Programming shape using kirigami tessellations

Programmable heterogeneous lamellar lattice architecture for dual mechanical protection

Shear bands frequently appear in lattice architectures subjected to compression, leading to an unstable stress-strain curve and global deformation. This deformation mechanism reduces their energy absorption and loading-bearing capacity and causes the architectures to prioritize mechanical protection of external components at the expense of the entire structure. Here, we leverage the design freedom offered by additive manufacturing and the geometrical relation of dual-phase nanolamellar crystals to fabricate heterogeneous lamellar lattice architectures consisting of body-centered cubic (BCC) and face-centered cubic (FCC) unit cells in alternating lamella. The lamellar lattice demonstrates more than 10 and 9 times higher specific energy absorption and energy absorption efficiency, respectively, compared to the BCC lattice. The drastic improvement arises as the nucleation of shear bands is inhibited by the discrete energy threshold for plastic buckling of adjacent heterogeneous lattice lamella during loading. Despite its lower density than the FCC lattice, the lamellar lattice exhibits significant enhancement in plateau stress and crushing force efficiency, attributed to the strengthening effect induced by simultaneous deformation of unit cells in the BCC lattice lamella and the resulting cushion shielding effect. The design improves the global mechanical properties, making lamellar lattices compare favorably against numerous materials proposed for mechanical protection. Additionally, it provides opportunities to program the local mechanical response, achieving programmable internal protection alongside overall external protection. This work provides a different route to design lattice architecture by combining internal and external dual mechanical protection, enabling a generation of multiple mechanical protectors in aerospace, automotive, and transportation fields.

Geometric mechanics of kiri-origami-based bifurcated mechanical metamaterials

We explore a new design strategy of leveraging kinematic bifurcation in creating origami/kirigami-based three-dimensional (3D) hierarchical, reconfigurable, mechanical metamaterials with tunable mechanical responses. We start from constructing three basic, thick, panel-based structural units composed of 4, 6 and 8 rigidly rotatable cubes in close-looped connections. They are modelled, respectively, as 4R, 6R and 8R (R stands for revolute joint) spatial looped kinematic mechanisms, and are used to create a library of reconfigurable hierarchical building blocks that exhibit kinematic bifurcations. We analytically investigate their reconfiguration kinematics and predict the occurrence and locations of kinematic bifurcations through a trial-correction modelling method. These building blocks are tessellated in 3D to create various 3D bifurcated hierarchical mechanical metamaterials that preserve the kinematic bifurcations in their building blocks to reconfigure into different 3D architectures. By combining the kinematics and considering the elastic torsional energy stored in the folds, we develop the geometric mechanics to predict their tunable anisotropic Poisson's ratios and stiffnesses. We find that kinematic bifurcation can significantly effect mechanical responses, including changing the sign of Poisson's ratios from negative to positive beyond bifurcation, tuning the anisotropy, and overcoming the polarity of structural stiffness and enhancing the number of deformation paths with more reconfigured shapes.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Blueprints of Architected Materials: A Guide to Metamaterial Design for Tissue Engineering

Mechanical metamaterials are rationally designed structures engineered to exhibit extraordinary properties, often surpassing those of their constituent materials. The geometry of metamaterials' building blocks, referred to as unit cells, plays an essential role in determining their macroscopic mechanical behavior. Due to their hierarchical design and remarkable properties, metamaterials hold significant potential for tissue engineering; however their implementation in the field remains limited. The major challenge hindering the broader use of metamaterials lies in the complexity of unit cell design and fabrication. To address this gap, a comprehensive guide is presented detailing the design principles of well-established metamaterials. The essential unit cell geometric parameters and design constraints, as well as their influence on mechanical behavior, are summarized highlighting essential points for effective fabrication. Moreover, the potential integration of artificial intelligence techniques is explored in meta-biomaterial design for patient- and application-specific design. Furthermore, a comprehensive overview of current applications of mechanical metamaterials is provided in tissue engineering, categorized by tissue type, thereby showcasing the versatility of different designs in matching the mechanical properties of the target tissue. This review aims to provide a valuable resource for tissue engineering researchers and aid in the broader use of metamaterials in the field.

Digital Mechanical Metamaterial with Programmable Functionality

Digitization has brought a new era to the world, liberating information from physical media. The material structure-property relation is high-dimensional and nonlinear, and the digitization of structure-property relations may bring unprecedented functional programmability and diversity. Here, a new concept of digital mechanical metamaterial (DMM) is presented, where property design is realized by programming the digital states of the DMM to decouple the design of the structure and property. Transforming the binary stable states of a curved beam to the digital bit, one unit cell of DMM manifests three distinct deformation responses under compression, i.e., compression-twist coupling (CTC), compression-shear coupling (CSC), and pure compression (PC). These deformation modes show notable differences in motion and stiffness, which, by digitally programming a series of DMM, can yield a spectrum of functionalities, including information encryption, stress-strain relation customization, energy absorption in cushioning, effective vibration isolation, and tunable force transmission. This study pioneers a versatile material design paradigm that provides much greater freedom for the property design of intelligent mechanical metamaterials.

Harnessing plasticity in sequential metamaterials for ideal shock absorption

Mechanical metamaterials exhibit interesting properties such as high stiffness at low density1-3, enhanced energy absorption3,4, shape morphing5-7, sequential deformations8-11, auxeticity12-14 and robust waveguiding15,16. Until now, metamaterial design has primarily relied on geometry, and materials nonlinearities such as viscoelasticity, fracture and plasticity have been largely left out of the design rationale. In fact, plastic deformations have been traditionally seen as a failure mode and thereby carefully avoided1,3,17,18. Here we embrace plasticity instead and discover a delicate balance between plasticity and buckling instability, which we term 'yield buckling'. We exploit yield buckling to design metamaterials that buckle sequentially in an arbitrary large sequence of steps whilst keeping a load-bearing capacity. We make use of sequential yield buckling to create metamaterials that combine stiffness and dissipation-two properties that are usually incompatible-and that can be used several times. Hence, our metamaterials exhibit superior shock-absorption performance. Our findings add plasticity to the metamaterial toolbox and make mechanical metamaterials a burgeoning technology with serious potential for mass production.

Electronically configurable microscopic metasheet robots

Shape morphing is vital to locomotion in microscopic organisms but has been challenging to achieve in sub-millimetre robots. By overcoming obstacles associated with miniaturization, we demonstrate microscopic electronically configurable morphing metasheet robots. These metabots expand locally using a kirigami structure spanning five decades in length, from 10 nm electrochemically actuated hinges to 100 μm splaying panels making up the ~1 mm robot. The panels are organized into unit cells that can expand and contract by 40% within 100 ms. These units are tiled to create metasheets with over 200 hinges and independent electronically actuating regions that enable the robot to switch between multiple target geometries with distinct curvature distributions. By electronically actuating independent regions with prescribed phase delays, we generate locomotory gaits. These results advance a metamaterial paradigm for microscopic, continuum, compliant, programmable robots and pave the way to a broad spectrum of applications, including reconfigurable micromachines, tunable optical metasurfaces and miniaturized biomedical devices.

Energy-efficient dynamic 3D metasurfaces via spatiotemporal jamming interleaved assemblies for tactile interfaces

Inspired by the natural shape-morphing abilities of biological organisms, we introduce a strategy for creating energy-efficient dynamic 3D metasurfaces through spatiotemporal jamming of interleaved assemblies. Our approach, diverging from traditional shape-morphing techniques reliant on continuous energy inputs, utilizes strategically jammed, paper-based interleaved assemblies. By rapidly altering their stiffness at various spatial points and temporal phases during the relaxation of the soft substrate through jamming, we enable the formation of refreshable, intricate 3D shapes with a desirable load-bearing capability. This process, which does not require ongoing energy consumption, ensures energy-efficient and lasting shape displays. Our theoretical model, linking buckling deformation to residual pre-strain, underpins the inverse design process for an array of interleaved assemblies, facilitating the creation of diverse 3D configurations. This metasurface holds notable potential for tactile displays, particularly for the visually impaired, heralding possibilities in visual impaired education, haptic feedback, and virtual/augmented reality applications.

A seamless auxetic substrate with a negative Poisson's ratio of -1

Auxetic metamaterials are a unique class of materials or structures with a negative Poisson's ratio and a wide array of functionalities. However, their inherent porosity presents challenges in practical applications. Filling the inherent perforations while preserving their unique auxeticity is difficult because it demands the seamless integration of components that have highly distinct mechanical characteristics. Here we introduce a seamless auxetic substrate film capable of achieving a negative Poisson's ratio of -1, the theoretical limit of isotropic materials. This breakthrough is realized by incorporating a highly rigid auxetic structure reinforced by glass-fabric, with surface-flattening soft elastomers. We effectively optimize the mechanical properties of these components by systematic experimental and theoretical investigations into the effects of relative differences in the moduli of the constituents. Using the developed auxetic film we demonstrate an image distortion-free display having 25 PPI resolution of micro-LEDs that is capable of 25% stretching without performance degradation.

Fabric soft pneumatic actuators with programmable turing pattern textures

This paper presents a novel computational design and fabrication method for fabric-based soft pneumatic actuators (FSPAs) that use Turing patterns, inspired by Alan Turing's morphogenesis theory. These inflatable structures can adapt their shapes with simple pressure changes and are applicable in areas like soft robotics, airbags, and temporary shelters. Traditionally, the design of such structures relies on isotropic materials and the designer's expertise, often requiring a trial-and-error approach. The present study introduces a method to automate this process using advanced numerical optimization to design and manufacture fabric-based inflatable structures with programmable shape-morphing capabilities. Initially, an optimized distribution of the material orientation field on the surface membrane is achieved through gradient-based orientation optimization. This involves a comprehensive physical deployment simulation using the nonlinear shell finite element method, which is integrated into the inner loop of the optimization algorithm. This continuous adjustment of material orientations enhances the design objectives. These material orientation fields are transformed into discretized texture patterns that replicate the same anisotropic deformations. Anisotropic reaction-diffusion equations, using diffusion coefficients determined by local orientations from the optimization step, are then utilized to create space-filling Turing pattern textures. Furthermore, the fabrication methods of these optimized Turing pattern textures are explored using fabrics through heat bonding and embroidery. The performance of the fabricated FSPAs is evaluated through three different deformation shapes: C-shaped bending, S-shaped bending, and twisting.

A mechanically robust and facile shape morphing using tensile-induced buckling

Inspired by the adaptive mechanisms observed in biological organisms, shape-morphing soft structures have emerged as promising platforms for many applications. In this study, we present a shape-morphing strategy to overcome existing limitations of the intricate fabrication process and the lack of mechanical robustness against mechanical perturbations. Our method uses tensile-induced buckling, achieved by attaching restraining strips to a stretchable substrate. When the substrate is stretched, the stiffness mismatch between the restraining strips and the substrate, and the Poisson's effect on the substrate cause the restraining strips to buckle, thereby transforming initially flat shapes into intricate three-dimensional (3D) configurations. Guided by an inverse design method, we demonstrate the capability to achieve complicated and diverse 3D shapes. Leveraging shape morphing, we further develop soft grippers exhibiting outstanding universality, high grasping efficiencies, and exceptional durability. Our proposed shape-morphing strategy is scalable and material-independent, holding notable potential for applications in soft robotics, haptics, and biomedical devices.

Flexible sensors with zero Poisson's ratio

Flexible sensors have been developed for the perception of various stimuli. However, complex deformation, usually resulting from forces or strains from multi-axes, can be challenging to measure due to the lack of independent perception of multiaxial stimuli. Herein, flexible sensors based on the metamaterial membrane with zero Poisson's ratio (ZPR) are proposed to achieve independent detection of biaxial stimuli. By deliberately designing the geometric dimensions and arrangement parameters of elements, the Poisson's ratio of an elastomer membrane can be modulated from negative to positive, and the ZPR membrane can maintain a constant transverse dimension under longitudinal stimuli. Due to the accurate monitoring of grasping force by ZPR sensors that are insensitive to curvatures of contact surfaces, rigid robotic manipulators can be guided to safely grasp deformable objects. Meanwhile, the ZPR sensor can also precisely distinguish different states of manipulators. When ZPR sensors are attached to a thermal-actuation soft robot, they can accurately detect the moving distance and direction. This work presents a new strategy for independent biaxial stimuli perception through the design of mechanical metamaterials, and may inspire the future development of advanced flexible sensors for healthcare, human-machine interfaces and robotic tactile sensing.

Optoelectronically navigated nano-kirigami microrotors

With the rapid development of micro/nanofabrication technologies, the concept of transformable kirigami has been applied for device fabrication in the microscopic world. However, most nano-kirigami structures and devices were typically fabricated or transformed at fixed positions and restricted to limited mechanical motion along a single axis due to their small sizes, which significantly limits their functionalities and applications. Here, we demonstrate the precise shaping and position control of nano-kirigami microrotors. Metallic microrotors with size of ~10 micrometers were deliberately released from the substrates and readily manipulated through the multimode actuation with controllable speed and direction using an advanced optoelectronic tweezers technique. The underlying mechanisms of versatile interactions between the microrotors and electric field are uncovered by theoretical modeling and systematic analysis. This work reports a novel methodology to fabricate and manipulate micro/nanorotors with well-designed and sophisticated kirigami morphologies, providing new solutions for future advanced optoelectronic micro/nanomachinery.

Adaptive plasticity of auxetic Kirigami hydrogel fabricated from anisotropic swelling of cellulose nanofiber film

Hydrogels are flexible materials that typically accommodate elongation with positive Poisson's ratios. Auxetic property, i.e., the negative Poisson's ratio, of elastic materials can be macroscopically implemented by the structural design of the continuum. We realize it without mold for hydrogel made of cellulose nanofibers (CNFs). The complex structural design of auxetic Kirigami is first implemented on the dry CNF film, i.e., so-called nanopaper, by laser processing, and the CNF hydrogel is formed by dipping the film in liquid water. The CNF films show anisotropic swelling where drastic volumetric change mainly originates from increase in the thickness. This anisotropy makes the design and fabrication of the emergent Kirigami hydrogel straightforward. We characterize the flexibility of this mechanical metamaterial made of hydrogel by cyclic tensile loading starting from the initial end-to-end distance of dry sample. The tensile load at the maximum strain decreases with the increasing number of cycles. Furthermore, the necessary work up to the maximum strain even decreases to the negative value, while the work of restoration to the original end-to-end distance increases from the negative value to the positive. The equilibrium strain where the force changes the sign increases to reach a plateau. This plastic deformation due to the cyclic loading can be regarded as the adaptive response without fracture to the applied dynamic loading input.

Engineering Kirigami Frameworks Toward Real-World Applications

The surge in advanced manufacturing techniques has led to a paradigm shift in the realm of material design from developing completely new chemistry to tailoring geometry within existing materials. Kirigami, evolved from a traditional cultural and artistic craft of cutting and folding, has emerged as a powerful framework that endows simple 2D sheets with unique mechanical, thermal, optical, and acoustic properties, as well as shape-shifting capabilities. Given its flexibility, versatility, and ease of fabrication, there are significant efforts in developing kirigami algorithms to create various architectured materials for a wide range of applications. This review summarizes the fundamental mechanisms that govern the transformation of kirigami structures and elucidates how these mechanisms contribute to their distinctive properties, including high stretchability and adaptability, tunable surface topography, programmable shape morphing, and characteristics of bistability and multistability. It then highlights several promising applications enabled by the unique kirigami designs and concludes with an outlook on the future challenges and perspectives of kirigami-inspired metamaterials toward real-world applications.

Anisotropic Morphing in Bistable Kirigami through Symmetry Breaking and Geometric Frustration

Shape morphing in bistable kirigami enables remarkable functionalities appealing to a diverse range of applications across the spectrum of length scale. At the core of their shape shifting lies the architecture of their repeating unit, where highly deformable slits and quasi-rigid rotating units often exhibit multiple symmetries that confer isotropic deployment obeying uniform scaling transformation. In this work, symmetry breaking in bistable kirigami is investigated to access geometric frustration and anisotropic morphing, enabling arbitrarily scaled deployment in planar and spatial bistable domains. With an analysis on their symmetry properties complemented by a systematic investigation integrating semi-analytical derivations, numerical simulations, and experiments on elastic kirigami sheets, this work unveils the fundamental relations between slit symmetry, geometric frustration, and anisotropic bistable deployment. Furthermore, asymmetric kirigami units are leveraged in planar and flat-to-3D demonstrations to showcase the pivotal role of shear deformation in achieving target shapes and functions so far unattainable with uniformly stretchable kirigami. The insights provided in this work unveil the role of slit symmetry breaking in controlling the anisotropic bistable deployment of soft kirigami metamaterials, enriching the range of achievable functionalities for applications spanning deployable space structures, wearable technologies, and soft machines.

Diffusive kinks turn kirigami into machines

Kinks define boundaries between distinct configurations of a material. In the context of mechanical metamaterials, kinks have recently been shown to underpin logic, shape-changing and locomotion functionalities. So far such kinks propagate by virtue of inertia or of an external load. Here, we discover the emergence of propagating kinks in purely dissipative kirigami. To this end, we create kirigami that shape-change into different textures depending on how fast they are stretched. We find that if we stretch fast and wait, the viscoelastic kirigami can eventually snap from one texture to another. Crucially, such a snapping instability occurs in a sequence and a propagating diffusive kink emerges. As such, it mimics the slow sequential folding observed in biological systems, e.g., Mimosa Pudica. We finally demonstrate that diffusive kinks can be harnessed for basic machine-like functionalities, such as sensing, dynamic shape morphing, transport and manipulation of objects.

Duality and Sheared Analytic Response in Mechanism-Based Metamaterials

Mechanical metamaterials designed around a zero-energy pathway of deformation known as a mechanism, challenge the conventional picture of elasticity and generate complex spatial response that remains largely uncharted. Here, we present a unified theoretical framework to showing that the presence of a unimode in a 2D structure generates a space of anomalous zero-energy sheared analytic modes. The spatial profiles of these stress-free strain patterns is dual to equilibrium stress configurations. We show a transition at an exceptional point between bulk modes in structures with conventional Poisson ratios (anauxetic) and evanescent surface modes for negative Poisson ratios (auxetic). We suggest a first application of these unusual response properties as a switchable signal amplifier and filter for use in mechanical circuitry and computation.

Spatially Programmable Architected Materials Inspired by the Metallurgical Phase Engineering

Programmable architected materials with the capabilities of precisely storing predefined mechanical behaviors and adaptive deformation responses upon external stimulations are desirable to help increase the performance and the organic integration of materials with surrounding environments. Here, a new approach inspired by the physical metallurgical principles is proposed to allow the materials designers to not only enhance the global strength but also precisely tune mechanical properties (such as strength, modulus, and plastic deformation) locally in architected materials to create a new class of intelligent mechanical metamaterials. Such programmable materials not only have high strength and plastic deformation stability but also the ability to regulate the local deformation states and spatially control the internal propagation of deformation. This innovative approach also provides new and effective ways to enhance the adaptivity of the materials thanks to responsive strengths that not only make the materials increasingly stronger but also allow threshold-based adaptive responses to external loading.

Flexible Metasurfaces for Multifunctional Interfaces

Optical metasurfaces, capable of manipulating the properties of light with a thickness at the subwavelength scale, have been the subject of extensive investigation in recent decades. This research has been mainly driven by their potential to overcome the limitations of traditional, bulky optical devices. However, most existing optical metasurfaces are confined to planar and rigid designs, functions, and technologies, which greatly impede their evolution toward practical applications that often involve complex surfaces. The disconnect between two-dimensional (2D) planar structures and three-dimensional (3D) curved surfaces is becoming increasingly pronounced. In the past two decades, the emergence of flexible electronics has ushered in an emerging era for metasurfaces. This review delves into this cutting-edge field, with a focus on both flexible and conformal design and fabrication techniques. Initially, we reflect on the milestones and trajectories in modern research of optical metasurfaces, complemented by a brief overview of their theoretical underpinnings and primary classifications. We then showcase four advanced applications of optical metasurfaces, emphasizing their promising prospects and relevance in areas such as imaging, biosensing, cloaking, and multifunctionality. Subsequently, we explore three key trends in optical metasurfaces, including mechanically reconfigurable metasurfaces, digitally controlled metasurfaces, and conformal metasurfaces. Finally, we summarize our insights on the ongoing challenges and opportunities in this field.

Rigid-foldable cylindrical origami with tunable mechanical behaviors

Rigid-foldable origami shows significant promise in advanced engineering applications including deployable structures, aerospace engineering, and robotics. It undergoes deformation solely at the creases during the folding process while maintaining rigidity throughout all facets. However, most types of cylindrical origami, such as Kresling origami, water-bomb origami, and twisted tower origami, lack rigid-foldability. Although shape transformation can be achieved through elastic folding, their limited rigid foldability constrains their engineering applications. To address this limitation, we proposed a type of cylindrical origami inspired by Kresling origami, named foldable prism origami (FP-ori), in this paper. FP-ori possesses not only rigid-foldability but also several tunable properties, including flat-foldability, self-locking, and bistability. The geometric properties of FP-ori were analyzed and the relationship between different parameters and tunable mechanical behaviors were verified through finite element method simulations, as well as experiments using paper models. Furthermore, we proposed stacked structures composed of multiple cubic FP-ori units, the rotation directions of which could be controlled through the combination arrangement. And drawing inspiration from kirigami, a negative Poisson's ratio tessellation structure was created. These results indicated that FP-ori has substantial potential for broad application in engineering fields.

Physics-aware differentiable design of magnetically actuated kirigami for shape morphing

Shape morphing that transforms morphologies in response to stimuli is crucial for future multifunctional systems. While kirigami holds great promise in enhancing shape-morphing, existing designs primarily focus on kinematics and overlook the underlying physics. This study introduces a differentiable inverse design framework that considers the physical interplay between geometry, materials, and stimuli of active kirigami, made by soft material embedded with magnetic particles, to realize target shape-morphing upon magnetic excitation. We achieve this by combining differentiable kinematics and energy models into a constrained optimization, simultaneously designing the cuts and magnetization orientations to ensure kinematic and physical feasibility. Complex kirigami designs are obtained automatically with unparalleled efficiency, which can be remotely controlled to morph into intricate target shapes and even multiple states. The proposed framework can be extended to accommodate various active systems, bridging geometry and physics to push the frontiers in shape-morphing applications, like flexible electronics and minimally invasive surgery.

Emergent nonlocal combinatorial design rules for multimodal metamaterials

Combinatorial mechanical metamaterials feature spatially textured soft modes that yield exotic and useful mechanical properties. While a single soft mode often can be rationally designed by following a set of tiling rules for the building blocks of the metamaterial, it is an open question what design rules are required to realize multiple soft modes. Multimodal metamaterials would allow for advanced mechanical functionalities that can be selected on the fly. Here we introduce a transfer matrix-like framework to design multiple soft modes in combinatorial metamaterials composed of aperiodic tilings of building blocks. We use this framework to derive rules for multimodal designs for a specific family of building blocks. We show that such designs require a large number of degeneracies between constraints, and find precise rules on the real space configuration that allow such degeneracies. These rules are significantly more complex than the simple tiling rules that emerge for single-mode metamaterials. For the specific example studied here, they can be expressed as local rules for tiles composed of pairs of building blocks in combination with a nonlocal rule in the form of a global constraint on the type of tiles that are allowed to appear together anywhere in the configuration. This nonlocal rule is exclusive to multimodal metamaterials and exemplifies the complexity of rational design of multimode metamaterials. Our framework is a first step towards a systematic design strategy of multimodal metamaterials with spatially textured soft modes.

A Review of Structure Engineering of Strain-Tolerant Architectures for Stretchable Electronics

Stretchable electronics possess significant advantages over their conventional rigid counterparts and boost game-changing applications such as bioelectronics, flexible displays, wearable health monitors, etc. It is, nevertheless, a formidable task to impart stretchability to brittle electronic materials such as silicon. This review provides a concise but critical discussion of the prevailing structural engineering strategies for achieving strain-tolerant electronic devices. Not only the more commonly discussed lateral designs of structures such as island-bridge, wavy structures, fractals, and kirigami, but also the less discussed vertical architectures such as strain isolation and elastoplastic principle are reviewed. Future opportunities are envisaged at the end of the paper.

Modular design of curved beam-based recyclable architected materials

Advances in manufacturing technologies have enabled architected materials with unprecedented properties. These materials are typically irreversibly designed and fabricated with characteristic geometries and specific mechanical properties, thus rendering them suitable for pre-specified requests. However, these materials cannot be recycled or reconstructed into different shapes and functionalities to economically adapt to various environments. Hence, we present a modular design strategy to create a category of recyclable architected materials comprising elastic initially curved beams and rigid cylindrical magnets. Based on numerical analyses and physical prototypes, we introduce an arc-serpentine curved beam (ASCB) and systematically investigate its mechanical properties. Subsequently, we develop two sets of hierarchical modules for the ASCB, thus expanding the constructable shape of architected materials from regular cuboids to complex curved surfaces. Furthermore, we demonstrate that the magnets attached to the centers of specific serpentine patterns of the modules allows the effective in-situ recycling of the designed materials, including sheet materials for non-damage storage, bulk materials for tunable stiffness, and protective package boxes for reshaping into decorative lampshades. We expect our approach to improve the flexibility of architected materials for multifunctional implementation in resource-limited scenarios.

A reprogrammable mechanical metamaterial with origami functional-group transformation and ring reconfiguration

Recent advancements in reprogrammable metamaterials have enabled the development of intelligent matters with variable special properties in situ. These metamaterials employ intra-element physical reconfiguration and inter-element structural transformation. However, existing mono-characteristic homo-element mechanical metamaterials have limited reprogramming functions. Here, we introduce a reprogrammable mechanical metamaterial composed of origami elements with heterogeneous mechanical properties, which achieves various mechanical behavior patterns by functional group transformations and ring reconfigurations. Through the anisotropic assembly of two heterogeneous elements into a functional group, we enable mechanical behavior switching between positive and negative stiffness. The resulting polygonal ring exhibits rotational deformation, zero Poisson's ratio stretching/compression deformation, and negative Poisson's ratio auxetic deformation. Arranging these rings periodically yields homogeneous metamaterials. The reconfiguration of quadrilateral rings allows for continuous fine-tunability of the mechanical response and negative Poisson's ratio. This mechanical metamaterial could provide a versatile material platform for reprogrammable mechanical computing, multi-purpose robots, transformable vehicles and architectures at different scales.

Image-Based OA-Style Paper Pop-Up Design via Mixed-Integer Programming

Origami architecture (OA) is a fascinating papercraft that involves only a piece of paper with cuts and folds. Interesting geometric structures 'pop up' when the paper is opened. However, manually designing such a physically valid 2D paper pop-up plan is challenging since fold lines must jointly satisfy hard spatial constraints. Existing works on automatic OA-style paper pop-up design all focused on how to generate a pop-up structure that approximates a given target 3D model. This article presents the first OA-style paper pop-up design framework that takes 2D images instead of 3D models as input. Our work is inspired by the fact that artists often use 2D profiles to guide the design process, thus benefited from the high availability of 2D image resources. Due to the lack of 3D geometry information, we perform novel theoretic analysis to ensure the foldability and stability of the resultant design. Based on a novel graph representation of the paper pop-up plan, we further propose a practical optimization algorithm via mixed-integer programming that jointly optimizes the topology and geometry of the 2D plan. We also allow the user to interactively explore the design space by specifying constraints on fold lines. Finally, we evaluate our framework on various images with interesting 2D shapes. Experiments and comparisons exhibit both the efficacy and efficiency of our framework.

Hydrogel muscles powering reconfigurable micro-metastructures with wide-spectrum programmability

Stimuli-responsive geometric transformations endow metamaterials with dynamic properties and functionalities. However, using existing transformation mechanisms to program a single geometry to transform into diverse final configurations remains challenging, imposing crucial design restrictions on achieving versatile functionalities. Here, we present a programmable strategy for wide-spectrum reconfigurable micro-metastructures using linearly responsive transparent hydrogels as artificial muscles. Actuated by the hydrogel, the transformation of micro-metastructures arises from the collaborative buckling of their building blocks. Rationally designing the three-dimensional printing parameters and geometry features of the metastructures enables their locally isotropic or anisotropic deformation, allowing controllable wide-spectrum pattern transformation with programmable chirality and optical anisotropy. This reconfiguration mechanism can be applied to various materials with a wide range of mechanical properties. Our strategy enables a thermally reconfigurable printed metalattice with pixel-by-pixel mapping of different printing powers and angles for displaying or hiding complex information, providing opportunities for encryption, miniature robotics, photonics and phononics applications.

Pneumatic cells toward absolute Gaussian morphing

On a flat map of the Earth, continents are inevitably distorted. Reciprocally, curving a plate simultaneously in two directions requires a modification of in-plane distances, as Gauss stated in his seminal theorem. Although emerging architectured materials with programmed in-plane distortions are capable of such shape morphing, an additional control of local bending is required to precisely set the final shape of the resulting three-dimensional surface. Inspired by bulliform cells in leaves of monocotyledon plants, we show how the internal structure of flat panels can be designed to program bending and in-plane distortions simultaneously when pressurized, leading to a targeted shell shape. These surfaces with controlled stiffness and fast actuation are manufactured using consumer-grade materials and open a route to large-scale shape-morphing robotics applications.

Transition of deformation modes from bending to auxetic compression in origami-based metamaterials for head protection from impact

For the protection of the human head by energy absorption structures, a soft mechanical response upon contact with the head is required to mitigate the effect of impact, while a hard mechanical response for highly efficient energy absorption is required to stop the movement of the head. This study realized the opposite mechanical properties during head protection by transitioning the deformation mode from bending to auxetic compression. First, non-linear finite element (FE) models were constructed to numerically reproduce the bending behavior. The calculated force responses agreed well with forces in bending tests. Using the FE models, the EA structures with proper transition of deformation modes were designed and installed in the seat headrests of real vehicles. Head protection was evaluated by dynamic loading in sled testing, in which the force on the head of the crash test dummy was measured. The head injury criterion improved from 274 to 155, indicating the superior performance of the tested structures compared to that achieved by energy absorption structures based on steel plates. Moreover, the deformation of auxetic structures prevented neck bending by holding the head. These findings present new possibilities for effectively protecting the human body by mitigating impact, facilitating energy absorption, and ensuring head stability.

Passively addressed robotic morphing surface (PARMS) based on machine learning

Reconfigurable morphing surfaces provide new opportunities for advanced human-machine interfaces and bio-inspired robotics. Morphing into arbitrary surfaces on demand requires a device with a sufficiently large number of actuators and an inverse control strategy. Developing compact, efficient control interfaces and algorithms is vital for broader adoption. In this work, we describe a passively addressed robotic morphing surface (PARMS) composed of matrix-arranged ionic actuators. To reduce the complexity of the physical control interface, we introduce passive matrix addressing. Matrix addressing allows the control of N2 independent actuators using only 2N control inputs, which is substantially lower than traditional direct addressing (N2 control inputs). Using machine learning with finite element simulations for training, our control algorithm enables real-time, high-precision forward and inverse control, allowing PARMS to dynamically morph into arbitrary achievable predefined surfaces on demand. These innovations may enable the future implementation of PARMS in wearables, haptics, and augmented reality/virtual reality.

Non-orientable order and non-commutative response in frustrated metamaterials

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking^1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices^5-8, confined colloidal suspensions⁹ and crumpled paper sheets¹⁰. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories^11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science⁸, photonics²⁰, magnetism⁷ and atomic physics²¹.

An additive framework for kirigami design

We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.

Auxetic Kirigami Metamaterials upon Large Stretching

Auxetic kirigami metamaterials (KMs) attain negative Poisson's ratios with periodic slender cuts on thin sheets. The existing thin auxetic KMs forfeit auxeticity under large tensions because their auxeticity mainly arises from in-plane deformation, but out-of-plane buckling could arise to cause large deviations, and thicker KMs would suffer from stress failure. This paper proposes a novel family of KMs that can realize and retain auxeticity for up to 0.50 applied strains by fully exploiting out-of-plane buckling in the design model. Numerical and experimental results show that the designed KMs possess unique properties that are not exhibited by existing KMs, including a wide range of negative Poisson's ratios with designable variation modes under different applied strains, sheet thickness-insensitive auxeticity, and excellent shape recoverability. A potential application is exemplified with a scenario that they are designed as a stretchable display without image distortions under large tensions. The proposed auxetic KMs open new opportunities for the design of specific functional devices in areas of compliant robotics, bio-medical devices, and flexible electronics.

Switchable Kirigami Structures as Window Envelopes for Energy-Efficient Buildings

Efficient regulation of thermal radiation is an effective way to conserve energy consumption of buildings. Because windows are the least energy-efficient part of buildings, their thermal radiation regulation is highly demanded, especially in the changing environment, but is still a challenge. Here, by employing a kirigami structure, we design a variable-angle thermal reflector as a transparent envelope of windows for their thermal radiation modulation. The envelope can be easily switched between heating and cooling modes by loading different pre-stresses, which endow the envelope windows with the ability of temperature regulation, and the interior temperature of a building model can be reduced by ~3.3 °C under cooling mode and increased by ~3.9 °C under heating mode in the outdoor test. The improved thermal management of windows by the adaptive envelope provides an extra heating, ventilation, and air-conditioning energy savings percentage of 13% to 29% per year for buildings located in different climate zones around the world, making the kirigami envelope windows a promising way for energy-saving utilization.

Programming 3D curved mesosurfaces using microlattice designs

Cellular microstructures form naturally in many living organisms (e.g., flowers and leaves) to provide vital functions in synthesis, transport of nutrients, and regulation of growth. Although heterogeneous cellular microstructures are believed to play pivotal roles in their three-dimensional (3D) shape formation, programming 3D curved mesosurfaces with cellular designs remains elusive in man-made systems. We report a rational microlattice design that allows transformation of 2D films into programmable 3D curved mesosurfaces through mechanically guided assembly. Analytical modeling and a machine learning-based computational approach serve as the basis for shape programming and determine the heterogeneous 2D microlattice patterns required for target 3D curved surfaces. About 30 geometries are presented, including both regular and biological mesosurfaces. Demonstrations include a conformable cardiac electronic device, a stingray-like dual mode actuator, and a 3D electronic cell scaffold.

Explosive rigidity percolation in kirigami

Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials.

Natural Shaping of Acellular Dermal Matrices for Implant-Based Breast Reconstruction via Expansile Kirigami

To complete a successful and aesthetic breast reconstruction for breast cancer survivors, tissue reinforcing acellular dermal matrices (ADMs) are widely utilized to create slings/pockets to keep breast implants or autologous tissue transfer secured against the chest wall in the desired location. However, ADM sheets are 2D and cannot completely cover the entire implant without wrinkles. Here, guided by finite element modeling, a kirigami strategy is presented to cut the ADM sheets with locally and precisely controlled stretchability, curvature, and elasticity. Upon expansion, a single kirigami ADM sheet can conformably wrap the implant regardless of the shape and size, forming a natural teardrop shape; contour cuts prescribe the topographical height and fractal cuts in the center ensures horizontal expandability and thus conformability. This kirigami ADM can provide support to the reconstructed breast in the desired regions, potentially offering optimal outcomes and patient-specific reconstruction, while minimizing operative time and cost.

Inverse design of 3D reconfigurable curvilinear modular origami structures using geometric and topological reconstructions

The recent development of modular origami structures has ushered in an era for active metamaterials with multiple degrees of freedom (multi-DOF). Notably, no systematic inverse design approach for 3D curvilinear modular origami structures has been reported. Moreover, very few modular origami topologies have been studied to design active metamaterials with multi-DOF. Herein, we develop an inverse design method for constructing 3D reconfigurable architected structures - we synthesize modular origami structures whose unit cells can be volumetrically mapped into a prescribed 3D curvilinear shape followed by volumetric shrinkage to construct modules. After modification of the tubular geometry, we search through all the possible geometric and topological combinations of the modular origami structures to attain the target mobility using a topological reconstruction of modules. Our inverse design using geometric and topological reconstructions can provide an effective solution to construct 3D curvilinear reconfigurable structures with multi-DOF. Our work opens a path toward 3D reconfigurable systems based on volumetric inverse design, such as 3D active metamaterials and 3D morphing devices for automotive, aerospace, and biomedical engineering applications.

Flexible Metamaterial Electronics

Over the last decade, extensive efforts have been made on utilizing advanced materials and structures to improve the properties and functionalities of flexible electronics. While the conventional ways are approaching their natural limits, a revolutionary strategy, namely metamaterials, is emerging toward engineering structural materials to break the existing fetters. Metamaterials exhibit supernatural physical behaviors, in aspects of mechanical, optical, thermal, acoustic, and electronic properties that are inaccessible in natural materials, such as tunable stiffness or Poisson's ratio, manipulating electromagnetic or elastic waves, and topological and programmable morphability. These salient merits motivate metamaterials as a brand-new research direction and have inspired extensive innovative applications in flexible electronics. Here, such a groundbreaking interdisciplinary field is first coined as "flexible metamaterial electronics," focusing on enhancing and innovating functionalities of flexible electronics via the design of metamaterials. Herein, the latest progress and trends in this infant field are reviewed while highlighting their potential value. First, a brief overview starts with introducing the combination of metamaterials and flexible electronics. Then, the developed applications are discussed, such as self-adaptive deformability, ultrahigh sensitivity, and multidisciplinary functionality, followed by the discussion of potential prospects. Finally, the challenges and opportunities facing flexible metamaterial electronics to advance this cutting-edge field are summarized.

Kirigami-Inspired Programmable Soft Magnetoresponsive Actuators with Versatile Morphing Modes

Untethered soft magnetoresponsive actuators (SMRAs), which can realize rapid shape transformation, have attracted widespread attention for their strategic applications in exploration, transportation, and minimally invasive medicine. It remains a challenge to fabricate SMRAs with complicated morphing modes (more than bending and folding), limiting their applications to simple shape-morphing and locomotion. Herein, a method integrating the ancient kirigami art and an advanced mechanical assembly method is proposed, which realizes 2D-to-3D and 3D-to-3D complicated shape-morphing and precise magnetization programming through cut-guided deformation. The kirigami-inspired SMRAs exhibit good robustness after actuating more than 10000 times. An integrated finite element analysis method is developed to quantitatively predict the shape transformation of SMRAs under magnetic actuation. By leveraging this method, a set of 3D curved responsive morphologies with programmed Gaussian curvature are fabricated (e.g., ellipsoid and saddle structures), specifically 3D multilayer structures and face-like shapes with different expressions, which are difficult to realize using previous approaches. Furthermore, a bionic-scaled soft crawling robot with significant obstacle surmounting ability is fabricated using the kirigami-inspired method. The ability of the method to achieve programmable SMRAs with versatile morphing modes may broaden its applications in soft robotics, color-switchable devices, and clinical treatments.

Engineering by Cuts: How Kirigami Principle Enables Unique Mechanical Properties and Functionalities

Kirigami, the ancient art of paper cutting, has evolved into a design and fabrication framework to engineer multi-functional materials and structures at vastly different scales. By slit cutting with carefully designed geometries, desirable mechanical behaviors-such as accurate shape morphing, tunable auxetics, super-stretchability, buckling, and multistability-can be imparted to otherwise inflexible sheet materials. In addition, the kirigami sheet provides a versatile platform for embedding different electronic and responsive components, opening up avenues for building the next generations of metamaterials, sensors, and soft robotics. These promising potentials of kirigami-based engineering have inspired vigorous research activities over the past few years, generating many academic publications. Therefore, this review aims to provide insights into the recent advance in this vibrant field. In particular, this paper offers the first comprehensive survey of unique mechanical properties induced by kirigami cutting, their underlying physical principles, and their corresponding applications. The synergies between design methodologies, mechanics modeling, advanced fabrication, and material science will continue to mature this promising discipline.

Shape-morphing into 3D curved surfaces with nacre-like composite architectures

Inhomogeneous in-plane deformation of soft materials or cutting and folding of inextensible flat sheets enables shape-morphing from two dimensional (2D) to three-dimensional (3D), while the resulting structures often have weakened mechanical strength. Shells like nacre are known for the superior fracture toughness due to the "brick and mortar" composite layers, enabling stress redistribution and crack stopping. Here, we report an optimal and universal cutting and stacking strategy that transforms composite plies into 3D doubly curved shapes with nacre-like architectures. The multilayered laminate exhibits staggered cut distributions, while the interlaminar shear mitigates the cut-induced mechanical weakness. The experimentally consolidated hemispherical shells exhibit, on average, 37 and 69% increases of compression peak forces, versus those with random cut distributions, when compressed in different directions. Our approach opens a previously unidentified paradigm for shape-conforming arbitrarily curved surfaces while achieving high mechanical performance.

A dynamically reprogrammable surface with self-evolving shape morphing

Dynamic shape-morphing soft materials systems are ubiquitous in living organisms; they are also of rapidly increasing relevance to emerging technologies in soft machines^1-3, flexible electronics^4,5 and smart medicines⁶. Soft matter equipped with responsive components can switch between designed shapes or structures, but cannot support the types of dynamic morphing capabilities needed to reproduce natural, continuous processes of interest for many applications^7-24. Challenges lie in the development of schemes to reprogram target shapes after fabrication, especially when complexities associated with the operating physics and disturbances from the environment can stop the use of deterministic theoretical models to guide inverse design and control strategies^25-30. Here we present a mechanical metasurface constructed from a matrix of filamentary metal traces, driven by reprogrammable, distributed Lorentz forces that follow from the passage of electrical currents in the presence of a static magnetic field. The resulting system demonstrates complex, dynamic morphing capabilities with response times within 0.1 second. Implementing an in situ stereo-imaging feedback strategy with a digitally controlled actuation scheme guided by an optimization algorithm yields surfaces that can follow a self-evolving inverse design to morph into a wide range of three-dimensional target shapes with high precision, including an ability to morph against extrinsic or intrinsic perturbations. These concepts support a data-driven approach to the design of dynamic soft matter, with many unique characteristics.

A Snakeskin-Inspired, Soft-Hinge Kirigami Metamaterial for Self-Adaptive Conformal Electronic Armor

A majority of soft-body creatures evolve armor or shells to protect themselves. Similar protection demand is for flexible electronics working in complex environments. Existing works mainly focus on improving the sensing capabilities such as electronic skin (E-skin). Inspired by snakeskin, a novel electronic armor (E-armor) is proposed, which not only possesses mechanical flexibility and electronic functions similar to E-skin, but is also able to protect itself and the underlying soft body from external physical damage. The geometry of the kirigami mechanical metamaterial (Kiri-MM) ensures auxetic stretchability and meanwhile large areal coverage for sufficient protection. Moreover, to suppress the inherent but undesired out-of-plane buckling of conventional Kiri-MMs for conformal applications, soft hinges are used to form a distinct soft (hinges)-rigid (tiles) configuration. Analytical, computational, and experimental studies of the mechanical behaviors of the soft-hinge Kiri-MM E-armor demonstrate the merits of this design, i.e., stretchability, conformability, and protectability, as applied to flexible electronics. Deploying a conductive soft material at the hinges enables facile wiring strategies for large-scale circuit arrays. Functional E-armor systems for controllable display and sensing purposes provide simple examples of a wide spectrum of applications of this concept.

Deployable mechanical metamaterials with multistep programmable transformation

Transformations in shape are critical to actuation in engineered metamaterials. Existing engineering metamaterials are typically limited to a small number of shape transformations that must be built-in during material synthesis. Here, inspired by the multistability and programmability of kirigami-based self-folding elements, a robust framework is introduced for the construction of sequentially programmable and reprogrammable mechanical metamaterials. The materials can be locked into multiple stable deployed configurations and then, using tunable bistability enabled by temperature-responsive constituent materials, return to their original reference configurations or undergo mode bifurcation. The framework provides a platform to design metamaterials with multiple deployable and reversible configurations in response to external stimuli.

Always-on photocatalytic antibacterial facemask with mini UV-LED array

The facemask is a device to protect yourself and others against pandemics, such as coronavirus disease 2019 (COVID-19), and adding a functional filter to the facemask could offer extra protection against infectious microbes (such as bacteria and viruses) to the wearer. Here, we designed and fabricated an always-on photocatalytic antibacterial facemask, which comprised a reusable polypropylene filter layer coated with the photocatalytic laminated ZnO/TiO₂ bilayer and a separate UV-LEDs layer to supply UV whenever necessary. The fabricated photocatalytic filter was able to be directly inserted into the reusable facemask together with the UV-LEDs layer. This facemask could be used repeatedly and sustainably anytime and anywhere regardless of solar illumination. The photocatalytic filter exhibited an excellent photocatalytic antibacterial effect likely due to recombination suppression of electrons and holes of ZnO/TiO₂ bilayer and wetting transition from hydrophilic to superhydrophilic state on the surface of the filter. Thanks to the kirigami pattern in both photocatalytic filter and UV-LEDs layer, full-face covering, breathability, flexibility, and the snug fit are believed to be improved. Although further in-depth studies are still needed and there is a long way to go, we expect our design idea on the facemask to be considered in various fields.

Continuum Field Theory for the Deformations of Planar Kirigami

Mechanical metamaterials exhibit exotic properties that emerge from the interactions of many nearly rigid building blocks. Determining these properties theoretically has remained an open challenge outside a few select examples. Here, for a large class of periodic and planar kirigami, we provide a coarse-graining rule linking the design of the panels and slits to the kirigami's macroscale deformations. The procedure gives a system of nonlinear partial differential equations expressing geometric compatibility of angle functions related to the motion of individual slits. Leveraging known solutions of the partial differential equations, we present an illuminating agreement between theory and experiment across kirigami designs. The results reveal a dichotomy of designs that deform with persistent versus decaying slit actuation, which we explain using the Poisson's ratio of the unit cell.

Kirigami-Inspired Biodesign for Applications in Healthcare

Mechanically flexible and conformable materials and integrated devices have found diverse applications in personalized healthcare as diagnostics and therapeutics, tissue engineering and regenerative medicine constructs, surgical tools, secure systems, and assistive technologies. In order to impart optimal mechanical properties to the (bio)materials used in these applications, various strategies have been explored-from composites to structural engineering. In recent years, geometric cuts inspired by the art of paper-cutting, referred to as kirigami, have provided innovative opportunities for conferring precise mechanical properties via material removal. Kirigami-based approaches have been used for device design in areas ranging from soft bioelectronics to energy storage. In this review, the principles of kirigami-inspired engineering specifically for biomedical applications are discussed. Factors pertinent to their design, including cut geometry, materials, and fabrication, and the effect these parameters have on their properties and configurations are covered. Examples of kirigami designs in healthcare are presented, such as, various form factors of sensors (on skin, wearable), implantable devices, therapeutics, surgical procedures, and cellular scaffolds for regenerative medicine. Finally, the challenges and future scope for the successful translation of these biodesign concepts to broader deployment are discussed.

Rigidly flat-foldable class of lockable origami-inspired metamaterials with topological stiff states

Origami crease patterns have inspired the design of reconfigurable materials that can transform their shape and properties through folding. Unfortunately, most designs cannot provide load-bearing capacity, and those that can, do so in certain directions but collapse along the direction of deployment, limiting their use as structural materials. Here, we merge notions of kirigami and origami to introduce a rigidly foldable class of cellular metamaterials that can flat-fold and lock into several states that are stiff across multiple directions, including the deployment direction. Our metamaterials rigidly fold with one degree of freedom and can reconfigure into several flat-foldable and spatially-lockable folding paths due to face contact. Locking under compression yields topology and symmetry changes that impart multidirectional stiffness. Additionally, folding paths and mixed-mode configurations can be activated in situ to modulate their properties. Their load-bearing capacity, flat-foldability, and reprogrammability can be harnessed for deployable structures, reconfigurable robots, and low-volume packaging.

While origami-inspired metamaterials can spatially fold, they usually collapse along the deployment direction limiting applicability. Here authors introduce a cellular structure that can be reprogrammed in-situ to not only deploy and rigidly flat-fold but also lock and offer rigidity across all directions.