Most materials — from rubber bands to steel beams — thin out as they are stretched, but engineers can use origami’s interlocking ridges and precise folds to reverse this tendency and build devices that grow wider as they are pulled apart.

Researchers increasingly use this kind of technique, drawn from the ancient art of origami, to design spacecraft components, medical robots and antenna arrays. However, much of the work has progressed via 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, remain unaffected, or thicken as they are stretched, pushed or bent.

Kon-Well Wang, a professor of mechanical engineering 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 paths for the technical community to harness and pursue that will further elevate the functionalities of advanced origami and metamaterials. The impact is tremendous.”

In a paper published Aug. 3 in the Proceedings of the National Academy of Sciences, Paulino and his colleagues lay out their general rule for the way a broad class of origami responds to stress. The rule applies to origami formed from parallelograms (such as a square, rhombus or rectangle) made of thin material. In their article, the researchers use origami to explore how structures respond to certain kinds of mechanical stress — for example, how a rectangular sponge swells in a bowtie shape when squeezed in the middle of its long sides. Of particular interest was how materials behave when stretched, like a stick of chewing gum that thins as it is pulled at both ends. The ratio of compression along one axis with stretching along the other is called the Poisson ratio.

“Most materials have a positive Poisson ratio. If, for example, you pick up a rubber band and stretch it, it will become thinner and thinner before it breaks,” said Glaucio Paulino, the Margareta Engman Augustine Professor of Engineering at Princeton. “Cork has a zero Poisson ratio, and that is the only reason you can put the cork back in a wine bottle. Otherwise, you would break the bottle.”

The researchers were able to write a set of equations to predict how origami-inspired structures will behave under this kind of stress. They then used the equations to create origami structures with a negative Poisson ratio — origami structures that grew wide instead of narrower when their ends were pulled, or structures that snapped into dome shapes when bent instead of sagging into a saddle shape.

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