Wei Ding

Tianning Chen

Chen Chen

Dimitrios Chronopoulos

Jian Zhu
email: jianzhuxj@xjtu.edu.cn

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

Thomson scattering-induced bandgap in planar chiral phononic crystals

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chirality, i.e., the spatial arrangement which does not possess centrosymmetry and rotational symmetry [START_REF] Gold | Compendium of chemical terminology[END_REF][START_REF] Zhu | Observation of chiral phonons[END_REF][START_REF] Kelvin | The Molecular Tactics of a Crystal[END_REF], makes macroscopic materials with negative Poisson's ratio effects [START_REF] Lakes | Foam Structures with a Negative Poisson's Ratio[END_REF][START_REF] Prall | Properties of a chiral honeycomb with a Poisson's ratio-1[END_REF][START_REF] Lakes | Deformation mechanisms in negative Poisson's ratio materials: structural aspects[END_REF]. In essence, the rotation-translation coupling in these lattices is the fundamental basis of the negative Poisson's ratio [START_REF] Lakes | Deformation mechanisms in negative Poisson's ratio materials: structural aspects[END_REF][START_REF] Alderson | Elastic constants of 3-, 4-and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading[END_REF]. Therefore, both axial and planar twist chiral lattices can achieve this unique property [START_REF] Ha | Chiral three-dimensional isotropic lattices with negative Poisson's ratio[END_REF][START_REF] Fu | A novel category of 3D chiral material with negative Poisson's ratio[END_REF].

Chirality in geometry can not only endow mechanical metastructures with negative

Poisson's ratio as well as twist [START_REF] Zhong | A novel three-dimensional mechanical metamaterial with compression-torsion properties[END_REF][START_REF] Zheng | A novel metamaterial with tension-torsion coupling effect[END_REF][START_REF] Frenzel | Three-dimensional mechanical metamaterials with a twist[END_REF][START_REF] Spadoni | Elasto-static micropolar behavior of a chiral auxetic lattice[END_REF], but also can produce phononic crystals (or elastic metastructures) with some unique dynamical properties, such as elastic wave polarization [START_REF] Spadoni | Phononic properties of hexagonal chiral lattices[END_REF][START_REF] Nassar | Polar Metamaterials: A New Outlook on Resonance for Cloaking Applications[END_REF][START_REF] Carta | Wave polarization and dynamic degeneracy in a chiral elastic lattice[END_REF][START_REF] Zhang | Tunable wave propagation in octa-chiral lattices with local resonators[END_REF][START_REF] Zhang | On the directional wave propagation in the tetrachiral and hexachiral lattices with local resonators[END_REF], nonlinear wave propagation [START_REF] Karathanasopoulos | Chiral and non-centrosymmetric effects on the nonlinear wave propagation characteristics of architectured cellular materials[END_REF], enhancing stiffness and damping [START_REF] Baravelli | Internally resonating lattices for bandgap generation and low-frequency vibration control[END_REF][START_REF] Agnese | Composite chiral shear vibration damper[END_REF], widening local resonance-induced bandgap [START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF][START_REF] Bacigalupo | Auxetic anti-tetrachiral materials: Equivalent elastic properties and frequency band-gaps[END_REF][START_REF] Bacigalupo | Optimal design of lowfrequency band gaps in anti-tetrachiral lattice meta-materials[END_REF][START_REF] Qi | Bandgap and wave attenuation mechanisms of innovative reentrant and anti-chiral hybrid auxetic metastructure[END_REF][START_REF] Li | Vibration characteristics of innovative reentrant-chiral elastic metamaterials[END_REF]. At present, the planar chiral lattice has been received intensive investigations [START_REF] Wu | Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review[END_REF]. In these researches, the coupling of the translation and rotation can generate a positive elastic module [START_REF] Bigoni | Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization[END_REF][START_REF] Liu | Wave propagation characterization and design of two-dimensional elastic chiral metacomposite[END_REF],

and embedding locally resonant substructure can achieve negative dynamic equivalent density [START_REF] Liu | Wave propagation characterization and design of two-dimensional elastic chiral metacomposite[END_REF][START_REF] Liu | Analytic model of phononic crystals with local resonances[END_REF][START_REF] Ding | Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density[END_REF]. When the double negative properties overlap in the same frequency band, it is beneficial to achieve negative refraction [START_REF] Bigoni | Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization[END_REF][START_REF] Ding | Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density[END_REF][START_REF] Zhu | Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial[END_REF]. However, the double negative properties are unbeneficial for opening a continuous bandgap due to the pass band with negative group velocity induced by the negative elastic module [START_REF] Liu | Wave propagation characterization and design of two-dimensional elastic chiral metacomposite[END_REF][START_REF] Ding | Metamaterial with Simultaneously Negative Bulk Modulus and Mass Density[END_REF].

Therefore, using chiral lattices to create a broad local resonance-induced bandgap mainly depends on the excellent stiffness-mass ratio (specific stiffness) of the matrix [START_REF] Baravelli | Internally resonating lattices for bandgap generation and low-frequency vibration control[END_REF][START_REF] Zhu | A chiral elastic metamaterial beam for broadband vibration suppression[END_REF][START_REF] Wu | Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review[END_REF][START_REF] Bacigalupo | Simplified modelling of chiral lattice materials with local resonators[END_REF]. It is well known that the lower boundary of the local resonance bandgap is negatively correlated with the ratio of the oscillator and matrix masses [START_REF] Bacigalupo | Optimal design of lowfrequency band gaps in anti-tetrachiral lattice meta-materials[END_REF][START_REF] Mousanezhad | Elastic properties of chiral, anti-chiral, and hierarchical honeycombs: A simple energy-based approach[END_REF][START_REF] Liu | Locally Resonant Sonic Materials[END_REF], while the bandwidth of the negative dynamic equivalent density is positively correlated with the ratio of the oscillator to the substrate mass [START_REF] Liu | Analytic model of phononic crystals with local resonances[END_REF][START_REF] Huang | Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density[END_REF]. Therefore, improving the specific stiffness of the matrix can optimize the local resonance-induced bandgap.

In axial twist chiral phononic crystals [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF][START_REF] Zheng | Mechanical logic switches based on DNA-inspired acoustic metamaterials with ultrabroad low-frequency band gaps[END_REF][START_REF] Orta | Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism[END_REF], motion coupling is a critical factor in bandgap formation [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. At deep subwavelength scales, because of this coupling, the outgoing wave will propagate in two polarization modes when the incident wave polarized in one mode propagates through this chiral substructure. This progress is analogous to Thomson scattering in classical electromagnetic fields. In classical Thomson scattering, the incident wave forces the electron to vibrate, thus generating a divergent wave field. The divergent outgoing field results from superimposing waves polarized in multiple directions [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. In this analogy, the chiral sub-unit cell is regarded as the electron; the two and more waves polarized in different freedoms were superimposed to form an outgoing wave field [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF].

More critically, a phononic crystal with the Thomson scattering-induced bandgap must have a second Thomson scattering ability to cause destructive interferences [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. Specifically, two polarization modes are produced through the first Thomson scattering.

After that, the scattered waves vibrating in these two modes each take on the role of secondary incident waves for a second Thomson scattering. These second scattered waves polarized in the same mode will have opposite phases, and thus cause destructive interferences [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. During the quantification of the analogous Thomson scattering, the inertia matrix will show the inertial amplification effect [START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF][START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF]. It has been demonstrated that the geometric and physical parameters, including lattice constant, equivalent stiffness, and density, are inextricably linked to the starting frequency and width of the bandgap [START_REF] Phani | Wave propagation in two-dimensional periodic lattices[END_REF][START_REF] Delpero | Inertia Amplification in Phononic Crystals for Low Frequency Band Gaps[END_REF][START_REF] Ding | A three-dimensional twisted phononic crystal with omnidirectional bandgap based on inertial amplification by utilizing translation-rotation coupling[END_REF]. Because of the inertial amplification [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF][START_REF] Orta | Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism[END_REF], whose dynamic inertia can be amplified without increasing the static inertia, theoretically, this bandgap can be lower than Bragg scattering for the same lattice constant, equivalent density, and stiffness [START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF][START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF], and this lattice will be lighter than the local resonance lattice for the same lattice constant, equivalent stiffness and bandgap width requirements [START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF][START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF][START_REF] Li | Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores[END_REF][START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF]. In other words, inertial amplification is a critical concept to release the dependence of the low-frequency and broad bandgaps on low stiffness and bulky masses. However, numerous extended research ideas are limited to the classical models to slow or stall the advances in this problem [START_REF] Li | Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores[END_REF][START_REF] Xi | Inertial amplification induced band gaps in corrugated-core sandwich panels[END_REF][START_REF] Schmied | Toward structurally integrated locally resonant metamaterials for vibration attenuation[END_REF][START_REF] Barys | Efficient attenuation of beam vibrations by inertial amplification[END_REF][START_REF] Li | Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects[END_REF][START_REF] Zeighami | Inertial amplified resonators for tunable metasurfaces[END_REF][START_REF] Zhou | A nonlinear resonator with inertial amplification for very low-frequency flexural wave attenuations in beams[END_REF][START_REF] Li | Multiple band gaps for efficient wave attenuation by inertial amplification in periodic functionally graded beams[END_REF][START_REF] Yuksel | Realization of an ultrawide stop band in a 2-D elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF]. The proposal of the Thomson scattering provides new insight into designing more phonon crystals with an inertial amplification effect.

Based on the Thomson scattering mechanism, this work proposes a single-phase planar torsional chiral phononic crystal to achieve a low-frequency and wide-band confinement. In contrast to previous works, we use a chiral lattice not because of its higher specific stiffness but because of its motion coupling. It is essential to state that our "kinematic coupling" is not the negative equivalent modulus but the coupling that allowed for Thomson scattering phenomena in elastodynamics. To generate destructive interferences, the chiral unit cell will be mirrored to be anti-chiral. Eventually, the antichiral super-unit cell can open a low-frequency and broad bandgap. In contrast to the conventional local resonance, this Thomson scattering lattice has a broader bandgap and nearly the same order of magnitude decay starting frequency for the same lattice constant, equivalent stiffness, and density conditions. The material damping can effectively suppress the high-frequency modes, so that the bandgap exhibits the effect of a low-pass filter rather than a band-stop filter in the axial chiral lattice.

The paper is organized as follows. Section 1 presents the background for planar chiral phononic crystals. In order to distinguish our design method from that based on local resonance and clear the superiority of this study, Section 2 is devoted to comparing classical local resonance-induced and Thomson scattering-induced planar chiral phononic crystals. Then, the design mechanism is discussed in Section 3. Section 4 presents a single-phase planar chiral phononic crystal with a more superior bandgap obtained according to the two-stage inertial amplification in the translation-rotation coupling system. Section 5 exhibits an experimental verification. Finally, Section 6 concludes the paper.

Design & Method

The low-frequency and broad bandgaps contradict equivalent physical parameters such as lattice constant, equivalent stiffness, equivalent density [START_REF] Phani | Wave propagation in two-dimensional periodic lattices[END_REF][START_REF] Delpero | Inertia Amplification in Phononic Crystals for Low Frequency Band Gaps[END_REF]. For a fair comparison and to clarify the innovation of our study, we have embedded local resonance substructures and steel pillars in an identical chiral matrix to form a local resonance lattice (Fig. 1(a)) and a composite lattice (Fig. 1(b)) with equal weight. The minimum period of the local resonance (Fig. 1(a)) and the composite structures (Fig. The composite unit cell (Fig. 1(b)) is polluted from the local resonance lattice, where the rubber is replaced by nylon. The rubber has the same density as the nylon to ensure that the two lattices have the same mass. In addition to the outer circle diameter 𝑑 " =20 mm of the rubber, the two super-unit cells have the same other geometric parameters, including the lattice constant 𝑎 =50 mm, the angle 𝜃 =30°, the width 𝑙 = 10 mm and thickness 𝑡 = 2.5 mm, and the parameters ℎ =8.32 mm and 𝑑 # =12.4 mm. Therefore, the two lattices have the same equivalent density. The ligaments are more easily deformed compared to the lumped mass, so the equivalent stiffness of the lattice is mainly determined by the ligaments. Since both lattices have the same ligaments, both have a quasi-same equivalent stiffness. [START_REF] Chen | A study of topological effects in 1D and 2D mechanical lattices[END_REF] on the super-unit cell. It can be found that the local resonance bandgap extends from about 800 to 1050 Hz, while in the composite lattice, we have a narrow bandgap, i.e., 3540-3760 Hz. According to the relative width of the bandgap to evaluate the two unit cells [START_REF] Taniker | Generating ultra wide vibration stop bands by a novel inertial amplification mechanism topology with flexure hinges[END_REF], it is 27% of the local resonance unit cell and 6% of the composite structure.

In short, the local resonance unit cell has better performance than the composite structure for both starting frequency and width of the bandgap.

To obtain the broad bandgap similar to that of Refs. [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF][START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF], it is indispensable to mirror unit cells shown in Fig. 1. The geometries of the mirrored super-lattices are presented in Fig. 2

The mechanism of the bandgap opening

Statics can illustrate the polarization mode and direction of the LMs in dynamics in deep subwavelength [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF][START_REF] Hibbeler | Engineering Mechanics: Statics & Dynamics[END_REF]. To understand the underlying physics, the static deformation of the mirrored super-unit cell (Fig. 3(a)) is computed numerically with COMSOL solid module. In the simulation, the material is linear, and geometric nonlinearity is not considered. As shown in Fig. 3, for easy differentiation, the first layer chiral units L $ are numbered ① and ②, and the second layer L % are numbered ③ and ④. The nodes between the ① and ③ as well as the ② and ④ are defined as 𝑝 $ and 𝑝 % . Both upper and lower plates are considered as rigid, and the rest of the structure is defined as nylon. A force 𝐹 along -𝑦 axis is applied at the top of the upper plate and a fixed boundary condition is set at the lower plate. In this chiral system, when the incident wave polarizing in translational harmonic motion goes through the first-layer unit cell L $ , the outgoing wave will have two polarizations, i.e., translation and rotation [START_REF] Bergamini | Tacticity in chiral phononic crystals[END_REF]. By analogy, each subsequent wave passing through a layer of chiral unit cells is equivalent to passing through Thomson scattering once more. Therefore, we defined that the first layer L $ refers to the first Thomson scattering and the second layer L % refers to the second Thomson scattering [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF].

Fig. 3(a) is the total deformation of the mirrored super-unit cell. One can observe from the deformation profiles that all chiral unit cells have significant translation along 𝑦axis and rotation around the 𝑧-axis, as well as slight translation along 𝑥-axis. For ease of understanding, we ignore the translational motion in the 𝑥 -axis in following discussion. The coupled phenomenon indicates that the unit cell has first two basic properties of Thomson scattering bandgap, i.e., two orthogonal motions and they are coupled on the same lumped mass. However, to realize the Thomson scattering bandgap, generating the opposite sub-motions in the same unit cell is crucial.

Although there is a significant reverse of the motion between the unit cells in L $ and L % , it is unclear whether the movements of the unit cells in L $ can create opposite submotions in the same polarization of the unit cells in L % . Consequently, for the sake of clarity, we have limited the 𝑧-axis rotational freedom and 𝑦-axis translational freedom of the L $ unit cells separately to distinguish the polarized directions of the L % unit cells drived by the two types of motion of the L $ unit cells. Fig. 3(b) is the deformation profile when the rotation of L $ is limited. In this condition, the unit cells ① and ② have only the translation along 𝑦 axis. It can be found that nodes 𝑝 $ and 𝑝 % will move along -𝑦 axis and close to each other. According to the movements of two nodes, from the perspective of the unit cells ③ and ④, the force direction of the node 𝑝 $ is denoted as the inset in the upper right of Fig. 3(b), and the ones of the node 𝑝 % is presented as the inset in the lower right of Fig. 3(b). When the translational degrees of freedom of the layer L $ are constrained, the deformation of the super-unit cell is shown in Fig. 3(c). For clarity, Fig. 3(d) shows the enlarged deformation profile of L % , and the force directions of nodes 𝑝 $ and 𝑝 % are illustrated in the right of Fig. 3(d). Comparing the force directions shown in Fig. 3(b) and Fig. 3(d), it can be found that the translation and rotation of the unit cells L $ result in node 𝑝 $ (or 𝑝 % ) having opposite force directions, respectively. Because of the symmetry of the system, we can take unit cells ① and ③ as analytical objectives. Regarding the unit cell ① shown in Fig. 3(b), it will have the translation along -𝑦 axis and rotation around -z axis when it is compressed; conversely, it will produce the translation +𝑦 axis and rotation around +𝑧 axis when it is stretched. When unit cell ① has only translation, it will compress unit cell ③, at which time unit cell ③ will produce -𝑦axis translation and -𝑧-axis rotation; when unit cell ① has only rotation, it will stretch unit cell ③, as a result, unit cell ③ will have +𝑦-axis translation and +𝑧-axis rotation.

In other words, when unit cell ① is compressed, it has two polarizations, i.e., translation and rotation. Further, these two polarizations drive unit cell ③ have two sub-translations and two sub-rotations simultaneously. The most crucial is that these sub-motions of the unit cells L % vibrating in the same polarization have opposite directions. Consequently, the chiral twist super-unit cell shown in Fig. 3(a) satisfies the three conditions of generating Thomson scattering bandgap, i.e., two and more orthogonal motions coupled on the same lumped mass. After the second Thomson scattering, the sub-motions polarized in the same mode have opposite initial motion directions, which is the underlying mechanism by which this super-unit cell can achieve a low-frequency and wide bandgap. The analytical approach was applied to the arrayed super-unit cell (in Fig. 4(a)) to clarify the difference between arrayed and mirrored super-unit cells. The arrayed superunit cell is unstable because it produces shear motion along the 𝑥 axis under the 𝑦direction load. The unstable phenomenon can be observed from the animations in Supplementary information. To realize the same input displacement condition as the mirrored phononic crystal, the translational motion of the upper top plate in the 𝑥direction is constrained in the calculation, while the other boundary conditions are the same as the ones applied on the mirrored super-unit cell. Fig. 4(a) is the total static deformation profile. Different from Fig. 3(a), the deformation is asymmetric. Some unit cells have a significant rotation, such as unit cell ②, while some units do not have significant rotation, such as unit cell ①, even if both translation and rotation are feeble, such as unit cell ④. In more details, when the rotation of the unit cells L $ around 𝑧 axis is constrained, the deformation of the unit cells L % exhibits the same direction but to a different extent, as illustrated in Fig. 4(b). Also, when the rotation of the unit cells L $ along -𝑦 axis is constrained, the deformation of the unit cells L % exhibits the same direction but to a different extent as well, as illustrated in Fig. 4(c). The right insets of Fig. 4(b) and Fig. 4(d) show the force directions from the viewpoint of the unit cells L % . One can observe that both nodes 𝑝 $ and 𝑝 % have opposite force directions in the y-direction, which seems to mean that there are destructive interferences in the arrayed super-unit cell. However, the force direction in 𝑥-direction is identical. Because of the identical force direction, the superunit cell has the asymmetric deformation, and some chiral unit cells have almost completely lost the coupled motion, which is one of the critical conditions for this bandgap generation. Consequently, the arrayed super-unit cell does not have the excellent bandgap of the mirrored super-unit cell, although both unit cells have the same equivalent density, lattice constant, and virtually identical stiffness and strength (Please refer to Fig. A1).

Single-phase planar phononic crystal

Our previous study [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF] has demonstrated that the coupling of the orthogonal motions is analogous to Thomson scattering. During the quantitative description of the Thomson scattering based on the analytical formalism, the inertia matrix exhibits inertial amplification [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. Interestingly, the involvement of the rotation allows the system to have two stages of inertial amplification, one of which depends on angle 𝜃, and the other on the rotational inertia of lumped masses [START_REF] Ding | A three-dimensional twisted phononic crystal with omnidirectional bandgap based on inertial amplification by utilizing translation-rotation coupling[END_REF]. The larger the rotational inertial is, the lower the starting frequency of the bandgap is.

Regarding the composite structure shown in Fig. 2(b), it is not conducive to manufacturing and controlling the robustness of the elastic dynamics in practice [START_REF] Matlack | Composite 3Dprinted metastructures for low-frequency and broadband vibration absorption[END_REF].

Besides, steel, as a material with high density, is not suitable for obtaining large rotational inertia. To overcome these limitations, we propose a single-phase chiral unit cell composed of a matrix Ⅰ and a lumped mass Ⅱ, as shown in Fig. 5(a). The parameter 𝐻 = 2ℎ + 𝑙 is designed to maintain the same height in z-axis as the composite unit cell.

For the meshing in the numerical calculation, the diameter 𝑑 & is equal to 𝐷 # -2 mm (please refer to Fig. 1(b) to find 𝐷 # ). In order to use the rotational inertia of the lumped mass to generate a low starting frequency of the bandgap, while giving enough space for the dynamical deformation of the system, we set the diameter as 𝐷 " =0.5𝑎 -2 mm.

Ultimately, for comparison purpose, the entire lattice (Fig. 5 Hz, with a decrease of 36%. The upper boundary of the bandgap is reduced from 5200 Hz to 5025 Hz. The relative width of the bandgap is improved from 110% to 135%. It is worth noting that these variations happened in the case of the same lattice constant, equivalent density, and stiffness.

Experimental verification and discussion

The FRFs of the finite structure (shown in Fig. 6 (a)), composed by 2×3 unit cells, is measured numerically and experimentally to validate the prediction of the created bandgap in the band structure. Two foams support the sample to levelling and isolate the vibration from the background [START_REF] D'alessandro | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF]. A cubic block was designed at both the excitation end 𝑆 # and the output end 𝑆 " to facilitate the installation of the transducer and the excitation during the experiment. The same geometry is used in the simulation to minimize the differences between experiments and simulations. In the COMSOL V6.0 solid mechanics module, a harmonic excitation force 𝐹 is applied to the input 𝑆 # , and the accelerations of 𝑆 # and 𝑆 " are recorded simultaneously by two domain point probes. The sweep frequency range covers 200 Hz to 7000 Hz with a frequency resolution of 5 Hz. The ratio of 𝑆 " to 𝑆 # is defined as the FRF. The experimental configuration (shown in Fig. 6) includes M+p Vibpilot, Modelshop K2007E01, and two acceleration transducers PCB353 B15.

The numerical and experimental results are illustrated in Fig. 6(b), where the blue plot corresponds to the simulation while the red plot to the experimental measurement. One can see that the attenuation of both results starts at about 710 Hz. The band gap type can be determined from the distribution of energy within the bandgap. In Fig. 6(c), we have presented the deformation profiles at 3000 Hz. It can be found that the energy is mainly concentrated on the two lumped masses close to the source. In more detail, the vibration mode of the lumped masses is mainly rotational. According to the Thomson scattering mechanism, the attenuation relies mainly on the destructive interference formed during the second Thomson scattering, so the energy is mainly localized in the first layer of lumped masses, as presented in our previous work [START_REF] Ding | Experimental Observation and Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering[END_REF]. In other words, this bandgap belongs to the Thomson scattering type.

Compared to the numerical FRF of Fig. 6(b), there are some discrepancies between numerical and experimental results. Exceptionally, the depth of the experimental FRFs does not exceed 1e-4, which is because the output signal is already at the same level as the background noise. Besides, some numerical peaks close to 6200 Hz should belong to pass bands, but their attenuation coefficient approximates that of the bandgap in the experimental result. To find the reason, the deformation profile at 6250 Hz is shown in Fig. 6(d). It is clear that the resonant modes are the local resonance of the lumped masses. It has been demonstrated that damping in nylon is an effective approach to suppressing high-frequency local resonance modes [START_REF] D'alessandro | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF][START_REF] D'alessandro | Mechanical lowfrequency filter via modes separation in 3D periodic structures[END_REF]. When the Rayleigh damping of 𝛼 =0.1 s '$ and 𝛽 = 4e-7 s [START_REF] D'alessandro | Modeling and experimental verification of an ultra-wide bandgap in 3D phononic crystal[END_REF][START_REF] D'alessandro | Mechanical lowfrequency filter via modes separation in 3D periodic structures[END_REF] is considered in simulation, as illustrated by the black dashed line in Fig. 6(b), the resonant peaks near 6250 Hz are compressed effectively, which is relative to a weak upward trend of the experimental result. In addition, manufacturing errors and the slight bending of the entire sample might be the causes of some peak shifts in experimental results, such as the difference between the numerical peak at 1750 Hz and experimental ones at 1400 Hz. Nevertheless, the consistent trend between the experiment and simulation can reveal the bandgap existence. Hz. The attenuation range shown in FRF is 710 Hz-6220 Hz. The starting frequency of the attenuation in FRF is lower by about 250 Hz than that of the bandgap, with a relative difference of more than 25%. The cutoff of the bandgap is more than 1000 Hz higher than the prediction in band structure; the relative bandgap width is 135% in the bandgap, but the result is 159% in FRF.

To analyze the causes of the lower boundary differences, it is necessary to analyze the resonant modes from 710 Hz to 970 Hz located at the lower boundary of the bandgap.

There are seven orders of passbands and three types of resonance modes. For simplicity, Fig. 7(a)-(c) show the three typical vibration modes (some repeated vibration modes are not mentioned). The first mode (in Fig. 7(a)) is the local deformation at the ligament; the second mode (in Fig. 7(b)) is the translation of the lumped mass in the z-axis; the third mode (in Fig. 7(c)) is the rotation of the lumped mass around an axis in the surface.

Because these modes are orthogonal to the excitation direction, they cannot be effectively excited under the current excitation conditions. In order to verify the above analysis, the out-of-plane input is applied in the simulation.

The numerical results are shown in Fig. 8. It can be found that the attenuation starts at 1000 Hz and the cutoff frequency is 5000 Hz, which coincides with the prediction in band structure shown in Fig. 5(c). This analysis confirms the explanation that the attenuation width in the experimental FRF is wider than the energy band prediction.

Conclusion

In summary, a single-phase planar chiral phononic crystal with a low-frequency and broad bandgap induced by Thomson Scattering has been proposed and investigated numerically and experimentally. Under the same lattice constant, stiffness and mass, the mirrored chirality only improves the width and starting frequency of the local resonance bandgap, while it can open a broad bandgap when the local resonant substructure is discarded. The superior bandgap performance depends on the Thomson scattering provided by translational and rotational coupling. According to the two-stage inertial amplification, we further simplified the material components of the unit cell and lowered the starting frequency of the bandgap as well, while the method is the changing in the appearance of the lumped masses only. The consequent result is that the starting frequency of the attenuation can be in the same order of magnitude as the local resonance bandgap and, moreover, the width is significantly better than the latter.

Notably, different from the axially chiral phononic crystals whose bandgap exhibits band-pass filter properties, the blueprint carried out in this work has low-pass filtering properties due to the material damping. One cannot deny that the local resonance can generate a low-frequency bandgap due to the relevance between the starting frequency and the elastic module of the coating material [START_REF] Liu | Locally Resonant Sonic Materials[END_REF][START_REF] Mizukami | Three-dimensional printing of[END_REF]. Of course, a new proposal in design may offer the possibility of increasing the inertial amplification factor, for the purpose of reducing the reliance on low stiffness and bulky masses for low-frequency and broad bandgaps [START_REF] Li | Phononic band gaps by inertial amplification mechanisms in periodic composite sandwich beam with lattice truss cores[END_REF]. These properties obtained in this work may make Thomson scattering a powerful approach and mechanism.

Appendix. A Fig. A1. Schematics of the ligaments. The red line is the symmetric line of the thickness, which is tangent to the black circle. Please note, the diameter of that circle is equal to 𝐷 # -𝑡.

Appendix. B

  1(b)) is one-quarter of the lattices shown in Fig. 1. Thus, we named the geometries shown in Fig. 1(a) and Fig. 1(b) as super-unit cells. The local resonance unit cell (Fig. 1(a)) is composed of three materials, including the nylon with the elastic module 𝐸 =1.6e9 Pa, density 𝜌 =1000 kg/m ! , Poison's ratio 𝑣 =0.4, the steel with the elastic module 𝐸 =2.1e11 Pa, density 𝜌 =7850 kg/m ! , Poison's ratio 𝑣 =0.3, and the rubber with the elastic module 𝐸 =5e6 Pa, density 𝜌 =1000 kg/m ! , Poison's ratio 𝑣 =0.47.

Fig. 1 .

 1 Fig. 1. (a)-(b) Schematics and (c)-(d) band structures of the arrayed local resonant super-unit cell and composite super-unit cell. The unit cell is one-quarter of the lattice.

Fig. 1 (

 1 Fig. 1(c) and Fig. 1(d) illustrate the band structures of the local resonance lattice (Fig. 1(c)) and composite lattice ( Fig. 1(d)), which are obtained in the solid mechanic module of COMSOL Multiphysics through applying the Bloch periodic boundary[START_REF] Chen | A study of topological effects in 1D and 2D mechanical lattices[END_REF]

  (a) and (b), and the band structures are shown in Fig. 2(c) and Fig. 2(d). One can observe that the bandgap of the local resonance extends from 700 to 1050 Hz. Compared with the previous results shown in Fig. 1(c), the lower boundary is shifted down by about 100 Hz, and the upper boundary is nearly invariable. Remarkably, for the composite super-unit cell, the narrow bandgap shown in Fig. 1(d) becomes a continuous and broad bandgap extending from 1600 to 5200 Hz, with the lower boundary significantly lowered from 3540 Hz to 1600 Hz. To sum up, it is effective that mirroring improves the lower boundary of the local resonance bandgap. Still, the mirroring here is an optimization without changing the mechanism of the bandgap opening. Unlike in the local resonance super-unit cell, mirroring in the composite super-unit cell activates a new mechanism, thus resulting in a broad bandgap, which sets our study and approaches apart from the traditional design ideas.

Fig. 2 .

 2 Fig. 2. (a)-(b) Schematics and (c)-(d) band structures of the mirrored local resonant super-unit cell and composite super-unit cell.

Fig. 3 .

 3 Fig. 3. Deformation profiles of the mirrored lattice under the condition of (a) free, (b) constrained 𝑧-axis rotation of the L ! , and (c) constrained 𝑦-axis translation of the L ! . (d) Enlargement of Fig.

  3(c). The black wireframe is the original shape of the lattice. The red arrow points to the deformation direction.

Fig. 4 .

 4 Fig. 4. Deformation profiles of the arrayed lattice under the condition of (a) free, (b) constrained zaxis rotation of the L ! , and (c) constrained 𝑦-axis translation of the L ! . (d) Enlargement of Fig.

  4(c). The black wireframe is the original shape of the lattice. The red arrow points to the deformation direction.

  Fig. 5(b) can be obtained by mirroring the lattice Ⅲ shown in Fig. 5(a). Fig. 5(c) is the band structure of the model exhibiting a broad bandgap within 965-5025Hz. Compared to the composite super-unit cell, the starting frequency is reduced from 1510 Hz to 965

Fig. 5 .

 5 Fig. 5. (a) Schematics of the single-phase unit cell composed by Ⅰ the matrix and Ⅱ the lumped mass. (c) Band structure of the (b) mirrored unit cell.

Fig. 6 .

 6 Fig. 6. (a) Photograph and configuration of the experiment. (b) Numerical and experimental FRFs. The red zone is the theoretical bandgap width predicted from the band structure. Deformation profiles at (c) 2968 Hz and (d) 6200Hz.

Fig. 7 .

 7 Fig. 7. (a)-(c) Three types of modes at the lower boundary (within 710-970 Hz) of the bandgap. (d)-(f) Three types of modes at the upper boundary (within approximately 5000-6100 Hz).

Fig. 8 .

 8 Fig. 8. Numerical FRF of the normal excitation. The red area refers to the bandgap width.

Fig. A2 .

 A2 Fig. A2. The photographs of the mirrored lattice (a), arrayed lattice (b), and compression experiment (c). In order to ensure that the sample will not deform significantly in the normal direction during compression, the thickness 𝑙 is 30 mm. Other geometric dimensions are the same as the matrix Ⅰ Fig. 5(a). The universal testing machine is Instron 5982 and the loading speed is 2 mm/min. (d) The experimental results of the force-displacement curves.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Nos. 516750402 and 12002258)

Supplementary data

The data that support the findings of this study are available from the corresponding author upon reasonable request. locally resonant carbon-fiber composite metastructures for attenuation of broadband vibration, Compos. Struct., 255 (2021) 112949.


Christian Chervin

A El-Kereamy

H Ibrahim

M Garcia

François Dedieu

Charles Romieu

Nathalie Ollat

J P Roustan

Ethanol application at veraison decreases acidity in Cabernet Sauvignon grapes

Keywords: Vitis vinifera, ripening, malate, tartrate, ethanol, acidity

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Introduction: As shown previously, spraying aqueous solutions of ethanol onto Cabernet Sauvignon clusters at mid-veraison increased the ethylene concentration in berries in comparison to controls sprayed with water [START_REF] Chervin | An ethanol spray at veraison enhances colour in red wines[END_REF]. This is similar to results obtained after spraying ethanol solutions onto tomatoes [START_REF] Beaulieu | Inhibition or promotion of tomato fruit ripening by acetaldehyde and ethanol is concentration dependent and varies with initial fruit maturity[END_REF]. Since ethylene is known to increase plant cell respiration [START_REF] Abeles | Ethylene in Plant Biology[END_REF] which is known to catabolise some of the malate pool in grapes [START_REF] Ruffner | Metabolism of tartaric and malic acids in Vitis: A review Part B[END_REF], we studied the influence of ethanol sprays on grape acidity.

Material and Methods:

The experiments were conducted over three consecutive years (1999)(2000)(2001) in a vineyard close to Toulouse, southwestern France. Cabernet Sauvignon vines (Vitis vinifera L.) grafted to 110 Richter rootstocks (Vitis Berlandieri x V. rupestris) were used. Bunches were sprayed with an aqueous solution of ethanol (5 % v/v) using a back-pack spray unit (200 l•ha -1 ). Controls were sprayed with water. This was done when approximately 50 % of the berries had started to change colour, 8 to 9 weeks after full bloom. Observations by [START_REF] Hale | Effects of ethylene and 2-chloroethylphosphonic acid on the ripening of grapes[END_REF] suggest that grapes respond to ethylene only after veraison is initiated. At harvest, 8 weeks later, batches of 60 berries were picked from a block of 5 vines. Three blocks were used per treatment. The experimental blocks were set up on different vines each year. For analyses, berries were weighed and then blended for 2 min in a Seb blender, 15 ml of this slurry were then centrifuged for 3 min at 2,000 g at room temperature. Brix values were obtained with a hand-held refractometer (Atago, Japan) using 200 µl of clear supernatant. Titratable acidity was assessed using 5 ml of supernatant with 0.1 NaOH (to a pH of 7.0). The concentrations of malic and tartaric acids were assessed by capillary electrophoresis on aliquots of the remaining supernatant as described by [START_REF] Arellano | Method development and validation for the simultaneous determination of organic and inorganic acids by capillary zone electrophoresis[END_REF].

Results and Discussion: Compared to controls titratable acidity of grape juice at harvest was reduced by 11 % due to ethanol sprayed at veraison (Table ). This reduction was significant at the 0.05 % level, even though calculation was based on sets of data from three consecutive seasons with different weather conditions and different sites in the vineyard with individual effects on vigour and ripening (data not shown). Detailed analyses of the two major grape acids showed that the decrease in acidity was due to a significant decrease (33 %) in the malic acid content in ethanol-treated grapes, compared to the controls (Table ). Tartaric acid concentration was not significantly altered and there was no significant change in the pH of the must.

The decrease in titratable acidity can be explained by the change in malic acid content. Calculations showed that cation contents were unaffected (Table ). This may indicate that there was no change in the rate of phloem loading between ethanol-treated and control plants. Potassium accumulation was shown to be a continuous process in ripening berries [START_REF] Possner | The localisation of acids, sugars, potassium and calcium in developing grape berries[END_REF]. The mean Brix values of the batches were not significantly different (19.2 ± 0.9 for controls and 20.0 ± 1.0 for treated grapes; means ± SE with n = 9 over three seasons; t-test P = 0.114). Thus the differences in acidity induced by ethanol cannot be explained by a global increase in maturity. They cannot be explained by a difference in berry weight either (1.30 ± 0.21 g for controls and 1.29 ± 0.19 g for treated grapes; means ± SE with n = 9 as above; t-test P = 0.904).

In a recent study, we reported that ethanol treatment enhanced colour of red wines and suggested that ethanol affects anthocyanin accumulation through an increased ethylene production [START_REF] Chervin | An ethanol spray at veraison enhances colour in red wines[END_REF]. The decrease in the malic acid content, observed here, agrees herewith, since the malate is a substrate of both respiration and gluconeogenesis. The rates of both of these pathways happen to be increased by ethylene [START_REF] Beaudry | Banana ripening: Implications of changes in glycolytic intermediate concentrations, glycolytic and gluconeogenic carbon flux, and fructose 2,6-bisphosphate concentration[END_REF][START_REF] Abeles | Ethylene in Plant Biology[END_REF]). However, ethanol may also act independently of ethylene. It is well known that malate respiration begins simultaneously with sugar accumulation in berries, just before colour changes [START_REF] Ruffner | Metabolism of tartaric and malic acids in Vitis: A review Part B[END_REF][START_REF] Coombe | Research on development and ripening of the grape berry[END_REF]. The onset of sugar storage is accompanied by a compartmentation breakdown of the apoplast, phloem unloading strongly increasing as a result of xylem rupture [START_REF] Lang | Partitioning control by water potential gradient: Evidence for compartmentation breakdown in grape berries[END_REF][START_REF] Coombe | Research on development and ripening of the grape berry[END_REF][START_REF] Dreier | Invertase activity, grape berry development and cell compartmentation[END_REF]. Regarding the effects of ethanol on phloem unloading we got no clear indication about changes in water, sugars and potassium (berry weight, Brix and calculated cations). Malate respiration may result from increased permeability of the vacuolar membrane (TER-RIER et al. 2001). Ethanol and its oxidation product, acetaldehyde may conceivably increase malate decompartmentation and respiration [START_REF] Terrier | Inhibition of vacuolar proton pumps by ethanol impairs grape berry compartmentation[END_REF].

Whether the treatment could have commercial applications or not will depend on further research on and development of commercial sprayers. Indeed the optimal ethanol concentration will vary according to the spraying equipment.

The authors are grateful to Dr. R. VAN HEESWIJCK (Adelaide, University, Australia) for fruitful discussions, to F. ATTIA for technical help and to Dr. C. DAVIES (CSIRO, Adelaide) and anonymous referees for valuable comments on the manuscript. They also thank the French Embassy in Australia for supporting this study with visiting fellowship grants.