Selective buckling via states of self-stress in topological metamaterials

Strut-Buckling Transformation Enabling Anomalous Density-Scaling Toughening Law in Ultralight Lattice Metamaterials

Lightweight lattice metamaterials attract considerable attention due to their exceptional and tunable mechanical properties. However, their practical application is ultimately limited by their tolerance to inevitable manufacturing defects. Traditional fracture mechanics of lattice metamaterials are confined to localized tensile failure of a crack-tip strut, overlooking the toughening effect of buckling instability in discrete struts around the crack front. Here, via a combination of additive manufacturing, numerical simulation, and theoretical analysis, this work identifies an anomalous power scaling law of specific fracture energy with relative density, where the scaling exponent shifts to negative values below a critical relative density. This anomalous toughening law stems from crack-tip blunting triggered by delocalized strut-buckling transformation at ultralow densities, which is universal across various lattice metamaterials with varying length scales, crack orientations, node connectivity, and component properties. By strategically harnessing strut buckling mechanisms, exceptionally high specific fracture toughness can be achieved at extremely low relative density, thereby addressing gaps in the material property design space. These findings not only provide physical insights into discrete lattice fracture but also offer design motifs for ultralight, ultra-tough lattice metamaterials.

Observation of Floppy Flexural Modes in a 3D Polarized Maxwell Beam

Beams are fundamental objects in solid mechanics, displaying flexural and torsional modes in three dimensions, and support important applications across all fields of engineering. Here, we introduce Maxwell lattice topological mechanics to beams and present a Maxwell beam model that supports topological floppy flexural modes, localized exclusively at one of its ends. We introduce a modified topological index for this Maxwell beam which lacks a complete band gap, and establish a relation between Maxwell topological polarization and frozen evanescent phonons, shedding new light on the bulk origin of the topological localization. The floppy eigenmodes and their exceptional robustness against defects are experimentally validated through vibration measurements on 3D laser-printed samples at kHz frequencies. This study opens new avenues in fields from mechanical and civil engineering to robotics by introducing topologically polarized mechanics in slender structures.

Geometrically Frustrated, Mechanical Metamaterial Membranes: Large-Scale Stress Accumulation and Size-Selective Assembly

We study the effect of geometric frustration on dilational mechanical metamaterial membranes. While shape frustrated elastic plates can only accommodate nonzero Gaussian curvature up to size scales that ultimately vanish with their elastic thickness, we show that frustrated metamembranes accumulate hyperbolic curvatures up to mesoscopic length scales that are ultimately independent of the size of their microscopic constituents. A continuum elastic theory and discrete numerical model describe the size-dependent shape and internal stresses of axisymmetric, trumpetlike frustrated metamembranes, revealing a nontrivial crossover to a much weaker power-law growth in elastic strain energy with size than in frustrated elastic membranes. We study a consequence of this for the self-limiting assembly thermodynamics of frustrated trumpets, showing a severalfold increase in the size range of self-limitation of metamembranes relative to elastic membranes.

Microstructure-Reconfigured Graphene Oxide Aerogel Metamaterials for Ultrarobust Directional Sensing at Human-Machine Interfaces

Graphene aerogels hold huge promise for the development of high-performance pressure sensors for future human-machine interfaces due to their ordered microstructure and conductive network. However, their application is hindered by the limited strain sensing range caused by the intrinsic stiffness of the porous microstructure. Herein, an anisotropic cross-linked chitosan and reduced graphene oxide (CCS-rGO) aerogel metamaterial is realized by reconfiguring the microstructure from a honeycomb to a buckling structure at the dedicated cross-section plane. The reconfigured CCS-rGO aerogel shows directional hyperelasticity with extraordinary durability (no obvious structural damage after 20 000 cycles at a directional compressive strain of ≤0.7). The CCS-rGO aerogel pressure sensor exhibits an ultrahigh sensitivity of 121.45 kPa-1, an unprecedented sensing range, and robust mechanical and electrical performance. The aerogel sensors are demonstrated to monitor human motions, control robotic hands, and even integrate into a flexible keyboard to play music, which opens a wide application potential in future human-machine interfaces.

Fully Polarized Topological Isostatic Metamaterials in Three Dimensions

Topological surface states are unique to topological materials and are immune to disturbances. In isostatic lattices, mechanical topological floppy modes exhibit softness depending on the polarization relative to the terminating surface. However, in three dimensions, the polarization of topological floppy modes is disrupted by the ubiquitous mechanical Weyl lines. Here, we demonstrate, both theoretically and experimentally, the fully polarized topological mechanical phases free of Weyl lines. Floppy modes emerge exclusively on a particular surface of the three-dimensional isostatic structure, leading to the strongly asymmetric stiffness between opposing boundaries. Additionally, uniform soft strains can reversibly shift the lattice configuration to Weyl phases, switching the stiffness contrast to a trivially comparable level. Our work demonstrates the fully polarized topological mechanical phases in three dimensions, and paves the way towards engineering soft and adaptive metamaterials.

Stress guides in generic static mechanical metamaterials

The confinement of waves within a waveguide can enable directional transmission of signals, which has found wide applications in communication, imaging, and signal isolation. Extending this concept to static systems, where material deformation is piled up along a spatial trajectory, remains elusive due to the sensitivity of localized deformation to structural defects and impurities. Here, we propose a general framework to characterize localized static deformation responses in two-dimensional generic static mechanical metamaterials, by exploiting the duality between space in static systems and time in one-dimensional non-reciprocal wave systems. An internal time-reverse symmetry is developed by the space-time duality. Upon breaking this symmetry, quasi-static load-induced deformation can be guided to travel along a designated path, thereby realizing a stress guide. A combination of time-reverse and inversion symmetries discloses the parity-time symmetry inherent in static systems, which can be leveraged to achieve directional deformation shielding. The tailorable stress guides can find applications in various scenarios, ranging from stress shielding and energy harvesting in structural tasks to information processing in mechanical computing devices.

Training physical matter to matter

Biological systems offer a great many examples of how sophisticated, highly adapted behavior can emerge from training. Here we discuss how training might be used to impart similarly adaptive properties in physical matter. As a special form of materials processing, training differs in important ways from standard approaches of obtaining sought after material properties. In particular, rather than designing or programming the local configurations and interactions of constituents, training uses externally applied stimuli to evolve material properties. This makes it possible to obtain different functionalities from the same starting material (pluripotency). Furthermore, training evolves a material in situ or under conditions similar to those during the intended use; thus, material performance can improve rather than degrade over time. We discuss requirements for trainability, outline recently developed training strategies for creating soft materials with multiple, targeted and adaptable functionalities, and provide examples where the concept of training has been applied to materials on length scales from the molecular to the macroscopic.

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.

Model-free characterization of topological edge and corner states in mechanical networks

Topological materials can host edge and corner states that are protected from disorder and material imperfections. In particular, the topological edge states of mechanical structures present unmatched opportunities for achieving robust responses in wave guiding, sensing, computation, and filtering. However, determining whether a mechanical structure is topologically nontrivial and features topologically protected modes has hitherto relied on theoretical models. This strong requirement has limited the experimental and practical significance of topological mechanics to laboratory demonstrations. Here, we introduce and validate an experimental method to detect the topologically protected zero modes of mechanical structures without resorting to any modeling step. Our practical method is based on a simple electrostatic analogy: Topological zero modes are akin to electric charges. To detect them, we identify elementary mechanical molecules and measure their chiral polarization, a recently introduced marker of topology in chiral phases. Topological zero modes are then identified as singularities of the polarization field. Our method readily applies to any mechanical structure and effectively detects the edge and corner states of regular and higher-order topological insulators. Our findings extend the reach of chiral topological phases beyond designer materials and allow their direct experimental investigation.

Complex Materials with Stochastic Structural Patterns: Spiky Colloids with Enhanced Charge Storage Capacity

Self-assembled materials with complex nanoscale and mesoscale architecture attract considerable attention in energy and sustainability technologies. Their high performance can be attributed to high surface area, quantum effects, and hierarchical organization. Delineation of these contributions is, however, difficult because complex materials display stochastic structural patterns combining both order and disorder, which is difficult to be consistently reproduced yet being important for materials' functionality. Their compositional variability make systematic studies even harder. Here, a model system of FeSe2 "hedgehog" particles (HPs) was selected  to gain insight into the mechanisms of charge storage n complex nanostructured materials common for batteries and supercapacitors. Specifically, HPs represent self-assembled biomimetic nanomaterials with a medium level of complexity; they display an organizational pattern of spiky colloids with considerable disorder yet non-random; this patternt is consistently reproduced from particle to particle. . It was found that HPs can accommodate ≈70× greater charge density than spheroidal nano- and microparticles. Besides expanded surface area, the enhanced charge storage capacity was enabled by improved hole transport and reversible atomic conformations of FeSe2 layers in the blade-like spikes associated with the rotatory motion of the Se atoms around Fe center. The dispersibility of HPs also enables their easy integration into energy storage devices. HPs quadruple stored electrochemical energy and double the storage modulus of structural supercapacitors.

Soft Mechanical Metamaterials with Transformable Topology Protected by Stress Caching

Maxwell lattices possess distinct topological states that feature mechanically polarized edge behaviors and asymmetric dynamic responses protected by the topology of their phonon bands. Until now, demonstrations of non-trivial topological behaviors from Maxwell lattices have been limited to fixed configurations or have achieved reconfigurability using mechanical linkages. Here, a monolithic transformable topological mechanical metamaterial is introduced in the form of a generalized kagome lattice made from a shape memory polymer (SMP). It is capable of reversibly exploring topologically distinct phases of the non-trivial phase space via a kinematic strategy that converts sparse mechanical inputs at free edge pairs into a biaxial, global transformation that switches its topological state. All configurations are stable in the absence of confinement or a continuous mechanical input. Its topologically-protected, polarized mechanical edge stiffness is robust against broken hinges or conformational defects. More importantly, it shows that the phase transition of SMPs that modulate chain mobility, can effectively shield a dynamic metamaterial's topological response from its own kinematic stress history, referred to as "stress caching". This work provides a blueprint for monolithic transformable mechanical metamaterials with topological mechanical behavior that is robust against defects and disorder while circumventing their vulnerability to stored elastic energy, which will find applications in switchable acoustic diodes and tunable vibration dampers or isolators.

Self-Ordering of Buckling, Bending, and Bumping Beams

A collection of thin structures buckle, bend, and bump into each other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system. How and when these patterns form, as well as the force required to pack these structures is not currently understood. Here we study the emergence of order in a canonical example of packing in slender structures, i.e., a system of parallel confined elastic beams. Using tabletop experiments, simulations, and standard theory from statistical mechanics, we predict the amount of confinement (growth or compression) of the beams that will guarantee a global system order, which depends only on the initial geometry of the system. Furthermore, we find that the compressive stiffness and stored bending energy of this metamaterial are directly proportional to the number of beams that are geometrically frustrated at any given point. We expect these results to elucidate the mechanisms leading to pattern formation in these kinds of systems and to provide a new mechanical metamaterial, with a tunable resistance to compressive force.

A hydrogel-based mechanical metamaterial for the interferometric profiling of extracellular vesicles in patient samples

The utility of mechanical metamaterials for biomedical applications has seldom been explored. Here we show that a metamaterial that is mechanically responsive to antibody-mediated biorecognition can serve as an optical interferometric mask to molecularly profile extracellular vesicles in ascites fluid from patients with cancer. The metamaterial consists of a hydrogel responsive to temperature and redox activity functionalized with antibodies to surface biomarkers on extracellular vesicles, and is patterned into micrometric squares on a gold-coated glass substrate. Through plasmonic heating, the metamaterial is maintained in a transition state between a relaxed form and a buckled state. Binding of extracellular vesicles from the patient samples to the antibodies on the hydrogel causes it to undergo crosslinking, induced by free radicals generated via the activity of horseradish peroxidase conjugated to the antibodies. Hydrogel crosslinking causes the metamaterial to undergo fast chiral re-organization, inducing amplified changes in its mechanical deformation and diffraction patterns, which are detectable by a smartphone camera. The mechanical metamaterial may find broad utility in the sensitive optical immunodetection of biomolecules.

Topological transformability and reprogrammability of multistable mechanical metamaterials

Concepts from quantum topological states of matter have been extensively utilized in the past decade to create mechanical metamaterials with topologically protected features, such as one-way edge states and topologically polarized elasticity. Maxwell lattices represent a class of topological mechanical metamaterials that exhibit distinct robust mechanical properties at edges/interfaces when they are topologically polarized. Realizing topological phase transitions in these materials would enable on-and-off switching of these edge states, opening opportunities to program mechanical response and wave propagation. However, such transitions are extremely challenging to experimentally control in Maxwell topological metamaterials due to mechanical and geometric constraints. Here we create a Maxwell lattice with bistable units to implement synchronized transitions between topological states and demonstrate dramatically different stiffnesses as the lattice transforms between topological phases both theoretically and experimentally. By combining multistability with topological phase transitions, this metamaterial not only exhibits topologically protected mechanical properties that swiftly and reversibly change, but also offers a rich design space for innovating mechanical computing architectures and reprogrammable neuromorphic metamaterials. Moreover, we design and fabricate a topological Maxwell lattice using multimaterial 3D printing and demonstrate the potential for miniaturization via additive manufacturing. These design principles are applicable to transformable topological metamaterials for a variety of tasks such as switchable energy absorption, impact mitigation, wave tailoring, neuromorphic metamaterials, and controlled morphing systems.

Omnimodal topological polarization of bilayer networks: Analysis in the Maxwell limit and experiments on a 3D-printed prototype

Periodic networks on the verge of mechanical instability, called Maxwell lattices, are known to exhibit zero-frequency modes localized to their boundaries. Topologically polarized Maxwell lattices, in particular, focus these zero modes to one of their boundaries in a manner that is protected against disorder by the reciprocal-space topology of the lattice's band structure. Here, we introduce a class of mechanical bilayers as a model system for designing topologically protected edge modes that couple in-plane dilational and shearing modes to out-of-plane flexural modes, a paradigm that we refer to as "omnimodal polarization." While these structures exhibit a high-dimensional design space that makes it difficult to predict the topological polarization of generic geometries, we are able to identify a family of mirror-symmetric bilayers that inherit the in-plane modal localization of their constitutive monolayers, whose topological polarization can be determined analytically. Importantly, the coupling between the layers results in the emergence of omnimodal polarization, whereby in-plane and out-of-plane edge modes localize on the same edge. We demonstrate these theoretical results by fabricating a mirror-symmetric, topologically polarized kagome bilayer consisting of a network of elastic beams via additive manufacturing and confirm this finite-frequency polarization via finite element analysis and laser-vibrometry experiments.

Transition fronts and their universality classes

Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

Deep Learning-Accelerated Designs of Tunable Magneto-Mechanical Metamaterials

Metamaterials are artificially structured materials with unusual properties, such as negative Poisson's ratio, acoustic band gap, and energy absorption. However, metamaterials made of conventional materials lack tunability after fabrication. Thus, active metamaterials using magneto-mechanical actuation for untethered, fast, and reversible shape configurations are developed to tune the mechanical response and property of metamaterials. Although the magneto-mechanical metamaterials have shown promising capabilities in tunable mechanical stiffness, acoustic band gaps, and electromagnetic behaviors, the existing demonstrations rely on the forward design methods based on experience or simulations, by which the metamaterial properties are revealed only after the design. Considering the massive design space due to the material and structural programmability, a robust inverse design strategy is desired to create the magneto-mechanical metamaterials with preferred tunable properties. In this work, we develop an inverse design framework where a deep residual network replaces the conventional finite-element analysis for acceleration, realizing metamaterials with predetermined global strains under magnetic actuations. For validation, a direct-ink-writing printing method of the magnetic soft materials is adopted to fabricate the designed complex metamaterials. The deep learning-accelerated design framework opens avenues for the designs of magneto-mechanical metamaterials and other active metamaterials with target mechanical, acoustic, thermal, and electromagnetic properties.

Edge betweenness centrality as a failure predictor in network models of structurally disordered materials

Network theoretical measures such as geodesic edge betweenness centrality (GEBC) have been proposed as failure predictors in network models of load-driven materials failure. Edge betweenness centrality ranks which links are significant, based on the fraction of shortest paths that pass through the links between network nodes. We study GEBC as a failure predictor for two-dimensional fuse network models of load transmission in structurally disordered materials. We analyze the evolution of edge betweenness centrality in the run-up to failure and the correlation between GEBC and failure propensity for both hierarchical and non-hierarchical networks exhibiting various degrees of disorder. We observe a non trivial relationship between GEBC and failure propensity, which suggests that the idea of GEBC as a useful failure predictor needs to be strongly qualified.

Responsive materials architected in space and time

Rationally designed architected materials have attained previously untapped territories in materials property space. The properties and behaviours of architected materials need not be stagnant after fabrication; they can be encoded with a temporal degree of freedom such that they evolve over time. In this Review, we describe the variety of materials architected in both space and time, and their responses to various stimuli, including mechanical actuation, changes in temperature and chemical environment, and variations in electromagnetic fields. We highlight the additive manufacturing methods that can precisely prescribe complex geometries and local inhomogeneities to make such responsiveness possible. We discuss the emergent physics phenomena observed in architected materials that are analogous to those in classical materials, such as the formation and behaviour of defects, phase transformations and topologically protected properties. Finally, we offer a perspective on the future of architected materials that have a degree of intelligence through mechanical logic and artificial neural networks.

Mechanical computing

Mechanical mechanisms have been used to process information for millennia, with famous examples ranging from the Antikythera mechanism of the Ancient Greeks to the analytical machines of Charles Babbage. More recently, electronic forms of computation and information processing have overtaken these mechanical forms, owing to better potential for miniaturization and integration. However, several unconventional computing approaches have recently been introduced, which blend ideas of information processing, materials science and robotics. This has raised the possibility of new mechanical computing systems that augment traditional electronic computing by interacting with and adapting to their environment. Here we discuss the use of mechanical mechanisms, and associated nonlinearities, as a means of processing information, with a view towards a framework in which adaptable materials and structures act as a distributed information processing network, even enabling information processing to be viewed as a material property, alongside traditional material properties such as strength and stiffness. We focus on approaches to abstract digital logic in mechanical systems, discuss how these systems differ from traditional electronic computing, and highlight the challenges and opportunities that they present.

Topological floppy modes in models of epithelial tissues

Recent advances in topological mechanics have revealed unusual phenomena such as topologically protected floppy modes and states of self-stress that are exponentially localized at boundaries and interfaces of mechanical networks. In this paper, we explore the topological mechanics of epithelial tissues, where the appearance of these boundary and interface modes could lead to localized soft or stressed spots and play a role in morphogenesis. We consider both a simple vertex model (VM) governed by an effective elastic energy and its generalization to an active tension network (ATN) which incorporates active adaptation of the cytoskeleton. By analyzing spatially periodic lattices at the Maxwell point of mechanical instability, we find topologically polarized phases with exponential localization of floppy modes and states of self-stress in the ATN when cells are allowed to become concave, but not in the VM.

Recent advances in topological elastic metamaterials

Topological elastic metamaterials have emerged as a new frontier in the quest of topological phases in condensed matter physics. Their exotic topological properties open a wealth of promising engineering-oriented applications that are difficult to realize with traditional elastic metamaterials, such as robust and defect insensitive waveguiding, signal sensing, and splitting. In this review, we retrospectively examine the underlying physical concept of topologically ordered states of elastic waves, starting from the one-dimensional example based on the Su-Schrieffer-Heeger model. We then move on to two-dimensional topological metamaterials, discussing elastic analogues of quantum Hall, pseudospin-Hall, valley-Hall phases. Finally, we survey the latest developments in the field including three-dimensional elastic topological phases and higher-order topological insulators. Altogether, this paper provides a comprehensive overview of the flourishing research frontier on topological elastic metamaterials, and highlights prominent future directions in this field.

Fractional Excitations in Non-Euclidean Elastic Plates

We show that minimal-surface non-Euclidean elastic plates share the same low-energy effective theory as Haldane's dimerized quantum spin chain. As a result, such elastic plates support fractional excitations, which take the form of charge-1/2 solitons between degenerate states of the plate, in strong analogy to their quantum counterpart. These fractional excitations exhibit properties similar to fractional excitations in quantum fractional topological states and in Haldane's dimerized quantum spin chain, including deconfinement and braiding, as well as unique new features such as holographic properties and diodelike nonlinear response, demonstrating great potentials for applications as mechanical metamaterials.

Topology in Nonlinear Mechanical Systems

Many advancements have been made in the field of topological mechanics. The majority of the work, however, concerns the topological invariant in a linear theory. In this Letter, we present a generic prescription to define topological indices that accommodates nonlinear effects in mechanical systems without taking any approximation. Invoking the tools of differential geometry, a Z-valued quantity in terms of a topological index in differential geometry known as the Poincaré-Hopf index, which features the topological invariant of nonlinear zero modes (ZMs), is predicted. We further identify one type of topologically protected solitons that are robust to disorders. Our prescription constitutes a new direction of searching for novel topologically protected nonlinear ZMs in the future.

Real-space origin of topological band gaps, localization, and reentrant phase transitions in gyroscopic metamaterials

Lattices of interacting gyroscopes naturally support band gaps and topologically protected wave transport along material boundaries. Recently the authors and their collaborators found that amorphous arrangements of such coupled gyroscopes also support nontrivial topological phases. In contrast to periodic systems, for which there is a comprehensive understanding and predictive framework for band gaps and band topology, the theory of spectral gaps and topology for amorphous materials remains less developed. Here we use experiments, numerics, and analytic tools to address the relationship between local interactions and nontrivial topology. We begin with a derivation of the equations of motion within the framework of symplectic mechanics. We then present a general method for predicting whether a gap exists and for approximating the Chern number using only local features of a network, bypassing the costly diagonalization of the system's dynamical matrix. Finally we study how strong disorder interacts with band topology in gyroscopic metamaterials and find that amorphous gyroscopic Chern insulators exhibit similar critical behavior to periodic lattices. Our experiments and simulations additionally reveal a topological Anderson insulation transition, wherein disorder drives a trivial phase into a topological one.

Exploring the property space of periodic cellular structures based on crystal networks

Finding genuine novelty in cellular structures is inherently difficult due to the numerous possible topological and geometrical configurations and their complex mechanical and physical interrelations. Here, we draw inspiration from the incredibly rich collection of crystallographic periodic networks that we interpret from a structural point of view to identify and design novel cellular structures with unique properties. We provide a ready-to-use catalog with more than 17,000 unique entries and show how crystallographic symmetries relate to their mechanical properties. Our work provides a foundation to support future applications in science and engineering, ranging from mechanical and optical metamaterials, over bone tissue engineering, to the design of electrochemical devices.

The properties of periodic cellular structures strongly depend on the regular spatial arrangement of their constituent base materials and can be controlled by changing the topology and geometry of the repeating unit cell. Recent advances in three-dimensional (3D) fabrication technologies more and more expand the limits of fabricable real-world architected materials and strengthen the need of novel microstructural topologies for applications across all length scales and fields in both fundamental science and engineering practice. Here, we systematically explore, interpret, and analyze publicly available crystallographic network topologies from a structural point of view and provide a ready-to-use unit cell catalog with more than 17,000 unique entries in total. We show that molecular crystal networks with atoms connected by chemical bonds can be interpreted as cellular structures with nodes connected by mechanical bars. By this, we identify new structures with extremal properties as well as known structures such as the octet-truss or the Kelvin cell and show how crystallographic symmetries are related to the mechanical properties of the structures. Our work provides inspiration for the discovery of novel cellular structures and paves the way for computational methods to explore and design microstructures with unprecedented properties, bridging the gap between microscopic crystal chemistry and macroscopic structural engineering.

Characterization, stability, and application of domain walls in flexible mechanical metamaterials

Domain walls, commonly occurring at the interface of different phases in solid-state materials, have recently been harnessed at the structural scale to enable additional modes of functionality. Here, we combine experimental, numerical, and theoretical tools to investigate the domain walls emerging upon uniaxial compression in a mechanical metamaterial based on the rotating-squares mechanism. We first show that these interfaces can be generated and controlled by carefully arranging a few phase-inducing defects. We establish an analytical model to capture the evolution of the domain walls as a function of the applied deformation. We then employ this model as a guideline to realize interfaces of complex shape. Finally, we show that the engineered domain walls modify the global response of the metamaterial and can be effectively exploited to tune its stiffness as well as to guide the propagation of elastic waves.

Non-Hermitian Band Topology and Skin Modes in Active Elastic Media

Solids built out of active components can exhibit nonreciprocal elastic coefficients that give rise to non-Hermitian wave phenomena. Here, we investigate non-Hermitian effects present at the boundary of two-dimensional active elastic media obeying two general assumptions: their microscopic forces conserve linear momentum and arise only from static deformations. Using continuum equations, we demonstrate the existence of the non-Hermitian skin effect in which the boundary hosts an extensive number of localized modes. Furthermore, lattice models reveal non-Hermitian topological transitions mediated by exceptional rings driven by the activity level of individual bonds.

Foundations for Soft, Smart Matter by Active Mechanical Metamaterials

Emerging interest to synthesize active, engineered matter suggests a future where smart material systems and structures operate autonomously around people, serving diverse roles in engineering, medical, and scientific applications. Similar to biological organisms, a realization of active, engineered matter necessitates functionality culminating from a combination of sensory and control mechanisms in a versatile material frame. Recently, metamaterial platforms with integrated sensing and control have been exploited, so that outstanding non-natural material behaviors are empowered by synergistic microstructures and controlled by smart materials and systems. This emerging body of science around active mechanical metamaterials offers a first glimpse at future foundations for autonomous engineered systems referred to here as soft, smart matter. Using natural inspirations, synergy across disciplines, and exploiting multiple length scales as well as multiple physics, researchers are devising compelling exemplars of actively controlled metamaterials, inspiring concepts for autonomous engineered matter. While scientific breakthroughs multiply in these fields, future technical challenges remain to be overcome to fulfill the vision of soft, smart matter. This Review surveys the intrinsically multidisciplinary body of science targeted to realize soft, smart matter via innovations in active mechanical metamaterials and proposes ongoing research targets that may deliver the promise of autonomous, engineered matter to full fruition.

Automatic design of mechanical metamaterial actuators

Mechanical metamaterial actuators achieve pre-determined input–output operations exploiting architectural features encoded within a single 3D printed element, thus removing the need for assembling different structural components. Despite the rapid progress in the field, there is still a need for efficient strategies to optimize metamaterial design for a variety of functions. We present a computational method for the automatic design of mechanical metamaterial actuators that combines a reinforced Monte Carlo method with discrete element simulations. 3D printing of selected mechanical metamaterial actuators shows that the machine-generated structures can reach high efficiency, exceeding human-designed structures. We also show that it is possible to design efficient actuators by training a deep neural network which is then able to predict the efficiency from the image of a structure and to identify its functional regions. The elementary actuators devised here can be combined to produce metamaterial machines of arbitrary complexity for countless engineering applications.

Efficient strategies to optimize metamaterial design for specific applications are urgently needed despite the rapid progress in this area. Here the authors propose a computational method combining an optimization algorithm with discrete element simulations for the automatic design of mechanical metamaterial actuators.

Symmetries and Dualities in the Theory of Elasticity

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden nonspatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such nonspatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of nonspatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena.

Continuum Theory for Topological Edge Soft Modes

Topological edge zero modes and states of self stress have been intensively studied in discrete lattices at the Maxwell point, offering robust properties concerning surface and interface stiffness and stress focusing. In this Letter, we present a topological elasticity theory for general continuous media where a gauge-invariant bulk topological index independent of microscopic details is defined. This index directly predicts the number of zero modes on edges at long length scales, and it naturally extends to media that deviate from the Maxwell point, depicting how topological zero modes turn into topological soft modes.

Metamaterials with engineered failure load and stiffness

Architected materials or metamaterials have proved to be a very effective way of making materials with unusual mechanical properties. For example, by designing the mesoscale geometry of architected materials, it is possible to obtain extremely high stiffness-to-weight ratio or unusual Poisson's ratio. However, much of this work has focused on designing properties like stiffness and density, and much remains unknown about the critical load to failure. This is the focus of the current work. We show that the addition of local internal prestress in selected regions of architected materials enables the design of materials where the critical load to failure can be optimized independently from the density and/or quasistatic stiffness. We propose a method to optimize the specific load to failure and specific stiffness using sensitivity analysis and derive the maximum bounds on the attainable properties. We demonstrate the method in a 2D triangular lattice and a 3D octahedral truss, showing excellent agreement between experimental and theoretical results. The method can be used to design materials with predetermined fracture load, failure location, and fracture paths.

Corner-Sharing Tetrahedra for Modeling Micro-structure

This paper introduces Corner-Sharing Tetrahedra (CoSTs), a minimalist, constraint-graph representation of micro-structure. CoSTs have built-in structural guarantees, such as connectivity and minimal rigidity. CoSTs form a space, fully accessible via local operations, that is rich enough to design regular or irregular micro-structure at multiple scales within curved objects. All operations are based on efficient local graph manipulation, which also enables efficient analysis and adjustment of static physical properties. Geometric and material detail, parametric or solid splines, can be added locally, on-demand, for example, for printing.

Reversible signal transmission in an active mechanical metamaterial

Mechanical metamaterials are designed to enable unique functionalities, but are typically limited by an initial energy state and require an independent energy input to function repeatedly. Our study introduces a theoretical active mechanical metamaterial that incorporates a biological reaction mechanism to overcome this key limitation of passive metamaterials. Our material allows for reversible mechanical signal transmission, where energy is reintroduced by the biologically motivated reaction mechanism. By analysing a coarse-grained continuous analogue of the discrete model, we find that signals can be propagated through the material by a travelling wave. Analysis of the continuum model provides the region of the parameter space that allows signal transmission, and reveals similarities with the well-known FitzHugh-Nagumo system. We also find explicit formulae that approximate the effect of the time scale of the reaction mechanism on the signal transmission speed, which is essential for controlling the material.

Signatures of Topological Phonons in Superisostatic Lattices

Soft topological surface phonons in idealized ball-and-spring lattices with coordination number z=2d in d dimensions become finite-frequency surface phonons in physically realizable superisostatic lattices with z>2d. We study these finite-frequency modes in model lattices with added next-nearest-neighbor springs or bending forces at nodes with an eye to signatures of the topological surface modes that are retained in the physical lattices. Our results apply to metamaterial lattices, prepared with modern printing techniques, that closely approach isostaticity.

Origami-based impact mitigation via rarefaction solitary wave creation

The principles underlying the art of origami paper folding can be applied to design sophisticated metamaterials with unique mechanical properties. By exploiting the flat crease patterns that determine the dynamic folding and unfolding motion of origami, we are able to design an origami-based metamaterial that can form rarefaction solitary waves. Our analytical, numerical, and experimental results demonstrate that this rarefaction solitary wave overtakes initial compressive strain waves, thereby causing the latter part of the origami structure to feel tension first instead of compression under impact. This counterintuitive dynamic mechanism can be used to create a highly efficient-yet reusable-impact mitigating system without relying on material damping, plasticity, or fracture.

Single-Molecule Study of Peptide Gel Dynamics Reveals States of Prestress

De novo peptide surfactant (I₃K) gels provide an ideal system to study the complex dynamics of lightly cross-linked semiflexible fibers because of their large contour lengths, simple chemistry, and slow dynamics. We used single-molecule fluorescence microscopy to record individual fibers and Fourier decomposition of the fiber dynamics to separate thermal contributions to the persistence length from compressive states of prestress (SPS). Our results show that SPS in the network depend strongly on peptide concentration, buffer, and pH and that the fibril energies in SPS follow a Lévy distribution. The presence of SPS in the network imply that collective states of self-stress are also present. Therefore, semiflexible polymer gels need to be considered as complex load-bearing structures and the mean field models for polymer gel elasticity and dynamics often applied to them will not be fully representative of the behavior at the nanoscale. We quantify the impact of cross-links on reptation tube dynamics, which provides a second population of tube fluctuations in addition to those expected for uncross-linked entangled solutions.

Topology and Geometry of Spin Origami

Kagome antiferromagnets are known to be highly frustrated and degenerate when they possess simple, isotropic interactions. We consider the entire class of these magnets when their interactions are spatially anisotropic. We do so by identifying a certain class of systems whose degenerate ground states can be mapped onto the folding motions of a generalized "spin origami" two-dimensional mechanical sheet. Some such anisotropic spin systems, including Cs_{2}ZrCu_{3}F_{12}, map onto flat origami sheets, possessing extensive degeneracy similar to isotropic systems. Others, such as Cs_{2}CeCu_{3}F_{12}, can be mapped onto sheets with nonzero Gaussian curvature, leading to more mechanically stable corrugated surfaces. Remarkably, even such distortions do not always lift the entire degeneracy, instead permitting a large but subextensive space of zero-energy modes. We show that for Cs_{2}CeCu_{3}F_{12}, due to an additional point group symmetry associated with the structure, these modes are "Dirac" line nodes with a double degeneracy protected by a topological invariant. The existence of mechanical analogs thus serves to identify and explicate the robust degeneracy of the spin systems.

Flexibility of nanolayers and stacks: implications in the nanostructuration of clays

The basic structural units of adsorbing microporous materials such as clays and cementitious materials are flexible nanolayers. The flexibility of these layers is reported to play a crucial role in the structuration of these materials, potentially affecting therefore the thermo-mechanical behavior of such materials. Adsorbed fluids are structured in a discrete number of layers within the space between the nanolayers in these materials. This discrete nature of adsorption states may lead to micro-instabilities due to non-convex energy profiles. The transition between adsorption states may involve the bending of layers. Bending contributes to metastability, which is reported to be a potential source of the irreversibilities notably in clay behavior. In this paper, we determine the bending modulus of clay nanolayers by the combination of plate theory with molecular simulations of sodium montmorillonite. The case of clays is illustrative of the behavior of phyllosilicates (i.e. sheet-silicates) which are ubiquitous minerals in the Earth's crust. We discuss the conditions in which clay particles, i.e. a stack of nanolayers, can be viewed as thin plates. Estimations of the bending modulus according to the hydration state and dimensions of clay particles are provided. We analyze the implications of the flexibility of the layers in the behavior of a stack of layers as well as in the transitions between adsorption states. The energy barrier associated with bending of clay layers and the characteristic length of bending in such transitions are provided. Our results contribute to a better understanding of the nanostructure of layered adsorbing materials.

Edge Modes and Asymmetric Wave Transport in Topological Lattices: Experimental Characterization at Finite Frequencies

Although topological mechanical metamaterials have been extensively studied from a theoretical perspective, their experimental characterization has been lagging. To address this shortcoming, we present a systematic, laser-assisted experimental characterization of topological kagome lattices, aimed at elucidating their in-plane phononic and topological characteristics. We specifically explore the continuum elasticity limit, which is established when the ideal hinges that appear in the theoretical models are replaced by ligaments capable of supporting bending deformation, as observed for instance in realistic physical lattices. We reveal how the zero-energy floppy edge modes predicted for ideal configurations morph into finite-frequency phonon modes that localize at the edges. By probing the lattices with carefully designed excitation signals, we are able to extract and characterize all the features of a complex, low-frequency acoustic regime in which bulk modes and topological edge modes overlap and entangle in response. The experiments provide unequivocal evidence of the existence of strong asymmetric wave transport regimes at finite frequencies.

A study of deformation localization in nonlinear elastic square lattices under compression

The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where a variety of localized deformations were found depending on loading configuration, lattice parameters and boundary conditions. These studies are conducted on other lattice structures, with the objective of identifying and investigating minimal models that exhibit localization, hysteresis and path-dependent behaviour. To this end, we first consider a two-dimensional square lattice consisting of point masses connected by in-plane axial springs and vertical ground springs, which may be considered as a discrete description of an elastic membrane supported by an elastic substrate. Results illustrate that, depending on the relative values of the spring constants, the lattice exhibits in-plane or out-of-plane instabilities leading to localized deformations. This model is further simplified by considering the one-dimensional case of a spring-mass chain sitting on an elastic foundation. A bifurcation analysis of this lattice identifies the stable and unstable branches and sheds light on the mechanism of transition from affine deformation to global or diffuse deformation to localized deformation. Finally, the lattice is further reduced to a minimal four-mass model, which exhibits a deformation qualitatively similar to that in the central part of a longer chain. In contrast to the widespread assumption that localization is induced by defects or imperfections in a structure, this work illustrates that such phenomena can arise in perfect lattices as a consequence of the mode shapes at the bifurcation points.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Topologically protected interface phonons in two-dimensional nanomaterials: hexagonal boron nitride and silicon carbide

We perform both lattice dynamics analysis and molecular dynamics simulations to demonstrate the existence of topologically protected phonon modes in two-dimensional, monolayer hexagonal boron nitride and silicon carbide sheets. The topological phonon modes are found to be localized at an in-plane interface that divides these systems into two regions of distinct valley Chern numbers. The dispersion of this topological phonon mode crosses over the frequency gap, which is opened through analogy with the quantum valley Hall effect by breaking the inversion symmetry of the primitive unit cells. Consequently, vibrational energy with frequency within this gap is topologically protected, resulting in wave propagation that exhibits minimal backscattering, is robust with regard to structural defects such as sharp corners, and exhibits excellent temporal stability. Our findings open up the possibility of actuating and detecting topological phonons in two-dimensional nanomaterials.

Solitons in one-dimensional mechanical linkage

It has been observed that certain classical chains admit topologically protected zero-energy modes that are localized on the boundaries. The static features of such localized modes are captured by linearized equations of motion, but the dynamical features are governed by its nonlinearity. We study quasiperiodic solutions of nonlinear equations of motion of one-dimensional classical chains. Such quasi-periodic solutions correspond to periodic trajectories in the configuration space of the discrete systems, which allows us to define solitons without relying on a continuum theory. Furthermore, we study the dynamics of solitons in inhomogeneous systems by connecting two chains with distinct parameter sets, where transmission or reflection of solitons occurs at the boundary of the two chains.

Geometry of Wave Propagation on Active Deformable Surfaces

Fundamental biological and biomimetic processes, from tissue morphogenesis to soft robotics, rely on the propagation of chemical and mechanical surface waves to signal and coordinate active force generation. The complex interplay between surface geometry and contraction wave dynamics remains poorly understood, but it will be essential for the future design of chemically driven soft robots and active materials. Here, we couple prototypical chemical wave and reaction-diffusion models to non-Euclidean shell mechanics to identify and characterize generic features of chemomechanical wave propagation on active deformable surfaces. Our theoretical framework is validated against recent data from contractile wave measurements on ascidian and starfish oocytes, producing good quantitative agreement in both cases. The theory is then applied to illustrate how geometry and preexisting discrete symmetries can be utilized to focus active elastic surface waves. We highlight the practical potential of chemomechanical coupling by demonstrating spontaneous wave-induced locomotion of elastic shells of various geometries. Altogether, our results show how geometry, elasticity, and chemical signaling can be harnessed to construct dynamically adaptable, autonomously moving mechanical surface waveguides.

Transition mechanism for a periodic bar-and-joint framework with limited degrees of freedom controlled by uniaxial load and internal stiffness

A specific periodic bar-and-joint framework with limited degrees of freedom is shown to have a transition mechanism when subjected to an external force. The static nonlinear elasticity of this framework under a uniaxial load is modelled with the two angular variables specifying the rotation and distortion of the linked square components. Numerically exploring the equilibrium paths then reveals a transition state of the structure at a critical value of the internal stiffness. A simplified formulation of the model with weak nonlinear terms yields an exact solution of its transition state. Load-displacement behaviour and stability for the two systems with or without approximation are analysed and compared.

Capillary-driven binding of thin triangular prisms at fluid interfaces

We observe capillary-driven binding between thin, equilateral triangular prisms at a flat air-water interface. The edge length of the equilateral triangle face is 120 μm, and the thickness of the prism is varied between 2 and 20 μm. For thickness to length (T/L) ratios of 1/10 or less, pairs of triangles preferentially bind in either tip-to-tip or tip-to-midpoint edge configurations; for pairs of prisms of thickness T/L = 1/5, the tip of one triangle binds to any position along the other triangle's edge. The distinct binding configurations for small T/L ratios result from physical bowing of the prisms, a property that arises during their fabrication. When bowed prisms are placed at the air-water interface, two distinct polarity states arise: prisms either sit with their center of mass above or below the interface. The interface pins to the edge of the prism's concave face, resulting in an interface profile that is similar to that of a capillary hexapole, but with important deviations close to the prism that enable directed binding. We present corresponding theoretical and numerical analysis of the capillary interactions between these prisms and show how prism bowing and contact-line pinning yield a capillary hexapole-like interaction that results in the two sets of distinct, highly-directional binding events. Prisms of all T/L ratios self-assemble into space-spanning open networks; the results suggest design parameters for the fabrication of building blocks of ordered open structures such as the Kagome lattice.

Soft self-assembly of Weyl materials for light and sound

Soft materials can self-assemble into highly structured phases that replicate at the mesoscopic scale the symmetry of atomic crystals. As such, they offer an unparalleled platform to design mesostructured materials for light and sound. Here, we present a bottom-up approach based on self-assembly to engineer 3D photonic and phononic crystals with topologically protected Weyl points. In addition to angular and frequency selectivity of their bulk optical response, Weyl materials are endowed with topological surface states, which allow for the existence of one-way channels, even in the presence of time-reversal invariance. Using a combination of group-theoretical methods and numerical simulations, we identify the general symmetry constraints that a self-assembled structure has to satisfy to host Weyl points and describe how to achieve such constraints using a symmetry-driven pipeline for self-assembled material design and discovery. We illustrate our general approach using block copolymer self-assembly as a model system.

Observation of elastic topological states in soft materials

Topological elastic metamaterials offer insight into classic motion law and open up opportunities in quantum and classic information processing. Theoretical modeling and numerical simulation of elastic topological states have been reported, whereas the experimental observation remains relatively unexplored. Here we present an experimental observation and numerical simulation of tunable topological states in soft elastic metamaterials. The on-demand reversible switch in topological phase has been achieved by changing filling ratio, tension, and/or compression of the elastic metamaterials. By combining two elastic metamaterials with distinct topological invariants, we further demonstrate the formation and dynamic tunability of topological interface states by mechanical deformation, and the manipulation of elastic wave propagation. Moreover, we provide a topological phase diagram of elastic metamaterials under deformation. Our approach to dynamically control interface states in soft materials paves the way to various phononic systems involving thermal management and soft robotics requiring better use of energy.

Here the authors present an experimental observation of topological states in soft elastic metamaterials. They show reversibility in topological phases by changing filling ratio, tension and/or compression, while also demonstrating tunability of topological interface states by mechanical deformation.

Amplitude-dependent topological edge states in nonlinear phononic lattices

This work investigates the effect of nonlinearities on topologically protected edge states in one- and two-dimensional phononic lattices. We first show that localized modes arise at the interface between two spring-mass chains that are inverted copies of each other. Explicit expressions derived for the frequencies of the localized modes guide the study of the effect of cubic nonlinearities on the resonant characteristics of the interface, which are shown to be described by a Duffing-like equation. Nonlinearities produce amplitude-dependent frequency shifts, which in the case of a softening nonlinearity cause the localized mode to migrate to the bulk spectrum. The case of a hexagonal lattice implementing a phononic analog of a crystal exhibiting the quantum spin Hall effect is also investigated in the presence of weakly nonlinear cubic springs. An asymptotic analysis provides estimates of the amplitude dependence of the localized modes, while numerical simulations illustrate how the lattice response transitions from bulk-to-edge mode-dominated by varying the excitation amplitude. In contrast with the interface mode of the first example studies, this occurs both for hardening and softening springs. The results of this study provide a theoretical framework for the investigation of nonlinear effects that induce and control topologically protected wave modes through nonlinear interactions and amplitude tuning.