are very simple. They are inspiring.”

meeting of minds

Since I got my PhD. In his work, Catifori was interested in the mechanics of how thin films bend. Although this remains intriguing, her research path veered toward fluid flow networks instead. Then, while collaborating on a project with fellow University of Pennsylvania Randall Kamen and then postdoctoral fellow Hillel Aharoni, Katifori noticed something she couldn’t explain at the time. “That is, we noticed wrinkles forming in the spheres,” she says.

In other words, when a file curved surface It becomes flat, ending with excess material and subsequent wrinkles. These wrinkles appear in patterns or segments. The question arises, why do you arrange wrinkles in this way? Katifori says. “We didn’t really understand how important domains are in wrinkles.”

At a conference in 2016, mathematician Ian Tobasco, assistant professor at UIC, heard Aharoni talk about the topic. “This was the first time I saw the introduction of this modular system,” says Tobasco. “I thought it was really cool.” In mid-2017, Katifori, Aharoni, and colleagues published findings on the topic in Nature Communicationsthen in a workshop later that year, Tobasco met Joseph Poulsen of Syracuse, who provided preliminary data on his group’s experiment on wrinkles.

In early 2018, Tobasco began working in earnest on a Mathematical Theory For wrinkles, and during lunch at a conference that summer, Catifori, Tobasco, and Paulsen agreed that they shared an interest in the problem. They decided to collaborate, focusing on analyzing how important the material’s physical shape and the curvature you start with are for wrinkle patterns.

Work to solve the problem

For some background, the curvature can be positive, such as the roundness of a baseball or globe, or negative, such as a horse’s saddle or the spot on a glass bottle where the neck meets the base. There is also flat material, such as a piece of paper.

In this work, researchers focused on positively and negatively curved shells.

Then they removed from each of them the basic shapes, such as triangles, squares, and ovals. “Think of a cookie cutter. Suppose I take something that has a positive or negative curvature, then cut out one of those shapes and put it on the liquid,” says Kativori. Is it possible to guess wrinkle patterns and calculate the direction of wrinkle flow? For each figure, Tobasco will solve the theorem based on Basic principles He worked on it and published it, then came up with predictions.

Based on these results, Katifori and Benn, postdoctoral researcher, Deslava Todorova, ran simulations, inserting individual shapes and parameters into a computer program. Similar work was happening in the lab directed by Paulsen, an assistant professor of physics at Syracuse, through experiments he was running on polystyrene film 1,000 times thinner than a piece of paper. “It’s made of the same packing material as the packing peanuts, but instead of the 3-D shape of the packed peanuts, imagine it’s flat like a plate,” says Tobasco.

Through simulations, experimentation, and lots of back-and-forth steps to improve the process and expand on the original theory, the trio began to realize that by applying direct engineering principles they could figure out what pattern the wrinkles would take early on and a subset. – what they describe as “tidy” wrinkles – whatever direction they run.

engineering principles

To explain one of these principles, Katifori uses a pentagon. “First, I score a circle in the polygon,” she says. “The points where the circle touches the edges of the polygon determine where I draw my lines.” Pause to create a second shape inside the first, this shape with four uneven sides; Each line begins where the circle meets the outer polygon, and connects all four inner lines. “Now I have one field, two, three, four, five areas,” she continues, pointing to the five of the newly cordoned off sections.

For simple shapes like this, the outer sections will have neat creases, neat and orderly, following the direction of the inner lines drawn by Kativori. Inside the new inner polygon, wrinkles are still forming, but they remain turbulent and unpredictable.

Tobasco points to another example, one of which he identified was universally true for shapes cut from negatively curved shells. “In the end, wrinkle patterns are very easy to predict. All you have to do is draw line segments that meet the stroke at right angles.” In other words, start from a point inside the shape and create a straight line at the edge of the shape, but only at the point where the right angle is formed.

That took a year for the team to understand. “The equations that define the shape of wrinkles are horrific to solve, and many of the patterns we observed in our experiments and simulations are very complex,” Paulsen says. “But it turns out that under a certain set of conditions, you can predict wrinkle layout with a simple set of rules. This means we now have a fast and efficient way to model wrinkle patterns.”

“Its simplicity is beautiful and useful, too,” he adds, especially for wrinkled surfaces that perform a function such as allowing for sticking or fluid flow.

Catevory cites similar examples. “Let’s say there is moisture or moisture in the air. Water will behave differently in valleys and hills on an excavated surface,” she says. “By controlling the wrinkle pattern, you can probably influence how the water thickens.”

What will come next

Researchers still have more to understand about these complex surfaces, such as how to pull patterns out of chaos wrinkleswhy ordered and turbulent domains can coexist, and why there is a “mutuality” that connects curved shells negatively and positively, meaning that once the pattern of one is determined, it is easy to predict the pattern of the other.

However, for now, they say they are excited by the possibility of what they have learned up to this point.

“You have a complex theory that ultimately boils down to relatively simple math that just about anyone can do with a compass and ruler,” says Kativori. “It’s an elegant and beautiful solution to a complex problem.”


Using mathematical proofs, experiments, and simulations to show how a material wrinkles when it is flattened


more information:
Ian Tobasco et al., Precision Wrinkle Pattern Solutions for Confined Elastic Shells, Nature Physics (2022). DOI: 10.1038 / s41567-022-01672-2

the quote: When Curved Materials Flatten, Simple Geometry Can Predict Wrinkle Patterns That Appear (2022, September 11) Retrieved September 11, 2022 from

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