Harnessing buckling to design tunable locally resonant acoustic metamaterials

Tunable elastic wave transmission and resonance in a periodically aligned tube-block structure

A tube-block structure is proposed to realize tunable elastic wave transmission and resonance, consisting of periodically aligned circular tubes sandwiched and joined by two blocks. Finite element simulations for a unit structure are carried out to reveal the frequency dependence of the transmission behavior for the normal incidence of longitudinal and transverse waves in the tube-block structure. As a result, the transmission ratios are found to take multiple local maxima at different peak frequencies. Eigenfrequency analysis shows that the local resonances of the tube and the block surfaces occur at the peak frequencies in the transmission ratios. The peak frequencies originating from the local resonance of the tube depend on its radius and thickness, while those from the resonance on the block surfaces are in good agreement with the theoretical relation between the interval of the periodically aligned tubes and the wavelength of the Rayleigh wave. Furthermore, when the tube-block structure is subjected to compressive loading, the deformation shifts the peak frequencies of the transmission ratio corresponding to the local resonance of the tube. This result implies that the proposed structure has the potential to serve as a tunable meta-interface between solid blocks.

Phase transitions in 2D multistable mechanical metamaterials via collisions of soliton-like pulses

In recent years, mechanical metamaterials have been developed that support the propagation of an intriguing variety of nonlinear waves, including transition waves and vector solitons (solitons with coupling between multiple degrees of freedom). Here we report observations of phase transitions in 2D multistable mechanical metamaterials that are initiated by collisions of soliton-like pulses in the metamaterial. Analogous to first-order phase transitions in crystalline solids, we observe that the multistable metamaterials support phase transitions if the new phase meets or exceeds a critical nucleus size. If this criterion is met, the new phase subsequently propagates in the form of transition waves, converting the rest of the metamaterial to the new phase. More interestingly, we numerically show, using an experimentally validated model, that the critical nucleus can be formed via collisions of soliton-like pulses. Moreover, the rich direction-dependent behavior of the nonlinear pulses enables control of the location of nucleation and the spatio-temporal shape of the growing phase.

Machine learning-enabled constrained multi-objective design of architected materials

Architected materials that consist of multiple subelements arranged in particular orders can demonstrate a much broader range of properties than their constituent materials. However, the rational design of these materials generally relies on experts' prior knowledge and requires painstaking effort. Here, we present a data-efficient method for the high-dimensional multi-property optimization of 3D-printed architected materials utilizing a machine learning (ML) cycle consisting of the finite element method (FEM) and 3D neural networks. Specifically, we apply our method to orthopedic implant design. Compared to uniform designs, our experience-free method designs microscale heterogeneous architectures with a biocompatible elastic modulus and higher strength. Furthermore, inspired by the knowledge learned from the neural networks, we develop machine-human synergy, adapting the ML-designed architecture to fix a macroscale, irregularly shaped animal bone defect. Such adaptation exhibits 20% higher experimental load-bearing capacity than the uniform design. Thus, our method provides a data-efficient paradigm for the fast and intelligent design of architected materials with tailored mechanical, physical, and chemical properties.

Buckling Metamaterials for Extreme Vibration Damping

Damping mechanical resonances is a formidable challenge in an increasing number of applications. Many passive damping methods rely on using low stiffness, complex mechanical structures or electrical systems, which render them unfeasible in many of these applications. Herein, a new method for passive vibration damping, by allowing buckling of the primary load path in mechanical metamaterials and lattice structures, is introduced, which sets an upper limit for vibration transmission: the transmitted acceleration saturates at a maximum value in both tension and compression, no matter what the input acceleration is. This nonlinear mechanism leads to an extreme damping coefficient tanδ ≈ 0.23 in a metal metamaterial-orders of magnitude larger than the linear damping coefficient of traditional lightweight structural materials. This principle is demonstrated experimentally and numerically in free-standing rubber and metal mechanical metamaterials over a range of accelerations. It is also shown that damping nonlinearities even allow buckling-based vibration damping to work in tension, and that bidirectional buckling can further improve its performance. Buckling metamaterials pave the way toward extreme vibration damping without mass or stiffness penalty, and, as such, could be applicable in a multitude of high-tech applications, including aerospace, vehicles, and sensitive instruments.

Bioinspired Soft Elastic Metamaterials for Reconstruction of Natural Hearing

Natural hearing which means hearing naturally like normal people is critical for patients with hearing loss to participate in life. Cochlear implants have enabled numerous severe hearing loss patients to hear voice functionally, while cochlear implant users can hardly distinguish different tones or appreciate music subject to the absence of rate coding and insufficient frequency channels. Here a bioinspired soft elastic metamaterial that reproduces the shape and key functions of the human cochlea is reported. Inspired by human cochlea, the metamaterials are designed to possess graded microstructures with high effective refractive index distributed on a spiral shape to implement position-related frequency demultiplexing, passive sound enhancements of 10 times, and high-speed parallel processing of 168-channel sound/piezoelectric signals. Besides, it is demonstrated that natural hearing artificial cochlea has fine frequency resolution up to 30 Hz, a wide audible range from 150-12 000 Hz, and a considerable output voltage that can activate the auditory pathway in mice. This work blazes a promising trail for reconstruction of natural hearing in patients with severe hearing loss.

Architected material with independently tunable mass, damping, and stiffness via multi-stability and kinematic amplification

We report on a class of architected material lattices that exploit multi-stability and kinematic amplification to independently adjust the local effective mass, damping, and stiffness properties, thereby realizing congruent alterations to the acoustic dispersion response post-fabrication. The fundamental structural tuning element permits a broad range in the effective property space; moreover, its particular design carries the benefit of tuning without altering the original size/shape of the emerging structure. The relation between the tuning element geometry and the achieved variability in effective properties is explored. Bloch's theorem facilitates the dynamic analysis of representative one- and two-dimensional (1D/2D) systems, revealing, e.g., bandgap formation, migration, and closure and positive/negative metadamping in accordance with the tuning element configuration. To demonstrate a utility, we improvise a waveguide by appropriately patterning the tuning element configuration within a 2D system. We believe that the proposed strategy offers a new way to expand the range of performance and functionality of architected materials for elastodynamics.

Propagation of fundamental Lamb modes along the non-principal axes of strain-stiffened soft compressible plates: A numerical investigation

This paper numerically investigates the propagation of elastic plate waves along the non-principal directions in a prestretched compressible material described by the Gent model of hyperelasticity. We formulate the elastic tensor and the underlying wave equations in the Lagrangian space by employing the theory of nonlinear elasticity together with the linearized incremental equations. An extension of the Semi-Analytical Finite Element (SAFE) method is discussed for computing the dispersion characteristics of the two fundamental guided wave modes. The predictive capabilities of the numerical framework are established using the previously published data for a weakly nonlinear as well as hyperelastic material models. Using the numerical framework, we then bring out the effects of applied prestretch, orientation of the propagation direction, and material parameters on the dispersion characteristics of the fundamental Lamb modes. A limiting case of the neo-Hookean material model is first considered for elucidating such implicit dependencies, which are further highlighted by considering the strain-stiffening effect captured through the Gent material model. Our results indicate the existence of a threshold prestretch for which the Gent-type material can encounter a snap-through instability; leading to the change in the dispersion characteristics of the fundamental symmetric Lamb mode.

Dynamic morphological transformations in soft architected materials via buckling instability encoded heterogeneous magnetization

The geometric reconfigurations in three-dimensional morphable structures have a wide range of applications in flexible electronic devices and smart systems with unusual mechanical, acoustic, and thermal properties. However, achieving the highly controllable anisotropic transformation and dynamic regulation of architected materials crossing different scales remains challenging. Herein, we develop a magnetic regulation approach that provides an enabling technology to achieve the controllable transformation of morphable structures and unveil their dynamic modulation mechanism as well as potential applications. With buckling instability encoded heterogeneous magnetization profiles inside soft architected materials, spatially and temporally programmed magnetic inputs drive the formation of a variety of anisotropic morphological transformations and dynamic geometric reconfiguration. The introduction of magnetic stimulation could help to predetermine the buckling states of soft architected materials, and enable the formation of definite and controllable buckling states without prolonged magnetic stimulation input. The dynamic modulations can be exploited to build systems with switchable fluidic properties and are demonstrated to achieve capabilities of fluidic manipulation, selective particle trapping, sensitivity-enhanced biomedical analysis, and soft robotics. The work provides new insights to harness the programmable and dynamic morphological transformation of soft architected materials and promises benefits in microfluidics, programmable metamaterials, and biomedical applications.

On dispersion curve coloring for mechanical metafilters

This paper formalizes smooth curve coloring (i.e., curve identification) in the presence of curve intersections as an optimization problem, and investigates theoretically properties of its optimal solution. Moreover, it presents a novel automatic technique for solving such a problem. Formally, the proposed algorithm aims at minimizing the summation of the total variations over a given interval of the first derivatives of all the labeled curves, written as functions of a scalar parameter. The algorithm is based on a first-order finite difference approximation of the curves and a sequence of prediction/correction steps. At each step, the predicted points are attributed to the subsequently observed points of the curves by solving an Euclidean bipartite matching subproblem. A comparison with a more computationally expensive dynamic programming technique is presented. The proposed algorithm is applied with success to elastic periodic metamaterials for the realization of high-performance mechanical metafilters. Its output is shown to be in excellent agreement with desirable smoothness and periodicity properties of the metafilter dispersion curves. Possible developments, including those based on machine-learning techniques, are pointed out.

Machine learning-based prediction and inverse design of 2D metamaterial structures with tunable deformation-dependent Poisson's ratio

With the aid of recent efficient and prior knowledge-free machine learning (ML) algorithms, extraordinary mechanical properties such as negative Poisson's ratio have extensively promoted the diverse designs of metamaterials with distinctive cellular structures. However, most existing ML approaches applied to the design of metamaterials are primarily based on a single property value with the assumption that the Poisson's ratio of a material is stationary, neglecting the dynamic variability of Poisson's ratio, termed deformation-dependent Poisson's ratio, during the loading process that a metamaterial other than conventional materials may experience. This paper first proposes a crystallographic symmetry-based methodology to build 2D metamaterials with complex but patterned topological structures, and then converts them into computational models suitable for molecular dynamics simulations. Then, we employ an integrated approach, consisting of molecular dynamics simulations capable of generating and collecting a large dataset for training/validation, in addition to ML algorithms (CNN and Cycle-GAN) able to predict the dynamic characteristics of Poisson's ratio and offer the inverse design of a metamaterial structure based on a target quasi-continuous Poisson's ratio-strain curve, to eventually unravel the underlying mechanism and design principles of 2D metamaterial structures with tunable Poisson's ratio. The close match between the predefined Poisson's ratio response and that from the generated structure validates the feasibility of the proposed ML model. Owing to the high efficiency and complete independence from prior knowledge, our proposed approach offers a novel and robust technique for the prediction and inverse design of metamaterial structures with tailored deformation-dependent Poisson's ratio, an unprecedented property attractive in flexible electronics, soft robotics, and nanodevices.

Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach

In this paper, a coherent boundary value problem to model metamaterials' behaviour based on the relaxed micromorphic model is established. This boundary value problem includes well-posed boundary conditions, thus disclosing the possibility of exploring the scattering patterns of finite-size metamaterial specimens. Thanks to the simplified model's structure (few frequency- and angle-independent parameters), we are able to unveil the scattering metamaterial's response for a wide range of frequencies and angles of propagation of the incident wave. These results are an important stepping stone towards the conception of more complex large-scale meta-structures that can control elastic waves and recover energy. This article is part of the theme issue 'Wave generation and transmission in multi-scale complex media and structured metamaterials (part 1)'.

Photo-responsive hydrogel-based re-programmable metamaterials

This paper explores a novel programmable metamaterial using stimuli-responsive hydrogels with a demonstration of bandgap formation and tuning. Specifically, a photo-responsive hydrogel beam that can achieve re-programmable periodicity in geometric and material properties through patterned light irradiation is designed. Hydrogels consist of polymeric networks and water molecules. Many unique properties of hydrogels, including bio-compatibility, stimuli-responsiveness, and low dissipation make them ideal for enabling re-programmable metamaterials for manipulating structural dynamic response and wave propagation characteristics. Bandgap generation and tunability in photo-responsive hydrogel-based metamaterial (in the form of a diatomic phononic chain) as well as the effects of system parameters such as light exposure pattern and photo-sensitive group concentration on the bandgap width and center frequency are systematically studied. In agreement with finite-element model simulations, it is observed that an increase in light exposure region size reduces both the bandgap width and center frequency, while an increase in the concentration of photo-sensitive group increases bandgap width, attenuation and reduces its center frequency. This work unveils the potential of stimuli-response hydrogels as a new class of low-loss soft metamaterials, unlike most other soft materials that are too lossy to sustain and exploit wave phenomena.

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.

Moth wings as sound absorber metasurface

In noise control applications, a perfect metasurface absorber would have the desirable traits of not only mitigating unwanted sound, but also being much thinner than the wavelengths of interest. Such deep-subwavelength performance is difficult to achieve technologically, yet moth wings, as natural metamaterials, offer functionality as efficient sound absorbers through the action of the numerous resonant scales that decorate their wing membrane. Here, we quantify the potential for moth wings to act as a sound-absorbing metasurface coating for acoustically reflective substrates. Moth wings were found to be efficient sound absorbers, reducing reflection from an acoustically hard surface by up to 87% at the lowest frequency tested (20 kHz), despite a thickness to wavelength ratio of up to 1/50. Remarkably, after the removal of the scales from the dorsal surface the wing's orientation on the surface changed its absorptive performance: absorption remains high when the bald wing membrane faces the sound but breaks down almost completely in the reverse orientation. Numerical simulations confirm the strong influence of the air gap below the wing membrane but only when it is adorned with scales. The finding that moth wings act as deep-subwavelength sound-absorbing metasurfaces opens the door to bioinspired, high-performance sound mitigation solutions.

Wave-packet spreading in disordered soft architected structures

We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice, which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of Deng et al. [Nat. Commun. 9, 1 (2018)]. This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi-Pasta-Ulam-Tsingou lattice and a linear Klein-Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial excitations on the rotational DoF, we evoke the nonlinear coupling between the system's translational and rotational DoFs. Our results reveal that such coupling induces rich wave-packet spreading behavior in the presence of strong disorder. In the weakly nonlinear regime, we observe energy spreading only due to the coupling of the two DoFs (per site), which is in contrast to what is known for KG lattices with a single DoF per lattice site, where the spreading occurs due to chaoticity. Additionally, for strong nonlinearities, we show that initially localized wave-packets attain near ballistic behavior in contrast to other known models. We also reveal persistent chaos during energy spreading, although its strength decreases in time as quantified by the evolution of the system's finite-time maximum Lyapunov exponent. Our results show that flexible, disordered, and strongly nonlinear lattices are a viable platform to study energy transport in combination with multiple DoFs (per site), also present an alternative way to control energy spreading in heterogeneous media.

Magnetoactive acoustic metamaterials based on nanoparticle-enhanced diaphragm

Magnetoactive membrane-type acoustic metamaterials are fabricated by coating a layer of magnetic nanoparticles on the polyethylene (PE) membranes and their vibration characters are investigated experimentally. From our experiments, we discovered that, under different magnetic fields by varying the distance between a magnet and the membranes, such membranes exhibit tunable vibration eigenfrequencies (the shift towards lower frequencies), which is caused by the variation of the effective mass density and effective tension coefficient resulted from the second derivative of the magnetic field. The strong magnetic force between the layer of magnetic nanoparticles and the magnet enhances the eigenfrequency shift. A spring oscillator model is proposed and it agrees well with the experimental results. We also experimentally observed that the vibration radius, effective mass density, and effective tension coefficient of the membranes can enormously affect the eigenfrequencies of the membranes. We believe that this type of metamaterials may open up some potential applications for acoustic devices with turntable vibration properties.

Programmable Multistable Perforated Shellular

Developing bistable metamaterials has recently offered a new design paradigm for deployable structures and reusable dampers. While most bistable mechanisms possess inclined/curved struts, a new 3D multistable shellular metamaterial is developed by introducing delicate perforations on the surface of Schwarz's Primitive shellular, integrating the unique properties of shellular materials such as high surface area, stiffness, and energy absorption with the multistability concept. Denoting the fundamental snapping part by motif, certain shellular motifs with elliptical perforations exhibit mechanical bistability. To bring the concept of multistability to a single motif, multistable shellular motifs are developed by introducing multilayer staggered perforations that form hinges and facilitate local instability. Adopting an n-layer staggered perforation (n hinges) design leads to a maximum 2^n-1 stable states within one shellular motif during loading and unloading. Three-directional multistable shellulars are attained by extending the perforation design in three orthogonal directions. Harnessing snap-through and snap-back behaviors and self-contact, the introduced multistable perforated shellulars exhibit strong rigidity both in loading and unloading, and enhanced energy dissipation. The introduced design strategy opens up new horizons for creating multidirectional multistable metamaterials with load bearing capabilities for applications in soft robotics, shape-morphing architectures, and reusable and deployable energy absorbers/dampers.

Buckling-Mediated Phase Transitions in Nano-Electromechanical Phononic Waveguides

Waveguides for mechanical signal transmission in the megahertz to gigahertz regimes enable on-chip phononic circuitry, which brings new capabilities complementing photonics and electronics. Lattices of coupled nano-electromechanical drumhead resonators are suitable for these waveguides due to their high Q-factor and precisely engineered band structure. Here, we show that thermally induced elastic buckling of such resonators causes a phase transition in the waveguide leading to reversible control of signal transmission. Specifically, when cooled, the lowest-frequency transmission band associated with the primary acoustic mode vanishes. Experiments show the merging of the lower and upper band gaps, such that signals remain localized at the excitation boundary. Numerical simulations show that the temperature-induced destruction of the pass band is a result of inhomogeneous elastic buckling, which disturbs the waveguide's periodicity and suppresses the wave propagation. Mechanical phase transitions in waveguides open opportunities for drastic phononic band reconfiguration in on-chip circuitry and computing.

Wavenumber-space band clipping in nonlinear periodic structures

In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumber shift depending on whether the excitation is prescribed as an initial condition or as a boundary condition, respectively. Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonic initial excitations. However, these models are not compatible with harmonic boundary excitations, which represent the conditions encountered in most practical applications. To overcome this limitation, we present a multiple scales framework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomic chains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersion correction effect, which we term wavenumber-space band clipping. We then extend the framework to locally resonant periodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuning capability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.

Metamaterial-like aerogels for broadband vibration mitigation

We report a mechanical metamaterial-like behavior as a function of the micro/nanostructure of otherwise chemically identical aliphatic polyurea aerogels. Transmissibility varies dramatically with frequency in these aerogels. Broadband vibration mitigation is provided at low frequencies (500-1000 Hz) through self-assembly of locally resonant metastructures wherein polyurea microspheres are embedded in a polyurea web-like network. A micromechanical constitutive model based on a discrete element method is established to explain the vibration mitigation mechanism. Simulations confirm the metamaterial-like behavior with a negative dynamic material stiffness for the micro-metastructured aerogels in a much wider frequency range than the majority of previously reported locally resonant metamaterials.

Nonlinear harmonic generation in two-dimensional lattices of repulsive magnets

In this work, we provide experimental evidence of nonlinear wave propagation in a triangular lattice of repulsive magnets supported by an elastic foundation of thin pillars, and we interpret all the individual features of the nonlinear wave field through the lens of a phonon band calculation that precisely accounts for the interparticle repulsive forces. We confirm the coexistence of two spectrally distinct components (homogeneous and forced) in the wave response that is induced via second harmonic generation (SHG) as a result of the quadratic nonlinearity embedded in the magnetic interaction. The detection of the forced component, specifically, allows us to attribute unequivocally the generation of harmonics to the nonlinear mechanisms germane to the lattice. We show that the spatial characteristics of the second harmonic components are markedly different from those exhibited by the fundamental harmonic. This endows the lattice with a functionality enrichment capability, whereby additional modal characteristics and directivity patterns can be triggered and tuned by merely increasing the amplitude of excitation.

A reprogrammable mechanical metamaterial with stable memory

Metamaterials are designed to realize exotic physical properties through the geometric arrangement of their underlying structural layout1,2. Traditional mechanical metamaterials achieve functionalities such as a target Poisson's ratio3 or shape transformation4-6 through unit-cell optimization7-9, often with spatial heterogeneity10-12. These functionalities are programmed into the layout of the metamaterial in a way that cannot be altered. Although recent efforts have produced means of tuning such properties post-fabrication13-19, they have not demonstrated mechanical reprogrammability analogous to that of digital devices, such as hard disk drives, in which each unit can be written to or read from in real time as required. Here we overcome this challenge by using a design framework for a tileable mechanical metamaterial with stable memory at the unit-cell level. Our design comprises an array of physical binary elements (m-bits), analogous to digital bits, with clearly delineated writing and reading phases. Each m-bit can be independently and reversibly switched between two stable states (acting as memory) using magnetic actuation to move between the equilibria of a bistable shell20-25. Under deformation, each state is associated with a distinctly different mechanical response that is fully elastic and can be reversibly cycled until the system is reprogrammed. Encoding a set of binary instructions onto the tiled array yields markedly different mechanical properties; specifically, the stiffness and strength can be made to range over an order of magnitude. We expect that the stable memory and on-demand reprogrammability of mechanical properties in this design paradigm will facilitate the development of advanced forms of mechanical metamaterials.

Moth wings are acoustic metamaterials

Metamaterials assemble multiple subwavelength elements to create structures with extraordinary physical properties (1-4). Optical metamaterials are rare in nature and no natural acoustic metamaterials are known. Here, we reveal that the intricate scale layer on moth wings forms a metamaterial ultrasound absorber (peak absorption = 72% of sound intensity at 78 kHz) that is 111 times thinner than the longest absorbed wavelength. Individual scales act as resonant (5) unit cells that are linked via a shared wing membrane to form this metamaterial, and collectively they generate hard-to-attain broadband deep-subwavelength absorption. Their collective absorption exceeds the sum of their individual contributions. This sound absorber provides moth wings with acoustic camouflage (6) against echolocating bats. It combines broadband absorption of all frequencies used by bats with light and ultrathin structures that meet aerodynamic constraints on wing weight and thickness. The morphological implementation seen in this evolved acoustic metamaterial reveals enticing ways to design high-performance noise mitigation devices.

A design strategy to match the band gap of periodic and aperiodic metamaterials

The focus of this paper is on elastic metamaterials characterised by the presence of wide sub-wavelength band gap. In most cases, such mechanical property is strictly connected to the periodic repetition of the unit cell. Nonetheless, the strict periodicity requirement could represent a drawback. In this paper, we present a design strategy for aperiodic elastic metamaterials in order to achieve the same performances as for the periodic counterparts. This is done by exploiting the concept of separation of modes for different building blocks, arranged in aperiodic fashion. A theoretical explanation is provided, as well as numerical simulations; the concept is validated by means of a set of experimental tests on prototypes that are realized via additive manufacturing.

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.

Instability-Induced Pattern Formations in Soft Magnetoactive Composites

Elastic instabilities can trigger dramatic microstructure transformations giving rise to unusual behavior in soft matter. Motivated by this phenomenon, we study instability-induced pattern formations in soft magnetoactive elastomer (MAE) composites deforming in the presence of a magnetic field. We show that identical MAE composites with periodically distributed particles can switch to a variety of new patterns with different periodicity upon developments of instabilities. The newly formed patterns and postbuckling behavior of the MAEs are dictated by the magnitude of the applied magnetic field. We identify the particular levels of magnetic fields that give rise to strictly doubled or multiplied periodicity upon the onset of instabilities in the periodic particulate soft MAE. Thus, the predicted phenomenon can be potentially used for designing new reconfigurable soft materials with tunable material microstructures remotely controlled by a magnetic field.

Designing complex architectured materials with generative adversarial networks

Architectured materials on length scales from nanometers to meters are desirable for diverse applications. Recent advances in additive manufacturing have made mass production of complex architectured materials technologically and economically feasible. Existing architecture design approaches such as bioinspiration, Edisonian, and optimization, however, generally rely on experienced designers' prior knowledge, limiting broad applications of architectured materials. Particularly challenging is designing architectured materials with extreme properties, such as the Hashin-Shtrikman upper bounds on isotropic elasticity in an experience-free manner without prior knowledge. Here, we present an experience-free and systematic approach for the design of complex architectured materials with generative adversarial networks. The networks are trained using simulation data from millions of randomly generated architectures categorized based on different crystallographic symmetries. We demonstrate modeling and experimental results of more than 400 two-dimensional architectures that approach the Hashin-Shtrikman upper bounds on isotropic elastic stiffness with porosities from 0.05 to 0.75.

Acoustic response for nonlinear, coupled multiscale model containing subwavelength designed microstructure instabilities

Nonperiodic arrangements of inclusions with incremental linear negative stiffness embedded within a host material offer the ability to achieve unique and useful material properties on the macroscale. In an effort to study such types of inclusions, the present paper develops a time-domain model to capture the nonlinear dynamic response of a heterogeneous medium containing a dilute concentration of subwavelength nonlinear inclusions embedded in a lossy, nearly incompressible medium. Each length scale is modeled via a modified Rayleigh-Plesset equation, which differs from the standard form used in bubble dynamics by accounting for inertial and viscoelastic effects of the oscillating spherical element and includes constitutive equations formulated with incremental deformations. The two length scales are coupled through the constitutive relations and viscoelastic loss for the effective medium, both dependent on the inclusion and matrix properties. The model is then applied to an example nonlinear inclusion with incremental negative linear stiffness stemming from microscale elastic instabilities embedded in a lossy, nearly incompressible host medium. The macroscopic damping performance is shown to be tunable via an externally applied hydrostatic pressure with the example system displaying over two orders of magnitude change in energy dissipation due to changes in prestrain. The numerical results for radial oscillations versus time, frequency spectra, and energy dissipation obtained from the coupled dynamic model captures the expected response for quasistatic and dynamic regimes for an example buckling inclusion for both constrained and unconstrained negative stiffness inclusions.

Sharkskin-Inspired Magnetoactive Reconfigurable Acoustic Metamaterials

Most of the existing acoustic metamaterials rely on architected structures with fixed configurations, and thus, their properties cannot be modulated once the structures are fabricated. Emerging active acoustic metamaterials highlight a promising opportunity to on-demand switch property states; however, they typically require tethered loads, such as mechanical compression or pneumatic actuation. Using untethered physical stimuli to actively switch property states of acoustic metamaterials remains largely unexplored. Here, inspired by the sharkskin denticles, we present a class of active acoustic metamaterials whose configurations can be on-demand switched via untethered magnetic fields, thus enabling active switching of acoustic transmission, wave guiding, logic operation, and reciprocity. The key mechanism relies on magnetically deformable Mie resonator pillar (MRP) arrays that can be tuned between vertical and bent states corresponding to the acoustic forbidding and conducting, respectively. The MRPs are made of a magnetoactive elastomer and feature wavy air channels to enable an artificial Mie resonance within a designed frequency regime. The Mie resonance induces an acoustic bandgap, which is closed when pillars are selectively bent by a sufficiently large magnetic field. These magnetoactive MRPs are further harnessed to design stimuli-controlled reconfigurable acoustic switches, logic gates, and diodes. Capable of creating the first generation of untethered-stimuli-induced active acoustic metadevices, the present paradigm may find broad engineering applications, ranging from noise control and audio modulation to sonic camouflage.

Digitally virtualized atoms for acoustic metamaterials

By designing tailor-made resonance modes with structured atoms, metamaterials allow us to obtain constitutive parameters outside their limited range from natural materials. Nonetheless, tuning the constitutive parameters depends on our ability to modify the physical structure or external circuits attached to the metamaterials, posing a fundamental challenge to the range of tunability in many real-time applications. Here, we propose the concept of virtualized metamaterials on their signal response function to escape the boundary inherent in the physical structure of metamaterials. By replacing the resonating physical structure with a designer mathematical convolution kernel with a fast digital signal processing circuit, we demonstrate a decoupled control of the effective bulk modulus and mass density of acoustic metamaterials on-demand through a software-defined frequency dispersion. Providing freely software-reconfigurable amplitude, center frequency, bandwidth of frequency dispersion, our approach adds an additional dimension to constructing non-reciprocal, non-Hermitian, and topological systems with time-varying capability as potential applications.

Microfluidic on-demand engineering of longitudinal dynamic self-assembly of particles

A viscoelasticity-induced self-assembling microfluidic system is proposed to increase the manoeuvrability and orderliness of longitudinal dynamic self-assembly of particles, achieving control of interparticle spacing's and particle frequency.

An unbounded approach to microfluidics using the Rayleigh-Plateau instability of viscous threads directly drawn in a bath

We study the droplet-forming instability of a thin jet extruded from a nozzle moving horizontally below the surface of an isoviscous immiscible fluid bath. While this interfacial instability is a classic problem in fluid mechanics, it has never been studied in the context of the deposition of a thread into a reservoir, an open-sky version of microfluidics. As the nozzle translates through the reservoir, drops may form at the nozzle (dripping) or further downstream (jetting). We first focus on rectilinear printing paths and derive a scaling law to rationalize the transition between dripping and jetting. We then leverage the flexibility of our system and study the dynamics of breakup when printing sinusoidal paths. We unravel a methodology to control both the size of the drops formed by the instability and the distance that separates them.

Active control of graphene-based membrane-type acoustic metamaterials using a low voltage

Membrane-type acoustic metamaterials for acoustic insulation applications have been attracting ever increasing attention. However, the first anti-resonant frequency of these acoustic metamaterials is fixed once the membrane type is chosen. Here, we propose a novel yet convenient strategy to actively adjust the anti-resonant frequency of the membrane. The poly(vinyl alcohol)/graphene (PVA/GR) nanocomposite membrane is introduced into the acoustic metamaterial, the effective modulus of which is tunable by applying an external electric field. As a result, the first anti-resonant frequency of membrane-type acoustic metamaterials can be actively tuned between 369.2 to 420 Hz, leading to excellent sound attenuation properties. The noise reduction frequency can be actively modulated by DC voltage. Moreover, the change in frequency is consistent with the modulus variation of the PVA/GR nanocomposite membrane when the graphene concentration is varied. In addition, the conductive PVA/GR nanocomposite membrane also exhibits good electromagnetic interference shielding performance in the frequency range of 8-12 GHz. Being actively tunable by an external electric field, this PVA/GR nanocomposite membrane-based acoustic metamaterial is very promising for use in frequency-tunable acoustic insulation applications.

Emergent subharmonic band gaps in nonlinear locally resonant metamaterials induced by autoparametric resonance

With the aim of developing high-performance locally resonant metamaterials, the effect of nonlinear hyperelastic interactions between a rubberlike elastomeric local resonator and the host matrix is investigated. The results reveal a new emergent physical phenomenon not previously reported within the framework of elastoacoustic metamaterials: The appearance of a half subharmonic attenuation zone complementing the local resonance band gap around the fundamental frequency. Evidence of the emergent attenuation zone is provided by direct numerical simulations as well as semianalytical developments via the method of multiple scales. The analyses demonstrate that, in the considered nonlinear locally resonant metamaterial, the combined effects of autoparametric and local resonance induce saturation of the primary wave at certain conditions and, subsequently, promote energy exchange from a primary propagating wave to an evanescent subharmonic wave, giving rise to an additional attenuation zone. This opens new possibilities for the design of passive filtering devices for elastoacoustic waves.

Tunable Dirac cones in two-dimensional acoustic metamaterials with matryoshka structure

Dirac cones of an acoustic system are the foundation of most topological phase transitions and topological states and have recently become a research hotspot. Although the Dirac cones, Dirac-like cones, double Dirac cones, and semi-Dirac points are all skillfully designed, it is still indispensable to realize a tunable Dirac cone in a novel acoustic structure. This paper proposes two-dimensional acoustic metamaterials with matryoshka structure to achieve tunable Dirac cones and topological spin states. Dirac points can be obtained on the dispersion curves owing to the high symmetry. The concentric circular scattering units of the matryoshka structure are arranged in honeycomb lattices. By a rotating-scatterer mechanism to break the symmetry, the Dirac cone at K (K^') is split and the topological spin states appear at the band valley. The existence of a topological transition with opposite Chern numbers as the rotating angle varies is also verified, and helical edge states are obtained along the interfaces separating the topologically opposite spin states insulators. Moreover, the frequency of the Dirac cone is tuned by rotating the inner structure in a double-layer matryoshka structure.

Frequency-doubling effect in acoustic reflection by a nonlinear, architected rotating-square metasurface

Nonlinear acoustic metamaterials offer the potential to enhance wave control opportunities beyond those already demonstrated via dispersion engineering in linear metamaterials. Managing the nonlinearities of a dynamic elastic system, however, remains a challenge, and the need now exists for new strategies to model and design these wave nonlinearities. Inspired by recent research on soft architected rotating-square structures, we propose herein a design for a nonlinear elastic metasurface with the capability to achieve nonlinear acoustic wave reflection control. The designed metasurface is composed of a single layer of rotating squares connected to thin and highly deformable ligaments placed between a rigid plate and a wall. It is shown that during the process of reflection at normal incidence, most of the incoming fundamental wave energy can be converted into the second harmonic wave. A conversion coefficient of approximately 0.8 towards the second harmonic is derived with a reflection coefficient of <0.05 at the incoming fundamental frequency. The theoretical results obtained using the harmonic balance method for a monochromatic pump source are confirmed by time-domain simulations for wave packets. The reported design of a nonlinear acoustic metasurface can be extended to a large family of architected structures, thus opening new avenues for realistic metasurface designs that provide for nonlinear or amplitude-dependent wave tailoring.

Binary-phase acoustic passive logic gates

The recent rapid development of acoustic logic devices has opened up the possibilities of sound computing and information processing. However, simultaneous realization of acoustic logic devices with subwavelength size, broad bandwidth and passive structure still poses a great challenge. To overcome it, we propose a subwavelength acoustic logic gate which consists of binary-phase passive unit cells placed into a multi-port waveguide. Based on the phase manipulations of the unit cells, we experimentally and numerically realize three basic logic gates OR, NOT and AND, and a composite logic gate XOR with a uniform threshold of 0.4 Pa based on linear acoustic interferences. More importantly, We also design a composite logic gate XNOR by a four-port waveguide, and composite logic gates NOR and NAND and a logic operation A⊙(B+C) based on two logic gates. We demonstrate a 0.6λ-length, 0.3λ-width, and 0.2-fractional bandwidth acoustic logic gate constructed by passive structures, which may lead to important advances in various applications, such as acoustic computing, acoustic information processing and integrated acoustics.

Domain Formations and Pattern Transitions via Instabilities in Soft Heterogeneous Materials

Experimental observations of domain formations and pattern transitions in soft particulate composites under large deformations are reported herein. The system of stiff inclusions periodically distributed in a soft elastomeric matrix experiences dramatic microstructure changes upon the development of elastic instabilities. In the experiments, the formation of microstructures with antisymmetric domains and their geometrically tailored evolution into a variety of patterns of cooperative particle rearrangements are observed. Through experimental and numerical analyses, it is shown that these patterns can be tailored by tuning the initial microstructural periodicity and concentration of the inclusions. Thus, these fully determined new patterns can be achieved by fine tuning of the initial microstructure.

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.

Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies

Nanoelectromechanical systems (NEMS) that operate in the megahertz (MHz) regime allow energy transducibility between different physical domains. For example, they convert optical or electrical signals into mechanical motions and vice versa¹. This coupling of different physical quantities leads to frequency-tunable NEMS resonators via electromechanical non-linearities^2-4. NEMS platforms with single- or low-degrees of freedom have been employed to demonstrate quantum-like effects, such as mode cooling⁵, mechanically induced transparency⁵, Rabi oscillation^6,7, two-mode squeezing⁸ and phonon lasing⁹. Periodic arrays of NEMS resonators with architected unit cells enable fundamental studies of lattice-based solid-state phenomena, such as bandgaps^10,11, energy transport^10-12, non-linear dynamics and localization^13,14, and topological properties¹⁵, directly transferrable to on-chip devices. Here we describe one-dimensional, non-linear, nanoelectromechanical lattices (NEML) with active control of the frequency band dispersion in the radio-frequency domain (10-30 MHz). The design of our systems is inspired by NEMS-based phonon waveguides^10,11 and includes the voltage-induced frequency tuning of the individual resonators^2-4. Our NEMLs consist of a periodic arrangement of mechanically coupled, free-standing nanomembranes with circular clamped boundaries. This design forms a flexural phononic crystal with a well-defined bandgap, 1.8 MHz wide. The application of a d.c. gate voltage creates voltage-dependent on-site potentials, which can significantly shift the frequency bands of the device. Additionally, a dynamic modulation of the voltage triggers non-linear effects, which induce the formation of a phononic bandgap in the acoustic branch, analogous to Peierls transition in condensed matter¹⁶. The gating approach employed here makes the devices more compact than recently proposed systems, whose tunability mostly relies on materials' compliance^17,18 and mechanical non-linearities^19-22.

Frequency-dependent behavior of media containing pre-strained nonlinear inclusions: Application to nonlinear acoustic metamaterials

One emerging research area within the fields of acoustic and elastic metamaterials involves designing subwavelength structures that display elastic instabilities in order to generate an effective medium response that is strongly nonlinear. To capture the overall frequency-dependent and dispersive macroscopic response of such heterogeneous media with subwavelength heterogeneities, a theoretical framework is developed that accounts for higher-order stiffnesses of a resonant, nonlinear inclusion that varies with a macroscopic pre-strain, and the inherent inertia associated with an inclusion embedded in a nearly incompressible elastic matrix material. Such a model can be used to study varying macroscopic material properties as a function of both frequency and pre-strain and the activation of such microscale instabilities due to an external, macroscopic loading, as demonstrated with a buckling metamaterial inclusion that is of interest due to its tunable and tailorable nature. The dynamic results obtained are consistent with similar static behavior reported in the literature for structures with elastic instabilities.

Recent Advances in Tunable and Reconfigurable Metamaterials

Metamaterials are composed of nanostructures, called artificial atoms, which can give metamaterials extraordinary properties that cannot be found in natural materials. The nanostructures themselves and their arrangements determine the metamaterials’ properties. However, a conventional metamaterial has fixed properties in general, which limit their use. Thus, real-world applications of metamaterials require the development of tunability. This paper reviews studies that realized tunable and reconfigurable metamaterials that are categorized by the mechanisms that cause the change: inducing temperature changes, illuminating light, inducing mechanical deformation, and applying electromagnetic fields. We then provide the advantages and disadvantages of each mechanism and explain the results or effects of tuning. We also introduce studies that overcome the disadvantages or strengthen the advantages of each classified tunable metamaterial.