Looking for the perfect fold? It’s frustrating.

Princeton engineers are twisting, stretching and creasing structures to create a new type of origami, one that changes its shape and properties in response to changing circumstances. The new method could be useful for prosthetics, antennas and other devices.

When a device needs to fit into a compact space — in a spacecraft or a surgical device — and then unfold into an intricate shape, origami often provides a solution. But most origami shapes are locked into a few set patterns once their folds are made.

A Princeton team led by Glaucio Paulino wanted to create structures that react to an outside stimulus in multiple ways, not just in a few patterned responses. To accomplish this, the team turned to a technique called geometric frustration.

An origami-based structure will fold and twist in certain ways based on the structure’s material properties and its geometry. When engineers prevent that natural motion, they call it “frustrating” the structure. Normally, engineers have to work around frustration, but in this case it expands their toolkit.

“Sometimes frustration is desirable,” said Paulino, the Margareta Engman Augustine Professor of Engineering at Princeton. Frustration allows designers to cause the origami to follow patterns not normally allowed by its geometry.  “This opens up many possibilities of things we could engineer that we could never do before.”

In an article published in the Proceedings of the National Academy of Sciences, the researchers described how they added elastic components to cylindrical origami structures called Kresling cells. The elastic sections act like springs. By controlling how the springs respond to a force, the researchers were able to execute precise folding patterns of the cells that were not feasible without the springs.

Paulino said springs allow designers to introduce internal energy into the folded structure using pre-stress. This pre-stress allows the origami to respond in ways that are not possible with ordinary materials. For example, engineers can introduce a twisting spring that rotates the origami in a specific fashion; they can add a spring along the structure’s main axis that either squeezes the structure into a compact shape or stretches it out.

By combining frustrated cells in stacks, the engineers were able to develop materials with fine control over material properties like stiffness. For example, a prosthetic leg built with this system can stiffen to provide support while walking on a flat surface but reconfigure into a more flexible state for climbing stairs. The designers could also create adjustable metasurfaces that are used in antennas and optics.

“Exploiting frustration lets us reprogram origami mechanics, for instance turning random Kresling folding into precise, controllable sequences and opening new possibilities for advanced applications,” said Diego Misseroni, a collaborator from the University of Trento.

“We can program any mechanical property that we wish, so this is quite unique,” said Tuo Zhao, a postdoctoral researcher in Paulino’s group.

The team sees potential impact for this type of structure in many fields. This frustrated origami system can combine with other techniques and materials that can change on demand, according to Shixi Zang, postdoctoral researcher and first author of the paper. One example is using frustrated origami to develop responsive, modular devices like a passive sunshade that opens and closes based on the ambient temperature.

The paper, “Origami Frustration and Its Influence on Energy Landscapes of Origami Assemblies,” was published in the Proceedings of the National Academy of Sciences, https://doi.org/10.1073/pnas.2426790122. In addition to Paulino, Zang and Zhao, Diego Misseroni of the University of Trento is a co-author. Support for this work was provided by the Margareta Engman Augustine Professorship of Engineering, the National Science Foundation (NSF grant 2323276), and the European Union (ERC grant HE GA 101086644 S-FOAM).