# Unfolding the secrets of origami structures

While most materials – from rubber bands to steel beams – thin out as they are stretched. The interlocking ridges and folds in origami structures behave differently, growing wider when pulled apart.

Researchers from Princeton University and Georgia Tech have developed a formula which can predict the behaviour of origami-inspired structures when they are stretched, pushed or bent. Their findings are published in the Proceedings of the National Academy of Sciences.

The structural properties of origami are increasingly used in the design of spacecraft componentsmedical robots and to improve solar cell efficiency. However much of this work has relied on trial and error, or instinct.

The researchers developed a set of equations which apply to the ways origami parallelograms (such as a square, rhombus or rectangle) made of thin material respond to certain kinds of mechanical stress.

They were particularly interested in the behaviour of materials and structures when stretched, like a stick of chewing gum which thins as it’s pulled at both ends. This property can be described by the “Poisson ratio” which is the ratio of compression along one axis, with stretching along the other.

“Most materials have a positive Poisson ratio”, this means they thin out when stretched, says paper co-author and Professor of Engineering Glaucio Paulino from Princeton.

“Cork has a zero Poisson ratio, and that is the only reason you can put the cork back in a wine bottle.”

Having developed a formula to predict how origami-inspired structures would behave under this kind of stress, they then used the equations to create origami structures with a negative Poisson ratio – able to grow wider instead of narrower when pulled, and structures which snapped into dome shapes when bent, instead of sagging into a saddle shape.

Zeb Rocklin, an assistant physics professor at Georgia Tech and a co-author of the paper, said that origami presented fascinating and contradictory behaviours.

“Usually, if you take a thin sheet or slab and you pull on it, 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 instead thicken when you pull on them, and those always form domes rather than saddles. The amount of thinning always predicts the amount of bending,” he says.

“The bending of these origami is exactly the opposite of all conventional materials,” Rocklin says.

Many researchers have spent years trying to define rules for different classes, folding patterns and shapes of origami. Rocklin says the research team discovered the class of origami was not important, rather it was the way the folds interacted which was key.

In the future, the research team intends to build on their work by examining more complex origami systems.

“We would like to try to validate this for different patterns, different configurations; to make sense of the theory and validate it,” Paulino says. “For example, we need to investigate patterns such as the blockfold pattern, which is quite intriguing.”