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In this work, the Schwarz primitive unit cell is used as the building block of different types of metastructures for steering and focusing elastic vibrations. The emergence of a Bragg-type bandgap when constructing a two-dimensional plate from such unit cells is experimentally validated. It is demonstrated that increasing both mass and porosity of the Schwarz primitive leads to a decrease in the frequency of the out-of-plane propagating wave targeted in this study. By arranging these modified Schwarz primitive unit cells in constant and graded layouts, two-dimensional plates with an embedded metabarrier and a metalens are numerically designed. The metabarrier protects an interior area of the plate from the propagating waves on a wide frequency band (approx. 1.4-3.4 kHz). Equally, the refractive index profile necessary for gradient index lenses is obtained via a progressive variation of the added mass or, alternatively, the porosity of the unit cell over a rectangular area. For the first time, bending of the out-of-plane mode towards the focusing point is practically validated in a challenging mesoscale experiment requiring the assembly of different three-dimensional printed sections of the plate. The increased porosity design is advantageous not only in terms of overall lightweight, but also towards additive manufacturing as it requires less material.

In this paper, we incorporate the effect of nonlinear damping with the concept of locally resonant metamaterials to enable vibration attenuation beyond the conventional bandgap range. The proposed design combines a linear host cantilever beam and periodically distributed inertia amplifiers as nonlinear local resonators. The geometric nonlinearity induced by the inertia amplifiers causes an amplitude-dependent nonlinear damping effect. Through the implementation of both modal superposition and numerical harmonic methods the finite nonlinear metamaterial is accurately modelled. The resulting nonlinear frequency response reveals the bandgap is both amplitude-dependent and broadened. Furthermore, the modal frequencies are also attenuated due to the nonlinear damping effect. The theoretical results are validated experimentally. By embedding the nonlinear damping effect into locally resonant metamaterials, wideband attenuation of the proposed metamaterial is achieved, which opens new possibilities for versatile metamaterials beyond the limit of their linear counterparts.

Nonlinearity has enabled energy harvesting to advance towards higher power output and broader bandwidth in monostable, bistable, and multistable systems. However, challenges in operating in the high energy orbit (HEO) rather than low energy orbit (LEO) have restricted their applications. Based on a monostable nonlinear system, this paper proposes a self-contained solution for time-sharing orbit jump and energy harvesting. The joint dynamics of an electromechanical assembly consisting of a nonlinear energy harvester and a switch-mode piezoelectric interface circuit is studied for high-capability energy harvesting. The proposed solution is carried out by utilizing a cutting-edge switch-mode bidirectional energy conversion circuit (BECC), which enables time-sharing dual functions of energy harvesting and vibration exciting. A theoretical model is established based on impedance analysis and multiple time scales method to analyze the stability, frequency response, and phase evolution of the autonomous and nonautonomous nonlinear energy harvesting systems. Particularly, the detailed dynamics for the orbit jumps with vibration exciting mode of BECC are studied. Experiments are performed to validate the full-range orbit jumps with the monostable nonlinear energy harvester. The harvested power after orbit jumps yields a nine-fold increase, compensating for the energy consumption under vibration exciting mode quickly. The proposed solution also refrains the system from extra mechanical or electrical energy sources for orbit jumps, which leads to the first self-contained solution for simultaneous energy harvesting and orbit jump in nonlinear piezoelectric energy harvesting. This work enhances the practical utility of nonlinear energy harvesting technologies toward engineering applications.

We experimentally implement a virtual geometric periodicity in an elastic metamaterial. First, unwanted boundary reflections at the domain ends are cancelled through the iterative injection of the polarity reversed, reflected wavefield. The resulting boundless experimental state allows for a much better analysis of the metamaterials influence on the propagating wavefield. Subsequently, the propagating wavefield exiting on one end of the structure is reintroduced at the opposite end, creating a virtual geometric periodicity. We find that the experimentally observed band gap converges to the analytical solution through the introduction of the virtual periodicity. The established workflow introduces a novel approach to the experimental investigation and validation of metamaterial prototypes in the presence of strongly dispersive wave propagation and internal scattering. The fully data driven, ad-hoc treatment of boundary conditions in metamaterial experimentation with arbitrary mechanical properties enables reflection suppression, virtual periodicity, and the introduction of more general fictitious boundaries.

This work studies a broadband graded metamaterial, which integrates the piezoelectric energy harvesting function targeting low-frequency structural vibrations, lying below 100 Hz. The device combines a graded metamaterial with beam-like resonators, piezoelectric patches and a self-powered piezoelectric interface circuit for energy harvesting. Based on the mechanical and electrical lumped parameters, an integrated model is proposed to investigate the power performance of the proposed design. Thorough numerical simulations were conducted to analyse the spatial frequency separation capacity and the slow-wave phenomenon of the graded metamaterial for broadband and high-capability piezoelectric energy harvesting. Experiments with realistic vibration sources show that the harvested power of the proposed design yields a five-fold increase with respect to conventional harvesting solutions based on single cantilever harvesters. Our results reveal the significant potential on exploitation of graded metamaterials for energy-efficient vibration-powered devices.

In this work, we investigate the dynamics and attenuation properties of a one-dimensional inertial amplified lattice with opposite chirality. The unit cell of the structure consists of a hollow-square plate connected to a ring through arch-like ligaments. The peculiar geometry and orientation of the links allow for coupling the axial and the torsional motion of the lattice, thus amplifying the inertia of the system. We develop both simplified analytical and numerical models of the building block to derive the complex dispersion relation of the infinite lattice. The structure supports a frequency-tailorable attenuation zone, whose lower bound is controlled by the second coupled axial-torsional mode. Laboratory measurements of the transmission spectrum on a 3D printed sample match very well with the analytical and numerical predictions, confirming the wide-band filtering properties of this lattice. We complete our investigation by developing and solving a constrained optimization model to obtain the optimized geometric parameters of the unit cell that minimize the bandgap opening frequency and, at the same time, fulfill structural requirements. In particular, the internal stresses induced by the self-weight of the structure are kept to a low by virtue of the employed design, with the aim to prevent plastic deformations and failure. The inertial amplification mechanism, proposed and investigated in this work, offers an efficient variant for the efficient design of materials and structures for vibration mitigation and shock protection., Comment: submitted to JSV

Motivated by the importance of lattice structures in multiple fields, we investigate the propagation of flexural waves in a thin woven plate augmented with two classes of metastructures for wave mitigation and guiding, namely metabarriers and metalenses. The cellular architecture of this plate invokes the well-known octet topology, while the metadevices rely on novel customized octets either comprising spherical masses added to the midpoint of their struts or variable node thickness. We numerically determine the dispersion curves of a doubly-periodic array of octets, which produce a broad bandgap whose underlying physics is elucidated and leveraged as a design paradigm, allowing the construction of a metabarrier effective for inhibiting the transmission of waves. More sophisticated effects emerge upon parametric analyses of the added masses and node thickness, leading to graded designs that spatially filter waves through an enlarged bandgap via rainbow trapping. Additionally, Luneburg and Maxwell metalenses are realized using the spatial modulation of the tuning parameters and numerically tested. Wavefronts impinging on these structures are progressively curved within the inhomogeneous media and steered toward a focal point. Our results yield new perspectives for the use of octet-like lattices, paving the way for promising applications in vibration isolation and energy focusing.

In this work, we experimentally and numerically investigate the propagation and attenuation of vertically polarized surface waves in an unconsolidated granular medium equipped with small-scale metabarriers of different depths, i.e., arrays composed of one, two, and three embedded layers of sub-wavelength resonators. Our findings reveal how such a multi-layer arrangement strongly affects the attenuation of the surface wave motion within and after the barrier. When the surface waves collide with the barriers, the wavefront is back-scattered and steered downward underneath the oscillators. Due to the stiffness gradient of the granular medium, part of the wavefield is then rerouted to the surface level after overcoming the resonant array. Overall, the in-depth insertion of additional layers of resonators leads to a greater and broader band wave attenuation when compared to the single layer case.

We amalgamate two fundamental designs from distinct areas of wave control in physics, and place them in the setting of elasticity. Graded elastic metasurfaces, so-called metawedges, are combined with the now classical Su-Schrieffer-Heeger (SSH) model from the field of topological insulators. The resulting structures form one-dimensional graded-SSH-metawedges that support multiple, simultaneous, topologically protected edge states. These robust, enhanced localised modes are leveraged for applications in elastic energy harvesting using the piezoelectric effect. The designs we develop are first motivated by applying the SSH model to mass-loaded Kirchhoff-Love thin elastic plates. We then extend these ideas to using graded resonant rods, and create SSH models, coupled to elastic beams and full elastic half-spaces.

We demonstrate that a rainbow-based metasurface, created by a graded array of resonant rods attached to an elastic beam, operates as a mechanical delay-line by slowing down surface elastic waves to take advantage of wave interaction with resonance. Experiments demonstrate that the rainbow effect reduces the amplitude of the propagating wave in the host structure. At the same time it dramatically increases both the period of interaction between the waves and the resonators, and the wavefield amplitude in the rod endowed with the harvester. Increased energy is thus fed into the resonators over time, we show the enhanced energy harvesting capabilities of this system.

In this paper, we describe the manifestation of localized states through coherent and incoherent analyses of a diffuse elastic wavefield inside a two-dimensional metamaterial made of a collection of vertical long beams glued to a thin plate. We demonstrate that localized states arise due to multi-wave interactions at the beam-plate attachment when the beams acts as coupled resonators for both compressional and flexural resonances on the metasurface. Due to the low-quality factor compressional resonance of the beams, inside the main bandgap the modal density of the systel drops to near a high-quality factor flexural resonance of the beams, and blocks the diffusion process of the wavefield intensity. This experiment physically highlights the tight-binding-like coupling in the localized regime for this two-dimensional metamaterial., Comment: 7 pages, 6 figures, regular paper

Elastic waves guided along surfaces dominate applications in geophysics, ultrasonic inspection, mechanical vibration, and surface acoustic wave devices; precise manipulation of surface Rayleigh waves and their coupling with polarized body waves presents a challenge that offers to unlock the flexibility in wave transport required for efficient energy harvesting and vibration mitigation devices. We design elastic metasurfaces, consisting of a graded array of rod resonators attached to an elastic substrate that, together with critical insight from Umklapp scattering in phonon-electron systems, allow us to leverage the transfer of crystal momentum; we mode-convert Rayleigh surface waves into bulk waves that form tunable beams. Experiments, theory and simulation verify that these tailored Umklapp mechanisms play a key role in coupling surface Rayleigh waves to reversed bulk shear and compressional waves independently, thereby creating passive self-phased arrays allowing for tunable redirection and wave focusing within the bulk medium., Comment: 5 pages of text with 3 figures plus 3 pages of supplemental text with 5 supplemental figures

In this article the physics of horizontally polarized shear waves travelling across a locally resonant metasurface in an unconsolidated granular medium is experimentally and numerically explored. The metasurface is comprised of an arrangement of sub-wavelength horizontal mechanical resonators embedded in silica microbeads. The metasurface supports a frequency-tailorable attenuation zone induced by the translational mode of the resonators. The experimental and numerical findings reveal that the metasurface not only backscatters part of the energy, but also redirects the wavefront underneath the resonators leading to a considerable amplitude attenuation at the surface level, when all the resonators have similar resonant frequency. A more complex picture emerges when using resonators arranged in a so-called graded design, e.g., with a resonant frequency increasing/decreasing throughout the metasurface. Unlike Love waves propagating in a bi-layer medium, shear waves localized at the surface of our metasurface are not converted into bulk waves. Although a detachment from the surface occurs, the depth-dependent velocity profile of the granular medium prevents the mode-conversion, steering again the horizontally polarized shear waves towards the surface. The outcomes of our experimental and numerical studies allow for understanding the dynamics of wave propagation in resonant metamaterials embedded in vertically inhomogeneous soils and, therefore, are essential for improving the design of engineered devices for ground vibration and seismic wave containment., Comment: Manuscript submitted to Phys. Rev. Appl

The article analyses two potential metamaterial designs, the metafoundation and the metabarrier, capable to attenuate seismic waves on buildings or structural components in a frequency band between 3.5 to 8 Hz. The metafoundation serves the dual purpose of reducing the seismic response and supporting the superstructure. Conversely the metabarrier surrounds and shields the structure from incoming waves. The two solutions are based on a cell layout of local resonators whose dynamic properties are tuned using finite element simulations combined with Bloch-Floquet boundary conditions. To enlarge the attenuation band, a graded design where the resonant frequency of each cell varies spatially is employed. If appropriately enlarged or reduced, the metamaterial designs could be used to attenuate lower frequency seismic waves or groundborne vibrations respectively. A sensitivity analysis over various design parameters including size, number of resonators, soil type and source directivity, carried out by computing full 3D numerical simulations in time domain for horizontal shear waves is proposed. Overall, the metamaterial solutions discussed here can reduce the spectral amplification of the superstructure between approx. 15 to 70% depending on several parameters including the metastructure size and the properties of the soil. Pitfalls and advantages of each configuration are discussed in detail. The role of damping, crucial to avoid multiple resonant coupling, and the analogies between graded metamaterials and tuned mass dampers is also investigated., Comment: Article submitted to Journal of Sound and Vibrations. Comments and suggestions are welcome

In elastic wave systems, combining the powerful concepts of resonance and spatial grading within structured surface arrays enable resonant metasurfaces to exhibit broadband wave focusing, mode conversion from surface (Rayleigh) waves to bulk (shear) waves, and spatial frequency selection. Devices built around these concepts allow for precise control of surface waves, often with structures that are subwavelength, and utilise rainbow trapping that separates the signal spatially by frequency. Rainbow trapping yields large amplifications of displacement at the resonator positions where each frequency component accumulates. We investigate whether this amplification, and the associated control, can be used to create energy harvesting devices; the potential advantages and disadvantages of using graded resonant devices as energy harvesters is considered. We concentrate upon elastic plate models for which the A0 mode dominates, and take advantage of the large displacement amplitudes in graded resonant arrays of rods, to design innovative metasurfaces that focus waves for enhanced piezoelectric sensing and energy harvesting. Numerical simulation allows us to identify the advantages of such graded metasurface devices and quantify its efficiency, we also develop accurate models of the phenomena and extend our analysis to that of an elastic half-space and Rayleigh surface waves.

It is well known in metamaterials that local resonance and hybridization phenomena dramatically influence the shape of dispersion curves; the metasurface created by a cluster of resonators, subwavelength rods, atop an elastic surface being an exemplar with these features. On this metasurface, band-gaps, slow or fast waves, negative refraction and dynamic anisotropy can all be observed by exploring frequencies and wavenumbers from the Floquet-Bloch problem and by using the Brillouin zone. These extreme characteristics, when appropriately engineered, can be used to design and control the propagation of elastic waves along the metasurface. For the exemplar we consider, two parameters are easily tuned: rod height and cluster periodicity. The height is directly related to the band-gap frequency, and hence to the slow and fast waves, while the periodicity is related to the appearance of dynamic anisotropy. Playing with these two parameters generates a gallery of metasurface designs to control the propagation of both flexural waves in plates and surface Rayleigh waves for half-spaces. Scalability with respect to the frequency and wavelength of the governing physical laws allows the application of these concepts in very different fields and over a wide range of lengthscales., Comment: Paper currently submitted to Frontiers in Mechanics of Materials

Recent years have heralded the introduction of metasurfaces that advantageously combine the vision of sub- wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. An important topic within metasurfaces is the tailored rainbow trapping and selective spatial frequency separation of electromagnetic and acoustic waves using graded metasurfaces. This frequency dependent trapping and spatial frequency segregation has implications for energy concentrators and associated energy harvesting, sensing and wave filtering techniques. Different demonstrations of acoustic and electromagnetic rainbow devices have been performed, however not for deep elastic substrates that support both shear and compressional waves, together with surface Rayleigh waves; these allow not only for Rayleigh wave rainbow effects to exist but also for mode conversion from surface into bulk waves. We design, and test experimentally, a device creating not only elastic Rayleigh wave rainbow trapping, but also selective mode conversion of surface Rayleigh waves to shear waves., Comment: Article currently submitted for review. Comments are welcome!