Harnessing plasticity in sequential metamaterials for ideal shock absorption.

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

Non-reciprocal topological solitons in active metamaterials.

From protein motifs1 to black holes2, topological solitons are pervasive nonlinear excitations that are robust and can be driven by external fields3. So far, existing driving mechanisms all accelerate solitons and antisolitons in opposite directions3,4. Here we introduce a local driving mechanism for solitons that accelerates both solitons and antisolitons in the same direction instead: non-reciprocal driving. To realize this mechanism, we construct an active mechanical metamaterial consisting of non-reciprocally coupled oscillators5-8 subject to a bistable potential9-14. We find that such nonlinearity coaxes non-reciprocal excitations-so-called non-Hermitian skin waves5-8,15-22, which are typically unstable-into robust one-way (anti)solitons. We harness such non-reciprocal topological solitons by constructing an active waveguide capable of transmitting and filtering unidirectional information. Finally, we illustrate this mechanism in another class of metamaterials that shows the breaking of 'supersymmetry'23,24 causing only antisolitons to be driven. Our observations and models demonstrate a subtle interplay between non-reciprocity and topological solitons, whereby solitons create their own driving force by locally straining the material. Beyond the scope of our study, non-reciprocal solitons might provide an efficient driving mechanism for robotic locomotion25 and could emerge in other settings, for example, quantum mechanics26,27, optics28-30 and soft matter31.

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

From atomic crystals to animal flocks, the emergence of order in nature is captured by the concept of spontaneous symmetry breaking1-4. However, this cornerstone of physics is challenged when broken symmetry phases are frustrated by geometrical constraints. Such frustration dictates the behaviour of systems as diverse as spin ices5-8, confined colloidal suspensions9 and crumpled paper sheets10. These systems typically exhibit strongly degenerated and heterogeneous ground states and hence escape the Ginzburg-Landau paradigm of phase ordering. Here, combining experiments, simulations and theory we uncover an unanticipated form of topological order in globally frustrated matter: non-orientable order. We demonstrate this concept by designing globally frustrated metamaterials that spontaneously break a discrete [Formula: see text] symmetry. We observe that their equilibria are necessarily heteregeneous and extensively degenerated. We explain our observations by generalizing the theory of elasticity to non-orientable order-parameter bundles. We show that non-orientable equilibria are extensively degenerated due to the arbitrary location of topologically protected nodes and lines where the order parameter must vanish. We further show that non-orientable order applies more broadly to objects that are non-orientable themselves, such as buckled Möbius strips and Klein bottles. Finally, by applying time-dependent local perturbations to metamaterials with non-orientable order, we engineer topologically protected mechanical memories11-19, achieve non-commutative responses and show that they carry an imprint of the braiding of the loads' trajectories. Beyond mechanics, we envision non-orientability as a robust design principle for metamaterials that can effectively store information across scales, in fields as diverse as colloidal science8, photonics20, magnetism7 and atomic physics21.

Elasticity and rheology of auxetic granular metamaterials.

The flowing, jamming, and avalanche behavior of granular materials is satisfyingly universal and vexingly hard to tune: A granular flow is typically intermittent and will irremediably jam if too confined. Here, we show that granular metamaterials made from particles with a negative Poisson's ratio yield more easily and flow more smoothly than ordinary granular materials. We first create a collection of auxetic grains based on a re-entrant mechanism and show that each grain exhibits a negative Poisson's ratio regardless of the direction of compression. Interestingly, we find that the elastic and yielding properties are governed by the high compressibility of granular metamaterials: At a given confinement, they exhibit lower shear modulus, lower yield stress, and more frequent, smaller avalanches than materials made from ordinary grains. We further demonstrate that granular metamaterials promote flow in more complex confined geometries, such as intruder and hopper geometries, even when the packing contains only a fraction of auxetic grains. Moreover, auxetic granular metamaterials exhibit enhanced impact absorption. Our findings blur the boundary between complex fluids and metamaterials and could help in scenarios that involve process, transport, and reconfiguration of granular materials.

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.

Model of Active Solids: Rigid Body Motion and Shape-Changing Mechanisms.

Active solids such as cell collectives, colloidal clusters, and active metamaterials exhibit diverse collective phenomena, ranging from rigid body motion to shape-changing mechanisms. The nonlinear dynamics of such active materials remains, however, poorly understood when they host zero-energy deformation modes and when noise is present. Here, we show that stress propagation in a model of active solids induces the spontaneous actuation of multiple soft floppy modes, even without exciting vibrational modes. By introducing an adiabatic approximation, we map the dynamics onto an effective Landau free energy, predicting mode selection and the onset of collective dynamics. These results open new ways to study and design living and robotic materials with multiple modes of locomotion and shape change.

Thermoresponsive oil-continuous gels based on double-interpenetrating colloidal-particle networks.

Gels composed of multicomponent building blocks offer promising opportunities for the development of novel soft materials with unique and useful structures. While interpenetrating polymer networks have been extensively studied and applied in the creation of these gels, equivalent strategies utilizing colloidal particles have received limited scientific and technological attention. This study presents a novel class of thermo-responsive apolar double gels from interpenetrating networks of attractive colloidal silica and lipid particles. These double gels are easily assembled and suitable for the fabrication of 3D-printed edible soft constructs. Emphasis is focused on the rheological properties and structure emerging on the dilute regime (ϕ ≲ 0.1). Rheological investigations demonstrate that double gels exhibit greater stiffness and resilience to yielding compared to their single lipid gel counterparts. The scaling behavior of the oscillatory linear shear moduli and the critical strain for yielding with volume fraction remain comparable between single and double gels. Creep yielding in double gels exhibits two exponential decay regimes, suggesting the presence of thicker gel strands undergoing flow. Visualization and quantification of the quiescent microstructure confirms the existence of such denser aggregates devoid of larger clusters due to steric hindrance of interpenetrating networks in double gels. This is in stark contrast to lipid single gels where aggregates grow unrestrictedly into larger clusters. Our study constitutes the first demonstration on the assembly of apolar double gel networks as a promising avenue for the design of novel soft materials and foods with tailored structure and mechanics.

Multi-step self-guided pathways for shape-changing metamaterials.

Multi-step pathways-which consist of a sequence of reconfigurations of a structure-are central to the functionality of various natural and artificial systems. Such pathways execute autonomously in self-guided processes such as protein folding1 and self-assembly2-5, but have previously required external control to execute in macroscale mechanical systems, provided by, for example, actuators in robotics6-9 or manual folding in origami8,10-12. Here we demonstrate shape-changing, macroscale mechanical metamaterials that undergo self-guided, multi-step reconfiguration in response to global uniform compression. We avoid the need for external control by using metamaterials that are made purely of passive components. The design of the metamaterials combines nonlinear mechanical elements with a multimodal architecture that enables a sequence of topological reconfigurations caused by the formation of internal self-contacts between the elements of the metamaterial. We realize the metamaterials by using computer-controlled water-jet cutting of flexible materials, and show that the multi-step pathway and final configuration can be controlled by rational design of the nonlinear mechanical elements. We also demonstrate that the self-contacts suppress errors in the pathway. Finally, we create hierarchical architectures to extend the number of distinct reconfiguration steps. Our work establishes general principles for designing mechanical pathways, opening up new avenues for self-folding media11,12, pluripotent materials9,13 and pliable devices14 in areas such as stretchable electronics and soft robotics15.

Oligomodal metamaterials with multifunctional mechanics.

Mechanical metamaterials are artificial composites that exhibit a wide range of advanced functionalities such as negative Poisson's ratio, shape shifting, topological protection, multistability, extreme strength-to-density ratio, and enhanced energy dissipation. In particular, flexible metamaterials often harness zero-energy deformation modes. To date, such flexible metamaterials have a single property, for example, a single shape change, or are pluripotent, that is, they can have many different responses, but typically require complex actuation protocols. Here, we introduce a class of oligomodal metamaterials that encode a few distinct properties that can be selectively controlled under uniaxial compression. To demonstrate this concept, we introduce a combinatorial design space containing various families of metamaterials. These families include monomodal (i.e., with a single zero-energy deformation mode); oligomodal (i.e., with a constant number of zero-energy deformation modes); and plurimodal (i.e., with many zero-energy deformation modes), whose number increases with system size. We then confirm the multifunctional nature of oligomodal metamaterials using both boundary textures and viscoelasticity. In particular, we realize a metamaterial that has a negative (positive) Poisson's ratio for low (high) compression rate over a finite range of strains. The ability of our oligomodal metamaterials to host multiple mechanical responses within a single structure paves the way toward multifunctional materials and devices.

Self-Oscillation and Synchronization Transitions in Elastoactive Structures.

The interplay between activity and elasticity often found in active and living systems triggers a plethora of autonomous behaviors ranging from self-assembly and collective motion to actuation. Among these, spontaneous self-oscillations of mechanical structures is perhaps the simplest and most widespread type of nonequilibrium phenomenon. Yet, we lack experimental model systems to investigate the various dynamical phenomena that may appear. Here, we introduce a centimeter-sized model system for one-dimensional elastoactive structures. We show that such structures exhibit flagellar motion when pinned at one end, self-snapping when pinned at two ends, and synchronization when coupled together with a sufficiently stiff link. We further demonstrate that these transitions can be described quantitatively by simple models of coupled pendula with follower forces. Beyond the canonical case considered here, we anticipate our work to open avenues for the understanding and design of the self-organization and response of active biological and synthetic solids, e.g., in higher dimensions and for more intricate geometries.

Static non-reciprocity in mechanical metamaterials.

Reciprocity is a general, fundamental principle governing various physical systems, which ensures that the transfer function-the transmission of a physical quantity, say light intensity-between any two points in space is identical, regardless of geometrical or material asymmetries. Breaking this transmission symmetry offers enhanced control over signal transport, isolation and source protection. So far, devices that break reciprocity (and therefore show non-reciprocity) have been mostly considered in dynamic systems involving electromagnetic, acoustic and mechanical wave propagation associated with fields varying in space and time. Here we show that it is possible to break reciprocity in static systems, realizing mechanical metamaterials that exhibit vastly different output displacements under excitation from different sides, as well as one-way displacement amplification. This is achieved by combining large nonlinearities with suitable geometrical asymmetries and/or topological features. In addition to extending non-reciprocity and isolation to statics, our work sheds light on energy propagation in nonlinear materials with asymmetric crystalline structures and topological properties. We anticipate that breaking reciprocity will open avenues for energy absorption, conversion and harvesting, soft robotics, prosthetics and optomechanics.

Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial.

Topological edge modes are excitations that are localized at the materials' edges and yet are characterized by a topological invariant defined in the bulk. Such bulk-edge correspondence has enabled the creation of robust electronic, electromagnetic, and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter, and geophysical flows. Recently, the advent of non-Hermitian topological systems-wherein energy is not conserved-has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Hermitian systems have been introduced. However, whether such phases can be experimentally observed, and what their properties are, have remained open questions. Here, we identify and observe a form of bulk-edge correspondence for a particular non-Hermitian topological phase. We find that a change in the bulk non-Hermitian topological invariant leads to a change of topological edge-mode localization together with peculiar purely non-Hermitian properties. Using a quantum-to-classical analogy, we create a mechanical metamaterial with nonreciprocal interactions, in which we observe experimentally the predicted bulk-edge correspondence, demonstrating its robustness. Our results open avenues for the field of non-Hermitian topology and for manipulating waves in unprecedented fashions.

Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials.

Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with conventional statistical and numerical methods. Here we show that convolutional neural networks can learn to recognize these boundaries for combinatorial mechanical metamaterials, down to finest detail, despite using heavily undersampled training sets, and can successfully generalize. This suggests that the network infers the underlying combinatorial rules from the sparse training set, opening up new possibilities for complex design of (meta)materials.

Edible mechanical metamaterials with designed fracture for mouthfeel control.

Metamaterials can display unusual and superior properties that come from their carefully designed structure rather than their composition. Metamaterials have permeated large swatches of science, including electromagnetics and mechanics. Although metamaterials hold the promise for realizing technological advances, their potential to enhance interactions between humans and materials has largely remained unexplored. Here, we devise a class edible mechanical metamaterials with tailored fracture properties to control mouthfeel sensory experience. Using chocolate as a model material, we first demonstrate how to create and control the fracture anisotropy, and the number of cracks, and demonstrate that these properties are captured in mouthfeel experience. We further use topology optimization to rationally design edible metamaterials with maximally anisotropic fracture strength. Our work opens avenues for the use of metamaterials to control fracture and to enhance human-matter interactions.

Combinatorial design of textured mechanical metamaterials.

The structural complexity of metamaterials is limitless, but, in practice, most designs comprise periodic architectures that lead to materials with spatially homogeneous features. More advanced applications in soft robotics, prosthetics and wearable technology involve spatially textured mechanical functionality, which requires aperiodic architectures. However, a naive implementation of such structural complexity invariably leads to geometrical frustration (whereby local constraints cannot be satisfied everywhere), which prevents coherent operation and impedes functionality. Here we introduce a combinatorial strategy for the design of aperiodic, yet frustration-free, mechanical metamaterials that exhibit spatially textured functionalities. We implement this strategy using cubic building blocks-voxels-that deform anisotropically, a local stacking rule that allows cooperative shape changes by guaranteeing that deformed building blocks fit together as in a three-dimensional jigsaw puzzle, and three-dimensional printing. These aperiodic metamaterials exhibit long-range holographic order, whereby the two-dimensional pixelated surface texture dictates the three-dimensional interior voxel arrangement. They also act as programmable shape-shifters, morphing into spatially complex, but predictable and designable, shapes when uniaxially compressed. Finally, their mechanical response to compression by a textured surface reveals their ability to perform sensing and pattern analysis. Combinatorial design thus opens up a new avenue towards mechanical metamaterials with unusual order and machine-like functionalities.

Programmable mechanical metamaterials: the role of geometry.

We experimentally and numerically study the role of geometry for the mechanics of biholar metamaterials, which are quasi-2D slabs of rubber patterned by circular holes of two alternating sizes. We recently showed how the response to uniaxial compression of these metamaterials can be programmed by lateral confinement. In particular, there is a range of confining strains εx for which the resistance to compression becomes non-trivial-non-monotonic or hysteretic-in a range of compressive strains εy. Here we show how the dimensionless geometrical parameters t and χ, which characterize the wall thickness and size ratio of the holes that pattern these metamaterials, can significantly tune these ranges over a wide range. We study the behavior for the limiting cases where the wall thickness t and the size ratio χ become large, and discuss the new physics that arises there. Away from these extreme limits, the variation of the strain ranges of interest is smooth with porosity, but the variation with size ratio evidences a cross-over at low χ from biholar to monoholar (equal sized holes) behavior, related to the elastic instabilities in purely monoholar metamaterials. Our study provides precise guidelines for the rational design of programmable biholar metamaterials, tailored to specific applications, and indicates that the widest range of programmability arises for moderate values of both t and χ.

Discontinuous Buckling of Wide Beams and Metabeams.

We uncover how nonlinearities dramatically alter the buckling of elastic beams. First, we show experimentally that sufficiently wide ordinary elastic beams and specifically designed metabeams-beams made from a mechanical metamaterial-exhibit discontinuous buckling, an unstable form of buckling where the postbuckling stiffness is negative. Then we use simulations to uncover the crucial role of nonlinearities, and show that beams made from increasingly nonlinear materials exhibit an increasingly negative postbuckling slope. Finally, we demonstrate that for sufficiently strong nonlinearity, we can observe discontinuous buckling for metabeams as slender as 1% numerically and 5% experimentally.

Programmable mechanical metamaterials.

We create mechanical metamaterials whose response to uniaxial compression can be programmed by lateral confinement, allowing monotonic, nonmonotonic, and hysteretic behavior. These functionalities arise from a broken rotational symmetry which causes highly nonlinear coupling of deformations along the two primary axes of these metamaterials. We introduce a soft mechanism model which captures the programmable mechanics, and outline a general design strategy for confined mechanical metamaterials. Finally, we show how inhomogeneous confinement can be explored to create multistability and giant hysteresis.

Diffusive kinks turn kirigami into machines.

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