Stable propagation of mechanical signals in soft media using stored elastic energy

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.

Universally bistable shells with nonzero Gaussian curvature for two-way transition waves

Multi-welled energy landscapes arising in shells with nonzero Gaussian curvature typically fade away as their thickness becomes larger because of the increased bending energy required for inversion. Motivated by this limitation, we propose a strategy to realize doubly curved shells that are bistable for any thickness. We then study the nonlinear dynamic response of one-dimensional (1D) arrays of our universally bistable shells when coupled by compressible fluid cavities. We find that the system supports the propagation of bidirectional transition waves whose characteristics can be tuned by varying both geometric parameters as well as the amount of energy supplied to initiate the waves. However, since our bistable shells have equal energy minima, the distance traveled by such waves is limited by dissipation. To overcome this limitation, we identify a strategy to realize thick bistable shells with tunable energy landscape and show that their strategic placement within the 1D array can extend the propagation distance of the supported bidirectional transition waves.

Curved elastic shells have unique mechanical behavior and multiple stable configurations, but these properties fade when the shell thickness increases. Here the authors report a strategy to realize bistable doubly curved shells with arbitrary thickness, and how to optimize the dynamic response of one-dimensional connected arrays of such doubly-curved bistable shells.

Artificial Venus Flytraps: A Research Review and Outlook on Their Importance for Novel Bioinspired Materials Systems

Bioinspired and biomimetic soft machines rely on functions and working principles that have been abstracted from biology but that have evolved over 3.5 billion years. So far, few examples from the huge pool of natural models have been examined and transferred to technical applications. Like living organisms, subsequent generations of soft machines will autonomously respond, sense, and adapt to the environment. Plants as concept generators remain relatively unexplored in biomimetic approaches to robotics and related technologies, despite being able to grow, and continuously adapt in response to environmental stimuli. In this research review, we highlight recent developments in plant-inspired soft machine systems based on movement principles. We focus on inspirations taken from fast active movements in the carnivorous Venus flytrap (Dionaea muscipula) and compare current developments in artificial Venus flytraps with their biological role model. The advantages and disadvantages of current systems are also analyzed and discussed, and a new state-of-the-art autonomous system is derived. Incorporation of the basic structural and functional principles of the Venus flytrap into novel autonomous applications in the field of robotics not only will inspire further plant-inspired biomimetic developments but might also advance contemporary plant-inspired robots, leading to fully autonomous systems utilizing bioinspired working concepts.

Non-Reciprocal Supratransmission in Mechanical Lattices with Non-Local Feedback Control Interactions

We numerically investigate the supratransmission phenomenon in an active nonlinear system modeled by the 1D/2D discrete sine-Gordon equation with non-local feedback. While, at a given frequency, the typical passive system exhibits a single amplitude threshold marking the onset of the phenomenon, we show that the inclusion of non-local feedback manifests additional thresholds that depend upon the specific boundary from which supratransmission is stimulated, realizing asymmetric (i.e., non-reciprocal) dynamics. The results illustrate a new means of controlling nonlinear wave propagation and energy transport for, e.g., signal amplification and mechanical logic.

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.

Nonreciprocal elasticity and the realization of static and dynamic nonreciprocity

The realization of the mechanical nonreciprocity requires breaking either the time-reversal symmetry or the material deformation symmetry. The time-reversal asymmetry was the commonly adopted approach to realize dynamic nonreciprocity. However, a static nonreciprocity requires-with no any other option-breaking the material deformation symmetry. By virtue of the Maxwell-Betti reciprocal theorem, the achievement of the static nonreciprocity seems to be conditional by the use of a nonlinear material. Here, we further investigate this and demonstrate a novel "nonreciprocal elasticity" concept. We investigated the conditions of the attainment of effective static nonreciprocity. We revealed that the realization of static nonreciprocity requires breaking the material deformation symmetry under the same kinematical and kinetical conditions, which can be achieved only and only if the material exhibits a nonreciprocal elasticity. By means of experimental and topological mechanics, we demonstrate that the realization of static nonreciprocity requires nonreciprocal elasticity no matter what the material is linear or nonlinear. We experimentally demonstrated linear and nonlinear metamaterials with nonreciprocal elasticities. The developed metamaterials were used to demonstrate that nonreciprocal elasticity is essential to realize static nonreciprocal-topological systems. The nonreciprocal elasticity developed here will open new venues of the design of metamaterials that can effectively break the material deformation symmetry and achieve, both, static and dynamic nonreciprocity.

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

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

Dome-Patterned Metamaterial Sheets

The properties of conventional materials result from the arrangement of and the interaction between atoms at the nanoscale. Metamaterials have shifted this paradigm by offering property control through structural design at the mesoscale, thus broadening the design space beyond the limits of traditional materials. A family of mechanical metamaterials consisting of soft sheets featuring a patterned array of reconfigurable bistable domes is reported here. The domes in this metamaterial architecture can be reversibly inverted at the local scale to generate programmable multistable shapes and tunable mechanical responses at the global scale. By 3D printing a robotic gripper with energy-storing skin and a structure that can memorize and compute spatially-distributed mechanical signals, it is shown that these metamaterials are an attractive platform for novel mechanologic concepts and open new design opportunities for structures used in robotics, architecture, and biomedical applications.

Propagation and attenuation of mechanical signals in ultrasoft 2D solids

The propagation of elastic waves in soft materials plays a crucial role in the spatiotemporal transmission of mechanical signals, e.g., in biological mechanotransduction or in the failure of marginal solids. At high Reynolds numbers Re ≫ 1, inertia dominates and wave propagation is readily observed. However, mechanical cues in soft and biological materials often occur at low Re, where waves are overdamped. Overdamped waves are not only difficult to observe experimentally, also theoretically their description remains incomplete. Here, we present direct measurements of the propagation and attenuation of mechanical signals in colloidal soft solids, induced by an optical trap. We derive an analytical theory for low Re wave propagation and damping, which is in excellent agreement with the experiments. Our results present both a previously unexplored method to characterize damped waves in soft solids and a theoretical framework showing how localized mechanical signals can provoke a remote and delayed response.

Snapping of hinged arches under displacement control: Strength loss and nonreciprocity

We investigate experimentally and numerically the response of hinged shallow arches subjected to a transverse midpoint displacement. We find that this simple system supports a rich set of responses, which, to date, have received relatively little attention. We observe not only the snapping of the arches to their inverted equilibrium configuration, but also an earlier dynamic transition from a symmetric to an asymmetric shape that results in a sudden strength loss. Moreover, we find that the response of plastically deformed arches is nonreciprocal with respect to the loading direction. Finally, we discover that, while elastically deformed arches always snap to the inverted stable configuration, for plastically deformed ones there is a critical rise below which the structures are monostable.

Inflatable soft jumper inspired by shell snapping

Fluidic soft actuators are enlarging the robotics toolbox by providing flexible elements that can display highly complex deformations. Although these actuators are adaptable and inherently safe, their actuation speed is typically slow because the influx of fluid is limited by viscous forces. To overcome this limitation and realize soft actuators capable of rapid movements, we focused on spherical caps that exhibit isochoric snapping when pressurized under volume-controlled conditions. First, we noted that this snap-through instability leads to both a sudden release of energy and a fast cap displacement. Inspired by these findings, we investigated the response of actuators that comprise such spherical caps as building blocks and observed the same isochoric snapping mechanism upon inflation. Last, we demonstrated that this instability can be exploited to make these actuators jump even when inflated at a slow rate. Our study provides the foundation for the design of an emerging class of fluidic soft devices that can convert a slow input signal into a fast output deformation.

Pulse-driven robot: Motion via solitary waves

The unique properties of nonlinear waves have been recently exploited to enable a wide range of applications, including impact mitigation, asymmetric transmission, switching, and focusing. Here, we demonstrate that the propagation of nonlinear waves can be as well harnessed to make flexible structures crawl. By combining experimental and theoretical methods, we show that such pulse-driven locomotion reaches a maximum efficiency when the initiated pulses are solitons and that our simple machine can move on a wide range of surfaces and even steer. Our study expands the range of possible applications of nonlinear waves and demonstrates that they offer a new platform to make flexible machines to move.

Viewpoint: From Responsive to Adaptive and Interactive Materials and Materials Systems: A Roadmap

Soft matter systems and materials are moving toward adaptive and interactive behavior, which holds outstanding promise to make the next generation of intelligent soft materials systems inspired from the dynamics and behavior of living systems. But what is an adaptive material? What is an interactive material? How should classical responsiveness or smart materials be delineated? At present, the literature lacks a comprehensive discussion on these topics, which is however of profound importance in order to identify landmark advances, keep a correct and noninflating terminology, and most importantly educate young scientists going into this direction. By comparing different levels of complex behavior in biological systems, this Viewpoint strives to give some definition of the various different materials systems characteristics. In particular, the importance of thinking in the direction of training and learning materials, and metabolic or behavioral materials is highlighted, as well as communication and information-processing systems. This Viewpoint aims to also serve as a switchboard to further connect the important fields of systems chemistry, synthetic biology, supramolecular chemistry and nano- and microfabrication/3D printing with advanced soft materials research. A convergence of these disciplines will be at the heart of empowering future adaptive and interactive materials systems with increasingly complex and emergent life-like behavior.

Design and Testing of Bistable Lattices with Tensegrity Architecture and Nanoscale Features Fabricated by Multiphoton Lithography

A bistable response is an innate feature of tensegrity metamaterials, which is a conundrum to attain in other metamaterials, since it ushers unconventional static and dynamical mechanical behaviors. This paper investigates the design, modeling, fabrication and testing of bistable lattices with tensegrity architecture and nanoscale features. First, a method to design bistable lattices tessellating tensegrity units is formulated. The additive manufacturing of these structures is performed through multiphoton lithography, which enables the fabrication of microscale structures with nanoscale features and extremely high resolution. Different modular lattices, comprised of struts with 250 nm minimum radius, are tested under loading-unloading uniaxial compression nanoindentation tests. The compression tests confirmed the activation of the designed bistable twisting mechanism in the examined lattices, combined with a moderate viscoelastic response. The force-displacement plots of the 3D assemblies of bistable tensegrity prisms reveal a softening behavior during the loading from the primary stable configuration and a subsequent snapping event that drives the structure into a secondary stable configuration. The twisting mechanism that characterizes such a transition is preserved after unloading and during repeated loading-unloading cycles. The results of the present study elucidate that fabrication of multistable tensegrity lattices is highly feasible via multiphoton lithography and promulgates the fabrication of multi-cell tensegrity metamaterials with unprecedented static and dynamic responses.

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.

Harnessing transition waves to realize deployable structures

Transition waves that sequentially switch bistable elements from one stable configuration to another have received significant interest in recent years not only because of their rich physics but also, for their potential applications, including unidirectional propagation, energy harvesting, and mechanical computation. Here, we exploit the propagation of transition waves in a bistable one-dimensional (1D) linkage as a robust mechanism to realize structures that can be quickly deployed. We first use a combination of experiments and analyses to show that, if the bistable joints are properly designed, transition waves can propagate throughout the entire structure and transform the initial straight configuration into a curved one. We then demonstrate that such bistable linkages can be used as building blocks to realize deployable three-dimensional (3D) structures of arbitrary shape.

Guided transition waves in multistable mechanical metamaterials

Transition fronts, moving through solids and fluids in the form of propagating domain or phase boundaries, have recently been mimicked at the structural level in bistable architectures. What has been limited to simple one-dimensional (1D) examples is here cast into a blueprint for higher dimensions, demonstrated through 2D experiments and described by a continuum mechanical model that draws inspiration from phase transition theory in crystalline solids. Unlike materials, the presented structural analogs admit precise control of the transition wave's direction, shape, and velocity through spatially tailoring the underlying periodic network architecture (locally varying the shape or stiffness of the fundamental building blocks, and exploiting interactions of transition fronts with lattice defects such as point defects and free surfaces). The outcome is a predictable and programmable strongly nonlinear metamaterial motion with potential for, for example, propulsion in soft robotics, morphing surfaces, reconfigurable devices, mechanical logic, and controlled energy absorption.

Solitary waves in a nonintegrable chain with double-well potentials

We study solitary waves in a one-dimensional lattice of identical masses that are connected in series by nonlinear springs. The potential of each spring is nonconvex, where two disjoint convex regions, phase I and phase II, are separated by a concave, spinodal region. Consequently, the force-strain relation of the spring is nonmonotonous, which gives rise to a bistable behavior. Based on analytical treatment, with some approximations, combined with extensive numerical simulations, we are able to reveal important insights. For example, we find that the solitary-wave solution is indifferent to the energy barrier that separates the two energy wells associated with phase I and phase II, and that the shape of the wave can be described by means of merely two scalar properties of the potential of the springs, namely, the ratio of stiffness in phase II and phase I, and the ratio between the Maxwell's force and corresponding transition strain. The latter ratio provides a useful measure for the significance of the spinodal region. Linear stability of the solitary-wave solution is studied analytically using the Vakhitov-Kolokolov criterion applied to the approximate solutions obtained in the first part. These results are validated by numerical simulations. We find that the solitary-wave solution is stable provided that its velocity is higher than some critical value. It is shown that, practically, the solitary waves are stable for almost the entire range of possible wave velocities. This is also manifested in the interaction between two solitary waves or between a solitary wave and a wall (rigid boundary). Such interaction results in a minor change of height and shape of the solitary wave along with the formation of a trail of small undulations that follow the wave, as expected in a nonintegrable system. Even after a significant number of interactions the changes in the wave height and shape are minor, suggesting that the bistable chain may be a useful platform for delivering information over long distances, even concurrently with additional information (other solitary waves) passing through the chain.

Non-reciprocal robotic metamaterials

Non-reciprocal transmission of motion is potentially highly beneficial to a wide range of applications, ranging from wave guiding to shock and vibration damping and energy harvesting. To date, large levels of non-reciprocity have been realized using broken spatial or temporal symmetries, yet mostly in the vicinity of resonances, bandgaps or using nonlinearities, thereby non-reciprocal transmission remains limited to narrow ranges of frequencies or input magnitudes and sensitive to attenuation. Here, we create a robotic mechanical metamaterials wherein we use local control loops to break reciprocity at the level of the interactions between the unit cells. We show theoretically and experimentally that first-of-their-kind spatially asymmetric standing waves at all frequencies and unidirectionally amplified propagating waves emerge. These findings realize the mechanical analogue of the non-Hermitian skin effect. They significantly advance the field of active metamaterials for non hermitian physics and open avenues to channel mechanical energy in unprecedented ways.

Cnoidal wave propagation in an elastic metamaterial

Advances in fabrication techniques have led to a proliferation of studies on new mechanical metamaterials, particularly on elastic and linear phenomena (for example, their phonon spectrum and acoustic band gaps). More recently, there has been a growing interest in nonlinear wave phenomena in these systems, and particularly how geometric parameters affect the propagation of high-amplitude nonlinear waves. In this paper, we analytically, numerically, and experimentally demonstrate the propagation of cnoidal waves in an elastic architected material. This class of traveling waves constitutes a general family of nonlinear waves, which reduce to phonons and solitons under suitable limits. Although cnoidal waves were first discovered as solutions to the conservation laws for shallow water, they have subsequently appeared in contexts as diverse as ion plasmas and nonlinear optics, but have rarely been explored in elastic solids. We show that geometrically nonlinear deformations in architected soft elastic solids can result in cnoidal waves. Insights from our analysis will be critical to controlling the propagation of stress waves in advanced materials.

Focusing and Mode Separation of Elastic Vector Solitons in a 2D Soft Mechanical Metamaterial

Soft mechanical metamaterials can support a rich set of dynamic responses, which, to date, have received relatively little attention. Here, we report experimental, numerical, and analytical results describing the behavior of an anisotropic two-dimensional flexible mechanical metamaterial when subjected to impact loading. We not only observe the propagation of elastic vector solitons with three components-two translational and one rotational-that are coupled together, but also very rich direction-dependent behaviors such as the formation of sound bullets and the separation of pulses into different solitary modes.

Reversible signal transmission in an active mechanical metamaterial

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

Origami-based impact mitigation via rarefaction solitary wave creation

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

Harnessing uniaxial tension to tune Poisson's ratio and wave propagation in soft porous phononic crystals: an experimental study

Exerting mechanical loads on soft periodic porous phononic crystals provides a unique opportunity to control the propagation of waves through the peculiar band gaps. However, it is quite difficult to experimentally confirm the band gaps in soft materials owing to their viscosity and instability-prone character. We investigate here via experiments the effect of regulation of uniaxial tension on the band gaps in a 2D soft phononic crystal with criss-crossed elliptical holes which was designed based on the contrarian thinking to our previous study. The results show that the soft phononic crystal has rich initial band gaps and can be tuned by harnessing uniaxial tension to achieve continuous control of elastic band gaps. Moreover, the effect of the uniaxial tension on the effective Poisson's ratio of the structure is also studied. The present study confirms the feasibility of the design of soft tunable phononic crystals and acoustic devices by harnessing uniaxial tension.

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.

Additively manufacturable micro-mechanical logic gates

Early examples of computers were almost exclusively based on mechanical devices. Although electronic computers became dominant in the past 60 years, recent advancements in three-dimensional micro-additive manufacturing technology provide new fabrication techniques for complex microstructures which have rekindled research interest in mechanical computations. Here we propose a new digital mechanical computation approach based on additively-manufacturable micro-mechanical logic gates. The proposed mechanical logic gates (i.e., NOT, AND, OR, NAND, and NOR gates) utilize multi-stable micro-flexures that buckle to perform Boolean computations based purely on mechanical forces and displacements with no electronic components. A key benefit of the proposed approach is that such systems can be additively fabricated as embedded parts of microarchitected metamaterials that are capable of interacting mechanically with their surrounding environment while processing and storing digital data internally without requiring electric power.

Mechanical computing based on the logic devices that utilize mechanical energy can be an alternative to conventional electronic computing. Here, Song et al. show a micromechanical logic gate, fabricated using multi-stable buckling flexures, which is capable of realizing all digital logic operations.

3D printing of twisting and rotational bistable structures with tuning elements

Three-dimensional (3D) printing is ideal for the fabrication of various customized 3D components with fine details and material-design complexities. However, most components fabricated so far have been static structures with fixed shapes and functions. Here we introduce bistability to 3D printing to realize highly-controlled, reconfigurable structures. Particularly, we demonstrate 3D printing of twisting and rotational bistable structures. To this end, we have introduced special joints to construct twisting and rotational structures without post-assembly. Bistability produces a well-defined energy diagram, which is important for precise motion control and reconfigurable structures. Therefore, these bistable structures can be useful for simplified motion control in actuators or for mechanical switches. Moreover, we demonstrate tunable bistable components exploiting shape memory polymers. We can readjust the bistability-energy diagram (barrier height, slope, displacement, symmetry) after printing and achieve tunable bistability. This tunability can significantly increase the use of bistable structures in various 3D-printed components.

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.

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.

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.

Metamaterials with amplitude gaps for elastic solitons

We combine experimental, numerical, and analytical tools to design highly nonlinear mechanical metamaterials that exhibit a new phenomenon: gaps in amplitude for elastic vector solitons (i.e., ranges in amplitude where elastic soliton propagation is forbidden). Such gaps are fundamentally different from the spectral gaps in frequency typically observed in linear phononic crystals and acoustic metamaterials and are induced by the lack of strong coupling between the two polarizations of the vector soliton. We show that the amplitude gaps are a robust feature of our system and that their width can be controlled both by varying the structural properties of the units and by breaking the symmetry in the underlying geometry. Moreover, we demonstrate that amplitude gaps provide new opportunities to manipulate highly nonlinear elastic pulses, as demonstrated by the designed soliton splitters and diodes.

Metastable modular metastructures for on-demand reconfiguration of band structures and nonreciprocal wave propagation

We present an approach to achieve adaptable band structures and nonreciprocal wave propagation by exploring and exploiting the concept of metastable modular metastructures. Through studying the dynamics of wave propagation in a chain composed of finite metastable modules, we provide experimental and analytical results on nonreciprocal wave propagation and unveil the underlying mechanisms that facilitate such unidirectional energy transmission. In addition, we demonstrate that via transitioning among the numerous metastable states, the proposed metastructure is endowed with a large number of bandgap reconfiguration possibilities. As a result, we illustrate that unprecedented adaptable nonreciprocal wave propagation can be realized using the metastable modular metastructure. Overall, this research elucidates the rich dynamics attainable through the combinations of periodicity, nonlinearity, spatial asymmetry, and metastability and creates a class of adaptive structural and material systems capable of realizing tunable bandgaps and nonreciprocal wave transmissions.

Origami mechanologic

Robots autonomously interact with their environment through a continual sense-decide-respond control loop. Most commonly, the decide step occurs in a central processing unit; however, the stiffness mismatch between rigid electronics and the compliant bodies of soft robots can impede integration of these systems. We develop a framework for programmable mechanical computation embedded into the structure of soft robots that can augment conventional digital electronic control schemes. Using an origami waterbomb as an experimental platform, we demonstrate a 1-bit mechanical storage device that writes, erases, and rewrites itself in response to a time-varying environmental signal. Further, we show that mechanical coupling between connected origami units can be used to program the behavior of a mechanical bit, produce logic gates such as AND, OR, and three input majority gates, and transmit signals between mechanologic gates. Embedded mechanologic provides a route to add autonomy and intelligence in soft robots and machines.

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

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

Harnessing bistability for directional propulsion of soft, untethered robots

In most macroscale robotic systems, propulsion and controls are enabled through a physical tether or complex onboard electronics and batteries. A tether simplifies the design process but limits the range of motion of the robot, while onboard controls and power supplies are heavy and complicate the design process. Here, we present a simple design principle for an untethered, soft swimming robot with preprogrammed, directional propulsion without a battery or onboard electronics. Locomotion is achieved by using actuators that harness the large displacements of bistable elements triggered by surrounding temperature changes. Powered by shape memory polymer (SMP) muscles, the bistable elements in turn actuate the robot's fins. Our robots are fabricated using a commercially available 3D printer in a single print. As a proof of concept, we show the ability to program a vessel, which can autonomously deliver a cargo and navigate back to the deployment point.

Demonstration of Dispersive Rarefaction Shocks in Hollow Elliptical Cylinder Chains

We report an experimental and numerical demonstration of dispersive rarefaction shocks (DRS) in a 3D-printed soft chain of hollow elliptical cylinders. We find that, in contrast to conventional nonlinear waves, these DRS have their lower amplitude components travel faster, while the higher amplitude ones propagate slower. This results in the backward-tilted shape of the front of the wave (the rarefaction segment) and the breakage of wave tails into a modulated waveform (the dispersive shock segment). Examining the DRS under various impact conditions, we find the counterintuitive feature that the higher striker velocity causes the slower propagation of the DRS. These unique features can be useful for mitigating impact controllably and efficiently without relying on material damping or plasticity effects.

Observation of elastic topological states in soft materials

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

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

Input-Independent Energy Harvesting in Bistable Lattices from Transition Waves

We demonstrate the utilisation of transition waves for realising input-invariant, frequency-independent energy harvesting in 1D lattices of bistable elements. We propose a metamaterial-inspired design with an integrated electromechanical transduction mechanism to the unit cell, rendering the power conversion capability an intrinsic property of the lattice. Moreover, focusing of transmitted energy to desired locations is demonstrated numerically and experimentally by introducing engineered defects in the form of perturbation in mass or inter-element forcing. We achieve further localisation of energy and numerically observe a breather-like mode for the first time in this type of lattice, improving the harvesting performance by an order of magnitude. Our approach considers generic bistable unit cells and thus provides a universal mechanism to harvest energy and realise metamaterials effectively behaving as a capacitor and power delivery system.

Snap-buckling in asymmetrically constrained elastic strips

When a flat elastic strip is compressed along its axis, it is bent in one of two possible directions via spontaneous symmetry breaking, forming a cylindrical arc. This is a phenomenon well known as Euler buckling. When this cylindrical section is pushed in the other direction, the bending direction can suddenly reverse. This instability is called "snap-through buckling" and is one of the elementary shape transitions in a prestressed thin structure. Combining experiments and theory, we study snap-buckling of an elastic strip with one end hinged and the other end clamped. These asymmetric boundary constraints break the intrinsic symmetry of the strip, generating mechanical behaviors, including largely hysteretic but reproducible force responses and switchlike discontinuous shape changes. We establish the set of exact analytical solutions to fully explain all our major experimental and numerical findings. Asymmetric boundary conditions arise naturally in diverse situations when a thin object is in contact with a solid surface at one end. The introduction of asymmetry through boundary conditions yields new insight into complex and programmable functionalities in material and industrial design.

A New 3D Printing Strategy by Harnessing Deformation, Instability, and Fracture of Viscoelastic Inks

Direct ink writing (DIW) has demonstrated great potential as a multimaterial multifunctional fabrication method in areas as diverse as electronics, structural materials, tissue engineering, and soft robotics. During DIW, viscoelastic inks are extruded out of a 3D printer's nozzle as printed fibers, which are deposited into patterns when the nozzle moves. Hence, the resolution of printed fibers is commonly limited by the nozzle's diameter, and the printed pattern is limited by the motion paths. These limits have severely hampered innovations and applications of DIW 3D printing. Here, a new strategy to exceed the limits of DIW 3D printing by harnessing deformation, instability, and fracture of viscoelastic inks is reported. It is shown that a single nozzle can print fibers with resolution much finer than the nozzle diameter by stretching the extruded ink, and print various thickened or curved patterns with straight nozzle motions by accumulating the ink. A quantitative phase diagram is constructed to rationally select parameters for the new strategy. Further, applications including structures with tunable stiffening, 3D structures with gradient and programmable swelling properties, all printed with a single nozzle are demonstrated. The current work demonstrates that the mechanics of inks plays a critical role in developing 3D printing technology.

Traveling waves in a spring-block chain sliding down a slope

Traveling waves are studied in a spring slider-block model. We explicitly construct front waves (kinks) for a piecewise-linear spinodal friction force. Pulse waves are obtained as the matching of two traveling fronts with identical speeds. Explicit formulas are obtained for the wavespeed and the wave form in the anticontinuum limit. The link with localized waves in a Burridge-Knopoff model of an earthquake fault is briefly discussed.

Elastic Vector Solitons in Soft Architected Materials

We demonstrate experimentally, numerically, and analytically that soft architected materials can support the propagation of elastic vector solitons. More specifically, we focus on structures comprising a network of squares connected by thin and highly deformable ligaments and investigate the propagation of planar nonlinear elastic waves. We find that for sufficiently large amplitudes two components-one translational and one rotational-are coupled together and copropagate without dispersion. Our results not only show that soft architected materials offer a new and rich platform to study the propagation of nonlinear waves, but also open avenues for the design of a new generation of smart systems that take advantage of nonlinearities to control and manipulate the propagation of large amplitude vibrations.

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.

Three-Dimensional Multiscale, Multistable, and Geometrically Diverse Microstructures with Tunable Vibrational Dynamics Assembled by Compressive Buckling

Microelectromechanical systems remain an area of significant interest in fundamental and applied research due to their wide ranging applications. Most device designs, however, are largely two-dimensional and constrained to only a few simple geometries. Achieving tunable resonant frequencies or broad operational bandwidths requires complex components and/or fabrication processes. The work presented here reports unusual classes of three-dimensional (3D) micromechanical systems in the form of vibratory platforms assembled by controlled compressive buckling. Such 3D structures can be fabricated across a broad range of length scales and from various materials, including soft polymers, monocrystalline silicon, and their composites, resulting in a wide scope of achievable resonant frequencies and mechanical behaviors. Platforms designed with multistable mechanical responses and vibrationally de-coupled constituent elements offer improved bandwidth and frequency tunability. Furthermore, the resonant frequencies can be controlled through deformations of an underlying elastomeric substrate. Systematic experimental and computational studies include structures with diverse geometries, ranging from tables, cages, rings, ring-crosses, ring-disks, two-floor ribbons, flowers, umbrellas, triple-cantilever platforms, and asymmetric circular helices, to multilayer constructions. These ideas form the foundations for engineering designs that complement those supported by conventional, microelectromechanical systems, with capabilities that could be useful in systems for biosensing, energy harvesting and others.

Integrated Design and Simulation of Tunable, Multi-State Structures Fabricated Monolithically with Multi-Material 3D Printing

Multi-material 3D printing has created new opportunities for fabricating deployable structures. We design reversible, deployable structures that are fabricated flat, have defined load bearing capacity, and multiple, predictable activated geometries. These structures are designed with a hierarchical framework where the proposed bistable actuator serves as the base building block. The actuator is designed to maximise its stroke length, with the expansion ratio approaching one when serially connected. The activation force of the actuator is parameterised through its joint material and joint length. Simulation and experimental results show that the bistability triggering force can be tuned between 0.5 and 5.0 N. Incorporating this bistable actuator, the first group of hierarchical designs demonstrate the deployment of space frame structures with a tetrahedron module consisting of three active edges, each containing four serially connected actuators. The second group shows the design of flat structures that assume either positive or negative Gaussian curvature once activated. By flipping the initial configuration of the unit actuators, structures such as a dome and an enclosure are demonstrated. A modified Dynamic Relaxation method is used to simulate all possible geometries of the hierarchical structures. Measured geometries differ by less than 5% compared to simulation results.