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(a) Refractive index profile of the Luneburg lens and the Maxwell lens. (b) Calculated interaction coefficient profile for a Luneburg lens and a Maxwell lens.
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Metasurfaces of mechanical resonators have been successfully used to control in-plane polarized surface waves for filtering, waveguiding and lensing applications across different length scales. In this work, we extend the concept of metasurfaces to anti-plane surface waves existing in semi-infinite layered media, generally known as Love waves. By m...
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Context 1
... consider two specific GRIN lenses, namely (i) the Luneburg lens and (ii) the Maxwell lens whose refractive index profiles are given in Table 1 and shown in Fig. ...
Context 2
... func- tion in order to obtain a continuous description of the interaction coefficient over the lens domain F(r/R). The interaction coefficient profiles F(r/R) calculated according the above described procedure for a layered medium with velocities c T,2 = 1, α = 0.3, H′ = 0.848, ω * = 0.89ω r for both a Luneburg and a Maxwell lens are shown in Fig. ...
Context 3
... F(r/R) is given in Fig. 5b. Since each truss-mass element has a mono-axial behavior, for each unit cell we utilize a couples of identical resonators in an orthogonal configuration to reconstruct the force P orthogonal to the wave- front direction, which varies along the metalens domain (see also inset Fig. 6). As such, the surface load P ...
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Citations
... While so far we only have focused on Rayleigh waves, layered soils will practically also allow the study of anti-plane waves, i.e., Love waves requiring horizontal resonators (Palermo and Marzani, 2018). Although the overall results are consistent, there are a number of experimental peculiarities which can be excluded in theoretical studies but which are unavoidable in the real-world experiments. ...
The deflection and the control of the effects of the complex urban seismic wavefield on the built environment is a major challenge in earthquake engineering. The interactions between the soil and the structures and between the structures strongly modify the lateral variability of ground motion seen in connection to earthquake damage. Here we investigate the idea that flexural and compressional resonances of tall turbines in a wind farm strongly influence the propagation of the seismic wavefield. A large-scale geophysical experiment demonstrates that surface waves are strongly damped in several distinct frequency bands when interacting at the resonances of a set of wind turbines. The ground-anchored arrangement of these turbines produces unusual amplitude and phase patterns in the observed seismic wavefield, in the intensity ratio between stations inside and outside the wind farm and in surface wave polarization while there is no metamaterial-like complete extinction of the wavefield. This demonstration is done by setting up a dense grid of 400 geophones and another set of radial broadband stations outside the wind farm to study the properties of the seismic wavefield propagating through the wind farm. Additional geophysical equipment (e.g., an optical fiber, rotational and barometric sensors) was used to provide essential explanatory and complementary measurements. A numerical model of the turbine also confirms the mechanical resonances that are responsible for the strong coupling between the wind turbines and the seismic wavefield observed in certain frequency ranges of engineering interest.
... In general, the ratio of the penetration depths δ 1 (ω), δ 2 (ω) is expressed by the dispersion equation (Equation (21)), i.e., δ 2 (ω)/δ 1 (ω) = −s (1) 44 (ω)/s (2) on ρ 1 /ρ 2 . On the other hand, by virtue of Equations (27) and (28), the product of the normalized penetration depths equals: ...
... Using Equation (33) in conjunction with Equations (27) and (28), one can demonstrate that if → 0 then ( ) ( ) ( ) ( ) →0 . On the other hand, if → , then ( ) ( ) ( ) ( ) →− 1 , (see Figure 14). ...
... Using Equation (33) in conjunction with Equations (27) and (28), one can demonstrate that if ω → 0 then P ...
The advent of elastic metamaterials at the beginning of the 21st century opened new venues and possibilities for the existence of new types of elastic (ultrasonic) surface waves, which were deemed previously impossible. In fact, it is not difficult to prove that shear horizontal (SH) elastic surface waves cannot exist on the elastic half-space or at the interface between two conventional elastic half-spaces. However, in this paper we will show that SH elastic surface waves can propagate at the interface between two elastic half-spaces, providing that one of them is a metamaterial with a negative elastic compliance s44(ω). If in addition, s44(ω) changes with frequency ω as the dielectric function ε(ω) in Drude’s model of metals, then the proposed SH elastic surface waves can be considered as an elastic analogue of surface plasmon polariton (SPP) electromagnetic waves, propagating at a metal-dielectric interface. Due to inherent similarities between the proposed SH elastic surface waves and SPP electromagnetic waves, the new results developed in this paper can be readily transferred into the SPP domain and vice versa. The proposed new SH elastic surface waves are characterized by a strong subwavelength confinement of energy in the vicinity of the guiding interface; therefore, they can potentially be used in subwavelength ultrasonic imaging, superlensing, and/or acoustic (ultrasonic) sensors with extremely high mass sensitivity.
... Surface waves with LRM have been the topic of many recent research papers. For example, Boechler et al. [10] studied Rayleigh waves with distributed mass-spring resonators on the surface, Maznev & Gusev [11] studied Love waves with distributed locally resonant oscillators on the surface for frequencies below the local resonant frequency, Maurel et al. [12] studied Love waves with a forest of trees on the surface, Palermo & Marzani [13] studied the control of Love waves of a two-layer elastic half-space with distributed local resonators on the surface, Zeigham et al. [14] studied Rayleigh waves with an elastic layer filled with local resonators, Skvortsov et al. [15] studied sound absorption by a metasurface comprising hard spheres, Guo and Chen [16] studied seismic metamaterials for energy attenuation of Love waves in transversely isotropic media, Maznev [17] developed an effective medium model for Rayleigh waves with mass-spring resonators on the surface, Pillarisetti et al. [18] studied Rayleigh waves with Mindlin-type boundary conditions on the surface, and Fang et al. [19] studied Rayleigh waves with vibration absorbers on the surface. To the best of our knowledge, Love surface wave of an elastic half-space coated with a hard sphere-filled elastic thin layer has not been addressed in literature. ...
... It is seen from (17) and (10,11,13) that the wave speed of Love waves c 0 for ω > 0 only if ε 0 and ω/ω 0 1 (then A ∞). In particular, the wave speed of Love waves cannot be zero with η > 0 (ε > 0). ...
... It is seen from (11,13,18) that the attenuation coefficient of Love waves is zero when the damping is ignored (η 0 or ε 0). Therefore, the damping effect (η > 0) is essential to study wave attenuation of Love waves. ...
Inspired by recent research interest in metasurfaces, an effective medium model is presented to study anti-plane (Love) surface waves of an elastic half-space coated with an elastic metasurface thin layer filled with coated or uncoated hard spheres. Explicit formulas are derived for phase velocity and attenuation coefficient, and the general implications of the derived results are discussed with specific examples without or with damping effect on local resonance of embedded hard spheres. Emphasis is placed on the effects of the metasurface and damping on the structure of bandgap for Love waves, with detailed comparison to the known results on Love waves of an elastic half-space coated with a thin elastic layer or elastic plate. The derived results and conclusions may offer new insights into the study of surface elastic waves in locally resonant metacomposites.
... Such an approach amounts to considering a physical setup in a plane parallel to the surface. Many numerical works followed, mostly considering the same approach, notably a proposal to reroute Love waves (Palermo, 2018) in the transverse plane thanks to a graded metasurface, and some conversion of Love waves into downward propagating antiplane shear bulk waves via a wedge effect in the vertical plane (Maurel, Marigo, Pham, & Guenneau, 2018). In parallel, small-scale experiments on the control of surface seismic waves (Colombi, Roux, Guenneau, Gueguen, & Craster, 2016;Colombi, Zaccherini, Aguzzi, Palermo, & Chatzi, 2020;Palermo et al., 2018) have shown that one can also act upon the deflection of Rayleigh waves in the vertical plane with an array of resonators atop, or buried, in the soil. ...
In this work we propose a strategy based on coordinate transformation to cloak Rayleigh waves. Rayleigh waves are in-plane elastic waves which propagate along the free surface of semi-infinite media.
They are governed by Navier equations that retain their form for an in-plane arbitrary coordinate transformation $\textbf{x}=\Xi(\textbf{X})$, upon choosing the specific kinematic relation $\textbf{U}(\Xi(\textbf{X}))=\textbf{u}(\textbf{x})$ between displacement fields in virtual, i.e. reference, ($\textbf{U}$) and transformed, i.e. cloaked, ($\textbf{u}$) domains. However, the elasticity tensor of the transformed domain is no longer fully symmetric, and thus, it is difficult to design with common materials. Motivated by this issue, we propose a symmetrization technique, based on the arithmetic mean, to obtain anisotropic, yet symmetric, elastic tensors for Rayleigh wave near-cloaking.
In particular, by means of time-harmonic numerical simulations and dispersion analyses, we compare the efficiency of triangular and semi-circular cloaks designed with the original non-symmetric tensors and the related symmetrized versions. In addition, different coordinate transformations, e.g. linear, quadratic and cubic, are adopted for the semi-circular cloaks. Through the analyses, we show that a symmetrized semi-circular cloak, obtained upon the use of a quadratic transformation, performs better than the other investigated designs. Our study provides a step toward the design of feasible and efficient broadband elastic metamaterial cloaks for surface waves.
... Compared to the Rayleigh-wave counterparts, the research on the manipulation of Love waves using metasurfaces or metabarriers is currently very limited. Palermo and Marzani [53] utilized the metasurface concept that had been successfully applied in controlling in-plane Rayleigh waves to manipulate anti-plane Love waves. An effective medium approach was adopted to derive the dispersion relation and the finite-element simulations were performed to evidence the wavefield. ...
... Although these works based on numerical simulations have made important contributions to the interaction of Love waves and metasurfaces, analytical investigations for modelling wavefields are still desirable for facilitating the understanding of wave phenomena and optimal design. In particular, dispersive properties are usually studied by analytical methods, for example, multiple-scale method [55], classic energy method (e.g., for complex-shape lattices) [56], plane wave expansion method [57] and effective medium approach [53]. Meanwhile, analytical methods are also performed for the wavefield analysis of incident waves interacting with various scatters, for example, multiple scattering theory [58] and homogenization techniques [59]. ...
... For this coupled system, we observe the transmission ratio T r calculated by our analytical formulation is in excellent agreement with that obtained by FE simulations (see black dots in Fig. 3(a)), and the accuracy of our formulation is thus verified. The details about the FE model are given in Appendix A. Besides, we also validate the proposed analytical formulation by comparing the dispersion curve with that reported in Ref. [53] (see Appendix B). Additionally, noting that the larger reduction of the coupled two-resonator case is not a simple superposition of reductions from single resonators 1 and 2, we can deduce that the interplay between two resonators plays a significant role in the coupled system response especially when their resonance frequencies are the same [69]. ...
Seismic Love waves are horizontally polarized surface waves propagating within an overlying layer, which have been shown to pose potential hazards to infrastructures. In this study, we propose an original analytical formulation to investigate the interaction between seismic Love waves and a metasurface that is composed of a cluster of tunable anti-plane resonators. The formulation utilizes the Green's function of a layered half-space subjected to a surface SH source, to describe both the source-induced incident field and the resonator-excited scattered field. Based on the multiple scattering theory, we obtain the total wavefield of the layered half-space equipped with surface resonators by coupling the incident field and the scattered field. Besides, a closed-form dispersion relation governing the hybrid Love waves is derived by utilizing the effective medium approximation. We demonstrate the capabilities of our analytical formulation by modelling Love waves propagating across a pair and a cluster of resonators, evidenced by some complex phenomena (e.g., surface-to-bulk wave conversion and higher-order hybridized surface modes). The hybridization features of different Love modes and metasurface resonances are clearly revealed. We anticipate the proposed tunable metasurface and the formulation can shed light on versatile control of Love wave propagation.
... That is, thin resonant interfaces arranged on the surfaces of elastic waveguides or at the junction of different media to manipulate surface waves, such as Rayleigh waves or Love waves, propagating in the elastic substrate [12]. Such metasurfaces exhibit rich wave phenomena, for example, amplitude decay via mode conversion [13,14], energy trapping [15], waveguiding [16], and lensing [17,18]. Revealing these phenomena will provide technical support for the applications of surface acoustic wave devices with functions including signal processing [19], energy harvesting, and ground vibration mitigation [20]. ...
To protect engineering structures from being destroyed by earthquakes, seismic metamaterials have been proposed for Rayleigh wave attenuation. Introducing nonlinearity may provide new design options for seismic metamaterials. Although nonlinear seismic metamaterials lying on homogeneous substrates have been reported, the real stratigraphy is a layered structure. In the present work, Rayleigh wave propagation through a nonlinear seismic metamaterial is studied. The seismic metamaterial consists of nonlinear resonators attaching on layered soils composed of clayey slit and sandy slit. First, an analytical model is established, and the closed-form solution for the dispersion of nonlinear Rayleigh wave is calculated via the leading-order harmonic balance approach. Then, finite element (FE) simulations are conducted to obtain the wave mode, dispersion, as well as transmission, and validate the analytical results. This study shows that the clayey slit reduces the phase velocity of Rayleigh wave. Due to the interaction of the resonators with Rayleigh wave propagating in the layered substrate, solutions in the form of surface wave vanish within a certain frequency band. This frequency band is the bandgap for Rayleigh wave. The FE simulations further show that in this frequency band, the energy of Rayleigh wave is converted into bulk wave. Moreover, the dispersion of nonlinear Rayleigh wave is amplitude-dependent, which enhances the designability of seismic metamaterials.
... Extensive work has been done in designing electromagnetic or acoustic metamaterials in recent years (Ahmed et al., 2021;Amirkulova et al., 2022;Jiang and Fan, 2020), such as designing acoustic metamaterials using deep learning, reinforce learning, or generative adversarial networks (Gurbuz et al., 2021;Lai et al., 2021;Shah et al., 2021). However, the state-of-the-art design of locally resonant elastodynamic metasurfaces still relies on arrays of simple resonator geometries, e.g., rods (Rupin et al., 2014), holes (Brûl e et al., 2014), cuboids, beams, trusses (Zaccherini et al., 2020), four-arm resonators (Hakoda et al., 2019), or mass-spring systems (Palermo and Marzani, 2018). These metasurface designs are accomplished through parametric tuning of dispersion curves empirically until the desired bandgap is achieved; a rational design process is lacking. ...
Control of guided waves has applications across length scales ranging from surface acoustic wave devices to seismic barriers. Resonant elastodynamic metasurfaces present attractive means of guided wave control by generating frequency stop-bandgaps using local resonators. This work addresses the systematic design of these resonators using a density-based topology optimization formulated as an eigenfrequency matching problem that tailors antiresonance eigenfrequencies. The effectiveness of our systematic design methodology is presented in a case study, where topologically optimized resonators are shown to prevent the propagation of the S 0 wave mode in an aluminum plate.
... Over the last couple of decades, the accessibility to the accurate values of the modal properties of the structures, as their intrinsic and exclusive characteristics, has attracted considerable attention of the researchers and engineers [1][2][3]. Several prominent applications of the modal properties of the structures (i.e., natural frequency, damping ratio, and mode shape) in the scope of the real-world engineering structures include, but are not limited to, the following employments: preventing of the resonance phenomena [4][5][6][7]; energy-dissipation and microwave-adsorption in broadband metamaterial [8][9][10][11]; soilstructure as well as hydraulic-structure interactions [12][13][14][15]; structural health monitoring [16,17]; structural damage identification [18][19][20]. ...
... The number of stories and the value of height of these four buildings were SS-1 (5, 15 m), SS-2 (5, 16 m), SS-3 (4, 12 m), and SS-4 (5, 16), respectively. Moreover, the values of the length and width of these buildings were SS-1 (21.5, 12) m, SS-2 (12, 10) m, SS-3 (14,9) m, and SS-4 (22, 10) m, respectively. The materials of the walls and facade in the entire buildings were hollow brick and stone, respectively. ...
Non-structural components (NSCs) in the majority of the FE models are commonly considered as a concentrated/distributed mass, and their stiffness along with damping ratio is ignored. Likewise, NSCs have significant efficacy on the accurate vulnerability assessment and select an appropriate structural system versus the seismicity of the area. The chief aim of this research is to present a simple theoretical method in conjunction with a practical empirical technique for assessing the influences of NSCs on the modal properties and vulnerability of buildings, exclusively steel structures. In this regard, the ambient vibrations were measured on four buildings with concentrically steel braced frame structural system under the construction process in three independent stages: (1) after the erection of the structural and skeletal systems; (2) after the construction of the interior and exterior partition walls; (3) after the installation of the facade and parapet elements. Utilizing two signal processing techniques (i.e., floor spectral ratio and random decrement method), the acquired microtremors were analyzed to extract the modal properties and compute the vulnerability indices of the structures. It is perceived that the values of the modal properties of the steel structures, namely, natural frequency and damping ratio, have considerably increased during the construction process with taking the influence of NSCs into account. In addition, the obtained results disclosed that the vulnerability index has significantly decreased in the second stage over the first one. However, it has not noticeably fluctuated in the third stage over the second one.
... The traditional ω(k) method applies to real dispersion relation, while k(ω) method applies to complex dispersion relation [29]. The former method is successfully used by Meng, Marzani and Palermo [30][31][32], defining the attenuation zone where no propagation modes exist. But it can not explain the decay form of waves in AZ [33]. ...
Topological insulator (TI) has received enormous attention in recent years due to its intriguing dynamic characterizes of topologically protected wave transport and defect immunity. Besides, the elastic waves can propagate along the pre-designed interface but body, which may dramatically promote practical applications of novel devices. This work introduces the concept of TI into the design of periodic pile barriers, which is expected to realize vibration mitigation and waveguiding simultaneously. By tuning the difference of pile radius, the inversion symmetry is broken and the attenuation zone (AZ) emerges. Then, the topological phase transitions happen when pile positions are interchanged in a unit cell and the opposite valley Chern numbers (Cv) are obtained meanwhile. Based on the complex dispersion (k(ω) method) analysis of supercell, the existence of topological edge states between two types of unit cells with distinct topological phases is confirmed at first. Second, some key parameters affecting the design of periodic pile barriers is discussed comprehensively, especially the influence of soil damping on attenuation zones and edge states. Compared to the traditional real dispersion (ω(k) method), the complex dispersion can describe the propagation property and attenuation property synchronously, providing useful guidances for this novel multifunctional periodic pile barriers. Subsequently, broadband wave attenuation and topological wave transport of novel periodic pile barriers are further validated by the analysis in both frequency domain and time domain, showing high vibration reduction and transmission efficiency. This new kind of wave barriers may have great potential for both vibration mitigation and elastic wave energy harvesting.
... However, a shear horizontal (SH) SW [4] cannot be supported by a homogenous classical elastic half-space (although supported by a half-space with micropolar elasticity [5]). If the elastic half-space is covered with a relatively soft strip, a well-known Love SW [6,7] (a kind of SH SW) can propagate along their interface with the energy trapped in the strip and evanescent far from the interface. If the boundary of the elastic half-space is structured by arrays of grooves, holes, or resonators [8][9][10], etc., an equivalent soft strip, namely a structured strip, is formed, and the structured strip supports the so-called Rayleigh-Bloch SW [11,12]. ...
A flexural edge wave (FEW) is a guided wave propagating along the free edge of a semi-infinite isotropic elastic thin plate. Here, we investigate the FEW in a plate with its free edge structured by an array of grooves, revealing the essence of a kind of modulated FEW. We analytically solve the dispersion relation of the modulated FEW by developing the coupled mode theory (usually used in diffraction optics and acoustics) that couples diffraction modes and high-order flexural waveguide modes, and discuss the propagation characteristics of the first- and second-order modulated FEWs (symmetric and antisymmetric modes, respectively). Based on the features of the dispersion curves corresponding to the unit cells of the plate with grooves of graded depths, we numerically and experimentally realize the rainbow trapping of the first-order modulated FEW and the mode conversion of the second-order modulated FEW to the bulk wave. Our work provides new ideas for the manipulation of flexural waves and has the potential to develop corresponding acoustic devices based on the FEW.
