Bistable Lattice Research Articles

Monotonicity of wave profiles for a bistable system on 2-D lattice

This paper is concerned with the monotonicity problem of traveling wave solutions for a class of nonlocal bistable system on two-dimensional lattice. We prove that any traveling wave solution, including the standing wave solution (wave speed c = 0 c=0 ), is strictly monotone increasing for this nonlocal bistable lattice system.

Design of Deployable Structures by Using Bistable Compliant Mechanisms.

A deployable structure can significantly change its geometric shape by switching lattice configurations. Using compliant mechanisms as the lattice units can prevent wear and friction among multi-part mechanisms. This work presents two distinctive deployable structures based on a programmable compliant bistable lattice. Several novel parameters are introduced into the bistable mechanism to better control the behaviour of bistable mechanisms. By adjusting the defined geometry parameters, the programmable bistable lattices can be optimized for specific targets such as a larger deformation range or higher stability. The first structure is designed to perform 1D deployable movement. This structure consists of multi-series-connected bistable lattices. In order to explore the 3D bistable characteristic, a cylindrical deployable mechanism is designed based on the curved double tensural bistable lattice. The investigation of bistable lattices mainly involves four types of bistable mechanisms. These bistable mechanisms are obtained by dividing the long segment of traditional compliant bistable mechanisms into two equal parts and setting a series of angle data to them, respectively. The experiment and FEA simulation results confirm the feasibility of the compliant deployable structures.

Topological wave energy harvesting in bistable lattices

In this paper, we present an input-independent energy harvesting mechanism exploiting topological solitary waves. This class of medium transforming solitons, or transition waves, entails energy radiation in the form of trailing phonons in discrete bistable lattices. We observe numerically and experimentally that the most dominant frequencies of these phonons are invariant to the input excitations as long as transition waves are generated. The phonon energy at each unit cell is clustered around a single invariant frequency, enabling input-independent resonant harvesting with conventional energy transduction mechanisms. The presented mechanism fundamentally breaks the link between the unit cell size and the metamaterial’s operating frequencies, offering a broadband solution to energy harvesting that is particularly robust for low-frequency input sources. We further investigate the effect of lattice discreteness on the energy harvesting potential, observing two performance gaps and a topological wave harvesting pass band where the potential for energy conversion increases almost monotonically. The observed frequency-invariant phonons are intrinsic to the discrete bistable lattices, enabling broadband energy harvesting to be an inherent metamaterial property.

Snaking bifurcations of localized patterns on ring lattices

Abstract We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, while all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph.

Controlled snapping sequence and energy absorption in multistable mechanical metamaterial cylinders

Mechancial metamaterials comprising bistable unit cells utilize a sinusoidal beam to enable significant deformation between stable states, which is associated with energy absorption. A stack of similar bistable unit cells, i.e., multistable metamaterials, will snap to a second stable state in a seemingly random sequence that is influenced by minute variations in manufacturing. However, it is desirable to tailor the energy absorbed and the snapping sequence to obtain predictable geometric reconfiguration in multistable cylindrical metamaterials. Additive Manufacturing (AM) processes, such as Fused Deposition Modeling (FDM), are able to produce these structures with flexible filaments, such as Thermoplastic Polyurethane (TPU), to enable flexible, multistable cylindrical structures. In this study, we investigated experimentally and computationally the mechanical behavior and controlled snapping sequence of bistable lattices incorporated into cylindrical shells. Individual layers are printed using FDM and assembled into a cylinder. Experimental and computational results for uniaxial tension and compression tests, applied along the length of the cylinder, are used to evaluate the non-linear mechanical behavior and quantify energy absorption of the system during both loading (tension) and unloading (compression). It was shown that energy absorption during loading was always greater than unloading and increased with an increasing number of layers. The energy absorbed by individual layer was approximately 1.25 J in loading and 0.52 J in unloading for a layer with sinusoidal beam thickness of 1.5 mm. The direction dependence of the energy absorption is attributed to the printed configuration of the lattices. Individual layers can be stacked to tailor the amount of energy absorption. In addition, the energy absorbed and the snapping sequence of the multistable cylinder could be controlled by changing the thickness of the sinusoidal beams in each layer. The variation of energy absorbed is strongly influenced by the thickness of the sinusoidal beam. By increasing the beam thickness from 1 mm to 2 mm, energy absorption increased by approximately a factor of three. Beam thickness accurately controlled the snap-through behavior in experimental samples, but the computational model experienced re-distribution of strain and snap-close forces that cause less predictable snap-through behavior in unloading (compression). The design, analysis and control of snapping sequence of cylindrical shells with multistable states provides new opportunities in practical application of control of elastic waves, energy absorption and reconfigurable structures.

A new uniqueness theorem of wave profiles for a 2-D bistable lattice dynamical system

The wave propagation dynamics of bistable lattice dynamical systems was extensively studied in literatures. It is a challenging problem if the wave profiles are unique up to translation. In this note, we establish a new uniqueness theorem of wave profiles for a 2-D bistable lattice dynamical system with global interaction. The obtained result greatly extends the known results in the literature, which need the monotonicity and smoothness restrictions on the wave profiles.

Isolas of multi-pulse solutions to lattice dynamical systems

This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

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.

Linking tracks in mica crystals with phase transitions in a bistable lattice

Since the middle of the last century, scientific research has been conducted to explain the nature of the tracks visible to the naked eye in mica muscovite crystals. In the present work, an attempt to link the appearance of tracks with a phase transition in a bistable medium is made using classical molecular dynamics method. For this purpose, a two-dimensional triangular lattice simulating one row of potassium atoms in mica is considered. Interactions between atoms are described via pairwise Morse potential and a local potential, whose minima create a hexagonal lattice. In order to create a bistability in the system, a mismatch between the equilibrium distance of the triangular lattice and the period of the local potential is artificially introduced. The phase transitions arising from a monotonic increase or decrease of the depth of the local potential are described. It is revealed that at lower temperatures the domains of different phases can coexist, but at higher temperatures the domain with lower potential energy grows with heat release by reducing the high energy domain. It is speculated that the considered model, which provides the possibility of coexistence of two different phases, can be used to explain at qualitative level the nature of dark tracks visible with the naked eye in transparent crystals of mica muscovite.

Solitary waves in bistable lattices with stiffness grading: Augmenting propagation control

In this study we introduce small perturbations in the forms of graded intersite stiffnesses and graded on-site potentials to a lattice composed of bistable unit cells under elastic interactions. Based on a known soliton solution in the $\ensuremath{\phi}$-4 model, we use a perturbation approach to approximate the effects of the perturbations on the propagation speeds of transition waves. Numerical validations follow on the exact discrete equations of motion, from which we observe eventual stoppage of transition waves in the periodic lattice under physical damping, unidirectional propagation of the waves in the direction of softening properties, and boomerang-like reflection of the waves in the stiffening direction. Finally, we present three-dimensional-printed experimental lattices, confirming the theoretical and numerical results. The observed behaviors imply the extreme controllability of solitary waves through slight engineering manipulations in material-level structures. We further find that both kink (rarefaction) and antikink (compression) waves are allowed at any site in the lattice, extending the functionality of the lattice in engineering applications such as energy harvesting.

Solitary waves in a bistable lattice

The design of architectured materials with bistable building blocks holds exciting possibilities. This new class of metamaterials exploits micro-level structural instabilities to obtain extraordinary physical and mechanical properties. Still, the dynamic behavior of these lattice structures is largely unexplored. Here, we study the dynamic response of a 1-D bistable lattice, i.e. a FPU chain with springs having a non-convex double-well energy potential. In addition to metamaterials, this model-problem is prototypical to a large number of systems, such as unfolding/refolding of proteins, crack propagation, plasticity, and mechanisms underlying martensitic phase transformations. We show that, depending only on the stiffness-ratio of the two energy wells of the bistable springs, the system exhibits two fundamentally different responses to impact; either the impact energy is (almost entirely) trapped in the form of large undulations of the first few springs, or the energy of the impact is (almost completely) transmitted along the chain in the form of a solitary wave that involves transition to the secondary energy-well and back. Focusing our attention to the latter, we reveal, based on analytical treatment and extensive numerical simulations, a universal feature of the solitary wave. Namely, the height of the solitary wave is indifferent to the energy barrier separating the two equilibria of the double well potential. This remarkable feature indicates that the spinodal region of the double-well potential affects the behavior only through its breadth.

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.

Entire solutions for bistable lattice differential equations with obstacles

We consider scalar lattice dierential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by \holes) are present in the lattice. Our work generalizes to the discrete spatial setting the results obtained in [9] for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.

Tunable Wave Propagation in Granular Crystals by Altering Lattice Network Topology

We develop a framework for wave tailoring by altering the lattice network topology of a granular crystal consisting of spherical granules in contact. The lattice topology can alternate between two stable configurations, with the spherical granules of the lattice held in stable equilibrium in each configuration by gravity. Under impact, the first configuration results in a wave with rapidly decaying amplitude as it propagates along a primary chain, while the second configuration results in a solitary wave propagating along the primary chain with no decay. The mechanism to achieve such tunability is by having energy diverted to the granules adjacent to the primary chain in the first case but not the second. The tunable design of the proposed network is validated using both numerical simulations and experiments. In terms of potential applications, the proposed bistable lattice network can be viewed either as a wave attenuator or as a device that allows higher amplitude wave propagation in one direction than in the opposite direction. The lattice is analogous to a crystal phase transformation due to the change in atomic configurations, leading to the change in properties at the macroscale.

Unidirectional Transition Waves in Bistable Lattices.

We present a model system for strongly nonlinear transition waves generated in a periodic lattice of bistable members connected by magnetic links. The asymmetry of the on-site energy wells created by the bistable members produces a mechanical diode that supports only unidirectional transition wave propagation with constant wave velocity. We theoretically justify the cause of the unidirectionality of the transition wave and confirm these predictions by experiments and simulations. We further identify how the wave velocity and profile are uniquely linked to the double-well energy landscape, which serves as a blueprint for transition wave control.

Competing interactions and traveling wave solutions in lattice differential equations

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj [1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.

Suppression of pattern complexity by discrete-space state feedback with equal intervals

The suppression of the pattern complexity of spatiotemporal chaos in Coupled Acousto-optic Bistable Map Lattice system (CABMLs) is investigated. The pattern complexity of spatiotemporal dynamics is exhibited quantitatively by employing the average Kolmogorov–Sinai (KS) entropy density according to the Lyapunov exponents of lattices. By designing a global control method with periodic state feedback, we provide the stability analysis for suppressing spatiotemporal chaos to a stable state. Then, a pattern control strategy using discrete-space state feedback with equal intervals is proposed. Numerical simulations demonstrate that the pattern complexity of spatiotemporal chaos is decreased to a low level by the proposed method. Finally, the dependence of the control effect on the control interval is discussed in detail by considering the control efficiency and control cost and the optimal control interval is obtained.

Orientation-Dependent Pinning and Homoclinic Snaking on a Planar Lattice

We study homoclinic snaking of one-dimensional, localized states on two-dimensional, bistable lattices via the method of exponential asymptotics. Within a narrow region of parameter space, fronts connecting the two stable states are pinned to the underlying lattice. Localized solutions are formed by matching two such stationary fronts back-to-back; depending on the orientation relative to the lattice, the solution branch may “snake” back and forth within the pinning region via successive saddle-node bifurcations. Standard continuum approximations in the weakly nonlinear limit (equivalently, the limit of small mesh size) do not exhibit this behavior, due to the resultant leading-order reaction-diffusion equation lacking a periodic spatial structure. By including exponentially small effects hidden beyond all algebraic orders in the asymptotic expansion, we find that exponentially small but exponentially growing terms are switched on via error function smoothing near Stokes lines. Eliminating these otherwise...

Propagation of Periodic Patterns in a Discrete System with Competing Interactions

In this paper we consider a bistable lattice differential equation with competing first and second nearest neighbor interactions. We construct heteroclinic orbits connecting the stable zero equilibrium state with stable spatially periodic orbits of period p=2,3,4 using transform techniques and a bilinear bistable nonlinearity. We investigate the existence, global structure, and multiplicity of such traveling wave solutions. We show that in a certain parameter range the p=2 patterns have a finite maximum propagation speed. Our analysis also shows that multiple solutions involving a p=4 pattern may coexist at some velocities. Numerical simulations suggest the stability of some of the obtained solutions.

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localized with respect to the strain variable. As a standard Lyapunov--Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterize the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive on a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relations.