Katia Bertoldi

Inflatable structures offer remarkable versatility due to their compact storage and rapid deployment, making them ideal for lightweight, quickly assembled, and deployable applications. These structures are typically made from membranes that are nearly inextensible yet highly flexible. Upon inflation, the membranes avoid energy-intensive stretching and instead deform primarily through bending, which results in the formation of localized surface crumples. While previous studies have largely focused on understanding the mechanics of crumple formation, here we take a different approach: we investigate how these surface crumples - traditionally viewed as a failure mode - can be harnessed to enable functionality. Specifically, we show how they can be used to design reconfigurable structures across scales and to develop advanced impact-mitigation systems.

Soft actuators, characterized by their compliance and flexibility, have tremendous potential for diverse applications, ranging from medical devices to submarine operations. However, significant challenges remain in the design of these actuators, specifically in maintaining precise control over their mechanical behavior and motion. To date, heuristic methods have been commonly used to design soft actuators, which are potentially incapable of producing designs that achieve specific target behaviors. We propose a gradient-based inverse design framework to synthesize three dimensional soft actuators with tailored mechanical responses. Our design framework utilizes gradient information that captures the inherent geometrical and material nonlinearities of the soft actuator to morph its shape. We exemplify the capabilities of the proposed framework by designing soft actuators with bespoke deformation patterns, making use of sophisticated deformation mechanisms to realize the target behavior. The capabilities of the proposed framework are validated via experimental testing of cast designs, which confirms a strong correlation between measurements and numerical simulations.

Soft actuators, characterized by their compliance and flexibility, have tremendous potential for diverse applications, ranging from medical devices to submarine operations. However, significant challenges remain in the design of these actuators, specifically in maintaining precise control over their mechanical behavior and motion. To date, heuristic methods have been commonly used to design soft actuators, which are potentially incapable of producing designs that achieve specific target behaviors. We propose a gradient-based inverse design framework to synthesize three dimensional soft actuators with tailored mechanical responses. Our design framework utilizes gradient information that captures the inherent geometrical and material nonlinearities of the soft actuator to morph its shape. We exemplify the capabilities of the proposed framework by designing soft actuators with bespoke deformation patterns, making use of sophisticated deformation mechanisms to realize the target behavior. The capabilities of the proposed framework are validated via experimental testing of cast designs, which confirms a strong correlation between measurements and numerical simulations.

ABSTRACT Curved elastic shells can be fabricated through molding or by harnessing residual stresses. These shells often exhibit snap‐through behavior and multistability when loaded. We present a unique way of fabricating curved elastic shells that exhibit multistability and snap‐through behavior, weft‐knitting. The knitting process introduces internal stresses into the textile sheet, which lead to complex 3D curvatures. We explore the relationship between the geometry and the mechanical response, identifying a parameter space where the textiles are multistable. We harness the snapping behavior and shape change through multistability to design soft conductive switches with built‐in haptic feedback and incorporate these textile switches into two wearable devices and one reconfigurable lamp. This work will allow us to harness the nonlinear mechanical behavior of textiles to create functional, soft, and seamless devices.

Flexible mechanical structures can undergo large deformations under small loads, enabling large, complex, and nonlinear wave responses under finite-frequency driving. Here, we study a dynamically driven canonical flexible mechanical metamaterial composed of rigid squares connected at their corners by flexible hinges. This metamaterial supports a uniform dilational mechanism and, in the limit of ideal joints, exhibits a Poisson ratio of -1. The presence of this dilational mode of deformation gives rise to a conformal symmetry, in which the dynamics are approximately invariant under a wide class of physical transformations -- conformal maps. We find that the low-frequency response of the system is dominated by conformal deformations consisting of spatially varying rotations and dilations concentrated at the boundary. Even at high frequencies, each conformal map implies a conserved spatially complex momentum. We explore how experimental parameters such as material stiffnesses and the geometry and number of unit cells allow experimental conformal momenta to approach this conservation, varying slowly compared to the non-conformal momenta of same order. These results constitute a new framework opening fundamental avenues for the study of conformal wave phenomena in dilational metamaterials as well as potential strategies for controlling nonlinear waves and vibrations.

Flexible mechanical structures can undergo large deformations under small loads, enabling large, complex, and nonlinear wave responses under finite-frequency driving. Here, we study a dynamically driven canonical flexible mechanical metamaterial composed of rigid squares connected at their corners by flexible hinges. This metamaterial supports a uniform dilational mechanism and, in the limit of ideal joints, exhibits a Poisson ratio of -1. The presence of this dilational mode of deformation gives rise to a conformal symmetry, in which the dynamics are approximately invariant under a wide class of physical transformations -- conformal maps. We find that the low-frequency response of the system is dominated by conformal deformations consisting of spatially varying rotations and dilations concentrated at the boundary. Even at high frequencies, each conformal map implies a conserved spatially complex momentum. We explore how experimental parameters such as material stiffnesses and the geometry and number of unit cells allow experimental conformal momenta to approach this conservation, varying slowly compared to the non-conformal momenta of same order. These results constitute a new framework opening fundamental avenues for the study of conformal wave phenomena in dilational metamaterials as well as potential strategies for controlling nonlinear waves and vibrations.

In a world where engineering innovation is growing in parallel with pollution and climate changes caused by its byproducts, researchers face an increasing urgency to develop new materials and systems that are environmentally friendly. While many robots have historically been designed to perform challenging or dangerous tasks in remote environments, deployed robots will contribute to pollution if they are lost or not retrieved in full. In this work, we propose a design scheme for replacing synthetic components of a known soft robot with entirely biodegradable alternatives to demonstrate the opportunities for systems that can be deployed without concern for robot failure or retrieval. Additionally, our robot demonstrates programmable end-of-life biodegradation, in this case when in contact with water. Our robust yet biodegradable robot remains operable for days under standard conditions. Upon contact with the programmed stimulus, water, the robot ceases operation and begins degrading, allowing future on-board payloads to be selectively deposited into a water-rich area. This robot demonstrates critical components for our long-term vision of future robot designs that can be rapidly prototyped and deployed to complete tasks autonomously without consideration for post-operation collection and failure contingencies.

ABSTRACT Achieving true mechanical reprogrammability — where structural functions can be dynamically defined, modified, and accessed on demand — requires the ability to arbitrarily set and alter the states of arrays of mechanical bits. Here, we introduce a new approach that accomplishes exactly this by exploiting asynchronous symmetry breaking in oscillator arrays driven by a single global actuator. Using an array of pendula as an experimental model, we show that intrinsic frequency separation enables arbitrary information writing and even makes possible the realization of a mechanical piano. Because the system is controlled exclusively through the timing of a global actuation signal, this strategy offers a scalable and efficient route toward reprogrammable matter, with applicability across elastic structures, chemical oscillators, and electronic circuits.

Transition waves in mechanical metamaterials manifest themselves as propagating interfaces between different stable states in lattices composed of arrays of coupled, intrinsically bistable elements. Here, we show experimentally and numerically that arrays of elastic unit cells that are individually monostable, yet whose energy landscapes can be programmed through interactions with neighboring units, provide a rich and largely unexplored platform for transition wave propagation. We implement this concept by designing a unit cell comprising a von Mises truss supported by two vertical elastic beams. In one-dimensional arrays of such units, we demonstrate that each cell's energy landscape can change from monostable to bistable depending on the state of its neighbors. This neighbor-programmable energy landscape enables the controlled initiation and propagation of transition waves, giving rise to highly discrete, directionally unbiased, domino-like wave propagation. Experiments and numerical simulations show that the existence and speed of the waves are governed by geometric design and mass distribution. Our results establish neighboring effects as a distinct mechanism for transition wave propagation, expanding the design space of mechanical metamaterials beyond architectures that rely on intrinsically multistable building blocks.

Transition waves in mechanical metamaterials manifest themselves as propagating interfaces between different stable states in lattices composed of arrays of coupled, intrinsically bistable elements. Here, we show experimentally and numerically that arrays of elastic unit cells that are individually monostable, yet whose energy landscapes can be programmed through interactions with neighboring units, provide a rich and largely unexplored platform for transition wave propagation. We implement this concept by designing a unit cell comprising a von Mises truss supported by two vertical elastic beams. In one-dimensional arrays of such units, we demonstrate that each cell's energy landscape can change from monostable to bistable depending on the state of its neighbors. This neighbor-programmable energy landscape enables the controlled initiation and propagation of transition waves, giving rise to highly discrete, directionally unbiased, domino-like wave propagation. Experiments and numerical simulations show that the existence and speed of the waves are governed by geometric design and mass distribution. Our results establish neighboring effects as a distinct mechanism for transition wave propagation, expanding the design space of mechanical metamaterials beyond architectures that rely on intrinsically multistable building blocks.

Squeaking is a constant companion in various aspects of our daily lives, whether we slide rubber-soled shoes across hardwood floors1, scrape chalk on a blackboard2, engage the brakes on a bicycle3 or walk with a hip replacement4,5. When two rigid bodies slide over each other, squeaking is widely understood to result from self-excited stick–slip oscillations, triggered by a decrease in the friction coefficient with increasing slip velocity6, 7, 8, 9–10. However, sliding of extended interfaces can involve crack or slip-pulse propagation11, 12, 13, 14, 15, 16, 17, 18, 19, 20–21. This distinction is amplified when a soft body slides on a rigid one, in which large deformations and material mismatch can cause detachment by opening slip pulses22, 23, 24, 25, 26–27. Previous studies focused mainly on slow sliding17,26,28, 29, 30, 31, 32, 33–34, in which pulses are slow and squeaking is absent. Although squeaking at soft–rigid interfaces has been linked to stick–slip oscillations35, 36–37, the mechanisms remain unclear. Here we experimentally investigate soft–rigid interfaces sliding at velocities that produce squeaking. High-speed imaging and acoustic analysis show that opening pulses propagate at approximately the shear wave speed of the soft material, mediating local slip across diverse materials. In flat samples, these pulses are irregular and generate broadband acoustic emissions. Introducing thin surface ridges confines pulse propagation, yielding a consistent repetition frequency matching the first shear mode of the sliding block and squeaking at that frequency. These findings show a structure-driven mechanism that stabilizes rupture in bimaterial friction. Geometric confinement suppresses competing modes, transforming irregular two-dimensional dynamics into coherent one-dimensional pulse trains, offering new insights into frictional rupture from engineered surfaces to geological faults. High-speed imaging reveals that the squeak of soft–rigid frictional interfaces, like sneakers sliding on a basketball court, arises from intersonic opening slip pulses—analogous to earthquake ruptures—that thin ridges on the rubber confine to repeat at a musical frequency.

Post-processing of all data in this dataset were completed using the code dynamic-conformal-metamaterials developed for the paper. This dataset can be loaded and analyzed using dynamic-conformal-metamaterials with the following steps: Install dynamic-conformal-metamaterials. Download data.zip from this dataset. Extract data.zip and place its content in the repository's root folder. Load the data associated with each analysis shown in the paper using the notebooks. Run cells in notebooks>nonlinear_notebooks>dynamics_of_conformal_metamaterials.ipynb to generate and save all reaction force data prior to running any cells in linear_notebooks and/or conformalanalysis_notebooks

Post-processing of all data in this dataset were completed using the code dynamic-conformal-metamaterials developed for the paper. This dataset can be loaded and analyzed using dynamic-conformal-metamaterials with the following steps: Install dynamic-conformal-metamaterials. Download data.zip from this dataset. Extract data.zip and place its content in the repository's root folder. Load the data associated with each analysis shown in the paper using the notebooks. Run cells in notebooks>nonlinear_notebooks>dynamics_of_conformal_metamaterials.ipynb to generate and save all reaction force data prior to running any cells in linear_notebooks and/or conformalanalysis_notebooks

Abstract The concept of cloaking—hiding objects from external detection—has seen wide success in linear systems. Yet, translating these advancements to nonlinear mechanical systems remains an open challenge. Here, we present a new approach to nonlinear mechanical cloaking that frames cloaking as an optimization problem aimed at replicating a target mechanical response. This problem is solved using a differentiable simulation framework coupled with gradient‐based optimization. This approach is implemented in a class of mechanical metamaterials constructed from rigid units with elastic couplings that support large deformation and contact interactions. Using both numerical simulations and physical experiments, optimal cloak structures are designed that effectively mask internal inhomogeneities and shield against external mechanical disturbances both in static and dynamic regimes. This approach provides a versatile design paradigm for creating mechanical systems with integrated cloaking functionality across a broad range of loading scenarios.

Aerodynamic Surface Dimpling Static aerodynamic surfaces cannot adapt to dynamic wind profiles, limiting performance under variable operating conditions. In article number 2505817, Katia Bertoldi and co-workers introduce a stretch-induced dimpling textile metamaterial that tunes aerodynamic properties even when body-conformed. Wind-tunnel tests and simulations show drag modulation up to 20%. Active stretching enables optimal performance across variable speeds, opening transformative applications in wearables, aerospace, maritime, and civil engineering.

Programming physical intelligence into mechanisms holds great promise for machines that can accomplish tasks such as navigation of unstructured environments while utilizing a minimal amount of computational resources and electronic components. In this study, we introduce a design approach for physically intelligent underactuated mechanisms capable of autonomously adjusting their motion in response to environmental interactions. Specifically, multistability is harnessed to sequence the motion of different degrees of freedom in a programmed order. A key aspect of this approach is that this order can be passively reprogrammed through mechanical stimuli arising from interactions with the environment. To showcase our approach, we construct a mechanism that passively sorts objects based on their mass and a four-degree-of-freedom robot capable of autonomously moving away from obstacles. Remarkably, these devices operate without relying on traditional computational architectures and utilize only a single linear actuator.

The concept of cloaking -- hiding objects from external detection -- has seen wide success in linear systems. Yet, translating these advancements to nonlinear mechanical systems remains an open challenge. Here, we present a new approach to nonlinear mechanical cloaking that frames cloaking as an optimization problem aimed at replicating a target mechanical response. We solve this problem using a differentiable simulation framework coupled with gradient-based optimization. We implement this approach in a class of mechanical metamaterials constructed from rigid units with elastic couplings that support large deformation and contact interactions. Using both numerical simulations and physical experiments, we design optimal cloak structures that effectively mask internal inhomogeneities and shield against external mechanical disturbances both in static and dynamic regimes. This approach provides a versatile design paradigm for creating mechanical systems with integrated cloaking functionality across a broad range of loading scenarios.

Mechanical metamaterials composed of bistable elements have recently emerged as promising platforms for mechanical memory. Traditional approaches to writing information in these systems typically rely on localized actuation or predefined coupling schemes, which are often labor-intensive or lack adaptability. In this work, we introduce a one-dimensional metamaterial consisting of mass-in-mass bistable units that are statically decoupled yet dynamically switchable, allowing arbitrary mechanical information to be encoded through nonlinear waves applied at the boundary of the system. Through a combination of experiments and simulations, we demonstrate that tailored input signals can selectively trigger state transitions deep within the structure, enabling remote and programmable bit writing. This approach opens a new avenue for mechanical memory, harnessing the robustness of bistable elements and the tunability of nonlinear wave-driven actuation.

Mechanical metamaterials use geometric design to achieve unconventional properties, such as high strength at low density, efficient waveguiding and complex shape morphing. The ability to control changes in shape builds on the complex relationship between geometry and nonlinear mechanics, and opens new possibilities for disruptive technologies across diverse fields, including wearable devices, medical technology, robotics and beyond. In this Review, we examine the current state of the field of shape-morphing metamaterials and propose a unified classification system for the mechanisms involved, as well as the design principles underlying them. Specifically, we explore two main categories of unit cells — those that exploit structural anisotropy and those that exploit internal rotations — and two potential approaches to tessellating these cells, based on kinematic compatibility or geometric frustration. We conclude by discussing the available design tools and highlighting emerging challenges in the development of shape-morphing metamaterials. Shape-morphing metamaterials use geometric design to achieve advantageous properties, enabling innovations in fields from robotics to wearable devices. This Review proposes a unified classification of the design principles underlying shape-morphing behaviour, discusses available design tools and highlights emerging challenges in the development of shape-morphing metamaterials.

Nearly all soft wearable robots rely on textiles to distribute actuation forces to the human body; however, the mechanical hysteresis of these materials significantly complicates device control. If not properly accounted for, this history-dependent behavior can result in substantial over-/under-support for which the human user must actively compensate. While a number of hysteresis modeling approaches have been proposed, these techniques are either (a) heuristic-driven and do not accurately reflect the observed physical behavior or (b) rely on complex benchtop calibration procedures that are not amenable to wearable applications where the complete human-robot system must be holistically considered. In this work, we present a new strategy to predict the complex hysteretic response of the combined human-robot system given its full state history using a mathematical technique known as a Preisach model. Our approach is directly personalized to each individual with data collected on the body in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>∼</mml:mo> <mml:mn>90</mml:mn> </mml:math> seconds. We demonstrate the technique with a previously proposed soft wearable robot for shoulder assistance, though the concept is applicable to any joint. To benchmark the efficacy of our approach against previously proposed strategies, we performed an open-loop trajectory tracking procedure with 12 human participants and an articulated mannequin. Our strategy achieved an average shoulder elevation angle tracking accuracy of 5.3° across human participants, representing a significant improvement compared to prior techniques. We anticipate that this new approach will facilitate significantly improved soft wearable robot control by providing reliable estimates of the full hysteretic system response, enabling more robust physical human-robot interaction and coordination.

This work describes a microfluidic high-throughput fabrication method for untethered soft microactuators which, while initially unspecific, develop distinct shapes, surface textures, and actuation modes based on various environmental cues. Analogous to the core concept of cell differentiation, the central idea of this technique is to apply controlled mechanical and chemical stimuli to a deformable hydrogel fiber and transmit the induced geometrical and textural changes to embedded droplets. Using liquid crystal (LC) monomer droplets as a core allows us to orthogonally program the geometric, textural, and molecular architecture of the resulting microactuators upon droplet polymerization. Fine-tuning of the microfluidic parameters yields microdroplets that dry and transform into microparticles with a variety of shapes, including spindle, rod, pancake, dumbbell, pyramid, and worm-like assemblies with a range of aspect ratios. Leveraging mechanical instability via rapid dehydration of hydrogel fibers allows us to generate and impart stable 3D patterns to the core, resulting in microparticles that vary both in global shape and surface texture. After polymerizing these precursor droplets in a magnetic field to encode the mesogenic orientation, LCE microactuators are realized with a rich library of shapes, surface patterns, and molecular structures, each displaying distinct deformations upon heating, validated via finite element analysis.

Static aerodynamic surfaces are inherently limited in their ability to adapt to dynamic velocity profiles or environmental changes, restricting their performance under variable operating conditions. This challenge is particularly pronounced in high-speed competitive sports, such as cycling and downhill skiing, where the properties of a static textile surface are mismatched with highly dynamic wind-speed profiles. Here, an textile metamaterial is introduced that is capable of variable aerodynamic profiles through a stretch-induced dimpling mechanism, even when tightly conformed to a body or object. Wind-tunnel experiments are used to characterize the variable aerodynamic performance of the dimpling mechanism, while Finite Element (FE) simulations efficiently characterize the design space to identify optimal textile metamaterial architectures. By controlling dimple size, the aerodynamic performance of the textile can be tailored for specific wind-speed ranges, resulting in an ability to modulate drag force at target wind-speeds by up to 20%. Furthermore, the potential for active control of a textiles' aerodynamic properties is demonstrated, in which controlled stretching allows the textile to sustain optimal performance across a dynamic wind-speed profile. These findings establish a new approach to aerodynamic metamaterials, with surface dimpling and thus variable fluid-dynamic properties offering transformative applications for wearables, as well as broader opportunities for aerospace, maritime, and civil engineering systems.

The ancient art of origami, traditionally used to transform simple sheets into intricate objects, also holds potential for diverse engineering applications, such as shape morphing and robotics. In this study, it is demonstrated that one of the most basic origami structures-a rigid, foldable degree-four vertex-can be engineered to create a crawler capable of navigating complex paths using only a single input. Through a combination of experimental studies and modeling, it is shown that modifying the geometry of a degree-four vertex enables sheets to move either in a straight line or turn. Furthermore, it is illustrated that leveraging the nonlinearities in folding allows the design of crawlers that can switch between moving straight and turning. Remarkably, these crawling modes can be controlled by adjusting the range of the folding angle's actuation. This study opens avenues for simple machines that can follow intricate trajectories with minimal actuation.

Elastic rods exhibit complex, nonlinear mechanical behaviors, especially under combined axial tension and twisting. Our study focuses on the nonlinear response of double-helix chiral rods, structures that combine a cylindrical core with helically coiled reinforcements. Through experiments, analytical modeling, and finite element simulations, we reveal that twisting induces mechanical instabilities, leading to complex deformation patterns. These instabilities are heavily influenced by the interplay between the core and the helical reinforcements, with the resulting deformations showing strong sensitivity to geometric and material characteristics. The findings enhance our understanding of chiral rods, with potential applications in soft robotics and tunable optical devices.

Multistable Kirigami Metamaterials In article number 2421638, A. Corvi, K. Bertoldi, and A. S. Meeusse present a systematic approach for designing highly multistable mechanical structures. This method, which features a generalized bistable kirigami unit cell along with a set of geometric compatibility rules, enables the creation of arbitrary space-filling tessellations. The resulting metamaterials' mechanics are described using energetic considerations. Unusual emergent resetting behavior as well as generalization to 3D materials are explored.

This study experimentally demonstrates the reprogrammability of a rotating-squares-based mechanical metamaterial with an embedded array of permanent magnets. How the orientation, residual magnetization, and stiffness of the magnets influence both the static and dynamic responses of the metamaterial is systematically investigated. It is showed that by carefully tuning the magnet orientation within the metamaterial, notable tunability of the metamaterial response can be achieved across static and dynamic regimes. More complex magnetic node configurations can optimize specific structural responses by decoupling the tunability of quasi-static stress-strain behavior from energy absorption under impact loading. Additionally, reprogrammability can be further enhanced by an external magnetic field, which modulates magnetic interactions within the structure. This work paves the way for developing engineered structural components with adaptable mechanical responses, reprogrammable through either the redistribution of magnetic elements or the application of an external magnetic field.

Based on flexible mechanical metamaterial platforms that have been proven to be experimentally realizable, that support nonlinear deformations and waves, and that possess a large number of adjustable geometrical design parameters, we propose here to realize mechanical metamaterials for cloaking in the nonlinear regime. Mechanical cloaking in the linear regime has been previously achieved using various strategies, from transformation elasticity to data-driven optimization approaches. However, cloaking in structures that support large-amplitude nonlinear elastic responses remains an open challenge due to the difficulty to design the nonlinear elastic properties of metamaterials. The key point is to implement differentiable simulations for our metamaterial dynamical response and solve an optimization problem to systematically find optimal cloaking structures. The latter is then fabricated and tested under both static and dynamic excitations. Specifically, we realize highly deformable structures capable of hosting “undetectable” inclusions, shielding point excitations, and creating stress-free regions within metamaterial domains. Potential applications include shielding against unwanted vibrations, protecting sensitive sensors, and generating desired haptic feedback in highly deformable robotic systems.

Architected structures such as checkerboard patterns of squares joined by thin ligaments at their corners are capable of nonlinear deformations in which the structure contracts globally in the plane while individual elements counter-rotate. However, recent work shows that under generic driving such structures display a wide variety of nonlinear waves rather than this single global mode. We show that such excitations can be understood in terms of conformal symmetry, in which a wide variety of nonlinear (conformal) deformations are nearly degenerate in energy. This symmetry leads to low-energy surface waves whose dispersion is controlled by the structure's bulk modulus and by the size of the architected unit cell. Simulations and experiments also provide evidence of new conserved quantities, the spatially complex generalizations of linear and angular momentum. This approach provides a new platform for creating and controlling novel nonlinear waves for energy harvesting, sensing, and mode conversion. Conformal symmetry grants a degree of analytic control and predictability for characterizing novel waves in complex structures. [Work supported by Army Research Office (MURI # W911NF2210219)]

Abstract Multistable architected materials are proposed as promising candidates for shock absorption, wave guiding, and shape transformation. However, current design strategies typically rely on one‐off unit cell designs with limited stable states, arranged in periodic tessellations. In this study, a systematic approach is presented for designing planar multistable kirigami metamaterials. This method is based on a bistable triangular kirigami building block, along with a set of compatibility rules that enable the creation of both periodic and aperiodic metamaterials. Guided by an analytical model that captures the nonlinear behavior of the building block, we successfully fabricate and test a diverse range of multistable kirigami metamaterials. Furthermore, the emergent resetting of deployed structures under random actuation is demonstrated, and a generalization to 3D kirigami designs is proposed. This work establishes a versatile design platform for resettable multistable architectures capable of extreme mechanical deformations.

Mechanical metamaterials leverage geometric design to achieve unconventional properties, such as high strength at low density, efficient wave guiding, and complex shape morphing. The ability to control shape changes builds on the complex relationship between geometry and nonlinear mechanics, and opens new possibilities for disruptive technologies across diverse fields, including wearable devices, medical technology, robotics, and beyond. In this review of shape-morphing metamaterials, we examine the current state of the field and propose a unified classification system for the mechanisms involved, as well as the design principles underlying them. Specifically, we explore two main categories of unit cells-those that exploit structural anisotropy or internal rotations-and two potential approaches to tessellating these cells: based on kinematic compatibility or geometric frustration. We conclude by discussing the available design tools and highlighting emerging challenges in the development of shape-morphing metamaterials.

Abstract Mechanical metamaterials—structures with unusual properties that emerge from their internal architecture—that are designed to undergo large deformations typically exploit large internal rotations, and therefore, necessitate the incorporation of flexible hinges. Kirigami structures, made by introducing ordered cuts in a planar material, are one such example. In the mechanism limit, these structures consist of rigid bodies connected by ideal hinges that deform at zero energy cost. However, fabrication in this limit has remained elusive. Here, we demonstrate that the integration of textile hinges provides a scalable platform for creating large kirigami metamaterials with mechanism‐like behaviors. Further, leveraging recently introduced kinematic optimization tools, we show that textile hinges enable extreme shape‐morphing responses, paving the way for the next generation of mechanism‐based metamaterials.

Programmable Instabilities The article number 2406611 by David Melancon, Katia Bertoldi, and co-workers focuses on the highly nonlinear response of elastomeric cylindrical shells during depressurization. Instability-driven deformations are harnessed to build soft machines capable of a programmable sequence of movements with a single actuation input.

The ancient art of origami, traditionally used to transform simple sheets into intricate objects, also holds potential for diverse engineering applications, such as shape morphing and robotics. In this study, we demonstrate that one of the most basic origami structures (i.e., a rigid, foldable degree-four vertex) can be engineered to create a crawler capable of navigating complex paths using only a single input. Through a combination of experimental studies and modeling, we show that modifying the geometry of a degree four vertex enables sheets to move either in a straight line or turn. Furthermore, we illustrate how leveraging the nonlinearities in folding allows the design of crawlers that can switch between moving straight and turning. Remarkably, these crawling modes can be controlled by adjusting the range of the actuation folding angle. Our study opens avenues for simple machines that can follow intricate trajectories with minimal actuation.

Mechanical meta-materials are solids whose geometric structure results in exotic nonlinear behaviors that are not typically achievable via homogeneous materials. We show how to drastically expand the design space of a class of mechanical meta-materials known as cellular solids, by generalizing beyond translational symmetry. This is made possible by transforming a reference geometry according to a divergence free flow that is parameterized by a neural network and equivariant under the relevant symmetry group. We show how to construct flows equivariant to the space groups, despite the fact that these groups are not compact. Coupling this flow with a differentiable nonlinear mechanics simulator allows us to represent a much richer set of cellular solids than was previously possible. These materials can be optimized to exhibit desirable mechanical properties such as negative Poisson's ratios or to match target stress-strain curves. We validate these new designs in simulation and by fabricating real-world prototypes. We find that designs with higher-order symmetries can exhibit a wider range of behaviors.

Harnessing the rich nonlinear dynamics of highly deformable materials has the potential to unlock the next generation of functional smart materials and devices. However, unlocking such potential requires effective strategies to spatially engineer material architectures within the nonlinear dynamic regime. Here we introduce an inverse-design framework to discover flexible mechanical metamaterials with a target nonlinear dynamic response. The desired dynamic task is encoded via optimal tuning of the full-scale metamaterial geometry through an inverse-design approach powered by a fully differentiable simulation environment. By deploying such a strategy, mechanical metamaterials are tailored for energy focusing, energy splitting, dynamic protection and nonlinear motion conversion. Furthermore, our design framework can be expanded to automatically discover reprogrammable architectures capable of switching between different dynamic tasks. For instance, we encode two strongly competing tasks—energy focusing and dynamic protection—within a single architecture, using static precompression to switch between these behaviours. The discovered designs are physically realized and experimentally tested, demonstrating the robustness of the engineered tasks. Our approach opens an untapped avenue towards designer materials with tailored robotic-like reprogrammable functionalities. A framework is presented to automate the design of flexible metamaterial structures that can execute desired nonlinear dynamic tasks and have reprogrammable functionality.

The substantial deformation exhibited by hyperelastic cylindrical shells under pressurization makes them an ideal platform for programmable inflatable structures. If negative pressure is applied, the cylindrical shell will buckle, leading to a sequence of rich deformation modes, all of which are fully recoverable due to the hyperelastic material choice. While the initial buckling event under vacuum is well understood, here, the post-buckling regime is explored and a region in the design space is identified in which a coupled twisting-contraction deformation mode occurs; by carefully controlling the geometry of our homogeneous shells, the proportion of contraction versus twist can be controlled. Additionally, bending as a post-buckling deformation mode can be unlocked by varying the thickness of our shells across the circumference. Since these soft shells can fully recover from substantial deformations caused by buckling, then these instability-driven deformations are harnessed to build soft machines capable of a programmable sequence of movements with a single actuation input.

Mechanical metamaterials -- structures with unusual properties that emerge from their internal architecture -- that are designed to undergo large deformations typically exploit large internal rotations, and therefore, necessitate the incorporation of flexible hinges. In the mechanism limit, these metamaterials consist of rigid bodies connected by ideal hinges that deform at zero energy cost. However, fabrication of structures in this limit has remained elusive. Here, we demonstrate that the fabrication and integration of textile hinges provides a scalable platform for creating large structured metamaterials with mechanism-like behaviors. Further, leveraging recently introduced kinematic optimization tools, we demonstrate that textile hinges enable extreme shape-morphing responses, paving the way for the development of the next generation of mechanism-based metamaterials.

Knitting interloops one-dimensional yarns into three-dimensional fabrics that exhibit behaviour beyond their constitutive materials. How extensibility and anisotropy emerge from the hierarchical organization of yarns into knitted fabrics has long been unresolved. We seek to unravel the mechanical roles of tensile mechanics, assembly and dynamics arising from the yarn level on fabric nonlinearity by developing a yarn-based dynamical model. This physically validated model captures the mechanical response of knitted fabrics, analogous to flexible metamaterials and biological fibre networks due to geometric nonlinearity within such hierarchical systems. Fabric anisotropy originates from observed yarn–yarn rearrangements during alignment dynamics and is topology-dependent. This yarn-based model also provides a design space of knitted fabrics to embed functionalities by varying geometric configuration and material property in instructed procedures compatible to machine manufacturing. Our hierarchical approach to build up a knitted fabric computationally modernizes an ancient craft and represents a first step towards mechanical programmability of knitted fabrics in wide engineering applications.

Motivated by the need to harness the properties of renewable and biodegradable polymers for the design and manufacturing of multi-scale structures with complex geometries, we have employed our additive manufacturing platform that leverages molecular self-assembly for the production of metre-scale structures characterized by complex geometries and heterogeneous material composition. As a precursor material, we used chitosan, a chemically modified form of chitin, an abundant and sustainable structural polysaccharide. We demonstrate the ability to control concentration-dependent crystallization as well as the induction of the preferred orientation of the polymer chains through the combination of extrusion-based robotic fabrication and directional toolpathing. Anisotropy is demonstrated and assessed through high-resolution micro-X-ray diffraction in conjunction with finite element simulations. Using this approach, we can leverage controlled and user-defined small-scale propagation of residual stresses to induce large-scale folding of the resulting structures.