Multiple Stable Configurations Research Articles

Seasonal refuge patterns of phytoplankton trigger irregular bloom events in a contaminated environment

Several experimental evidences and field data documented that zooplankton may alter its behavioral response in the presence of toxic phytoplankton, reducing its consumption to the point of starvation. This paper is devoted to the mathematical study of such interactions of toxic phytoplankton with grazer zooplankton. The non-toxic phytoplankton is assumed to adopt a density-dependent refuge strategy to avoid over-predation by zooplankton. Both groups of phytoplankton are assumed to suffer direct harm from anthropogenic toxicants, while zooplankton is affected indirectly by ingesting contaminated phytoplankton. We calibrate the proposed model with the field data from Talsari and Digha Mohana, India, and estimate some crucial model parameters consistent with the behavior of the observed data. Our results demonstrate that zooplankton grazing on toxic phytoplankton plays a key role in the emergence or mitigation of plankton blooms. We also highlight the system’s potential to exhibit multiple stable configurations under the same ecological conditions. The plankton system experiences significant regime shifts, which are explored through various bifurcation scenarios, such as transcritical and saddle-node bifurcations. These shifts are influenced by changes in refuge capacity, species growth rates, and environmental carrying capacity. Furthermore, we incorporate environmental variations due to seasonal periodic or almost periodic changes, allowing the refuge parameter to be time-dependent. We observe that the forced system exhibits double periodic solutions. Moreover, stronger seasonal variations in the refuge pattern lead to irregular chaotic blooms. In conclusion, the results offer valuable insights into the sustainability of biodiversity, potentially shedding light on the origin of diverse plankton bloom phenomena.

Shape memory polymer lattice structures with programmable thermal recovery time

Shape memory polymer (SMP) lattice structures have garnered considerable attention due to their intrinsic capability for self-recovery and mechanical reconfiguration. The temporal path, encompassing aspects such as recovery time and deployment sequence, of the shape recovery process in SMP lattice structures holds paramount significance across various domains, including but not limited to medical devices and space deployable structures. Nonetheless, the programming of shape recovery time or deployment sequences in SMP lattice structures, particularly in scenarios devoid of external controllers, remains a challenge. In addressing these challenges, this study presents a novel class of SMP structures endowed with customizable thermal recovery times and programmable deployment sequences, leveraging the influence of structural geometry. Notably, the programmable recovery time and serialized deployment behavior of the proposed SMP lattice structure are achieved within a constant temperature environment, obviating the need for external time-varying stimuli. Finite element simulations and experimental validations corroborate that the proposed SMP structures can be programmed to exhibit recovery times spanning from mere seconds to several hundred seconds. Moreover, a three-stage sequential recovery behavior is attained without necessitating prior local configuration programming. Furthermore, exploiting the distinctive sequential reversibility inherent in a constant high-temperature environment, the designed lattice structure showcases the ability to transition to multiple distinct stable configurations by modulating the duration of high-temperature exposure. The proposed recovery time programmable SMP lattice structure thus presents a viable avenue for realizing intricate multistage controllable shape-shifting structures devoid of external control equipment.

Reusable energy-absorbers design harnessing snapping-through buckling of tailored multistable architected materials

Abstract Multistable metamaterials are artificially engineered materials that possess microarchitectures capable of maintaining multiple stable configurations. However, the realization of mechanical metamaterials with numerous programmable stable configurations using Double-Curved Beam (DCB) elements remains an ongoing challenge. In this study, we exploit the snapping-through buckling phenomenon exhibited by architected DCB structures to devise a mechanical metamaterial with a unique deformation mode, encompassing Multi-stability, Multi-path, Multi-platform, and Multi-step characteristics, hence referred to as a 4M mechanical metamaterial. By employing DCB as fundamental building block elements, architected metamaterials with two-dimensional (2D) series or parallel lattices are successfully constructed, as well as three-dimensional (3D) tubular geometries, denoted as DCB-n-m-C and DCB-n-m-M metamaterials, respectively. These metamaterials exhibit reversible energy absorption characteristics and the stiffness can be transformed from positive to negative under both small and large elastic deformations. Functional gradient design and tailored deformation capability are given by adjusting the wall thickness of each layer of double-curved beams, thereby demonstrating the multi-path deformation features inherent in 4M metamaterials. DCB-n-m-M metamaterials has multiple energy platforms in the process of snapping-through, which reflects the multi-platform characteristics of 4M metamaterial. Consequently, novel properties such as multistability, programmability, and reusable energy absorption characteristics are achieved. To comprehensively understand the mechanical response of the metamaterials, a thorough investigation into the influence of geometric parameters is conducted, including the number of polygonal edges, the number of the layers, and the aspect ratio Q. This investigation involves a combination of theoretical analyses, numerical simulations, and experimental verifications. The introduced design strategy paves a way for the innovative design of multistable, multi-step, tailored, and reversible deformation metamaterials.

Mechanical logic gate design based on multi-stable metamaterial with multi-step deformation

Mechanical metamaterials exhibit remarkable design versatility, enabling a series of unconventional mechanical properties via structural modifications. However, it is worth noting that many existing mechanical metamaterials either possess a singular deformation path or only multi-step deformation paths without reusability. This study proposes a novel multi-step deformation metamaterial that possesses multiple stable configurations. This metamaterial produces a unique two-step deformation mode under displacement loading, where the curved beams of the unit cells undergo snap-through and the vertical beams buckling. The influence of geometrical parameters on the multistability of the structure is thoroughly examined, and a theoretical prediction model for the buckling load is formulated. To validate the accuracy of the model, experimental tests and finite element simulations are conducted on the two-step deformation mode of periodic array of the unit cell. Additionally, a bi-material design approach for the unit cell is further introduced to enhance the specific energy absorption of the structure while reinforce the negative stiffness effect. Leveraging the multistable characteristics of the structure, mechanical logic gates such as AND, OR, NOT and NAND are conceptualized and experimentally validated. Through the implementation of rational designs and combinatorial connections within the logic gate structure, more complex logic operations can be realized.

A seven-parameter high-order finite element model for multi-stable analysis of variable stiffness laminated shells

Variable angle fiber composites provide large design space for multi-stable structures. However, they require reliable numerical models to accurately predict their mechanical behavior characterized by a complex material response and large deformation. To this end, we propose an accurate yet efficient model for stability analysis of variable angle fiber laminated shells. In this model, first a seven-parameter shell formulation with Equivalent Single Layer theory, which is suitable for three-dimensional constitutive laws, is utilized to accurately characterize the large deformation. The fiber orientation angle is described by a linear variation from the center to the end of the shell. Then, to improve the accuracy and to reduce the computational cost, the spectral/hp element with high-order polynomial basis functions is adopted for discretization. Finally, complex nonlinear equilibrium paths are effectively tracked by the Riks method. By comparison with ABAQUS solutions, the proposed model shows to be efficient and accurate for variable angle fiber laminates under various conditions, in which both the buckling, the snap-through and tristability behaviors are studied. Moreover, variable angle fiber shells provide more design possibilities than straight fiber shells. By opportunely varying the fibers angle of each lamina, it is possible to control the stable states as well as to obtain multiple stable configurations (bi- and tri-stability).

Reprogrammable multistable ribbon kirigami with a wide cut

Upon stretching, a ribbon kirigami with parallel major cuts exhibits multistable behaviors that can maintain at multiple stable configurations. This work investigates the phenomena of a ribbon kirigami with one of the major cuts replaced by a wide cut, of which the multistable behaviors of the wide-cut located cell, the stable configurations, and the energy barriers between the stable configurations are explored. It is observed that the introduction of the wide cut results in local symmetry breaking, enabling bidirectional transition of the stable configurations in such a kirigami. The results also reveal that the geometries of major cuts and the stretch level enable reprogramable dynamic behaviors, such as the number of transitions once triggered. A kirigami-Morse code system is hereby presented, utilizing dynamic reconfiguration and showing a refreshable mechanical readout utilizing reprogrammability. The kirigami has potential for developing metamaterials with unique dynamic features.

Tuning of Multi-stability Profile and Transition Sequence of Stacked Miura-Origami Metamaterials

Multi-stable origami structures and metamaterials possess unique advantages and could exhibit multiple stable three-dimensional configurations, which have attracted widespread research interest and held promise for applications in many fields. Although a great deal of attention has been paid to the design and application of multi-stable origami structures, less knowledge is available about the transition sequence among different stable configurations, especially in terms of the fundamental mechanism and the tuning method. To fill this gap, with the multi-stable dual-cell stacked Miura-ori chain as a platform, this paper explores the rules that govern the configuration transition and proposes effective methods for tuning the transition sequence. Specifically, by correlating the energy evolution, the transition paths, and the associated force–displacement profiles, we find that the critical extension/compression forces of the component cells play a critical role in governing the transition sequence. Accordingly, we summarize the rules for predicting the transition sequence: the component cell that first reaches the critical force during quasi-static extension or compression will be the first to undergo a configuration switch. Based on these findings, two methods, i.e., a design method based on crease-stiffness assignment and an online method based on internal pressure regulation, are proposed to tune the stability profile and the transition sequence of the multi-stable origami structure. The crease-stiffness design approach, although effective, cannot be employed for online tuning once the prototype has been fabricated. The pressure-based approach, on the other hand, has been shown experimentally to be effective in adjusting the constitutive force–displacement profiles of the component cells and, in turn, tuning the transition sequence according to the summarized rules. The results of this study will advance the state of the art of origami mechanics and promote the engineering applications of multi-stable origami metamaterials.

Uncertainty quantification of bistable variable stiffness laminate using machine learning assisted perturbation approach

Morphing structures have received growing interest in aerospace structures and wind turbines due to their rapid shape-changing ability in response to the change in operating conditions. Bistable laminates using variable stiffness composites are considered potential candidates in morphing structures for their ability to tailor the design space with a plethora of multiple stable configurations and satisfy the conflicting requirements of load-carrying capacity and deformability. Even though extensive works have been reported on the analysis and design of the cured shape of unsymmetrical variable stiffness laminates for morphing application, the effect of uncertainty in design variables on the behavior of bistable laminates is not profoundly assessed in the literature. In particular, uncertainty propagation through a highly non-linear map can lead to a significant discrepancy between the numerically predicted and experimental observations. Therefore, for adaptability in practical application, it is imperative to quantify the uncertainty as well as to characterize the non-linearity present near the design point of interest. In this work, a general purpose machine learning assisted uncertainty quantification (MLAUQ) framework is developed and demonstrated on unsymmetrical bistable laminate. The study considers three different variants of the approach based on the order of approximation (O(hk)) used for the training purpose. It is found that the MLAUQ−3 approach performs better than other approaches, and a theoretical justification for the same is provided. The method relies on the fact that expensive computation of the Hessian required for standard perturbation approaches can be bypassed by training a neural network while retaining accurate gradient information near the design point of interest. In the case of bistable laminate, a network trained with a few training samples can capture the local model non-linearity. Further, numerical investigations reveal that the proposed approach is computationally efficient and accurate compared to traditional uncertainty quantification (UQ) approaches.

On the frequency selection mechanism of the low-Re flow around rectangular cylinders

In the flow past elongated rectangular cylinders at moderate Reynolds numbers, vortices shedding from the leading- and trailing-edge corners are frequency locked by the impinging leading-edge vortex instability. The present work investigates how the chord-based Strouhal number varies with the aspect ratio of the cylinder at a Reynolds number (based on the cylinder thickness and the free-stream velocity) of $Re=400$ , i.e. when locking is strong. Several two-dimensional, nonlinear simulations are run for rectangular and D-shaped cylinders, with the aspect ratio ranging from $1$ to $11$ , and a global linear stability analysis of the flow is performed. The shedding frequency observed in the nonlinear simulations is predicted fairly well by the eigenfrequency of the leading eigenmode. The inspection of the structural sensitivity confirms the central role of the trailing-edge vortex shedding in the frequency locking, as already assumed by other authors. Surprisingly, however, the stepwise increase of the Strouhal number with the aspect ratio reported by several previous works is not fully reproduced. Indeed, with increasing aspect ratio, two distinct flow behaviours are observed, associated with two flow configurations where the interaction between the leading- and trailing-edge vortices is different. These two configurations are fully characterised, and the mechanism of selection of the flow configuration is discussed. Lastly, for aspect ratios close to the jump between two consecutive shedding modes, the Strouhal number is found to present hysteresis, implying the existence of multiple stable configurations. Continuing the lower shedding-mode branch by increasing the aspect ratio, we found that the periodic configuration loses stability via a Neimark–Sacker bifurcation leading to different Arnold tongues. This hysteresis can explain, at least partially, the significant scatter of existing experimental and numerical data.

A bio-inspired robotic climbing robot to understand kinematic and morphological determinants for an optimal climbing gait

Robotic systems for complex tasks, such as search and rescue or exploration, are limited for wheeled designs, thus the study of legged locomotion for robotic applications has become increasingly important. To successfully navigate in regions with rough terrain, a robot must not only be able to negotiate obstacles, but also climb steep inclines. Following the principles of biomimetics, we developed a modular bio-inspired climbing robot, named X4, which mimics the lizard’s bauplan including an actuated spine, shoulders, and feet which interlock with the surface via claws. We included the ability to modify gait and hardware parameters and simultaneously collect data with the robot’s sensors on climbed distance, slip occurrence and efficiency. We first explored the speed-stability trade-off and its interaction with limb swing phase dynamics, finding a sigmoidal pattern of limb movement resulted in the greatest distance travelled. By modifying foot orientation, we found two optima for both speed and stability, suggesting multiple stable configurations. We varied spine and limb range of motion, again showing two possible optimum configurations, and finally varied the centre of pro- and retraction on climbing performance, showing an advantage for protracted limbs during the stride. We then stacked optimal regions of performance and show that combining optimal dynamic patterns with either foot angles or ROM configurations have the greatest performance, but further optima stacking resulted in a decrease in performance, suggesting complex interactions between kinematic parameters. The search of optimal parameter configurations might not only be beneficial to improve robotic in-field operations but may also further the study of the locomotive evolution of climbing of animals, like lizards or insects.

Rotational snap-through behavior of multi-stable beam-type metastructures

Metastructures consisting of planar arrangements of bi-stable snap-through beams are able to exhibit multiple stable configurations. Apart from the expected translational state transition, when all beam elements snap through, rotational states may exist as well. In this paper we explore the rotational properties of multi-stable metastructures on the basis of both experimental and theoretical investigations, and define the conditions for achieving rotational stable states. Results show that the metastructure is able to realize both translational and rotational states, while the rotational transitions require less energy as compared to their translational counterparts. The influence of geometric parameters on rotational stability is investigated via parametric studies. Furthermore, to determine the design criteria for rotational stability, a theoretical investigation based on mode superposition principle is performed to predict the nonlinear-deformation of a unit cell. The theoretical analysis predicts well the rotational snap-through transitions that are observed in finite element simulations. It is found that the rotational stability is determined by setting proper values for h/L and t/L (h, t, L represent apex height, thickness and span of the bi-stable beam structure, respectively). Finally, we experimentally demonstrate that the proposed metastructure with multiple layers is able to achieve large rotations and translations.

Enhanced models for the nonlinear bending of planar rods: localization phenomena and multistability.

We deduce a one-dimensional model of elastic planar rods starting from the Föppl-von Kármán model of thin shells. Such model is enhanced by additional kinematical descriptors that keep explicit track of the compatibility condition requested in the two-dimensional parent continuum, that in the standard rods models are identically satisfied after the dimensional reduction. An inextensible model is also proposed, starting from the nonlinear Koiter model of inextensible shells. These enhanced models describe the nonlinear planar bending of rods and allow to account for some phenomena of preeminent importance even in one-dimensional bodies, such as formation of singularities and localization (d-cones), otherwise inaccessible by the classical one-dimensional models. Moreover, the effects of the compatibility translate into the possibility to obtain multiple stable equilibrium configurations.

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.

Target Bubbles in Fe3Sn2 Nanodisks at Zero Magnetic Field.

We report a vortex-like magnetic configuration in uniaxial ferromagnet Fe3Sn2 nanodisks using differential phase contrast scanning transmission electron microscopy. This magnetic configuration is transferred from a conventional magnetic vortex using a zero-magnetic-field warming process and is characterized by a series of concentric cylinder domains. We termed them as "target bubbles" that are identified as three-dimensional depth-modulated magnetic objects in combination with numerical simulations. Target bubbles have room-temperature stability even at zero magnetic field and multiple stable magnetic configurations. These advantages render the target bubble an ideal bit to be an information carrier and can advance magnetic target bubbles toward functionalities in the long term by incorporating emergent degrees of freedom and purely electrically controllable magnetism.

Geometrically reconfigurable 3D mesostructures and electromagnetic devices through a rational bottom-up design strategy.

Microelectronic devices with reconfigurable three-dimensional (3D) microarchitecture that can be repetitively switched among different geometrical and/or working states have promising applications in widespread areas. Traditional approaches usually rely on stimulated deformations of active materials under external electric/magnetic fields, which could potentially introduce parasitic side effects and lower device performances. Development of a rational strategy that allows access to high-performance 3D microdevices with multiple stable geometric configurations remains challenging. We introduce a mechanically guided scheme to build geometrically reconfigurable 3D mesostructures through a bottom-up design strategy based on a class of elementary reconfigurable structures with the simplest ribbon geometries. Quantitative mechanics modeling of the structural reconfigurability allows for the development of phase diagrams and design maps. Demonstrations of ~30 reconfigurable mesostructures with diverse geometric topologies and characteristic dimensions illustrate the versatile applicability. The multimode nature enables customized distinct beamforming and discrete beam scanning using a single antenna capable of on-demand reconfiguration.

Resolving the adsorption of molecular O2 on the rutile TiO2(110) surface by noncontact atomic force microscopy

Interaction of molecular oxygen with semiconducting oxide surfaces plays a key role in many technologies. The topic is difficult to approach both by experiment and in theory, mainly due to multiple stable charge states, adsorption configurations, and reaction channels of adsorbed oxygen species. Here we use a combination of noncontact atomic force microscopy (AFM) and density functional theory (DFT) to resolve [Formula: see text] adsorption on the rutile [Formula: see text](110) surface, which presents a longstanding challenge in the surface chemistry of metal oxides. We show that chemically inert AFM tips terminated by an oxygen adatom provide excellent resolution of both the adsorbed species and the oxygen sublattice of the substrate. Adsorbed [Formula: see text] molecules can accept either one or two electron polarons from the surface, forming superoxo or peroxo species. The peroxo state is energetically preferred under any conditions relevant for applications. The possibility of nonintrusive imaging allows us to explain behavior related to electron/hole injection from the tip, interaction with UV light, and the effect of thermal annealing.

Nonlinear wave propagation and dynamic reconfiguration in two-dimensional lattices with bistable elements

This paper examines the propagation of elastic waves of large amplitudes in two-dimensional square lattices that include an alternating pattern of linear springs and nonlinear bistable springs. Because of the presence of bistable springs, these lattices have multiple stable configurations. In this paper, a theoretical model that takes into account the non-linearity of the bistable springs while neglecting geometric nonlinearities is used to study non-linear wave propagation in these lattices. Results from numerical simulations demonstrate that, for a lattice that is initially in a deformed stable equilibrium configuration, stimuli of large amplitudes are able to cause a reconfiguration of the lattice to a configuration of lower potential energy. Even for initial stable configurations that exhibit omnidirectional or directional bandgaps for infinitesimal wave propagation, waves of large amplitudes can propagate in the lattice due to its reconfiguration; however, due to the nonlinearity of bistable springs, the transmitted response tends to have multiple spectral peaks or be broadband for narrow-band stimuli of large amplitude. Simulations are used to study how the reconfiguration of the lattice depends on the amplitude and duration of the stimulus, on damping, and on the energy barrier between the two stable equilibria of the bistable springs. These numerical results suggest that these multistable lattices can serve as reconfigurable wave guides for which the amplitude of the stimulus offers opportunities for enhanced tunability.

Energy dissipation in functionally two-dimensional phase transforming cellular materials

Phase Transforming Cellular Materials (PXCMs) are periodic cellular materials whose unit cells exhibit multiple stable or meta-stable configurations. Transitions between the various (meta-) stable configurations at the unit cell level enable these materials to exhibit reusable solid state energy dissipation. This energy dissipation arises from the storage and non-equilibrium release of strain energy accompanying the limit point traversals underlying these transitions. The material deformation is fully recoverable, and thus the material can be reused to absorb and dissipate energy multiple times. In this work, we present two designs for functionally two-dimensional PXCMs: the S-type with four axes of reflectional symmetry based on a square motif and, the T-type with six axes of symmetry based on a triangular motif. We employ experiments and simulations to understand the various mechanisms that are triggered under multiaxial loading conditions. Our numerical and experimental results indicate that these materials exhibit similar solid state energy dissipation for loads applied along the various axes of reflectional symmetry of the material. The specific energy dissipation capacity of the T-type is slightly greater and less sensitive to the loading direction than the S-type under the most of loading directions. However, both types of material are shown to be very effective in dissipating energy.

Ferroelectric ZrO2 Monolayers as Buffer Layers between SrTiO3 and Si

A monolayer of ZrO$_{2}$ has recently been grown on the Si(001) surface and shown to have ferroelectric properties, which signifies the realization of the lowest possible thickness in ferroelectric oxides [M. Dogan et al., Nano Lett., 18 (1) (2018)]. In our previous computational study, we reported on the multiple (meta)stable configurations of ZrO$_{2}$ monolayers on Si, and how switching between a pair of differently polarized configurations may explain the observed ferroelectric behavior of these films [M. Dogan and S. Ismail-Beigi, arXiv:1902.01022 (2019)]. In the current study, we conduct a DFT-based investigation of (i) the effect of oxygen content on the ionic polarization of the oxide, and (ii) the role of zirconia monolayers as buffer layers between silicon and a thicker oxide film that is normally paraelectric on silicon, e.g. SrTiO$_{3}$. We find that (i) total energy-vs-polarization behavior of the monolayers, as well as interface chemistry, is highly dependent on the oxygen content; and (ii) SrTiO$_{3}$/ZrO$_{2}$/Si stacks exhibit multiple (meta)stable configurations and polarization profiles, i.e. zirconia monolayers can induce ferroelectricity in oxides such as SrTiO$_{3}$ when used as a buffer layer. This may enable a robust non-volatile device architecture where the thickness of the gate oxide (here strontium titanate) can be chosen according to the desired properties.

Multistable two-dimensional spring-mass lattices with tunable band gaps and wave directionality

In this paper, we consider elastic wave propagation in two dimensional periodic lattices that include an alternating pattern of linear springs and nonlinear bistable springs. Because of the bistable springs, these lattices have multiple stable equilibrium configurations. An analytical model that takes into account both nonlinear geometrical effects and nonlinearity of the springs is developed for the total potential energy of the system. After finding the stable equilibrium configurations of the unit cell by minimizing its potential energy, the propagation of waves of small amplitude is analyzed in each of these stable configurations using Bloch theorem. Examination of the band diagrams demonstrates that, depending on the model parameters, directional or complete band gaps can be observed in some of the deformed configurations, particularly for deformed configurations that are close to a critical point. Furthermore, analysis of the iso-frequency contours of the dispersion surfaces and of polar plots of the phase and group velocities indicates that the low frequency wave directionality of the lattice is tunable. For some designs, switching from one configuration to another configuration makes it possible to dramatically alter the preferred direction of wave propagation. Simulations of the response of lattices of finite size confirm that these lattices can be used as reconfigurable phononic crystals with tunable directivity, both in the low frequency range and within the directional band gaps.