Multistable Mechanical Metamaterials Research Articles

Design and analysis of a tunable multistable mechanical metamaterial

Multistable mechanical metamaterials can have unusual properties that facilitate promising applications, including reusable vibration isolation and efficient energy absorption. In this paper, we propose a tunable multistable mechanical metamaterial (TMMM) that is capable of independent and continuous adjustment of its post-fabrication mechanical properties. The unit cell of the TMMM consists of a sinusoidal beam and several supporting beams, with its mechanical behavior finely tuned by inserting specially designed connecting beams into the gaps of the supporting beams. We adopt a combined theoretical and numerical approach to explore the effect of geometric parameters and identify the design space of monostability and bistability. Based on the unit cell, a multilayer TMMM is designed and fabricated and its tunable mechanical response is investigated. The results show that not only the force-displacement curves but also the snap-through behavior and deformation order can be tuned in a straightforward manner. The proposed design has the potential to enhance the development of multistable mechanical metamaterials and broaden their applications in engineering scenarios with varying demands.

Topology optimization design of recoverable bistable structures for energy absorption with embedded shape memory alloys

In this work, multistable mechanical metamaterials with recoverable capabilities are designed for high energy absorption and resistance to repetitive impacts. Based on the known curved beam with energy dissipation characteristics, the shape memory alloys (SMAs) curved beam is added and optimized to achieve recoverability of the bi-layer beam structure. To simultaneously achieve bistable characteristics and recoverability in the designed metamaterial, the optimization model is formulated to maximize the energy absorption capacity of the structure while constraining the peak and valley forces. To solve this complex and highly nonlinear topological optimization problem with stringent constraints, the topological design of the SMAs layer is represented using a limited number of design variables along with the material-field series expansion strategy, and the optimization problem is subsequently solved using a non-gradient-based optimization algorithm. Finite element simulation results demonstrated that the optimized design possesses a substantial capacity for energy absorption. Additionally, the design structure can recover to its initial state from buckling induced by the initial load by regulating the external temperature. Different multistable metamaterials, including those designed to prevent repetitive impacts, have been further developed based on the designed recoverable high-energy-absorbing bistable unit cells, utilizing combinations of varying energy absorption capacities.

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.

Dynamic Behavior of Bistable Shallow Arches: From Intrawell to Chaotic Motion

Abstract Bistable shallow arches are ubiquitous in many engineering systems ranging from compliant mechanisms and biomedical stents to energy harvesters and passive fluidic controllers. In all these scenarios, the bistable states of the arch and the sudden transitions between them via snap-through instability are harnessed. However, bistable arches have been traditionally studied and characterized by triggering snap-through instability using quasi-static forces. Here, we analytically examine the effect of oscillatory loads on bistable arches and investigate the dynamic behaviors ranging from intrawell motion to periodic and chaotic interwell motion. The linear and nonlinear dynamic responses of both elastically and plastically deformed shallow arches are presented. Introducing an energy potential criterion, we classify the structure’s behavior within the parameter space. This energy-based approach allows us to explore the parameter space for high-dimensional models of the arch by varying the force amplitude and excitation frequency. Bifurcation diagrams, Lyapunov exponents, and maximum critical energy plots are presented to characterize the dynamic response of the system. Our results reveal that unstable solutions admitted through higher modes govern the critical energy required for interwell motion. This study investigates the rich nonlinear dynamic behavior of the arch element and it introduces an energy potential criterion that can scale easily to classify motion of arrays of bistable arches for future developments of multistable mechanical metamaterials.

Fractal patterns in the parameter space of a bistable Duffing oscillator.

We study the dissipative bistable Duffing oscillator with equal energy wells and observe fractal patterns in the parameter space of driving frequency, forcing amplitude, and damping ratio. Our numerical investigation reveals the Hausdorff fractal dimension of the boundaries that separate the oscillator's intrawell and interwell behaviors. Furthermore, we categorize the interwell behaviors as three steady-state types: switching, reverting, and vacillating. While fractal patterns in the phase space are well known and heavily studied, our results point to another research direction about fractal patterns in the parameter space. Another implication of this study is that the vibration of a continuous bistable system modeled using a single-mode approximation also manifests fractal patterns in the parameter space. In addition, our findings can guide the design of next-generation bistable and multistable mechanical metamaterials.

Phase patterning in multi-stable metamaterials: Transition wave stabilization and mode conversion

This Letter proposes a design strategy leveraging tunable structural defects in multi-stable mechanical metamaterials for manipulating the propagation of the supported transition waves toward the endowment of a multi-phase patterning capability. The defect reversibly adjusts the on-site potential in order to affect the motion of the transition waves which traverse it, either prohibiting wave transmission (i.e., stabilization) or permitting transmission of specific modes, possibly converting one mode into another. Thus, the defect is able to control the occurrence and distribution of the structural phases and realize the desired phase patterns. Although the metamaterial model for our analytical and numerical study is a one-dimensional (1D) architecture comprising tri-stable elements, the proposed method is shown to apply to 2D architectures and is amenable to elements possessing more than three stable states, demonstrating greater flexibility in metamaterial design than current approaches. The proposed method expands the configuration space of phase-transforming metamaterials, which contributes to efforts aimed at re-programmable mechanical/dynamic performance.

Programmable and multistable metamaterials made of precisely tailored bistable cells

This study proposes a systematic inverse design framework for constructing multistable mechanical metamaterials with programmable gradients. Herein, we designed the tailored bistable cells with precisely controlled maximum instability forces through the topology optimization approach. Then, the designed bistable structures were programmed to construct the multistable mechanical metamaterials with different target gradient snapping sequences and deformation models. Consequently, the simulation and experimental results showed the feasibility of the design method, which successfully produced two- and three-dimensional mechanical metamaterial structures with different functions. Finally, we verified the expected deformation sequences and multistable behaviors of mechanical metamaterials by testing the designed specimens prepared via additive manufacturing. Overall, our findings show that the proposed design strategy offers a new paradigm for developing precisely tailored and programmable mechanical metamaterials.

Topological transformability and reprogrammability of multistable mechanical metamaterials

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

Snap-fit mechanical metamaterials

Reusability and multi-functionality are challenges in the design of energy-absorbing structures, which may realize through special metamaterials. Multistable mechanical metamaterials with different energy barriers are one of the most promising candidates to achieve this goal. Here, a novel snap-fit based modular multistable mechanical metamaterial was proposed for energy absorbing, together with reprogrammable, reconfigurable and reusable properties. The multistable states of metamaterials was achieved by connecting the snap-fit units in series and the modular design was realized with the mechanical response decoupling of these assembled units connected in parallel. The mechanical performances of single snap-fit unit and metamaterials are theoretically, numerically and experimentally investigated. It is evident that the designed metamaterials exhibit excellent impact resistance and energy absorption capacity in the demonstrative experiments, which can be considered as a potential candidate in the development of shock absorbers. The metamaterials can also be adapted to different non-planar protectors with flexibly deployed and easily replaced in complex environments. This strategy provides new mentality and methods for the design of reusable impact protection devices, material constitutive simulators, mechanical information storage and logic gates, robots and many other fields and devices.

Multistability, symmetry and geometric conservation in eightfold waterbomb origami

The nonlinearities inherent in the mechanics of origami make it a rich design space for multistable structures and mechanical metamaterials. Here, we investigate the multistability of a classic origami base: the symmetric eightfold waterbomb. We prove that the waterbomb is bistable for certain crease properties, and derive bounds on, and closed-form approximations of, its stable states. We introduce a simplified form of the waterbomb kinematics and present a design procedure for tuning the depth and the symmetry/asymmetry of its energy wells. By incorporating the concept of pretensioned torsional springs, we also demonstrate the existence of tristable cases for the waterbomb fold pattern. We then apply the analysis of a single waterbomb to study quasi-one-dimensional arrays of waterbombs, where we discover a conserved geometric-kinematic quantity in which the number of popped-up and popped-down vertices is determined uniquely through analysis of the origami structure’s boundaries. This culminates with a discussion of how the quasi-one-dimensional array may be designed to achieve stable states with various degeneracies, kinematics and gaps between energy levels. Collectively, this work presents an alternative approach for characterizing origami multistability properties and reveals an origami design motif that has potential applications in physically reconfigurable structures, mechanical energy absorption and metamaterials.

Energy dissipation in multistable auxetic mechanical metamaterials

Multistable mechanical metamaterials are periodic structures whose unit cells show bistable configurations. The snap-through behavior of the unit cell enhances the energy dissipation of the metamaterials. The materials are reusable and can dissipate energy multiple times since the deformation is within the elastic region. This paper presents two types of two-dimensional auxetic metamaterials: the shuriken-shaped motif with four axes of symmetry and the triangle motif with six axes of symmetry. The cylindrical and three-dimensional (3D) configurations are designed with the geometric topology method. We carry out experiments with 3D printed samples, finite element analysis (FEA), and theoretical investigation to reveal the various mechanisms of energy dissipation. The results demonstrate that the planar metamaterials exhibit different energy dissipation along various loading directions. Each type of metamaterial exhibits discrepant force-displacement characteristics. The triangle motif performs better than the shuriken-shaped motif in terms of energy dissipation along some loading directions. The proposed auxetic metamaterials perform better than the conventional multistable structures with zero Poisson’s ratio. Energy dissipation and auxetic behavior of the typically cylindrical and 3D metamaterials are verified numerically and experimentally.

Programming Multistable Metamaterials to Discover Latent Functionalities.

Using multistable mechanical metamaterials to develop deployable structures, electrical devices, and mechanical memories raises two unanswered questions. First, can mechanical instability be programmed to design sensors and memory devices? Second, how can mechanical properties be tuned at the post-fabrication stage via external stimuli? Answering these questions requires a thorough understanding of the snapping sequences and variations of the elastic energy in multistable metamaterials. The mechanics of deformation sequences and continuous force/energy-displacement curves are comprehensively unveiled here. A 1D array, that is chain, of bistable cells is studied to explore instability-induced energy release and snapping sequences under one external mechanical stimulus. This method offers an insight into the programmability of multistable chains, which is exploited to fabricate a mechanical sensor/memory with sampling (analog to digital-A/D) and data reconstruction (digital to analog-D/A) functionalities operating based on the correlation between the deformation sequence and the mechanical input. The findings offer a new paradigm for developing programmable high-capacity read-write mechanical memories regardless of thei size scale. Furthermore, exotic mechanical properties can be tuned by harnessing the attained programmability of multistable chains. In this respect, a transversely multistable mechanical metamaterial with tensegrity-like bistable cells is designed to showcase the tunability ofchirality.

Valley-protected topological interface state of the elastic wave: From discrete model to multistable mechanical metamaterials

Topological metamaterials provide a new strategy to guide wave energy and exhibit unprecedented robustness. In this study, a novel design strategy is proposed to realize programmable topological metamaterials with local resonant eigenstates. Firstly, in the discrete spring-mass model, two resonant masses and two base masses are introduced into the hexagonal lattice, and a local resonant eigenstate-induced Dirac cone can be formed at the high symmetry point of the Brillouin zone. By introducing the spring stiffness difference, a topological bandgap is opened near the Dirac degeneracy frequency. Thereafter, the nontrivial nature of the bandgap is verified by the theoretical evaluation of the Berry curvature and Chern number. Secondly, the symmetry-breaking configuration is extended to a continuum plate-resonator model. The numerical results demonstrate that the interface state wave propagates along the interface path at a frequency located at the edge-bulk band. Finally, by using the asymmetric effective stiffness of the bistable structure between tension and compression, a programmable topological interface path is realized. The proposed reconfigurable design based on asymmetry significantly expands the design space of metamaterials.

Effect of beam configuration on its multistable and negative stiffness properties

Multistable and negative stiffness mechanical metamaterials are promising and gain great attention recently. Beam structure is the most commonly used negative stiffness element in these metamaterials’ designs. The tilted beam element (V-shape) and the curved beam element are frequently exploited, but their difference is almost neglected. In this study, we investigated the mechanical properties of the beam structures with different configurations (the tilted, the curved, and the arched shapes) and the influence of the variable cross-section design using a combination of analytical, numerical and experimental methods. We draw the geometric parameter boundary of positive stiffness, negative stiffness, and bi-stable regions for these beam structures, and compare their basic mechanical properties as well as energy trapping capacity. Research results indicate that the curved beam and the tilted beam respectively show better performance in different parameter regions, and are both more suitable than the arched beam in constructing the metamaterials. The variable cross-section design can be applied to tune the basic mechanical properties and shift the boundary for various mechanical behaviors. The obtained results are also verified in different material systems and can be a reference to the design of multistable and negative stiffness mechanical metamaterials.

Programming nonreciprocity and reversibility in multistable mechanical metamaterials

Nonreciprocity can be passively achieved by harnessing material nonlinearities. In particular, networks of nonlinear bistable elements with asymmetric energy landscapes have recently been shown to support unidirectional transition waves. However, in these systems energy can be transferred only when the elements switch from the higher to the lower energy well, allowing for a one-time signal transmission. Here, we show that in a mechanical metamaterial comprising a 1D array of bistable arches nonreciprocity and reversibility can be independently programmed and are not mutually exclusive. By connecting shallow arches with symmetric energy wells and decreasing energy barriers, we design a reversible mechanical diode that can sustain multiple signal transmissions. Further, by alternating arches with symmetric and asymmetric energy landscapes we realize a nonreciprocal chain that enables propagation of different transition waves in opposite directions.

SMP-based multi-stable mechanical metamaterials: From bandgap tuning to wave logic gates

The elastic metamaterials have exceptional physical properties and functions unavailable in natural materials, a major challenge is how to realize the dynamic and stable adjustment of working frequency to control elastic waves. In this work, we propose a method of creating geometrically reconfigurable and mechanically tunable multi-stable metamaterials to realize a kind of tunable elastic metamaterial with a stable state during the active regulation without continuous-consuming energy based on 4D printing technology. The underlying mechanical mechanism of dynamic and stable adjustment is investigated through theoretical model, finite element analysis and experiment. The tunable elastic metamaterial can adjust the starting/ending frequencies and broaden the frequency ranges of bandgaps and control the elastic wave propagation. The method of intelligent and active manipulation of elastic wave can be used in vibration-isolated steady-state adjustable equipment and smart wave device. Inspired by electronic technology, we implement a bi-stable logic-gate elastic metamaterial to correctly execute simple wave logic operations.

Multistable mechanical metamaterials with highly tunable strength and energy absorption performance

A novel multistable mechanical metamaterial showing highly tunable strength and energy absorption performance has been designed in this paper. A theoretical model is established to describe mechanical properties of the unit cell. The specimens of unit cells and metamaterials have been fabricated by fused deposition modeling additive manufacturing technology. Quasi-static compression and tensile experiments and the corresponding simulation analyses on these specimens are carried out. Results obtained from experiment and simulation methods demonstrate that mechanical properties of the presented bistable unit cell and multistable metamaterial can be tunable in a wide range with little change in total mass.

Transition waves in multi-stable metamaterials with space-time modulated potentials

This Letter introduces a strategy for transition wave (soliton) management in multi-stable mechanical metamaterials, enabling on-demand, post-fabrication control of the associated phase transformation kinetics and distribution. Specifically, the wave dynamics are controlled by a small, kinematically prescribed spatiotemporal variation in the elastic potential, constituting a driving force. The stability of the wave profile under slow-propagation conditions and the characteristic spatial localization of the Hamiltonian energy support an analogy with a Newtonian particle traversing a viscous medium under forcing. The theoretical analysis adopts this particle perspective, describing the soliton dynamics through ordinary, rather than partial, differential equations. While myriads of definitions for the potential modulation are possible, a traveling sinusoid assists the development of analytical solutions. Following this prescription, two wave propagation regimes are revealed: in one, the soliton is carried by the modulation with a commensurate velocity; in the other, the soliton is out-paced by the modulation and, thus, travels at reduced velocity. To illustrate the utility of this method, we demonstrate both the tractor and repulsor effects in multi-stable systems away from equilibrium: as a tractor (repulsor), the potential variation attracts (repels) the transition wave front in opposition to the system's energy-minimizing tendency. This method provides greater flexibility to the transformation performance of multi-stable metamaterials and supports the adoption of such systems in applications demanding multi-functionality.

Additively manufactured negative stiffness structures for shock absorber applications

Negative stiffness structures (NSS), as a branch of multi-stable mechanical metamaterials, exhibit multiple stable configurations. Their characteristics, such as bistability, snap-through and negative stiffness, make them particularly suitable for shock absorber applications. The majority of NSS is designed in a cuboidal shape and only recently few studies focused on cylindrical NSS. Lately, Fraunhofer for Additive Manufacturing Technologies IAPT, has started some studies to optimize these structures and to profit by their features in different applications, such as InspectionCopter project. During this study, three types of special-shaped NSS were designed, produced and tested. To determine the influence of dimensional parameters and materials on the functionality of these flexible structures, for each one of three concepts, five different versions in two different materials and techniques were realized. The specimens were fabricated in PEBA (PolyEther Block Amide) and TPU (Thermoplastic PolyUrethane) using, respectively, Selective Laser Sintering (SLS) and MultiJet Printing (MJP) technologies; the design freedom of Additive Manufacturing (AM) allows the production of complex structures and the possibility of functional integration, such as shock absorber functionality. To investigate the mechanical and NS properties of these structures and their deformation mechanisms, quasi-static compression tests were performed according to ASTM D695 − 15 regulation. The results, analyzed through force–displacement curves, highlighted the energy recovery of the specimens during deformation and the influence of dimensional parameters on the response to the applied loads. During the tests, it was also evident how the usage of different dimensions and materials can lead, for the same structure, to a symmetric or asymmetric buckling mode in the collapse of the layers and to prevent the structure from returning to its original shape once the load has been removed.

Transition Waves and Formation of Domain Walls in Multistable Mechanical Metamaterials

We experimentally, numerically, and analytically investigate transition waves in a mechanical metamaterial comprising a chain of tristable elements. Transition waves can be initiated from different stable phases within the highly tunable tristable energy landscape. These waves propagate via coupled translational and rotational motion that leads to formation of stationary domain walls if transition waves of incompatible type collide. Since domain formation and propagation is fully reversible, the unique nonlinear behavior could be exploited in new classes of reconfigurable metamaterials.