Complex ordered patterns in mechanical instability induced geometrically frustrated triangular cellular structures

Emergent disorder and mechanical memory in periodic metamaterials

Ordered mechanical systems typically have one or only a few stable rest configurations, and hence are not considered useful for encoding memory. Multistable and history-dependent responses usually emerge from quenched disorder, for example in amorphous solids or crumpled sheets. In contrast, due to geometric frustration, periodic magnetic systems can create their own disorder and espouse an extensive manifold of quasi-degenerate configurations. Inspired by the topological structure of frustrated artificial spin ices, we introduce an approach to design ordered, periodic mechanical metamaterials that exhibit an extensive set of spatially disordered states. While our design exploits the correspondence between frustration in magnetism and incompatibility in meta-mechanics, our mechanical systems encompass continuous degrees of freedom, and thus generalize their magnetic counterparts. We show how such systems exhibit non-Abelian and history-dependent responses, as their state can depend on the order in which external manipulations were applied. We demonstrate how this richness of the dynamics enables to recognize, from a static measurement of the final state, the sequence of operations that an extended system underwent. Thus, multistability and potential to perform computation emerge from geometric frustration in ordered mechanical lattices that create their own disorder.

Anisotropic Morphing in Bistable Kirigami through Symmetry Breaking and Geometric Frustration

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

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.

Self-driven configurational dynamics in frustrated spring-mass systems

Various physical systems relax mechanical frustration through configurational rearrangements. We examine such rearrangements via Hamiltonian dynamics of simple internally stressed harmonic four-mass systems. We demonstrate theoretically and numerically how mechanical frustration controls the underlying potential energy landscape. Then, we examine the harmonic four-mass systems' Hamiltonian dynamics and relate the onset of chaotic motion to self-driven rearrangements. We show such configurational dynamics may occur without strong precursors, rendering such dynamics seemingly spontaneous.

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

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

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.

Kirigami-Inspired Biodesign for Applications in Healthcare

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

Hierarchical mechanical metamaterials built with scalable tristable elements for ternary logic operation and amplitude modulation

Multistable mechanical metamaterials are artificial materials whose microarchitectures offer more than two different stable configurations. Existing multistable mechanical metamaterials mainly rely on origami/kirigami-inspired designs, snap-through instability, and microstructured soft mechanisms, with mostly bistable fundamental unit cells. Scalable, tristable structural elements that can be built up to form mechanical metamaterials with an extremely large number of programmable stable configurations remains illusive. Here, we harness the elastic tensile/compressive asymmetry of kirigami microstructures to design a class of scalable X-shaped tristable structures. Using these structure as building block elements, hierarchical mechanical metamaterials with one-dimensional (1D) cylindrical geometries, 2D square lattices, and 3D cubic/octahedral lattices are designed and demonstrated, with capabilities of torsional multistability or independent controlled multidirectional multistability. The number of stable states increases exponentially with the cell number of mechanical metamaterials. The versatile multistability and structural diversity allow demonstrative applications in mechanical ternary logic operators and amplitude modulators with unusual functionalities.

Buckling and metastability in membranes with dilation arrays

We study periodic arrays of impurities that create localized regions of expansion, embedded in two-dimensional crystalline membranes. These arrays provide a simple elastic model of shape memory. As the size of each dilational impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for each of the M impurities to buckle up or down into the third dimension, thus allowing for of order 2^{M} metastable surface configurations corresponding to different impurity "spin" configurations. With both discrete simulations and the nonlinear continuum theory of elastic plates, we explore the buckling of both isolated dilations and dilation arrays at zero temperature, guided by analogies with Ising antiferromagnets. We conjecture ground states for systems with triangular and square impurity superlattices, and comment briefly on the possible behaviors at finite temperatures.

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.

Topology Restricts Quasidegeneracy in Sheared Square Colloidal Ice

Recovery of ground-state degeneracy in two-dimensional square ice is a significant challenge in the field of geometric frustration with far-reaching fundamental implications, such as realization of vertex models and understanding the effect of dimensionality reduction. We combine experiments, theory, and numerical simulations to demonstrate that sheared square colloidal ice partially recovers the ground-state degeneracy for a wide range of field strengths and lattice shear angles. Our method could inspire engineering a novel class of frustrated microstructures and nanostructures based on sheared magnetic lattices in a wide range of soft- and condensed-matter systems.

Isotropic Chiral Acoustic Phonons in 3D Quasicrystalline Metamaterials

The elastic properties of three-dimensional (3D) crystalline mechanical metamaterials, unlike those of amorphous structures, are generally strongly anisotropic-even in the long-wavelength limit and for highly symmetric crystals. Aiming at isotropic linear elastic wave propagation, we therefore study 3D periodic approximants of 3D icosahedral quasicrystalline mechanical metamaterials consisting of uniaxial chiral metarods. Considering the increasing order of the approximants, we approach nearly isotropic effective speeds of sound and isotropic acoustical activity. The latter is directly connected to circularly polarized 3D metamaterial chiral acoustic phonons-for all propagation directions in three dimensions.

Frustration and Atomic Ordering in a Monolayer Semiconductor Alloy

Frustrated interactions can lead to short-range ordering arising from incompatible interactions of fundamental physical quantities with the underlying lattice. The simplest example is the triangular lattice of spins with antiferromagnetic interactions, where the nearest-neighbor spin-spin interactions cannot simultaneously be energy minimized. Here we show that engineering frustrated interactions is a possible route for controlling structural and electronic phenomena in semiconductor alloys. Using aberration-corrected scanning transmission electron microscopy in conjunction with density functional theory calculations, we demonstrate atomic ordering in a two-dimensional semiconductor alloy as a result of the competition between geometrical constraints and nearest-neighbor interactions. Statistical analyses uncover the presence of short-range ordering in the lattice. In addition, we show how the induced ordering can be used as another degree of freedom to considerably modify the band gap of monolayer semiconductor alloys.

Programmable and robust static topological solitons in mechanical metamaterials

Solitary, persistent wave packets called solitons hold potential to transfer information and energy across a wide range of spatial and temporal scales in physical, chemical, and biological systems. Mechanical solitons characteristically emerge either as a single wave packet or uncorrelated propagating topological entities through space and/or time, but these are notoriously difficult to control. Here, we report a theoretical framework for programming static periodic topological solitons into a metamaterial, and demonstrate its implementation in real metamaterials computationally and experimentally. The solitons are excited by deformation localizations under quasi-static compression, and arise from buckling-induced kink-antikink bands that provide domain separation barriers. The soliton number and wavelength demonstrate a previously unreported size-dependence, due to intrinsic length scales. We identify that these unanticipated solitons stem from displacive phase transitions with periodic topological excitations captured by the well-known [Formula: see text] theory. Results reveal pathways for robust regularizations of stochastic responses of metamaterials.

Electrochemically reconfigurable architected materials

Architected materials can actively respond to external stimuli-such as mechanical forces, hydration and magnetic fields-by changing their geometries and thereby achieve novel functionalities. Such transformations are usually binary and volatile because they toggle between 'on' and 'off' states and require persistent external stimuli. Here we develop three-dimensional silicon-coated tetragonal microlattices that transform into sinusoidal patterns via cooperative beam buckling in response to an electrochemically driven silicon-lithium alloying reaction. In situ microscopy reveals a controllable, non-volatile and reversible structural transformation that forms multiple ordered buckling domains separated by distorted domain boundaries. We investigate the mechanical dynamics of individual buckling beams, cooperative coupling among neighbouring beams, and lithiation-rate-dependent distributions of domain sizes through chemo-mechanical modelling and statistical mechanics analysis. Our results highlight the critical role of defects and energy fluctuations in the dynamic response of architected materials. We further demonstrate that domain boundaries can be programmed to form particular patterns by pre-designing artificial defects, and that a variety of reconfigurational degrees of freedom can be achieved through micro-architecture design. This framework enables the design, fabrication, modelling, behaviour prediction and programming of electrochemically reconfigurable architected materials, and could open the way to beyond-intercalation battery electrodes, tunable phononic crystals and bio-implantable devices.

Propagation of pop ups in kirigami shells

Kirigami-inspired metamaterials are attracting increasing interest because of their ability to achieve extremely large strains and shape changes via out-of-plane buckling. While in flat kirigami sheets, the ligaments buckle simultaneously as Euler columns, leading to a continuous phase transition; here, we demonstrate that kirigami shells can also support discontinuous phase transitions. Specifically, we show via a combination of experiments, numerical simulations, and theoretical analysis that, in cylindrical kirigami shells, the snapping-induced curvature inversion of the initially bent ligaments results in a pop-up process that first localizes near an imperfection and then, as the deformation is increased, progressively spreads through the structure. Notably, we find that the width of the transition zone as well as the stress at which propagation of the instability is triggered can be controlled by carefully selecting the geometry of the cuts and the curvature of the shell. Our study significantly expands the ability of existing kirigami metamaterials and opens avenues for the design of the next generation of responsive surfaces as demonstrated by the design of a smart skin that significantly enhances the crawling efficiency of a simple linear actuator.

Evidence for hidden order in a nonlinear model elastic system

Hidden order may arise in strongly correlated systems even if there is an apparent lack of long-range order as measured by local order parameters. This phenomenon has been essentially associated with topological order in quantum systems. Here, we demonstrate the emergence of hidden order in a 1D non-linear classical mechanical system that supports rotational degrees of freedom. The potential energy of the model system creates a bistable system for which hidden order emerges with the introduction of a biquadratic term. To our surprise, we discover that varying the strength of the biquadratic term leads to four distinct phases quantified by the behaviors of the Néel and string order parameters. Three of these phases are locally disordered. Hidden order is identified by a string order parameter that shows correlations with significantly longer range than the Néel order parameter. The hidden order correlation length diverges as the kinetic energy of the system is lowered with a critical exponent ~0.5. The observation of hidden order in a mechanical system reveals that instability and non-linearity may play critical roles in the generation of nonlocal long-range correlations in apparently locally disordered systems.

Bifurcation-based embodied logic and autonomous actuation

Many plants autonomously change morphology and function in response to environmental stimuli or sequences of stimuli. In contrast with the electronically-integrated sensors, actuators, and microprocessors in traditional mechatronic systems, natural systems embody these sensing, actuation, and control functions within their compositional and structural features. Inspired by nature, we embody logic in autonomous systems to enable them to respond to multiple stimuli. Using 3D printable fibrous composites, we fabricate structures with geometries near bifurcation points associated with a transition between bistability and monostability. When suitable stimuli are present, the materials swell anisotropically. This forces a key geometric parameter to pass through a bifurcation, triggering rapid and large-amplitude self-actuation. The actuation time can be programmed by varying structural parameters (from 0.6 to 108 s for millimeter-scale structures). We demonstrate this bioinspired control strategy with examples that respond to their environment according to their embodied logic, without electronics, external control, or tethering.

Compliant rolling-contact architected materials for shape reconfigurability

Architected materials can achieve impressive shape-changing capabilities according to how their microarchitecture is engineered. Here we introduce an approach for dramatically advancing such capabilities by utilizing wrapped flexure straps to guide the rolling motions of tightly packed micro-cams that constitute the material’s microarchitecture. This approach enables high shape-morphing versatility and extreme ranges of deformation without accruing appreciable increases in strain energy or internal stress. Two-dimensional and three-dimensional macroscale prototypes are demonstrated, and the analytical theory necessary to design the proposed materials is provided and packaged as a software tool. An approach that combines two-photon stereolithography and scanning holographic optical tweezers is demonstrated to enable the fabrication of the proposed materials at their intended microscale.

The authors introduce a compliant rolling-contact architected material as a class of shape reconfigurable mechanical metamaterial. They also devise an approach that combines two photon stereolithography and scanning holographic optical tweezers, which allows them to fabricate these architected materials on the microscale.

Autonomous Actuation of Zero Modes in Mechanical Networks Far from Equilibrium

A zero mode, or floppy mode, is a nontrivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this Letter, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by nonequilibrium activity. We find that a correlated active bath actuates an infinitesimal zero mode while simultaneously suppressing fluctuations in higher modes compared to thermal fluctuations, which we experimentally mimic by high frequency shaking of a physical network. Furthermore, self-propulsive dynamics spontaneously mobilize finite mechanisms as exemplified by a self-propelled topological soliton. Nonequilibrium activity thus enables autonomous actuation of coordinated mechanisms engineered through network topology.

Pop-Up Conducting Large-Area Biographene Kirigami

We demonstrate the rapid, large-area transformation of bioenabled graphene laminates into multidimensional geometries for pop-up and stretchable applications. Water-vapor annealing facilitates the controlled plasticization of the multilayered silk-graphene morphologies, allowing highly localized kirigami cuts by programmable drag knife with diverse type and depth of cuts. By adjusting drag-knife depth, we can generate a microscale array of full and partial cuts, enabling a purely topological approach toward the control of metastable fold-unfold states and crack fracture paths in kirigami structures. Through orthogonal control over the graphene-silk composite's nanoscale morphology, cut pattern, and semimetal-like conductivity, we showcase bioenabled laminates as a platform for prospective soft and shape-transforming electronics as flexible interconnects and stretchable energy harvesters.

Analytic analysis of auxetic metamaterials through analogy with rigid link systems

In recent years, many structural motifs have been designed with the aim of creating auxetic metamaterials. One area of particular interest in this subject is the creation of auxetic material properties through elastic instability. Such metamaterials switch from conventional behaviour to an auxetic response for loads greater than some threshold value. This paper develops a novel methodology in the analysis of auxetic metamaterials which exhibit elastic instability through analogy with rigid link lattice systems. The results of our analytic approach are confirmed by finite-element simulations for both the onset of elastic instability and post-buckling behaviour including Poisson's ratio. The method gives insight into the relationships between mechanisms within lattices and their mechanical behaviour; as such, it has the potential to allow existing knowledge of rigid link lattices with auxetic paths to be used in the design of future buckling-induced auxetic metamaterials.

Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks

We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state configurations. By using a real-space shell-expansion method, we compute the exact expression of the entropy for harmonic interactions, while for repulsive harmonic interactions we obtain the entropy arising from a limited subset of the system by numerical integration. We compare these results with a three-dimensional triangular-network model, which shows the same attraction-mediated selection mechanism of the stable phase, and conclude that this effect is general with respect to the dimensionality of the system.

Forming three-dimensional closed shapes from two-dimensional soft ribbons by controlled buckling

Conventional manufacturing techniques-moulding, machining and casting-exist to produce three-dimensional (3D) shapes. However, these industrial processes are typically geared for mass production and are not directly applicable to residential settings, where inexpensive and versatile tools are desirable. Moreover, those techniques are, in general, not adequate to process soft elastic materials. Here, we introduce a new concept of forming 3D closed hollow shapes from two-dimensional (2D) elastic ribbons by controlled buckling. We numerically and experimentally characterize how the profile and thickness of the ribbon determine its buckled shape. We find a 2D master profile with which various elliptical 3D shapes can be formed. More complex natural and artificial hollow shapes, such as strawberry, hourglass and wheel, can also be achieved via strategic design and pattern engraving on the ribbons. The nonlinear response of the post-buckling regime is rationalized through finite-element analysis, which shows good quantitative agreement with experiments. This robust fabrication should complement conventional techniques and provide a rich arena for future studies on the mechanics and new applications of elastic hollow structures.

Creation and Annihilation of Skyrmions in the Frustrated Magnets with Competing Exchange Interactions

In triangular-lattice magnets, the coexistence of third-neighbor antiferromagnetic and nearest-neighbor ferromagnetic exchange interactions can induce rich magnetic phases including noncoplanar skyrmion crystals. Based on Monte Carlo simulation, we studied the dependence of magnetic phase transition on exchange interaction strength. Under the consideration of uniaxial anisotropy and magnetic field both perpendicular to the film plane, a large antiferromagnetic exchange interaction induces a high frustration. When the value of antiferromagnetic exchange interaction is one and a half times larger than the ferromagnetic one, a magnetic phase composed of canting spin stripes, never observed in the chiral magnets, forms. Interestingly, different canting spin stripes along three 120 degree propagation directions may coexist randomly in a magnetic phase, attesting that the canting spin stripes are three-fold degenerate states akin to helices and the multiple state of canting spin stripes is a circular configuration with zero skyrmion charge number. Moreover, skyrmions and antiskyrmions can be observed simultaneously in the configuration at the low temperature nearly close to 0 K, and their configuration and diameter properties are discussed. Finally, the mechanisms of skyrmion creation and annihilation are properly interpreted by comparing exchange and Zeeman energy terms.

Macroscopic chiral symmetry breaking in monolayers of achiral nonconvex platelets

The fabrication of chiral structures using achiral building blocks is a fundamental problem that remains a challenge in materials science. In this work we present a molecular dynamics simulation study of nonconvex polygonal platelets, interacting via soft-repulsive interactions, that are confined in two-dimensional space. These particle models are designed to promote, even at moderate densities, a natural offset displacement between the edges of neighbouring particles. In particular we demonstrate that nonconvex platelets exhibit macroscopic chiral symmetry breaking when the symmetry of the particles equals (or is multiple of) the number of nearest neighbours in the condensed crystalline phase, corresponding to the situation of platelets with 4-, 6-, and 12-fold symmetries.

A class of diatomic 2-D soft granular crystals undergoing pattern transformations

We propose a class of diatomic 2-D soft granular crystals, which features pattern transformation under compression with lateral confinement. The proposed granular crystals are composed of two different types of cylinders: large soft cylinders and small hard cylinders. The pattern-transformable granular crystals are obtained by exploring perturbed packing patterns as potential configurations, and compression with lateral confinement as the driving force of the transition. As a demonstration of the proof-of-concept, we first show the results of desktop-scaled experiments and finite element simulations for a representative case. Then, we present the procedure to obtain these new pattern transformations in soft granular crystals based on the compact packing theory of diatomic circles. The scale-independent compact packing theory serves as an important part of the veiled underlying mechanism of the observed pattern transformations, so the proposed granular crystals can open new avenues in the microstructural design of functional materials towards practical applications.

Cuts Guided Deterministic Buckling in Arrays of Soft Parallel Plates for Multifunctionality

Harnessing buckling instability in soft materials offers an effective strategy to achieve multifunctionality. Despite great efforts in controlling the wrinkling behaviors of film-based systems and buckling of periodic structures, the benefits of classical plate buckling in soft materials remain largely unexplored. The challenge lies in the intrinsic indeterminate characteristics of buckling, leading to geometric frustration and random orientations. Here, we report the controllable global order in constrained buckling of arrays of parallel plates made of hydrogels and elastomers on rigid substrates. By introducing patterned cuts on the plates, the randomly phase-shifted buckling in the array of parallel plates transits to a prescribed and ordered buckling with controllable phases. The design principle for cut-directed deterministic buckling in plates is validated by both mechanics model and finite element simulation. By controlling the contacts and interactions between the buckled parallel plates, we demonstrate on-demand reconfigurable electrical and optical pathways, and the potential application in design of mechanical logic gates. By varying the local stimulus within the plates, we demonstrate that microscopic pathways can be written, visualized, erased, and rewritten macroscopically into a completely new one for potential applications such as soft reconfigurable circuits and logic devices.

Attraction Controls the Inversion of Order by Disorder in Buckled Colloidal Monolayers

We show how including attraction in interparticle interactions reverses the effect of fluctuations in ordering of a prototypical artificial frustrated system. Buckled colloidal monolayers exhibit the same ground state as the Ising antiferromagnet on a deformable triangular lattice, but it is unclear if ordering in the two systems is driven by the same geometric mechanism. By a real-space expansion we find that, for buckled colloids, bent stripes constitute the stable phase, whereas in the Ising antiferromagnet straight stripes are favored. For generic pair potentials we show that attraction governs this selection mechanism, in a manner that is linked to local packing considerations. This supports the geometric origin of entropy in jammed sphere packings and provides a tool for designing self-assembled colloidal structures.

Design and application of 'J-shaped' stress-strain behavior in stretchable electronics: a review

A variety of natural biological tissues (e.g., skin, ligaments, spider silk, blood vessel) exhibit 'J-shaped' stress-strain behavior, thereby combining soft, compliant mechanics and large levels of stretchability, with a natural 'strain-limiting' mechanism to prevent damage from excessive strain. Synthetic materials with similar stress-strain behaviors have potential utility in many promising applications, such as tissue engineering (to reproduce the nonlinear mechanical properties of real biological tissues) and biomedical devices (to enable natural, comfortable integration of stretchable electronics with biological tissues/organs). Recent advances in this field encompass developments of novel material/structure concepts, fabrication approaches, and unique device applications. This review highlights five representative strategies, including designs that involve open network, wavy and wrinkled morphologies, helical layouts, kirigami and origami constructs, and textile formats. Discussions focus on the underlying ideas, the fabrication/assembly routes, and the microstructure-property relationships that are essential for optimization of the desired 'J-shaped' stress-strain responses. Demonstration applications provide examples of the use of these designs in deformable electronics and biomedical devices that offer soft, compliant mechanics but with inherent robustness against damage from excessive deformation. We conclude with some perspectives on challenges and opportunities for future research.

Atomimetic Mechanical Structures with Nonlinear Topological Domain Evolution Kinetics

A mechanical metamaterial, a simple, periodic mechanical structure, is reported, which reproduces the nonlinear dynamic behavior of materials undergoing phase transitions and domain switching at the structural level. Tunable multistability is exploited to produce switching and transition phenomena whose kinetics are governed by the same Allen-Cahn law commonly used to describe material-level, structural-transition processes. The reported purely elastic mechanical system displays several key features commonly found in atomic- or mesoscale physics of solids. The rotating-mass network shows qualitatively analogous features as, e.g., ferroic ceramics or phase-transforming solids, and the discrete governing equation is shown to approach the phase field equation commonly used to simulate the above processes. This offers untapped opportunities for reproducing material-level, dissipative and diffusive kinetic phenomena at the structural level, which, in turn, invites experimental realization and paves the road for new active, intelligent, or phase-transforming mechanical metamaterials bringing small-scale processes to the macroscopically observable scale.

Effect of internal architecture on microgel deformation in microfluidic constrictions

The study of how soft particles deform to pass through narrow openings is important for understanding the transit of biological cells, as well as for designing deformable drug delivery carriers. In this work, we systematically explore how soft microparticles with various internal architectures deform during passage through microfluidic constrictions. We synthesize hydrogel particles with well-defined internal structure using lithography-based UV polymerization in microfluidic channels (stop-flow lithography). Using this in situ technique, we explore a range of 2D particle architectures and their effect on particle deformation. We observe that particles undergo buckling of internal supports and reorient at the constriction entrance in order to adopt preferred shapes that correspond to minimum energy configurations. Using finite element simulations of elastic deformation under compression, we accurately predict the optimal deformation configuration of these structured particles.

Programmable Kiri-Kirigami Metamaterials

Programmable kirigami metamaterials with controllable local tilting orientations on demand through prescribed notches are constructed through a new approach of kiri-kirgami, and their actuation of pore opening via both mechanical stretching and temperature, along with their potential application as skins for energy-saving buildings, is discussed.

Harnessing Geometric Frustration to Form Band Gaps in Acoustic Channel Lattices

We demonstrate both numerically and experimentally that geometric frustration in two-dimensional periodic acoustic networks consisting of arrays of narrow air channels can be harnessed to form band gaps (ranges of frequency in which the waves cannot propagate in any direction through the system). While resonant standing wave modes and interferences are ubiquitous in all the analyzed network geometries, we show that they give rise to band gaps only in the geometrically frustrated ones (i.e., those comprising of triangles and pentagons). Our results not only reveal a new mechanism based on geometric frustration to suppress the propagation of pressure waves in specific frequency ranges but also open avenues for the design of a new generation of smart systems that control and manipulate sound and vibrations.

Identical band gaps in structurally re-entrant honeycombs

Structurally re-entrant honeycomb is a sort of artificial lattice material, characterized by star-like unit cells with re-entrant topology, as well as a high connectivity that the number of folded sheets jointing at each vertex is at least six. In-plane elastic wave propagation in this highly connected honeycomb is investigated through the application of the finite element method in conjunction with the Bloch's theorem. Attention is devoted to exploring the band characteristics of two lattice configurations with different star-like unit cells, defined as structurally square re-entrant honeycomb (SSRH) and structurally hexagonal re-entrant honeycomb (SHRH), respectively. Identical band gaps involving their locations and widths, interestingly, are present in the two considered configurations, attributed to the resonance of the sketch folded sheets, the basic component elements for SSRH and SHRH. In addition, the concept of heuristic models is implemented to elucidate the underlying physics of the identical gaps. The phenomenon of the identical bandgaps is not only beneficial for people to further explore the band characteristics of lattice materials, but also provides the structurally re-entrant honeycombs as potential host structures for the design of lattice-based metamaterials of interest for elastic wave control.

Harnessing Buckling to Design Architected Materials that Exhibit Effective Negative Swelling

Inspired by the need to develop materials capable of targeted and extreme volume changes during operation, numerical simulations and experiments are combined to design a new class of soft architected materials that achieve a reduction of projected surface-area coverage during swelling.

Elastic metamaterials for tuning circular polarization of electromagnetic waves

Electromagnetic resonators are integrated with advanced elastic material to develop a new type of tunable metamaterial. An electromagnetic-elastic metamaterial able to switch on and off its electromagnetic chiral response is experimentally demonstrated. Such tunability is attained by harnessing the unique buckling properties of auxetic elastic materials (buckliballs) with embedded electromagnetic resonators. In these structures, simple uniaxial compression results in a complex but controlled pattern of deformation, resulting in a shift of its electromagnetic resonance, and in the structure transforming to a chiral state. The concept can be extended to the tuning of three-dimensional materials constructed from the meta-molecules, since all the components twist and deform into the same chiral configuration when compressed.

A nonlinear mechanics model of bio-inspired hierarchical lattice materials consisting of horseshoe microstructures

Development of advanced synthetic materials that can mimic the mechanical properties of non-mineralized soft biological materials has important implications in a wide range of technologies. Hierarchical lattice materials constructed with horseshoe microstructures belong to this class of bio-inspired synthetic materials, where the mechanical responses can be tailored to match the nonlinear J-shaped stress-strain curves of human skins. The underlying relations between the J-shaped stress-strain curves and their microstructure geometry are essential in designing such systems for targeted applications. Here, a theoretical model of this type of hierarchical lattice material is developed by combining a finite deformation constitutive relation of the building block (i.e., horseshoe microstructure), with the analyses of equilibrium and deformation compatibility in the periodical lattices. The nonlinear J-shaped stress-strain curves and Poisson ratios predicted by this model agree very well with results of finite element analyses (FEA) and experiment. Based on this model, analytic solutions were obtained for some key mechanical quantities, e.g., elastic modulus, Poisson ratio, peak modulus, and critical strain around which the tangent modulus increases rapidly. A negative Poisson effect is revealed in the hierarchical lattice with triangular topology, as opposed to a positive Poisson effect in hierarchical lattices with Kagome and honeycomb topologies. The lattice topology is also found to have a strong influence on the stress-strain curve. For the three isotropic lattice topologies (triangular, Kagome and honeycomb), the hierarchical triangular lattice material renders the sharpest transition in the stress-strain curve and relative high stretchability, given the same porosity and arc angle of horseshoe microstructure. Furthermore, a demonstrative example illustrates the utility of the developed model in the rapid optimization of hierarchical lattice materials for reproducing the desired stress-strain curves of human skins. This study provides theoretical guidelines for future designs of soft bio-mimetic materials with hierarchical lattice constructions.

Geometry-based control of instability patterns in cellular soft matter

Rationally designing of geometrical features can control the functionality of cellular soft matter.

Topological mechanics of gyroscopic metamaterials

Topological mechanical metamaterials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. Here, we present an experimental and theoretical study of an active metamaterial--composed of coupled gyroscopes on a lattice--that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes that propagate in only one direction and are unaffected by disorder. We present a mathematical model that explains how the edge mode chirality can be switched via controlled distortions of the underlying lattice. This effect allows the direction of the edge current to be determined on demand. We demonstrate this functionality in experiment and envision applications of these edge modes to the design of one-way acoustic waveguides.

Snapping mechanical metamaterials under tension

A snapping mechanical metamaterial is designed, which exhibits a sequential snap-through behavior under tension. The tensile response of this mechanical metamaterial can be altered by tuning the architecture of the snapping segments to achieve a range of nonlinear mechanical responses, including monotonic, S-shaped, plateau, and non-monotonic snap-through behavior.

Point group symmetry and deformation-induced symmetry breaking of superlattice materials

The point group symmetry of materials is closely related to their physical properties and quite important for material modelling. However, superlattice materials have more complex symmetry conditions than crystals due to their multi-level structural feature. Thus, a theoretical framework is proposed to characterize and determine the point group symmetry of non-magnetic superlattice materials systematically. A variety of examples are presented to show the symmetry features of superlattice materials in different dimensions and scales. In addition, the deformation-induced symmetry-breaking phenomenon is also studied for superlattice materials, which has potential application in tuning physical properties by imposing a strain field.

Selective buckling via states of self-stress in topological metamaterials

States of self-stress--tensions and compressions of structural elements that result in zero net forces--play an important role in determining the load-bearing ability of structures ranging from bridges to metamaterials with tunable mechanical properties. We exploit a class of recently introduced states of self-stress analogous to topological quantum states to sculpt localized buckling regions in the interior of periodic cellular metamaterials. Although the topological states of self-stress arise in the linear response of an idealized mechanical frame of harmonic springs connected by freely hinged joints, they leave a distinct signature in the nonlinear buckling behavior of a cellular material built out of elastic beams with rigid joints. The salient feature of these localized buckling regions is that they are indistinguishable from their surroundings as far as material parameters or connectivity of their constituent elements are concerned. Furthermore, they are robust against a wide range of structural perturbations. We demonstrate the effectiveness of this topological design through analytical and numerical calculations as well as buckling experiments performed on two- and three-dimensional metamaterials built out of stacked kagome lattices.

Chiral structures from achiral liquid crystals in cylindrical capillaries

We study chiral symmetry-broken configurations of nematic liquid crystals (LCs) confined to cylindrical capillaries with homeotropic anchoring on the cylinder walls (i.e., perpendicular surface alignment). Interestingly, achiral nematic LCs with comparatively small twist elastic moduli relieve bend and splay deformations by introducing twist deformations. In the resulting twisted and escaped radial (TER) configuration, LC directors are parallel to the cylindrical axis near the center, but to attain radial orientation near the capillary wall, they escape along the radius through bend and twist distortions. Chiral symmetry-breaking experiments in polymer-coated capillaries are carried out using Sunset Yellow FCF, a lyotropic chromonic LC with a small twist elastic constant. Its director configurations are investigated by polarized optical microscopy and explained theoretically with numerical calculations. A rich phenomenology of defects also arises from the degenerate bend/twist deformations of the TER configuration, including a nonsingular domain wall separating domains of opposite twist handedness but the same escape direction and singular point defects (hedgehogs) separating domains of opposite escape direction. We show the energetic preference for singular defects separating domains of opposite twist handedness compared with those of the same handedness, and we report remarkable chiral configurations with a double helix of disclination lines along the cylindrical axis. These findings show archetypally how simple boundary conditions and elastic anisotropy of confined materials lead to multiple symmetry breaking and how these broken symmetries combine to create a variety of defects.

Engineering the shape and structure of materials by fractal cut

In this paper we discuss the transformation of a sheet of material into a wide range of desired shapes and patterns by introducing a set of simple cuts in a multilevel hierarchy with different motifs. Each choice of hierarchical cut motif and cut level allows the material to expand into a unique structure with a unique set of properties. We can reverse-engineer the desired expanded geometries to find the requisite cut pattern to produce it without changing the physical properties of the initial material. The concept was experimentally realized and applied to create an electrode that expands to >800% the original area with only very minor stretching of the underlying material. The generality of our approach greatly expands the design space for materials so that they can be tuned for diverse applications.

Nonlinear conduction via solitons in a topological mechanical insulator

Networks of rigid bars connected by joints, termed linkages, provide a minimal framework to design robotic arms and mechanical metamaterials built of folding components. Here, we investigate a chain-like linkage that, according to linear elasticity, behaves like a topological mechanical insulator whose zero-energy modes are localized at the edge. Simple experiments we performed using prototypes of the chain vividly illustrate how the soft motion, initially localized at the edge, can in fact propagate unobstructed all of the way to the opposite end. Using real prototypes, simulations, and analytical models, we demonstrate that the chain is a mechanical conductor, whose carriers are nonlinear solitary waves, not captured within linear elasticity. Indeed, the linkage prototype can be regarded as the simplest example of a topological metamaterial whose protected mechanical excitations are solitons, moving domain walls between distinct topological mechanical phases. More practically, we have built a topologically protected mechanism that can perform basic tasks such as transporting a mechanical state from one location to another. Our work paves the way toward adopting the principle of topological robustness in the design of robots assembled from activated linkages as well as in the fabrication of complex molecular nanostructures.

Mechanical response of a creased sheet

We investigate the mechanics of thin sheets decorated by noninteracting creases. The system considered here consists of parallel folds connected by elastic panels. We show that the mechanical response of the creased structure is twofold, depending both on the bending deformation of the panels and the hingelike intrinsic response of the crease. We show that a characteristic length scale, defined by the ratio of bending to hinge energies, governs whether the structure's response consists in angle opening or panel bending when a small load is applied. The existence of this length scale is a building block for future works on origami mechanics.