Laryssa Carvalho De Araujo
email: laryssa.carvalho-dearaujo@unicaen.fr

Jacques Junior Manguele
email: jacques-junior.manguele@ensicaen.fr

Bertrand Vilquin
email: bertrand.vilquin@ec-lyon.fr

Zhe Wang

Carolina Adamo
email: carolinaadamo30@gmail.com

Pedro Rojo Romeo
email: pedro.rojo-romeo@ec-lyon.fr

Christophe Cibert
email: christophe.cibert@ensicaen.fr

Gilles Poullain
email: gilles.poullain@ensicaen.fr

Bernadette Domengès
email: bernadette.domenges@ensicaen.fr

Victor Pierron
email: victor.pierron@ensicaen.fr

Laryssa Mirelly

Carvalho De Araujo

Darrell G Schlom
email: schlom@cornell.edu

Laurence Méchin
email: laurence.mechin@ensicaen.fr

Use of epitaxial PZT thin films for La2/3Sr1/3MnO3 based MEMs

Keywords: Epitaxial PZT, piezoelectric MEMS, Oxide electrodes, PFM

teaching and research institutions in France or abroad, or from public or private research centers.

I. INTRODUCTION

Micro-ElectroMechanical systems (MEMS) are micrometer devices with movable parts fabricated through microtechnology tools and are widely used in various sensors and actuators applications. La2/3Sr1/3MnO3 (LSMO) presents a ferromagnetic-to-paramagnetic transition at about 360 K that is accompanied by a metal-to-insulator transition above room temperature, which is very promising for uncooled sensors, such as anisotropic magnetoresistances [1] or infrared bolometers [START_REF] Nascimento | Electro-thermal and optical characterization of an uncooled suspended bolometer based on an epitaxial La0.7Sr0.3MnO3 film grown on CaTiO3/Si[END_REF]. In addition, the high sensitivity of epitaxial oxides to strain, hence to mechanical effects, makes these materials promising and encourage us to investigate the potentialities of LSMO based MEMS resonators [START_REF] Huong Giang | Fabrication and characterization of PZT string based MEMS devices[END_REF].

The objective of the present paper is to demonstrate the integration of c-axis oriented Pb(Zr,Ti)O3 (PZT) thin films on LSMO/SrTiO3(STO)/Si free-standing doubly-clamped structures. Epitaxial growth of PZT is required in order to allow lower voltage actuation and higher sensitivity detection, through inverse and direct piezoelectric effects, due to the larger electro-mechanical coupling than those obtained with polycrystalline films [START_REF] Yin | Epitaxial growth and electrical measurement of single crystalline Pb(Zr0.52Ti0.48)O3 thin film on Si(001) for micro-electromechanical systems[END_REF]. Besides epitaxy, the integration of PZT on a conductive oxide electrode is expected to provide better fatigue performances and suppression of dead-layers in the PZT film [START_REF] Ramesh | Fatigue and retention in ferroelectric Y-Ba-Cu-O/Pb-Zr-Ti-O/Y-Ba-Cu-O heterostructures[END_REF][6] [START_REF] Sama | On the Influence of the Top and Bottom Electrodes-A Comparative Study between Pt and LNO Electrodes for PZT Thin Films[END_REF], leading to long-lasting final devices. During polarization cycles, the interface between PZT and LSMO, with its characteristic of a wide bandgap p-type semiconductor, is free of charge accumulation, such that, with no carrier injection leading to accumulation of O2 vacancies [8][9], the fatigue problems are expected to be further minimized [START_REF] Leufke | Ferroelectric vs. structural properties of large-distance sputtered epitaxial LSMO/PZT heterostructures[END_REF] [START_REF] Funakubo | Epitaxial PZT films for MEMS printing applications[END_REF].

After a brief description of the experimental methods in section II, we will present the structural and morphological properties of the deposited layers in section III and the electrical properties in section IV. Conclusions are given in section V.

II. EXPERIMENTAL METHODS

A. Thin film deposition

The PZT/LSMO/STO/Si heterostructures were obtained thanks to the successive use of 3 different deposition methods. First 20 nm thick STO layers were deposited by molecular-beam epitaxy on 3-inch (001) silicon wafers following the same process as reported previously [12]. STO covered Si wafers were cut into 10 × 10 mm 2 squares and about 2 nm thick STO followed by 45 nm thick LSMO films were deposited by pulsed laser deposition at 750°C and 20 Pa of O2.

We selected the chemical composition Zr:Ti=52:48 of PZT, since it exhibits the material's largest piezoelectric coefficient, near the so called morphotropic phase boundary. 500 nm thick PZT thin films were deposited on the LSMO/STO/Si samples in a muti-target RF magnetron sputtering system from 3 metallic targets using power during deposition of Pb : 0.7 W•cm -² / Zr : 2.7 W•cm -² / Ti : 3.3 W•cm -²), in a pressure of 0.7 Pa of 90% Ar / 10% O2 atmosphere at 600°C.

B. Characterization methods

The crystallinity of the samples was characterized by performing X-ray diffraction (XRD) measurements in θ-2θ configuration as well as rocking curves around the 002 peak of PZT, where the full-width at half-maximum (FWHM) of the peak could be estimated. The data were collected using a PANalytical X'Pert PRO diffractometer with Cu-Kα radiation.

Atomic Force Microscopy (AFM) measurements using a Nano-Observer system from Concept Scientific Instruments were performed to characterize the surface morphology of the films in resonant mode with standard Al-coated Si probes showing nominal resonant frequency of 300 kHz. In addition Piezoresponse Force Microscopy (PFM) employing doped diamond probes with an elastic constant of about 2.8 N•m -1 was used to investigated the piezoelectric properties of the as-grown PZT films on LSMO/STO/Si (001).

Pt layers were finally deposited on the PZT films by sputtering at room temperature and patterned using a lift-off process in order to design top electrodes of 250 × 250 µm² size and to perform polarization hysteresis (P -E) loop measurements using a Sawyer-Tower (ST) circuit, driven by a sine wave of amplitude 20 V and frequency 500 kHz. The capacitors were contacted on a Suss MicroTec PM5 probe station. A standard capacitor Cst of 22 pF serving as a charge holder, was used during the hysteresis measurements.

III. STRUCTURAL AND MORPHOLOGICAL PROPERTIES

The epitaxial growth of LSMO and PZT on STO/Si was confirmed by XRD, as shown in the -2 scan of Fig. 1, where the 00 peaks of the aforementioned layers were observed. No parasitic phases of the PZT were observed. The FWHM of the rocking curve around the 002 peak of PZT presented in Fig. 2 was equal to 0.35°, confirming the high structural quality of the deposited layers.

The 2 × 2 µm² AFM images of LSMO and PZT layers are presented in Fig. 3. Both LSMO and PZT showed smooth surfaces. LSMO and PZT rms roughness was equal to 0.3 ± 0.04 nm and 1 ± 0.1 nm, respectively.

IV. ELECTRICAL

The piezoelectric and ferroelectric properties were first characterized by PFM measurements through the creation of artificial domains by poling a 5 × 5 µm² surface through a scanning with +10V dc voltage applied on the cantilever tip (domain down), followed by -10V dc voltage to reverse the piezoelectric dipoles in a 2 × 2 µm² surface central area (domain up). The out-of-plane piezoelectric response was then collected with a 4 V ac drive voltage at 330 kHz in a 15 × 15 µm² scan. As exhibited in Fig. 4, the poling was uniformly and completely achieved. The two stable states of polarization were maintained showing good ferroelectric properties of the deposited PZT film at low voltage compatible with a final MEMS device actuation. Spectroscopy measurements during which the area underneath the PFM tip is switched between -10 V and +10 V and back were performed inside the previously polarized area in order to acquire series of hysteresis loops using the same tip as used for imaging. The displacement versus applied voltage was measured with a drive voltage of 5 V and is displayed in Fig. 5. We could observe the typical butterfly-shaped curves measured in piezoelectric materials showing ferroelectric switching effect.

V. CONCLUSIONS

The successful integration of c-axis PZT deposited by magnetron sputtering at 600°C on LSMO/STO/Si was confirmed by XRD showing the 00 peaks of the PZT layer, as well as those of LSMO and STO with FWHM of the rocking curve around the 002 peak of PZT equal to 0.35°.

The artificial domains created by PFM were stable for several hours. The PFM displacement versus tip voltage curves showed the typical butterfly-shaped curves measured in piezoelectric materials showing ferroelectric switching effect and P(E) curves have exhibited good electrical properties, overall showing that these PZT thin films can be used for MEMS actuation.

These PZT layers will now be integrated in suspended LSMO bridges using a combination of ion beam etching in argon and reactive ion etching in SF6 gas using a fabrication process adapted from [START_REF] Liu | Free-standing La0.7Sr0.3MnO3 suspended micro-bridges on buffered silicon substrates showing undegraded low frequency noise properties[END_REF].

Fig. 1 .

 1 Fig. 1. X-Ray Diffraction θ-2θ scan of epitaxial PZT/LSMO/STO/Si heterostructure showing the 00 peaks of all layers.

Fig. 2 .

 2 Fig. 2. Rocking curve (ω-scan) around the 002 peak of PZT showing a FWHM of 0.35°.

Fig. 3 .

 3 Fig. 3. AFM images of (a) 2 × 2 µm² sacn of the 45 nm thick LSMO layer on STO/Si (001) and (b) 5 × 5 µm² sacn of the 500 nm thick PZT layer on LSMO/STO/Si (001).

Fig. 4 .

 4 Fig. 4. 15×15 µm 2 phase image in PFM mode for a 500 nm thick PZT film deposited by sputtering at 600 °C. Squares of 2×2 µm 2 and 5×5 µm 2 size were first written with -10 V and +10 V tip voltage, respectively.

Fig. 5 .

 5 Fig. 5. The typical displacement versus tip voltage curve measured with a drive voltage of 5 VThe macroscopic polarization hysteresis curves, P(E), of the PZT films were performed on samples with top Pt electrodes. A decentralized cycle in the x axis was observed due to the different top and bottom electrode materials, being Pt and LSMO, respectively. The cycle exhibited a low coercive field and remanent polarization of about 22 µC•cm - ², in accordance with typical values found in the literature[START_REF] Bouregba | Sawyer-Tower hysteresis measurements on micron sized Pb(Zr,Ti)O3 capacitors[END_REF] indicating excellent ferroelectric properties.

Fig. 6 .

 6 Fig. 6. Polarization versus electric field hysteresis loop measured with bottom LSMO electrodes and top Pt electrodes.

ACKNOWLEDGMENTS L.M.C.A. was supported by a PhD grant RIN Doctorants 100% from Région Normandie.


Yi Zeng

Liyun Cao

Sheng Wan

Tong Guo

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

Qiu-Jiao Du

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Yue-Sheng Wang

Seismic metamaterials: Generating low-frequency bandgaps induced by inertial amplification

Keywords: seismic metamaterials, bandgap, surface waves, inertial amplification * " +(,,.)

In the last two decades, seismic metamaterials have attracted significant attention of researchers because of the characteristic of the bandgap which can be used to control seismic surface waves. The inertial amplification mechanism has been used to design metamaterials capable of isolating elastic waves in rods, beams, and plates at low frequencies. In this work, we propose an alternative type of seismic metamaterial with a low-frequency bandgap induced by inertial amplification for isolating seismic surface waves. The characteristics of the bandgap induced by inertial amplification are experimentally demonstrated by using a metamaterial plate composed of 25 unit cells.

The propagation of the flexural waves imaged by scanning laser Doppler vibrometer shows strong attenuation effects induced by the metamaterial plate in the bandgap. The broadband attenuation is investigated by using two kinds of unit cells. Finally, similar structures with inertial amplification are introduced to design the seismic metamaterials to isolate seismic surface waves at low frequencies.

Introduction

Seismic Rayleigh waves at the frequencies below 20 Hz are the greatest threat to the man-made buildings located in the mid and far fields of the seismic source when earthquakes happen [START_REF] Arnold | EARTHQUAKE EFFECTS ON BUILDINGS 4, Des. Earthquakes, A Man. Archit[END_REF][START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF]. Seismic metamaterials (SMs) [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Finocchio | Seismic metamaterials based on isochronous mechanical oscillators[END_REF][START_REF] Aravantinos-Zafiris | Large scale phononic metamaterials for seismic isolation[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF][START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Achaoui | Clamped seismic metamaterials: ultra-low frequency stop bands[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF] have been proposed by researchers to isolate the seismic Rayleigh waves before they arrive the critical infrastructures and buildings. This is due to the bandgap characteristics of the SMs composed of the periodic arrangement of artificial unit cells. The seismic waves will be attenuated significantly by the SMs in the frequency range of the bandgap [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Wu | Frequency band-gap measurement of two-dimensional air/silicon phononic crystals using layered slanted finger interdigital transducers[END_REF][START_REF] Laude | Full band gap for surface acoustic waves in a piezoelectric phononic crystal[END_REF][START_REF] Ding | Metamaterial with simultaneously negative bulk modulus and mass density[END_REF][START_REF] Gao | Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer: Effect of substrate[END_REF][START_REF] Ma | Polarization bandgaps and fluid-like elasticity in fully solid elastic metamaterials[END_REF][START_REF] Colombi | Elastic wave control beyond band-gaps: shaping the flow of waves in plates and half-spaces with subwavelength resonant rods[END_REF][START_REF] An | Band gap characteristics of radial wave in a twodimensional cylindrical shell with radial and circumferential periodicities[END_REF][START_REF] Wang | Tunable and Active Phononic Crystals and Metamaterials[END_REF].

In 1999, the experiments [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF] in a marble with prepared phononic crystals constituted of a periodic array of cylindrical holes showed the existence of the bandgaps for the surface waves. Therefore, the possible applications were obtained to attenuate seismic surface waves. During the last two decades, various kinds of SMs [START_REF] Finocchio | Seismic metamaterials based on isochronous mechanical oscillators[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Diatta | Control of Rayleigh-like waves in thick plate Willis metamaterials[END_REF][START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF][START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF][START_REF] Brule | Experimental evidence of auxetic features in seismic metamaterials: Ellipticity of seismic Rayleigh waves for subsurface architectured ground with holes[END_REF][START_REF] Palermo | Metabarriers with multi-mass locally resonating units for broad band Rayleigh waves attenuation[END_REF][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Zhang | Wave propagation in onedimensional fluid-saturated porous phononic crystals with partial-open pore interfaces[END_REF][START_REF] Liu | Partially Embedded Gradient Metabarrier: Broadband Shielding from Seismic Rayleigh Waves at Ultralow Frequencies[END_REF] have been designed to isolate surface waves in seismic movements, especially at low frequencies.

It is worth mentioning that the plate model was validated to open an attenuation zone (AZ) around 50 Hz by using large-scale experiments on the phononic crystal in the ground [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF]. This work demonstrated that the SMs have the ability to attenuate the propagation and control the transmission of the elastic waves in the earth's surface, although the working frequency is not low enough in the experiments. Therefore, when researchers found that the forests can attenuated seismic waves in earthquakes, the SMs constituted of an array of pillars on a soil substrate were proposed to achieve lowfrequencies bandgap based on the local resonance of the pillars [START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF][START_REF] Brûlé | Flat lens effect on seismic waves propagation in the subsoil[END_REF][START_REF] Colquitt | Seismic metasurfaces: Subwavelength resonators and Rayleigh wave interaction[END_REF][START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF]. In addition, for better integration of SMs in the foundation of a building for seismic isolation purposes, Yan et al. [START_REF] Yan | Seismic isolation of two dimensional periodic foundations[END_REF] proposed a 2D periodic foundation constituted of ductile cast iron, supper soft rubber and reinforced concrete, while Casablanca et al. [START_REF] Casablanca | Seismic isolation of buildings using composite foundations based on metamaterials[END_REF] designed a periodic mass-in-mass foundation placed underneath buildings. To obtain wider bandgaps, the pillars with different kinds of structures [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF][START_REF] Roux | Toward seismic metamaterials: The METAFORET project[END_REF][START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF], and the "rainbow trapping effects" [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF][START_REF] Gan | Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings[END_REF][START_REF] Hu | Rainbow trapping in hyperbolic metamaterial waveguide[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures[END_REF][START_REF] Wu | Broadband Rayleigh wave attenuation by gradient metamaterials[END_REF][START_REF] Zeng | Coupling the first and second attenuation zones in seismic metasurface[END_REF] were utilized. However, the ultra-low-frequency (< 2 Hz) bandgaps are hard to obtain by using small-size structures.

Fortunately, the inertial amplification proposed by Yilmaz et al. [START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF][START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF], which induces low-frequency bandgaps by using a small mass, was utilized to design metamaterials.

The lever-arm structures were introduced to amplify the motion of the resonate small mass, which in turn effectively reduces the resonance frequency. Based on the similar structures, Li et al. [START_REF] Li | Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects[END_REF] proposed a phononic beam to isolate transverse waves utilizing inertial amplification resonators. A wide bandgap induced by inertial amplification in a continuous elastic rod was obtained and a general 2D realization of the inertially amplified system in a plate was proposed in Ref. [START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF]. In addition, Acar et al. [START_REF] Acar | Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in two-dimensional solid structures[END_REF] experimentally and numerically proposed a periodic solid structure which exists wide and deep bandgaps induced by embedded inertial amplification mechanisms. The 3D structures was experimentally demonstrated by using embedded inertial amplification mechanisms, which can isolate elastic waves in a wide frequency range [START_REF] Taniker | Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands[END_REF]. Recently, a 2D solid structure with embedded inertial amplification mechanisms was designed by using topology optimization to obtain an ultrawide bandgap at low frequencies [START_REF] Yuksel | Realization of an ultrawide stop band in a 2-d elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF]. To enlarge the bandwidth of the bandgap and the specific stiffness simultaneously, Mizukami et al. [START_REF] Mizukami | Design and three-dimensional printing of carbon-fibercomposite elastic metamaterials with inertial amplification mechanisms[END_REF] proposed an inertial amplification-type metamaterial by using continuous carbon fibers. Furthermore, the rotary motion conversion mechanism of an inertial amplification of continuous structure was used to generate a wide bandgap at low frequencies [START_REF] Orta | Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism[END_REF]. A four-bar inertial amplification mechanism on corrugated-core sandwich panels was proposed to achieve strong vibration attenuation in the range of low frequencies [START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF]. However, there is almost no research of the SMs with inertial amplification to achieve the low-frequency bandgaps for surface waves.

In this work, we propose a kind of the SM with a low-frequency bandgap induced by inertial amplification to attenuate seismic surface waves. We first experimentally demonstrate the bandgap characteristics of inertial amplification effects in a metamaterial plate (MMP) sample formed by 5 × 5 periodic unit cells. The band structure of the simplified model of the MMP is also analytically calculated by using the transfer-matrix method. Then, to achieve the broadband attenuation, two kinds of unit cells with different positions of hinge joints are chosen to construct a new complex metamaterial plate (CMP). Finally, the inertial amplification mechanisms are introduced to design the SMs which are capable of isolating seismic surface waves at low frequencies. It is very known that inertial amplification mechanisms with small mass can achieve strong vibration attenuation at low frequencies, which property the SMs most need. This work demonstrates that the inertial amplification mechanisms can also induce the bandgap for seismic surface waves at ultra-low frequencies even with using the simple structures.

Method and Models

Figure 1(a) illustrates the unit cell of the MMP based on inertial amplification in the experiments and utilized in the simulation calculations. The simplified model of the structure is also given in Fig. 1(b). The unit cell is constituted of the lever, the support bar, mass, spring, hinge joint and the base plate. In the MMP, the lattice constant a is 3 cm. The thickness t of the plate is 1 mm. The other geometric parameters of the structure are shown in Fig. 1(c). The resonant frequency of the resonator in the model is [START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF][START_REF] Wang | Adjustable low-frequency bandgap of flexural wave in an Euler-Bernoulli meta-beam with inertial amplified resonators[END_REF]]

𝑓 = ! "# $ $ (&'!) ! ) , (1)
where μ = l1/ l2 is the lever ratio with l1 being the length of the lever and l2 being the distance between the spring and support bar; k is the stiffness of the spring; and m is the mass. The inertial mass will be amplified (μ-1) 2 times and the resonance frequency will be reduced (μ-1) times when the lever is rigid and has no mass. A low resonance frequency f can be achieved in the following cases. First, when the stiffness of the spring k is small, which means the soft material is used. Second, when the lever ratio μ is large, which means the long lever (i.e., l1 is large) is used or the position of the support bar is near the spring (i.e., l2 is small). Third, when large mass (i.e., m is large) is used.

Therefore, there are a lot of options to design elastic metamaterials with low-frequency bandgaps, including SMs which we will mention later. ×5 periodic unit cells. In this sample, all parts are printed using 3D printer. The PLA rod is fabricated and the lever and the support bar are assembled with PLA rod horizontally installed through the holes to form the hinged part. More enlarged photograph of the hinged part can be found in Appendix A. The source vibrating in z

direction is excited at one end of the MMP. Figure 1(e) shows the experimental layout.

Label A is a computer, which is utilized to set the output signal of the source on sample B and processes the input signal from the laser vibrometer C. In order to obtain clear and perfect propagation effects of flexural waves in the MMP, the laser vibrometer (Polytec PSV-500) is used to measure on the back of the MMP. In addition, a bare plate D, which is the same plate as in the MMP, is also measured to provide the control results. In order to facilitate the realization of the 3D printing of the MMP, the material of the whole structure is chosen as PLA whose Young's modulus is 3.4398 GPa, density is 1086.3 kg/m 3 , and Poisson's ratio is 0.35 in the both simulation and experiment [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF].

The connections between the support bar and the base plate, the base plate and the spring, the lever and the mass are perfect continuous. The spring and the lever are perfectly glued together. It is worth noting that the material of the spring part is PLA.

Of course, according to the Eq. ( 1), the spring part can be replaced by a softer material to obtain a less k and a lower-frequency bandgap. The connection between the lever and the support bar is hinge joint. The thickness of the lever along z axis is set as 0.3a to enhance the bending stiffness of the lever along z axis. At low frequencies, the lever can be regarded as a rigid part to achieve the inertial amplification.

The simulation results in this work are all calculated by using the multibody dynamics

TS = 20 × log10 (R1/R0), (2)
where R0 and R1 are the root mean square (RMS) of the out-of-plane velocities at the two points a1 (a2) and b1 (b2) shown in Fig. 1(d), respectively. The transfer-matrix method can be used to analytically calculate the band structure of the one-dimensional periodic simplified model. As shown in Fig. 2(a), the simplified model used for analytical solution can be assumed as the Euler-Bernoulli beam with inertial amplified resonators. So, the governing equation for the flexural waves in the beam is

Analytical solutions of the simplified model

𝐸𝐼 = 0, (3)
where E, I, ρ and A are Young's modulus, section moment of inertia, density and section area of the beam, respectively. z(x,t) is the out-plane displacement of the beam. The physical quantities of the beam can be expressed in the following forms [START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF]:

𝑧(𝑥, 𝑡) = 1𝐴𝑒 '/0, + 𝐵𝑒 /0, + 𝐶𝑒 '0, + 𝐷𝑒 0, 6𝑒 /1. , ( 4)

𝜃(𝑥, 𝑡) = 𝑦(𝑥, 𝑡) 2 , (5)
𝑀(𝑥, 𝑡) = 𝐸𝐼𝑦(𝑥, 𝑡) 22 , ( 6)

𝑄(𝑥, 𝑡) = -𝐸𝐼𝑦(𝑥, 𝑡) 222 , (7)
where A, B, C and D are complex coefficients; λ = =𝜌𝐴𝜔 " 𝐸𝐼 ⁄ "

with ω = 2πf being the circular frequency is the flexural wavenumber; and θ, M and Q are the slope, bending moment and shear force, respectively.

In the nth unit cell, the out-plane displacement of section 1 is

𝑧 3,! (𝑥 ! , 𝑡) = 1𝐴 3,! 𝑒 '/0, # + 𝐵 3,! 𝑒 /0, # + 𝐶 3,! 𝑒 '0, # + 𝐷 3,! 𝑒 0, # 6𝑒 /1. , (8)
where 𝑥 ! = 𝑥 -(𝑛 -1)𝑎 and (𝑛 -1)𝑎 ≤ 𝑥 ≤ (𝑛 -1)𝑎 + 𝑙 " . So, from the left side to the right side of section 1 of the nth unit cell, we obtain the transfer matrix in this field by following:

𝚿 3,! (0, 𝑡) = 𝐇 !4 J𝐴 3,! 𝐵 3,! 𝐶 3,! 𝐷 3,! K 5 𝑒 /1. , (9)
𝚿 3,! (𝑙 " , 𝑡) = 𝐇 !6 J𝐴 3,! 𝐵 3,! 𝐶 3,! 𝐷 3,! K 5 𝑒 /1. , (10)
where 𝚿(𝑥, 𝑡) = [𝑧(𝑥, 𝑡) 𝜃(𝑥, 𝑡) 𝑀(𝑥, 𝑡) 𝑄(𝑥, 𝑡)] 5 is the state vector; and J𝐴 3,! 𝐵 3,! 𝐶 3,! 𝐷 3,! K 5 is the coefficient vector. Substituting Eq. ( 10) into Eq. ( 9), we obtain

𝚿 3,! (𝑙 " , 𝑡) = 𝐇 !6 𝐇 !4 '! 𝚿 3,! (0, 𝑡) ≜ 𝐇 ! 𝚿 3,! (0, 𝑡), (11)
where 𝐇 ! is the transfer matrix. It is shown in Appendix B.

Similarly, the out-plane displacement of section 2 is 𝑧 3," (𝑥 " , 𝑡) = 1𝐴 3," 𝑒 '/0, ! + 𝐵 3," 𝑒 /0, ! + 𝐶 3," 𝑒 '0, ! + 𝐷 3," 𝑒 0, ! 6𝑒 /1. , [START_REF] Liu | Locally resonant sonic materials[END_REF] where 𝑥 " = 𝑥 -(𝑛 -1)𝑎 and (𝑛 -1)𝑎 + 𝑙 " ≤ 𝑥 ≤ 𝑛𝑎. So, from the left interface to the right one of section 2 of the nth unit cell, we obtain 𝚿 3," (𝑎, 𝑡) = 𝐇 "6 𝐇 "4 '! 𝚿 3," (𝑙 " , 𝑡) ≜ 𝐇 " 𝚿 3," (𝑙 " , 𝑡),

where 𝐇 " is the transfer matrix. It is shown in Appendix B.

As for the nth lever-arm structure on the beam, the dynamic equilibrium equation is

𝑓 3! (𝑡) + 𝑓 3" (𝑡) + 𝑚𝑍 ̈3(𝑡) = 0, (14A) 𝑙 " 𝑓 3! (𝑡) + (𝑙 ! -𝑙 " )𝑚𝑍 ̈3(𝑡) = 0, ( 14B
)
where 𝑍 3 (𝑡) is the displacement of the nth mass of the lever-arm structure; and 𝑓 3! (𝑡)

and 𝑓 3" (𝑡) are the forces of the spring and support bar on the lever, respectively. Due to the Hooke's Law, we obtain

𝑓 3! (𝑡) = 𝑘 S 7 $ (.) &'! + 𝑧 3,! (0, 𝑡) -𝑧 3," (𝑙 " , 𝑡)T, ( 15
)
where k is the spring stiffness; and μ = l1/ l2 is lever ratio. Substituting Eq. ( 15) into Eq. ( 14), we obtain

𝑓 3! (𝑡) = $)8 ! (&'!) ! )8 ! (&'!) ! '$ [𝑧 3,! (0, 𝑡) -𝑧 3," (𝑙 " , 𝑡)] ≜ 𝐺 ! [𝑧 3,! (0, 𝑡) -𝑧 3," (𝑙 " , 𝑡)], (16)
𝑓 3" (𝑡) = $)8 ! 9'& ! :;&'"< )8 ! (&'!) ! '$ [𝑧 3,! (0, 𝑡) -𝑧 3," (𝑙 " , 𝑡)] ≜ 𝐺 " [𝑧 3,! (0, 𝑡) -𝑧 3," (𝑙 " , 𝑡)]. ( 17
)
Due to the continuities at the interface between (n -1)th and nth unit cells, we obtain 𝑧 3'!," (𝑎, 𝑡) = 𝑧 3,! (0, 𝑡), (18A)

𝜃 3'!," (𝑎, 𝑡) = 𝜃 3,! (0, 𝑡), (18B) 𝑀 3'!," (𝑎, 𝑡) = 𝑀 3,! (0, 𝑡), (18C) 𝑄 3'!," (𝑎, 𝑡) = 𝑄 3,! (0, 𝑡) -𝑓 3! (𝑡). ( 18D
)
Substituting Eq. ( 16) into Eq. ( 18), we obtain

𝚿 3'!," (𝑎, 𝑡) = 𝐇 ; 𝚿 3,! (0, 𝑡) + 𝐇 = 𝚿 3," (𝑙 " , 𝑡), (19)
where 𝐇 ; and 𝐇 = are shown in Appendix B. Due to the continuities at the interface between sections 1 and 2 of the nth unit cell, we obtain

𝑧 3,! (𝑙 " , 𝑡) = 𝑧 3," (𝑙 " , 𝑡), (20A) 𝜃 3,! (𝑙 " , 𝑡) = 𝜃 3," (𝑙 " , 𝑡), (20B) 𝑀 3,! (𝑙 " , 𝑡) = 𝑀 3," (𝑙 " , 𝑡), (20C) 𝑄 3,! (𝑙 " , 𝑡) = 𝑄 3," (𝑙 " , 𝑡) -𝑓 3" (𝑡). (20D)
Substituting Eq. ( 17) into Eq. ( 20), we obtain

𝚿 3,! (𝑙 " , 𝑡) = 𝐇 > 𝚿 3," (𝑙 " , 𝑡) + 𝐇 ? 𝚿 3," (0, 𝑡), (21)
where 𝐇 > and 𝐇 ? are shown in Appendix B. Substituting Eqs. ( 11), ( 13), ( 19) and ( 21), the transfer equation of the flexural waves from the right interface of the (n -1)th unit cell to the right interface of the nth unit cell is obtain

𝚿 3," (𝑎, 𝑡) = 𝐓 @3/. 𝚿 3'!," (𝑎, 𝑡), (22)
where

𝐓 @3/. = {𝐇 ; 𝐇 ! '! [(𝐈 A -𝐇 ? 𝐇 ! '! ) '! 𝐇 > 𝐇 " '! ] + 𝐇 = 𝐇 " '! } '! is the transfer matrix in which 𝐈 A is a 4×4 identity matrix.
Considering the Bloch-Floquet theory, we obtain

𝚿 3," (𝑎, 𝑡) = 𝑒 /BC 𝚿 3'!," (𝑎, 𝑡), ( 23
)
where q is the wavenumber. Substituting Eq. ( 23) into Eq. ( 22), we obtain

Z𝐓 @3/. -𝑒 /BC 𝐈 A Z = 0. (24)
By solving the eigenvalue equation, the band structure can be obtained [START_REF] Liu | Design guidelines for flexural wave attenuation of slender beams with local resonators[END_REF].

For instance, Fig. 2 Figure 3 shows the band structure of the MMP and the transmission of the flexural waves in the MMP and the bare plate along the ΓX direction (i.e. x direction). In Fig. 3(a), the out-of-plane polarization in the band structure of the MMP is given. The parameter of the out-of-plane polarization 𝜉 is defined as

Results and Discussions of the MMPs

𝜉 = ∫ |𝑤| 𝟐 𝑑𝑠 E / ∫ (|𝑢| 𝟐 + |𝑣| 𝟐 + |𝑤| 𝟐 )𝑑𝑆 E , ( 25
)
where u, v and w are the displacement components in the three directions (x, y and z);

and S is the whole unit cell. A wide bandgap for the flexural waves can be found from 385 to 1200 Hz. It is worth noting that, the end parts of the second and third dispersion curves have high out-of-plane polarization value. However, these two bands almost do not affect the wide bandgap in Fig. 3(a). We can easily find these two bands are almost flat, especially when out-of-plane polarization value is large. It means it is just the local resonance at a single frequency, which can be ignored when we determine the bandgap.

The vibration modes at these two bands are illustrated in Appendix C. As shown in Fig.

3(b)

, there is a significant AZ at these frequencies in the simulated and experimental results. Compared with the simulated results, the attenuation effects in the experimental results are not particularly obvious at low frequencies in the bandgap. In addition, in the whole bandgap, the attenuation effects in the experimental results are weaker than those in the simulated results. This is due to the fact that the hinge joint in the experiments is not completely perfect like how it is considered in the simulation. In the experiments, the friction and backlash in the hinged part are difficult to eliminate.

Instead, in the simulations, all boundaries in the hinge joint are rigid and smooth.

However, it is easy to find that the MMP in the experiments can obviously attenuate the flexural waves in the bandgap compared with the bare plate. For a better comparison, as shown in Fig. 3(c), a periodic structure composed of the same plate, springs and masses as in the MMP is considered. In the band structure and the transmission of the metamaterial, the first bandgap and AZ is very narrow and its lowest frequency is around 520 Hz in Fig. 3(d). In Fig. 1, the lever ratio μ of the proposed unit cell is about 2, so the bandgap range of the inertial amplification mechanism is similar to the that of the spring-mass structure in Fig. 3(c). When the position of the support bar is near the spring, which means the l2 becomes smaller and μ becomes larger, the bandgap range significantly reduces.

The vibration modes at the marked points in the band structures are illustrated in Fig. 4. Points A1 (A2) and B1 (B2) are the two boundaries of the bandgaps shown in Figs. Comparing with the vibrations at points a1 and b1, we can find that the flexural waves are strongly attenuated in the MMP. Because there is no damping at the boundaries of the plate, the reflected waves can be found from Fig.

3(a) and 3(d)

5(d).

Therefore, the velocity of point b2 is larger than that of point a2. The results also demonstrate the strong attenuation ability of the MMP. This is due to the fact that the magnification of inertia is constantly decreasing [START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF][START_REF] Wang | Adjustable low-frequency bandgap of flexural wave in an Euler-Bernoulli meta-beam with inertial amplified resonators[END_REF]. The AZ is attenuation zone.

To achieve the broadband attenuation, different kinds of unit cells are selected from Fig. The harmonic vibration in z direction on the line source is applied to achieve out-ofplane displacement in the plate. Figure 6(c) shows the transmission of the flexural waves in the range of 200 -1800 Hz. Compared with the transmission in Fig. 3(b), the range of the AZ of the CMP is wider. Two transmission peaks can be found at 750 Hz and 950 Hz, which is due to the coupling effects between the different unit cells. We believe that more unit cells are added in the supercells, better attenuation effects and further improvement of the bandwidth we will have [START_REF] Donda | Extreme low-frequency ultrathin acoustic absorbing metasurface[END_REF][START_REF] Quan | Passive acoustic metasurface with unitary reflection based on nonlocality[END_REF].

Figures 6(d) and 6(e) show the propagations of flexural waves in the CMP at 400 Hz and 1100 Hz. These two frequencies are in the first bandgap of the MMPs whose hinge joints are at -0.15a and 0.15a, respectively. Therefore, the flexural waves can hardly travel through the CMP at these two frequencies. In addition, at high frequencies, the attenuation effects of the CMP are strong.

Results and Discussions of the SMs

In this section, the similar structure with inertial amplification is utilized on a soil halfspace to induce low-frequency bandgap for surface waves. As we know, the artificial structure on a plate, such as the MMP, has an excellent effect in attenuating flexural waves (out-of-plane vibration). The SMs' target is attenuation of surface waves (xzplane vibration) on a half-space. Due to the huge difference between flexural waves and surface waves, the artificial structure in the MMPs that can be used to attenuate flexural waves are not necessarily suitable for a half-space to attenuate surface waves. However, some artificial structures in the MMPs can play a key role in a half-space. For instance, drilling holes periodically in a half-space [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF], or burying pillars periodically in a halfspace [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF], or erecting pillars periodically on a half-space [START_REF] Brûlé | Flat lens effect on seismic waves propagation in the subsoil[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] all can generate bandgaps for surface waves. Moreover, some artificial structures of the MMPs covering a half-space can exert unexpected effects for surface waves. For instance, twocomponent [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF][START_REF] Wang | Periodic in-filled pipes embedded in semi-infinite space as seismic metamaterials for filtering ultra-low-frequency surface waves[END_REF] and three-component [START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF] MMPs on a half space can achieve ultralow-frequency bandgaps for surface waves. Therefore, due to the excellent performance of the lever-arm structure on the plate as shown above, we introduce it into the soil halfspace to target seismic surface waves. Similar as the lever-arm structure shown in Fig. 1, the unit cell of the SM shown in Fig. 7(a) is constituted of the lever, support bar, mass, spring, hinge joint and soil substrate. Comparing with the lattice constant in Fig. 1, the lattice constant a in Fig. 7(a) is scaled to 1.5 m. The relationship between other geometric parameters and lattice constant a is consistent with those in Fig. 1.

Theoretically, the depth of the half-space should be infinitely deep. In order to facilitate simulation calculations and obtain relatively accurate convergence results, the depth of the soil substrate is selected as H = 500a [START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Zeng | Coupling the first and second attenuation zones in seismic metasurface[END_REF]. A part of the spring is replaced by a thin rubber layer with thickness of t1 to achieve bandgaps at lower frequencies. In addition, the SM is supposed to be constructed by the ordinary building materials, for instance, steel (mass), rubber (spring) and aluminum (support bar and lever). The material properties [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF] are shown in Table 1. The mass of the steel part of the unit cell is 709.317 kg. Rayleigh waves cannot appear in the frequency range of the second band [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF]. However, in the range of the third band, Rayleigh waves appear again with the particles mainly moving along x direction, see Figs. 8(e) and (f). So, the bandgap region shown in Fig. 7(b) for the Rayleigh waves is between the first and third bands [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF].

It is known that seismic metamaterials with small mass can hardly achieve bandgaps below 10 Hz. However, the inertial amplification mechanisms with small mass can easily induce low-frequency bandgap, even for surface waves. As the first time of introducing inertial amplification mechanisms into seismic metamaterials, our goal is the existence of this bandgap rather than the enormous width of the bandgap. Therefore, we believe that narrow bandgap in this work is acceptable. To achieve the broadband attenuation, the effects of the parameters of the hinge joint and rubber are calculated below. The effects of the hinge joint's position, the rubber layer's thicknesses and the Young's modulus of the rubber on the bandgaps are illustrated in Fig. 9. The frequencies of the bandgap reduce significantly when the hinge joint moves from right to left along x direction. When the hinge joint moves from right to left, the μ increases and the f in Eq.

1 decreases. In addition, the frequencies of the bandgap reduce quickly when the rubber is provided, and keep decreasing when the thickness of the rubber increases. However, the relative bandwidth of the bandgap decreases with the frequency. The frequencies and the relative bandwidth of the bandgaps all increase when the Young's modulus of the rubber increases from 50 to 350 kPa. When the thickness of the rubber decreases or the Young's modulus of the rubber increases, the translational stiffness k in the vertical direction increases [START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF] and the f in Eq. 1 decreases. Comparing with the bandgaps in Fig. 3(a) or Fig. 6(a), the bandgaps in Fig. 9 or Fig. 7(b) are narrow. This is an inevitable result. In Fig. 3(a) and Fig. 6(a), it is the flexural waves (vibration only along the z axis) that the metamaterials attenuate in a wide frequency range. In Fig. 9 and Fig. 7(b), it is the Rayleigh surface waves, which include both longitudinal (x axis) and transverse (z axis) motions, that the SMs are designed to attenuate at ultra-low frequencies. These two kinds of bandgaps cannot be directly compared. In addition, at ultra-low frequencies, the bandgap induced by inertial amplification is quite narrow, as shown in Fig. 2. Of course, the broadband attenuation effects demonstrated by using different kinds of unit cells in Figs. 6(b)-(e), can be achieved by using the SM. Suitable unit cells can be selected from Fig. 9 to construct a new seismic metamaterial to isolate seismic waves in a wide frequency range.

Conclusion

An alternative type of seismic metamaterial based on inertial amplification mechanism is presented in this work to open bandgaps for seismic surface waves at low frequency regime. A conceived metamaterial plate with lever-arm structures has been experimentally investigated and discussed. It is shown that a wide bandgap for flexural waves appears in a low-frequency region. The propagation analysis measured on the back side of the MMP has demonstrated that the lever-arm structures strongly attenuate the flexural waves. In addition, a new complex metamaterial plate with two kinds of unit cells has been constructed to enlarge the attenuation zone. Finally, the similar leverarm structures are introduced into a soil half-space to obtain the bandgaps induced by inertial amplification for seismic surface waves in a low-frequency range. The significant advantage is that inertial amplification mechanisms with small mass can achieve strong vibration attenuation at low frequencies, a property that should be realized for seismic metamaterials. Comparing with the seismic metamaterials proposed in the literature, the width of the bandgap here is not remarkable. However, since it is the first time the inertial amplification mechanism is introduced and considered in seismic metamaterials, our goal is the creation of this bandgap rather than obtaining a large width of the bandgap. Therefore, the narrow bandgap in this work is acceptable. This work provides an alternative way to control seismic surface waves at low frequencies by using a small mass. In addition, the embedded inertial amplification mechanisms have great potential to be used to attenuate seismic surface waves in a wide range at ultra-low frequencies which will be the focus of future researches.

𝐇 "6 = i 𝑒 '/0C -𝑖𝜆𝑒 '/0C -𝐸𝐼𝜆 " 𝑒 '/0C -𝑖𝐸𝐼𝜆 ; 𝑒 '/0C 𝑒 /0C 𝑖𝜆𝑒 /0C -𝐸𝐼𝜆 " 𝑒 /0C 𝑖𝐸𝐼𝜆 ; 𝑒 /0C 𝑒 '0C -𝜆𝑒 '0C 𝐸𝐼𝜆 " 𝑒 '0C 𝐸𝐼𝜆 ; 𝑒 '0C 𝑒 0C 𝜆𝑒 0C 𝐸𝐼𝜆 " 𝑒 0C -𝐸𝐼𝜆 ; 𝑒 0C

Figure 1 (

 1 Figure 1(d) illustrates the experimental sample of the MMP. The sample is formed by 5

Figure 1 (

 1 Figure 1 (a) The unit cell of the MMP considered in the numerical simulation and experiment. (b)

module of COMSOL 5 . 4 .

 54 The periodic boundary conditions are set on the plate of the unit cell along x and y directions. It is worth noting that both boundary similarity and pointwise constraint are utilized to achieve the periodic boundary condition in the multibody dynamics module. The hinge joint between the lever and the support bar shown in Fig. 1 is set as hinge-connection condition. The same structure as the experimental sample shown in Fig. 1(d) is used in simulation of the flexural wave propagation in the MMP. The transmission spectrums in the experiments and simulations are all defined as

Figure 2 (

 2 Figure 2 (a) The simplified model used for analytical solution. (b) the numerical simulation results

  (b) shows the simulated results and analytical results of the band structure of the simplified model, when the lattice constant is a = 1 m; the height, breadth, sectional area, section moment of inertia, Young's modulus and density of the beam is h = 0.05 m, b = 1 m, A = bh, I = bh 3 /12, 207 GPa and 7784 kg/m 3 , respectively; l1 = 0.9 m, l2 = 0.3 m; m = 15.2838 kg and k = 100,000 N/m. It is worth noting that the support bar and the lever are rigid. The simulation results are calculated by COMSOL 5.4 (Multibody Dynamics Module) to evaluate the accuracy of the analytical solution.The simulated results are almost in agreement with the analytical results, especially for the position of the bandgap. It is worth noting that the frequencies of the two bandgaps are consistent with the predicted result from Eq. (1). Obviously, the frequency of the bandgap can be controlled easily in this model by changing one or more parameters of the structure. For instance, an ultra-low frequency bandgap can be achieved when the mass m or lever ratio μ is larger, or when the spring stiffness k is smaller than the example shown in Fig.2.

Figure 3 (

 3 Figure 3 (a) The band structure of the MMP along the ΓX direction. (b) The transmissions of the

Figure 4 (

 4 a) shows the strong resonance of the mass at point A1. At point B1, most of the displacement is around the spring part shown in Fig. 4(b). It is clear that the large bandgap is due to the inertial amplification system and the strong resonance of the mass. In Fig. 4(c), most of the displacement is only at the top of the mass, which increases the frequency of the bottom boundary of the first bandgap. In addition, the strong resonance of the mass shown in Fig. 4(d) reduces the frequency of the top boundary of the first bandgap. Therefore, in Fig. 3(d), the frequency of the first bandgap is higher and the width of the first bandgap is narrower than that in Fig. 3(a).

Figure 4

 4 Figure 4 Vibration modes corresponding to points (a) A1, (b) B1, (c) A2 and (d) B2 marked in Figs.

Figure 5

 5 Figure 5 Experimentally measured flexural waves on the back sides of (a) the MMP and (b) the bare

Figure 6 (

 6 Figure 6 (a) Effects of the hinge joint's position along x direction on the first and second bandgaps

6

  (a) to construct a new CMP. The selected criterion is obtaining a wider AZ with minimal kinds of unit cells. Two kinds of unit cells whose hinge joints are at -0.15a and 0.15a are selected, due to the AZ of the them together is from 320 to 1800 Hz (Of course, there are many other choices). Figure6(b)shows the schematics of the CMP. The periodic boundary conditions are applied on the edges along the y direction. The perfectly matched layer (PML) is utilized to reduce reflections of the free boundaries.

Figure 7

 7 Figure 7 (a) The unit cell of the SM for surface waves. (b) The band structure of the SM for surface

Figure 7 (

 7 Figure 7(b) shows the band structure of the SM along the ΓX direction when the

Figure 8

 8 Figure 8 Vibration modes and their central section along the xz plane of the SM for surface waves

Figure 9

 9 Figure 9 Effects of the hinge joint's position along x direction (a), the rubber layer's thickness (b)

  Figure A11 The vibration modes at (a) 2 nd and (b) 3 rd bands in Fig. 3(a).

Table 1 :

 1 The material parameters used in this paper.

	Material	Density (kg/m 3 )	Young's modulus (Pa)	Poisson's ratio
	Steel	7784	2.07 x 10 11

Acknowledgements

This work is supported by la Région Grand Est, the Institut CARNOT ICEEL, and the National Natural Science Foundation of China (Grant Nos. 12021002, 12122207, 11991031, 11991032 and 41974059). The first author is grateful to the support of China Scholarship Council (Grant No. 202006250084).

𝐇 "4 = 𝐇 !6 ,

Data Availability

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

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. In the experiments, all parts are printed by using 3D printer. The PLA rod and the hole in the lever and the support bar shown in Fig. A10(a) are designed. The lever and the support bar are assembled with PLA rod horizontally installed through the holes to form the hinged part shown in Fig. A10(b).

Appendix A. The enlarged photograph of the hinged part

Appendix B. The matrixes used in the analytical calculation

All the matrixes in the analytical calculation of the band structure of the simplified model are shown below:

transmission results. Figure A12 shows the unit cell, band structure and transmission of the 1D seismic metamaterial. Similar as the unit cell shown in Fig. 7(a), the yellow, red, silver and dark grey parts are soil, rubber, aluminum and steel, respectively. The lattice constant a is 1.5 m. The depth of the substrate H is 500a. The thickness of the rubber is 0.02a. The size of the steel is 0.1a * 0.1a. The length of the lever l1 and the distance between the spring and support bar l2 are 0.8a and 0.35a, respectively. 10 unit cells are used to calculated the transmission curve. The band structure and transmission show the bandgap and attenuation zone around 7.5 Hz.

Appendix D. 1D periodic seismic metamaterial


Si-Min Yuan

A-Li Chen
email: alchen@bjtu.edu.cn

Xing-Yue Du

Hua-Wei Zhang

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Yue-Sheng Wang
email: yswang@bjtu.edu.cn

Reconfigurable Flexural Waves Manipulation by Broadband Elastic Metasurface

Keywords: Elastic metasurfaces, Flexural waves, Reconfigurability, Broadband, Phase modulation, Wavefront control

We propose a reconfigurable elastic metasurface for broadband plate wave control where the flexural-dominated plate mode is concerned. The structure consists of a plate with nuts and screws perpendicular to the plate. With the continuous changes of nut positions on the screws, the phase shift can cover 2π span with high transmittance over broadband frequency. The mechanism of the tunability of phase shift caused by the screw-in depth is discussed based on the dispersion theory of local resonant phononic crystals. Then, subwavelength elastic metasurfaces are designed based on the generalized Snell's law demonstrating numerically and experimentally abnormal refraction and wave focusing functionalities without remanufacturing the metasurface. Furthermore, we show asymmetric transmission with a structure containing two parallel metasurfaces. Different functionalities can be achieved by changing the screw-in depth at different frequencies. The proposed metasurface designs could provide a novel and feasible route to control flexural waves in broadband frequency range and take a critical step in the pragmatic development of elastic metasurfaces.

Introduction

Metamaterials can give unique characteristics that cannot exist in conventional materials [START_REF] Park | Recent advances in non-traditional elastic wave manipulation by macroscopic artificial structures[END_REF]. However, further applications of metamaterials have been restricted because of their bulky and heavy characteristics [START_REF] Hyun | Partitioned gradient-index phononic crystals for full phase control[END_REF]. Metasurfaces, which can be considered as a two-dimensional (2D) counterpart of metamaterials, have opened up possibilities for realizing extraordinary wave manipulation with lightweight and thin planar designs [START_REF] Holloway | An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials[END_REF][START_REF] Chen | A review of metasurfaces: physics and applications[END_REF][START_REF] Ding | Gradient metasurfaces: a review of fundamentals and applications[END_REF]. After optic [START_REF] Bharadwaj | Optical antennas[END_REF][START_REF] Novotny | Antennas for light[END_REF] and acoustic metasurfaces [START_REF] Assouar | Acoustic metasurfaces[END_REF][START_REF] Chen | Multifunction switching by a flat structurally tunable acoustic metasurface for transmitted waves[END_REF][START_REF] Peng | Efficient scattering-free wavefront transformation with power flow conformal bianisotropic acoustic metasurfaces[END_REF][START_REF] Zhang | A reconfigurable active acoustic metalens[END_REF][START_REF] Song | Research progress and development trends of acoustic metamaterials[END_REF], elastic metasurfaces have been gradually developed because of their excellent manipulation ability of elastic waves.

Most studies on elastic metasurfaces focus on the plate waves, and so does this paper. The phase shift of elastic waves can be tuned mainly by changing their velocities through a resonant or non-resonant way. Usually, the former can be achieved by introducing resonant units such as pillars or columns [START_REF] Zhu | Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces[END_REF][START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF][START_REF] Wang | Experimental realization of a pillared metasurface for flexural wave focusing[END_REF][START_REF] Jin | Pillar-type acoustic metasurface[END_REF][START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF][START_REF] Jin | Elastic metasurfaces for deep and robust subwavelength focusing and imaging[END_REF][START_REF] Wang | Experimental realization of a pillared metasurface for flexural wave focusing[END_REF][START_REF] Cao | Disordered elastic metasurfaces[END_REF], zigzag units [START_REF] Liu | Source illusion devices for flexural lamb waves using elastic metasurfaces[END_REF][START_REF] Li | Efficient asymmetric transmission of elastic waves in thin plates with lossless metasurfaces[END_REF][START_REF] Su | Steering flexural waves by amplitude-shift elastic metasurfaces[END_REF][START_REF] Hu | Realization of ultrathin waveguides by elastic metagratings[END_REF][START_REF] Zhang | Vibration control of flexural waves in thin plates by 3D-printed metasurfaces[END_REF], and blocks [START_REF] Lin | Modular elastic metasurfaces with mass oscillators for transmitted flexural wave manipulation[END_REF][START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF][START_REF] Lee | Broad-angle refractive transmodal elastic metasurface[END_REF][START_REF] Lee | Full-wave tailoring between different elastic media: a double-unit elastic metasurface[END_REF][START_REF] Rong | Frequency-coded passive multifunctional elastic metasurfaces[END_REF] into the structure. The latter, however, can be realized by grooves [START_REF] Xu | Anomalous refraction manipulation of Lamb waves using single-groove metasurfaces[END_REF], notched units [START_REF] Jiang | Multiparameter independent manipulation for flexural wave by notched metasurface[END_REF], strips [START_REF] Su | Focusing, refraction, and asymmetric transmission of elastic waves in solid metamaterials with aligned parallel gaps[END_REF][START_REF] Qiu | Manipulation of shear horizontal guided wave with arbitrary wave fronts by using metasurfaces[END_REF] or composite strips [START_REF] Zeng | Asymmetric transmission of elastic shear vertical waves in solids[END_REF][START_REF] Su | Design of metasurfaces to enable shear horizontal wave trapping[END_REF], etc. Full phase control can also be achieved by gradient-index phononic crystals [START_REF] Hyun | Partitioned gradient-index phononic crystals for full phase control[END_REF]. However, the dimension of the structure is much larger than the wavelength. According to the abovementioned methods, researchers have proposed various metasurfaces to steer the wavefronts of refracted plate waves. Anomalous refraction of Lamb wave [START_REF] Zhu | Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces[END_REF] was first realized by a metasurface composed of tapered resonance units. By erecting pillars or columns perpendicularly on a plate, metasurfaces were designed with changing the heights of the pillars or columns to manipulate flexural waves [START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF][START_REF] Wang | Experimental realization of a pillared metasurface for flexural wave focusing[END_REF][START_REF] Jin | Pillar-type acoustic metasurface[END_REF][START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF][START_REF] Jin | Elastic metasurfaces for deep and robust subwavelength focusing and imaging[END_REF][START_REF] Wang | Experimental realization of a pillared metasurface for flexural wave focusing[END_REF]. Cao et al. [START_REF] Cao | Disordered elastic metasurfaces[END_REF] also discussed the identical pillared resonators with a random arrangement. In-plate [START_REF] Liu | Source illusion devices for flexural lamb waves using elastic metasurfaces[END_REF][START_REF] Li | Efficient asymmetric transmission of elastic waves in thin plates with lossless metasurfaces[END_REF][START_REF] Su | Steering flexural waves by amplitude-shift elastic metasurfaces[END_REF][START_REF] Hu | Realization of ultrathin waveguides by elastic metagratings[END_REF] and out-of-plate [START_REF] Zhang | Vibration control of flexural waves in thin plates by 3D-printed metasurfaces[END_REF] zigzag units were proposed to design metasurfaces for source illusion, refraction, focus, asymmetric transmission and ultrathin waveguide of A0 Lamb waves. Changing the sizes of resonant blocks added to beams was also able to manipulate the phase of the transmitted flexural wave [START_REF] Lin | Modular elastic metasurfaces with mass oscillators for transmitted flexural wave manipulation[END_REF]. Making slits or openings on a plate can form resonant units for the deflection of longitudinal wave [START_REF] Lee | Mass-stiffness substructuring of an elastic metasurface for full transmission beam steering[END_REF] and broadangle modal conversion between P-and SV-waves [START_REF] Lee | Broad-angle refractive transmodal elastic metasurface[END_REF] in plates. With the similar structure of Ref. [START_REF] Lee | Broad-angle refractive transmodal elastic metasurface[END_REF], a double-unit elastic metasurface composed of a phase modulator and an impedance matcher was proposed to realize full-wave tailoring of longitudinal wave [START_REF] Lee | Full-wave tailoring between different elastic media: a double-unit elastic metasurface[END_REF]. Topology optimization was utilized to obtain resonant units for the deflection and focus of a plate longitudinal wave [START_REF] Rong | Frequency-coded passive multifunctional elastic metasurfaces[END_REF].

As for the non-resonant way, the phase manipulation and different functions of S0 Lamb wave were realized by changing the groove positions in the groove array structure [START_REF] Xu | Anomalous refraction manipulation of Lamb waves using single-groove metasurfaces[END_REF]. A notched metasurface provides an innovative way to manipulate flexural waves without resonators, which supports high transmittance and multi-parameter modulation [START_REF] Jiang | Multiparameter independent manipulation for flexural wave by notched metasurface[END_REF]. Various functionalities of in-pane waves [START_REF] Su | Focusing, refraction, and asymmetric transmission of elastic waves in solid metamaterials with aligned parallel gaps[END_REF] and S0 waves [START_REF] Qiu | Manipulation of shear horizontal guided wave with arbitrary wave fronts by using metasurfaces[END_REF] in plates were achieved via designing the thickness of the straight beam unit. The straight beam can be replaced by a composite beam or strip consisting of two different materials [START_REF] Zeng | Asymmetric transmission of elastic shear vertical waves in solids[END_REF][START_REF] Su | Design of metasurfaces to enable shear horizontal wave trapping[END_REF].

However, the aforementioned metasurfaces can only work for a certain function at a fixed frequency or an extremely narrow frequency range, which limits their practical applications [START_REF] Wang | Tunable and active phononic crystals and metamaterials[END_REF]. To break these restrictions, tunable metasurfaces have been proposed with the help of piezoelectric materials or mechanical reconfigurability of the structure. In fact, piezoelectric units with feedback circuits have been widely applied in smart wave control in phononic crystals and metamaterials, cf. Ref. [START_REF] Marconi | Experimental observation of nonreciprocal band gaps in a space-time-modulated beam using a shunted piezoelectric array[END_REF] among others. In Ref. [START_REF] Li | Tunable modulation of refracted lamb wave front facilitated by adaptive elastic metasurfaces[END_REF], piezoelectric resonant units were proposed to design a tunable metasurface which can realize directional refraction, wave focusing and source illusion of A0 Lamb wave by adjusting the negative capacitance. The similar method was used to design a tunable metasurface for SH0 wave modulation [START_REF] Xia | In situ steering of shear horizontal waves in a plate by a tunable electromechanical resonant elastic metasurface[END_REF]. And the real time control of a broadband flexural wave was achieved via a programmable metasurface based on the piezoelectric unit and feedback circuit control [START_REF] Chen | A programmable metasurface for real time control of broadband elastic rays[END_REF]. Recently, we proposed a fish-bone mechanically reconfigurable elastic metasurface to manipulate the wave front of flexural [START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF] and inplane waves [START_REF] Yuan | Tunable multifunctional fish-bone elastic metasurface for the wavefront manipulation of the transmitted inplane waves[END_REF] based on the "screw-and-nut" working principle. By changing the positions of the nuts on the screws, the phase shift of the transmitted wave can be manipulated continuously with high transmittance. Mechanical reconfiguring methods were usually used in design linear or nonlinear meta-structured materials [START_REF] Bae | Nonlinear elastic metamaterial for tunable bandgap at quasi-static frequency[END_REF].

Recent researches on the structure-based reconfigurable elastic metasurfaces without a complex system are relatively scarce. In this paper, the "screw-nut" structure is improved to make the metasurface easy to fabricate; and then different functionalities for the refracted flexural-dominated wave are demonstrated numerically and experimentally. The proposed broadband elastic metasurface could make a significant progress in the practical applications.

Element design and phase shift analysis

Unlike the previous fish-bone model for which the screws are paralleled to the plate [START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF][START_REF] Yuan | Tunable multifunctional fish-bone elastic metasurface for the wavefront manipulation of the transmitted inplane waves[END_REF], in this paper we improve the structure by making the screws perpendicular to the plate which will be more efficient for steering the flexure wave and can facilitate the fabrication of metasurfaces. The calculation models of basic unit and functional element are shown in Figs. 1(a) and (b). The gray and green regions represent the substrate (aluminum) and vibrators (steel), respectively. The material parameters of aluminum are as follows: mass density ρ = 2700 kg/m 3 , elastic modulus E = 68.9 GPa and Poisson's ratio v = 0.33, and those of steel are ρ = 7670 kg/m 3 , E = 199.4 GPa and v = 0.31. The influences of the geometric parameters shown in Fig. 1(a) on the modulation of phase shift and transmittance are discussed in Appendix A. Base on the results therein, we select a model of the functional element with 6 basic units in the following analysis and take the values of other geometric parameters as shown in Table 1. The screw-in depth (s) refers to the distance that the nut moves inward from the outer end of the column, as shown in Fig. 1(a). Notably, the distance between the front and back rows of the nuts and the plate is always keeping the same. Fig. 1(c) is the photograph of a functional element corresponding to Fig. 1(b).

For harmonic waves propagating in the inhomogeneous isotropic solids, the wave equation can be written as the following form by omitting the time harmonic term: ,

where is the differential operator; λ and μ are the Lamé constants; is the position vector; is the displacement vector and ω = 2πf is the circular frequency. In this paper, the finite element method is used to study the wave properties. A line source with the out-of-plane excitation is applied on the left side of the functional element. The phase shift and transmittance can be obtained by taking the averaged values along the detection line of the right side. Continuous periodic conditions are applied on the upper and lower boundaries, i.e. ,

where and are the displacements of the upper and lower surfaces, respectively. In order to attenuate reflection, perfectly matched layers (PMLs) are set on the left and right ends. The phase shift of the transmitted wave is regulated by changing the screwin depth (s) of the nuts. The phase shift and transmittance are calculated by the finite element method with the software COMSOL Multiphysics 5.4; and their variations with the frequency and screw-in depth are illustrated in the nephograms shown in Figs. 1(d) and (e). It is noted that the step of screw-in depth is 0.01mm which can be realized by rotate the nut by a certain angle (a). The relation between s and a is s = t´a/360 where t is the thread pitch.

( ) ( ) ( )( ) ( ) ( ) 2 2 l µ µ r w é ù Ñ + Ñ× -Ñ´Ñ´=-é ù ë û ë û r r u r u r u ( ) / , / , / x y z Ñ = ¶ ¶ ¶ ¶ ¶ ¶ ( , , ) x y z = r ( , , )
In the present design, we choose t = 0.5 mm. Thus the precision of 0.01mm for the screw-in depth corresponds to a = 7.2° which can also be easily realized in the experiments. The results shown in Fig. 1(d) demonstrate that the phase shift can cover a 2π period in broadband when s is changed continuously. It should be noticed that the appearance of "sudden changes" in the phase shift nephogram is due to the "phase fold" at f = 2mπ when the phase shift exceeds 2mπ (m = 1, 2, 3…) [START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF][START_REF] Yuan | Tunable multifunctional fish-bone elastic metasurface for the wavefront manipulation of the transmitted inplane waves[END_REF].

The transmittance ξf is calculated by

, ( 3
)
where A1 is the transmitted out-of-plane displacement amplitude of the strip model with a functional element shown in Fig. 1(b); and A0 is that for the uniform structure without the "screw-nuts". The transmittance within the two white dotted lines (the black zone in Fig. 1 (e)) is very low, so we should avoid this zone when designing the metasurface.

To interpret the mechanism of phase manipulation of the functional element shown in Fig. 1(b), we consider it as a finite structure intercepted from a one-dimensional locally resonant phononic crystal with the periodically distributed basic units (Fig. 1(a)) along x direction. Based on the Bloch's theory, the wave field satisfies:

, ( 4
)
where kx is the Bloch wave vector; and a is the lattice constant (here it is 9 mm), as shown in Fig. 1(a)). Then, the dispersion curves of different screw-in depths are calculated; and the results for s = 1.3, 2.9 and 4.5 mm are shown in Fig. 2(a). The color of the bands represents the proportion of the out-of-plane displacement to the total displacement defined as ,

where the integrals are taken over the volume (Vp) of the rectangular plate of basic unit. By observing Band I dominated by out-of-plane displacement, it can be clearly seen that with the increase of s, the band is getting steeper indicating the increase of the phase velocity cp. For a given frequency, the phase velocity can be calculated from the dispersion curve through cp = 2πf/k where k is the wave number. For example, cp varying with s at f = 6.4 kHz can be calculated from the crosspoints of the dot-line and Band I; and the results are shown in Fig. 2(b). Then, when the wave propagates through the functional element, the phase shift f can be calculated by

1 0 f A A x = ( ) ( )exp( i ) x x a x k a + = - u u 2 2 2 d d p p y V x y z V u V u u u V h = + + ò ò , ( 6
)
where l is the length of the "screw-nut" structure i.e. the thickness of the metasurface (here it is 54 mm). The phase shifts (the blue dots) calculated from the phase velocity curve are close to the simulated ones (the black curve), as shown in Fig. 2(c). By the way, the low transmittance region mentioned above can also be interpreted by the dispersion curves. As shown in Fig. 2(a), there is a band gap (gray region) between the two red passbands. Taking s = 1.3 mm as an example, the band gap for the flexural wave shown in Fig. 2 The phase regulating capacity of the functional element is also examined experimentally in this paper. The experiment setups and the specimen (1054 mm×10 mm×4 mm) are shown in Figs. 3(a) and (b). The Polytec PSV-500 generated a sinusoidal signal of f = 6.4 kHz which was transmitted to the piezoelectric patch (10 mm×10 mm×1 mm) shown in Fig. 3(c) through the power amplifier (AG1020). The 2π / p fl c f = phase values on the line (see Fig. 3(d)) in the transmitted field were picked up by the laser scanning vibrometer Polytec PSV-500. The big enough homogeneous regions with the blue-tacks glued on the upper and lower boundaries of the specimen [START_REF] Zhu | Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces[END_REF][START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF] can reduce the wave reflection well. Changing the screw-in depths of the nuts and repeating the above operations, we can get the phase shifts which are shown by the scattered green triangles in Fig. 2(c). The experimental results match well with the theoretical and numerical ones when s £ 4 mm. The large difference for s > 4 mm may be owing to the fact that the sample is not strictly symmetrical due to processing errors. With the increase of s, the resonance of the basic unit (screw with a nut) gradually weakens, and other wave modes (e.g. the torsional modes) may be relatively enhanced, which affects the experimental results. Fortunately, it can be seen from Appendix B (Tables B1-B3) that most values of s are smaller than 4 mm. So this phenomenon has little impact on the ultimate goal of the present design of the metasurface to achieve different functionalities. Anyway, the sample processing and measurement deserve improvement in future studies.

Metasurfaces design and numerical simulations of multiple functionalities in broadband

In this section, we will design the metasurface based on the GSL [START_REF] Yu | Light propagation with phase discontinuities: generalized laws of reflection and refraction[END_REF] to realize the manipulation of transmitted flexural waves. First, abnormal refraction with arbitrary refracted angles and wave focusing with arbitrary focus positions are realized numerically in broadband by a single metasurface. Then, a double-layered metasurface system is constructed to achieve the asymmetric transmission under different operating frequencies. The calculation model of a single metasurface with 40 functional elements is shown in Fig. 4(a). The boundaries around the model are set as PMLs. An edge load along the y-direction with the amplitude of (N/m) is applied at the 4(b)-(d). Sparse meshes are first built for calculation, and then the mesh density gradually increases until the results no longer change, that is, the results are convergent. The strips which contain resonant units are meshed by triangular prism elements and divided into 5 layers in y-direction (Fig. 4(c)). The remaining parts of substrate are also divided into 5 layers in y-direction by cuboid elements (Fig. 4(d)). The screws and nuts are meshed by tetrahedral elements (Fig. 4(c)) with the extremely fine size, which predefined in COMSOL Multiphysics.

In this paper, for the transmitted wave propagating along the designated direction, the required phase distribution can be calculated from the following refracted GSL:

, ( 7
)
where θt and θi are refraction and incident angles, respectively; ϕ(z) is the phase distribution along the metasurface which should be discretized during the design of the metasurface; and λ0 = c/f is the wavelength with c being the phase velocity of the flexural wave [START_REF] Su | Focusing, refraction, and asymmetric transmission of elastic waves in solid metamaterials with aligned parallel gaps[END_REF]:

, ( 8
)
where h2 is the thickness of the plate (here it is 4 mm). Once the phase shifts are known, we can obtain the corresponding screw-in depth of each functional element according to Fig. 1(d). Then the metasurface with the function of abnormal refraction can be constructed. For different refracted angles or operating frequencies, the metasurfaces can be reconfigured just by tuning the positions of the nuts. The screw-in depths of the nth functional element (n = 1-40) for different refraction angles in case of f = 6.4, 8.6 and 11.2 kHz (the corresponding wavelengths are 77.9, 67.2 and 58.9 mm, respectively) are listed in Tables B1 (θt = 15°, 50°) and B3 (θt = 90°) in Appendix B. The results are shown in Fig. 5 in which the out-of-plane displacement fields are presented. It can be seen that for different operating frequencies, the transmitted waves can propagate along the designated directions (θt = 15°, 50° and 90°) as marked by the arrows in Fig. 5. In conclusion, the designed tunable metasurface can realize the directional refraction of an arbitrary refracted angle under different frequencies. Besides, we notice that some undesired beams appear in addition to the designed angles, especially for the cases of 50°-and 90°-refractions. These unwanted beams stem from the high-order diffractions which become pronounced in the case of large angle refraction [START_REF] Larouche | Reconciliation of generalized refraction with diffraction theory[END_REF][START_REF] Zhou | High-efficiency Ultrathin Nonlocal Waterborne Acoustic Metasurface[END_REF]. Next, the wave focusing is demonstrated by adjusting the screw-in depths of the nuts of the designed metasurface. Given the desired focal positions (x, z), the required phase shifts are thereupon determined by [START_REF] Yuan | Switchable multifunctional fish-bone elastic metasurface for transmitted plate wave modulation[END_REF][START_REF] Yuan | Tunable multifunctional fish-bone elastic metasurface for the wavefront manipulation of the transmitted inplane waves[END_REF] , n = 2, 3, 4…

0 d ( ) sin sin 2 d t i z z l f q q p - = 2 2 1 2 4 2 1 1 12 (1 ) E h c w r n = -
where ϕ1 and ϕn are the phase shifts of the first and nth (n>1) functional elements of the metasurface, respectively; z1 and zn are the z coordinates of the first and nth functional elements of the metasurface, respectively. Likewise, by combining Fig. 1(d), the screwin depth for each functional element of the metasurface can be obtained. The corresponding screw-in depths of different focal positions (x, z) = (150 mm, 0) and (200 mm, 0) in case of f = 6.4, 8.6 and 11.2 kHz can be found in Table B2 in Appendix B. Fig. 6 shows the intensity of the out-of-plane wave fields. The positions of the focal points are basically matched with the default positions (150 mm, 0) and (200 mm, 0). All results show good focus capability of the proposed reconfigurable elastic metasurface. At last, we designed a structure with two metasurfaces to achieve asymmetric transmission of the flexural wave. As shown in Fig. 7, two elastic metasurfaces (EM1 and EM2) are set parallel with the distance of d=250 mm. The refraction angle of EM2 is designed as 90° in case of normal incidence. When the flexural wave is incident normally from the negative direction (ND), the propagating wave will be converted into an evanescent wave which cannot propagate through the structure. While when the wave comes from the positive direction (PD), the propagation direction can be tuned by EM1 to make sure that it can propagate through EM2.

2 2 2 2 1 1 0 2 ( ( ) ( ) ) n n z z x z z x p f f l -= - + - - +
First consider the wave coming from the ND. The phase shift of EM2, ϕ2(z), can be easily determined by the GSL [START_REF] Peng | Efficient scattering-free wavefront transformation with power flow conformal bianisotropic acoustic metasurfaces[END_REF] with θi = 0° and θt = 90°. Then consider the wave propagation along the PD. The phase shift of EM1, ϕ1(z), can be expressed as , [START_REF] Zhang | A reconfigurable active acoustic metalens[END_REF] At the same time ϕ2(z) should satisfy the following expression

, ( 12
)
where θt2 and θt1 are the two propagating directions modulated by EM1 and EM2 as shown in Fig. 7(a). So for a given θt2 ( ≤1), θt1 can be obtained from Eqs. [START_REF] Peng | Efficient scattering-free wavefront transformation with power flow conformal bianisotropic acoustic metasurfaces[END_REF] and [START_REF] Song | Research progress and development trends of acoustic metamaterials[END_REF]; and then substituting θt1 into Eq. ( 11), we can get ϕ1(z). Taking θt2 = 30° as an example, Figs. 7(b)-(g) exhibit the results of numerical simulation for f = 6.4, 8.6 and 11.2 kHz, respectively. When the flexural wave is incident from the PD, the transmitted wave propagates along the designated angle. When the

0 2 d ( ) sin sin 1 2π d t i z z l f q q - = = 0 1 1 d ( ) sin sin 2π d t i z z l f q q - = 0 2 2 1 d ( ) sin sin 2π d t t z z l f q q - = 2 sin 1 t q -
flexural wave is incident from the ND, it is converted into an evanescent wave by the EM2 and cannot propagate through the structure. It is noticed that the incident wave is not completely converted to the surface wave modes in Fig. 7(g). This can be interpreted by two reasons. The first one is that the incident wave beam is of finite width (as shown by the dotted lines in the figures), so scattering may appear at the edges of the beam. Secondly, as mentioned before, high-order diffractions may occur for large angle refraction [START_REF] Larouche | Reconciliation of generalized refraction with diffraction theory[END_REF][START_REF] Zhou | High-efficiency Ultrathin Nonlocal Waterborne Acoustic Metasurface[END_REF]. Even though, the asymmetric transmission can be well achieved by the proposed elastic metasurface in a wide frequency range. The screw-in depths of all units of EM1 and EM2 are listed in Table B3 in Appendix B.

The experimental demonstration

In this section, the functions of abnormal refraction and wave focusing are testified experimentally. The experiment platform with the manufactured plate (1650 mm×1100 mm×4 mm) is shown in Fig. 3(a). Figs. 3(e) and (f) show the zoomed-in diagrams of the metasurface. The signal generator (AFG 3102) generated a burst signal and the operation steps are similar as those of the functional element which are not stated here for brevity. An array of 18 piezoelectric patches (20 mm×20 mm×1 mm) are bonded on the surface of the plate as excitations. In order to suppress the symmetrical wave mode, the out-of-plane excitation is applied only on one-side of the plate. As well known, the wave mode excited in this way is flexure-dominated. The out-of-plane displacements of scan areas 1 and 2 (corresponding to the same regions of Fig. 4(a)) are picked up. Blue-tacks are glued on the boundaries of the plate [START_REF] Zhu | Anomalous refraction of acoustic guided waves in solids with geometrically tapered metasurfaces[END_REF][START_REF] Cao | Deflecting flexural wave with high transmission by using pillared elastic metasurface[END_REF], and the homogeneous regions are big enough; all these could well reduce the wave reflection.

For abnormal refraction, we consider the normal incident flexural wave of 6.4 kHz refracted along 15°. Figs. 8(a) and (c) are the simulated and experimental results, respectively. The polar directivity diagrams of Figs. 8(b) and (d) can be obtained from the out-of-plane displacement values of the vertical black lines in Figs. 8(a) and(c). It can be seen that the experimental results agree well with the simulation ones and the flexural waves propagate well along 15°. Due to the robustness of the structure, in a narrow frequency range of 6.3-6.5 kHz, 15°-refraction can still be achieved without changing the screw-in depth s, as shown in Figs. 8(e)-(l). But for other frequencies, s should be changed. Next, the focusing position is still (x, z) = (150 mm, 0), but the frequency is changed to 6.4 kHz. The simulated and experimental results are shown in Fig. 10. Fig. 11 shows the results for f=6.4 kHz and focusing position being (x, z) = (200 mm, 0). We can clearly see that the transmitted waves can be well focused on the expected positions, and the simulated results are well consistent with the experimental ones. All these validate the excellent focus performance of the metasurface.

Conclusion

In this research, a broadband reconfigurable metasurface has been conceived, designed and realized to control the wavefront of flexural waves based on the "screw-and-nut" operating mechanism. By changing the screw-in depths of the nuts, full 2π phase shift and high transmittance have been be obtained over a large frequency range. Abnormal refraction and wave focusing functionalities have been demonstrated numerically and experimentally. In addition, a structure with two parallel metasurfaces is constructed to demonstrate the asymmetric transmission under different operating frequencies. The proposed broadband elastic metasurface with continuous tunability proposed in this work is of great significance, making an important progress to achieve pragmatic engineering applications on elastic wave manipulation. It is should be noticed that the "broadband wave control" of this paper refers to the active control. It is necessary to change the screw-in depths of the nuts when switching between different functions and different frequencies. Finally, we would like to mention that the present design of the metasurface is based on the GSL which cannot eliminate higher order diffractions in unwanted directions. In order to solve this problem, the lattice diffraction theory can be adopted in designing metasurfaces. However, it is a non-trivial task, especially for complex functionalities, such as wave focusing, beam bending, etc. Generally, an optimization method should be employed to completely suppress high-order diffractions [START_REF] Zhou | High-efficiency Ultrathin Nonlocal Waterborne Acoustic Metasurface[END_REF]. This should be an interesting topic in a future work.

Appendix B The screw-in depths of the nuts

The following three tables show the screw-in depths of the nuts for various functions at different frequencies.

Table B1. The screw-in depths of the nuts in the nth functional element for the 15° and 50° directional refractions. (unit: mm).

Fig. 1

 1 Fig. 1 The basic unit (a) and the strip model with functional element (b) in numerical calculations; (c) the photograph of the strip specimen; the calculated nephograms of the phase shift (d) and transmittance (e) as functions of the screw-in depth s and frequency.

h ® 1 (

 1 the dark red color) represents the flexure-dominated wave modes. The displacement modes of the basic unit are shown in the insets of Fig.2(a). The flexural mode of the plate is mainly caused by the symmetric vibration mode (kx = 0.5π/a) of the resonant structure (screw-nut), corresponding to the red part of band I. There are also a few antisymmetric vibration modes (kx = π/a), corresponding to the blue part of Band I.

  (a) is (7.2 -8.0 kHz) which is exactly the same region of the black zone shown in Fig. 1(e).

Fig. 2 (

 2 Fig. 2 (a) The dispersion curves and the displacement modes for the screw-in depth s = 1.3, 2.9 and 4.5 mm; (b) the phase velocity curve varying with s for f = 6.4 kHz; and (c) the phase shifts obtained by numerical simulation (the black curve), analytical calculation (the blue dots) and experiment (the green triangles) for f = 6.4 kHz.

Fig. 3 .

 3 Fig. 3. (a) The experimental setup; (b) the functional element suspended on a vibration isolation table; the zoomed-in diagrams of (c) piezoelectric patch, and (d) measuring line of the phase shift; (e) the manufactured plate; and (f) the zoomed-in diagram of the metasurface structure and piezoelectric patches array.

Fig. 4 .

 4 Fig. 4. The calculation model of the metasurface (a) and its finite element meshes (b) established in COMSOL Multiphysics. (c) and (d) are the partial enlargements of (b).

  to excite a Gaussian beam, where H = 10 mm is the height of the functional element, as shown in Fig.1(b). Multi-functionalities such as directional refraction (α = 23.5 and w0 = 0.1), wave focusing (α = 0 and w0 = 0.4) and asymmetric transmission (α = 19.5 and w0 = 0.1) will be realized numerically.The finite element meshes ensuring the convergence of calculation are shown in Figs.

Fig. 5 .

 5 Fig. 5. The out-of-plane displacement fields for f = 6.4, 8.6 and 11.2 kHz. The refraction angles are (a)-(c) 15°, (d)-(f) 50° and (g)-(i) 90°.

Fig. 6

 6 Fig. 6 The intensity of the out-of-plane wave fields with f = 6.4, 8.6 and 11.2 kHz. The focal positions are (a-c) (150 mm, 0) and (d-f) (200 mm, 0).

Fig. 7 (

 7 Fig. 7 (a) Schematic diagram of asymmetric transmission with red arrows representing PD and green arrows representing ND. The out-of-plane displacement fields of (b)-(c) 6.4 kHz, (d)-(e) 8.6 kHz and (f)-(g) 11.2 kHz. (b), (d), (f) are the results of PD, and (c), (e), (g) are those of ND.

Fig. 8

 8 Fig. 8 The out-of-plane displacement fields of 15° refraction for 6.4 kHz obtained by (a) numerical simulation and (c) experiment; (b), (d) The polar directivity graphs correspond to the results of (a), (c), respectively. (e)-(h) and (i)-(l) are the results of 6.3 kHz and 6.5 kHz with the

Fig. 9

 9 Fig. 9 The wave focusing results of (x, z) = (150 mm, 0) for f = 6 kHz. (a) The out-of-plane displacement fields of numerical simulation. The values along the transverse (b) and vertical (c) black dashed lines of Fig. 9(a). (d)-(f) are the experimental counterparts of (a)-(c).

Fig. 10

 10 Fig. 10 The wave focusing results of (x, z) = (150 mm, 0) for f = 6.4 kHz. (a) The out-of-plane displacement fields of numerical simulation. The values along the transverse (b) and vertical (c) black dashed lines of Fig. 10(a). (d)-(f) are the experimental counterparts of (a)-(c).

Fig. 11

 11 Fig. 11 The wave focusing results of (x, z) = (200 mm, 0) for f = 6.4 kHz. (a) The normalized outof-plane displacement fields of numerical simulation. The values along the transverse (b) and vertical (c) black dashed lines of Fig. 11(a). (d)-(f) are the experimental counterparts of (a)-(c).

Fig. A1

 A1 Fig. A1 Phase shift (black solid line) and transmittance (red dotted line) as functions of screw-in depth s for different n1 at f = 5.6 kHz.

Table 1 .

 1 Geometric parameters (unit: mm).

	h1	h2	h3	h4	h5	h6	h7
	3	4	9	6	2.5	3	8

Table B2 .

 B2 The screw-in depths of the nuts in the nth functional element for the wave focusing. (unit: mm).

	s2 s1	3.98 3.30	4.18 0.88		3.50 2.52	3.83	0.79 3.69	2.33 2.37	3.53 3.61
	s3 s2	4.81 2.18	4.55 0.80		3.51 2.44	3.72	0.85 2.62	2.40 2.31	3.57 3.56
	s4 s3	0.79 1.63	2.33 3.98		3.53 2.38	3.62	0.93 2.09	2.46 3.76	3.64 3.52
	s5 s4	0.80 1.28	2.35 2.68		3.54 4.65	3.57	1.04 1.74	2.54 3.31	3.73 4.69
	s6 s5	0.83 1.04	2.38 2.08		3.56 3.75	3.52	1.16 1.50	2.66 3.04	3.85 4.31
	s7 s6	0.85 0.87	2.40 1.72		3.57 3.27	4.64	1.31 1.33	2.83 2.86	4.04 4.07
	s8 s7	0.88 0.75	2.42 1.47		3.59 2.99	4.25	1.52 1.21	3.06 2.72	4.34 3.91
	s9 s8	0.90 0.66	2.44 1.29		3.61 2.82	4.02	1.84 1.12	3.42 2.62	4.80 3.80
	s10 s9	0.94 0.57	2.46 1.17		3.64 2.68	3.86	2.31 1.04	4.08 2.54	3.54 3.73
	s11 s10	0.97 0.50	2.49 1.08		3.67 2.57	3.77	3.16 0.97	2.35 2.49	3.59 3.67
	s12 s11	1.01 0.46	2.51 1.00		3.70 2.50	3.69	0.78 0.91	2.41 2.45	3.67 3.62
	s13 s12	1.05 0.42	2.55 0.92		3.74 2.46	3.63	0.84 0.87	2.48 2.41	3.76 3.58
	s14 s13	1.09 3.83	2.58 0.87		3.77 2.42	3.59	0.91 0.83	2.57 2.38	3.90 3.56
	s15 s14	1.13 2.90	2.62 0.83		3.81 2.38	3.56	1.02 0.80	2.70 2.35	4.12 3.54
	s16 s15	1.17 2.39	2.67 0.80		3.86 2.35	3.54	1.14 0.79	2.88 2.33	4.47 3.52
	s17 s16	1.22 2.07	2.73 0.78		3.91 4.79	3.52	1.28 4.75	3.12 4.52	3.51 3.51
	s18 s17	1.27 1.87	2.79 4.41		3.98 4.37	3.50	1.48 4.10	3.54 4.24	3.55 3.50
	s19 s18	1.32 1.74	2.85 3.84		4.06 4.11	3.50	1.77 3.72	4.30 4.05	3.60 4.78
	s20 s19	1.39 1.67	2.92 3.53		4.15 3.97	3.49	2.21 3.49	2.36 3.95	3.70 4.70
	s21 s20	1.47 1.63	2.99 3.40		4.25 3.90	3.49	2.97 3.40	2.43 3.90	3.80 4.66
	s22 s21	1.55 1.63	3.08 3.40		4.37 3.90	3.49	4.78 3.40	2.49 3.90	3.96 4.66
	s23 s22	1.65 1.67	3.19 3.53		4.52 3.97	3.49	0.83 3.49	2.59 3.95	4.21 4.70
	s24 s23	1.75 1.74	3.32 3.84		4.71 4.11	3.50	0.90 3.72	2.74 4.05	4.62 4.78
	s25 s24	1.88 1.87	3.47 4.41		3.50 4.37	3.50	1.00 4.10	2.92 4.24	3.52 3.50
	s26 s25	2.02 2.07	3.67 0.78		3.51 4.79	3.52	1.12 4.75	3.20 4.52	3.57 3.51
	s27 s26	2.19 2.39	3.90 0.80		3.53 2.35	3.54	1.26 0.79	3.67 2.33	3.63 3.52
	s28 s27	2.39 2.90	4.19 0.83		3.55 2.38	3.56	1.44 0.80	4.56 2.35	3.73 3.54
	s29 s28	2.63 3.83	2.31 0.87		3.56 2.42	3.59	1.72 0.83	2.38 2.38	3.84 3.56
	s30 s29	2.94 0.42	2.33 0.92		3.58 2.46	3.63	2.13 0.87	2.44 2.41	4.03 3.58
	s31 s30	3.34 0.46	2.36 1.00		3.59 2.50	3.69	2.81 0.91	2.51 2.45	4.31 3.62
	s32 s31	3.98 0.50	2.38 1.08		3.62 2.57	3.77	4.34 0.97	2.62 2.49	4.80 3.67
	s33 s32	4.81 0.57	2.40 1.17		3.65 2.68	3.86	0.81 1.04	2.78 2.54	3.54 3.73
	s34 s33	0.79 0.66	2.42 1.29		3.68 2.82	4.02	0.89 1.12	2.98 2.62	3.58 3.80
	s35 s34	0.80 0.75	2.44 1.47		3.71 2.99	4.25	0.99 1.21	3.29 2.72	3.66 3.91
	s36 s35	0.83 0.87	2.46 1.72		3.75 3.27	4.64	1.10 1.33	3.83 2.86	3.76 4.07
	s37 s36	0.85 1.04	2.49 2.08		3.78 3.75	3.52	1.23 1.50	4.80 3.04	3.89 4.31
	s38 s37	0.88 1.28	2.51 2.68		3.82 4.65	3.57	1.41 1.74	2.39 3.31	4.10 4.69
	s39 s38	0.90 1.63	2.55 3.98		3.87 2.38	3.62	1.67 2.09	2.46 3.76	4.44 3.52
	s40 s39	0.94 2.18	2.58 0.80		3.93 2.44	3.72	2.05 2.62	2.53 2.31	3.50 3.56
	s40	3.30	0.88		2.52	3.83	3.69	2.37	3.61
			15°					50°
	si	6.4 kHz	8.6 kHz (150 mm, 0)	11.2 kHz	6.4 kHz	8.6 kHz (200 mm, 0)	11.2 kHz
	s1 si	3.39 6 kHz	3.89 6.4 kHz	3.49 8.6 kHz	11.2 kHz	3.39 6.4 kHz	3.89 8.6 kHz	3.49 11.2 kHz

Table B3 .

 B3 The screw-in depths of the nuts in the nth functional element for the asymmetric transmission. (unit: mm).

		EM1 (30°)	EM2 (90°)	EM1 (30°)	EM2 (90°)	EM1 (30°)	EM2 (90°)
	s1	3.39	3.39	3.89	3.89	4.66	4.66
	s2	2.67	0.80	3.48	2.35	4.35	3.54
	s3	2.23	0.89	3.20	2.44	4.13	3.60
	s4	1.92	1.03	3.01	2.53	3.98	3.72
	s5	1.68	1.19	2.86	2.70	3.86	3.88
	s6	1.50	1.41	2.75	2.94	3.78	4.18
	s7	1.36	1.77	2.64	3.35	3.71	3.49
	s8	1.25	2.40	2.56	4.21	3.65	3.55
	s9	1.16	3.85	2.50	2.38	3.59	3.62
	s10	1.08	0.82	2.45	2.46	3.56
	s11	1.00	0.92	2.41	2.57	3.53	3.91
	s12	0.93	1.06	2.37	2.76	3.50	4.22
	s13	0.87	1.23	2.33	3.03	4.58	4.84
	s14	0.83	1.48	4.13	3.52	4.29	3.55
	s15	0.79	1.88	3.64	4.62	4.09	3.63
	s16	4.06	2.59	3.31	2.40	3.95	3.75
	s17	3.04	4.48	3.08	2.48	3.84	3.94
	s18	2.46	0.84	2.92	2.61	3.76	4.27
	s19	2.09	0.95	2.79	2.82	3.70	3.50
	s20	1.81	1.10	2.68	3.13	3.64	3.56
	s21	1.60	1.27	2.59	3.74	3.59	3.64
	s22	1.44	1.55	2.52	2.34	3.56	3.77
	s23	1.30	1.99	2.47	2.42	3.53	3.97
	s24	1.20	2.83	2.43	2.51	3.50	4.32
	s25	1.12	0.78	2.39	2.67	4.50	3.51
	s26	1.04	0.86	2.35	2.89	4.25	3.57
	s27	0.97	0.98	2.30	3.26	4.06	3.66
	s28	0.90	1.13	3.83	4.02	3.92	3.78
	s29	0.85	1.32	3.44	2.36	3.82	3.99
	s30	0.81	1.62	3.17	2.45	3.75	4.38
	s31	0.78	2.12	2.99	2.55	3.68	3.51
	s32	3.54	3.12	2.85	2.72	3.62	3.57
	s33	2.76	0.79	2.73	2.98	3.58	3.67
	s34	2.29	0.88	2.63	3.42	3.55	3.80
	s35	1.96	1.01	2.55	2.30	3.52	4.03
	s36	1.72	1.17	2.49	2.39	3.50	4.44
	s37	1.53	1.37	2.45	2.47	4.44	3.52
	s38	1.38	1.71	2.41	2.59	4.20	3.58
	s39	1.26	2.27	2.37	2.79	4.02	3.68
	s40	1.17	3.49	2.32	3.07	3.89	3.82
	si	6.4 kHz	8.6 kHz	11.2 kHz

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant Nos. 11872101, 11991031 and 12021002).

Appendix A Effects of structure parameters in design of elastic metasurfaces

The influences of different structural parameters (n1, h1, h5, h6 and h7, where n1 is the number of basic units in a functional element) on the variation of phase shift and transmittance with the screw-in depth are discussed in this appendix. We first obtain the results for a set of parameters: n1 = 6, h1 = 3 mm, h5 = 2.5 mm, h6 =3 mm and h7 = 8 mm. Then we alter the value of n1 (here we take n1 = 4, 5, 6 and 7) with other parameters unchanged, calculate the curves, and illustrate them in Fig. A1. We can observe from the figure that with the increase of n1, the coverage interval of phase shift ϕ (black solid line) gradually increases and cover a full 2π range with high transmittance ξf (red dotted line) until n1 = 6. In order to keep the thickness of metasurface as small as possible, we take n1 = 6. Similarly, we change other parameters, h1, h5, h6 or h7, one by one to discuss their influences on phase shift and transmittance. The results are presented in Figs. A2-A5. The shadowed areas represent the cases with low transmittance. It can be clearly seen that with the gradual increase of n1, h1, h5 and h7, the resonance is gradually enhanced, and the phase shift can gradually cover a 2π range with high transmittance. With the increase of h6, the resonance becomes weaker and the coverage interval of phase shift decreases. In design of a metasurface, it should be ensured that the phase shift with high transmittance should cover a full 2π range.