The physicists also created an “odd ball” that always bounces to one side and an “odd wall” that controls where it absorbs energy from an impact. The objects all stem from the same equation describing an asymmetric relationship between stretching and squashing motions that the researchers identified two years ago.
“These are indeed behaviors you would not expect,” said Auke Ijspeert, a bioroboticist at the Swiss Federal Institute of Technology Lausanne. Coulais and Vitelli declined to comment while their latest paper is under peer review.
In addition to guiding the design of more robust robots, the new research may prompt insights into the physics of living systems and inspire the development of novel materials.
Odd Matter
The odd wheel grew out of Coulais and Vitelli’s past work on the physics of “active matter” — an umbrella term for systems whose constituent parts consume energy from the environment, such as swarms of bacteria, flocks of birds and certain artificial materials. The energy supply engenders rich behavior, but it also leads to instabilities that make active matter difficult to control.
Physicists have historically focused on systems that conserve energy, which must obey principles of reciprocity: If there’s a way for such a system to gain energy by moving from A to B, any process that takes the system from B back to A must cost an equal amount of energy. But with a constant influx of energy from within, this constraint no longer applies.
In a 2020 paper in Nature Physics, Vitelli and several collaborators began to investigate active solids with nonreciprocal mechanical properties. They developed a theoretical framework in which nonreciprocity manifested in the relationships between different kinds of stretching and squashing motions. “That to me was just a beautiful mathematical framework,” said Nikta Fakhri, a biophysicist at the Massachusetts Institute of Technology.
Suppose you squash one side of a solid, causing it to bulge outward in a perpendicular direction. You can also stretch and squash it along an axis rotated by 45 degrees, distorting it into a diamond shape. In an ordinary, passive solid, these two modes are independent; deforming the solid in one direction does not deform it along either diagonal.
In an active solid, the researchers showed that the two modes can instead have a nonreciprocal coupling: Squashing the solid in one direction will also squash it along the axis rotated by 45 degrees, but squashing along this diagonal will stretch it, not squash it, along the original axis. Mathematically, the number describing the coupling between these two modes is positive going one way and negative going the other way. Because of the sign difference, the physicists call the phenomenon “odd elasticity.”
In an odd elastic solid, undoing a deformation isn’t as simple as reversing the stretching and squashing motions that produced it; instead, the cycle of deformations that returns the solid to its starting configuration can leave it with some excess energy. This has striking consequences, such as enabling uphill locomotion of the odd wheel.
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