Researchers develop methodology for streamlined material deformation control


The researchers demonstrate that new physical theories provide accurate predictions of the deformations of certain structures, revealing that a flexible mechanical structure is governed by some of the same mathematics as electromagnetic waves and even black holes.

A conformal deformation of Kagome’s metamaterial provides an example of the dramatic possibilities.

Can you crumple two sheets of paper in exactly the same way? Probably not – the very flexibility that allows flexible structures from paper to biopolymers and membranes to undergo many types of large deformations makes them notoriously difficult to control. Researchers from the Georgia Institute of Technology, Universiteit van Amsterdam and Universiteit Leiden have shed new light on this fundamental challenge, demonstrating that new physical theories provide accurate predictions of the deformations of certain structures, as recently published in Communication Nature.

In the article, Michael Czajkowski and D. Zeb Rocklin of Georgia Tech, Corentin Coulais of Universiteit van Amsterdam and Martin van Hecke of AMOLF and Universiteit Leiden discuss a highly studied exotic elastic material, discover a geometric description intuitive of pronounced – or non-linear – soft deformations, and show how to activate one of these deformations on demand with a minimum of inputs. This new theory reveals that a flexible mechanical structure is governed by some of the same mathematics as electromagnetic waves, phase transitions and even black holes.

“So many other systems struggle to be strong and solid in some ways, but flexible and compliant in others, from the human body and microorganisms to clothing and industrial robots,” Rocklin said. “These structures solve this problem in an incredibly elegant way that allows a single bending mechanism to generate a large family of deformations. We have shown that a single bending mode can transform a structure into an infinite family of shapes.

A brief history of metamaterials

Metamaterials rely on the use of hinges, folds, cuts, and “flexible” ingredients to display the variety of counter-intuitive physics that have been steadily revealed over the past decade of intense research. Many of these new behaviors emerged from the development of auxetics, materials that tend to shrink in all directions when compressed in any direction rather than expanding outward. Although the structure chosen by the researchers, “Rotating Squares”, is already one of the most studied metamaterials, they have discovered an entirely new and powerful physics hidden in its deformations.

“Normally complex real-world structures defy analytical physics, which made it even more exciting when Michael discovered that his consistent predictions could explain 99.9% of the variance in Corentin’s structure,” Rocklin said. “This new approach could allow us to predict and control robust and flexible structures, from the size of skyscrapers to the microscopic scale.”

Compliant findings

The results of this paper are based on the new observation that these maximal auxetic metamaterials deform conformally, which the researchers confirmed with a high degree of accuracy. This means that any angle drawn on the material before and after the deformation will always look like the same angle. This seemingly banal observation activates powerful mathematical structures.

This conformal insight enables a variety of pen-and-paper analytical advances: a nonlinear energy functional, strain-fitting methods, new prediction methods, and more. This results in a recipe for choosing one of these conformal deformations in an exact, reversible, and mathematically simple way via boundary manipulation. By choosing how much the boundary is stretched, the overall shape can be chosen from endless possibilities.

Such deformation control is further limited by the essential character of conformal deformations. However, the underlying principles are quite general, and researchers are working to apply these new principles to more varied and complex structures.

“Our results are very promising for soft microscopic robotics that are being developed for non-invasive surgical purposes,” Czajkowski said. “In this effort, scalability and precise external control are two of the primary goals, and our style of deformation control seems perfectly suited to the task.”

The leap to more provocative applications is probably not far off, as the realm of metamaterials has gradually become populated with manipulable faces, a variety of new claws and hands, and even an elastic worm that can thread a series of needles. These advances will become essential in the effort to develop soft microscopic robots, which must be externally manipulated to move around a body and perform non-invasive surgeries.


Michael Czajkowski, Corentin Coulais, Martin van Hecke and D. Zeb Rocklin, “Conformal elasticity of mechanism-based metamaterials”, Nature Communication (January 11, 2022).


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