Formula for stretching origami structures

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Whereas most materials – from rubber bands to steel beams – thin out when stretched. Interlocking ridges and folds in origami structures behave differently, widening when separated.

Researchers from Princeton University and Georgia Tech have developed a formula that can predict how origami-inspired structures behave when 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 components, medical robots and to improve the efficiency of solar cells. However, much of this work relied on trial and error, or instinct.

The researchers developed a set of equations that apply to the ways in which origami parallelograms (like a square, diamond, or rectangle) made of thin material respond to certain types of mechanical stress.


Read more: The technical potential of origami and kirigami


They were particularly interested in how materials and structures behave when stretched, like chewing gum that thins when pulled from 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’s ratio,” meaning they thin when stretched, says paper co-author and Princeton engineering professor Glaucio Paulino.

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

After developing a formula to predict how origami-inspired structures would behave under this type of stress, they then used the equations to create origami structures with negative Poisson’s ratio – able to expand instead to shrink when pulled, and structures that snap into the dome form when bent, instead of sag in a saddle shape.

Zeb Rocklin, assistant professor of physics at Georgia Tech and co-author of the paper, said origami exhibits fascinating and contradictory behaviors.

“Usually if you take a sheet or a thin slab and pull on it, it will retract in the middle. If you take the same sheet and fold it upwards, it will usually form a Pringle – or saddle – shape. Some materials thicken instead when you pull on them, and these always form domes rather than saddles. The amount of thinning always predicts the amount of bending,” he says.

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

Many researchers have spent years trying to come up with rules for different classes, folding patterns, and origami shapes. Rocklin says the research team found that origami class wasn’t important, rather it was how the folds interacted that 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 models, different configurations; to make sense of the theory and validate it,” says Paulino. “For example, we need to study patterns like the blockfold pattern, which is quite intriguing.”

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