Most materials – from rubber bands to steel beams – thin when stretched, but engineers can use the interlocking edges and precise folds of origami to reverse this trend and build devices that widen to as they are separated.
Researchers are increasingly using this type of technique, derived from the ancient art of origami, to design components for spacecraft, medical robots and antenna arrays. However, much of the work progressed by instinct and trial and error. Now researchers from Princeton Engineering and Georgia Tech have developed a general formula that analyzes how structures can be configured to thin, stay the same, or thicken when stretched, pushed, or bent.
Kon-Well Wang, a mechanical engineering professor at the University of Michigan who was not involved in the research, called the work “elegant and extremely intriguing.”
Wang said the paper “creates new tools and avenues for the technical community to leverage and pursue that will further elevate the functionality of origami and advanced metamaterials. The impact is enormous.”
In an article published August 3 in the Proceedings of the National Academy of Sciences, Paulino and colleagues outline their general rule of how a large origami class responds to stress. The rule applies to origami formed from parallelograms (such as a square, rhombus, or rectangle) of thin material. In their paper, the researchers use origami to explore how structures respond to certain types of mechanical stress – for example, how a rectangular sponge swells into a bow tie shape when pressed down the middle of its long sides. Of particular interest was the way the materials behave when stretched, like chewing gum that thins when pulled at both ends. The ratio of compression along one axis with stretching along the other is called Poisson’s ratio.
“Most materials have a positive Poisson’s ratio. If, for example, you take a rubber band and stretch it, it will get thinner and thinner before it breaks,” said engineering professor Glaucio Paulino. Margareta Engman Augustine at Princeton. “Cork has zero Poisson’s ratio, and that’s the only reason you can put the cork back in a wine bottle. Otherwise, you’d break the bottle.”
The researchers were able to write a set of equations to predict how the origami-inspired structures will behave under this kind of stress. They then used the equations to create origami structures with negative Poisson’s ratio – origami structures that expand instead of shrink when their ends are pulled, or structures that snap into a dome shape. when bent instead of collapsing in a saddle shape.
“With origami you can do that,” said Paulino, who is a professor of civil and environmental engineering and of the Institute of Materials at Princeton. “It’s an amazing effect of geometry.”
James McInerney, the study’s first author and a postdoctoral researcher at the University of Michigan, said the team created the equations to understand the property of symmetry in structures. Symmetry means something that remains the same under certain transformations. For example, if you rotate a square 180 degrees around an axis passing between the centers of two sides, its shape remains the same.
“Things that are symmetrical deform in the expected way under certain conditions,” McInerney said. By finding these symmetries in the origami, the researchers were able to create a system of equations that governed how the structure would respond to stress.
McInerney said the process was more complex than defining the symmetry rules because some of the folds resulted in deformations that violated the rules. He said that generally deformations made in the same plane as the paper (or thin folded material) obeyed the rules, and those out of plane broke the rules. “They broke symmetry, but they broke symmetry in a way that we could predict,” he said.
Zeb Rocklin, assistant professor of physics at the Georgia Tech School of Physics and co-author, said the origami exhibited fascinating and contradictory behavior.
“Usually, if you take a thin sheet or slab and pull it out, it will retract in the middle. If you take the same sheet and bend it upwards, it will usually form a Pringle – or saddle shape. Some materials thicken when you pull on them, and these always form domes rather than saddles. The amount of thinning always predicts the amount of flex,” he said. “The folding of these origami is exactly the opposite of all conventional materials. Why?”
Researchers have spent years trying to define rules governing different classes of origami, with different folding patterns and shapes. But Rocklin said the research team found the origami class didn’t matter. It was how the folds interacted that was key. To understand why the origami seemed to defy motion typically defined by Poisson’s ratio — expanding when pulled, for example — researchers needed to understand how the interaction affected the motion of the whole structure. When artists bend the sheet so that it moves along its plane—for example, waving it so it can expand and contract—they also introduce curvature that moves the sheet in a saddle shape.
“It’s a hidden mode that comes with the ride,” Rocklin said.
Rocklin said that by looking at this hidden connection, the researchers were able to explain “this weird mode of the leaf doing the opposite of what was expected.”
“And we have a symmetry of that which is why it does the exact opposite,” he said.
In the future, the researchers intend to continue their work by examining more complex systems.
“We’d like to try to validate that for different models, different setups; to make sense of the theory and validate it,” Paulino said. “For example, we need to study patterns like the blockfold pattern, which is quite intriguing.”