Liyun Cao
email: liyun.cao@univ-lorraine.fr&badreddine.assouar@univ-lorraine.fr

Yifan Zhu

Sheng Wan

Yi Zeng

Yong Li

Badreddine Assouar

Perfect Absorption of Flexural Waves Induced by Bound State in the Continuum

Keywords: Bound state in the continuum (BIC), Flexural wave, Perfect absorption

Conventional vibration absorber is challenging to use in the extreme environment of high (low) temperature, due to the low tolerance of its additional damping material to temperature. To adapt to the extreme environment, here we propose an elastic metaabsorber (EMA) based on the quasi-bound states of the continuum (BICs) physical approach. The proposed absorber does not have any additional damping material, depends only on the extreme-low structural loss, and is capable of achieving perfect absorption of the flexural waves propagating in an elastic beam. The proof-of-concept we provide consists of three parallel subwavelength sub-beams sharing a single-port radiating channel in a host beam, and supports elastic Friedrich-Wintgen quasi-BICs.

Based on the latter, the subwavelength EMA can achieve a high quality factor in the elastic wave system to enhance, in an unprecedented way, the wave energy dissipation.

Using this physical mechanism, we experimentally and numerically validate and demonstrate the perfect absorption of flexural waves at subwavelength regime. Our work opens a new route to deal with vibration absorption with subwavelength structures in specific and extreme environments.

Introduction

Mechanical vibration absorption of beam-like or plate-like structures is of considerable significance in many engineering applications, such as preventing vibration hazards and reducing noise. Conventional vibration absorption methods [START_REF] Kerwin | Damping of flexural waves by a constrained viscoelastic layer[END_REF][START_REF] Sun | Passive, adaptive and active tuned vibration absorbers-a survey[END_REF][START_REF] Warburton | Optimum absorber parameters for various combinations of response and excitation parameters[END_REF], which are represented by attaching a large number of damping on the beam-like or plate-like structures, do not fulfill the requirements of lightweight and low-frequency features. As a newly generated method of vibration absorption, the acoustic black hole (ABH) [START_REF] Mccormick | Design optimization and performance comparison of three styles of one-dimensional acoustic black hole vibration absorbers[END_REF][START_REF] Tang | Characterization of acoustic black hole effect using a onedimensional fully-coupled and wavelet-decomposed semi-analytical model[END_REF][START_REF] Ma | Topological optimization of damping layout for minimized sound radiation of an acoustic black hole plate[END_REF][START_REF] Krylov | Experimental investigation of the acoustic black hole effect for flexural waves in tapered plates[END_REF][START_REF] Pelat | The acoustic black hole: A review of theory and applications[END_REF] can absorb propagating waves (dominated by flexural waves) by little attached damping material, which promotes the development of vibration absorption technique with a lightweight structure. However, the ABH is not useful for technological applications at low frequencies due to the weak damping effect. Recently, vibration absorption techniques based on the new method of designing open lossy resonators in one-dimensional elastic beams have been proposed [START_REF] Leng | Limits of flexural wave absorption by open lossy resonators: reflection and transmission problems[END_REF]. Afterwards, we have proposed the idea of lossy gradient elastic metasurface [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF] in two-dimensional elastic plates and experimentally demonstrated broadband vibration absorption in low frequency. However, almost all of the vibration absorption methods mentioned above require additional damping to enhance the loss of the designed structures. These damping materials have a low tolerance to temperature, making these absorption methods challenging to apply to extreme environments with high or low temperature.

On the other hand, bound states in the continuum (BICs), a general wave phenomenon identified in different materials and wave systems, are non-decaying localized modes embedded within the continuous spectrum of radiating waves [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF]. These states, defying the conventional bound states located outside the continuum, are also considered as trapped modes with infinite quality factors (infinite-Q) residing inside the continuum [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF]. Structures with BICs supporting natural high-Q have opened the route to many practical applications in optics and photonics, such as lasers [START_REF] Noda | Polarization Mode Control of Two-Dimensional Photonic Crystal Laser by Unit Cell Structure Design[END_REF][START_REF] Matsubara | GaN Photonic-Crystal Surface-Emitting Laser at Blue-Violet Wavelengths[END_REF][START_REF] Hirose | Watt-class high-power, high-beam-quality photonic-crystal lasers[END_REF][START_REF] Imada | Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure[END_REF], sensors [START_REF] Yanik | Seeing protein monolayers with naked eye through plasmonic Fano resonances[END_REF][START_REF] Romano | Tuning the exponential sensitivity of a bound-state-in-continuum optical sensor[END_REF][START_REF] Zhen | Enabling enhanced emission and low-threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals[END_REF], and filters [START_REF] Foley | Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating[END_REF][START_REF] Doskolovich | Integrated flat-top reflection filters operating near bound states in the continuum[END_REF][START_REF] Ju | Optical quantum frequency filter based on generalized eigenstates[END_REF]. For the subsequent acoustic wave systems, parallel plates in a wind tunnel supporting symmetry-protected BICs [START_REF] Parker | Low frequency resonance effects in wake shedding from parallel plates[END_REF][START_REF] Koch | Resonant acoustic frequencies of flat plate cascades[END_REF][START_REF] Cumpsty | The excitation of acoustic resonances by vortex shedding[END_REF] and waveguides with side-branched cavities supporting the Fabry-Pérot BICs [START_REF] Lyapina | Trapped modes in a non-axisymmetric cylindrical waveguide[END_REF][START_REF] Hein | Trapped modes and Fano resonances in two-dimensional acoustical duct-cavity systems[END_REF] have been investigated. Recently, sound confinement with an arbitrarily high quality factor was realized by designing two adjacent resonators supporting a Friedrich-Wintgen quasi-BIC [START_REF] Huang | Extreme Sound Confinement From Quasibound States in the Continuum[END_REF]. So far, BICs in elastic wave systems [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF] are rarely recognized and concerned due to the high sophisticated polarization states [START_REF] Rose | Ultrasonic waves in solid media[END_REF][START_REF] Graff | Wave motion in elastic solids[END_REF] varying with the complex media structures.

In the present research, by realizing elastic Friedrich-Wintgen quasi-BIC, we introduce the concept of elastic meta-absorber (EMA) capable of perfectly absorbing the flexural waves in an extreme-low-loss elastic beam. First, we establish the theoretical model by using the temporal coupled-mode method in the elastic wave system to obtain the elastic quasi-BIC. Second, based on the quasi-BIC, we theoretically design the EMA. In addition, the physical mechanism of the perfect absorption of the EMA is uncovered and discussed. Finally, the experimental and simulation results are presented to validate the perfect absorption functionality.

Theoretical model and design of EMA

Figure 1(a) shows the proposed EMA which consists of three parallel sub-beams on the right edge of the host beam. They are classified into two types, i.e., sub-beam A and sub-beam B, which are the outer ones with the length dA and the middle one with the length dB, respectively, as shown in Fig. 1(a). The whole structure is symmetrical to the central axis [the purple dotted line in Fig. 1(a)]. Note that if the two outer sub-beams are different, i.e., asymmetric to the central axis, the torsion mode in the host beam will be induced by the moment from the different shear forces between the two outer subbeams. In our system with symmetrical structure, there is no fundamental difference between three or more sub-beams sharing a single-port radiating channel in the host beam with non-reflection boundary on the left edge. The material parameters of all models are consistent with that of the experimental 3D-printed material PLA. The density and Poisson's ratio are [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF] and , respectively.

Young's modulus is Pa, and the small imaginary part, which is obtained by fitting the simulated resonance peaks with the experimental ones, corresponds to the low structural loss. Only flexural waves in these elastic beams are considered in the whole study, because there is no mode conversion. as [START_REF] Hsu | Bound states in the continuum[END_REF] (1)

A a B a A w B w ( ) ( ) i A B A B = t a e w × a ( ) ( ) A A A A B A A A B B d i + i i i i d i S t w g k g g g = --G + + + a a a a a (2)
where represents the incident wave, is the corresponding radiative decay rate, and represents the dissipative decay rate associated with the low structural loss. The In the absence of incident wave and intrinsic loss, Eqs. ( 1) and ( 2) can be derived as , where includes two types of mode amplitudes. The Hamiltonian H can be expressed as

The two eigenvalues of H can be obtained as

When the lengths of two resonators satisfy , they have the same resonance frequency of , the same radiative decay rate of , and the vanished near-field interaction . Therefore, according to Eq. ( 4), one eigenvalue becomes with more loss, while the other becomes a purely real without any loss. The latter indicates that the system turns into the BIC [START_REF] Hsu | Bound states in the continuum[END_REF] with an infinite Q. To intuitively display the BIC, the reflection coefficient of the system, r, can be expressed as

( ) ( ) B B B B A B A A d i i i 2 i d i S t w g k g g = --G + + + a a a a i S ( ) A B g G A ika B ika A A i g a B B i g a A A 2i g a i A t A - ¶ ¶ = H [ ] T A B , A = a a A AB A B A B B 2 i 0 = i 0 2 i g g g k w w g g k g ae ö - ae ö ç ÷ + ç ÷ ç ÷ - è ø è ø H ( ) ( )( ) 2 A A B B A B A B A A B B +2i i 4 i 2 i +2i +i 2 2 w g w g g g k g g k w g w g w -- - - - + = ± B A = 0 d d d D -= A B w w w = = A B g g g = = 0 k = w +3 i w g w (5)
The first term on the right-hand side of Eq. ( 5 ). This discrepancy is expected, because the intrinsic structural loss is considered in calculating to show VL. ) with infinite-Q is completely bound and cannot be directly coupled with the radiation channel [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF]. To achieve perfect absorption of incident flexural waves from the external channel, we introduce a slight deviation to the length of resonator B ( is tun to 0.7). The deviation will make the resonance system radiates a small amount of energy and couples it to the external channel. This process turns the ideal BIC into the quasi-BIC [START_REF] Huang | Extreme Sound Confinement From Quasibound States in the Continuum[END_REF][START_REF] Liu | High-$Q$ Quasibound States in the Continuum for Nonlinear Metasurfaces[END_REF][START_REF] Koshelev | Asymmetric Metasurfaces with High-$Q$ Resonances Governed by Bound States in the Continuum[END_REF] with a high Q.

A B A B 1+4i +2i i i a a r S S g g = A i A a g ( ) B B i a g A 2i A a g ( ) B B 2i a g 2 r B A d d d = +D A d 0.15 d h D = 0.15 d h D = 0 d h D = A B d d = 2 r
In this case, the real eigenvalue of the ideal BIC turns to complex with a tiny imaginary part indicated as the radiative decay rate of the system, . The perfect absorption mode, manifested by the absorption coefficient of , appears at the frequency ( = 3.67 kHz) between and with a critical low structure loss ( /ω0 = 5 × 10 -3 ), as shown in Fig. 2(b).

In addition, in a complex frequency plane, the distribution of can be plotted for the system with lossless, the structural loss , and the structural loss , respectively, as shown in Fig. 2(c). For the system with the low structure loss , "Zero" point intersects the real frequency axis. It indicates that the dissipative decay rate of the system ( ) is tuned to the same value as , which is equivalent to the so-called critical coupling [START_REF] Cai | Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System[END_REF][START_REF] Huang | Compact broadband acoustic sink with coherently coupled weak resonances[END_REF], however here it is fulfilled by the small and . The incident flexural waves can be completely absorbed without any backscattering. Note that, theoretically, we can obtain a perfect absorption for the specific frequency in the extreme environment with any high (low) temperature by adjusting the structural parameters to make equal to a different . Furthermore, we extract the phase distribution of the neutral plane of the structure for the three resonance frequencies, i.e.,

d h D

= d h D S g 2 =1 1 r a -» 0 2 w p A w B w 1 G 10 log r 1 G 2 G 1 G 1 G S g 1 G S g S g G , , and
, respectively, as shown in Fig. 2(d). For the resonance frequency of , we can see from dotted box in Fig. 2(d) that the phase difference between the adjacent sub-beams is π, which manifests the strong coupling behavior between these sub-beams [START_REF] Li | A sound absorbing metasurface with coupled resonators[END_REF]. The directions and sizes of the pink arrows in Fig. 2(d) indicate the directions and amplitudes of the particle displacements, respectively. It can be observed for the resonance frequency of that the displacement amplitudes inside the whole sub-beams are greatly enhanced, which is caused by the strong coupling. The large enhanced displacements in the sub-beams lead to the perfect absorption of the EMA, although the structural loss is low.

Experimental demonstration of perfect absorption of flexural waves

To confirm our theory, we have printed the specimens by 3D-printer, as shown in Fig. 3(a). A 15-mm-diameter circular PZT patch is bonded on the beam surface. The PZT patch is driven by a signal generator (Tektronix AFG3022C). The blue-tack layer is bonded in the left boundary of the specimen, which is equivalent to a non-reflection boundary. Point 1 and point 2 marked in Fig. 3(a) are two measuring points. The distance between the two points is less than an eighth wavelength. Simultaneously, their locations and the boundary of the resonance structure meet the far-field assumption [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF].

The assumption ensures that the measurement wavefield of the two points can approximate the sum of incident and reflection propagating waves.

Using the measurement mode "FFT" of Polytec Scanning Vibrometer 500 (PSV-500), the complex velocities of flexural waves in points 1 and 2 can be measured. These measured velocities are expressed as and

A w B w 0 w 0 w 0 w ( ) ( ) 1 2 1 2 ˆî i 1 I R b b s s k s s k v V e V e + × -+ × = × + ×
, respectively. and are the distance, as shown in Fig. 3(a).

is the measured wavenumber of flexural wave. The transfer function H12 of the total wavefield in the two points can be obtained from the measured complex velocities of v 1 and v 2 . The transfer function can be expressed as

, where the reflection coefficient is . Then, we can calculate the reflection coefficient of flexural waves as

The absorption coefficient of flexural waves can be calculated by . the whole sub-beams are greatly enhanced, which is consistent with the enhanced displacement amplitude in Fig. 2(d). In addition, in the test area, the incident wave is a perfect traveling wave, whose intuitive evidence can be found in Supplementary Video 1. The existence of the perfect traveling wave means that the EMA, indeed, perfectly absorbs flexural waves. For comparison, we also test the dynamic full wavefield at 3.2 kHz (away from the central frequency), as shown in Supplementary Video 2. The intuitive evidence from Supplementary Video 2 is the incident wave is a standing wave and the displacement deformations inside the sub-beams are not enhanced.

+ - + - é ù = = + + ë û ! ! R I / r V V = ! ( ) ( ) 1

The influence of the structural parameters on the absorption

We have also investigated the of the structural parameters (p and l), which are not discussed above, on the absorption spectra. Figure 4(a) shows the absorption spectrum as a function of the sub-beam width, p, with other parameters fixed.

It is found that with the increase in p, the absorption effect has strong robustness, and the absorption peak shifts to low frequency. The reason is that the resonant frequency of the sub-beams becomes lower when increasing p, leading to a lower frequency of quasi-BIC from the coupling between sub-beams.

Conclusion

We

Fig. 1 (

 1 Fig. 1(a) The elastic meta-absorber (EMA) consists of three parallel sub-beams on the right edge of

First

  , the EMA model in Fig.1(a) degenerates to the simplified ones with the same sub-beam length, i.e., including only sub-beam A or B. They are marked as the degenerated model A (DMA) and the degenerated model B (DMB), respectively. We obtain the absorption curves of these degenerated models by experiments [orange balls and blue-green balls in Fig.3(b)] and frequency-domain simulations [orange line and blue-green line in Fig.3(b)] based on solid mechanics module of COMSOL Multiphysics 5.4 software [10]. The corresponding resonance frequencies and are extracted from the peaks of the absorption curves. These resonances are from the coupling between the scattering fields of the sub-beams and the host beam boundaries. In classical mechanics, elastic beams are often simplified as onedimensional models, but our system is not suitable for this simplification due to the coupling. Note that, when the host beam equivalent to a plate model by applying periodic boundary conditions, the scattering fields from the host beam boundaries disappear, thus these resonances will disappear. For the EMA model in Fig. 1(a), each sub-beam resonator (A and B) can be described by the temporal coupled-mode equation [schematic illustration in Fig. 1(b)] for the lowest mode with amplitude

  terms and describe the near-field coupling between two types of resonators emanating from their evanescent flexural wave modes. The terms and in Eq. (1) represent the reradiation field of resonator A caused by the other two resonators. The term in Eq. (2) represents the reradiation field of resonator B caused by the other two resonators.

  ) describes the of the incident wave reflected by the right boundary in the background field. The second and third terms describe the reflections of flexural waves from resonator A and resonator B, respectively. Factors 4 and 2 occur in Eq. (5) because one resonator A(B) has the same left-radiation and right-radiation, i.e., . Due to the reflection of rightradiation in the right boundary, as shown in Fig. 1(a), the resonator A(B) will have the left-radiation of . Based on Eqs. (1), (2), and (5), the dimensionless reflected energy coefficient of the system varying with the length of resonator B ( is fixed to 7 mm) is shown in Fig. 2(a). Near and 4.14 kHz, it shows a vanishing linewidth (VL), which is a typical characterization of the ideal BIC with infinite-Q [11, 12]. The VL position corresponding to has a small discrepancy with its theory position corresponding to (i.e,

Fig. 2 (

 2 Fig. 2 (a) Dimensionless reflected energy coefficient of the system varying with the length

Fig. 3 (

 3 Fig. 3 (a) The specimen and test set-up. (b) For the absorption coefficient , the experiment results

  line and blue-green line), respectively. (c) The snapshots of Supplementary Video 1 from PSV 500 for the central of 3.66 kHz.To ensure the reliability of the data, we tested each experimental data three times. The experiment results with error bars are shown by red balls in Fig. 3(b). It can be seen that the peak values at 3.66 kHz are approximately one, that is, the flexural waves are almost completely absorbed. The absorption coefficient curves from the theory [solid black lines in Fig. 3(b)] and simulations [blue circles in Fig. 3(b)] present an excellent agreement with that from the experiments. To intuitively show the absorption, we obtain the dynamic full wavefield in the specimen at the resonance frequency of 3.66 kHz by the ''time'' measurement mode. Figs. 3(c) shows a snapshot of Supplementary Video 1 at 44.424 ms. It can be clearly seen that the displacement deformations inside

Figure 4 (

 4 Figure 4(b) illustrates the absorption spectrum as a function of the separation

Fig. 4 (

 4 Fig. 4 (a) The absorption spectrum as a function of the sub-beam width, p, with other parameters

  have theoretically designed, numerically, and experimentally demonstrated the sub-wavelength EMA composed of parallel sub-beam resonators. By realizing Friedrich-Wintgen quasi-BIC in the elastic wave system, the EMA can obtain a high-Q to enhance, in an unprecedented way, the energy dissipation. The EMA perfectly absorbs the incident flexural waves in the elastic beam when the low structure loss compensates an identically low radiation decay rate. The length of the EMA is only a quarter of the target wavelength and can be further reduced by increasing the width of the sub-beams or the separation between adjacent sub-beams. Significantly, the EMA does not require any additional damping material, which could open up possibilities in the families of elastic subwavelength meta-absorber to realize low-frequency perfect absorption in the extreme environment of high (low) temperature.

Acknowledgements

This project is supported by the Institute CARNOT (ICEEL) and la Région Grand Est.


Yi Zeng

Liyun Cao

Yifan Zhu

Yan-Feng Wang
email: yswang@tju.edu.cn

Qiu-Jiao Du

Yue-Sheng Wang

Badreddine Assouar
email: assouar@univ-lorraine.fr

Yan-Feng Wang

Yue-Sheng Wang

Coupling the first and second attenuation zones in seismic metasurface

Keywords: Elastic metasurfaces, Seismic surface waves, Attenuation zone, Rainbow trapping effect

published or not. The documents may come   L'archive ouverte pluridisciplinaire

In the past two decades, in the fields of applied physics, mechanics and civil engineering, the use of metamaterials and/or metasurfaces to control wave propagation has attracted great attention. [1][2][3][4][5][6][7][8][9][10][11][12][13] Regarding surface waves, the lowest-frequency attenuation zone (LFAZ) of the elastic metasurfaces composed of an array of artificial structures on a substrate is the most appealing and interesting, since their working wavelengths are well above their characteristic size. 11,[14][15][16][17] In the LFAZ, a number of experiments 15,18,19 and numerical studies 11,12,14,16 have demonstrated that surface waves can be strongly attenuated because of the local resonance of the artificial structures on the substrate. It is worth mentioning that this conclusion is always valid whether the materials of the substrate and the artificial structures are the same or not. 11,[14][15][16] However, the width of the LFAZ is usually narrow. Therefore, the second attenuation zone (SAZ) whose frequency is slightly higher than the LFAZ attracted our attention. In the frequencies of the SAZ, Achaoui et al. 15 experimentally demonstrated that surface waves are almost reflected in the elastic metasurface and the transmission is similar to the case of phononic crystals constituted of periodic holes in a substrate. In addition, the attenuation capacity of the elastic metasurfaces made of pillar-shaped artificial structure on a substrate significantly increases as the number of rows of the artificial structure increases. 11,16 However, the frequency range of the SAZ is high. The problems in these two attenuation zones are the biggest bottleneck encountered in the current research on seismic metasurfaces (SMS). In 1999, phononic crystals composed of a periodic array of holes in a marble experimentally reported the existence of bandgap to attenuate elastic surface waves. 20 It showed a possibility of application of the SMS on attenuation of seismic waves. In 2014, SMS composed of cylindrical holes arranged periodically in soil has been demonstrated in large-scale experiments to attenuate seismic waves at 50 Hz. 21 However, the lower-frequency attenuation zones of the SMS require the local resonance mechanism 22,23 . Therefore, the SMS composed of periodic pillars on a substrate has attracted more attention, especially the LFAZ and the SAZ of the SMSs.

These two attenuation zones are both located at low frequencies, i.e., the elastic metasurfaces are the subwavelength structures used to control surface waves. However, these two attenuation zones have both their own limitations. As mentioned earlier, although the LFAZ is at a lowest frequency, its relative bandwidth is usually very small.

The SAZ has, on the other hand, a larger relative bandwidth than the LFAZ, but is located at a higher frequency range. Therefore, the shape of the pillars have been optimized to enlarge the relative bandwidth of the LFAZ and the SAZ. 16,24 It also worth mentioning that hue and complex artificial structures have been introduced to achieve other attenuation zones in lower frequency range. [24][25][26] Hence, the attenuation zone which can produce an ultra-low-frequency range of the LFAZ and a wide bandwidth of the SAZ at the same time, is worth to investigate.

On the other hand, the "rainbow trapping effect", i.e., using frequency selective structures with graded geometry or properties to filter waves in wide frequency ranges, has been observed in optics 27 and acoustics 28 . Now, it has been extended to elastodynamics 17,[29][30][31][32] . The broadband attenuation effect was experimentally demonstrated by using a scaled experimental model with an array of different resonant structures. 17 In addition, by using the longitudinal-resonance bandgaps of pillars with varying heights, the proposed elastic metasurface not can only strongly attenuate surface waves in a wide frequency range, but the mode conversion of surface waves to bulk waves can also be found. 29,30 Recently, a kind of metabridge made of bridges of various sizes was proposed to protect bridges from earthquake efficiently by using destructive interferences and "rainbow trapping effect". 31 However, the manufacturing process becomes complicated because of various resonance units or structures. In the present research, we propose alternative approaches to overcome these challenges.

We report the properties of the LFAZ of a pillar-shaped elastic metasurface. Then, we experimentally demonstrate the "rainbow trapping effect" of the SAZ of an elastic metasurface composed of one kind of pillars with various lattice constants on a substrate. These two features of these two attenuation zones allow us to connect lower bound of the SAZ and the upper bound of the LFAZ, and design elastic metasurfaces using one kind of artificial structures to attenuate surface waves over a wide low-frequency range.

Figure 1 shows the schematic diagram of the elastic metasurface. Finally, we design a seismic metasurface (SMS) just using one kind of artificial structure on a half-space to attenuate Rayleigh waves in a wide frequency range.

An elastic metasurface composed of steel pillars (width w = 0.3 m, height l = 0.75 m) periodically arranged with a lattice constant of 1.5 m on a soil substrate is taken as an example to investigate the properties of the LFAZ. The depth of the substrate is H = 500a, which is much larger than two times of the wavelength (2λ) of the surface waves.

The bottom surface of the unit cell is set as fixed boundary condition. The considered material parameters of steel and soil are same as the literature 33,34 .

Figure 2(a) shows the complex band structures of the elastic metasurface calculated by the PDE module of the COMSOL Multiphysics software. The dotted line of cs is the shear wave speed of the substrate material, i.e., the boundary of the sound cone. 14,26 To draw all the surface modes in the band structure, the parameter ξ is introduced, i.e.

ξ = ∫ |𝒖|𝑑𝑠

!(#$) / ∫ |𝒖|𝑑𝑠 !(&) (1)
which is the ratio of the displacement (u) in a depth of twice (2λ) of the Rayleigh wave wavelength to the entire depth (H) of the substrate. 11,35 The full complex band structure (1 > ξ > 0) is shown on the left of the Fig. 2(a). Surface mode is defined as having parameter ξ > 0.9. The complex band structure for surface modes (ξ > 0.9) is shown on the right of the Fig. 2(a). The first bandgap is found from 5.0 to 5.4 Hz. Although there is a band of surface modes from 5.0 to 5.2 Hz, the surface waves are continuously attenuated as the distance traveled increases in this frequency range. Because the imaginary part of the wave vector of this band is positive, i.e., the wave mode of this band is evanescent 36,37 with strong attenuation. The surface vibration mode, which falls before the cs line on this band, is shown in the far-right of Fig. 2 Table 1: The geometric parameters of the metasurface in Fig. 3. The LFAZ generated by the elastic metasurfaces composed of an array of pillars on substrate is usually narrow. 11,22,38,39 Therefore, to obtain a wider attenuation zone in the low-frequency range, the SAZ with a higher frequency than the LFAZ is investigated and discussed by using numerical simulations and experiments. Figure 3(a) shows the unit cell of the elastic metasurface whose band structure can be numerically simulated by using the solid module of COMSOL Multiphysics. In this work, the bandgaps only in the ΓX direction of the first irreducible Brillouin zone are investigated. The numerical computations are described in Refs. [ 14,26 ]. The material of the unit cell is PLA (its Young's modulus is 3.4398 GPa, density is 1086.3 kg/m 3 , Poisson's ratio is 0.35) 40 and the geometric parameters are shown in Table 1. The characteristics of the first and second bandgaps are always valid and do not change, whether the materials of the substrate and the artificial structures are the same or not. 11,[14][15][16] Therefore, the proposed PLA models can accurately explore the basic properties of the bandgaps.

The effect of lattice constant a on the upper and lower frequency bounds of the first and second bandgaps is shown in Fig. 3(b). The first bandgap is quite stable with various lattice constants, which is consistent with the previous results. However, the central frequency of the second bandgap keeps decreasing as the lattice constant increases.

Therefore, this feature of the second bandgap is suitable to be used to obtain a wide attenuation zone at frequencies higher than the LFAZ through the "rainbow trapping effect" 17,29,41,42 .

Metasurface samples composed of three rows of PLA pillars with different lattice constants of 13.2, 12.2 and 11 mm on a PLA substrate have been considered for simulations and fabricated for experiments. Supplementary Information (S2) shows the detail information of the used geometric model in the simulation and experiment, and the setup of the experimental platform. From Fig. 3(b), it can be found that the SAZ of the considered sample extends from 37 kHz to 50 kHz due to the "rainbow trapping effect" of the second bandgap. observe that in the range of 37 -50 kHz, the elastic metasurface can significantly attenuate surface waves. We observe that there is a transmission peak at around 45 kHz.

We do believe that the experimental transmission peak at around 45 kHz is due to the imperfect absorption of the damping material covered the substrate for elastic waves. Indeed, when an artificial structure on the substrate is determined, the LFAZ is immobile even if the distances between the structures are changed. However, the SAZ whose frequency is higher than the LFAZ becomes wider when choosing the suitable distances between the structures due to the "rainbow trapping effect". Therefore, for a properly designed elastic metasurface, there is an opportunity to connect lower bound of the SAZ and the upper bound of the LFAZ. It is possible to propose an elastic metasurface with a wide attenuation zone from an ultra-low frequency for surface waves just using one kind of artificial structures on a substrate.

For instance, in this work, a kind of SMS with a wide attenuation zone for seismic Rayleigh waves is designed. Considering the wide range of the evanescent waves in the LFAZ of the seismic metasurface, the inverted T-shaped structure is chosen. 43 As shown in Fig. 5 of the structures and distances can also be selected. This paper only gives one possibility. The figure in Supplementary Information (S4) shows the schematic illustration of one possible three-dimensional configuration of the unit cell of the SMS.

In summary, the mechanical property of the LFAZ of the elastic metasurface made of pillars on a substrate has been investigated numerically. We found that the LFAZ is immobile with the changed distances between pillars. The "rainbow trapping effect" in the SAZ has then been demonstrated by numerical simulation and experiments. We have observed that the SAZ becomes wider when suitable distances between pillars are selected. Therefore, for a properly designed elastic metasurface, there is an opportunity to connect lower bound of the SAZ and the upper bound of the LFAZ. We then can obtain a wide attenuation zone from an ultra-low frequency for surface waves by just using one kind of artificial structures on a substrate. In the case of the seismic Rayleigh waves, these two characteristics of the LFAZ and the SAZ are used to design a seismic metasurface by just using one kind of artificial structure on a substrate. The proposed seismic metasurface can attenuation the Rayleigh waves from 5 to 22 Hz. Different with the interplay of the local resonance and Bragg bandgaps 44,45 , the LFAZ and the SAZ of the SMS hardly interfere with each other and together create a common attenuation zone.

Figure 1

 1 Figure 1 The schematic diagram of a kind of the elastic metasurface for surface waves. The

  (a). Most of the vibration exist in the substrate and decrease with increasing depth. The vibration mode of the pillars on substrate is similar as the one on the first band below 5.0 Hz.11 It is worth noting that the curve of the imaginary part should start from the end of the real part at the high symmetry Brillouin points from 5 to 5.15 Hz. However, the ideal results are difficult to calculate because the results are extremely dense at the junction.The most interesting feature of the first bandgap is its immobility nature. The system composed of the substrate and an artificial structure is regarded as a spring-mass model.The resonance frequency of the spring-mass model is fixed if the substrate and the artificial structure are both not changed. So, the first bandgap due to the resonance of the spring-mass models is unchanged, even if the distribution of the artificial structures in the elastic metasurface varies. Figure 2(b) shows the upper and lower frequency bounds of the first bandgap when only the lattice constant of the elastic metasurface is changed. It is found that the first bandgap is absolutely unchanged due to the immobility of the lower frequency bound at 5.0 Hz. The effects of other material parameters on the first bandgap are all calculated and shown in Supplementary Information (S1).

Figures 2 (

 2 Figures 2(c) shows a part of the elastic metasurfaces composed by 30 steel pillars

Figure 2 (

 2 Figure 2 (a) The complex band structures of the elastic metasurface around the first bandgap.

Figure 3

 3 Figure 3 (a) The unit cell of the elastic metasurface for surface waves, (b) the effect of the lattice

Figure 4 (

 4 Figure 4 (a) The transmissions of surface waves on the prepared samples in simulations and

Figure 4 (

 4 Figure 4(a) illustrates the simulation and experimental measurements of the

Figures 4 (

 4 Figures 4 (b)-(c) show the out-of-plane acceleration fields on the surface of the samples

Figure 5 (

 5 Figure 5 (a) Schematic picture of the finite system used for transmission calculations of the

Table 2 :

 2 The geometric parameters of the inverted T-shaped structure As shown in Fig. 5(b), when Rayleigh waves are incident from left to right (LTR) or from right to left (RTL), the SMS can all attenuate the energy well from 5.0 to 22.0 Hz. The attenuation of the Rayleigh waves in the SMSs is complex because the attenuation zone includes the frequency ranges of the first and second bandgaps. There are two mechanisms of the attenuation of the Rayleigh waves in the seismic metasurface. The first one is the attenuation due to the local resonance of the artificial structures, while the second one is due to the conversion of the Rayleigh waves to bulk waves in the SMS. These two mechanisms of the attenuation of the Rayleigh waves in the SMS are shown in Supplementary Information (S3). Comparing with the SMS consisted of a periodic arrangement of inverted T-shaped steel structures with a lattice constant of 1.5 m on soil substrate, the "rainbow trapping effect" of the SMS works well to block the passband from 8.0 to 16.0 Hz. It is worth noting that there are many kinds of threedimensional configurations coming from the two-dimensional one as shown in Fig. 5(a).

  (a), 15 identical inverted T-shaped steel structures are used. The geometric parameters are given in Table2. In the numerical simulation, the ideally perfect

	continuity is assumed between the inverted T-shaped structure and the substrate. The
	15 steel structures are placed with appropriate distances. The distances between the
	symmetry axes of each steel structure in this research are 4.9, 4.7, 4.5, 4.3, 4.1, 3.9, 3.7,
	3.5, 3.3, 3.1, 2.8, 2.4, 2, 1.65 m. Of course, other suitable artificial structures, number

Acknowledgement

This work has been supported by la Région Grand Est, the Institut CARNOT ICEEL, and the National Natural Science Foundation of China (NNSFC) under Grant Nos. 11702017, 11991031, 11991032, 12021002 and 41974059. The first author is grateful for the support of China Scholarship Council (CSC Grant No. 202006250084).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supplementary Information

See the Supplementary Information for supporting content.