The Physics of ... Wrinkles

Renaissance artists worked hard on their wrinkles. Rich garments cascading in complex folds over knee, elbow, and buttock evoked the bodies they concealed—and sometimes something more. “The restless activity of the drapery, like the quivering of the fingers and the rippling descent of the curls, carries a new kind of feverish emotion,” the art historian Frederick Hart once wrote about Andrea del Verrocchio’s sculpture of Doubting Thomas. Wrinkles in the Renaissance were frozen lines of energy, psychic as well as kinetic; getting the wrinkles right in a scene helped the artist get the energy right. Not incidentally, it also showed what an accomplished artist he was. Faces, fruit, and fabric are all played upon by similar forces. The wrinkles in skin or in an apple have short wavelengths, while the wrinkles in drapes, which reflect the substantial power of gravity pulling on them, are much longer and wider. Photographs by Jonathan Kantor Prescientifics that they were, however, the Renaissance masters did not fully understand the physical laws of wrinkling. “They were depicting reality, which is very different from trying to understand it,” says Lakshminarayanan Mahadevan, a physicist at Harvard University. Mahadevan thinks he has a handle on wrinkles—he and physicist Enrique Cerda at Cambridge University in England recently proposed a “general theory of wrinkling.” If you know the dimensions of the fabric and the forces acting on it, the theory predicts the amplitude and the wavelength of the resulting wrinkles—that is, how big and how far apart they will be. And it works not only for cloth, plastic wrap, and other fabrics but also for shriveled apples and people’s faces. A thin sheet of fabric or tissue, Mahadevan says, is like a spring: When you deform it, it stores elastic energy. The sheet can deform either by stretching or by bending. Being thin, it is typically less resistant to bending. The energy in a spring or in a stretched sheet is proportional to the strain from stretching squared, but the energy in a bent sheet is proportional to the curvature squared. A sheet with lots of little wrinkles contains more curvature and thus more bending energy than a sheet with one big wrinkle. It was the 18th-century Swiss mathematician Leonhard Euler who first realized this, and who also had the fundamental insight that underlies all wrinkle physics today: A deformed sheet adopts the shape that minimizes its total bending energy. One way to deform a thin sheet of paper, for example, is to compress it from all sides. Technically that’s called crumpling, and a team at the University of Chicago led by physicist Tom Witten has been studying the process for years. The energy you add to a sheet as you crumple it, they have found, causes it to buckle along a series of sharp ridges. Conversely, as you uncrumple the paper—or a candy wrapper—some of the elastic energy comes out again as a series of cracking sounds. In neither direction does the process go smoothly. As you crumple a piece of paper, it takes quantum jumps into new conformations, each one close to the lowest-energy state it can adopt under the applied force. The process can continue for a surprisingly long time. Witten and his colleague Sidney Nagel recently stuffed a sheet of Mylar eight inches in diameter into a tube four inches in diameter, then set a half-pound piston on top. After three weeks, the sheet was crumpled into a disk a tenth of an inch thick—and it still hadn’t reached a stable state. The reason crumpling takes so long, says Witten, is because the ridges are strong and put up a fight. Virtually all the elastic energy in the sheet is concentrated in the ridges. Eighty percent of the energy, Witten’s team has calculated, comes from the bending of the sheet, and the other 20 percent comes from stretching. Even a piece of paper stretches a bit as you crumple it. If it didn’t, its surfaces would all be perfectly flat and the ridges all perfectly sharp—not the lowest-energy state. “To avoid that, it does a trade-off between extreme bending and a little bit of stretching,” Witten says. That same trade-off is the essence of Mahadevan and Cerda’s theory of wrinkling—which Mahadevan, to distinguish it from brute crumpling, defines as “a roughly periodic arrangement of short-wavelength deformations of a thin sheet.” He and Cerda considered a simple model system: a thin sheet of polyethylene, longer than it is wide, clamped at both ends and stretched. What happens to the sheet, says Mahadevan, is “very bizarre: You’re pulling in one direction, and not only does the sheet get stretched in that direction, but in the other direction it actually gets compressed.” As the sheet narrows in the middle, the middle is deformed by a band of parallel wrinkles running lengthwise. You may have noticed this when you’ve tried to fold bedsheets with a partner. Why not just one wrinkle? If you hold a sheet of paper, one side in each hand, and then compress it by bringing your hands together, it will form one large bend; this minimizes the curvature and thus the bending energy. But the total energy of a deformed sheet includes stretching as well as bending energy; and because Mahadevan’s plastic sheet, unlike the paper, is clamped at both ends, it would have to stretch enormously in the middle to accommodate a large wrinkle. To minimize stretching, the sheet should make lots of tiny wrinkles. “Because you have these two competing effects, the optimal solution is a compromise,” Mahadevan says. That is, the sheet makes a medium amount of medium-size wrinkles. The wavelength and amplitude, Mahadevan and Cerda found, are proportional to the square root of the length of the sheet times its thickness. A clamped plastic sheet may sound like an idealized example, but in nature there are lots of thin sheets that are clamped—not at the ends, but to a foundation. Our skin, for instance, is attached to the underlying flesh. As we age, our skin begins to sag as the fatty tissue beneath it loses its stiffness. Mahadevan can’t predict exactly what your face will look like as you age, nor can he design a better Botox. “Our theory is only physics; it doesn’t include biology,” he says. But it does explain why wrinkles are so prominent in bony places like the face, hands, and knees: The skin is thin there and thus easy to bend, and the underlying fatty tissue is thin and thus hard to stretch. As a result the energy compromise favors minimal stretching, which means lots of highly visible, relatively long-wavelength wrinkles. Lately Mahadevan and Cerda have turned their attention to the physics of drapes, including the question of how cloth drapes a body. It’s a convoluted one. Whereas the polyethylene sheet in the model was merely bent and stretched, a drape is subject to gravity as well, so the energy trade-off becomes three-way. The sheet wants to hang flat to minimize its bending energy, unextended to minimize its stretching energy, and straight down to minimize its gravitational energy—except that it can’t penetrate the body it is draping, and it can’t do all that at once anyway. “The reason you see these fantastic patterns,” says Mahadevan, “is precisely because you’re trying to drape a curved surface with a flat sheet, which is impossible to do smoothly.” Confirming our everyday experience with dresses and suits, he and Cerda have shown that there is no single energy-minimizing solution: A given fabric can drape a given body in a range of stable states. The researchers’ equations describing that range just might be of use to artists today—at least to artists trying to animate realistically clothed characters for video games or cartoons. The Renaissance masters had it easy by comparison; their feverish and gloriously draped saints and deities didn’t have to move. Modern computer animation, Mahadevan suggests, might profit from a judicious application of cutting-edge physics. “I’m sure you’ve noticed that most of the characters [in video games] wear armor,” he says. “They never wear real clothes.”

The Physics of ... Wrinkles

A general theory of wrinkling puts your face in touch with the universe

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Robert Kunzig

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