Scientists have developed a new framework for understanding how active materials behave when they are forced to live on curved surfaces, revealing that geometry can do far more than simply bend an object. According to the study, curvature can determine where mechanical energy is injected, concentrate vibrations around defects, and make the boundaries of a material dramatically more unstable than its interior. The findings, published in Physical Review Letters, could help explain unusual patterns seen in living systems and provide a new strategy for designing artificial materials whose functions are programmed through shape.
Active materials are systems that generate internal forces and continuously consume energy to move, deform, or reorganize themselves. They include biological tissues made from motile cells, swarms of microorganisms, cytoskeletal networks inside cells, and engineered metamaterials containing motors or miniature actuators. Unlike passive materials, which generally respond to an external force by deforming near the point of impact, active materials can amplify, redirect, or redistribute mechanical stresses. Their internal energy supply allows them to produce motion and deformation that would be impossible in an ordinary material at equilibrium.
Most theories of active materials have been developed for flat surfaces or simple geometries. That assumption is often convenient, but it does not reflect the environments in which many active systems actually exist. Cells grow on curved tissues, organisms move across shells and membranes, and engineered devices may be built into tubes, domes, spheres, or other three-dimensional forms. Even a slight curvature can alter how neighboring components fit together and how forces travel through the material. The new work examines this overlooked interaction between internal activity and the geometry of the surface supporting it.
The research was partly motivated by biological experiments involving starfish embryos. In those experiments, embryos assembled themselves into crystal-like arrangements on the surface of water. The pattern was not perfectly regular because the water surface was slightly curved, rising near the walls of the container in a way familiar from a glass of water. That curvature prevented the embryos from maintaining a flawless repeating arrangement and produced localized irregularities known as defects. In a passive crystal, defects are already important because they influence how the structure stores strain. In an active crystal, however, they can become dynamic centers of motion and energy.
The researchers describe their theory using the concept of odd elasticity, a form of mechanical response associated with systems that break the usual reciprocity between applied forces and resulting deformations. In a conventional elastic material, pushing in one direction and measuring the response in another generally obeys symmetry relations connected to energy conservation and equilibrium. Active materials can violate those expectations because their microscopic constituents continuously consume energy. A force can produce a response that is not simply the reverse of the deformation generated by an opposite force. This nonreciprocal behavior allows active materials to pump energy into mechanical motion rather than merely store and release it.
When odd elasticity is placed on a curved surface, the effects of activity and geometry become inseparable. Curvature changes the local directions along which stresses and strains are defined, meaning that a deformation that appears uniform in flat space may vary from place to place on a curved one. The framework developed by the team shows that this geometric variation can influence where active work is performed and where energy accumulates. Instead of being distributed evenly across the material, mechanical activity may become concentrated in particular regions determined by the surface geometry.
Curvature also makes certain defects unavoidable. On a surface with nontrivial geometry, it is often impossible to arrange directional elements into a perfectly uniform pattern everywhere. The system must introduce disruptions in orientation or spacing, much as a map of a spherical Earth cannot represent every direction without distortion. In the active materials described by the researchers, these defects are not merely static imperfections. They can host localized vibrations, acting as mechanical hotspots where activity produces especially strong and persistent motion. Computer simulations carried out by the team supported the theoretical prediction that defects on curved surfaces behave as sources of distinctive vibrational modes.
The theory further predicts that boundaries are unusually sensitive to active forces. In many passive materials, the interior contains most of the material and therefore dominates the mechanical response. In the curved active systems studied here, the opposite can occur: edges and boundaries may oscillate more strongly than the bulk. The geometry near a boundary changes how stresses are transmitted and can allow active forces to reinforce one another instead of cancelling out. This boundary amplification could help explain why active systems often display waves, oscillations, and coordinated movements that begin or remain strongest at their edges.
The implications extend beyond the original biological observations. If curvature controls the location of energy injection, vibrational activity, and mechanical concentration, then shape could become an active design tool rather than a passive constraint. Engineers might create surfaces that focus motion in a selected region, suppress unwanted oscillations, or guide energy along a prescribed path. A curved metamaterial could be designed to vibrate around specific defects, move more intensely at its perimeter, or respond differently depending on how it is bent. Such systems could eventually be useful in soft robotics, adaptive structures, micromechanical devices, and materials that change behavior without conventional electronic control.
The researchers emphasize that the framework is intended to connect theory with experiments in both living and engineered matter. Biological tissues already combine active force generation with complex curvature, while artificial metamaterials can be fabricated with carefully controlled surface shapes and embedded actuators. Testing the predictions in these settings could reveal whether the same geometric principles govern systems made from cells, particles, or mechanical components. The broader message is that active materials cannot be fully understood by studying their ingredients alone. Their behavior also depends on the spaces in which they operate, suggesting that future materials may be programmed not only through composition and architecture, but through curvature itself.
Subject of Research: Active materials, odd elasticity, curvature, defects, vibrations, and curved-surface mechanics
Article Title: Curved Odd Elasticity
Web References: https://journals.aps.org/prl/abstract/10.1103/fhwd-lmgk
References: Yuan Zhou, Lazaros Tsaloukidis, Jack Binysh, Yuchao Chen, Nikta Fakhri, Corentin Coulais, and Piotr Surówka, “Curved Odd Elasticity,” Physical Review Letters 137, 088301 (2026).
Keywords
Active materials, odd elasticity, curved surfaces, curvature, mechanical metamaterials, biological tissues, starfish embryos, crystal defects, localized vibrations, active matter, soft robotics, nonreciprocal mechanics
