Engineering zero modes in transformable mechanical metamaterials

Mechanical metamaterials, with rationally designed microstructures, exhibit counter-intuitive properties like negative Poisson's ratio and graded stiffness. Recent advancements focus on tunability, allowing property adjustments through deformation. Origami and modular origami structures provide platforms for designing 3D flexible metamaterials with multiple deformation paths and tunable properties. Large deformation in flexible metamaterials often arises from elastic joints, creating zero modes—deformation modes with minimal elastic energy cost. While zero modes are utilized for tunability, their most direct application is in designing acoustic "invisibility" cloaks. The number of zero modes in 3D structures can alter the elasticity tensor, ranging from null-mode to hexa-mode, resulting in extremal static and dynamic properties. However, systematic design of these zero modes has been lacking, hindering the realization of materials with specific zero-mode counts or transformation capabilities. This research proposes a transformable mechanical metamaterial based on modular origami, reconfigurable among null, uni-, bi-, tri-, quadra-, penta-, and hexa-modes, leading to tunable mechanical properties and reprogrammable wave functionalities.

Existing literature extensively explores the design and applications of mechanical metamaterials, highlighting their unique properties and potential. Studies have demonstrated the creation of 2D and 3D metamaterials with tunable properties, often leveraging origami-inspired structures for flexibility and reconfigurability. Several works discuss the use of zero modes in achieving large deformations and tunable negative Poisson's ratios. The concept of using zero modes for acoustic cloaking through penta-mode materials has also been investigated. However, a systematic design approach for engineering a specific number of zero modes or the capability to transform between different zero-mode states in 3D metamaterials remained largely unexplored before this study.

This study utilizes a modular origami design approach to create a 3D transformable metamaterial. The design begins with a 2D tessellation of equilateral 4-bar linkages, where different configurations (completely extended (CE), completely folded (CF), partially folded (PF)) determine the number of zero modes. A homogenization method under the Cauchy-Born hypothesis is used to categorize these states (null-, uni-, bi-, tri-mode). The 2D tessellation is then extended to 3D using a modular origami technique with decoupled 4-bar linkages in orthogonal planes, enabling independent control of configurations along each direction. This cubic unit cell is tessellated to form a 3D metamaterial. A 4x4x4 tessellation is 3D-printed using Thermoplastic Polyurethanes (TPU). Static experiments are conducted to characterize the material's properties in different configurations (tri-mode, penta-mode) using a fixture and heat treatment for reconfiguration. The transformation process is shown to be reversible. A 2x2x2 tessellation is then analyzed to illustrate the transformability from null-mode to hexa-mode by changing folding angles along orthogonal directions. The total number of zero modes (N) is calculated as the sum of zero modes in each direction (Nx + Ny + Nz). This results in ten distinctive extremal metamaterials. Dynamic experiments are performed to investigate 1D, 2D, and 3D wave manipulations. The experiments demonstrate wave polarization control (longitudinal (L) and transverse (T) waves) and wave direction control (wave splitting and filtering), validated by finite element (FE) simulations and homogenized effective metamaterial models. The study also includes supplementary notes detailing the homogenization methods, kinematic properties, effects of geometric parameters, and the dynamic experiments.

The researchers successfully designed and fabricated a 3D transformable mechanical metamaterial capable of transitioning between seven types of extremal metamaterials (null-mode to hexa-mode). Static experiments confirmed the tunable mechanical properties across these states. The 4x4x4 TPU metamaterial prototype exhibited reversible transformations between tri-mode and penta-mode configurations, showing a significant difference in compressive and shear moduli. Analysis of a 2x2x2 tessellation revealed the possibility of achieving all seven states by manipulating folding angles. The study also experimentally demonstrated the reprogrammable wave functionalities of the metamaterial in 1D, 2D, and 3D systems. In 1D, wave polarization could be controlled by switching between uni-mode, bi-mode, and null-mode. In 2D, the all-CE configuration showed a unique 90° wave-splitting function, while partial folding in the y-direction enabled x-directional wave propagation only. In 3D, the tri-mode configuration exhibited a three-wave-splitting phenomenon, and the penta-mode configuration showed z-directional wave propagation only. The FE simulations and homogenized effective metamaterial models strongly supported the experimental findings.

This research successfully addressed the challenge of systematically designing 3D metamaterials with controllable zero modes, enabling transformations between different extremal states. The demonstration of reversible transitions between various zero-mode configurations highlights the potential for creating reconfigurable materials with tailored mechanical and wave-propagation properties. The observed tunable wave manipulation capabilities open new avenues for designing advanced devices with programmable wave functionalities. The good agreement between experimental results and FE simulations validates the design methodology and provides a reliable framework for future metamaterial designs.

This study provides a blueprint for designing 3D transformable mechanical metamaterials with controllable zero modes. The experimental validation of tunable static and dynamic properties, including reprogrammable wave control, showcases the potential of this approach. Future research could explore the integration of magneto-, electro-, or thermo-mechanical coupling to achieve faster and more efficient transformations. Further investigations into the 3D metamaterial's wave control abilities and the development of materials with reprogrammable nonlinearities are also promising directions.

The current transformation method requires multiple fixtures and a lengthy heat treatment process, limiting real-time transformation capabilities. The slight differences observed in repeated penta-mode experiments suggest a potential for material softening due to repeated heating. Future research should focus on developing more efficient and rapid transformation techniques.