Liyun Cao

Yifan Zhu

Yanlong Xu

Shi-Wang Fan

Zhichun Yang
email: yangzc@nwpu.edu.cn

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

Elastic Bound State in the Continuum with Perfect Mode Conversion

The partial or complete confinement of waves in an open system is omnipresent in nature and in wave-based materials and technology. Here, we theoretically analyze and experimentally observe the formation of a trapped mode with perfect mode conversion (TMPC) between flexural waves and longitudinal waves, by achieving a quasi-bound state in the continuum (BIC) in an open elastic wave system. The latter allows a quasi-BIC in a semi-infinite background plate when Fano resonance hybridizes flexural and longitudinal waves and balances their radiative decay rates. We demonstrate that when the Fabry-Pérot resonance of the longitudinal wave is realized simultaneously, the TMPC formed by the elastic BIC approaches infinite quality factor. Furthermore, we show that quasi-BIC can be tuned continuously to BIC through the critical frequency of mode conversion, which offers the possibility of TMPC with an arbitrarily high quality factor. Our reported concept and physical mechanism open new routes to achieve perfect mode conversion with tunable high quality factor in elastic systems.

Introduction

Bound states in the continuum (BICs), defying the conventional bound states located outside the continuum, are non-decaying localized states embedded within the continuous spectrum of radiating waves [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF]. The states can sometimes be regarded as embedded eigenvalues [START_REF] Hsu | Bound states in the continuum[END_REF] or embedded trapped modes (Hsu et al., 2013b) with infinite quality factors. The BIC concept was first proposed in quantum mechanics [START_REF] Von Neumann | Über merkwürdige diskrete Eigenwerte[END_REF] by mathematically constructing a 3D potential extending to infinity. Since this initial proposal, the wave phenomenon of BICs has been identified in different material and wave systems [START_REF] Hsu | Bound states in the continuum[END_REF], such as electromagnetic waves (Hsu et al., 2013b;[START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF][START_REF] Koshelev | Meta-optics and bound states in the continuum[END_REF][START_REF] Marinica | Bound States in the continuum in photonics[END_REF][START_REF] Minkov | Zero-Index Bound States in the Continuum[END_REF][START_REF] Plotnik | Experimental Observation of Optical Bound States in the Continuum[END_REF][START_REF] Zhen | Topological Nature of Optical Bound States in the Continuum[END_REF], acoustic waves in the air [START_REF] Cumpsty | The excitation of acoustic resonances by vortex shedding[END_REF][START_REF] Huang | Extreme Sound Confinement From Quasibound States in the Continuum[END_REF][START_REF] Lyapina | Trapped modes in a non-axisymmetric cylindrical waveguide[END_REF][START_REF] Parker | Low frequency resonance effects in wake shedding from parallel plates[END_REF], water waves [START_REF] Callan | Trapped modes in two-dimensional waveguides[END_REF][START_REF] Cobelli | Experimental study on water-wave trapped modes[END_REF] and surface acoustic waves [START_REF] Kawachi | Optimal cut for leaky SAW on LiTaO/sub 3/ for high performance resonators and filters[END_REF][START_REF] Lim | Character of Pseudo Surface Waves on Anisotropic Crystals[END_REF][START_REF] Trzupek | Isolated True Surface Wave in a Radiative Band on a Surface of a Stressed Auxetic[END_REF] in semiinfinite media. Structures supporting BICs or quasi-BICs with high quality factor (high-Q) have been widely studied and opened the route to numerous applications in different fields, especially in optics and photonics, such as lasers [START_REF] Hirose | Watt-class high-power, high-beam-quality photonic-crystal lasers[END_REF][START_REF] Imada | Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure[END_REF][START_REF] Lin | Ultrafast vortex microlasers based on bounded states in the continuum[END_REF][START_REF] Matsubara | GaN Photonic-Crystal Surface-Emitting Laser at Blue-Violet Wavelengths[END_REF][START_REF] Noda | Polarization Mode Control of Two-Dimensional Photonic Crystal Laser by Unit Cell Structure Design[END_REF], sensors [START_REF] Romano | Tuning the exponential sensitivity of a bound-state-in-continuum optical sensor[END_REF][START_REF] Yanik | Seeing protein monolayers with naked eye through plasmonic Fano resonances[END_REF][START_REF] Zhen | Enabling enhanced emission and low-threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals[END_REF], and filters [START_REF] Doskolovich | Integrated flat-top reflection filters operating near bound states in the continuum[END_REF][START_REF] Foley | Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating[END_REF][START_REF] Ju | Optical quantum frequency filter based on generalized eigenstates[END_REF].

Compared to the optical and acoustic BICs, the elastic counterpart is rarely concerned due to its more sophisticated polarization states (Graff, 1975;[START_REF] Rose | Ultrasonic waves in solid media[END_REF].

Recently some works have investigated elastic BICs in elastodynamics. Elastic BIC of bulk waves [START_REF] Haq | Bound States in the Continuum in Elasticity[END_REF] was studied using the Fabry-Perot mirroring mechanism in an infinite solid with mixed in-plane polarizations. Non-symmetryprotected BICs of surface acoustic waves [START_REF] Maznev | Bound acoustic modes in the radiation continuum in isotropic layered systems without periodic structures[END_REF] and symmetryprotected ones [START_REF] Lim | Character of Pseudo Surface Waves on Anisotropic Crystals[END_REF][START_REF] Trzupek | Isolated True Surface Wave in a Radiative Band on a Surface of a Stressed Auxetic[END_REF] have also been investigated in semi-infinite solids with surface polarization. However, most of those BIC-supporting models have never been experimentally witnessed. So far, in structural dynamics, especially for flexural and longitudinal waves (Graff, 1975;[START_REF] Rose | Ultrasonic waves in solid media[END_REF] in common plate-like structures, the researches on elastic BICs have still not been reported.

On the other hand, perfect mode conversion [START_REF] Giurgiutiu | Structural health monitoring with piezoelectric wafer active sensors[END_REF]Graff, 1975;[START_REF] Rose | Ultrasonic waves in solid media[END_REF] between different polarization modes has grown into a burgeoning research area in the elastic wave system, due to its potential wide applications in nondestructive testing, medical ultrasonography, and earthquake resistance in civil engineering. For example, a mode-coupled layer with the balanced mode excitations and diagonal polarizations [START_REF] Kweun | Transmodal Fabry-Perot Resonance: Theory and Realization with Elastic Metamaterials[END_REF] can achieve maximum mode conversion between longitudinal and shear modes, promoting the development of sensors in industrial and biomedical testing [START_REF] Yang | Asymptotic theory of bimodal quarter-wave impedance matching for full mode-converting transmission[END_REF][START_REF] Yang | Monolayer metamaterial for full mode-converting transmission of elastic waves[END_REF][START_REF] Zheng | Non-resonant metasurface for broadband elastic wave mode splitting[END_REF]. Pillared seismic metamaterials (Colombi et al., 2016a;Colombi et al., 2016b;[START_REF] Colquitt | Seismic metasurfaces: Subwavelength resonators and Rayleigh wave interaction[END_REF] can create effective bandgaps to achieve mode conversion from surface waves to bulk waves, promoting civil structures against seismic risk. Here, we propose a theoretical method to achieve perfect mode conversion between flexural waves (out-plane vibration) and longitudinal waves (in-plane vibration) (Graff, 1975;[START_REF] Rose | Ultrasonic waves in solid media[END_REF] in two perpendicular polarization planes, which has never been reported. The new elastic-wave mode conversion mechanism enriches the form of elastic wave energy flow, which can perfectly convert an in-plane test signal or in-plane vibration energy into an out-plane one for the simplifications of nondestructive testing or energy harvesting, perfectly convert an out of plane vibration into in-plane one for reducing vibration.

In this research, we present theoretical analysis and experimental evidence of achieving a trapped mode with perfect mode conversion (TMPC) in a quasi-BICsupporting elastic wave system. In the proposed scheme, Fano resonance hybridizes flexural and longitudinal waves in two perpendicular polarization planes and converts one mode into another. When the radiative decay rate of the converted mode balances that of the directly reflected wave from the incident wave entering the system, the incident wave will be completely trapped by the hybrid Fano resonance, i.e., quasi-BIC.

The resonant system only allows the converted mode to leak, leading to perfect mode conversion. Furthermore, we prove that TMPC can support infinite-Q BIC by simultaneously achieving the hybrid Fano and Fabry-Pérot resonances. We demonstrate that all quasi-BICs can be continuously tuned to BICs when the critical frequency of the mode conversion, depending on the incident angle, approaches the Fano resonance one.

Theory and results

To clearly show the unique TMPC in the quasi-BIC-supporting and BIC-supporting elastic wave systems, we investigate a simple structure with typical resonance characteristics. The structure is composed of a periodic waveguide resonator with the length s on the edge of a background plate with infinite left boundary, as shown in Fig. 1

where and are scattering vectors.

is the scattering matrix. The symbols t and r denote the transmission and reflection coefficients, respectively. The subscripts b and l represent the flexural wave (or bending wave) and longitudinal wave. The subscripts bl and lb represent the conversion from flexural wave to longitudinal wave and the reverse process, respectively. Analytical expressions of all scattering coefficients in the scattering matrix are obtained by the transfer matrix method (see detailed derivation in Appendix A).

In the following, we study the case that the pillared resonator and the waveguide resonator have the same flexural rigidity, i.e., the same thickness, providing a strong hybridization coupling [START_REF] Colquitt | Seismic metasurfaces: Subwavelength resonators and Rayleigh wave interaction[END_REF]. The material parameters of the model are obtained based on the experimental measurement of 3D printing material PLA (see Appendix F). Figs. 2(a) and 2(b) display the dimensionless scattering energy curves for

1 l -2 l - P = b b lb lb b b lb lb bl bl l l bl bl l l t r t r r t r t t r t r r t r t - + + - é ù ê ú - ê ú = × ê ú - ê ú - ë û S S M S [ ] 2 1 2 1 , , , T b b l l - - - - - = S [ ] + 1+ 2+ 1+ 2+
, , , Based on the analytical model, we analyze its band structure to intuitively show the hybridization coupling between flexural waves and longitudinal waves from the Fano resonance. Bloch boundary condition can be expressed as

, ( 3
)
where I is the identity matrix and is a typical period with a quarter wavelength in In the above intuitive understanding of the physical origin of the destructive interference, the coupling between the scattering modes of the resonance is an essential element. To theoretically explore the destructive interference in the elastic wave system, we expand the temporal coupled-mode theory [START_REF] Fan | Temporal coupled-mode theory for the Fano resonance in optical resonators[END_REF]Hsu et al., 2013a;Hsu et al., 2013b) in optics, which provides a simple analytical description for resonant objects weakly coupled to incoming and outgoing propagation modes. As shown in Fig.

* * + - + - + - é ù = ë û R 2 2 2 2 2 2 , , ,
1 (b), the amplitude a of the localized mode evolves as in the absence of input powers. is the radiative decay rate of the localized mode. The temporal coupled-mode theory for coupling between the resonance and incoming and outgoing propagation modes can be described [START_REF] Fan | Temporal coupled-mode theory for the Fano resonance in optical resonators[END_REF] as As shown in Fig. 1(b), according to Eq. ( 5), we can understand that the radiation of is from the coupling between the localized mode and the incoming propagation flexural modes at the right port. The radiation will leak into the left port ( )

0 0 d d i a t a w g = -- 0 g C a - + = × + × s s d [ ] T 1 2 1 2 = b b l l + + + + + s [ ] T 1 2 1 2 = b b l l - - - - - s T 1 2 3 4 = d d d d é ù ë û d 1 d 2 d 3 d 4 d 0 1 0 0 1 0 0 0 = 0 0 0 1 0 0 1 0 C ae ö ç ÷ ç ÷ ç ÷ ç ÷ è ø 1 d a ×
and includes the leakages of the flexural wave and the longitudinal one from mode conversion. Based on the scattering coefficient in Eq. ( 1), the leakages of the flexural wave and the converted longitudinal one in the left port can be expressed as and , respectively. Therefore, we can get the relationship between the localized mode and the leakages in the left port:

. ( 6)

Similarly, we can get the relationship in the right port:

. ( 7)

The incident propagating modes are not taken into account [START_REF] Hsu | Bound states in the continuum[END_REF].

With the propagation relation of , according to Eqs. ( 5)-( 7) and (A17),

we can get the analytical expression of the destructive interference relation as ,

where is a phase shift of one round-trip of flexural waves in the waveguide, and is the wavenumber of the flexural wave. On the other hand, considering an incident positive-going flexural wave of , the transmitted and reflected longitudinal waves from the mode conversion have a reverse polarization direction. These longitudinal waves will interfere and cancel at the scattering source (the resonator), based on Eq. (A10). Therefore, at the scattering source, the total scattering field is the superposition of the transmitted and reflected flexural waves with the same polarization direction, i.e., . According to the energy conservation, the energy of the total scattering field equal to that of the incident wave field. We can obtain the amplitude relationship of the total scattering field as . ( 9)

2 2 b t b b + + - 2 lb t l + 1 2 2 2 b lb d a t b b t l + + + × = -+ 2 2 2 b lb d a r b r l + + = + 1 b + i 2 2 b e b f - - + = × i b b e t r f -= + =2 3 2 b k s f p - b k i 1 t e w × b b r t + b b r t + 2 1 b b r t + =
In such case, the magnitude condition of Eq. ( 8) is always satisfied, and Eq. ( 8) can be reduced as its phase relation:

, ( 10
)
where n is a positive integer.

We should point out that hybridization coupling is an essential component of these unique local states from destructive interference. The reason is that the wave fields in the system only have one mode without mode hybridization, then any state must be leaky. The radiative energy from mode conversion in our system is similar to dissipative loss in the optical resonance system [START_REF] Hsu | Bound states in the continuum[END_REF]. But the ''dissipative loss'' in our study can be adjusted by the waveguide resonator, which will be discussed in detail later.

Analytical model of the whole resonator system

To evaluate all possible localized states in the parameter space, we establish the analytical model of the whole resonator system. As shown in Fig. 1(a), the background plate, the pillared resonator, and the waveguide resonator are divided into region (I) and region (II) by the dividing line . For the sake of universality to study the vertically and obliquely incident waves in the background plate, the governing equations for the flexural and longitudinal waves in the region (I) should be written in the two-dimension form:

,

, ( 12)

3 2 a r g ( + ) 2 2 b b s r t n p b w p - = + l ! ( ) Î i i i i i 0 ( , ) y y y x x x b b b b b b k y k y k y k x k x k x b b w x y A e e R e e R e e -× -× -× -× × × * = + + ( ) I i i ( , )
k = =sin y y b l i b k k k q × i q ( ) ( ) 2 2 = x y b b b k k k - ( ) ( ) 2 2 ˆ= i x y b b b k k k -- - ( ) ( ) 2 2 = x y l l l k k k - ( ) T I r = b b bl R R R * é ù ë û Τ l ! ( ) [ ] T II 2 2 2 2 2 2 = a b c d e f Τ 2 a 2 b 2 c 2 d 2 e 2 f j , (13)
, ( 14) , ( 15)

, ( 16
)
where , and j denotes the j th subunit.

In the same way, by applying the continuous boundary condition of the longitudinal wave displacement at the interface, we get

, ( 19
)
where and .

By applying the continuous boundary condition of x-components of bending moment at the interface x=0, and integrating along y-direction in the region , we can get

( ) ( ) I I x w x j ¶ = ¶ ( ) ( ) ( ) I I I 2 2 2 2 ( ) 2 0 + 2 ( ) 2 j x j w w D yy p M x y y y L L p u ì ¶ ¶ + -£ ï = ¶ ¶ í ï - £ - î ( ) ( ) ( ) I I I 3 3 3 2 ( ) 2 0 + 2 ( ) 2 j x j w w D yy p V x x y y y L L p ì ¶ ¶ - + -£ ï = ¶ ¶ ¶ í ï - £ - î ( ) ( ) I I 2 2 1 0 + 2 ( ) 2 j x j Ed u y y p F x y y L L p u ì ¶ × -£ ï = - ¶ í ï - £ - î ( ) 1 j y j L = - 2 j y y p -£ [ ] ( ) [ ] ( ) I I I 0 r 0 1 1 1 1 0 0 g A g g µ µ µ × + = Τ Τ sinc( / 2) y b g k p = × i y j b k y e µ -× = ( ) [
] ( ) Eqs. ( 17)-( 22) and Eq. ( 24), in turn, can be rewritten as

I I I 0 r î i 0 i i 0 0 x x x b b b b b b b k gA k g k g k k k k µ µ µ é ù - + = - - ë û Τ Τ [ ] ( ) [ ] ( ) I I I r ˆ0 0 0 0 0 0 1 1 g µ × = Τ Τ ˆsinc( / 2)
I I I 5 0 1 2 r 1 1 1 1 0 00 E A E E V V V V + =- - Τ Τ ( ) ( ) 2 2 1 x y b b E L k k u é ù = × - - ê ú ë û ( ) ( ) 2 2 2 ˆx y b b E L k k u é ù = - ê ú ë û 2 1 = / b pk g V µ ( ) ( ) 2 2 5 x y b b E L k k u é ù = × - - ê ú ë û u [ ] ( ) [ ] ( ) I I I 6 0 3 4 r 2 2 2 2 0 i i 0 0 E A E E V V V V + = × -× - Τ Τ ( ) ( ) 3 2 3 i (2 )i x x y b b b E L k k k u é ù = - -- ê ú ë û ( ) ( ) 3 2 6 i (2 )i x x y b b b E L k k k u é ù = - -- ê ú ë û 3 2 = b pk g V µ ( ) ( ) ( ) 3 2 4 ˆ2 x x y b b b E L k k k u é ù = -- ê ú ë û ( ) [ ] ( ) I I I r ˆˆˆ0 0 0 0 0 0 x l l l Lk pk g pk g µ µ é ù = - ë û Τ Τ ( ) ( ) ( ) P P I I 1 1 = R L N N = T T T ( ) ( ) ( ) P I I I I 4 3 4 3 1 5 0 R N N N N N N = = =
. ( 27
)
According to Eq. ( 27), we can get ,

.

Finally, according to Eq. ( 28), we can get the scattering equation of the whole resonator system , (29) equals the first element of the column vector .

Based on Eq. ( 29), the dimensionless reflected energy coefficient of the vertically incident flexural wave, varying with the waveguide resonator length s/d and

4 N 4 0 0 0 0 i i i i 1 1 0 0 1 1 1 1 0 0 N - é ù ê ú = - - ê ú ê ú -- ë û ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
= = = H T Q T H T Q T H T Q T H T Q T H T Q T H T Q T Q T ( ) ( ) I r II ae ö × = ç ÷ ç ÷ è ø 1 2 Τ G G Τ T T T T T T T 1 2 3 4 5 6 T T T T T T T 7 1 2 3 4 5 6 ae ö - - - - - - = ç ÷ è ø 1 0 H H H H H H G Q Q Q Q Q Q Q ( ) T 2 1 2 4 5 0 0 0 0 0 P P P P = G ( ) ( ) T 1 s II = b b b l R R R * - é ù ê ú = × ê ú ë û 1 2 G G M T b R s M 2 b R frequency
, is shown in Fig. 3

Elastic BIC and elastic quasi-BIC physics

As shown in Fig. 3(a), the white dashed lines and green dotted lines, obtained by Eq. ( 2) and Eq.( 10), correspond to the Fano resonance and destructive interference, respectively. Their intersections accurately predict the location of the vanishing linewidths, marked by the white circle. These vanishing linewidths are ideal local states with zero leakage in the continuous spectrum of radiating waves, i.e., the ideal BICs with infinite Q factor [START_REF] Kodigala | Lasing action from photonic bound states in continuum[END_REF][START_REF] Marinica | Bound States in the continuum in photonics[END_REF]. (d) The total dimensionless energy of all scattering modes in the whole resonance system.

These ideal BICs at the vanishing linewidths are entirely isolated and have no

2 b R 2 b R w
access to the external radiation channel [START_REF] Hsu | Bound states in the continuum[END_REF][START_REF] Huang | Extreme Sound Confinement From Quasibound States in the Continuum[END_REF]. To get the TMPC, we need to tune the waveguide resonator length s to weaken the destructive interference described by Eq. ( 10). This weakening makes the resonance system leak small energy of the flexural wave (with radiative decay rate ), which is radiating into the background plate. The leaked energy of the flexural wave will balance that of the longitudinal wave (with radiative decay rate ) from the hybridization coupling of Fano resonance. This balance condition supporting trapped modes is , equivalent to the so-called critical coupling [START_REF] Cai | Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System[END_REF], however here it is fulfilled by different mode conversion physics. The critical coupling can be demonstrated by the distribution of in the complex frequency plane, based on Eq. ( 29) (the detail see Appendix B). In this way, the incident flexural waves can be entirely trapped without any backscattering ( =0), i.e., tuning into the trapped mode (quasi-BIC). In addition, the system only allows the converted longitudinal waves to leak, leading to perfect mode conversion. Here, we only analyze these TMPCs near the second Fano resonance, which are marked from point A to point I [see discrete green crosses in Fig.

3(a)

]. Among these points, the position deviation between point A and the adjacent vanishing linewidth is the largest. The reason is that the radiative energy of longitudinal wave around point A is the largest (explained in the next paragraph), which needs more weakening of the destructive interference to leak more flexural wave energy to balance.

The more energy leakage leads to a wider reflection band.

For these marked TMPCs, the total radiative decay rate of the system corresponds to the Q factor of the system , where

where m is a positive integer. According to Eq. ( 30 is, the greater its Q factor is, and the lower its is. We chose two TMPCs with different radiative decay rates , corresponding to point A and point E in Fig. 4(a), to calculate their dimensionless energy distributions of x-direction displacement in the unit structure by full wavefield simulations, as shown in the insets of Fig. 4(a). It can be clearly seen that the flexural wave energy is trapped at the top of the pillared resonator. In addition, for point E with a lower , the system captures more flexural wave energy (the multiple of the dimensionless energy enhancement arrives up to 282, relativing to incident energy). The reason is that the lower is, the more trapped flexural wave energy in the resonance system with a higher Q is, which is similar to sound confinement from acoustic quasi-BIC [START_REF] Huang | Extreme Sound Confinement From Quasibound States in the Continuum[END_REF]. We note that by optimizing the parameters to satisfy Eqs. ( 2), (10), and (30) simultaneously, the Q of the system can be further increased. Therefore, our system can support the elastic BIC with an infinite Q. According to Eqs. ( 29), (A24), and (A25), we also show the curves of energy conversion ratio (see Appendix A) for these TMPCs (from point A to point F) to verify their perfect mode conversion, as shown in Fig. 4(b).

All peak values of the curves approach one. For point F, the linewidth of the curve at its half-maximum is less than 1 Hz, which is consistent with its high Q.

3(a)

. Further, for all these TMPCs (quasi-BIC), we can continuously tune them into the infinite-Q BIC by making their radiative decay rate close to zero, depending on tuning the critical frequency of mode conversion.

The critical frequency is decided by that the radiation angle of the converted longitudinal wave equal to 90 o . The radiation angle can be obtained as based on Snell's law. From the above equation, we get the critical frequency , which is decided by the non-zero incident angle . When the frequency is lower than fc, the calculated is imaginary, that is, Theoretically, when the critical frequency intersects the Fano resonance frequency, TMPC coincides with the zero-linewidth ideal local state and supports an infinite-Q BIC.

g 2 3 bl b R k d d = l g r bl q ( ) r arcsin sin i bl b l k k q q = ( ) 2
s d

77 h d q = 76 h d q = 78 h d q = F A F 2 b R F d A F F

Experimental evidence

In the experiment, we need to make great efforts to excite the ideal plane wave through the optimized array composed of several piezoelectric patches. To simplify the excitation, we simplify the model in Fig. Therefore, we confirm the existence of the trapped modes with different Q factors.

Furthermore, based on the experimental data, we calculate that the converted energy ratios of longitudinal waves are more than 0.93 and 0.9 for peak points in Figs. 6(b) and 6(c), respectively, which confirm the perfect mode conversion of the trapped modes, neglecting small structural losses. The detailed calculation process can be seen in V from one frequency to another. Therefore, the TMPC near the vanishing linewidth shift their positions and do not disappear. For example, theoretical capture coefficients, varying the pillared resonator height h/d and frequency f, are shown in Fig. 6(f), when the waveguide length is fixed as 10 mm. The green lines are the contour lines of 0.9.

High capture coefficients over a wide range indicate that the TMPCs are stable for various geometric parameters. Further, we have printed specimen three and specimen four with the same waveguide length of 10 mm. Their pillar heights h/d are 14.2 and 16. We can find the measured trapped modes ( ) for specimens three and four at around 8516.7 Hz and 6848 Hz, respectively. These measured points are added to Fig.

6(f).

It can be seen that these points are inside the contour line of 0.9, which are consistent with the theory results. The consistency confirms that the TMPCs are robust to changes in system parameters. Similar robustness of the trapped modes to system parameters can also be found in other BIC systems (Hsu et al., 2013a;[START_REF] Marinica | Bound States in the continuum in photonics[END_REF].

Conclusions

We have theoretically predicted and experimentally demonstrated a TMPC in an elastic wave system. The system supports elastic BIC by achieving simultaneous hybrid Fano and Fabry-Pérot resonances. We prove that the TMPC supporting quasi-BIC can be tuned to the TMPC supporting infinite-Q BIC by bringing the critical frequency of mode conversion closer to the Fano resonance frequency. This work paves the way for the investigation of the intriguing physics related to the interaction between elastic

1 V » D d w xt x t u x t d x t r b ae ö ¶ ¶ + = ç ÷ ¶ ¶ è ø ae ö ¶ ¶ - = ç ÷ ¶ ¶ è ø r D ( ) 3 2 1 1 2 d E D u - = E d u b 4 d D b r = ( ) ( ) i i i , b b b b k x k x k x k x t w x t Ae Be Ce De e w - - = + + + ! ! ! ! A ! B ! C ! D ! i b k x Ae - ! i b k x Be ! b k x Ce - ! b k x De ! b k b w = 2 f w p = frequency.
The general solution of displacement for the second-order partial derivative governing equation of longitudinal waves in the background plate is:

, (A3)
where and are complex coefficients. The longitudinal wavenumber is .

Similarly, the general solutions of the displacements for the flexural wave and longitudinal wave equations in the pillared resonator are (A4)

, respectively. The superscript indicates the pillared resonator.

We make an incident positive-going flexural wave of at the left interface of the pillared resonator bottom, so the wave fields at the left interface and the right interface of the pillared resonator bottom can be expressed as follows:

,

where the symbols t and r in and denote the transmission and reflection complex coefficients of the resonator, respectively. The subscript represents an evanescent flexural wave. The subscripts b and l represent the flexural mode (or bending wave) and longitudinal mode. The subscripts bl and lb represent the conversion from flexural wave to longitudinal wave and the reverse process, respectively. The ( ) ( )

i i i , l l k x k x t u x t Pe Qe e w - = + ! ! P ! Q ! l k ( ) 2 3 6 l k d b w = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
P P P P P i i i P P P i i i , , b b b b l l k z k z k z k z t k z k z t w z t A e B e
ì = + + + ï ï í ï = + ï î ! ! ! ! ! ! ( ) P i 1 t e w × ( ) L L , w u ( ) R R , w u ( ) ( ) ( ) ( ) ( ) ( ) * * i L i L i R i R , 1 , , , t b
ì = + + ï ï = ï í = + ï ï = ï î b t b r *
subscripts bb * and b * b represent the conversion from propagating flexural wave to evanescent one and the reverse process, respectively. Therefore, the vectors and for both sides of the bottom of the pillared resonator can be written as:

. (A6)

The relationships between slope and the displacement w, between shear force

V and the displacement w, between bending moment M and the displacement w, between axial force F and the displacement u are , , , and , respectively. Among both side interfaces of the pillared resonator bottom, the boundary conditions of displacement, slope, bending moment, shear force, and axial force must be satisfied [START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF]. According to these boundary conditions, we obtain the propagation equation:

. ( A7
)
where N1 is the transfer matrix.

From Eq. (A7), we can obtain , (A8) , ( A9)

, (A10) , ( 11
) L k R k * * T R T L = 1 0 0 = 0 0 0 b b l bb b b l bb r r r t t t ì é ù ï ë û í é ù ï ë û î k k j w j ¢ = V Dw¢¢¢ = - = M Dw¢¢ ( ) 4 2 =12 F u d r b ¢ R L 1 N = k k ( ) ( ) (
)

1 1 1 1 2 2 1 2 3 1 2 3 8 8 4 +8 + 4 +8 4 4i 2i 2i +4 4 b r i c c c c c c -G -G × -G G × G G × -× + × = -× - ( ) ( ) 1 1 1 4 2 4 3 0 1 2 3 4i 4i 4 4i 2i 2i 4 4i b t d d c c c c c c --× -G × + G × +G -× -× + × + - = ( ) ( ) ( ) ( ) 4 1 2 3 2 2i 4 4i+ 2 2i 2i 2i 2 2i 4 4i bl bl t r k c k c k k c k k c k - × = = - × + + -× + + + × + -- ( ) ( ) ( ) ( ) ( ) * 3 1 2 3 4 2 3 1 2 3 8 4 4 4 4 4 8 8 8 (1 i) 8 2 2 4 2 2i 4 8 bb r i i i i i i i c c k c c c c k c k c G + + -+ + + +G - é ù = ë û - + -+ --- + é ù ë û sin = sin 2 2i cos b b b k h k h k h k k k k G + - ( ) ( ) ( ) ( ) 4 cos = sin 2 2i cos b b b k h k h k h k k k k k G + - ( ) ( ) 1 cos cosh b b k h k h c = ( ) ( ) 2 cos sinh b b k h k h c = ( ) ( ) 3 sin cosh b b k h k h c = ( ) ( ) 4 sin sinh b b k h k h c = i 1 t e w × ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * i L i L i R i R , , 1+
ì = + ï ï = × ï í = + ï ï = ï î ! ! ! ! L k R k * * R T T L = 0 0 1 = 0 0 0 lb l lb lb l lb r r r t t t ì é ù ï ë û í é ù ï ë û î k k ( ) ( ) ( ) ( ) ( ) 1 3 2 1 2 3 2 2 i 2 2 i + 4 4i 2 2 i 2i 2 2i 2 i 4 4i l r kc k k c k k c k k c k k c k k c k - - + - + = + - + + - + + + + -- ( ) ( ) ( ) ( ) 4 3 1 2 1 i 4 4i 2 2 2 i 2 2 2 i 4 4i lb a a a a r c c c c * + -+ -- + + - -- = + ( ) ( ) 1
b + 2 b + 1 l + 2 l + 1 b -2 b - 1 l - 2 l - P = b b lb lb b b lb lb bl bl l l bl bl l l t r t r r t r t t r t r r t r t - + + - é ù ê ú - ê ú = × ê ú - ê ú - ë û S S M S [ ] 2 1 2 1 , , , T b b l l - - - - - = S [ ] + 1+ 2+ 1+ 2+
, , , Therefore, the scattering matrix including the phase origins [START_REF] Botten | Photonic band structure calculations using scattering matrices[END_REF] and evanescent flexural modes can be obtained as ,

(A23)

For the flexural wave with the amplitude of , its energy can be expressed as .

(A24)

For the longitudinal wave with the amplitude of , its energy can be expressed as .

(A25)

For the incident flexural wave of , the dimensionless energy of the reflected and transmitted flexural waves can be expressed as and , respectively.

According to Eqs. (A24) and (A25), the dimensionless energy of the reflected and When the resonant structure of the system is a side-coupled pillared resonator in Fig. 1(a), the hybrid coupling will induce the mode conversion to leak longitudinal wave. The leakage energy of the flexural wave is similar to dissipative loss in the optical resonance system. But the ''dissipative loss'' is from the hybridization coupling in our study. Fig. 7(a) shows the complex frequency graphs, corresponding to the system with TMPC marked by point C in Fig. 3(a). "Zero" point and "Pole" point have shifted to the left, and "Zero" point intersects the real frequency axis. The intersection is the socalled critical coupling, however here it is fulfilled by different mode conversion physics. The flexural wave is completely trapped by the system. For the critical coupling [START_REF] Cai | Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System[END_REF], the energy leakage of the flexural wave will balance approximate the sum of incident and reflection propagating waves. An ensemble average with 20 samples is used at every measurement point to ensure the signal quality. Using the measurement mode "FFT" of PSV-500, the complex velocities of flexural waves in points 1 and 2 can be measured and marked as and , respectively. They are expressed as and , respectively. and are the distance marked in Fig. 9.

- é ù ê ú - ê ú ê ú - = ê ú - ê ú ê ú - - ê ú - - ê ú ë û M ! P P * = ¡ ¡ M M ! î 2 î 2 ˆ2 ˆ2 ˆ2
( ) Imag( ) f f f ¢ ¢ ¢ = + Real( ) f ¢ Imag( ) f ¢ f ¢ b R Real( ) f ¢ Imag( ) f ¢ 10 log b R
is the flexural wave number. The transfer function H12 of the total wavefield in the two points can be obtained from the measured complex velocities of v

  (a). These waveguide resonators are connected to side-coupled pillared resonators with the height h. The thicknesses d of the waveguide resonator and pillared resonator are in deep subwavelength scale relative to the wavelength in the whole considered frequency range. Only the fundamental modes of flexural waves and longitudinal waves are involved.

Fig. 1 . 1 .

 11 Fig. 1. (a) A Fano-resonance-supporting elastic wave model: periodic waveguide with the length s

  wave, based on the scattering coefficients in Eq. (1) (see details in Appendix A) and the full wavefield simulations (see details in Appendix H), respectively. This consistency between Figs. 2(a) and 2(b) verifies the correctness of the analytical model. Another important verification is the power flow balance, i.e., the total energy of all scattering modes is unit (grey dot line). We note that the reflection energy curve has a clear Fano profile. The Fano profile is a typical feature of the Fano resonance (Fano, 1961; Rupin et al., 2014), which is the result of the interference between the flexural resonances of the pillared resonator and the incident flexural waves. The Fano resonance frequency ω0 [marked by yellow dashed lines in Fig. 2(a)], dominated by the flexural resonance, approximately satisfies the transcendental equation (Colquitt et al., 2017; Graff, constant, , and D is the bending rigidity.

Fig. 2 .=

 2 Fig. 2. (a) and (b) The dimensionless scattering energy curves for the incident flexural wave in the

  the second Fano resonance frequency. and are the transmission vectors in the left and right of the pillared resonator, respectively, where the subscript represents the evanescent flexural wave in the near field. According to Eqs. (1) and (3), the band structure can matrix, including the phase origins and evanescent flexural modes (see Appendix A). Based on Eq. (4), Fig. 2(c) displays the band structure (blue lines), which is very consistent with the simulated one in Fig. 2(d). The

Fig. 1 ,

 1 Fig. 1, reshapes the interference of the incident flexural waves. The length s is greater

  mode and the incoming propagation flexural wave at the right and left ports, respectively. and represent coupling coefficients between the localized mode and the incoming longitudinal wave at the right and left ports, respectively. The scatting matrix describes the incoming and outgoing waves in the ports couple through a direct pathway, i.e., ignoring the resonance-assisted coupling.

  where A0 is the amplitude of the incident flexural wave, , , and are the complex coefficients of the reflected propagating flexural wave, reflected evanescent flexural wave, and reflected longitudinal wave, respectively. and are y-component wave vectors of propagating flexural waves and longitudinal waves, respectively. According to the conservation of parallel wave vectors, we can get , where is the incident angle. and are x-component wave vectors of propagating and evanescent flexural waves, respectively. is x-component wave vector of longitudinal waves. The coefficient vector of the reflected scattering field in the region (I) can be defined as . The hybrid resonant system consists of the pillared and waveguide resonators. Only the fundamental mode needs to be taken into account since the widths p and thicknesses d of the waveguide and pillar are in deep subwavelength scale. The coefficient vector of the wavefield at the left interface of the waveguide resonator (the right of the dividing line ) can be defined as , where and correspond to the complex coefficients of the positive-going propagating and evanescent flexural waves in the waveguide resonator. and correspond to the complex coefficients of the negative-going propagating and evanescent flexural waves. and correspond to the complex coefficients of the positive-going and negative-going propagating longitudinal waves. In the x-direction of Fig. 1(a), the slope , bending moment M, shear force V, and axial force F in the two-dimension plate model of the region (I) are expressed as

  By applying the continuous boundary condition of x-components of the flexural wave displacement at the left interface of waveguide x=0 and integrating along ysame way, applying the continuous boundary condition of the slop at the interface of x=0, we can get .

  are the subunit period and Poisson's ratio, respectively. In the same way, by applying the continuous boundary condition of shear force at the interface, we can get , the same way, by applying the continuous boundary condition of axial force at the interface, we can get . (22) According to Eq. (A7), we obtain the coefficient vector of the wavefield in the right interface of the pillared resonance . (23) Since bending moment, shear force, and axial force are zero at the right free boundary of the waveguide resonant, and we can get ,

  Fig. 3(a), as shown in Fig. 3(c). According to Eqs. (29), (A24), and (A25), the total

  Fig. 3. (a) The analytical dimensionless reflected energy of the incident flexural wave in the

  second Fano resonance frequency. Therefore, decides the Q factor.The smaller the value of is, the greater the Q factor is. When one round-trip of the converted longitudinal wave between the right waveguide edge and the left one (the interface ) satisfies Fabry-Pérot resonance condition, approaches zero due to the resonance, which induces the TMPC with infinite Q factor. This Fabry-Pérot resonance condition is obtained by enforcing field relations at the waveguide resonator, and accordingly, ,

  ), we can obtain Fabry-Pérot resonance frequency as . We calculate the first resonance frequency (m = 1) varying with waveguide lengths s, as shown by the red dashed line in Fig. 3(a). The red line approximately crosses over the vanishing linewidth near point F, which indicates that near the ideal local state, approaches zero due to the Fabry-Pérot resonance. Therefore, the flexural wave energy leakage from the trapped mode, balanced with the longitudinal wave energy leakage, also approaches zero. In this way, the position of TMPC almost overlaps with that of the ideal local state, approaching the BIC. At point F, the system with TMPC has a very narrow reflection band, which cannot be resolved in the scale of Fig. 3(a). Based on in the analytical model, we obtain the Q factor and of these TMPCs to quantitatively analyze the resonance characteristics (the detailed calculation can be found in Appendix C). It can be seen from Fig. 4(a) that the Q factor at point F exceeds 8000 (indeed approaches infinite-Q BIC). The closer to point F the other TMPC

  Fig. 4. (a) Bar charts represent total radiative decay rates of these TMPCs from point A to point

  disappears. For the TMPC marked by point B in Fig. 3(a), the dimensionless incident angle is changed as , as an example. The interface impedance varying with frequency is shown in Fig. 5(a). At the critical frequency of = 8384 Hz marked by the light blue dashed lines, is infinite, which will suppress longitudinal wave radiation. In addition, the corresponding for the obliquely incident flexural wave, calculated by Eq. (29), is shown in Fig. 5(b). The critical frequency closes to the Fano resonance frequency (marked by white dashed lines), the large reduces of the TMPC (marked by green crosses). The decreased leads to a narrower reflection band of the TMPC, compared with that of the TMPC marked by point B in Fig. 3(a).

Fig. 5 .

 5 Fig. 5. (a) The interface impedance of the interface for the converted longitudinal wave varying

  1(a) by a strip-like model (beam), as shown in the inset of Fig.6(a). Then, we can use a single piezoelectric patch attached on the strip-like model to excite the ideal incident wave, which significantly increases the accuracy of the experiment. We have proved that for vertically incident waves, the results of the simplified model are consistent with those of the original model by theoretical and simulation methods (see details in Appendix D). To confirm our theory, based on the simplified models, we print specimen one and specimen two with different waveguide lengths (10 and 29 mm) to verify different-Q-factor TMPCs based on quasi-BIC. These TMPCs correspond to point A and point C in Fig. 3(a).

FIG. 6

 6 FIG. 6. (a) Experimental set-up. The specimen and the clamped test specimen are illustrated at the

  a quarter wavelength in the second Fano resonance frequency. Due to the subwavelength period, the evanescent flexural modes need to be considered in the

  Fig. 7(a). "Zero" and "Pole" are two extreme points referring to the lowest and the

Fig. 7 .

 7 Fig. 7. (a) The complex frequency graphs illustrating the distribution of in the complex

  longitudinal wave from hybridization coupling of Fano resonance. theory and simulated results show that the capture coefficients of the flexural waves for the 1D simplified models and the 2D ones are consistent, as shown in Figs. 8(b) and 8(c). We can print specimens to verify supporting the trapped modes corresponding to point A and point C in Fig.3(a). The 3D-printer Ultimaker 3, with a manufacturing precision of 0.06 mm, is adopted. The corresponding geometric parameters of the two specimens are shown in TABLE1. The left pillared resonator and the right one are symmetrical about the specimen center. The purpose of designing the left resonator is to reconvert the longitudinal wave into the flexural wave, which will be described in detail later.

Fig. 8 .

 8 Fig. 8. (a) 1D simplified specimen marked by geometric parameter symbol. (b) The analytical and

Fig. 9 .

 9 Fig. 9. Test set-up.

  of flexural wave energy can be obtained by . The amplitude of the longitudinal wave cannot be measured directly by the single scanning head of our PSV-500. But we can measure the capture coefficient of to characterize the energy capture of the flexural wave for the trapped modes.

Fig

  Fig. 10. (a) Specimen 1 with blue-tacks on the middle, the left end, and the right end. (b) Specimen

TABLE 1 .

 1 The geometric parameters of the specimen. All parameters are in units of mm.

	d	h	sw	st	s (Specimen 1)	s (Specimen 2)
	1	15	8	160	10	29

Acknowledgments

The authors acknowledge Professor Yong Li for fruitful discussions. This work is supported by the Air Force Office of Scientific Research under award number FA9550-18-1-7021, la Région Grand Est and Institut Carnot ICEEL. It is also supported by the National Natural Science Foundation of China (Grant No.11972296), the 111 Project (No. BP0719007), and the Fundamental Research Funds for the Central Universities (No. 310201901A005).

Appendix G.

To intuitively show the TMPC, we obtain the dynamic full wavefield in specimen one at the central frequency of 7580 Hz by the ''time'' measurement mode of PSV 500. Fig. 6(d) shows a snapshot of Supplementary Video one at 40.232 ms. In the right test area, the wavefield is almost a traveling wave, as observed in the Supplementary Video 1. The existence of the traveling wave means flexural waves are completely captured by the right resonance system, that is, there is almost no reflection. The absorption area in the middle can completely absorb the flexural wave (relevant confirmation can be found in Appendix G), but we still clearly see the flexural wave field in the left test area.

The abnormal phenomenon is caused by the mode conversion in the resonance system.

For comparison, we test the dynamic full wavefield in specimen one at 6100 Hz (away from the central frequency). Figs. 6(e) shows a snapshot of Supplementary Video two.

In the right test area, the incident wave is a standing wave. In the left test area, the amplitude of the flexural wave is close to zero, as observed in the Supplementary Video 2. The wave phenomenon in Supplementary Video two means that flexural waves are not captured by the right resonance system, and mode conversion does not happen either.

We note a subtle but essential difference between the TMPC in our system and the conventional trapped mode. The latter disappears when the parameters of the system slightly deviate from the designed structure, making it very difficult to observe experimentally (Hsu et al., 2013a;[START_REF] Hsu | Bound states in the continuum[END_REF]. In our theory, based on Eqs. ( 2) and ( 10), when the parameters of the system change, the vanishing linewidth jumps wave energies with different polarization planes, based on elastic BICs. The elastic BIC can obtain an unprecedented high quality factor in the elastic system, which may facilitate various applications, e.g., high-sensitivity elastic wave sensor and highresolution elastic wave filters.

Appendix A. Analytical expression of the scattering coefficient for the pillared resonator

First, we consider an incident wave perpendicular to the model without the right boundary, as shown in Fig. 1 Poisson's ratio, respectively. is the propagation constant, .

The general solution of displacement for the fourth-order partial derivative governing equation of the flexural wave in the background plate is:

where , , , and are complex coefficients. and correspond to the positive-going and negative-going propagating flexural waves, whereas and correspond to the positive-going and negative-going evanescent flexural waves. The flexural wavenumber is , where is the circular

In the same way, we make an incident positive-going longitudinal wave of at the left interface of the pillared resonator bottom, so the wave fields at the left interface and the right interface of the pillared resonator bottom can be expressed as follows:

, (A13)

The vectors and for both sides of the pillared resonator bottom can be rewritten as:

. (A14)

According to Eq. (A7), we can obtain

.

(A19)

In the same way, we make an incident evanescent flexural wave of at the left interface of the pillared resonator bottom, and we can obtain the coefficients of , , ,

As shown in Fig. 1 is the scattering matrix.

We note that the scattering matrix can also be derived directly from the transfer matrix.

The negative sign in Eq. ( A20) is to make the scattering matrix and transfer matrix keep consistent in mathematical form.

Based on the analytical model, we can analyze its band structure to intuitively show the hybridization coupling between flexural waves and longitudinal waves from the Fano resonance. The unit structure is shown in the illustration of Fig. 2(c). is a

factor and of these TMPCs

Since the energy leaks out from the resonator, each TMPC can be characterized by "Pole" point in the complex frequency graphs, as shown in Figs. 7(a) and 7(b). The frequency of the Pole is , with the position at the real axis . The TMPC has the decay rate [START_REF] Krasnok | Anomalies in light scattering[END_REF], which determines the linewidth of TMPC at its half-maximum. The Pole is close to the real frequency axis, the Q of the TMPC can be defined as , (C1)

In our system, the decay rate includes both flexural wave radiative and converted longitudinal wave radiative (similar to dissipation) [START_REF] Bliokh | Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media[END_REF].

In Fig. 5(f), for the TMPCs with a steep rise from 0 to 1, their Q factor can be calculated by , where is the central frequency of the resonance peak, and Δf is the full width at half maximum for the peak.

Appendix D. Simplified specimens and their manufacture

For the theoretical model in Fig. 1(a), the filling ratio of a slot in a subunit is a small value of 1/17, and the periodic L is 85 mm. When considering an incident wave perpendicular to the model, we can simplify the two-dimensional (2D)

periodic structure to a one-dimensional (1D) strip-like model shown in Fig. 8(a). The

Experimental set-up and measurements

As shown in Fig. 9, a 6-mm-diameter circular PZT patch, with a distance of from the right pillared resonator, is bonded on the plate surface. The PZT patch is driven by a signal generator (Tektronix AFG3022C). The blue-tack layer is bonded in the middle of the strip-like specimen to absorb the left propagating flexural wave excited by the PZT patch and the reflected flexural wave from the right resonance structures.

The left pillared and waveguide resonators can reconvert all the longitudinal waves, which are from the mode conversion in the right resonators, into flexural waves due to wave reciprocity. Then the reconverted flexural waves propagating to the right are absorbed by the middle blue-tack. In this way, the blue-tack can be equivalent to a nonreflection boundary of flexural waves.

The specimen surface is held perpendicular to the laser beam from PSV-500 scanning laser Doppler vibrometer, through a C-clamp and supporting rod applied to the lower edge, as shown in Fig. 9. The PSV-500 scanning head records the out-ofplane complex velocity in point 1 and point 2, marked in Fig. 9. The distance between the two points is less than an eighth wavelength. Simultaneously, their locations and the boundary of the resonance structure meet the far-field assumption [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF].

The assumption ensures that the measurement wavefield of the two points can 53d

Appendix F. Material parameter test of 3D-printed specimens

We note that the wavenumber in Eq. (E1) needs to be calculated by material parameters of 3D-printed specimens. All theories are also based on these material parameters. In the following, we will test the parameters. Although related material parameters of PLA, adopt in our 3D printer, have been reported in some literature.

However, the material properties of practical printed specimens could be different due to various printing processes. To make the experiment and theory result more matching, we set the fixed printing mode, for example, infill density and infill pattern, to print all specimens. Then we print a beam specimen ( ) without pillared resonators. We first measure the mass and volume of the beam specimen to obtain its


Yi Zeng

Shu-Yan Zhang

Hong-Tao Zhou

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

Liyun Cao

Yifan Zhu

Qiu-Jiao Du

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

Yue-Sheng Wang

Broadband inverted T-shaped seismic metamaterial

Keywords: seismic metamaterials, surface waves, inverted T-shaped, first bandgap, relative bandwidth, large-scale field experiments

Seismic metamaterials (SMs) are expected to assist or replace traditional isolation systems owing to their strong attenuation of seismic waves. In this work, a one-dimensional inverted T-shaped SM (1D ITSM) composed of arrays of inverted T-shaped structures on a half space is proposed, which have an ultra-wide first bandgap (FBG) from 6.7 to 17.2 Hz. We find that the FBG is composed of two parts; part 1 with surface evanescent waves from 6.7 to 11.0 Hz and part 2 with no surface modes from 11.0 to 17.2 Hz by using the complex band structures. The propagation of seismic surface waves in the 1D ITSM is different in these two frequency ranges of the FBG. In part 1, the seismic surface waves are significantly attenuated in the 1D ITSM because of the surface evanescent waves, while in part 2, the surface waves are converted into bulk waves because surface waves cannot exist in the ITSM. Finally, the ultra-wide FBG is verified by using a kind of the twodimensional ITSM in large-scale field experiments.

Introduction

Metamaterials [START_REF] Ding | Metamaterial with simultaneously negative bulk modulus and mass density[END_REF][START_REF] Liu | An elastic metamaterial with simultaneously negative mass density and bulk modulus[END_REF][START_REF] Deymier | Acoustic metamaterials and phononic crystals[END_REF][START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF][START_REF] Wickeler | Novel origami-inspired metamaterials: Design, mechanical testing and finite element modelling[END_REF][START_REF] Kollmann | Deep learning for topology optimization of 2D metamaterials[END_REF] are a kind of composite material composed of periodically arranged artificial unit cells. One of the most significant characteristics of metamaterials is the bandgap (BG). [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[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] Gao | Study of bandgap property of a bilayer membrane-type metamaterial applied on a thin plate[END_REF][START_REF] Yin | Bandgap characteristics of a tensegrity metamaterial chain with defects[END_REF] In the frequency range of the BG, waves cannot propagate in the metamaterials. This characteristic is used by physicists to control the propagation of seismic waves, thereby preventing casualties and property damage caused by the destruction of buildings. This kind of metamaterial used to control seismic waves is called seismic metamaterial (SM). [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Finocchio | Seismic metamaterials based on isochronous mechanical oscillators[END_REF][START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Aravantinos-Zafiris | Large scale phononic metamaterials for seismic isolation[END_REF][START_REF] Wang | Tunable and Active Phononic Crystals and Metamaterials[END_REF] The SM can be used to protect critical infrastructures and most buildings located in the mid-and far-fields of the earthquake source. It is worth noting that the natural frequency of these buildings is below 20 Hz. [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Fema | Designing for Earthquakes[END_REF] Therefore, the realization of the SM with ultra-low BGs is an important step to push the SM to practical application.

The conjecture of the SM was proposed after the experiment of the phononic crystal in marble in 1999. [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF] The attenuation of the elastic surface waves in the two-dimensional crystals identified the existence of the bandgaps. However, a large amount of research on the SMs began in 2014. In this year, cylindrical holes arranged periodically in soil were experimentally proved to attenuate seismic waves at 50 Hz. [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF] Previous works have demonstrated the propagation of seismic waves generated by the probe in the phononic crystals. Results show that it was consistent with the numerical simulation based on the approximate plate model. These large-scale experiments on the SM have

shown that it has unlimited potential in controlling seismic waves. Later, researchers changed the shape of the holes, and filled different materials into the holes to obtain the SMs with a wide BG. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF] However, the structure of this kind of SM based on the Bragg scattering mechanism is very huge to control ultra-low-frequencies seismic waves, which makes it difficult for practical application.

Fortunately, the SMs based on local resonance can solve this problem, which can use small structures to control long waves. [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Sheng | Locally resonant sonic materials[END_REF][START_REF] Bigoni | Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization[END_REF][START_REF] Achaoui | Experimental observation of locallyresonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars[END_REF] Therefore, most of these SMs are also called subwavelength SMs. [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] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Colquitt | Seismic metasurfaces: Subwavelength resonators and Rayleigh wave interaction[END_REF] The subwavelength SMs can be roughly divided into two categories. The first kind of subwavelength SMs can also be called the underground barrier of locally resonant metamaterial. [START_REF] Aravantinos-Zafiris | Large scale phononic metamaterials for seismic isolation[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures[END_REF][START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF][START_REF] Xu | An underground barrier of locally resonant metamaterial to attenuate surface elastic waves in solids[END_REF] This kind of underground SMs with small scale is based on resonance characteristics to attenuate long-wavelength seismic waves. [START_REF] Liu | Locally resonant sonic materials[END_REF] However, their BGs are generally narrow. On the other hand, these resonant unit cells should be placed where the elastic waves propagate to better attenuate the seismic waves. For example, the depth of these SMs burial into the ground should be roughly equal to twice the wavelength of the seismic surface wave to better attenuate them [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF]. In this way, although the dimensions of every unit cells are very small, the depth of burial is isolating. The second kind of subwavelength SMs are usually composed of periodic pillar-like structure and substrate, which can be called the pillar-like SM (PSM). A major characteristic of the PSM is that it mainly attenuates the propagation of seismic surface waves, [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF] especially in the low-frequency range of the first BG (FBG), and even if the PSM only has three rows. [START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] The PSM is found because seismic surface waves in a part of the frequency range cannot pass through forest. [START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF] When the seismic waves interact with a forest, some frequencies are strongly attenuated due to the resonance of trees. When trees are considered as steel pillars or pillars with different shapes, the attenuation of surface waves in the SMs still exists, and better results can be obtained. So, I-shaped, [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] fractal structure [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF] and Matrayoshka-like [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] PSMs are proposed to obtain a wider complete BG in the sound cone. However, the relative bandwidth of the FBG is currently too small to meet the needs of practical applications.

In this paper, a one-dimensional inverted T-shaped SM (1D ITSM) composed of pillars and plates with ultra-wide FBG is proposed. The ITSM is just a simple modification based on the PSM: a plate is added at the bottom of each pillar. The FBGs of the two different SMs are compared. The effects of the ITSM's geometric and material parameters on the FBG are discussed. The complex band structure of the ITSM for surface waves is calculated and compared with that of the PSM. The propagation properties of seismic surface waves in the ITSM are also given. Finally, we propose a kind of the two-dimensional (2D) ITSM with the ultra-wide FBG.

Model of the 1D ITSM

In this section, the classical PSM and the proposed 1D ITSM are compared. As shown in Fig. 1, the PSM and the ITSM have the same steel pillar, i.e., l2 = l, w2 = w. But the bottom of every pillar in the ITSM is clamped by a steel plate marked as gray part in Fig. 1(b). The geometric and material parameters of the unit cells are given in Tables 1 and2, respectively. In numerical simulation, the left and right boundaries of the substrate of the unit cell are set as Bloch periodic boundaries, and the bottom of the substrate is set as a fixed boundary condition to avoid surface modes on the bottom. [START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] It is worth noting that the calculation results will converge well at ultra-low frequency when the geometric parameter H is enough large. In this paper, the COMSOL Multiphysics is used to calculate the band structures and the propagation of Rayleigh waves in the SMs.

(a) (b) Although the FBG of the PSM is very low, it is extremely narrow (relative bandwidth is about 0.1) and thus unsuitable for seismic shielding. In contrast, although the minimum value of the FBG of the ITSM is 6.7 Hz, its relative bandwidth can reach an astonishing value as 0.88. We believe that this way of abandoning the BG with ultra-low frequency in exchange for the ultra-wide BG is desirable in the practical application of SMs. The vibration modes of the PSM and the ITSM at marked points are illustrated in Figs. 2 (c -d).

The marked points A1 and B1 (A2 and B2) are the lower and upper boundaries of the FBG of the PSM (ITSM), respectively. The direction of the arrows represents the direction of the particle movement, and the color indicates the normalized total displacement. It is easy to find that the vibration modes of the lower, upper boundaries of the two FBGs (points A1 and A2, points B1 and B2) are similar. This indicates that the generation mechanism of these two FBGs is also similar. On the first bands of two SMs, i.e., at points A1 and A2, the maximum displacement appears on the top of the steel structure. On the second bands of two SMs, i.e., at points B1 and B2, the displacement is on the entire steel structure and on the surface of substrate. The energy is evenly distributed on the entire structure. The pillar or the inverted T-shaped structure (ITS) can be considered as a whole this time. This shows that the structure of the ITS leads to a qualitative leap of the FBG. It is worth noting that although the second bandgap in Fig. 2

Effects of geometric parameters

After comparing the band structures of the PSM and the ITSM, it is obvious that the appearance of the plate at the bottom of the pillar makes a huge change in the frequency range and relative bandwidth of the FBG. In this section, the effects of geometric parameters of the plate on the FBG are calculated, when the pillar is unchanged. In addition, the effect of the height of the pillar on the FBG is calculated to obtain the optimal geometric parameters of the ITSM, when the total volume of the pillar (i.e., l2 × w2) is constant. increase from 0.6a to 1.7a, the upper and lower boundaries of the FBG continue to decline, and its relative bandwidth continues to increase. But when the pillar height is greater than 1.8a, the upper boundary of the FBG drops rapidly, leading to the decrease of the relative bandwidth. Therefore, when the pillar height is about 1.7a, the relative bandwidth of the FBG of the ITSM is the largest, about 0.93.

The results show that the geometric parameters of the plate and pillar in the ITS have an obvious effect on the relative bandwidth of the FBG. With the appearance of the plate and the increase in its length, although the center frequency of the FBG of the ITSM continues to increase, its relative bandwidth increases linearly. However, there is an optimal value for the thickness of the steel plate, about 0.05a. When the height of the pillar continues to increase within a certain range, the center frequency of the FBG continues to decrease, but the relative bandwidth continues to increase.

However, when the pillar is too high and its width is too small (l2 × w2 is constant), the decrease in the flexural resonance frequency of the pillar causes the relative bandwidth to decrease continuously.

Effects of material parameters

In this section, the influence of material parameters of the ITS on the FBG is analyzed. Material parameters of the ITS are swept only for qualitative comparison. [START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF] Only one parameter is changed at a time, and other parameters are consistent with those in Fig. 1(b).

The effect of mass density of the ITS (ρITS) on the FBG is shown in Fig. 4

Complex band structure

In order to deeply explore the wide FBG of the ITSM, the complex band structures shown in Fig. 5(a), are calculated by using the PDE module of the COMSOL. The complex band structures of seismic metamaterials are considered when the wave vector is complex. The negative wave vector means that the amplitude of elastic wave is attenuated in the space, i.e., there is a surface evanescent wave. [START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF][START_REF] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF] In order to draw only the surface modes in the complex band structure, the following parameter ξ is introduced, [START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF]

ξ = ∫ |𝒖|𝑑𝑠 !(#$) / ∫ |𝒖|𝑑𝑠 !(&) , (1)
that is, in the displacement fields of the ITSM, the ratio of the integral of the displacement (u) in the range of depth of twice the Rayleigh wave wavelength (2λ) below the surface to the integral of the displacement over the entire depth (H) of the substrate. The parameter ξ can be used to filter all surface modes from the complex band structure by using the displacement fields calculated with the finite element method (COMSOL Multiphysics). In this paper, the surface mode is defined as ξ > 0.9. This is because when the parameter ξ > 0.9 in one mode, i.e., more than 90% of the energy in a unit cell all exists near the surface, this mode can be deemed as a surface mode. In Fig. 5(a), the color bar on the far right represents the value of ξ. The black dotted line is the boundary of the sound cone.

In the complex band structure of the ITSM, the FBG can be divided into two completely different parts: part 1 with surface evanescent waves (light gray); part 2 with no surface mode (heavy gray).

From the real part of the complex band structure of the ITSM, there is a surface band in the FBG in part 1, with a frequency range from 6.7 Hz to 11.0 Hz. This band partially overlaps with the first band at about 6.7 Hz. In the imaginary part of the surface band in part 1 are all greater than zero.

They are surface evanescent waves in this frequency range and cannot be displayed in the classical band structure. The vibration mode at the end of the band is plotted. It can be found that most of the vibration exists in the substrate and decreases with increasing depth. We speculate that as the frequency increases, most of the vibration is transferred to the substrate thus the surface mode disappears. In part 2 with the frequency range of 11.0 Hz to 17.2 Hz, there is no surface band in the complex band structure of ITSM. Similar phenomenon is currently only found in the inverse dispersion SMs (Notes A and B in Supplementary Information (SI)). We speculate that in this frequency range, the steel structure can be equivalent to a layer of uniform medium on the substrate due to its longitudinal resonance (vibration perpendicular to the surface of the substrate). The layered media structure composed of this medium and the substrate has an inverse dispersion BG.

In this frequency range of the BG, the surface waves will be converted into bulk waves (Note B of SI).

(a) (b) Similarly, as shown in Fig. 5(b), as a comparison, the complex band structure of the FBG of the PSM shown in Fig. 1(a) is calculated. It is easy to find that the situation in these two FBGs is almost identical. So, we believe that the formation mechanism of these two FBGs is almost the same.

Frequency domain analysis

The transmission spectrum is calculated in this section. The finite model used for calculation is shown in Fig. 6(a), and consisting of two parts: the substrate represented by the yellow part, and the periodic ITSs represented by the gray part and the green part. The material and geometric parameters of the ITS are consistent with those in Fig. 1(b). The boundaries of the left, right, and bottom of the substrate are set as low reflection boundary condition to eliminate the reflection of elastic waves. [START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF] The number of rows of the ITSM is 10. Point A is set as the source to generate Rayleigh waves.

The polarization direction of the source is the xz direction. [START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[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] Accordingly, the data is collected at point B from the right side of the ITSM. Similarly, in order to obtain an accurate transmission spectrum at a sufficiently low-frequency range, the height of the substrate is set as H = 300a. The acceleration at point B is collected with and without the ITSM. When there is the ITSM, the acceleration at point B is A1. When there is no ITSM, the acceleration at point B is A0. The transmission is defined as T = 20 × log10 (A1 / A0).

As shown in Fig. 6(b), the ITSM has a very significant attenuation effect on the Rayleigh waves in the frequency range of the FBG. There is a huge attenuation zone from 6.7 Hz to 17.2 Hz, consistent with the range of the FBG in Fig. 2(b). This result shows that the ITSM can drastically attenuate the Rayleigh waves at a sufficiently low frequency and in a wide range. In addition, it is easy to find that in the FBG's two different parts identified in complex band structure (Fig. 5 As shown in Fig. 6(c), the distribution of displacement amplitude in the ITSM is given at two different but representative frequencies in the FBG: 7.0 Hz and 12.0 Hz. In the frequency range of part 1, for example, at 7.0 Hz, the violent resonance of the ITSM causes the surface waves to be significantly attenuated and almost unable to propagate to the third unit cell. In the frequency range of part 2, for example, at 12.0 Hz, after the surface waves enter the ITSM, we can observe that the surface waves are converted into bulk waves. These two results are easily reminiscent of the literature, [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF] where surface waves are incident from two ends of the metawedge. When the surface wave is incident from the short edge, the surface waves first encounter part 1 of the FBG of the shorter PSM. So, the surface wave is quickly attenuated because of the evanescent waves. When the surface wave is incident from the taller end, the surface waves first encounter part 2 of the FBG of the taller PSM. So, the surface waves are converted into bulk waves because of the inversedispersion effect (Note A in SI). This also indirectly proves that the formation mechanism of the FBG of the PSM and the ITSM is almost the same. So, in this work, the bandgap from 6.7 to 11 Hz is used to stop the propagation of the Rayleigh waves. And the bandgap from 11 to 17.2 Hz is used to convert Rayleigh waves into bulk waves. In conclusion, the ultra-wide bandgap from 6.7 to 17.2

Hz is useful to attenuate Rayleigh waves to protect the aimed buildings. Moreover, the propagation of the Rayleigh waves is studied when the bottoms of the steel structures are buried in the substrate.

The result is shown in Note C of SI.

The 2D ITSM and large-scale field experiments

For the SM, a 1D periodic model is a general choice for exploring mechanism of the BGs. [START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] This is because it is a common method to study the SMs for surface waves, especially for Rayleigh waves, whose displacement components on the surface only exist in the x and z directions [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF]. These two directions are both considered in the 1D simulations. Therefore, the 1D simulation is the basic study of the elastic metamaterials for surface waves. We can easily investigate the bandgap characteristic and the mechanism of bandgaps to easily design the 2D elastic metamaterials.

However, in practical applications, the design of the 2D periodic SM is more important and necessary. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF] In addition, due to some uncontrollable factors in actual situations, such as the difficulty of achieving perfect continuity between different materials, large-scale field experiments are still needed to verify the SMs with ultra-wide and low-frequency bandgaps for surface waves. Therefore, based on the design concept and the mechanism of the above-mentioned 1D ITSM, a kind of 2D ITSM and a large-scale field experiments are designed for isolating Rayleigh waves in this section. It is worth noting that due to the experimental equipment and test site, the 2D ITSM with smaller dimensions than the 1D ITSM are chosen. However, the operating frequency and frequency range of the 2D ITSM are lower and much wider than the SMs that has been experimentally verified. [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF] 3.1 Designs of the 2D ITSM and experimental setup Fig. 7(a) shows the unit cell of the 2D ITSM designed according to the structure of the 1D ITSM.

In order to facilitate field experiments and attenuate seismic surface waves in the lower frequency range, the lattice constant a1 of the unit cell is set as 0.25m. And the unit cell is composed of a thumbtack-shaped structure (purple part) and a soil substrate (gray part). Among them, the thumbtack-shaped structure (TS) is composed of a steel tube and a steel plate. The geometric parameters of the unit cell are shown in Table 3. As shown in Fig. 7 [START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF], where AA and AB are the accelerations in the vertical direction of points A and B, respectively. The material parameters of the tests is referenced in these literatures [START_REF] Krödel | Wide band-gap seismic metastructures[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF],

which is same as in Tables 2. It is worth noting that in the tests, the TS is simply placed on the substrate. But in the simulation mentioned above, the TSs and soil are perfectly continuous.

Results of the 2D ITSM and field experiments

The band structure of the 2D ITSM only in the ГX direction is calculated to facilitate comparison with the experiments, as shown in Fig. 8(a). [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Badreddine Assouar | Dispersion curves of surface acoustic waves in a twodimensional phononic crystal[END_REF] The sound cone is clearly surrounded by . This is consistent with the results in Fig. 6(c): the attenuation in the bandgap generated by the local resonance is stronger than that by inverse-dispersion characteristics when the rows of the unit cell are fewer (5 rows in this work). Regarding the application, the ITSM is proposed for a metropolis with different buildings. The ITSM with an ultra-wide bandgap can be used around the metropolis to strongly attenuate seismic surface waves to protect almost all the buildings in the metropolis. If the number of the rows of the ITSM is large, for example, the ITSM with 30 rows can theoretically attenuate the energy of seismic waves hundreds of times. Also, the ITSM can be used to protect critical infrastructures, like nuclear power station.

In order to intuitively show the novelty of the 2D ITSM, a comparison table (

Conclusion

In this research, an 1D ITSM with an ultra-wide FBG has been proposed by improving the pillarlike SM. With the total mass of the pillar unchanged, the relative bandwidth of the FBG of the ITSM is about 1 at maximum. This result is much better than that of the PSM. The effects of the geometrical and material parameters of the ITSM on the FBG have been also discussed. It is found that the appearance of the plate obviously increases the center frequency and relative bandwidth of the FBG of the ITSM. The complex band structure of the ITSM has been calculated to analyze the attenuation mechanism of surface waves. It can be found that the FBG of the ITSM is composed of two parts: part 1 with surface evanescent waves and part 2 with no surface mode. We also found that this result is consistent with the PSM. The propagation of seismic surface waves in the ITSM has been also investigated. It was found that the propagation modes of surface waves in the ITSM are completely different in different parts of the FBG. In the frequency range of part 1, the seismic surface waves are significantly attenuated and almost unable to propagate to the third unit cell in the ITSM. In the frequency range of part 2, the surface waves are converted into bulk waves. Finally, we have proposed a kind of the 2D ITSM with 25 unit cells in large-scale field experiments, which can attenuate Rayleigh waves in a ultra-wide frequency range. Moreover, the lattice constant in this work is much smaller than the wavelength of the surface waves in the ultra-low frequency range. A lower-frequency FBG can be obtained by appropriately increasing the lattice constant of the ITSM.

This work not only provides new options for controlling seismic surface waves at ultra-low frequency, but also provides new design ideas for steering surface wave.

Figure 1 :

 1 Figure 1: The unit cells of (a) the PSM and (b) the 1D ITSM.

Figure 2 :

 2 Figure 2: The classical band structure of (a) the PSM and (b) the 1D ITSM, the vibration modes of (c) the PSM and

  (a) is similar to the FBG in Fig.2(b), these two bandgaps have different generation physical mechanisms, which have been identified by many experiments and numerical simulations.[START_REF] Achaoui | Experimental observation of locallyresonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars[END_REF][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF]

Figures 3 (Figure 3 :

 33 Figures3(a -b) show that the variation of geometric parameters of the plate with the position and

Figure 3 (

 3 Figure 3(c) shows the effect of the pillar height on the FBG. When the pillar height continues to

Figure 4 :

 4 Figure 4: Variation of the upper and lower frequency boundaries, and the relative bandwidth of the FBG (a) the mass

Figure 5 :

 5 Figure 5: The complex band structures of (a) the 1D ITSM and (b) the PSM. The first bandgap composes of two

Figure 6 :

 6 Figure 6: (a) Schematic model of the 1D ITSM with 10 unit cells for transmission calculations. (b) Transmission

Table 3 :

 3 (b), the central garden of the Department of Mechanics on the Beiyangyuan Campus of Tianjin University (China) was selected for a large-scale field experiments in August 2020. The geometric parameters of the TS used in large-scale filed experiments are consistent with those in Fig. 7(a). Among them, the steel tube and steel plate are rigidly connected by high-strength bolts. 25 identical TSs are customized and periodically arranged on the substrate. The geometric parameters of the unit cell in Fig. 7(a)

Figure 7 :

 7 Figure 7: (a) Schematic of the unit cell and (b) top view of the experimental setup. Points A and B are signal

  the light gray areas. The whole band structure along Γ-X-M-Γ direction in the first Brillouin zone can be found in the Note D of SI. The vibrating mode of the highest point of the first band is given in the insert of Fig.8(a). The displacement is in x direction and mainly concentrated on the top of the steel tube. The vibrating mode of the second band exhibits also a displacement distribution mainly concentrated on the top of the steel tube but in y direction. It is worth noting that these two bands exist in all directions shown in the Note D of SI. The surface waves considered in this work are Rayleigh waves, whose displacement components on the surface only exist in the x and z directions[START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF]. The Rayleigh waves can excite the ITSM to vibrate in the frequency range of the first band, but not the second band, i.e., they can propagate in the ITSM in the frequency range of the first band, but not the second band[START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF][START_REF] Khelif | Locally resonant surface acoustic wave band gaps in a two-dimensional phononic crystal of pillars on a surface[END_REF]. Therefore, the bandgap starts from the highest point of the first band. The bandgap of the ITSM for Rayleigh waves is in the range of 34-130 Hz, which is marked by the light gray part in Fig.8(a).

  between the TSs and the soil in the field test is weaker than the perfect connection in the simulation. In addition, it is worth noting that in the bandgap, experimental and simulated results all show that the attenuation of Rayleigh waves is stronger at the band of 34-70 Hz than the one of 70-130 Hz. In the cross-sectional and top views of simulation results with PC, it is easy to find that the propagation of Rayleigh waves in the ITSM is completely different at 44 and 100 Hz, respectively. (The propagation of Rayleigh waves without the ITSM is shown in Note E of SI)

Table 2 :

 2 The material parameters used in this paper[START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF]

	a	H	l1	w1	l2(l)		w2(w)
	1.5 m	300 a	0.95 a	0.05 a	1.4285 a	0.07 a
	Table 1: The geometric parameter of the PSM and the 1D ITSM
	Material	Density (kg/m 3 )	Young's modulus (Pa)	Poisson's ratio
	Steel	7784		2.07 x 10 11			0.3
	Soil	1800		2 x 10 7			0.3

2. Results and Discussion of the 1D ITSM 2.1 Classical band structure

	Figures 2 (a -b) show the classical band structures of the PSM and the ITSM, respectively. The
	classical band structures are calculated by using the Solid Mechanics module of COMSOL. The
	sound cones are clearly surrounded by the light gray areas. There are only surface waves inside the
	sound cone. Therefore, surface waves cannot exist in the SMs in the frequency range where no band
	appears, i.e., the bandgap for surface waves. Moreover, the bandgaps at the lowest frequency, i.e.,
	the FBGs, are marked by the dark gray areas in Figs. 2 (a -b). In this work, the frequency range of
	the FBG is from 0.72 Hz to 0.80 Hz for the PSM, and from 6.7 Hz to 17.2 Hz for the ITSM.

Table 4 )

 4 is prepared based on the relative widths of the first attenuation zone of the 2D ITSM we are reporting in this work, and the existing similar designs in literature. It is easy to find that the 2D ITSM has a largest relative width of the first attenuation zone just by using one kind of man-made structure. It is worth noting that the low height of the 2D ITSM makes it more competitive.

		First attenuation zone
	Structures		Remarks
		(Relative width)
	2D ITSM	34 -130 Hz (1.17)	Height less than 1 m
	Trees in forest [34]	32 -40 Hz (0.22)	Trees with an average height of 14 m

Table 4 :

 4 The comparison of the ITSM with the existing similar designs in literatures

Acknowledgments

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

Data availability

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

In Fig. 8(b), the black line is the simulation results when there is perfect continuity (PC) between the man-made structures and substrate. The red dash line is the simulation results when there is imperfect continuity (IPC) between the man-made structures and substrate. And blue dot line experimental results. The model used in the simulation is the same as the experimental site.

Moreover, the influence of the continuity between the man-made structures and the substrate on the attenuation of Rayleigh waves is analyzed. A very thin PML (t = 0.02a) with the scaling factor s = 0.1 is added underneath all man-made structures. [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] In the range of 34-130 Hz, Rayleigh waves are attenuated significantly. It can be found that the attenuation effect in the simulated results with the PC is generally stronger than the results of the field test. However, the simulated results with the IPC are closer to the experimental results, especially in the range from 40 to 60 Hz. This is because