List of Figures

List of Tables

 Table 3-
Chapter 1 State of the Art

Introduction Earthquake refers to the vibration of the earth's surface caused by the vibration of a place or a certain area on the earth's surface or below the earth's surface. [1][2][3][4][5] It is usually caused by some natural phenomena such as crustal movement, meteorite impact and volcanic eruption. It can also be caused by human activities such as nuclear weapon explosions. The catastrophic earthquakes known so far are all caused by violent movement of the earth's crust. Earthquakes can cause surface faults, surface tremors, surface liquefaction, landslides, aftershocks and even tsunamis. [2,[4][5][6][7] The time of earthquake occurrence, the number and intensity of aftershocks will make the disaster more serious. Compared with other natural disasters, earthquakes cause a great threat to the safety of people's lives and properties. In ancient times, such as the Jiajing Earthquake that occurred on January 1556, 830,000 people were killed and immeasurable economic losses were caused. [8] Modern cities are characterized by dense buildings and high occupancy rates, as well as problems such as underdeveloped early warning systems and lack of public awareness of earthquake risks, which all exacerbate the severity of earthquake disasters.

After the founding of the PRC, the Haicheng Earthquake occurred in Liaoning Province. [9][START_REF] Aki | Temporal change in coda Q before the Tangshan earthquake of 1976 and the Haicheng earthquake of 1975[END_REF][START_REF] Wang | Predicting the 1975 Haicheng earthquake [J][END_REF][START_REF] Wyss | How many lives were saved by the evacuation before the M 7.3 Haicheng earthquake of 1975?[END_REF] This is the only successful case where the government evacuated people in an orderly manner and significantly reduced the damage based on an accurate prediction of an earthquake. The number of casualties from the earthquake was 18,308, representing only 0.22% of the total number of people in the quake area, with only 0.02% of the total number of fatalities. A year later, the Tangshan Earthquake claimed the lives of more than 242,000 people. [START_REF] Aki | Temporal change in coda Q before the Tangshan earthquake of 1976 and the Haicheng earthquake of 1975[END_REF][START_REF] Sheng | Medical support in the Tangshan earthquake: a review of the management of mass casualties and certain major injuries [J][END_REF][START_REF] Liu | The great Tangshan earthquake of 1976[END_REF] Although there were numerous seismic observatories in the area where the earthquake occurred, the short-term forecasts completely failed. Because there was little foreshock activity prior to the Tangshan Earthquake and other precursor phenomena appeared late. As a result, the Tangshan earthquake was also listed as one of the top ten natural disasters of the 20th century. As of today, accurate earthquake prediction is still difficult to achieve.

Seismic resistant technology is an important method of protecting target buildings by designing and arranging them before earthquakes. [START_REF] Hüffmann | Full base isolation for earthquake protection by helical springs and viscodampers[END_REF][START_REF] Coburn | Earthquake Protection[END_REF][START_REF] Alhan | Reliability of base isolation for the protection of critical equipment from earthquake hazards[END_REF] Conventional seismic resistant technology resists the forces of earthquakes by increasing the strength, stiffness and ductility of the structural elements of the building, specifically by increasing the cross-sectional size of the structural elements and adding reinforced concrete walls to resist earthquakes. [START_REF] Hüffmann | Full base isolation for earthquake protection by helical springs and viscodampers[END_REF][START_REF] Coburn | Earthquake Protection[END_REF][START_REF] Alhan | Reliability of base isolation for the protection of critical equipment from earthquake hazards[END_REF][START_REF] Taranath | Wind and Earthquake Resistant Buildings: Structural Analysis and Design[END_REF][START_REF] Agrawal | Earthquake Resistant Design of Structures[END_REF][START_REF] Kmar E R, Mathur | A study on earthquake resistant construction techniques[END_REF] In recent years, new seismic resistant technology have become more "flexible", using seismic isolation and damping devices such as rubber bearings [START_REF] Grant | Bidirectional modelling of highdamping rubber bearings [J][END_REF][START_REF] Kalpakidis | Effects of heating on the behavior of lead-rubber bearings. I: Theory [J[END_REF][START_REF] Kalpakidis | Effects of heating on the behavior of lead-rubber bearings. II: Verification of theory[END_REF][START_REF] Kalpakidis | Modeling strength degradation in lead-rubber bearings under earthquake shaking[END_REF][START_REF] Kelly | Mechanics of Rubber Bearings for Seismic and Vibration Isolation[END_REF] and dampers [START_REF] Hüffmann | Full base isolation for earthquake protection by helical springs and viscodampers[END_REF][START_REF] Fu | Comparative study of frames using viscoelastic and viscous dampers [J][END_REF][START_REF] Pekcan | Fundamental considerations for the design of nonlinear viscous dampers[END_REF][START_REF] Youssef | Viscous dampers at multiple levels for the historic preservation of Los Angeles City Hall [J][END_REF][START_REF] Lin | Earthquake response of elastic SDF systems with non-linear fluid viscous dampers[END_REF] , as shown in Fig. 1-1, to isolate and dissipate seismic energy. When an earthquake occurs, the damping devices dissipate the seismic wave energy in the building by compressing, bending and deflecting it in a way that will change its structure and shape, thus protecting the building.

Figure 1-1 Seismic isolations [START_REF] Tafheem | Seismic Isolation Systems in Structures-the State of Art Review[END_REF] (a) Rubber bearing. (b) Viscous dampers In contrast, seismic metamaterials, [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] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Casablanca | Seismic isolation of buildings using composite foundations based on metamaterials[END_REF][START_REF] Geng | Issues in design of one-dimensional metamaterials for seismic protection[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Liu | Achieving ultra-broadband and ultra-lowfrequency surface wave bandgaps in seismic metamaterials through topology optimization[END_REF] proposed in recent years, can control seismic wave based on local resonance and Bragg scattering due to the bandgap properties. [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 twodimensional 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] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Norris | Phononic band gap crystals with periodic fractal inclusions: Theoretical study using numerical analysis[END_REF][START_REF] Kuo | Fractal phononic crystals in aluminum nitride: An approach to ultra high frequency bandgaps[END_REF][START_REF] Li | Tuning of acoustic bandgaps in phononic crystals with Helmholtz resonators[END_REF][START_REF] Gao | A hollow inclusion self-similarity phononic crystal with an ultra-low-frequency bandgap[END_REF][START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF] In the frequency range of the bandgap, seismic metamaterials can isolate seismic wave or even change the direction of seismic wave propagation. Seismic metamaterials can be placed around or at the base of a protected building to strongly attenuate seismic waves. [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Brule | Experimental evidence of auxetic features in seismic metamaterials: Ellipticity of seismic Rayleigh waves for subsurface architectured ground with holes[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Wang | Periodic in-filled pipes embedded in semi-infinite space as seismic metamaterials for filtering ultra-low-frequency surface waves[END_REF][START_REF] Liu | Radial seismic metamaterials based on layering theory: Broadband shielding of ultra-low frequency seismic surface waves[END_REF] Because the resonant frequencies of most man-made buildings are below 20 Hz, [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Agency | Risk Management Series: Designing for Earthquakes-a Manual for Architects[END_REF] the bandgaps of most seismic metamaterials are in this range. However, there is still relatively little research into the design and theory of seismic metamaterials that can operate in the wide low-frequency range.

This thesis uses theoretical model analyses, numerical simulations and experimental measurements to study seismic metamaterials with a wide low-frequency bandgap, which further advances seismic metamaterials towards practical applications. This work is supported by the National Natural Science Foundation of China (NSFC), the China Scholarship Council (CSC) and la Région Grand Est.

Earthquake prediction, early warning and resistant technology

Earthquake prediction and early warning technology

Earthquake prediction [9,[START_REF] Rikitake | Earthquake prediction[END_REF][START_REF] Mogi | Earthquake Prediction[END_REF][START_REF] Rikitake | Earthquake prediction: an empirical approach[END_REF][START_REF] Keilis-Borok | Earthquake prediction [J][END_REF][START_REF] Kanamori | Earthquake prediction: An overview[END_REF][START_REF] Harrington | Bam earthquake prediction & space technology[END_REF][START_REF] Berhich | A location-dependent earthquake prediction using recurrent neural network algorithms[END_REF][START_REF] Saini | An integrated framework for smart earthquake prediction: IoT, fog, and cloud computing[END_REF] refers to the prediction of the time, location and magnitude of earthquakes that will occur. It is specifically subdivided into long-term, medium-term and short-term forecasts. Although the scientific community can identify and focus on monitoring and prevention of "key earthquake risk areas" for long-term predictions on a decadal scale, and strengthen preparations for earthquake prevention and mitigation in key areas for medium-term predictions, there is still an urgent need to solve the problems of scientific methods and accuracy of earthquake predictions, especially short-term predictions. In particular, short-term prediction is still a worldwide scientific challenge. Current methods for predicting earthquakes are mainly based on empirical analysis rather than physical mechanisms, mainly because of three main reasons: [START_REF] Rikitake | Earthquake prediction[END_REF][START_REF] Kanamori | Earthquake prediction: An overview[END_REF][START_REF] Harrington | Bam earthquake prediction & space technology[END_REF][START_REF] Berhich | A location-dependent earthquake prediction using recurrent neural network algorithms[END_REF] 1. Although man-made space probes have flown more than 20 billion kilometers in space and may even fly out of the solar system in the future, the deepest excavation on earth is only 10 kilometers.

2. Earthquakes are small probability events. Although the annual number of earthquakes is nearly one million, the time interval between two major earthquakes in the same region is long. In some areas, the interval is even as high as tens of thousands of years, and the observation of geophysics is only a few decades, so it is difficult to generalize a reliable law.

3. The difficulty of earthquake prediction experiments. Real earthquake sources are difficult to be accurately excited in experiments, so it is difficult to obtain accurate experimental data for summarizing experience.

The earthquake early warning technology [START_REF] Allen | The potential for earthquake early warning in southern California[END_REF][START_REF] Gasparini | Earthquake Early Warning Systems[END_REF][START_REF] Allen | The status of earthquake early warning around the world: An introductory overview[END_REF][START_REF] Allen | Earthquake early warning: Advances, scientific challenges, and societal needs [J][END_REF][START_REF] Cremen | Investigating the potential effectiveness of earthquake early warning across Europe [J][END_REF][START_REF] Böse | Testing the shakealert earthquake early warning system using synthesized earthquake sequences[END_REF] , which has been commonly used in recent years, can also be realized under certain conditions to send warning messages to areas at a certain distance from the earthquake source several seconds or even tens of seconds in advance, thus reducing human casualties. The principle is that near-field seismic waves include longitudinal and transverse waves, where longitudinal waves are faster than transverse waves, but longitudinal waves are much less destructive than transverse waves. When the seismograph near the source detects the longitudinal wave and the trailing transverse wave, the time difference between the two can be used to calculate the time it takes for the transverse wave to reach the nearby areas. The time information is sent to various terminals, including but not limited to cell phones, TVs and smart watches, using developed communication networks. Thus, the earthquake warning can be sent to the nearby areas several seconds or even tens of seconds in advance.

On September 14, 1995, this technology enabled the release of earthquake alerts 72 seconds in advance for the Guerrero earthquake, which was the first time an earthquake alert was released to the public. [START_REF] Allen | The status of earthquake early warning around the world: An introductory overview[END_REF][START_REF] Böse | Testing the shakealert earthquake early warning system using synthesized earthquake sequences[END_REF][START_REF] Lee | Earthquake Early Warning Systems: Current Status and Perspectives[END_REF] In China, after the Wenchuan earthquake, the Chinese government has increased the construction of earthquake alerts in areas with high seismicity. As a result, research and technology in earthquake warning has advanced rapidly. In Sichuan Province, the earthquake warning system can be sent to the people nearby through village radios, TVs and mobile phone apps. When a 6.1 magnitude earthquake struck Lushan County, Ya'an City, Sichuan Province on 1 June 2022, the system issued an alert at the fourth second after the earthquake had occurred. The city of Ya'an and the city of Chengdu received the alert 9 and 29 seconds before the arrival of the seismic waves, respectively. [START_REF] Peng | Chinese nationwide earthquake early warning system and its performance in the 2022 Lushan M 6.1 earthquake[END_REF] However, there are certain problems with earthquake warning technology, i.e., the closer the epicenter, the greater the hazard but the shorter the warning time.

Earthquake resistant technology

Earthquake resistant technology has been an important method since ancient times to protect target buildings. For example, during the Ming Dynasty in China, the rulers designed the construction of the Forbidden City in Beijing to incorporate glutinous rice layer with damping effect to achieve earthquake resistance. This allowed the Forbidden City in Beijing to remain intact after many earthquakes over hundreds of years. [START_REF] Shi | The main factors and mechanism effecting on the earthquake resistance stability of grottoes relics and countermeasures[END_REF][START_REF] Hu | The earthquake-resistant properties of Chinese traditional architecture[END_REF][START_REF] Chen | A seismic capacity evaluation approach for architectural heritage using finite element analysis of three-dimensional model: A case study of the limestone hall in the ming dynasty[END_REF] Conventional seismic resistant technologies are designed to resist earthquake forces by increasing the strength, stiffness, and ductility of structural elements in buildings. [START_REF] Hüffmann | Full base isolation for earthquake protection by helical springs and viscodampers[END_REF][START_REF] Coburn | Earthquake Protection[END_REF][START_REF] Alhan | Reliability of base isolation for the protection of critical equipment from earthquake hazards[END_REF][START_REF] Taranath | Wind and Earthquake Resistant Buildings: Structural Analysis and Design[END_REF][START_REF] Agrawal | Earthquake Resistant Design of Structures[END_REF][START_REF] Kmar E R, Mathur | A study on earthquake resistant construction techniques[END_REF] Although they are effective, they have the following problems:

1. For large earthquakes, the safety of buildings needs urgent consideration.

2. It is difficult to apply to buildings that have suffered earthquake damage or other equipment and instruments inside the buildings that need to be protected.

3. The increase in the cross-sectional size of structural members and the thickening of reinforced concrete walls will make the building cost much higher. 4. The current building technology uses more high-strength and lightweight materials, which makes the cross-section of the structural members of the building smaller, the height of the building higher and the structural span of the building larger, thus making it difficult for the traditional Earthquake resistant technology to be adopted.

In recent years, the new earthquake resistant technologies (vibration isolation and damping technologies) have become more " use softness to conquer strength". The seismic isolation technology [START_REF] Tafheem | Seismic Isolation Systems in Structures-the State of Art Review[END_REF][START_REF] Zayas | A simple pendulum technique for achieving seismic isolation[END_REF][START_REF] Mohammed | Seismic isolation in buildings to be a practical reality: Behaviour of structure and installation technique[END_REF][START_REF] Martelli | Seismic isolation and other antiseismic systems: recent applications in Italy and worldwide[END_REF][START_REF] Tsang | Seismic isolation for low-to-medium-rise buildings using granulated rubbersoil mixtures: numerical study[END_REF][START_REF] Warn | A review of seismic isolation for buildings: historical development and research needs[END_REF] refers to the use of seismic isolation devices for fixing between the building foundation and the main body of the building. Seismic isolation devices can extend the self-oscillation period of a building to reduce the seismic acceleration response of the building. In earthquakes, seismic isolation devices can isolate seismic waves in the building foundation from the main body of the building, thus achieving protection for the building. [START_REF] Tafheem | Seismic Isolation Systems in Structures-the State of Art Review[END_REF][START_REF] Mohammed | Seismic isolation in buildings to be a practical reality: Behaviour of structure and installation technique[END_REF] The seismic damping technology [START_REF] Johnson | An investigation into the explosive deep drawing of circular blanks using the plug-cushion technique[END_REF][START_REF] Peng | The cushion scanning technique: a method of dynamic tongue sonography and its comparison with the transducer-skin coupling scanning technique during swallowing[END_REF][START_REF] Uraoka | Carbon dioxide submucosal injection cushion: an innovative technique in endoscopic submucosal dissection[END_REF][START_REF] Mokha | Further results on frictional properties of teflon bearings[END_REF][START_REF] Makris | Analysis of motion resisted by friction. II. Velocity-dependent friction [J][END_REF][START_REF] Fenz | Behaviour of the double concave friction pendulum bearing[END_REF][START_REF] Fenz | Modeling triple friction pendulum bearings for response-history analysis[END_REF][START_REF] Kumar | Characterizing friction in sliding isolation bearings[END_REF][START_REF] Guerreiro | Seismic tests and numerical modeling of a rolling-ball isolation system [J][END_REF][START_REF] Oliveira | Semi-active structural vibration control of base-isolated buildings using magnetorheological dampers [J][END_REF] refers to the use of seismic damping devices (e.g., connecting elements) to fix and connect certain locations in the building. Seismic damping devices can dissipate seismic wave energy in buildings by changing their structure and shape through compression, bending, and deflection to protect the buildings. [START_REF] Peng | The cushion scanning technique: a method of dynamic tongue sonography and its comparison with the transducer-skin coupling scanning technique during swallowing[END_REF][START_REF] Uraoka | Carbon dioxide submucosal injection cushion: an innovative technique in endoscopic submucosal dissection[END_REF] The excellent earthquake resistant devices should have the following characteristics: [START_REF] Tafheem | Seismic Isolation Systems in Structures-the State of Art Review[END_REF][START_REF] Mohammed | Seismic isolation in buildings to be a practical reality: Behaviour of structure and installation technique[END_REF] 1. Sufficient load-bearing capacity. Because earthquake resistant devices are generally placed between building structures, they need to carry most or even the entire mass of the building structure in the vertical direction.

2. Excellent seismic isolation capacity. As an earthquake resistant device, this performance is essential, especially the horizontal stiffness should be variable.

3. Better resetting ability. Good resetting ability can make the building automatically return to its original position after the occurrence of flat motion. 4. Good damping ability. Earthquake resistant devices need to have damping energy dissipation characteristics to lose the seismic wave energy transmitted from the bottom of the building to the superstructure.

At present, there are some limitations of earthquake resistant devices:

1. The earthquake resistant effect for high-rise buildings is not obvious.

2. It needs to be specifically analyzed according to the vibration direction of the building structure. The propagation direction of seismic waves will have a significant impact on the vibration direction of the building structure.

3. High-rise buildings are susceptible to wind effects. The horizontal stiffness of seismically isolated buildings is generally small, but high-rise buildings are susceptible to large wind loads and require high shear resistance.

4. The service life of some earthquake resistant devices is well below the life cycle of the building, requiring replacement at a later stage.

5. Earthquake resistant devices can take up a lot of space and can affect the aesthetics and functionality of a building.

6. Most earthquake resistant device is costly.

Seismic metamaterials

In the early 1990s, Sigalas, Economou [START_REF] Sigalas | Band structure of elastic waves in two dimensional systems[END_REF] and Kushwaha [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF] introduced the concept of phononic crystals based on photonic crystals. Acoustic waves or elastic waves in a phononic crystal will not propagate in certain frequency range due to Bragg scattering. Theoretical studies on periodic structures can be traced back to 1883, when Floquet [START_REF] Floquet | Sur les équations différentielles linéaires à coefficients périodiques[END_REF] studied wave propagation in one-dimensional periodic spring oscillators. In 1928, Bloch [START_REF] Bloch | Quantum mechanics of electrons in crystal lattices[END_REF] proposed Bloch's theorem, which extended Floquet's conclusion to three dimensions. Already in 1946, Brillouin suggested that mechanical and electromagnetic waves are analogous and uniform. Wave propagation in periodic discrete structures was studied and discussed in the book Wave Propagation in Periodic Structures. [START_REF] Brillouin | Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices[END_REF] In 1995, Martinez-Sala et al. [START_REF] Martínez-Sala | Sound attenuation by sculpture [J][END_REF] tested the acoustic transmission properties of a periodically arranged sculpture "Flowing Melody". The existence of an acoustic bandgap was experimentally demonstrated for the first time. The most distinctive feature of a phononic crystal is the bandgap property. The band structure of a phononic crystal consists of bandgaps and bands. In the frequency range of the bandgap, waves cannot propagate in phononic crystals. In the frequency range of the passband, waves can propagate normally in phononic crystals.

In 2000, Liu et al. [START_REF] Liu | Locally resonant sonic materials[END_REF] proposed an acoustic metamaterial based on the local resonance mechanism. The lowest bandgap of this acoustic metamaterial corresponds to a wavelength 300 times the lattice constant, achieving the purpose of "small size control of large wavelength". The frequency of the bandgap of the acoustic metamaterial is almost independent of the periodicity of the unit cell, but closely related to the resonance frequency of the resonance unit cell. The flat bands and the bandgaps can be found in the band structures. The group velocity of the wave in the acoustic metamaterial is zero at this time, so the wave cannot propagate in the acoustic metamaterial. Since then such artificial structures composed of periodic materials have been widely studied and designed for engineering applications such as subwavelength imaging, [START_REF] Luo | Subwavelength imaging in photonic crystals[END_REF][START_REF] Lu | Three-dimensional subwavelength imaging by a photonic-crystal flat lens using negative refraction at microwave frequencies[END_REF][START_REF] Xia | Negative refraction and subwavelength imaging in a hexagonal two-dimensional annular photonic crystal[END_REF] cloak, [START_REF] Torrent | Acoustic cloaking in two dimensions: a feasible approach[END_REF][START_REF] Vanbésien | Photonic-crystal-based cloaking device at optical wavelengths[END_REF][START_REF] Farhat | Ultrabroadband elastic cloaking in thin plates[END_REF][START_REF] Norris | Elastic cloaking theory[END_REF][START_REF] Vasić | Self-focusing media using graded photonic crystals: Focusing, Fourier transforming and imaging, directive emission, and directional cloaking[END_REF] energy harvesting, [START_REF]Light trapping and solar energy harvesting with 3D photonic crystals[END_REF][START_REF] Chen | Metamaterials-based enhanced energy harvesting: A review[END_REF][START_REF] Chang | Transparent polymer photovoltaics for solar energy harvesting and beyond[END_REF] and vibration & noise reduction. [START_REF] Liu | Achieving ultra-broadband and ultra-lowfrequency surface wave bandgaps in seismic metamaterials through topology optimization[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Shen | Evaluation of optical properties for real photonic crystal fiber based on total variation in wavelet domain[END_REF][START_REF] Lu | Determining the effective electromagnetic parameters of photonic crystal by phase unwrapping and denoising method[END_REF][START_REF] Lim | From photonic crystals to seismic metamaterials: A review via phononic crystals and acoustic metamaterials[END_REF][START_REF] Meng | 3D rainbow phononic crystals for extended vibration attenuation bands[END_REF][START_REF] Wang | Large bandgaps of two-dimensional phononic crystals with cross-like holes [J][END_REF][START_REF] Ma | Polarization bandgaps and fluid-like elasticity in fully solid elastic metamaterials[END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF] Based on the photonic crystals, phononic crystals and metamaterials, the metasurface [START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF][START_REF] Pu | A multiple scattering formulation for finitesize flexural metasurfaces[END_REF][START_REF] Habibullah | The role of nonlocal response in second harmonic generation at metasurfaces with triangular metaatoms[END_REF][START_REF] Yang | Dual-stimulus control for ultra-wideband and multidimensional modulation in terahertz metasurfaces comprising graphene and metal halide perovskites[END_REF][START_REF] Yuan | Manipulation of acoustic wave reflection for arbitrary reflecting surfaces based on acoustic metasurfaces[END_REF][START_REF] Seyed | Spatial domain communication technique for future chipless ID sensors based on vortex terahertz beams generated by metasurfaces[END_REF][START_REF] Liu | Metasurfaces enabled dual-wavelength decoupling of near-field and far-field encoding[END_REF] and topological phononic crystals [START_REF] Li | Topological design of phononic crystals for multiple wide band gaps[END_REF][START_REF] Chaunsali | Stress wave isolation by purely mechanical topological phononic crystals[END_REF][START_REF] Tang | Topological phononic crystals with tunable interface state based on local resonance[END_REF][START_REF] Laforge | Acoustic topological circuitry in square and rectangular phononic crystals[END_REF][START_REF] Hu | Local resonator stimulated polarization transition in metamaterials and the formation of topological interface states[END_REF][START_REF] Zhao | Elastic valley spin controlled chiral coupling in topological valley phononic crystals[END_REF] have been proposed in recent years.

In recent years, researchers inspired by the phononic crystals and acoustic metamaterials, have proposed seismic metamaterials. Its protection of buildings is achieved in a new way: Seismic metamaterials are placed around buildings to reflect and attenuate seismic waves for protection purposes.

An overview of seismic metamaterials

After an earthquake occurs, mechanical vibrations generated at the source propagate inside and on the surface of the Earth thus forming seismic waves. [1][2][3][4][5] Seismic waves can be simply distinguished into two main categories: bulk waves and surface waves. In the mid-and far-field regions of an earthquake, bulk waves are coupled on the half-space surface to form Rayleigh surface waves (also known as Rayleigh waves). [1,[3][4][5][START_REF] Coburn | Earthquake Protection[END_REF][START_REF] Mogi | Earthquake Prediction[END_REF][START_REF] Keilis-Borok | Earthquake prediction [J][END_REF][START_REF] Kanamori | Earthquake prediction: An overview[END_REF] The amplitude of its mass vibration decays exponentially along the ground depth direction, so that the energy of Rayleigh waves is almost entirely concentrated near the ground. [START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF] Rayleigh waves are usually found far from the earthquake source. Rayleigh waves have large amplitude, slow decay, and can propagate in the earth's surface many times after a large earthquake and still have great energy. [START_REF] Tanaka | Breaking pattern and critical breaking condition of Japanese pine trees on coastal sand dunes in huge tsunami caused by Great East Japan Earthquake [J][END_REF][START_REF] Ogawa | Effects of the great east Japan earthquake and huge tsunami on glycaemic control and blood pressure in patients with diabetes mellitus [J][END_REF] Therefore, in the mid-and far-field regions of an earthquake, Rayleigh waves are more destructive to buildings than bulk waves and can threaten buildings at a greater distance. So, the design and research of seismic metamaterials for Rayleigh waves is of great importance.

Seismic metamaterials, usually based on local resonance and Bragg scattering, use the bandgap properties [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 twodimensional 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] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Norris | Phononic band gap crystals with periodic fractal inclusions: Theoretical study using numerical analysis[END_REF][START_REF] Kuo | Fractal phononic crystals in aluminum nitride: An approach to ultra high frequency bandgaps[END_REF][START_REF] Li | Tuning of acoustic bandgaps in phononic crystals with Helmholtz resonators[END_REF][START_REF] Gao | A hollow inclusion self-similarity phononic crystal with an ultra-low-frequency bandgap[END_REF][START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF] to isolate and attenuate seismic waves or even change the direction of seismic waves propagation. Seismic metamaterials can be placed around or at the bottom of a building to control seismic waves. The resonant frequency of most man-made buildings is below 20 Hz, [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Agency | Risk Management Series: Designing for Earthquakes-a Manual for Architects[END_REF] so most of the current seismic metamaterials are design for this range. In addition, researchers often use common materials on construction sites to reduce the cost of seismic metamaterials, such as steel, industrial rubber, and concrete.

Research background of seismic metamaterials

There are a large number of theoretical studies and experimental verifications on seismic metamaterials. Seismic metamaterials can be divided into three categories according to the type of seismic waves they mainly target: seismic metamaterials that mainly target seismic bulk waves (referred to as bulk wave seismic metamaterials), seismic metamaterials that mainly target seismic Lamb waves (referred to as Lamb wave seismic metamaterials), and seismic metamaterials that mainly target seismic Rayleigh waves (referred to as Rayleigh wave seismic metamaterials).

Bulk wave seismic metamaterials are mostly composed of three-dimensional periodic structures and generally have a complete bandgap, which can attenuate seismic transverse and longitudinal waves from any direction. [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] Casablanca | Seismic isolation of buildings using composite foundations based on metamaterials[END_REF][START_REF] Kim | Seismic negative belt of acoustic metamaterials[END_REF] Lamb wave seismic metamaterials are mostly composed of one-and two-dimensional periodic structures and generally have a complete bandgap. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] Seismic Lamb waves are coupled by seismic transverse and longitudinal waves in plate-like geological layers and are mostly found in earthquakes with sources close to the Earth's surface. Seismic Lamb waves are also of great interest because they can carry large amounts of energy to propagate to longer distances, thus threatening buildings in the mid and far fields. [START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] When the thickness of the plate layer is large enough, both symmetric and antisymmetric waves of Lamb waves will propagate as Rayleigh waves on the surface of the plate. Therefore, Lamb wave seismic metamaterials can also be used to control Rayleigh waves under certain conditions. [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Brule | Experimental evidence of auxetic features in seismic metamaterials: Ellipticity of seismic Rayleigh waves for subsurface architectured ground with holes[END_REF][START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] Rayleigh wave seismic metamaterials are also mostly composed of one-and two-dimensional periodic structures, generally with directional bandgaps or complete bandgaps, and are mainly used to attenuate seismic Rayleigh waves in the low-frequency range. [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF][START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF]

Bulk wave seismic metamaterials

As shown in Fig. 1-2, [START_REF] Wang | Complete bandgaps in two-dimensional phononic crystal slabs with resonators[END_REF][START_REF] Wang | Two-dimensional locally resonant elastic metamaterials with chiral comb-like interlayers: Bandgap and simultaneously double negative properties[END_REF][START_REF] Wang | Tunable and active phononic crystals and metamaterials[END_REF][START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF][START_REF] Bursi | Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment[END_REF] the bulk wave seismic metamaterial is generally placed at the base of a building as a building foundation or around the building to isolate seismic waves.

Bao et al. [START_REF] Bao | Dynamic responses of a structure with periodic foundations[END_REF] designed a one-dimensional seismic metamaterial composed of concrete and rubber. The band structure of this seismic metamaterial was calculated using the transfer matrix method and the parameter optimization was given. As shown in Fig. 1-3(a), to verify the effectiveness of this seismic metamaterial, they used the finite element method to compare the dynamic response of a seven-story frame structure on a concrete foundation, a seismic foundation and the seismic metamaterial, considering the seismic waves propagating from any direction. The results show that the seismic metamaterial can well attenuate the seismic waves in horizontal, vertical and oblique incidence directions and efficiently reduce the dynamic response in the seven-story frame structure due to the bandgap property of the seismic metamaterial.

Figure 1-2 Placement of bulk wave seismic metamaterials [START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF][START_REF] Bursi | Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment[END_REF] (a) At the bottom of the building and (b) around the building Xiang et al. [START_REF] Xiang | Periodic materials-based vibration attenuation in layered foundations: experimental validation[END_REF] proposed a one-dimensional seismic metamaterial consisting of a periodic arrangement of reinforced concrete and rubber. They [START_REF] Bao | Dynamic responses of a structure with periodic foundations[END_REF] (b) Experimental layout of the seismic metamaterial. [START_REF] Xiang | Periodic materials-based vibration attenuation in layered foundations: experimental validation[END_REF] (c) study on the transmission of bulk waves in the seismic metamaterial [START_REF] Geng | Issues in design of one-dimensional metamaterials for seismic protection[END_REF] A one-dimensional seismic metamaterial composed of two elastic materials was also proposed by Geng et al. [START_REF] Geng | Issues in design of one-dimensional metamaterials for seismic protection[END_REF] The variation of the seismic metamaterial under different material combination conditions was analyzed using an analytical model. It was found that the larger the width of the bandgap, the smaller the center frequency of the seismic metamaterial when the difference between the mass density and wave velocity of the two materials is large. Moreover, reducing the filling rate of the softer material is also beneficial to the attenuation of seismic waves. This is consistent with the findings of Zhou et al. [START_REF] Zhou | Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals[END_REF] in phononic crystals. In addition, as shown in Fig. 1-3(c), they constructed numerical models of seismic metamaterials using the finite element method to calculate the responses of seismic metamaterials to seismic waves. The results show that the increase in the number of unit cells of seismic metamaterials leads to the enhancement of the ability of seismic metamaterials to attenuate seismic waves.

The combination of unit cells with different lattice constants leads to an increase in the attenuation frequency range of the seismic metamaterials.

Yan et al. [START_REF] Yan | Three dimensional periodic foundations for base seismic isolation[END_REF] designed a three-dimensional three-component seismic metamaterial consisting of steel, rubber and concrete from the concept of the phononic metamaterial.

The unit cell of the seismic metamaterial is shown in the left of Fig. 1-4(a). Its central section is shown in the right of Fig. 1-4(a). They calculated the band structure of the seismic metamaterial using the finite element software ABAQUS and found that it has a complete bandgap in the range of 32.9-35.6 Hz. In addition, they used the seismic metamaterial for field experimental verification. It is found that the seismic metamaterial can well attenuate the seismic transverse and longitudinal waves due to the bandgap property. Compared with the traditional reinforced concrete foundation, the seismic metamaterial can reduce the acceleration by more than 90%. The experimental results are basically consistent with the simulation results.

Aravantinos-Zafiris et al. [START_REF] Sigalas | Large scale phononic metamaterials for seismic isolation[END_REF] proposed a three-dimensional seismic metamaterial consisting of periodically arranged concrete pillars. The unit cell is shown in Fig. 1-4(b). The left shows the three-dimensional view and the right shows the central section of the unit cell. The six concrete pillars are connected at the center of the unit cell by a solid with a very small cross section. They calculated the band structure of the seismic metamaterial by the time-domain finite difference method. It was found it can be used to attenuate seismic waves below 50 Hz when the structural parameters of the concrete pillars are appropriate. When the lattice constant is 10 m, the complete bandgap is 7.9-124 Hz.

Achaoui et al. [START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF] proposed a three-dimensional seismic metamaterial with a unit cell consisting of a rubber pillar, a steel sphere and concrete. The unit cell is shown in the lower left corner in Fig. 1234). The steel sphere and concrete are connected by six rubber pillars. Each sphere is free to translate and rotate in the cavity in which it is located. This seismic metamaterial can be placed around a building or used as the foundation of a building. They used asymptotic formulas for the eigenfrequencies associated with the steel sphere to infer the frequency of the low-frequency bandgap.

The band structure is also calculated using the finite element software COMSOL Multiphysics. A wide bandgap of 8-49 Hz can be obtained for seismic metamaterials with a lattice constant of 1 m. The transmission results show that the seismic metamaterial based on the local resonance mechanism can attenuate seismic waves well.

Figure 1234The 3D bulk waves seismic metamaterials (a) The unit cell of the 3D seismic metamaterial consisting of concrete, rubber, and iron. [START_REF] Yan | Three dimensional periodic foundations for base seismic isolation[END_REF] (b) the unit cell of the 3D seismic metamaterial consisting of six concrete pillars. [START_REF] Sigalas | Large scale phononic metamaterials for seismic isolation[END_REF] (c) the unit cell of the 3D seismic metamaterial consisting of concrete, rubber pillars and iron ball [START_REF] Achaoui | Seismic waves damping with arrays of inertial resonators[END_REF] The literature mentioned in this section is all about bulk wave seismic metamaterials. It can be found that such seismic metamaterials can separate the building from the building foundation where seismic waves can propagate to. Thus, the building is protected from the seismic waves during an earthquake. However, the seismic metamaterials arranged at the bottom of the building need to hold the weight of the building. So, the structural strength of seismic metamaterials should be higher. These seismic metamaterials can isolate most seismic waves and can be an excellent seismic technology for major infrastructures, such as nuclear power plants and important military facilities.

Rayleigh wave seismic metamaterials

As shown in Fig. 1-5, Rayleigh wave seismic metamaterials are typically placed around buildings to protect them by isolating seismic wave propagation. This seismic metamaterial generally has a half-space substrate, so the sound cone is found in the band structure. The sound cone is usually necessary to study the surface waves. The bandgaps for surface waves are within the sound cone. [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF][START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF][START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF][START_REF] Basone | Finite locally resonant metafoundations for the seismic protection of fuel storage tanks[END_REF][START_REF] Colquitt | Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF] These seismic metamaterials can be simply divided into two categories, one in which the artificial structure is placed on the ground (including half-buried), and one in which the artificial structure is completely buried in the soil. The first one is the Rayleigh wave seismic metamaterials formed by placing artificial structures on the ground (including half-buried). These seismic metamaterials originate from phononic crystals that control surface waves in devices such as MEMS.

The phononic crystals can be used to filter Rayleigh waves at the millimeter-micron scale. [START_REF] Laude | Full band gap for surface acoustic waves in a piezoelectric phononic crystal[END_REF][START_REF] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Wu | Surface and bulk acoustic waves in twodimensional phononic crystal consisting of materials with general anisotropy[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][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] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF] Colombi et al. [START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF] and Roux et al. [START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF] found that forest can be considered as a complex resonator for attenuating low-frequency seismic Rayleigh waves. The test site is shown in Fig. 1-6(a), [START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF] where the periodically arranged yellow dots are seismometers distributed on the boundaries of rapeseed fields (upper area) and forests (lower area). They found that the good attenuation effect in the experiments depends on the longitudinal resonance and flexural resonance of trees with different heights, shapes and thicknesses. Meanwhile, the simulation results using the finite element method demonstrated that the seismic metamaterial composed of trees and the soil substrate produces low-frequency bandgaps due to its longitudinal resonance and flexural resonance of the trees.

Colombi et al. [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF] introduced the Rainbow trapping effect to propose a seismic metawedge composed of trees of different heights arranged in a gradient on the soil as shown in Fig. 123456). The different resonant frequencies of trees of different heights make the seismic metawedge can attenuate seismic Rayleigh waves in a large widefrequency range. The simulation results show that the seismic metawedge can completely attenuate the Rayleigh waves when Rayleigh waves are incident from the left side; when incident from the right side, the seismic metamaterial converts the Rayleigh waves into bulk waves.

Muhammad et al. [START_REF] Lim | Natural seismic metamaterials: the role of tree branches in the birth of Rayleigh wave bandgap for ground born vibration attenuation[END_REF] considered that trees would have different branches as shown in Fig. 123456). Seismic metamaterials consisting of these two types of trees arranged periodically were studied. They obtained the bandgap of Rayleigh waves using the sound cone based on the finite element method for calculations. The results show that both types of trees can well attenuate Rayleigh waves in the bandgap. In addition, they introduced the metawedge and also found that the seismic metawedges completely attenuate the Rayleigh waves when Rayleigh waves are incident from the left side; when incident from the right side, the Rayleigh waves are converted to bulk waves. Therefore, they suggested that afforestation could be an effective way to resist earthquakes.

To obtain wide low-frequency bandgaps, Zeng et al. [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] replaced trees with steel pillars. They compared seismic metamaterials made of circular, rectangular, and Ishaped pillars arranged periodically on the soil. They found that seismic metamaterials composed of I-shaped pillars are easy to obtain wider bandgaps for low-frequencies Rayleigh waves. Du et al. [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF] and Zeng et al. [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] also proposed seismic metamaterials consisting of periodic H-shaped fractal pillars and Matryoshka-like pillars on a half space, respectively. These metamaterials can obtain bandgaps for Rayleigh wave in multiple frequencies due to that there are multilevel resonance structure in each unit cell. As the number of resonance levels increases, the bandgaps at lower frequencies are also obtained. As shown in Fig. 1-6(d), Wu et al. [START_REF] Wu | Broadband Rayleigh wave attenuation by gradient metamaterials[END_REF] also proposed to replace trees with concrete pillars. They compared seismic metamaterials consisting of periodic concrete pillars standing on the ground, rubber-coated concrete pillars buried in the ground, and a combination of these two. They found that the half-buried seismic metamaterial is easy to obtain a bandgap at low frequency. In addition, after introducing the metawedge, they also found that the seismic metamaterials completely attenuate the Rayleigh waves when they are incident from the left side; when they are incident from the right side, the Rayleigh waves are converted into bulk waves. (a) Experimental layout of forest seismic metamaterials. [START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF] (b) Comparison of three seismic metamaterial. [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF] (c) Effects of tree collaterals and bifurcations on seismic metamaterials. [START_REF] Lim | Natural seismic metamaterials: the role of tree branches in the birth of Rayleigh wave bandgap for ground born vibration attenuation[END_REF] (d) Comparison of three seismic metamaterial [START_REF] Wu | Broadband Rayleigh wave attenuation by gradient metamaterials[END_REF] Colquitt et al. [START_REF] Colquitt | Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction[END_REF] presented the typicality problem of standing rods on thin plates and half-space substrates. They theoretically explained the interaction between the waves in the substrate (in the plate and half-space) and the resonator structure (rods) on the substrate. They calculated the band structures of a metamaterial composed of periodic standing rods on a plate and half-space as shown in Fig. 1234567(a) by the improved Spectral Element Method. They found that the bending and compression resonance of the rod caused by the interaction of the elastic wave with the rod attenuates the wave propagation, both in the plate and in the half space. Pu et al. [START_REF] Pu | A multiple scattering formulation for finitesize flexural metasurfaces[END_REF] proposed a multiple scattering formulation to study the interaction between Rayleigh waves and a plate standing on an elastic half-space surface as shown in Fig. 1234567. They found that the scattered wave field caused by the bending resonance of the plate has an attenuating effect on Rayleigh waves. These works all provide a theoretical basis for the study of Rayleigh wave seismic metamaterials.

Liu et al. [START_REF] Liu | Achieving ultra-broadband and ultra-lowfrequency surface wave bandgaps in seismic metamaterials through topology optimization[END_REF] proposed that topology optimization can be used to design Rayleigh wave seismic metamaterials. As shown in Fig. 1-7(c), part of the region above the halfspace can be set as a design domain consisting of many small pixels. A "0" in a pixel means no solid material, while a "1" means a solid material is present. Finally, all "1" pixels form the topology of the seismic metamaterial. Rayleigh wave seismic metamaterials with an ultra-wide and ultra-low frequency bandgap composed of two materials were realized by topology optimization with set constraints and boundary restrictions. They found that the bandgap of the topologically optimized seismic metamaterial can have a minimum frequency as low as 1.6 Hz and a relative bandwidth of 150%. They also found that optimized structures consisting of "1" pixels generally have large masses and slim connecting axes. This kind of structure can enhance the local resonance capability of the structure and reduces the frequency of the local resonance bandgap.

The second one is Rayleigh wave seismic metamaterials formed by burying artificial structure in the soil. In 1999, Meseguer et al. [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF] drilled two-dimensional honeycomb lattice and triangular lattice cylindrical holes in the marble surface to measure the attenuation effects of such holes on Rayleigh waves. They demonstrated experimentally that this seismic metamaterial can attenuate Rayleigh waves in a frequency range. They demonstrated that this is due to the bandgap property of the seismic metamaterial by using theoretical calculations. They proposed for the first time that phononic crystals can be used to attenuate seismic Rayleigh waves, and also provided an experimental basis for the study of seismic metamaterials. (a) Periodic rods on an elastic thin plate and a half space. [START_REF] Colquitt | Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction[END_REF] (b) Periodic plates on an elastic half space. [START_REF] Pu | A multiple scattering formulation for finitesize flexural metasurfaces[END_REF] (c) Topology optimization design of the pillar-shaped seismic metamaterial [START_REF] Liu | Achieving ultra-broadband and ultra-lowfrequency surface wave bandgaps in seismic metamaterials through topology optimization[END_REF] Krödel et al. [START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF] proposed an elastic metamaterial based on a kind of local resonance structure. This resonant structure is shown in the left of Fig. 12345678, where soft material connects the upper and lower ends of the steel mass to the aluminum box. This elastic metamaterial can obtain a wide attenuation zone using multiple resonant structures. Using simulations and experiments shown in the right of 1-8(a), they found that the attenuation effect of this elastic metamaterial increases with the number of resonant structures. The buried seismic metamaterials and experimental layout. [START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF] (b) The transmission of the Rayleigh waves in the buried seismic metamaterials. [START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF] (c) The buried seismic metamaterials with multiple mass resonance. [START_REF] Palermo | Metabarriers with multi-mass locally resonating units for broad band Rayleigh waves attenuation[END_REF] (d) The simple buried seismic metamaterials and experimental layout [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] Palermo et al. [START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF] introduced a similar resonant structure to design a Rayleigh wave seismic metamaterial as shown in Fig. 1-8(b). They found that the seismic metamaterial can convert seismic Rayleigh waves to bulk waves and proposed an analytical model based on effective medium theory to explore the mechanism of this mode conversion. They used model experiments to demonstrate the attenuation effect of this seismic metamaterial on Rayleigh waves. The results show that this seismic metamaterial can reduce the surface motion by up to 50% in the bandgap. Palermo et al. [START_REF] Palermo | Metabarriers with multi-mass locally resonating units for broad band Rayleigh waves attenuation[END_REF] proposed a Rayleigh wave seismic metamaterial composed of multi-mass resonators as shown in Fig. 12345678). This seismic metamaterial can attenuate seismic Rayleigh waves in multiple frequency ranges. They designed a multi-mass resonator with minimum total mass to attenuate seismic Rayleigh waves in an ultra-wide frequency range by using genetic algorithm.

Zeng et al. [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] proposed a Rayleigh wave seismic metamaterial using rubber covered steel blocks and periodically buried in the ground as shown in the left of Fig. 12345678) for a unit cell. This seismic metamaterial has a simple and easy-to-implement structure with a wide low-frequency bandgap with a relative bandwidth of about 1. The 1:30 lab-scale experiments shown in the right of Fig. 12345678. Experimental results and simulations results show that this seismic metamaterial can attenuate Rayleigh waves well within the bandgap.

Liu et al. [START_REF] Liu | Topology optimization of periodic barriers for surface waves[END_REF] proposed that a topological optimization framework based on genetic algorithm and finite element method can be used to design Rayleigh wave seismic metamaterials. As shown in Fig. 123456789, this seismic metamaterial is composed of a periodic artificial structure buried in the ground. The part of the area below the half space can be set as a design domain consisting of many small pixels. The "light blue area" within the pixel indicates the substrate material (soil), while the "red area" indicates that it contains the design material (steel, concrete or lead). The "light blue area" and the "red area" form the topology of the seismic metamaterial. The topological optimization results show that as the mass density of the scatterer represented by the "red region" increases, the surface modes of the seismic metamaterial become richer and the bandgap becomes larger. The bandgap also widens as the depth of the design domain increases, especially the high-frequency bandgap. They found a feature of the optimal artificial structure with the first bandgap: there is a main scatterer and multiple small scatterers in the structure, with the main scatterer located at the center of the design domain within the unit cell and the multiple small scatterers near the surface.

The main scatterer determines the lower boundary of the first bandgap, and the small scatterers determine the upper boundary of the first bandgap. [START_REF] Liu | Topology optimization of periodic barriers for surface waves[END_REF] This subsection presents some of the research on Rayleigh wave seismic metamaterials. There is also a large amount of related work, [START_REF] Casablanca | Seismic isolation of buildings using composite foundations based on metamaterials[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Brule | Experimental evidence of auxetic features in seismic metamaterials: Ellipticity of seismic Rayleigh waves for subsurface architectured ground with holes[END_REF][START_REF] Wang | Periodic in-filled pipes embedded in semi-infinite space as seismic metamaterials for filtering ultra-low-frequency surface waves[END_REF][START_REF] Liu | Radial seismic metamaterials based on layering theory: Broadband shielding of ultra-low frequency seismic surface waves[END_REF][START_REF] Kim | Seismic negative belt of acoustic metamaterials[END_REF][START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF][START_REF] Zhang | Evanescent surface acoustic waves in 1D viscoelastic phononic crystals[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF] which is not detailed here.

Lamb wave seismic metamaterials

As shown in Fig. 12345, Lamb wave seismic metamaterials are generally placed around buildings to isolate the seismic waves. [START_REF] Wang | Precise and target-oriented control of the lowfrequency Lamb wave bandgaps[END_REF][START_REF] Gao | Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer: Effect of substrate[END_REF] As mentioned earlier, when the thickness of the plate waveguide is large enough, both symmetric and antisymmetric Lamb waves modes will propagate as Rayleigh waves on the surface of the plate. Therefore, Lamb wave seismic metamaterials can also be used to control seismic Rayleigh waves under certain conditions. Brûlé et al. [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF] proposed a seismic metamaterial composed of periodic holes in the ground as shown in Fig. 1-10(a). They calculated the band structure of the seismic metamaterial using the Mindlin plate model. [START_REF] Mindlin | Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates [J][END_REF] The first large-scale field experiments on seismic metamaterials were designed. As shown in the right of Fig. 1-10(a), a large exciter is used to generate the seismic source on one side of the seismic metamaterial, and three-component velocimeters are used to measure the elastic waves on the surface.

The results show that the experimental results are consistent with simulations using the Mindlin plate model: this seismic metamaterial can shield elastic waves around 50 Hz.

Miniaci et al. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF] comparatively studied three Lamb waves seismic metamaterials with different unit cells as shown in Fig. 1-10(b). Using the finite element software COMSOL Multiphysics, they calculated and compared the band structures of these three seismic metamaterials for different geometrical parameters. The attenuation of seismic metamaterials to seismic Lamb and Rayleigh waves are analyzed respectively.

In addition, they discussed the effect of viscoelasticity of soil on seismic wave attenuation based on simulation analysis of large-scale seismic metamaterials. They pointed out that considering the wavelength of low-frequency seismic waves, the depth of the structure buried in the soil substrate can be deepened to increase the attenuation effect. The viscoelasticity of the soil plays a positive role in the effect of seismic wave attenuation.

Figure 1-10 The Lamb waves seismic metamaterial (a) The seismic metamaterial formed by periodically arranging holes in soil. [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF] (b) Comparison of three seismic metamaterial. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF] (c) The seismic metamaterials formed by periodically arranging pillars on bedrock [START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF] As shown in Fig. 1-10(c), Achaoui et al. [START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF] introduced a plate structure model with the bottom being fixed to design a two-dimensional seismic metamaterial. In the Kirchhoff-Love thin-plate model with the bottom surface being fixed, the periodic structure easily obtains a zero-frequency bandgap, i.e., a bandgap starting from 0 Hz. In practical applications, the seismic metamaterials can attenuate ultra-low frequency seismic waves by periodically inserting pillars into the soil and clamping pillars in a giant bedrock. In a sedimentary basin environment with a giant bedrock at the bottom and a soil depth of 15 meters, this seismic metamaterial can attenuate seismic waves below 30 Hz.

Du et al. [START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] proposed a two-dimensional seismic metamaterial composed of periodically arranging square concrete pillars in soil. Attenuation of Lamb and Rayleigh waves can be achieved. For Lamb waves, they found that the square-shape pillar is easier to obtain a wider complete bandgap compared to cylindrical pillar. For Rayleigh waves, they found that this seismic metamaterial also has a low-frequency bandgap after introducing a half-space substrate. In addition, they discussed the effects of geometrical and material parameters on the width of the bandgap.

Figure 1-11 The Lab-scale experiments [START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF] Chen et al. [START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF] proposed a two-dimensional seismic metamaterial that can attenuate low-frequency seismic waves in multilayered soils. They demonstrated the attenuation effect of the seismic metamaterial constituted of periodic pillars in soil by using the lab-scale experiments shown in Fig. 1-11. The attenuation frequency range coincides with the bandgap calculated by numerical simulation. They introduced multiple layered soil. It was found that zero-frequency bandgap can be achieved when the underlying material can clamp the pillars.

The literature mentioned in this section are Lamb wave seismic metamaterials, which can also attenuate Rayleigh waves at sufficient depth. These seismic metamaterials can improve the seismic performance of existing buildings such as ancient buildings. For Lamb waves, these seismic metamaterials can have a significant effect. For Rayleigh waves, these seismic metamaterials need to be buried at sufficient depth to be effective. To achieve attenuation of seismic waves in the low frequency range, specific conditions need to be met, such as a giant bedrock.

Conclusions

The state of art and literature review introduces the main hazards of earthquakes to humans, as well as earthquake prediction, early warning methods, earthquake resisting techniques, and points out the shortcomings in these methods and techniques.

Then the seismic metamaterials, detailing bulk waves, Lamb waves, and Rayleigh wave seismic metamaterials are presented in this chapter.

Chapter 2 Methods

Seismic metamaterials are composed of periodic artificial structures, similar to crystals in solid state physics. [START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF][START_REF] Wang | Lumped-mass method for the study of band structure in two-dimensional phononic crystals[END_REF][START_REF] Hsu | Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates[END_REF][START_REF] Wang | Wave band gaps in three-dimensional periodic piezoelectric structures[END_REF][START_REF] Oudich | Complex band structures and evanescent Bloch waves in two-dimensional finite phononic plate[END_REF][START_REF] Du | Multiband and polarization-independent left-handed metamaterial with cross fractal structure[END_REF][START_REF] Liu | Band structure characteristics of T-square fractal phononic crystals[END_REF] Therefore, seismic waves in seismic metamaterials have dispersive properties, also known as dispersion curves or band structures. [START_REF] Sigalas | Band structure of elastic waves in two dimensional systems[END_REF][START_REF] Kushwaha | Acoustic band structure of periodic elastic composites[END_REF][START_REF] Wang | Lumped-mass method for the study of band structure in two-dimensional phononic crystals[END_REF] Usually there is a special phenomenon in the band structure, which is the bandgap. [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 twodimensional 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] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Norris | Phononic band gap crystals with periodic fractal inclusions: Theoretical study using numerical analysis[END_REF][START_REF] Li | Tuning of acoustic bandgaps in phononic crystals with Helmholtz resonators[END_REF][START_REF] Wang | Complete bandgaps in two-dimensional phononic crystal slabs with resonators[END_REF][START_REF] Zhou | Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals[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][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] Gao | Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer: Effect of substrate[END_REF][START_REF] Wang | Wave band gaps in three-dimensional periodic piezoelectric structures[END_REF][START_REF] Yan | Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals[END_REF][START_REF] Khelif | Complete band gaps in twodimensional phononic crystal slabs[END_REF][START_REF] Wu | Evidence of complete band gap and resonances in a plate with periodic stubbed surface[END_REF] When the seismic wave frequency falls inside the bandgap, it is difficult for the seismic waves to propagate in the seismic metamaterial. When it falls outside the bandgap, seismic waves can propagate in it.

Most of the proposed ones in this thesis are Rayleigh wave seismic metamaterials, so there are deep substrates in the model and sound cone in the band structure to obtain the surface wave bands. [START_REF] Laude | Full band gap for surface acoustic waves in a piezoelectric phononic crystal[END_REF][START_REF] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Wu | Surface and bulk acoustic waves in twodimensional phononic crystal consisting of materials with general anisotropy[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][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] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF][START_REF] Wu | Evidence of complete band gap and resonances in a plate with periodic stubbed surface[END_REF][START_REF] Clorennec | Analysis of surface acoustic wave propagation on a cylinder using laser ultrasonics[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Khelif | Surface acoustic waves in pillars-based two-dimensional phononic structures with different lattice symmetries[END_REF][START_REF] Achaoui | Local resonances in phononic crystals and in random arrangements of pillars on a surface[END_REF] This thesis focuses on the study of the properties of seismic metamaterials using the finite element method. The results of the finite element method are also demonstrated by the analytical method of the transfer matrix to the simplified model. Therefore, this chapter presents the physical principles, finite element method and transfer matrix method involved in the calculation of the properties of seismic metamaterials.

Basic theory

The basic theory involved in the calculation of the properties of metamaterials have been described in detail in several works [START_REF] Wang | Tunable and active phononic crystals and metamaterials[END_REF][START_REF] Wu | Surface and bulk acoustic waves in twodimensional phononic crystal consisting of materials with general anisotropy[END_REF][START_REF] Wang | Lumped-mass method for the study of band structure in two-dimensional phononic crystals[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Wang | Two-dimensional locally resonant phononic crystals with binary structures[END_REF][START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF][START_REF] Khelif | Phononic Crystals[END_REF] and are only briefly described in this section.

Elastic wave equation and Rayleigh waves

In solid media that is continuous, linearly elastic with small deformation, homogeneous and isotropic, the elastic wave equation is: [START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF][START_REF] Khelif | Phononic Crystals[END_REF]

𝜌𝒖̈= [𝜆 + 2𝜇]∇(∇ • 𝒖) -∇ × [𝜇∇ × 𝒖],

(2-1)

where 𝜌 is the mass density; 𝒖 is the displacement vector; 𝜆 and 𝜇 are the Lamé

𝜆 = 𝜈𝐸 (1 + 𝜈)(1 -2𝜈) , (2-2) 𝜇 = 𝐸 2(1 + 𝜈) . (2-3)
When the frequency of the displacement field of a wave is ω, the Eq. (2-1) can be written as

𝜔 2 𝒖 = 𝜌 -1 {[𝜆 + 2𝜇]∇(∇ • 𝒖) -∇ × [𝜇∇ × 𝒖]}. (2-4)
Rayleigh waves are a type of wave that propagate along a surface. In a half-infinite homogeneous elastic medium, the wave speed 𝑣 𝑅 of Rayleigh waves is constant. It can be considered as a coupling of two inhomogeneous plane waves (P and SV waves).

The Rayleigh wave velocity 𝑣 𝑅 is related to the longitudinal wave velocity 𝑣 𝑃 and the transverse wave velocity 𝑣 𝑆 by the following equation: [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF][START_REF] Rayleigh | On waves propagated along the plane surface of an elastic solid[END_REF][START_REF] Buchwald | Rayleigh waves in transversely isotropic media[END_REF][START_REF] Alterman | Propagation of Rayleigh waves in the earth[END_REF][START_REF] Xia | Estimation of near-surface shear-wave velocity by inversion of Rayleigh waves[END_REF]

0 = (2 - 𝑣 𝑅 2 𝑣 𝑆 2 ) 2 -4√1 - 𝑣 𝑅 2 𝑣 𝑃 2 √1 - 𝑣 𝑅 2 𝑣 𝑆 2 , (2-5)
when

𝑘 = 𝑣 𝑅 2 𝑣 𝑆 2 ⁄ and 𝑛 = 𝑣 𝑆 2 𝑣 𝑃 2 ⁄ , Eq. (2-5
) can be rewritten as the Rayleigh equation: [186,187,189]

𝑘 3 -8𝑘 2 + (24 -16𝑛)𝑘 -16(1 -𝑛) = 0. (2-6)
The parameter k is obtained by solving Eq. (2-6). The 𝑣 𝑅 is obtained when the 𝑣 𝑆 and the 𝑣 𝑃 (or Poisson's ratio 𝜈) of the substrate material are known. The equation has real roots in the interval 0 < k < 1, meaning that the 𝑣 𝑅 is less than the 𝑣 𝑆 . In halfinfinite homogeneous elastic media, the 𝑣 𝑅 is independent of frequency, i.e., the Rayleigh wave is a non-dispersive wave. If there is a cover layer of other media on the surface of the medium, the Rayleigh wave is a dispersive wave.

Periodic condition and Bloch's theorem

Seismic metamaterials are usually periodic structures, similar to crystals in solid state physics. The position vector of a unit cell of a periodic structure is {𝒂 1 , 𝒂 2, 𝒂 3 }.

The linear overlap of these three basis vectors can form a periodic structure in space.

The lattice vector is

𝑹 𝑛 = 𝑛 1 𝒂 1 + 𝑛 2 𝒂 2 + 𝑛 3 𝒂 3 (2-7)
where 𝑛 1 , 𝑛 2 and 𝑛 3 are integers, all physical parameters 𝑓(𝒓) at any 𝒓 also have periodicity: [START_REF] Deymier | Acoustic Metamaterials and Phononic Crystals[END_REF][START_REF] Khelif | Phononic Crystals[END_REF] 𝑓(𝒓 + 𝑹 𝑛 ) = 𝑓(𝒓).

(2-8)

The Fourier expansion of 𝑓(𝒓) is:

𝑓(𝒓) = ∑ 𝐹(𝑮 𝒉 )e i𝑮 ℎ 𝑟 ℎ , (2-9)
where 𝑮 ℎ is the inverse lattice vector in the inverse space (Fourier space). 𝑮 ℎ is related to 𝑹 𝑛 as follows:

𝑮 ℎ • 𝑹 𝑛 = 2𝑚𝜋, (2-10)
where m is an integer. In the inverse space, the inverted lattice vector satisfies

𝑮 ℎ = ℎ 1 𝒃 1 + ℎ 2 𝒃 2 + ℎ 3 𝒃 3 , (2-11)
where

𝒃 1 = 2𝜋 𝒂 2 × 𝒂 3 𝒂 1 • (𝒂 2 × 𝒂 3 ) , (2-12
)

𝒃 2 = 2𝜋 𝒂 3 × 𝒂 1 𝒂 1 • (𝒂 2 × 𝒂 3 ) , (2-13
)

𝒃 3 = 2𝜋 𝒂 1 × 𝒂 2 𝒂 1 • (𝒂 2 × 𝒂 3 ) . (2-14)
When an elastic wave propagates in a periodic structure, the solution of the elastic wave equation can be written in the following form: [START_REF] Bloch | Quantum mechanics of electrons in crystal lattices[END_REF] 𝒖(𝒓, 𝑡) = e -i𝜔𝑡 ∑ 𝒖 𝒌+𝑮 𝒉 e i(𝒌+𝑮 𝒉 )•𝒓 = e i(𝒌•𝒓-𝜔𝑡) 𝒖 𝒌 (𝒓), (2-15) where k is the wave vector; ω is the eigenfrequency. From Eq. (2-15), the result remains the same when the wave vector k adds an inverted lattice vector 𝑮 ℎ . Thus, the results within the whole periodic structure can be represented by calculating the eigenvalues corresponding to the wave vectors k in a small region. This small region is called the first Brillouin zone. The first Brillouin zone can be obtained as an irreducible Brillouin zone due to its symmetry. The first irreducible Brillouin zone for the seismic metamaterials with square lattice: M = (π/a, π/a), Γ = (0, 0), X = (π/a, 0)

For example, the irreducible Brillouin zone of a periodic structure with a twodimensional square lattice arrangement and a high symmetry internal structure is the Γ -X -M -Γ as shown in Fig. 2-1. [START_REF] Maurin | Probability that a band-gap extremum is located on the irreducible Brillouin-zone contour for the 17 different plane crystallographic lattices[END_REF] The band structure is obtained by scanning the wave vector k within this irreducible Brillouin zone.

Real band structures

The "band structure" in this thesis refers specifically to the "real band structure".

In the band structure, the frequency range where the band exists is called the pass band, and the frequency range where the band does not exist is called the bandgap (or stop band). [START_REF] Zhang | Wave propagation in one-dimensional fluid-saturated porous phononic crystals with partial-open pore interfaces[END_REF] In the frequency range of the pass band, the wave is able to pass through the periodic structure. Waves cannot pass through the periodic structure in the bandgap. [START_REF] Li | Tuning of acoustic bandgaps in phononic crystals with Helmholtz resonators[END_REF][START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Wang | Large bandgaps of two-dimensional phononic crystals with cross-like holes [J][END_REF][START_REF] Wang | Complete bandgaps in two-dimensional phononic crystal slabs with resonators[END_REF][START_REF] Wang | Two-dimensional locally resonant elastic metamaterials with chiral comb-like interlayers: Bandgap and simultaneously double negative properties[END_REF] For example, in the band structure shown in Fig. 2-2(a), [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] there are dispersion curves (or bands) in the frequency range less than 8 Hz, so there are pass bands. In the range of 8-14 Hz there is no band, so it is a bandgap. Therefore, based on the bandgap characteristics, the seismic metamaterial can isolate the seismic waves to protect the building.

The finite element method

Solid mechanics module

This subsection presents the calculations of the band structure and transmission spectrum of seismic metamaterials by the solid mechanics module of COMSOL Multiphysics 5.4. As shown in Fig. 2-3(a), [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] the seismic metamaterial generally consists of a deep substrate (gray part) and an artificial structure (blue part). The periodic boundary conditions are set on the four surfaces perpendicular to the ground according to Bloch's theorem. A fixed constraint is applied at the bottom of the substrate.

It is worth noting that the results of low-frequency surface waves in band structures are more convergent when the thickness of the substrate is deep enough.

Substituting the displacement condition of the periodic boundary [START_REF] Wang | Large bandgaps of two-dimensional phononic crystals with cross-like holes [J][END_REF][START_REF] Wang | Complete bandgaps in two-dimensional phononic crystal slabs with resonators[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][START_REF] Achaoui | Experimental observation of locallyresonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars[END_REF] 𝒖(𝒓 + 𝐚) = e i(𝒌•𝐚) 𝒖(𝒓), (2-16) into Eq. (2-1), the standard eigenvalue equation is obtained:

(𝐊 + 𝜔 2 𝐌)𝒖 = 0, (2-17)
where 𝐚 is the lattice constant vector; K and M are the stiffness and mass matrices of the unit cell, respectively. Solving the Eq. (2-17) and scanning the wave vector k in the irreducible Brillouin zone, the dispersion relation between the frequency ω and the wave vector k can be obtained, i.e., the band structure as shown in Fig. 23). Due to the deep substrate, it is easy to find sound cones in the band structure, i.e., outside the dark gray region in Fig. 2-3(b). It is often found in the study of Rayleigh wave metamaterials. [START_REF] Lim | Natural seismic metamaterials: the role of tree branches in the birth of Rayleigh wave bandgap for ground born vibration attenuation[END_REF][START_REF] Li | Ultrafast Rayleigh-like streaming in a sub-wavelength slit between two phononic crystal plates[END_REF][START_REF] Liu | Partially embedded gradient metabarrier: Broadband shielding from seismic Rayleigh waves at ultralow frequencies[END_REF] In all bands within the sound cone, the vibration modes of the seismic metamaterial are shown in Fig. 2-3(c), all of which are surface wave modes, i.e., the vibration exists only near the surface. So, the bandgaps for surface waves can be found within the sound cone. Outside the sound cone, there are complex vibration modes, including bulk wave modes and pseudo-surface wave modes. [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF] Figure 2-3 Calculation of band structure of the seismic metamaterial [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] ( The calculation of the transmission spectrum of seismic Rayleigh waves is a common method to study the bandgap properties of seismic metamaterials. [START_REF] Finocchio | Seismic metamaterials based on isochronous mechanical oscillators[END_REF][START_REF] Geng | Issues in design of one-dimensional metamaterials for seismic protection[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Wang | Periodic in-filled pipes embedded in semi-infinite space as seismic metamaterials for filtering ultra-low-frequency surface waves[END_REF][START_REF] Kim | Seismic negative belt of acoustic metamaterials[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF] spectrum of the seismic metamaterial is [START_REF] Liu | Achieving ultra-broadband and ultra-lowfrequency surface wave bandgaps in seismic metamaterials through topology optimization[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Bursi | Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment[END_REF][START_REF] Zeng | Coupling the first and second attenuation zones in seismic metasurface[END_REF][START_REF] Shi | Feasibility of reducing three-dimensional wave energy by introducing periodic foundations[END_REF][START_REF] Yan | Seismic isolation of two dimensional periodic foundations[END_REF][START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF] 𝑇 𝑠 = 20 log 10 (𝐴 1 /𝐴 0 ).

(2-18)

Multibody dynamics module

This subsection presents the calculations of the band structure of seismic metamaterials with hinge joints by the multibody dynamics module of COMSOL Multiphysics 5.4.

Figure 2345The unit cell of the beam metamaterial [START_REF] Li | Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects[END_REF] As shown in Fig. 2345, a unit cell of an inertially amplified metamaterial is shown.

According to Bloch's theorem, periodic boundary conditions are added to the boundary on both sides of the unit cell. It is worth noting that both boundary similarity and pointwise constraint are utilized to achieve the periodic boundary condition in the multibody dynamics module. Similarly, the band structure can be obtained by scanning the wave vector.

Partial differential equation module

This subsection presents the calculations of the complex band structure of seismic metamaterials by the partial differential equation (PDE) module of COMSOL Multiphysics 5.4.

The governing equation in the PDE module is [START_REF] Wang | Large bandgaps of two-dimensional phononic crystals with cross-like holes [J][END_REF][START_REF] Wang | Wave band gaps in three-dimensional periodic piezoelectric structures[END_REF][START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF][START_REF] Wang | Coupling of evanescent and propagating guided modes in locally resonant phononic crystals[END_REF]]

Λ 2 𝑒 𝑎 𝒖 -Λ𝑑 𝑎 𝒖 + ∇ • (-𝑐∇𝒖 -𝛼𝒖 + 𝛾) + 𝛽 • ∇𝒖 + 𝑏𝒖 = 𝑓, (2-19)
where Λ , 𝑒 𝑎 , 𝑑 𝑎 , 𝑐 , 𝛼 , 𝛾 , 𝛽 , 𝑏 and 𝑓 are the eigenvalues, mass coefficients, damping or mass coefficients, diffusion coefficients, conservative flux convection coefficients, convection coefficients, conservative flux source term, absorption coefficients and source terms, respectively.

For seismic metamaterials, the non-zero coefficients in Eq. (2-19) are the mass coefficient 𝑒 𝑎 , the diffusion coefficient 𝑐 , the conservative flux convection coefficients 𝛼, the conservative flux source term 𝛽, and the absorption coefficients 𝑏.

They are

𝑒 𝑎 = ( -(𝜆 + 2𝜇) 0 0 -𝜇 ) , (2-20)
𝑐 = ( [ (𝜆 + 2𝜇) 0 0 𝜇 ] [ 0 𝜆 𝜇 0 ] [ 0 𝜇 𝜆 0 ] [ 𝜇 0 0 (𝜆 + 2𝜇) ] ) , (2-21
)

𝛼 = ( [ (𝜆 + 2𝜇)Λ 0 ] [ 0 𝜇Λ ] [ 0 𝜇Λ ] [ 𝜇Λ 0 ] ) , (2-22
)

𝛽 = ( [ -(𝜆 + 2𝜇)Λ 0 ] [ 0 -𝜆Λ ] [ 0 -𝜆Λ ] [ -𝜆Λ 0 ] ) , (2-23) 𝑏 = ( -𝜌𝜔 2 0 0 -𝜌𝜔 2 ) . (2-24)
Transfer matrix method

Dispersion relations of Rayleigh waves

Figure 2-6 shows a layered model which consists of a homogeneous isotropic layer of thickness h overlying a homogeneous half space. For the two-layered structure, the dispersion relation of the Rayleigh wave can be calculated using the transfer matrix method. [START_REF] Yang | Dispersion curves of Rayleigh wave in threelayer media[END_REF][START_REF] Zhang | Elastic wave and excitation mechanism of surface waves in multilayered media[END_REF] Figure 2-6 Layered media structure

In the transfer matrix method, five scalars are passed from the bottom layer to the surface layer, which are 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 and 𝑥 5 : [START_REF] Yang | Dispersion curves of Rayleigh wave in threelayer media[END_REF][START_REF] Zhang | Elastic wave and excitation mechanism of surface waves in multilayered media[END_REF] 𝑥 1 = 1 + 𝛶 P2 𝛶 S2

(2-25)

𝑥 2 = 𝛶 2 + 𝛶 P2 𝛶 S2 (2-26)
𝑥 3 = i𝛶 S2 𝑔 2 (2-27) 𝑥 4 = -i𝛶 S2 𝑔 2 (2-28) 𝑥 5 = -𝛶 2 2 -𝛶 P2 𝛶 S2 (2-29)
where

𝛶 2 = 𝑣 𝑅 2 /(2𝑣 S2 ) 2 , 𝛶 P2 = √-1 + 𝑣 𝑅 2 𝑣 P2 2 ⁄ , 𝛶 S2 = √-1 + 𝑣 𝑅 2 𝑣 S2 2 ⁄
and 𝑔 2 = 1 -𝛶 2 ; where 𝑣 S2 and 𝑣 P2 are the transverse and longitudinal wave velocities of the bottom medium, respectively. When these five scalars are transferred from bottom medium to surface layer, they change sequentially as follows:

{ 𝑥 1 = 𝑥 1 𝑙 ⁄ 𝑥 5 = 𝑙𝑥 5 , (2-30)
where

𝑙 = 𝜌 2 𝑣 S2 2 /𝜌 1 𝑣 S1 2 , { 𝑝 1 = -𝛶 1 𝑥 1 + (1 + 𝛶 1 )𝑥 2 + 𝑥 5 𝑝 2 = 𝑥 1 -2𝑥 2 -𝑥 5 𝑝 3 = 𝑔 1 𝑥 3 𝑝 4 = -𝑔 1 𝑥 4 𝑝 5 = -𝛶 1 2 𝑥 1 + 2𝛶 1 𝑥 2 + 𝑥 3 , (2-31)
where 𝛶 1 = 𝑣 𝑅 2 /(2𝑣 S1 ) 2 and

𝑔 1 = 1 -𝛶 1 , { 𝑞 1 = 𝑎𝑏𝑝 2 + 𝑐𝑑𝑝 5 -𝑎𝑑𝑝 3 -𝑏𝑐𝑝 4 𝑞 2 = 𝑎𝑑𝑠𝑝 2 -𝑏𝑐𝑝 5 + 𝑎𝑏𝑝 3 -𝑐𝑑𝑠𝑝 5 𝑞 3 = 𝑏𝑐𝑟𝑝 2 -𝑎𝑑𝑝 5 -𝑐𝑑𝑟𝑝 3 + 𝑎𝑏𝑝 4 𝑞 4 = 𝑐𝑑𝑟𝑠𝑝 2 + 𝑎𝑏𝑝 5 + 𝑏𝑐𝑟𝑝 3 + 𝑎𝑑𝑠𝑝 4 , (2-32)
where 𝑎 = cos(ℎ𝑘𝛶 P1 ) , 𝑏 = cos(ℎ𝑘𝛶 S1 ) , 𝑐 = sin(ℎ𝑘𝛶 P1 ) 𝛶 P1 ⁄ , 𝑑 = sin(ℎ𝑘𝛶 S1 ) 𝛶 S1 ⁄ , 𝑟 = 𝛶 P1 2 and 𝑠 = 𝛶 s1 2 ;where 𝑘 = 𝜔/𝑣 𝑅 is the is the number of waves in the Rayleigh wave on the free surface,

𝑥 5 = -𝑞 1 𝛶 1 2 + 𝑞 4 -2𝑝 1 𝛶 1 .
(2-33)

When 𝑥 5 = 0 , the dispersion equation is obtained. [START_REF] Yang | Dispersion curves of Rayleigh wave in threelayer media[END_REF][START_REF] Zhang | Elastic wave and excitation mechanism of surface waves in multilayered media[END_REF] The relationship between Rayleigh wave velocity and frequency in a two-layered structure can be obtained by solving this dispersion equation. As shown in Fig. 234567, in the metamaterial beam consisting of a beam and spring oscillators, the governing equation for the flexural waves in the beam is [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF][START_REF] Guo | Interplay of local resonances and Bragg band gaps in acoustic waveguides with periodic detuned resonators[END_REF] 𝐸𝐼 𝜕 4 𝑧(𝑥, 𝑡)

Dispersion relations of flexural waves

𝜕𝑥 4 + 𝜌𝐴 𝜕 2 𝑧(𝑥, 𝑡) 𝜕𝑡 2 = 0, (2-34)
where E, I, ρ and A are Young's modulus, section moment of inertia, density and section area of the beam, respectively. z(x,t) is the out-plane displacement of the beam. The physical quantities of the beam can be expressed in the following forms [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF] 𝑧(𝑥, 𝑡) = (𝐴e -i𝜆𝑥 + 𝐵e i𝜆𝑥 + 𝐶e -𝜆𝑥 + 𝐷e 𝜆𝑥 )e -i𝜔𝑡 , (2-35)

𝜃(𝑥, 𝑡) = ∂𝑧(𝑥, 𝑡) ∂𝑥 , (2-36) 𝑀(𝑥, 𝑡) = 𝐸𝐼 ∂ 2 𝑧(𝑥, 𝑡) ∂𝑥 2 , (2-37) 𝑄(𝑥, 𝑡) = -𝐸𝐼 ∂ 3 𝑧(𝑥, 𝑡) ∂𝑥 3 , (2-38)
where A, B, C and D are complex coefficients; λ = √𝜌𝐴𝜔 2 𝐸𝐼 ⁄ 4

with ω = 2πf being the circular frequency is the flexural wavenumber; and θ, M and Q are the slope, bending moment and shear force, respectively. The out-plane displacement of the nth unit cell is [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF] 𝑧 𝑛 (𝑥 1 , 𝑡) = (𝐴 𝑛 e -i𝜆𝑥 1 + 𝐵 𝑛 e i𝜆𝑥 1 + 𝐶 𝑛 e -𝜆𝑥 1 + 𝐷 𝑛 e 𝜆𝑥 1 )e -i𝜔𝑡 , (2-39) where 𝑥 1 = 𝑥 -(𝑛 -1)𝑎 and (𝑛 -1)𝑎 ≤ 𝑥 ≤ 𝑛𝑎. So, from the left side to the right side of the nth unit cell, we obtain the transfer matrix in this field by following:

𝚿 𝑛 (0, 𝑡) = 𝐇 1𝛼 [𝐴 𝑛 𝐵 𝑛 𝐶 𝑛 𝐷 𝑛 ] 𝑇 e -i𝜔𝑡 , (2-40
)

𝚿 𝑛 (𝑎, 𝑡) = 𝐇 1𝛽 [𝐴 𝑛 𝐵 𝑛 𝐶 𝑛 𝐷 𝑛 ] 𝑇 e -i𝜔𝑡 , (2-41)
where 𝚿(𝑥, 𝑡) = [𝑧(𝑥, 𝑡) 𝜃(𝑥, 𝑡) 𝑀(𝑥, 𝑡) 𝑄(𝑥, 𝑡)] 𝑇 is the state vector; and

[𝐴 𝑛,1 𝐵 𝑛,1 𝐶 𝑛,1 𝐷 𝑛,1 ] 𝑇 is the coefficient vector. Substituting Eq. (2-41) into Eq. (2-40),
we obtain

𝚿 𝑛 (𝑎, 𝑡) = 𝐇 1𝛽 𝐇 1𝛼 -1 𝚿 𝑛 (0, 𝑡) ≜ 𝐇 1 𝚿 𝑛 (0, 𝑡), (2-42)
where 𝐇 1 is the transfer matrix. As for the nth spring oscillators on the beam, the dynamic equilibrium equation is

𝑓 𝑛 (𝑡) + 𝑚 ∂ 2 𝑍 𝑛 (𝑡) ∂𝑡 2 = 0, (2-43)
where 𝑓 𝑛 (𝑡) is the force of the nth spring and 𝑓 𝑛 (𝑡) = 𝐹 𝑛 e -i𝜔𝑡 , 𝑍 𝑛 (𝑡) is the displacement of the nth mass; where 𝑍 𝑛 (𝑡) = 𝑉 𝑛 e -i𝜔𝑡 and ∂ 2 𝑍 𝑛 (𝑡) ∂𝑡 2 ⁄ = -𝜔 2 𝑉 𝑛 e -i𝜔𝑡 ; where 𝑉 𝑛 is the amplitude of the nth mass. Due to the Hooke's Law, we obtain

𝑓 𝑛 (𝑡) = 𝑘[𝑍 𝑛 (𝑡) -𝑧 𝑛 (0, 𝑡)], (2-44)
where k is the spring stiffness. Substituting Eq. (2-44) into Eq. (2-43), we obtain

𝑍 𝑛 (𝑡) = 𝑘 𝑘 -𝑚𝜔 2 𝑧 𝑛 (0, 𝑡), (2-45)
𝑓 𝑛 (𝑡) = 𝑚𝜔 2 𝑘 𝑘 -𝑚𝜔 2 𝑧 𝑛 (0, 𝑡), (2-46) Due to the continuities at the interface between (𝑛 -1)th and nth unit cells, we obtain

𝑧 𝑛-1 (𝑎, 𝑡) = 𝑧 𝑛 (0, 𝑡), (2-47A) 𝜃 𝑛-1 (𝑎, 𝑡) = 𝜃 𝑛 (0, 𝑡), (2-47B) 𝑀 𝑛-1 (𝑎, 𝑡) = 𝑀 𝑛 (0, 𝑡), (2-47C) 𝑄 𝑛-1 (𝑎, 𝑡) = 𝑄 𝑛 (0, 𝑡) -𝑓 𝑛 (𝑡). ( 2
𝚿 𝑛 (𝑎, 𝑡) = 𝐓 𝑢𝑛𝑖𝑡 𝚿 𝑛-1 (𝑎, 𝑡), (2-49)
where 𝐓 𝑢𝑛𝑖𝑡 = 𝐇 1 𝐇 2 -1 is the transfer matrix. Considering the Bloch-Floquet theory, there is 𝚿 𝑛 (𝑎, 𝑡) = e iλ𝑎 𝚿 𝑛-1 (𝑎, 𝑡).

Chapter 3 Inverse-dispersion properties and wide bandgap design of three-component seismic metamaterials

A subwavelength SM consisting of three-component seismic metamaterial plate (SMP) on a half-space is proposed to attenuate ultra-low frequency seismic surface waves in a variety of geological situations in this chapter. We first review the surface wave transmission experiments of the two-component SM composed of twocomponent SMP on a half-space. [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] This is the experimental basis of the three- This explains that the ultra-low frequency BG and the wide BG can be more universally implemented theoretically than the previous works. [START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF] The subwavelength SM uses traditional building materials, making it a very practical design.

For the one-dimensional periodic SM, the reduced wave vector from 0 to π/a (a is the period of the SM) is used to obtain the eigenvalues. [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] For the two-dimensional periodic SM, which is highly symmetrical, the wave vector along Γ -X direction [START_REF] Maurin | Probability that a band-gap extremum is located on the irreducible Brillouin-zone contour for the 17 different plane crystallographic lattices[END_REF] is used to calculate the band structures. The schematic of the setup is in the upper left corner of Fig. 3-1(e). Among them, A is the signal generator and signal collector, B is a signal amplifier, C is a vibration exciter, D and E are two accelerometers, and F is a PC. The details of the experimental instruments used in the present work are consistent with this reference [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] . A thick layer of high-damping plastic foam is placed under the glass box to achieve a similar effect to the vibration isolation platform.

Two-component seismic metamaterials

Model and experimental setup

The accelerations are carried out by using two accelerometers placed on two symmetrical points (labels D and E in Fig. 3-1(e)). One (label D) is placed in the free zone where there is no interaction between the SM and the seismic wave, and the other ((label E)) is located behind the SM. Suppose that the acceleration values of points D and E are AD and AE, respectively. Then the attenuation spectrum of SM on the surface wave is 20 x log10(AE/ AD) shown in Fig. 3-1(f).

Results and Discussion

In order to verify if the new design of SMs can attenuate seismic surface waves, we have performed lab-scale experiment for a supercell of eight unit cells (four rows) of two-component SM to measure the transmission spectrum. The geometric parameters are: a = 5 cm, b = 1 cm, c = 4 cm, h = 15 cm and H = 20a. The material parameters [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] are illustrated in Table 3-1. All parameters used in the simulation are consistent with those in the lab-scale experiment.

Comparing the two band structures, we can find that when the SMP is placed on the half space, the original band is basically unchanged, but the sound cone (dark gray area in Fig. 3-1(d)) is added due to the existence of the half space. So, we can not only keep the BG of the SMP itself in Fig. 3 This lab-scale experiment demonstrates that the two-component SM can attenuate the surface waves in the frequency range of the BG well, but this SM is powerless for the lower frequency surface waves. However, our previous work [START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] shows that this SM composed of SMP and substrate can strongly attenuate surface waves in numerical simulation and lab-scale experiments. In addition, this kind of SM only needs to be laid on the base layer, which reduces costs and simplifies construction. The functional shortcomings of the two-component SM prompt us to further improve the SM to possess the above-mentioned advantages and also attenuate surface waves at ultra-low frequencies. In the following, the effect of SM composed of the three-component SMP and substrate is systematically investigated on the attenuation of surface waves in the ultra-low frequency range.

Three-component seismic metamaterials

From the above simulations and lab-scale experiment of the SM consisting of the two-component SMP and half space, it can be seen that the band structure of the SM is almost equal to the band structure of the SMP plus the sound cone. This SM can effectively attenuate the seismic surface waves to protect the buildings. Next, we will use FEM calculations to discuss the band structure and attenuation effect of the subwavelength SM consisting of the three-component SMP and a half space on the seismic surface waves. The ultra-low frequency BG of the subwavelength SM is discussed under the one-dimensional (1D) and two-dimensional (2D) periodic models. To prevent unwanted reflections, we add a perfectly matched layer (PML) with a thickness of 50a which is about the same as the incoming wavelength at low frequencies, as shown in Fig. 3-2(c). Among them, the PML scaling factor and the PML scaling curvature parameter are 1. Some works [START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF][START_REF] Huang | Multiple band gaps of phononic crystals with quasi-Sierpinski carpet unit cells[END_REF] also use low reflection boundaries to achieve similar results.

Models of 1D seismic metamaterials

Results and Discussion

For a 1D three-component SMP based on local resonance, a BG can easily occur.

As shown in Fig. 3-3(a), a narrow BG appears in the range of 10.0 -14.0 Hz. Even if the unit cells in Fig. 3-2(a) are periodically arranged in two dimensions, the BG is very narrow. [START_REF] Yan | Seismic isolation of two dimensional periodic foundations[END_REF] It is conceivable that such a narrow BG cannot meet our needs. However, when we place this SMP on a half space (i.e. the SM), we are surprised to find two wide BGs (0 -9.0 Hz and 9. proves that this work is completely different from the previous ones [START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Chen | Harnessing multi-layered soil to design seismic metamaterials with ultralow frequency band gaps[END_REF] which do not have modes in the ultra-low frequency BG. In Fig. 3-3(d), the effective density, shear modulus and bulk modulus of the SMP are obtained by using numerical methods. [START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF] We can see that the BG of SMP is due to the appearance of its negative effective mass density.

Through the transmission spectrum of the subwavelength SM, we can also see the existence of these two wide BGs. In the range of the ultra-low frequency BG, we see that the seismic surface waves are well attenuated in the range of 0.5 -9.0 Hz. However, for the seismic surface waves close to zero frequency, the attenuation effect needs to be improved. Because the number of unit cells in the proposed subwavelength SM is infinite because of the periodic boundary condition when calculating the band structure, while the number of the unit cells in the subwavelength SM is finite (1600 unit cells) when calculating the transmission spectrum. Therefore, we hypothesize that the number of the unit cells of the SM and the lower limit of the attenuation zone in the transmission spectrum are closely related.

To discover the relationship between the number of the unit cells of the subwavelength SM and the lower limit of the attenuation zone in the transmission spectrum, we simulate and plot the displacements of surface waves of the system varying with the frequency by calculating the transmission spectrum. As shown in Fig. We also use the subwavelength SM of 100 unit cells placed from 0 m to 150 m with for the calculation of the transmission spectrum in Fig. 34. We find that the deflection of Rayleigh waves is almost the same as the subwavelength SM with 1600 unit cells. When the Rayleigh waves propagate to the position of X = 60 m (presumably at the position of the 40th unit cell), most of the energy has turned from the surface to the bulk below the surface. This also shows that when the number of the unit cells of the subwavelength SM is sufficient, the transmission distance and deflection angle of the Rayleigh waves in the subwavelength SM are constant at the same frequency. In practical applications, the method of arranging the number of the unit cells of the subwavelength SM according to the specified frequency range is effective.

In order to investigate the influence of the material parameters of the substrate on the ultra-low frequency BG, we consider the effect of the Young's modulus and mass density of the substrate on the upper frequency bounds of the ultra-low frequency BG for the 1D subwavelength SM. The material parameters of the substrate are selected within the actual soil parameters [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF] . However, when the Young's modulus of the substrate increases from 10 MPa to 100 MPa, and the mass density of the substrate is between 1600 kg/m 3 and 2200 kg/m 3 , the upper boundary (9.0 Hz) of the ultra-low frequency BG does not change. In this range of the Young's modulus and the mass density, the upper boundary of the ultra-low frequency BG is determined by the band of the local resonance of the SMP, which is invariable. From the effect of the Young's modulus and mass density of the substrate on the upper frequency bounds of the ultralow frequency BG for the 1D SM, we can find that the subwavelength SM is suitable for different soils in different regions. This also shows that the subwavelength SM is universal.

The values for material parameters of the substrate are swept and are intended for qualitative comparison only [START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF] . Through the simulation calculation and discussion, we find this important universal equation: the band structure of the SM composed of the SMP and a half space is almost identical to band structure of the SMP plus the sound cone. It is noteworthy that this conclusion is incorrect when the heavy core of the unit cell of the SMP connects the substrate. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] The above 1D SWSM can be modelled as a two-layered medium as shown in Fig. For the layered medium shown in Fig. 3-5(a), the dispersion curve of the Rayleigh wave can be calculated by the fast scalar method. [START_REF] Zhang | Elastic wave and excitation mechanism of surface waves in multilayered media[END_REF] As shown in Fig. 3-5(b), the dispersion curve of the layered medium only supports a single waveguide mode and exhibits a typical inverse dispersion. [START_REF] Yang | Forward Modeling of Zigzag Dispersion and Pavement Systems Dispersion Curves[END_REF] The wave velocity of Rayleigh waves increases with the frequency increasing in the two-layered medium in the range of 0 -0.0125 Hz. And there is a cut-off frequency (0.0125 Hz). It means that, when the frequency is greater than the 0.0125 Hz, Rayleigh waves will not exist in the effective two-layered medium. This also verifies the correctness of the ultra-low frequency BG because 0.0125 Hz is close to zero.

Models of 2D seismic metamaterials

In order to verify that the above the subwavelength SM still has an ultra-low frequency BG in the three-dimensional space, we have studied the 2D SM as shown in 3-2(c), so it is not illustrated here. [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF]

Results and Discussion

Metamaterials based on the local resonance is a subject of debate and discussion.

Although they can obtain BGs at low frequencies, the width of the BGs needs to be expanded. [START_REF] Liu | Locally resonant sonic materials[END_REF] As shown in Fig. 34567(a), the BG of the 2D three-component SMP is very narrow and only exists around 10.0 -12.0 Hz. Even if the unit cells of such threecomponent SMP are three-dimensionally periodically arranged, the width of the BG is also narrow. [START_REF] Yan | Three dimensional periodic foundations for base seismic isolation[END_REF] However, the 2D SM has two wide BG for surface waves, and one of them is an ultra-low BG. By comparing the band structures in Figs. Appendix: Inverse-dispersion effect and a simple example method. [START_REF] Yang | Dispersion curves of Rayleigh wave in threelayer media[END_REF][START_REF] Haskell | The dispersion of surface waves on multilayered media[END_REF] When the material parameters of the two-layered structure are identical, i.e., the homogeneous half-space structure, the velocity of the Rayleigh waves is constant. [START_REF] Colquitt | Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction[END_REF] When the material parameters of the two-layered structure are different, situations are complicated, and only two representative cases are discussed here: Ⅰ (Vs2>Vr2>Vs1>Vr1) and Ⅱ (Vs1>Vr1>Vs2>Vr2). The assumed parameters are shown in For case Ⅰ, the dispersion curves of the Rayleigh waves of this two-layered layer structure are shown in Figure 3-10(a). This is a very common dispersion curves in earth science. [START_REF] Beaty | Simulated annealing inversion of multimode Rayleigh wave dispersion curves for geological structure[END_REF] There are multiple guided waves in the layered structure, and the maximum Rayleigh wave velocity (phase velocity) is equal to the shear wave velocity of the substrate (Vs2), and the minimum Rayleigh wave velocity (phase velocity) is equal to the Rayleigh wave velocity of the cover layer (Vr1). Vs2 and Vr1 are underlined in Table 3-2. For any one of these guided waves, the velocity of the Rayleigh waves gradually decreases with increasing frequency and finally becomes flat.

For case Ⅱ, the dispersion curves of the Rayleigh waves of this two-layered layer structure are shown in Fig. 3-10(b). We can find the inverse-dispersion effect, [START_REF] Rydén | Surface waves in inversely dispersive media[END_REF] that is, the wave velocity (phase velocity) of the Rayleigh waves increases with increasing The TSM can be roughly equivalent to a layered medium composed of two materials as shown in Fig. 3456789. Considering that the cover layer is a steel structure composed of pillars and a plate, we calculate the effective Young's modulus and effective density of this cover layer at a low frequency (0.1 Hz) through effective medium theory: Young's modulus E = 207 GPa, density ρ=4000 kg/m 3 . Considering that the materials of the plates and pillar are all steel, the effective Poisson's ratio of this cover layer is still 0.3. And the thickness of the cover layer is h1 = 6.25a, i.e. the height of pillar h plus thickness of steel plate D. This layered medium has only a single guided wave mode and there is a cutoff frequency of about 0.0023 Hz in Fig. 3-12. In this layered medium, when the frequency is higher than 0.0023 Hz, there is no longer any Rayleigh wave. This is the reason for the BG shown in Fig. 3-11(b). This phenomenon has also been found in the SM with a similar structure. [START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF] This is the second characteristic of the BG based on inverse-dispersion effect: SM can be equivalent to a two-layered medium, and it satisfies Vs1>Vr1>Vs2>Vr2. The frequency domain analysis of the TSM is conducted to verify the existence of the BG, according to the model of the finite system shown in Fig. 3-13(a). Because the ultra-low frequency Rayleigh waves has a longer wavelength, the parameters we set are larger when designing the finite system to ensure the accuracy of the simulation results.

Here, the number of unit cells of the TSM is set to 5000, mainly for lower frequency

Rayleigh waves, such as below 0.1 Hz. The problem of the number of unit cells of TSM will be discussed in detail later. As shown in Fig. 3 the substrate is set as a fixed boundary condition to avoid surface modes on the bottom. [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF][START_REF] Pu | A novel method for identifying surface waves in periodic structures[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. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] Material Density (kg/m experiments and numerical simulations. [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][START_REF] Achaoui | Experimental observation of locallyresonant and Bragg band gaps for surface guided waves in a phononic crystal of pillars[END_REF]

Classical band structures

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. height continues to 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. 4-1(b). The effect of mass density of the ITS (ρITS) on the FBG is shown in Fig. 4-4(a).

When ρITS increases from 4000 kg/m 3 to 10000 kg/m 3 , the lower boundary of the FBG decreases from 9.1 Hz to 6.0 Hz; and the upper boundary reduces from 21.0 Hz to 14.5

Hz. The center frequency of the FBG keeps dropping. However, the relative bandwidth first increases and then decreases, and its maximum value is about 0.9 when ρITS is about 7000 kg/m 3 . The effect of Young's modulus of the ITS (EITS) on the FBG is shown in when ρITS is about 7000 kg/m 3 and EITS is larger than 180 GPa. In reality, steel is the best choice for the ITS.

Complex band structures

In order to deeply explore the wide FBG of the ITSM, the complex band structures shown in Fig. 4-5(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] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF][START_REF] Wang | Wave propagation in two-dimensional 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] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Pu | A novel method for identifying surface waves in periodic structures[END_REF]

𝜉 = ∫ |𝒖|𝑑𝑠 𝑆(2𝜆) / ∫ |𝒖|𝑑𝑠 𝑆(𝐻) , (4-1)
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. 4-5(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. 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.

Similarly, as shown in Fig. 4-5(b), as a comparison, the complex band structure of the FBG of the PSM shown in Fig. 4-1(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. 4-6(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. 4-1(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 | Surface-wave attenuation by periodic pile barriers in layered soils[END_REF][START_REF] Pu | A novel method for identifying surface waves in periodic structures[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] Brule | Experimental evidence of auxetic features in seismic metamaterials: Ellipticity of seismic Rayleigh waves for subsurface architectured ground with holes[END_REF][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] 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. 4-6(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. 4-2(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. 4-5(a)), the attenuation effects of the ITSM on Rayleigh waves are completely different. Because there are evanescent surface waves in part 1, the attenuation effect in part 1 is more pronounced than that in part 2. In part 1, the larger the value of the imaginary part is, the stronger the attenuation effect of the ITSM on Rayleigh waves is. Moreover, there is a strong attenuation of the surface waves from 19.1 to 22.0 Hz shown in Fig. 4-6(b). This is because that, in the second bandgap of the ITSM, there are evanescent waves shown in Fig. 4-5(a), and the surface waves are converted into bulk waves [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF] . 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 inverse-dispersion effect. 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. improved to strongly attenuate more energy of Rayleigh waves at higher frequencies than the original ITSM. However, when the steel structure is buried deeper into the substrate, the lower boundary of the attenuation area is also rising.

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] 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][START_REF] Pu | Surface-wave attenuation by periodic pile barriers in layered soils[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 [J][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, largescale field experiments are still needed to verify the SMs with ultra-wide and lowfrequency 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 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] The 2D ITSM and experimental setup Considering the viscoelasticity and non-uniformity of the soil, we use the controlled variable method: using two sensors (uT, A21D100) with the same distance to the source to measure the acceleration of the two points (points A and B shown in of Rayleigh waves by the TSM is defined as 20*log10(AA/AB) [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 [J][END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] , which is same as in Tables 4-2. 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

The band structure of the 2D ITSM only in the ГX direction is calculated to facilitate comparison with the experiments, as shown in Fig. 4-9(a). [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF] The sound 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

Chapter 5 Rainbow trapping effect of gradient seismic metamaterials with ultra-wide frequency isolation design

In the past two decades, in the fields of applied physics, mechanics and civil engineering, the use of metamaterials to control wave propagation has attracted great attention. [START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF][START_REF] Zhou | Effects of material parameters on elastic band gaps of two-dimensional solid phononic crystals[END_REF][START_REF] Colquitt | Seismic metasurfaces: Sub-wavelength resonators and Rayleigh wave interaction[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][START_REF] Liu | Topology optimization of periodic barriers for surface waves[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Wang | Lumped-mass method for the study of band structure in two-dimensional phononic crystals[END_REF][START_REF] Khelif | Complete band gaps in twodimensional phononic crystal slabs[END_REF][START_REF] Pennec | Two-dimensional phononic crystals: Examples and applications[END_REF][START_REF] Wang | Two-dimensional locally resonant phononic crystals with binary structures[END_REF][START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF][START_REF] Zhou | Three-dimensional acoustic characteristic study of porous metasurface[END_REF][START_REF] Diatta | Control of Rayleigh-like waves in thick plate Willis metamaterials[END_REF] Regarding surface waves, the lowestfrequency attenuation zone (LFAZ) of the elastic metamaterials composed of an array of artificial structures on a substrate is the most appealing and interesting, since their working wavelengths are well above their characteristic size. [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][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][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] In the LFAZ, a number of experiments [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] Achaoui | Local resonances in phononic crystals and in random arrangements of pillars on a surface[END_REF] and numerical studies [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[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][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] have demonstrated that surface waves can be strongly attenuated because of the local resonance of the artificial structures on the substrate. It is worth mentioning that this conclusion is always valid whether the materials of the substrate and the artificial structures are the same or not. [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[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][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] However, the width of the LFAZ is usually narrow. Therefore, the second attenuation zone (SAZ) whose frequency is slightly higher than the LFAZ attracted our attention. In this chapter, we propose alternative approaches to overcome these challenges.

We report the properties of the LFAZ of a pillar-shaped elastic metamaterial. Then, we experimentally demonstrate the "rainbow trapping effect" of the SAZ of an elastic metamaterial composed of one kind of pillars with various lattice constants on a substrate. These two features of these two attenuation zones allow us to connect lower bound of the SAZ and the upper bound of the LFAZ, and design elastic metamaterials using one kind of artificial structures to attenuate surface waves over a wide lowfrequency range. Figure 5-1 shows the schematic diagram of the elastic metamaterial.

We consider the seismic Rayleigh waves as an example, and design a seismic metamaterial (SMS) just using one kind of artificial structure on a half-space, which can attenuate Rayleigh waves in the frequency range from 5 Hz to 22 Hz.

First attenuation zone

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

The bottom surface of the unit cell is set as fixed boundary condition. The considered material parameters of steel are 207 GPa, 7784 kg/m 3 and 0.3 for the Young's modulus, the density and the Poisson's ratio, respectively. For the soil substrate, the material parameters are 20 MPa, 1800 kg/m 3 and 0.3 for the Young's modulus, the density and the Poisson's ratio, respectively. [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF] Figure 5-2(a) shows the complex band structures of the elastic metamaterial for surface waves from 4.5 to 6.0 Hz calculated by the PDE module of the COMSOL Multiphysics software. The dotted line of cs is the shear wave speed of the substrate material, i.e., the boundary of the sound cone. [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF] All the modes at the right side of the dotted line are surface waves, but there are also some other surface modes, i.e., pseudo surface waves, at the left side of the dotted line. [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF] To draw all the surface modes in the band structure, the parameter ξ is introduced, i.e.

𝜉 = ∫ |𝒖|𝑑𝑠

𝑆(2𝜆) / ∫ |𝒖|𝑑𝑠 𝑆(𝐻) , (5-1)
which is the ratio of the displacement (u) in a depth of twice (2λ) of the Rayleigh wave wavelength to the entire depth (H) of the substrate. [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Liu | Partially embedded gradient metabarrier: Broadband shielding from seismic Rayleigh waves at ultralow frequencies[END_REF] The full complex band structure (1 > ξ > 0) is shown on the left of the Fig. 5-2(a). Surface mode is defined as having parameter ξ > 0.9. The complex band structure for surface modes (ξ > 0.9) is shown on the right of the Fig. 5-2(a). The first bandgap is found from 5.0 to 5.4 Hz.

Although there is a band of surface modes from 5.0 to 5.2 Hz, the surface waves are continuously attenuated as the distance traveled increases in this frequency range.

Because the imaginary part of the wave vector of this band is positive, i.e., the wave mode of this band is evanescent [START_REF] Wang | Wave propagation in two-dimensional viscoelastic metamaterials[END_REF][START_REF] Wang | Coupling of evanescent and propagating guided modes in locally resonant phononic crystals[END_REF] with strong attenuation. The surface vibration mode, which falls before the cs line on this band, is shown in the far-right of Fig. 5-2(a).

Most of the vibration exist in the substrate and decrease with increasing depth. The vibration mode of the pillars on substrate is similar as the one on the first band below 5.0 Hz. [START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] It is worth noting that the curve of the imaginary part should start from the end of the real part at the high symmetry Brillouin points from 5 to 5.15 Hz. However, the ideal results are difficult to calculate because the results are extremely dense at the junction.

The most interesting feature of the first bandgap is its immobility nature. The system composed of the substrate and an artificial structure is regarded as a spring-mass model. The resonance frequency of the spring-mass model is fixed if the substrate and the artificial structure are both not changed. So, the first bandgap due to the resonance of the spring-mass models is unchanged, even if the distribution of the artificial structures in the elastic metamaterial varies. phenomenon still exists, which has been experimentally demonstrated. [START_REF] Achaoui | Local resonances in phononic crystals and in random arrangements of pillars on a surface[END_REF] It is worth noting that the strong attenuation of the surface waves in the SMS with the lattice constant of 3 m from 9 to 11 Hz is due to the existence of the second and third bandgaps.

Second attenuation zone

The LFAZ generated by the elastic metamaterials composed of an array of pillars on substrate is usually narrow. [START_REF] Benchabane | Evidence for complete surface wave band gap in a piezoelectric phononic crystal[END_REF][START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Wu | Surface and bulk acoustic waves in twodimensional phononic crystal consisting of materials with general anisotropy[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] Therefore, to obtain a wider attenuation zone in the low frequency range, the SAZ with a higher frequency than the LFAZ is investigated and discussed by using numerical simulations and experiments. [ [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF] ]. The material of the unit cell is PLA (its Young's modulus is 3.4398 GPa, density is 1086.3 kg/m 3 , Poisson's ratio is 0.35) [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF] and the geometric parameters are shown in Table 1. The characteristics of the first and second bandgaps are always valid and do not change, whether the materials of the substrate and the artificial structures are the same or not. [START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[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][START_REF] Assouar | Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF] Therefore, the proposed PLA models can accurately explore the basic properties of the bandgaps.

The effect of lattice constant a on the upper and lower frequency bounds of the first and second bandgaps is shown in Fig. 5-4(b). The first bandgap is quite stable with various lattice constants, which is consistent with the previous results. However, the central frequency of the second bandgap keeps decreasing as the lattice constant increases. Therefore, this feature of the second bandgap is suitable to be used to obtain a wide attenuation zone at frequencies higher than the LFAZ through the "rainbow trapping effect" [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF][START_REF] Gan | Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings[END_REF][START_REF] Hu | Rainbow trapping in hyperbolic metamaterial waveguide[END_REF] . The propagation of elastic waves is measured on both sides of the elastic metamaterial by using the laser vibrometer (Polytec PSV-500). Then the accelerations of two points A and B are extracted. The method is the same as that in simulation calculations. Figure 5-9 shows the experimental layout. Label A is a computer, which is used to set the output signal of the piezoelectric sensor on sample B and process the input signal from the laser vibrometer C. can observe that in the range of 37k -50 kHz, the elastic metamaterial can significantly attenuate surface waves. We observe that there is a transmission peak at around 45 kHz.

We do believe that the experimental transmission peak at around 45 kHz is due to the imperfect absorption of the damping material covered the substrate for elastic waves.

Figures 5678910) show the out-of-plane acceleration fields on the surface of the samples in the experiment and simulation at 49 kHz, respectively. Limited to the number of rows of the pillars that compose the elastic metamaterial, the attenuation effects do not seem obvious in these fields. However, when the number of rows of the pillars increases and more suitable lattice constants are chosen, better attenuation ability and a wider SAZ can be obtained. Therefore, the elastic metamaterial constituted of pillars with different distances on a substrate presents a wide SAZ to attenuate surface waves. It is worth noting that these pillars are all the same.

Broadband seismic metamaterials

When an artificial structure on the substrate is determined, the LFAZ is immobile even if the distances between the structures are changed. However, the SAZ whose frequency is higher than the LFAZ becomes wider when choosing the suitable distances between the structures due to the "rainbow trapping effect". Therefore, for a properly designed elastic metamaterial, there is an opportunity to connect lower bound of the SAZ and the upper bound of the LFAZ. It is possible to propose an elastic metamaterial with a wide attenuation zone from an ultra-low frequency for surface waves just using one kind of artificial structures on a substrate. For instance, in this work, a kind of SMS with a wide attenuation zone for seismic Rayleigh waves is designed. Considering the wide range of the evanescent waves in the LFAZ of the seismic metamaterial, the inverted T-shaped structure is chosen. [START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF] As shown in Fig. 5-11(a), 15 identical inverted T-shaped steel structures are used. The geometric parameters are given in Table 5-2. In the numerical simulation, the ideally perfect continuity is assumed between the inverted T-shaped structure and the substrate.

The 15 steel structures are placed with appropriate distances. The distances between the symmetry axes of each steel structure in this research are 4.9, 4. Table 5-3: The parameters of the layered soil substrate [START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Gao | Field experiment and numerical study on active vibration isolation by horizontal blocks in layered ground under vertical loading[END_REF] Layers Depth(m) Density(kg/m 3 ) Young's modulus(MPa) Poisson's ratio

Conclusion

In summary, the mechanical property of the LFAZ of the elastic metamaterial made of pillars on a substrate has been investigated numerically. We found that the LFAZ is immobile with the changed distances between pillars. The "rainbow trapping effect" in the SAZ has then been demonstrated by numerical simulation and experiments. We have observed that the SAZ becomes wider when suitable distances between pillars are selected. Therefore, for a properly designed elastic metamaterial, there is an opportunity to connect lower bound of the SAZ and the upper bound of the LFAZ. We then can obtain a wide attenuation zone from an ultra-low frequency for surface waves by just using one kind of artificial structures on a substrate. In the case of the seismic Rayleigh

Chapter 6 Design of the inertially amplified lever-type seismic metamaterials with a low-frequency bandgap

Seismic Rayleigh waves at the frequencies below 20 Hz are the greatest threat to the man-made buildings located in the mid and far fields of the seismic source when earthquakes happen. [START_REF] Kim | Artificial seismic shadow zone by acoustic metamaterials[END_REF][START_REF] Arnold | EARTHQUAKE EFFECTS ON BUILDINGS 4[END_REF] Seismic metamaterials (SMs) [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] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Achaoui | Clamped seismic metamaterials: ultralow frequency stop bands[END_REF][START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF][START_REF] Colombi | Forests as a natural seismic metamaterial: Rayleigh wave bandgaps induced by local resonances[END_REF][START_REF] Sigalas | Large scale phononic metamaterials for seismic isolation[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] have been proposed by researchers to isolate the seismic Rayleigh waves before they arrive the critical infrastructures and buildings. This is due to the bandgap characteristics of the SMs composed of the periodic arrangement of artificial unit cells. The seismic waves will be attenuated significantly by the SMs in the frequency range of the bandgap. [START_REF] Wu | Frequency band-gap measurement of twodimensional 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] Liu | Locally resonant sonic materials[END_REF][START_REF] Ma | Polarization bandgaps and fluid-like elasticity in fully solid elastic metamaterials[END_REF][START_REF] Wang | Tunable and active phononic crystals and metamaterials[END_REF][START_REF] Gao | Band gaps of lower-order Lamb wave in thin plate with one-dimensional phononic crystal layer: Effect of substrate[END_REF][START_REF] Ding | Metamaterial with simultaneously negative bulk modulus and mass density[END_REF][START_REF] Colombi | Elastic wave control beyond bandgaps: shaping the flow of waves in plates and half-spaces with subwavelength resonant rods[END_REF][START_REF] An | Band gap characteristics of radial wave in a twodimensional cylindrical shell with radial and circumferential periodicities[END_REF] To obtain wider bandgaps, the pillars with different kinds of structures, [START_REF] Lim | Built-up structural steel sections as seismic metamaterials for surface wave attenuation with low frequency wide bandgap in layered soil medium[END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Roux | Toward seismic metamaterials: The metaforet project[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] and the "rainbow trapping effects" [START_REF] Colombi | A seismic metamaterial: The resonant metawedge[END_REF][START_REF] Liu | Ultra-wide band gap metasurfaces for controlling seismic surface waves[END_REF][START_REF] Pu | Seismic metasurfaces on porous layered media: Surface resonators and fluid-solid interaction effects on the propagation of Rayleigh waves[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF][START_REF] Zeng | Coupling the first and second attenuation zones in seismic metasurface[END_REF][START_REF] Gan | Experimental verification of the rainbow trapping effect in adiabatic plasmonic gratings[END_REF][START_REF] Hu | Rainbow trapping in hyperbolic metamaterial waveguide[END_REF][START_REF] Wu | Broadband Rayleigh wave attenuation by gradient metamaterials[END_REF] were utilized. However, the ultra-low-frequency (< 2 Hz) bandgaps are hard to obtain by using small-size structures.

Fortunately, the inertial amplification proposed by Yilmaz et al. [START_REF] Yilmaz | 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][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF] , which induces low-frequency bandgaps by using a small mass, was utilized to design metamaterials. The lever-type structures were introduced to amplify the motion of the resonate small mass, which in turn effectively reduces the resonance frequency. Based on the similar structures, Li et al. [START_REF] Li | Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects[END_REF] proposed a phononic beam to isolate transverse waves utilizing inertial amplification resonators. A wide bandgap induced by inertial amplification in a continuous elastic rod was obtained and a general 2D realization of the inertially amplified system in a plate was proposed in Ref. [START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF] . In addition, Acar et al. [START_REF] Acar | Experimental and numerical evidence for the existence of wide and deep phononic gaps induced by inertial amplification in twodimensional solid structures[END_REF] experimentally and numerically proposed a periodic solid structure which exists wide and deep bandgaps induced by embedded inertial amplification mechanisms. The 3D structures was experimentally demonstrated by using embedded inertial amplification mechanisms, which can isolate elastic waves in a wide frequency range. [START_REF] Taniker | Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands[END_REF] Recently, a 2D solid structure with embedded inertial amplification mechanisms was designed by using topology optimization to obtain an ultrawide bandgap at low frequencies. [START_REF] Yuksel | Realization of an ultrawide stop band in a 2-d elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF] However, there is almost no research of the SMs with inertial amplification to achieve the low-frequency bandgaps for surface waves.

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

Models and experimental setup

1 2𝜋 √ 𝑘 (𝜇-1) 2 𝑚 , (6-1)
where μ = l1/ l2 is the lever ratio with l1 being the length of the lever and l2 being the distance between the spring and support bar; k is the stiffness of the spring; and m is the mass. The inertial mass will be amplified (μ-1) 2 times and the resonance frequency will be reduced (μ-1) times when the lever is rigid and has no mass. A low resonance frequency f can be achieved in the following cases. First, when the stiffness of the spring k is small, which means the soft material is used. Second, when the lever ratio μ is large, which means the long lever (i.e., l1 is large) is used or the position of the support bar is near the spring (i.e., l2 is small). Third, when large mass (i.e., m is large) is used.

Therefore, there are a lot of options to design elastic metamaterials with low-frequency bandgaps, including SMs which we will mention later. The PLA rod is fabricated and the lever and the support bar are assembled with PLA rod horizontally installed through the holes to form the hinged part. More enlarged photograph of the hinged part can be found in Figure 6-2. The source vibrating in z

direction is excited at one end of the MMP. Figure 6-1(e) shows the experimental layout.

Label A is a computer, which is utilized to set the output signal of the source on sample B and processes the input signal from the laser vibrometer C. In order to obtain clear and perfect propagation effects of flexural waves in the MMP, the laser vibrometer (Polytec PSV-500) is used to measure on the back of the MMP. In addition, a bare plate D, which is the same plate as in the MMP, is also measured to provide the control results.

In order to facilitate the realization of the 3D printing of the MMP, the material of the whole structure is chosen as PLA whose Young's modulus is 3.4398 GPa, density is 1086.3 kg/m 3 , and Poisson's ratio is 0.35 in the both simulation and experiment. [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF] The connections between the support bar and the base plate, the base plate and the spring, the lever and the mass are perfect continuous. The spring and the lever are perfectly glued together. It is worth noting that the material of the spring part is PLA.

Of course, according to the Eq. (6-1), the spring part can be replaced by a softer material to obtain a less k and a lower-frequency bandgap. The connection between the lever and the support bar is hinge joint. The thickness of the lever along z axis is set as 0.3a to enhance the bending stiffness of the lever along z axis. At low frequencies, the lever can be regarded as a rigid part to achieve the inertial amplification. waves in the beam is [START_REF] Cao | Flexural wave absorption by lossy gradient elastic metasurface[END_REF][START_REF] Guo | Interplay of local resonances and Bragg band gaps in acoustic waveguides with periodic detuned resonators[END_REF] 𝐸𝐼 𝜕 4 𝑧(𝑥,𝑡)

𝜕𝑥 4 + 𝜌𝐴 𝜕 2 𝑧(𝑥,𝑡) 𝜕𝑡 2 = 0, (6-3)
where E, I, ρ and A are Young's modulus, section moment of inertia, density and section area of the beam, respectively. z(x,t) is the out-plane displacement of the beam. The physical quantities of the beam can be expressed in the following forms: [START_REF] Cao | Pillared elastic metasurface with constructive interference for flexural wave manipulation[END_REF] 𝑧(𝑥, 𝑡) = (𝐴𝑒 -𝑖𝜆𝑥 + 𝐵𝑒 𝑖𝜆𝑥 + 𝐶𝑒 -𝜆𝑥 + 𝐷𝑒 𝜆𝑥 )𝑒 𝑖𝜔𝑡 , ( In the nth unit cell, the out-plane displacement of section 1 is 𝑧 𝑛,1 (𝑥 1 , 𝑡) = (𝐴 𝑛,1 𝑒 -𝑖𝜆𝑥 1 + 𝐵 𝑛,1 𝑒 𝑖𝜆𝑥 1 + 𝐶 𝑛,1 𝑒 -𝜆𝑥 1 + 𝐷 𝑛,1 𝑒 𝜆𝑥 1 )𝑒 𝑖𝜔𝑡 , (

where 𝑥 1 = 𝑥 -(𝑛 -1)𝑎 and (𝑛 -1)𝑎 ≤ 𝑥 ≤ (𝑛 -1)𝑎 + 𝑙 2 . So, from the left side to the right side of section 1 of the nth unit cell, we obtain the transfer matrix in this field by following: where k is the spring stiffness; and μ = l1/ l2 is lever ratio. Substituting Eq. (6-15) into Eq. (6-14), we obtain

𝚿 𝑛,
𝑓 𝑛1 (𝑡) = 𝑘𝑚𝜔 2 (𝜇-1) 2 𝑚𝜔 2 (𝜇-1) 2 -𝑘
[𝑧 𝑛,1 (0, 𝑡) -𝑧 𝑛,2 (𝑙 2 , 𝑡)] ≜ 𝐺 1 [𝑧 𝑛,1 (0, 𝑡) -𝑧 𝑛,2 (𝑙 2 , 𝑡)], (6-16)

𝑓 𝑛2 (𝑡) = 𝑘𝑚𝜔 2 (-𝜇 2 +3𝜇-2)
𝑚𝜔 2 (𝜇-1) 2 -𝑘

[𝑧 𝑛,1 (0, 𝑡) -𝑧 𝑛,2 (𝑙 2 , 𝑡)] ≜ 𝐺 2 [𝑧 𝑛,1 (0, 𝑡) -𝑧 𝑛,2 (𝑙 2 , 𝑡)]. (6-17) Due to the continuities at the interface between (𝑛 -1)th and nth unit cells, we where 𝐇 5 and 𝐇 6 are shown in Appendix of this chapter. Substituting Eqs. (6-11), (6-13), (6-19) and (6-21), the transfer equation of the flexural waves from the right interface of the (𝑛 -1)th unit cell to the right interface of the nth unit cell is obtain 𝚿 𝑛,2 (𝑎, 𝑡) = 𝐓 𝑢𝑛𝑖𝑡 𝚿 𝑛-1,2 (𝑎, 𝑡), (6-22) where 𝐓 𝑢𝑛𝑖𝑡 = {𝐇 3 𝐇 1 -1 [(𝐈 A -𝐇 6 𝐇 1 -1 ) -1 𝐇 5 𝐇 2 -1 ] + 𝐇 4 𝐇 2 -1 } -1 is the transfer matrix in which 𝐈 A is a 4×4 identity matrix.

Considering the Bloch-Floquet theory, we obtain

𝚿 𝑛,2 (𝑎, 𝑡) = 𝑒 𝑖𝑞𝑎 𝚿 𝑛-1,2 (𝑎, 𝑡), (6-23)
where q is the wavenumber. Substituting Eq. (6-23) into Eq. (6-22), we obtain

|𝐓 𝑢𝑛𝑖𝑡 -𝑒 𝑖𝑞𝑎 𝐈 A | = 0. (6-24)
By solving the eigenvalue equation, the band structure can be obtained. For a better comparison, as shown in Fig. 6-4(c), a periodic structure composed of the same plate, springs and masses as in the MMP is considered. In the band structure and the transmission of the metamaterial, the first bandgap and AZ is very narrow and its lowest frequency is around 520 Hz in Fig. 6-4(d). In Fig. 6-3, the lever ratio μ of the proposed unit cell is about 2, so the bandgap range of the inertial amplification mechanism is similar to the that of the spring-mass structure in Fig. 6-4(c). When the position of the support bar is near the spring, which means the l2 becomes smaller and μ becomes larger, the bandgap range significantly reduces. and surface waves, the artificial structure in the MMPs that can be used to attenuate flexural waves are not necessarily suitable for a half-space to attenuate surface waves.

Vibration modes

However, some artificial structures in the MMPs can play a key role in a half-space.

For instance, drilling holes periodically in a half-space [START_REF] Meseguer | Rayleigh-wave attenuation by a semi-infinite two-dimensional elastic-band-gap crystal[END_REF] , or burying pillars periodically in a half-space [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF] , or erecting pillars periodically on a half-space [START_REF] Zeng | A Matryoshka-like seismic metamaterial with wide band-gap characteristics[END_REF][START_REF] Brûlé | Flat lens effect on seismic waves propagation in the subsoil[END_REF] all can generate bandgaps for surface waves. Moreover, some artificial structures of the MMPs covering a half-space can exert unexpected effects for surface waves. For instance, two-component [START_REF] Wang | Periodic in-filled pipes embedded in semi-infinite space as seismic metamaterials for filtering ultra-low-frequency surface waves[END_REF][START_REF] Zeng | A broadband seismic metamaterial plate with simple structure and easy realization[END_REF] and three-component [START_REF] Zeng | Subwavelength seismic metamaterial with an ultra-low frequency bandgap[END_REF] MMPs on a half space can achieve ultra-low-frequency bandgaps for surface waves. Therefore, due to the excellent performance of the lever-type structure on the plate as shown above, we introduce it into the soil half-space to target seismic surface waves. Similar as the levertype structure shown in Fig. 6-1, the unit cell of the SM shown in Fig. 6-8(a) is constituted of the lever, support bar, mass, spring, hinge joint and soil substrate.

Comparing with the lattice constant in Fig. 6-1, the lattice constant a in Fig. 6-8(a)

is scaled to 1.5 m. The relationship between other geometric parameters and lattice constant a is consistent with those in Fig. 6-1. Theoretically, the depth of the half-space should be infinitely deep. In order to facilitate simulation calculations and obtain relatively accurate convergence results, the depth of the soil substrate is selected as H = 500a [START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Zeng | Coupling the first and second attenuation zones in seismic metasurface[END_REF] . A part of the spring is replaced by a thin rubber layer with thickness of t1 to achieve bandgaps at lower frequencies. In addition, the SM is supposed to be constructed by the ordinary building materials, for instance, steel (mass), rubber (spring)

and aluminum (support bar and lever). The material properties [START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF] are shown in Table 6-1. Therefore, Rayleigh waves cannot appear in the frequency range of the second band [134] . However, in the range of the third band, Rayleigh waves appear again with the particles mainly moving along x direction, see Figs. 6789(e -f). So, the bandgap region shown in Fig. 6-8(b) for the Rayleigh waves is between the first and third bands [START_REF] Du | H-fractal seismic metamaterial with broadband lowfrequency bandgaps[END_REF] . Therefore, we believe that narrow bandgap in this work is acceptable.

Effects of the parameters

The effects of the hinge joint's position, the rubber layer's thicknesses and the Young's modulus of the rubber on the bandgaps are illustrated in Fig. 6-10. The frequencies of the bandgap reduce significantly when the hinge joint moves from right to left along x direction. When the hinge joint moves from right to left, the μ increases and the f in Eq. (6-1) decreases. In addition, the frequencies of the bandgap reduce quickly when the rubber is provided, and keep decreasing when the thickness of the rubber increases. However, the relative bandwidth of the bandgap decreases with the frequency. The frequencies and the relative bandwidth of the bandgaps all increase when the Young's modulus of the rubber increases from 50k to 350 kPa. When the thickness of the rubber decreases or the Young's modulus of the rubber increases, the translational stiffness k in the vertical direction increases [START_REF] Palermo | Engineered metabarrier as shield from seismic surface waves[END_REF] and the f in Eq. (6-1)

decreases. Comparing with the bandgaps in Fig. 6-4(a), the bandgaps in Fig. 6 with the seismic metamaterials proposed in the literature, the width of the bandgap here is not remarkable. However, since it is the first time the inertial amplification mechanism is introduced and considered in seismic metamaterials, our goal is the creation of this bandgap rather than obtaining a large width of the bandgap. Therefore, the narrow bandgap in this work is acceptable. This work provides an alternative way to control seismic surface waves at low frequencies by using a small mass. In addition, the embedded inertial amplification mechanisms have great potential to be used to attenuate seismic surface waves in a wide range at ultra-low frequencies which will be the focus of future researches.

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

𝐇 1𝛼 = [ 1 -𝑖𝜆 -𝐸𝐼𝜆 2 -𝑖𝐸𝐼𝜆 3 1 𝑖𝜆 𝐸𝐼𝜆 2 𝑖𝐸𝐼𝜆 3 1 -𝜆 𝐸𝐼𝜆 2 𝐸𝐼𝜆 3 1 𝜆 𝐸𝐼𝜆 2 -𝐸𝐼𝜆 3 ], 𝐇 1𝛽 = [ 𝑒 -𝑖𝜆𝑙 2 -𝑖𝜆𝑒 -𝑖𝜆𝑙 2 -𝐸𝐼𝜆 2 𝑒 -𝑖𝜆𝑙 2 -𝑖𝐸𝐼𝜆 3 𝑒 -𝑖𝜆𝑙 2 𝑒 𝑖𝜆𝑙 2 𝑖𝜆𝑒 𝑖𝜆𝑙 2 -𝐸𝐼𝜆 2 𝑒 𝑖𝜆𝑙 2 𝑖𝐸𝐼𝜆 3 𝑒 𝑖𝜆𝑙 2 𝑒 -𝜆𝑙 2 -𝜆𝑒 -𝜆𝑙 2 𝐸𝐼𝜆 2 𝑒 -𝜆𝑙 2 𝐸𝐼𝜆 3 𝑒 -𝜆𝑙 2 𝑒 𝜆𝑙 2 𝜆𝑒 𝜆𝑙 2 𝐸𝐼𝜆 2 𝑒 𝜆𝑙 2 -𝐸𝐼𝜆 3 𝑒 𝜆𝑙 2 ], 𝐇 1 = 𝐇 1𝛽 𝐇 1𝛼 -1 , 𝐇 2𝛼 = 𝐇 1𝛽 , 𝐇 2𝛽 = [ 𝑒 -𝑖𝜆𝑎 -𝑖𝜆𝑒 -𝑖𝜆𝑎 -𝐸𝐼𝜆 2 𝑒 -𝑖𝜆𝑎 -𝑖𝐸𝐼𝜆 3 𝑒 -𝑖𝜆𝑎 𝑒 𝑖𝜆𝑎 𝑖𝜆𝑒 𝑖𝜆𝑎 -𝐸𝐼𝜆 2 𝑒 𝑖𝜆𝑎 𝑖𝐸𝐼𝜆 3 𝑒 𝑖𝜆𝑎 𝑒 -𝜆𝑎 -𝜆𝑒 -𝜆𝑎 𝐸𝐼𝜆 2 𝑒 -𝜆𝑎 𝐸𝐼𝜆 3 𝑒 -𝜆𝑎 𝑒 𝜆𝑎 𝜆𝑒 𝜆𝑎 𝐸𝐼𝜆 2 𝑒 𝜆𝑎 -𝐸𝐼𝜆 3 𝑒 𝜆𝑎 ], 𝐇 2 = 𝐇 2𝛽 𝐇 2𝛼 -1 , 𝐇 3 = [ 1 0 0 -𝐺 1 0 1 0 0 0 0 1 0 0 0 0 1 ], 𝐇 4 = [ 0 0 0 𝐺 1 0 0 0 0 0 0 0 0 0 0 0 0 ], 𝐇 5 = [ 1 0 0 𝐺 2 0 1 0 0 0 0 1 0 0 0 0 1 ], 𝐇 6 = [ 0 0 0 -𝐺 2 0 0 0 0 0 0 0 0 0 0 0 0 ].
and IAM are all 55 mm. All the samples are fabricated by 3D printer based on polylactic acid (PLA). The material parameters of the PLA are shown in Table 7-1. [START_REF] Cao | Elastic Bound State in the Continuum with Perfect Mode Conversion[END_REF][START_REF] Zeng | Seismic metamaterials: Generating low-frequency bandgaps induced by inertial amplification[END_REF] It is noted that the position of the points a and b are all at the back of the samples. [START_REF] Zeng | Seismic metamaterials: Generating low-frequency bandgaps induced by inertial amplification[END_REF] [START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF] designed a passive low-pass filter-type vibration with low stop-band frequencies by using the lever-type structures without hinge joint. However, the hinge joint is a simple way to achieve the inertial amplification mechanism in the metamaterials. Such as in the other literature, Frandsen et al. [START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF] introduced the inertial amplification mechanism into a continuous elastic rod to obtain wide bandgap by using the hinge joints.

Analysis of the simplified model

The simplified model of the inertially amplified mechanism of the unit cell of the IAM is illustrated in Fig. 7-1(d). The ma takes the value of the mass of one arm in the middle of the arm. The k and ka mean the stiffness of the beam and arm, respectively.

Here, the ka can be regard as rigid at ultra-low frequencies. The angle θ between the arm and the beam is 5°. All the geometric parameters are shown in Figs. 7-1(c) and (e).

It is noted that the lattice constant a is 55 mm and the unit of all the geometric parameters is millimeter.

In Fig. 7-1(d), the x1 and x2 represent the displacement of the input and output, respectively. At ultra-low frequencies, the horizontal and vertical displacement of the two ma are

𝑥 𝑎 = (3𝑥 1 +𝑥 2 ) 4 , (7-1)
𝑧 𝑎 = cot (𝜃) (𝑥 1 -𝑥 2 ) 4 , (7-2)
𝑥 𝑏 = (𝑥 1 +3𝑥 2 ) 4 , (7-3
) -4) Therefore, the kinetic energy (T) and potential energy (V) of the system are

𝑧 𝑏 = cot (𝜃) (𝑥 1 -𝑥 2 ) 4 . ( 7
𝑇 = 𝑚 𝑎 (𝑥̇𝑎 2 +𝑧̇𝑎 2 ) 2 + 𝑚 𝑎 (𝑥̇𝑏 2 +𝑧̇𝑏 2 ) 2 , ( 7-5)
experiments and simulations are defined as 𝑇𝑆 = 20 × log 10 (𝑅 𝑏 𝑅 𝑎 ⁄ ), (7-12) where Ra and Rb are the root mean square (RMS) of the out-of-plane velocities at the points a and b shown in Fig. 7-1(b), respectively.

Results of the inertially amplified metamaterial

Band structure and transmission Module of the COMSOL, [START_REF] Brûlé | Experiments on seismic metamaterials: Molding surface waves[END_REF][START_REF] Miniaci | Large scale mechanical metamaterials as seismic shields [J][END_REF][START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Ding | Metamaterial with simultaneously negative bulk modulus and mass density[END_REF] is not illustrated in this chapter. in the beam and around the hinge joints between beam and arms. Due to the similar vibration modes and slopes (in the band structure) of these two points, it is clear that these two points come from the same band which is opened by the bandgap. This is common in the band structures of the metamaterials based on local resonance. [START_REF] Wang | Precise and target-oriented control of the lowfrequency Lamb wave bandgaps[END_REF][START_REF] Wang | Wave propagation in one-dimensional fluid-saturated porous metamaterials[END_REF] At points B and D, almost all the displacement appears around the hinge joint between two arms, which means that there is a strong resonance only on the two arms. Based on this information, it is not difficult to infer that the low-frequencies bandgap is opened by the local resonance of the inertial amplification mechanism. The lattice constant a in Fig. 7-3(a) is 2 m and the depth of the soil substrate is H = 500a [START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Zeng | Seismic metamaterials: Generating low-frequency bandgaps induced by inertial amplification[END_REF] . The angle θ is also 5° and other geometer parameters are shown in Fig. 7-3(a). The material parameters of the steel and soil used in this letter are shown in table 7-1. It is noted that the SM based on the embedded inertial amplification mechanism with smaller mass can achieve a bandgap with larger relative bandwidth than the SM based on the inertial amplification mechanism [START_REF] Zeng | Seismic metamaterials: Generating low-frequency bandgaps induced by inertial amplification[END_REF] at low frequencies. As shown in Fig. 7-3(c), it is the transmission spectrum of the surface waves in the SM with 7 unit cells. [START_REF] Zeng | Broadband inverted T-shaped seismic metamaterial[END_REF][START_REF] Graczykowski | Finite element analysis of true and pseudo surface acoustic waves in one-dimensional phononic crystals[END_REF][START_REF] Cai | Propagation and attenuation of Rayleigh and pseudo surface waves in viscoelastic metamaterials[END_REF] An attenuation peak can be found around at 4 Hz. The band structure of the SM and the attenuation effects of the surface waves in the SM all demonstrate that the existence of the ultra-low-frequencies bandgap.

Vibration modes

Results of the inertially amplified seismic metamaterial

Vibration modes

Prospects

On the basis of this thesis, there are still many issues that need to be further researched, such as:

1. The studies in this thesis are mostly based on the elastic dynamics, which is not exactly the same as reality. The soil half-space commonly used in the thesis instead of the geological environment of the Earth, although it can be approximated as a homogeneous half-space medium in some cases, the geological environment inside the Earth is very complex. For example, the phenomenon of layering of soils and the fact that soils are viscoelastic and porous plasticity are not considered.

2. The simulations and theoretical studies show a perfect continuity between different materials, while in reality it is different. It is difficult to achieve perfect continuity between different materials with vastly different properties. Perfect continuity between soil and steel, for example, needs to be achieved using other means, which have not been considered in this thesis either.

3. At present, academic research on seismic metamaterials is abundant, but practical applications of seismic metamaterials are absent. Efforts should be made to promote seismic metamaterials into practical applications.

4. In this thesis, point source excitation is used to generate surface waves, but the situation in real earthquakes is more complex. Data collected in real earthquakes should be used to verify the practicality of seismic metamaterials. On constate que l'apparition de la plaque augmente manifestement la fré quence centrale et la largeur de bande relative du FBG de l'ITSM. La structure de bande complexe de l'ITSM a é té calculé e pour analyser le mé canisme d'atté nuation des ondes de surface.

On constate que le FBG de l'ITSM est composé de deux parties. On constate que les modes de propagation des ondes de surface dans l'ITSM sont complè tement diffé rents dans les diffé rentes parties du FBG. Enfin, un type d'ITSM 2D avec 25 cellules unitaires est proposé dans des expé riences de terrain à grande é chelle, qui peut atté nuer les ondes de Rayleigh dans une gamme de fré quences ultra large.

Figure 1 - 1

 11 Figure 1-1 Seismic isolations .....................................................................................

Figure 1 - 2

 12 Figure 1-2 Placement of bulk wave seismic metamaterials .......................................

Figure 1 - 3

 13 Figure 1-3 The 1D bulk waves seismic metamaterials ............................................

Figure 1 - 4

 14 Figure 1-4 The 3D bulk waves seismic metamaterials ............................................

Figure 1 - 5

 15 Figure 1-5 Placement of seismic metamaterial ........................................................

Figure 1 - 6

 16 Figure 1-6 The Rayleigh waves seismic metamaterial ............................................

Figure 1 - 7

 17 Figure 1-7 Study of the pillar-shaped seismic metamaterials ..................................

Figure 1 - 8

 18 Figure 1-8 The buried seismic metamaterial ............................................................

Figure 1 - 9

 19 Figure 1-9 Topological optimization of buried seismic metamaterials ....................

Figure 1 -

 1 Figure 1-10 The Lamb waves seismic metamaterial ...............................................

Figure 1 -

 1 Figure 1-11 The Lab-scale experiments ...................................................................

Figure 2 - 1

 21 Figure 2-1 The first irreducible Brillouin zone ........................................................

Figure 2 - 2

 22 Figure 2-2 The band structure and the transmission ................................................

Figure 2 - 3

 23 Figure 2-3 Calculation of band structure of the seismic metamaterial ....................

Figure 2 - 4

 24 Figure 2-4 Calculation of transmission of surface waves ........................................

Figure 2 - 5

 25 Figure 2-5 Calculation of band structure of the metamaterial .................................

Figure 2 - 6

 26 Figure 2-6 Layered media structure .........................................................................

Figure 2 - 7

 27 Figure 2-7 Metamaterial beam consisting of a beam and spring oscillators ............

Figure 3 - 1

 31 Figure 3-1 The two-component seismic metamaterial .............................................

Figure 3 - 2

 32 Figure 3-2 The three-component seismic metamaterial ...........................................

Figure 3 - 3

 33 Figure 3-3 The results of three-component seismic metamaterial ...........................

Figure 3 - 4

 34 Figure 3-4 Transmission of the surface waves in the seismic metamaterial ............

Figure 3 - 5

 35 Figure 3-5 Effective two-layered medium ...............................................................

Figure 3 - 6

 36 Figure 3-6 The 2D three-component seismic metamaterial .....................................

Figure 3 - 7

 37 Figure 3-7 Transmission of the surface waves .........................................................

Figure 3 - 8

 38 Figure 3-8 Propagations of the Rayleigh waves ......................................................

Figure 3 - 9 Figure 3 -

 393 Figure 3-9 Layered media structure .........................................................................

Figure 3 -

 3 Figure 3-11 The T-shaped seismic metamaterial .....................................................

Figure 3 -

 3 Figure 3-12 The dispersion curves of Rayleigh waves ............................................

Figure 3 -

 3 Figure 3-13 The transmission results .......................................................................

Figure 3 -

 3 Figure 3-14 The propagations of the surface waves ................................................

Figure 4 - 1

 41 Figure 4-1 The unit cells of the seismic metamaterials ...........................................

Figure 4 - 2

 42 Figure 4-2 The band structures and vibration modes ..............................................

Figure 4 - 3

 43 Figure 4-3 Effects of geometric parameters on first bandgap .................................

Figure 4 - 4

 44 Figure 4-4 Effects of material parameters on first bandgap ....................................

Figure 4 - 5

 45 Figure 4-5 The complex band structures .................................................................

Figure 4 - 6

 46 Figure 4-6 The transmission of Rayleigh waves in the ITSM .................................

Figure 4 - 7

 47 Figure 4-7 Transmission of Rayleigh waves in different ITSMs ............................

Figure 4 - 8

 48 Figure 4-8 The experimental setup ..........................................................................

Figure 4 - 9

 49 Figure 4-9 The results of the 2D ITSM ...................................................................

Figure 4 -

 4 Figure 4-10 The simulated results of propagation ...................................................

Figure 5 - 1

 51 Figure 5-1 The schematic diagram of a kind of the elastic metamaterial ................

Figure 5 - 2

 52 Figure 5-2 Study of the first bandgap ......................................................................

Figure 5 - 3

 53 Figure 5-3 The effects of the parameters on the first bandgap ................................

Figure 5 - 4

 54 Figure 5-4 Study of the second bandgap .................................................................

Figure 5 - 5

 55 Figure 5-5 Geometric model for simulation calculations ........................................

Figure 5 - 6

 56 Figure 5-6 The top view of the geometric model for simulation calculations ........

Figure 5 - 7

 57 Figure 5-7 The sample for experiments ...................................................................

Figure 5 - 8

 58 Figure 5-8 The top view of the sample for experiments ..........................................

Figure 5 - 9

 59 Figure 5-9 The experimental layout ........................................................................

Figure 5 -

 5 Figure 5-10 The transmissions of surface waves in the sample ..............................

Figure 5 -

 5 Figure 5-11 The transmissions of surface waves in the ITSM ................................

Figure 5 -

 5 Figure 5-12 The propagations of the Rayleigh waves in the ITSM ........................

Figure 5 -Figure 6 - 1

 561 Figure 5-13 The transmissions of surface waves on the ITSM with layered substrate ....................................................................................................................

Figure 6 - 2

 62 Figure 6-2 The enlarged photograph of the hinged part ..........................................

Figure 6 - 3

 63 Figure 6-3 The analytical results of the simplified model .......................................

Figure 6 - 4

 64 Figure 6-4 The band structure and transmission ......................................................

Figure 6 - 5

 65 Figure 6-5 The vibration modes at 2 nd and 3 rd bands ...............................................

Figure 6 - 6

 66 Figure 6-6 Vibration modes of the metamaterial plates ...........................................

Figure 6 - 7

 67 Figure 6-7 The propagations of the flexural waves .................................................

Figure 6 - 8

 68 Figure 6-8 The unit cell and band structure ...........................................................

Figure 6 - 9

 69 Figure 6-9 Vibration modes of the seismic metamaterial ......................................

Figure 6 -

 6 Figure 6-10 Effects of parameters on the first bandgap .........................................

Figure 7 - 1

 71 Figure 7-1 The metamaterials and experimental setup ..........................................

Figure 7 - 2

 72 Figure 7-2 The band structure and transmission ....................................................

Figure 7 - 3

 73 Figure 7-3 Study of the inertially amplified seismic metamaterial ........................

1

 1 Material parameters of the unit cell .........................................................Table 3-2 The assumed material parameters when VS2>VR2>VS1>VR1 .................... Table 3-3 The assumed material parameters when VS1>VR1>VS2>VR2 .................... Table 4-1 The geometric parameter of the SMs ...................................................... Table 4-2 The material parameters .......................................................................... Table 4-3 The geometric parameters ....................................................................... Table 5-1 The geometric parameters of the metamaterial ....................................... Table 5-2 The geometric parameters of the ITSM................................................... Table 5-3: The parameters of the layered soil substrate .......................................... Table 6-1 The material parameters used in this chapter ........................................ Table 7-1 The material parameters used in this chapter ........................................

  calculated the band structure of the seismic metamaterial and analyzed the generation mechanism of the bandgap based on the Bloch-Floquet theorem. As shown in Fig. 1-3(b), they designed a lab-scaled sample based on the finite element model and experimentally demonstrated its attenuation effect on seismic waves. The results of the dynamic response of the upper frame show that the seismic metamaterials are able to attenuate environmental vibrations, seismic loads and harmonic excitations well in the bandgap. They pointed out that although it is difficult to design seismic metamaterials with low-frequency bandgaps, the attenuation of seismic waves by seismic metamaterials is obvious. Experimental data show that in the bandgap, seismic metamaterials can reduce the horizontal acceleration of seismic waves by about 50% and can reduce the vertical acceleration by about 16%.

Figure 1 - 3

 13 Figure 1-3 The 1D bulk waves seismic metamaterials (a) Comparison of concrete foundation, seismic isolations and seismic metamaterial.[START_REF] Bao | Dynamic responses of a structure with periodic foundations[END_REF] (b)

Figure 1 - 5

 15 Figure 1-5 Placement of seismic metamaterial

Figure 1 - 6

 16 Figure 1-6 The Rayleigh waves seismic metamaterial

Figure 1 - 7

 17 Figure 1-7 Study of the pillar-shaped seismic metamaterials

Figure 1 - 8

 18 Figure 1-8 The buried seismic metamaterial (a)The buried seismic metamaterials and experimental layout.[START_REF] Roux | Toward seismic metamaterials: The metaforet project[END_REF][START_REF] Krödel | Wide band-gap seismic metastructures [J][END_REF] (b) The transmission

Figure 1 - 9

 19 Figure1-9 Topological optimization of buried seismic metamaterials[START_REF] Liu | Topology optimization of periodic barriers for surface waves[END_REF]

  is the differential operator. The first term on the right-hand side of Eq. (2-1) corresponds to the longitudinal wave and the second term to the transverse wave. The Lamé constants are related to Young's modulus E and Poisson's ratio ν as follows:

Figure 2 - 1

 21 Figure 2-1 The first irreducible Brillouin zone

Figure 2 - 2

 22 Figure 2-2 The band structure and the transmission[START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF] (a) The band structure. (b) The transmission of waves in seismic metamaterials

Figure 2 - 4

 24 Figure 2-4 Calculation of transmission of surface waves[START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF] (a) Schematic model of the finite system used for calculating transmission. (b) The transmission of surface waves in the seismic metamaterial

Figure 2 -

 2 4(a) shows the seismic metamaterial with periodicity along the y direction. The bottom and sides of the substrate are set as PML to prevent the reflection of waves from the boundary. The wave source is excited farther to the left of the seismic metamaterial. As shown in Fig.2-4(b), the attenuation effect of Rayleigh waves can be observed directly from the top of the seismic metamaterial. In this thesis, a part of the right surface of the seismic metamaterial is often selected point to obtain the amplitude (or acceleration) A1, and then the amplitude (or acceleration) A0 of the same point is measured after moving out the artificial structure. The transmission

Figure 2 - 7

 27 Figure 2-7 Metamaterial beam consisting of a beam and spring oscillators

  -47D) Substituting Eq. (2-46) into Eq. (2-47), we obtain 𝚿 𝑛-1 (𝑎, 𝑡) = 𝐇 2 𝚿 𝑛 (0, 𝑡). (2-48) Substituting Eqs. (2-42) and (2-48), the transfer equation of the flexural waves from the right interface of the (𝑛 -1)th unit cell to the right interface of the nth unit cell is obtain

(2- 50 )

 50 Substituting Eq. (2-50) into Eq. (2-49), we obtain |𝐓 𝑢𝑛𝑖𝑡 -e iλ𝑎 𝐈 A | = 0, (2-51) where 𝐈 A is an identity matrix. By solving the eigenvalue equation, the band structure can be obtained. Conclusions In this chapter, the important methods used in this thesis are presented. This chapter first introduces the basic theory in the study of seismic metamaterials, including elastic wave equation, Rayleigh waves, periodic condition, Bloch's theorem and band structure. Then the finite element method study based on the solid mechanics module, multibody dynamics module and PDE module of the software COMSOL Multiphysics is presented. Finally, the analytical calculations of dispersion relation of Rayleigh waves in a two-layered half-space, and dispersion relation of flexural waves in a metamaterial beam are given based on the transfer matrix method.

  SMs in this paper. Then we calculate the band structures of the onedimensional (1D) and two-dimensional (2D) SMs, check their attenuation ability to Rayleigh waves, and specifically discuss the propagation of Rayleigh waves in the SM in the range of the ultra-low frequency BG. The subwavelength SM does not need bedrock or fixed boundary condition to clamp the bottom of the pillars in metamaterials.

Figure 3 -

 3 Figure 3-1(a) shows that the unit cell of the two-component SMP consists of a steel core wrapped with a flexible foam plate.Figure 3-1(b) shows the unit cell of the two-

Figure 3 - 1

 31 Figure 3-1 The two-component seismic metamaterial (a) The unit cell of two-component seismic metamaterial plate (SMP). (b) The unit cell of two-component seismic metamaterial (SM) consisting of two-component SMP and half space. Band structures of (c) two-component SMP and (d) two-component SM. (e) Experimental setup and the design of unit cell. (f) Experimental and numerical transmission spectrums for supercell of eight unit cells (four rows) of two-component SM

Figure 3 -

 3 Figure 3-2(a) shows a unit cell of 1D three-component SMP, which is set to a Bloch periodic boundary condition in the x direction and a free boundary condition in the z direction.Figure 3-2(b) shows a unit cell of the 1D SM consists of a unit cell of

  Figure 3-2(b) shows a unit cell of the 1D SM consists of a unit cell of 1D three-component SMP and a half space. It is set to the Bloch period boundary condition in the x direction, the upper boundary is a free boundary, and the lower boundary is a fixed boundary. The geometric parameters are: a = 1.5 m, a1 = 0.6a, a2 = 0.5a, H = 20a. The material parameters are illustrated in Table3-1.[START_REF] Du | Elastic metamaterial-based seismic shield for both Lamb and surface waves[END_REF]

Figure 3 - 2

 32 Figure 3-2 The three-component seismic metamaterial (a) The unit cell of the 1D three-component SMP. (b) The unit cell of 1D SM. (c) A schematic diagram of a system for calculating transmission spectra

Figure 3 -

 3 Figure 3-2(c) shows a schematic diagram of the system used to calculate the transmission spectrum. Since the ultra-low frequency BG of the subwavelength SM is mainly studied and discussed in this paper, and the wavelength of the seismic surface wave is large in the range of the ultra-low frequency BG, the values of the geometric parameters involved here are all large, especially the depth (500a) of the substrate, the

  4 -22.0 Hz) for surface waves. And one of them is an ultra-low frequency BG. By comparing the band structures in Figs. 3-3(a) and 3-3(b), we can easily find that the band structure of the 1D SM consisting of the 1D three-component SMP and a half space is almost equal to the band structure of the 1D three-component SMP plus the sound cone. Therefore, the ultra-low frequency BG and the wide BG are generated shown in Fig. 3-3(b). It is worth noting that there are still bulk modes in the frequency range of the ultra-low frequency BG without any surface mode. This result

3- 4

 4 (a), we use the subwavelength SM of 1600 unit cells placed from 0 m to 2400 m for calculation. The transmission distance of the Rayleigh waves increases in the subwavelength SM exponentially with the decrease of frequencies from 5.0 Hz to around 0.5 Hz. It is found that in a higher frequency range between 3 Hz to 5 Hz, we can also attenuate seismic surface waves with very few seismic metamaterial units.

Figure 3 - 3

 33 Figure 3-3 The results of three-component seismic metamaterial (a) Band structure of the 1D three-component SMP. (b) Band structure of the 1D SM. (c) Transmission spectrum of the system in Fig. 3-2(c). (d) The effective parameters of the 1D threecomponent SMP

Figure 3 - 4

 34 Figure 3-4 Transmission of the surface waves in the seismic metamaterial (a) The displacements of surface waves of the system varying with the frequency by calculating the transmission spectrum. (b) The propagation of Rayleigh waves in subwavelength SM with 1600 unit cells at 2.2 Hz. (c) The propagation of Rayleigh waves in subwavelength SM with 100 unit cells at 2.2 Hz. The colors indicate the total displacement (u) ranging from 0 to maximum

3- 5

 5 (a). The green part is the effective medium of the SMP. Its upper boundary is the free boundary, and the thickness is a = 1.5 m. In the low frequency region (0 -5 Hz) away from local resonance, as shown in Fig. 3-3(d), the effective mass density of the SMP is about 2000 kg/m 3 . And the effective Young's modulus Ee = 27.12 GPa, Poisson's ratio υe = 0.3. The substrate shown in Fig. 3-5(a) is a half-space of soil.

Figure 3 - 5

 35 Figure 3-5 Effective two-layered medium (a) Effective two-layered medium of 1D SM, (b) Rayleigh waves dispersion curve of the two-

Fig. 3 -

 3 Fig. 3-6(b). Similarly, we also calculate the band structure of the unit cell of the 2D three-component SMP shown in Fig. 3-6 (a).

Figure 3 - 6

 36 Figure 3-6 The 2D three-component seismic metamaterial (a) The unit cell of the 2D three-component SMP. (b) The unit cell of 2D SM

  3-7(a) and (b), we can easily find that the band structure of the subwavelength SM consisting of the 2D three-component SMP and a half space is almost equal to the band structure of the 2D three-component SMP plus the sound cone. This is also the reason for the wide ultralow BG.We calculate the transmission spectrums of Rayleigh waves in the 1D and 2D SMs (along ΓX direction) with 100 unit cells at 0.5 -5.0 Hz which is within the ultra-low frequency BG. The attenuation effect of the two kinds of subwavelength SM is almost the same, and the 2D SM is slightly better than the 1D SM at low frequencies (0.5 -4.0 Hz). This shows that the 2D SM can also attenuate ultra-low frequency surface waves in the three-dimensional space.As shown in Fig.3-8, by comparing the propagations of Rayleigh waves in the 1D and 2D SMs (xz section diagram) with 100 unit cells at 4.0 Hz, the Rayleigh waves in the 2D SM also convert from surface waves to bulk waves, and the deflection angle of the Rayleigh wave is almost the same as that of the 1D SM. This shows that the 2D SM can attenuate surface waves in the frequency range of its ultra-low frequency BG to protect the target building. Its mechanism is consistent with that of the 1D SM.

Figure 3 - 7 Figure 3 - 8

 3738 Figure 3-7 Transmission of the surface waves (a) Band structure of the 2D three-component SMP. (b) Band structure of the 2D SM. (c) Transmission spectrums of 1D and 2D subwavelength SMs (SWSMs) with 100 unit cells in the frequency range of 0.5 Hz -5.0 Hz

Figure 3 -

 3 Figure 3-9 shows a layered model which consists of a homogeneous isotropic layer of thickness h overlying a homogeneous half space. For the two-layered structure, the dispersion relation of the Rayleigh wave can be calculated using the transfer matrix

Tables 3 -

 3 2 and 3-3. Where Vp1, Vs1, Vr1, and ρ1 (Vp2, Vs2, Vr2, and ρ2) are the longitudinal wave velocity, shear wave velocity, Rayleigh wave velocity and density of the cover layer (substrate), respectively. The thickness of the cover layer is fixed at h = 6 m.

Figure 3 - 9

 39 Figure 3-9 Layered media structure

  frequency. This layered structure has only a unique Rayleigh guided wave: the minimum Rayleigh wave velocity (phase velocity) is equal to the Rayleigh wave velocity of the substrate (Vr2), and the maximum Rayleigh wave velocity (phase velocity) is equal to the shear wave velocity of the substrate (Vs2). Also, Vr2 and Vs2 are underlined in Table3-3. And the most interesting thing is that this Rayleigh guided wave has a cutoff frequency. When the frequency is higher than this cutoff frequency, about 25 Hz as shown in Fig. 3-10(b), there is no Rayleigh wave in such two-layered structure.

Figure 3 - 2 Figure 3 -

 323 Figure 3-10 The dispersion curves of the Rayleigh waves The dispersion curves of the Rayleigh waves of the two-layered layer structure when (a) Vs 2 >Vr 2 >Vs 1 >Vr 1 and (b) Vs 1 >Vr 1 >Vs 2 >Vr 2

Figure 3 -

 3 Figure 3-11(b) shows the complex band structures of the surface modes of the

Figure 3 -Figure 3 -

 33 Figure 3-11 The T-shaped seismic metamaterial (a) The unit cell and (b) complex band structure of the T-shaped seismic metamaterial

Figure 3 -

 3 Figure 3-13 The transmission results (a) A schematic diagram of the finite system for calculating transmission spectra. (b) Distribution of displacement amplitude of the finite system varying with the frequency

  -13(b), we plot the displacement of the surface of this finite system at different frequencies. The TSM is placed at 0 < X < 7500 m. From Fig.3-13(b), we can find that the TSM can attenuate Rayleigh waves well in the wide frequency range of 0.1 -4.8 Hz.In order to discuss the effect of the number of unit cells of the TSM on the attenuated Rayleigh waves, the attenuation of the Rayleigh waves by the TSM with 200 unit cells is calculated. For a slightly lower frequency, such as 1.3 Hz, the TSM with 200 unit cells can convert a portion of the Rayleigh waves into a bulk waves, but the rest of the Rayleigh wave also passes through the TSM in Fig.3-14(a). Obviously, the attenuation effect of the TSM with 200 unit cells is limited. As shown in Fig.3-14(b), the TSM with 5000 unit cells are used can well convert most of the Rayleigh waves into bulk waves. At a slightly higher frequency, such as 3.5 Hz in Fig.3-14(c-d), the attenuation effect of the Rayleigh waves by the TSM of 200 and 5000 unit cells is almost the same. Rayleigh waves are almost converted into bulk waves by the TSMs. This is the third characteristic of the BG based on inverse-dispersion effect: the SM with enough unit cells can convert most of surface waves to bulk waves.

Figure 3 -

 3 Figure 3-14 The propagations of the surface waves Distribution of displacement amplitude at 1.3 Hz for TSM with (a) 200 and (b) 5000 unit cells and at 3.5 Hz for (c) 200 and (d) 5000 unit cells.

Figure 4 - 1

 41 Figure 4-1 The unit cells of the seismic metamaterials The unit cells of (a) the pillar-like seismic metamaterial and (b) the inverted T-shaped SM

Figures 4 -

 4 Figures 4-2(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

Figure 4 - 2

 42 Figure 4-2 The band structures and vibration modes The classical band structures of (a) the PSM and (b) the 1D ITSM, the vibration modes of (c) the PSM and (d) the 1D ITSM at the points marked the classical band structures.

Figures 4 -

 4 Figures 4-3(a -b) show that the variation of geometric parameters of the plate with the position and relative bandwidth of the FBG. As shown in Fig. 4-3(a), when the length l1 of the plate increases from 0.5a to 0.99a, the lower boundary of the FBG slowly increases from 4.4 Hz to 7.0 Hz, and the upper boundary increases rapidly from 5.7 Hz to 20.7 Hz. Therefore, the relative bandwidth of the FBG increases from 0.27 to 0.99. As shown in Fig. 4-3(b), when the width of the plate increases from 0.02a to 0.18a, the lower boundary of the FBG first grows slowly from 5.5 Hz to 6.7 Hz, and then slowly drops to 6.4 Hz. However, the upper boundary of the FBG increases rapidly from 9.0 Hz to 17.2 Hz, and then slowly decreases to 13.7 Hz. Therefore, the relative

Figure 4 -

 4 Figure 4-3(c) shows the effect of the pillar height on the FBG. When the pillar

Figure 4 - 4

 44 Figure 4-4 Effects of material parameters on first bandgap Variation of the upper and lower frequency boundaries, and the relative bandwidth of the FBG (a) the mass density ρ ITS and (b) the Young's modulus E ITS . The black solid line is the upper and lower boundaries of the FBG, and the red dashed line is the relative bandwidth

Fig. 4 -

 4 Fig. 4-4(b). When EITS increases from 50 GPa to 350 GPa, the lower boundary increases slightly. When EITS increases from 50 GPa to 180 GPa, the upper boundary rapidly increases from 10.0 Hz to 17.0 Hz. Finally, the frequency slowly increases to 18.2 Hz, when EITS increases from 180 GPa to 350 GPa. Therefore, the relative bandwidth rapidly increases from 0.54 to 0.88, and it has just little change when EITS is larger than 180 GPa. These results show the ITSM has the FBG with largest relative bandwidth

Figure 4 - 5

 45 Figure 4-5 The complex band structures The complex band structures of (a) the 1D ITSM and (b) the PSM.

Figure 4 - 6

 46 Figure 4-6 The transmission of Rayleigh waves in the ITSM (a) Schematic model of the 1D ITSM with 10 unit cells for transmission calculations. (b) Transmission spectrum of the ITSM at the frequency range from 4 Hz to 22 Hz. (c) Distribution of displacement amplitude in the ITSM at 7 Hz and 12 Hz

Figure 4 - 7

 47 Figure 4-7 Transmission of Rayleigh waves in different ITSMs

Fig. 4 -

 4 Fig. 4-8(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 4-3. As shown in Fig. 4-8(b), the central garden of the Department of Mechanics on the Beiyangyuan Campus of Tianjin University (China) was selected for large-scale field experiments in August 2020. The geometric parameters of the TS used in largescale filed experiments are consistent with those in Fig. 4-8(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.

Fig. 4 -

 4 Fig. 4-8(b)). The vibration source is generated by the rammer (Mingtu Mechanics, C90t)

Figure 4 - 8

 48 Figure 4-8 The experimental setup (a) Schematic of the unit cell and (b) top view of the experimental setup

  cone is clearly surrounded by the light gray areas. The vibrating mode of the highest point of the first band is given in the insert of Fig. 4-9(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. The surface waves considered in this work are

Fig. 4 -

 4 Fig. 4-9(a).In Fig.4-9(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 (t = 0.02a) low-density material (Young's modulus is 20 GPa, density is 1.8 kg/m 3 and Poisson's ratio is 0.3) is added underneath all man-made structures to simulate the IPC between soil and the structures. It is worth noting that the inexistent low-density material is used just to qualitatively investigate the effect of the IPC on the attenuation of Rayleigh waves. 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 the simple connection 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. Figure4-10 shows the simulated results of propagation of Rayleigh waves without the 2D ITSM at 44 and 100 Hz, respectively. The boundary conditions of the system in the simulation are all low-reflection boundary condition, except the free boundary condition for the upper surface.

Figure 4 - 9

 49 Figure 4-9 The results of the 2D ITSM (a) the band structure in the ГX direction of the first irreducible Brillouin zone of the 2D ITSM. (b) The transmission spectrum of the simulated and experimental results along ГX direction through the 2D ITSM.

Figure 4 -

 4 Figure 4-10 The simulated results of propagation

Figure 5 - 1

 51 Figure 5-1 The schematic diagram of a kind of the elastic metamaterial

Figure 5 -

 5 2(b) shows the upper and lower frequency bounds of the first bandgap when only the lattice constant of the elastic metamaterial is changed. It is found that the first bandgap is absolutely unchanged due to the immobility of the lower frequency bound at 5.0 Hz, although the upper one decreases from 5.5 Hz to 5.2 Hz.

Figures 5 -

 5 Figures 5-2(c) shows a part of the elastic metamaterials composed by 30 steel pillars periodically arranged with a lattice constant of 1.5 m, 3 m, and randomly arranged within 75 m on soil substrate, respectively. The geometric parameters of the steel pillars are consistent with those in Figs. 5-2(a). A perfectly matched layer (PML) with a thickness of 50a is added around the substrate to prevent reflections. The transmission of Rayleigh waves in these three elastic metamaterials are shown in Fig. 5-2(d). It is clearly found that these three different arrangements have the same LFAZ around 5.0 Hz. This shows that the LFAZ is immobile with the changed distances between the pillars. The spring-mass model consisting of the pillar and substrate has not been changed. Therefore, the resonance frequency of the spring-mass model remains the same. Even if the same material is used for the pillars and the substrate, this

Figure 5 - 2 Figure 5 - 3

 5253 Figure 5-2 Study of the first bandgap (a) The complex band structures of the elastic metamaterial around the first bandgap. (b) The upper and lower frequency bounds of the first bandgap with various lattice constants of the elastic metamaterial. (c) Partial illustrations of two periodic arrangements of steel pillars with the lattice constants of 1.5 m, 3 m on a soil substrate and random arrangement of steel pillars on a soil substrate within 75 m. (d) The transmission of Rayleigh waves in the three elastic metamaterial

Figure 5 -

 5 3(a) shows that the center frequency of the first bandgap increases when the Young's modulus of the pillars increases from 0.1 GPa to 10 GPa. And when the Young's modulus of the pillars is larger than 10 GPa, the first bandgap is hardly changed. The position the first bandgap decreases continuously as the density of the pillar increases in Fig.5-3(b). Figures5-3(c -f) show the height and width of the pillar all play an important role in the position of the first bandgap whenever the density or the mass of the pillar is immobile. It is easy to find that the mass of the pillar on the substrate affects a lot on the first bandgap from Figs.5-3(bf).

Figure 5 -

 5 3(g) shows that the center frequency of the first bandgap increases rapidly when the Young's modulus of the substrate increases from 1 MPa to 1000 MPa.However, the density of the substrate is hardly to change the position of the first bandgap shown in Fig.5-3(h). We find that except the density of the substrate, all these parameters can significantly move the position of the first bandgap.

Figure 5 - 4 (

 54 a) shows the unit cell of the elastic metamaterial whose band structure can be numerically simulated by using the solid module of COMSOL Multiphysics. In this work, the bandgaps only in the ΓX direction of the first irreducible Brillouin zone are investigated. The numerical computations are described in Refs.

Figure 5 - 4 Figure 5 - 5 Figure 5 - 6

 545556 Figure 5-4 Study of the second bandgap (a) The unit cell of the elastic metamaterial for surface waves, (b) the effect of the lattice constant a on the first and second bandgaps in the ΓX direction of the elastic metamaterial

Figure 5 - 7 Figure 5 - 8 Figure 5 - 9

 575859 Figure 5-7 The sample for experiments

Figure 5 -

 5 Figure 5-10 The transmissions of surface waves in the sample (a) The transmissions of surface waves on the prepared samples in simulations and experiments. Propagations of surface waves on the sample (b) in the experiment and (c) in the simulation at 49 kHz. The color indicates the acceleration field on the surface of samples

Figure 5 -

 5 Figure 5-10(a) illustrates the simulation and experimental measurements of the transmission of the surface waves through the considered metamaterial sample. One

  Of course, other suitable artificial structures, number of the structures and distances can also be selected. This paper only gives one possibility. As shown in Fig. 5-11(b), when Rayleigh waves are incident from left to right (LTR) or from right to left (RTL), the SMS can all attenuate the energy well in the range from 5.0 to 22.0 Hz. The attenuation of the Rayleigh waves in the SMSs is complex because the attenuation zone includes the frequency ranges of the first and second bandgaps. There are two mechanisms of the attenuation of the Rayleigh waves in the seismic metamaterial.

Figure 5 -

 5 Figure 5-11 The transmissions of surface waves in the ITSM (a) Schematic picture of the finite system used for transmission calculations of the seismic metamaterial, (b) the transmission of Rayleigh waves propagating in the seismic metamaterial from different directions, and the transmission of Rayleigh waves propagating in a periodic arrangement of the 15 inverted T-shaped structures with a lattice constant of 1.5 m. The gray areais the attenuation zone of the seismic metamaterial. The LTR, RTL and Per. mean "from left to right", "from right to left" and "periodic", respectively

Figure 5 -

 5 Figure 5-12 The propagations of the Rayleigh waves in the ITSM

FillFigure 5 -

 5 Figure 5-13 The transmissions of surface waves on the ITSM with layered substrate (a) Schematic picture of the finite system. (b) The transmission of Rayleigh waves propagating in the seismic metamaterials

Figure 6 -

 6 Figure 6-1(a) illustrates the unit cell of the MMP based on inertial amplification in the experiments and utilized in the simulation calculations. The simplified model of the structure is also given in Fig. 6-1(b). The unit cell is constituted of the lever, the support bar, mass, spring, hinge joint and the base plate. In the MMP, the lattice constant a is 3 cm. The thickness t of the plate is 1 mm. The other geometric parameters of the structure are shown in Fig.6-1(c). The resonant frequency of the resonator in the model is[START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF]

Figure 6 - 1

 61 Figure 6-1 The metamaterial plate and experimental setup (a) The unit cell of the metamaterial plate (MMP) considered in the numerical simulation and experiment. (b) The simplified model of the (a). (c) The geometric parameters of the unit cell. (d) The experimental sample of the MMP and (e) the experimental layout with a computer (Label A), the MMP (Label B), the laser vibrometer (Label C) and the same plate (Label D) as in the MMP

Figure 6 - 2

 62 Figure 6-2 The enlarged photograph of the hinged part (a) The PLA rod, lever and support bar used at hinged part. (b) The assembled hinged part by using lever, the support bar and PLA rod

)

  𝜃(𝑥, 𝑡) = 𝑦(𝑥, 𝑡) ′ , (6-5) 𝑀(𝑥, 𝑡) = 𝐸𝐼𝑦(𝑥, 𝑡) ′′ , (6-6) 𝑄(𝑥, 𝑡) = -𝐸𝐼𝑦(𝑥, 𝑡) ′′′ , (6-7) where A, B, C and D are complex coefficients; λ = √𝜌𝐴𝜔 2 𝐸𝐼 ⁄ 4 with ω = 2πf being the circular frequency is the flexural wavenumber; and θ, M and Q are the slope, bending moment and shear force, respectively.

Figure 6 - 3

 63 Figure 6-3 The analytical results of the simplified model (a) The simplified model used for analytical solution. (b) the numerical simulation results (COMSOL) and analytical results of the band structure of the simplified model

For instance, Fig. 6 -

 6 3(b) shows the simulated results and analytical results of the band structure of the simplified model, when the lattice constant is a = 1 m; the height, breadth, sectional area, section moment of inertia, Young's modulus and density of the beam is h = 0.05 m, b = 1 m, A = bh, I = bh 3 /12, 207 GPa and 7784 kg/m 3 , respectively; l1 = 0.9 m, l2 = 0.3 m; m = 15.2838 kg and k = 100,000 N/m. It is worth noting that the support bar and the lever are rigid. The simulation results are calculated by COMSOL 5.4 (Multibody Dynamics Module) to evaluate the accuracy of the analytical solution.The simulated results are almost in agreement with the analytical results, especially for the position of the bandgap. It is worth noting that the frequencies of the two bandgaps are consistent with the predicted result from Eq. (6-1). The effective inertial mass is 4m due to the μ = l1/ l2 = 3. Obviously, the frequency of the bandgap can be controlled easily in this model by changing one or more parameters of the structure. For instance, an ultra-low frequency bandgap can be achieved when the mass m or lever ratio μ is larger, or when the spring stiffness k is smaller than the example shown in Fig.6-3.

Figure 6 - 4 Figure 6 - 5

 6465 Figure 6-4 The band structure and transmission (a) The band structure of the MMP along the ΓX direction. (b) The transmissions of the flexural waves in the MMPs and the bare plate along the ΓX direction. (c) The unit cell consisted of the same plate, spring and mass as in MMP. (d) The band structure and transmission of the metamaterial constituted of periodic (c)

Figure 6 - 6 Fig. 6 - 6 .

 6666 Figure 6-6 Vibration modes of the metamaterial plates Vibration modes corresponding to points (a) A 1 , (b) B 1 , (c) A 2 and (d) B 2 marked in 6-4(d). The color bar represents the degree of displacement

Figure 6 -TransmissionFigure 6 - 7

 667 Figs.6-4(a -d).Figure 6-6(a) shows the strong resonance of the mass at point A1. At point B1, most of the displacement is around the spring part shown in Fig. 6-6(b). It is clear that the large bandgap is due to the inertial amplification system and the strong

Figure 6 - 8

 68 Figure 6-8 The unit cell and band structure (a) The unit cell of the SM for surface waves. (b) The band structure of the SM for surface waves along the ΓX direction in the range from 4 Hz to 6 Hz

Figure 6 -Figure 6 -

 66 Figure 6-8(b) shows the band structure of the SM along the ΓX direction when the thickness of the rubber is 0.02a. The area outside the dark gray area is the sound cone. Three bands can be found under the sound cone, which are illustrated by the blue, red and green lines. A bandgap which is marked by the light gray area is shown in the frequency range from 4 Hz to 6 Hz.

Figures 6 -

 6 Figures 6-9(a -b) show that the vibration mainly appears at the two ends of the lever which rotates around the hinge joint in the xz plane. This implies that the bandgap for surface waves is induced by the inertial amplification. The vibration modes illustrated in Figs. 6-9(c -d) show that there is almost no vibration in the xz plane.

Figure 6 - 9

 69 Figure 6-9 Vibration modes of the seismic metamaterial Vibration modes and their central section along the xz plane of the SM for surface waves corresponding to points (a-b) A 3 , (c-d) B 3 and (e-f) C 3 marked in Fig. 6-8(b). The arrows in (b), (d) and (f) indicate the displacement eigenvector; and its length and direction indicate the magnitude and trend of the particle movement, respectively. The color bar represents the degree of the displacement

  -10 or Fig.6-8(b) are narrow. This is an inevitable result. In Fig.6-4(a), it is the flexural waves (vibration only along the z axis) that the metamaterials attenuate in a wide frequency range. In Fig.6-10 and Fig.6-8(b), it is the Rayleigh surface waves, which include both longitudinal (x axis) and transverse (z axis) motions, that the SMs are designed to attenuate at ultra-low frequencies. These two kinds of bandgaps cannot be directly compared. In addition, at ultra-low frequencies, the bandgap induced by inertial amplification is quite narrow, as shown in Fig.6-2. Suitable unit cells can be selected from Fig.6-10 to construct a new seismic metamaterial to isolate seismic waves in a wide frequency range.

Figure 6 -

 6 Figure 6-10 Effects of parameters on the first bandgap Effects of the hinge joint's position along x direction (a), the rubber layer's thickness (b) and the Young's modulus of the rubber (c) on the bandgaps. The blue and red solid lines are the center frequencies and relative bandwidth of the bandgaps, respectively

Figure 7 -

 7 1(b) illustrates the schematic diagram of the experimental setup. The computer outputs the source signal to the piezoelectric patches fixed at the end of the sample. The out-of-plane accelerations at points a and b are measured by the laser vibrometer (Polytec PSV-500). The collected data is input back to the computer and is processed.

Figure 7 - 1

 71 Figure 7-1 The metamaterials and experimental setup (a) The samples used in the experiments (TM: traditional metamaterial; IAM: inertially amplified metamaterial). (b) Schematic diagram of the experimental setup including laser vibrometer (PSV), the computer and the sample. Points a and b are two data collection points. (c) The unit cell of the IAM. (d) The simplified model of the inertially amplified mechanism in the unit cell and (e) the geometric parameters of the unit cell of the IAM

Figure 7 -

 7 Figure 7-1(c) illustrates the unit cell of the IAM considered in the numerical simulations and experiments. The unit cell includes one base beam, two arms and three hinge joints. The thickness t of the beam is 1 mm. The angle θ, which determines the magnitude of the amplification generated by the inertially amplified mechanisms, inthis work is set at 5°. It is worth noting that hinge joint is not the necessary part for inertial amplification mechanisms. As shown in the literature, Yilmaz et al.[START_REF] Yilmaz | Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations[END_REF]

Figure 7 -

 7 Figure 7-2(a) illustrates the band structure of the IAM along x direction and the color bar represents the degree of the out-of-plane polarization. The parameter 𝜉 is used to defined the degree of the out-of-plane polarization:

Figure 7 - 2

 72 Figure 7-2 The band structure and transmission (a) The band structure of the IAM. The color bar represents the degree of the out-of-plane polarization. (b) The transmission spectrums of the flexural waves in the IAM, the TM and the BB (bare beam). The AZ is attenuation zone. (c) Vibration modes corresponding to points A -D shown in (a). The color bar in (c) represents the degree of displacement

Figure 7 -

 7 Figure 7-2(c) illustrates the vibration modes at the marked points A -D in the band

Model and band structure

  It is noted that the simplified model of embedded inertial amplification mechanism shown in Fig.7-1(d) can also theoretically work on a half-space when the substrate provides the stiffness k. Therefore, in theory, the embedded inertial amplification mechanism is very suitable to design surface wave metamaterials, which, just with small mass, can induce low-frequency bandgap for surface waves. As shown in Fig.7-3(a), the embedded inertial amplification mechanism is introduced on a half-space to attenuate seismic surface waves. The unit cell of the seismic metamaterial (SM) is constituted of three parts: two steel arms represented by gray area, three hinge joints represented by dotted circle and soil substrate represented by yellow area. The blue and red lines represent Floquet-Bloch boundary condition and fixed constraint, respectively.

Figure 7 -

 7 Figure 7-3(b) illustrates the band structure of the one-dimension periodic SM. The sound cone and the bandgap for surface waves are represented by the dark grey and light grey zones, respectively. The bandgap is in the range from 4.0 to 4.7 Hz and the relative bandwidth is 0.16. Despite the narrow width of this bandgap, the definitive existence of this bandgap is encouraging.

Figure 7 - 3

 73 Figure 7-3 Study of the inertially amplified seismic metamaterial

Figure 7 -Chapter 8 2 . 3 . 2 . 4 .

 782324 Figure 7-3(d) illustrates the vibration modes at the marked points E -G in the band structure shown in Fig. 7-3(b). The arrows shown in the unit cell indicate the displacement eigenvector, whose length and direction represent the magnitude and trend of the particle movement, respectively. At points E and G, the significant displacement appears both in the soil substrate and the arms. At point F, almost all the displacement appears around the hinge joint between the two arms, which is similar as the bandgap shown in Fig. 7-2. The strong local resonance appears only on the two arms of the inertial amplification mechanism. It demonstrates the bandgap for surface waves is induced by the inertial amplification.

5 .

 5 The design methods and theories of phononic crystals and other related metamaterials can be used to achieve a systematic design of seismic metamaterials. tremblement de terre de Tangshan et que d'autres phénomènes précurseurs sont apparus tardivement. En conséquence, le tremblement de terre de Tangshan a également été classé parmi les dix principales catastrophes naturelles du 20e siècle. À ce jour, il est toujours difficile de prévoir avec précision les tremblements de terre.La technologie de résistance sismique est une méthode importante pour protéger les bâtiments cibles en les concevant et en les aménageant avant les tremblements de terre. La technologie conventionnelle de résistance sismique résiste aux forces des tremblements de terre en augmentant la force, la rigidité et la ductilité des éléments structurels du bâtiment, plus précisément en augmentant la taille de la section transversale des éléments structurels et en ajoutant des murs en béton armé pour résister aux tremblements de terre. Ces dernières années, les nouvelles technologies de résistance aux séismes sont devenues plus "souples", utilisant des dispositifs d'isolation et d'amortissement sismiques tels que des paliers en caoutchouc et des amortisseurs pour isoler et dissiper l'énergie sismique. Lorsqu'un tremblement de terre se produit, les dispositifs d'amortissement dissipent l'énergie des ondes sismiques dans le bâtiment en le comprimant, le pliant et le déviant de manière à modifier sa structure et sa forme, protégeant ainsi le bâtiment. Au début des années 1990, le concept de cristaux phononiques a été proposé sur la base des cristaux photoniques. Les ondes acoustiques ou les ondes élastiques dans un cristal phononique constitué de milieux périodiques élastiques ne se propagent pas dans une certaine gamme de fréquences en raison de la diffusion de Bragg. En 1995, des chercheurs ont testé les propriétés de transmission acoustique d'une sculpture disposée périodiquement "Flowing Melody". L'existence d'une bande interdite acoustique a été démontrée expérimentalement pour la première fois. La caractéristique la plus distinctive d'un cristal phononique est la propriété de la bande interdite. La structure de bande d'un cristal phononique se compose de bandes interdites et de bandes. Dans la gamme de fréquences de la bande interdite, les ondes ne peuvent pas se propager dans les cristaux phononiques. Dans la gamme de fréquences de la bande passante, les ondes peuvent se propager normalement dans les cristaux phononiques. En 2000, un métamatériau acoustique a été proposé sur la base du mécanisme de résonance locale. La bande interdite la plus basse de ce métamatériau acoustique correspond à une longueur d'onde de 300 fois la constante de réseau, ce qui permet d'atteindre l'objectif de "contrôle de petite taille de grande longueur d'onde". La fréquence de la bande interdite du métamatériau acoustique est presque indépendante de la périodicité de la cellule unitaire, mais étroitement liée à la fréquence de résonance de la cellule unitaire de résonance. Les bandes plates et les bandes interdites peuvent être trouvées dans les structures de bande. La vitesse de groupe de l'onde dans le métamatériau acoustique est nulle à ce moment-là, l'onde ne peut donc pas se propager dans le cristal phononique. Ces dernières années, des chercheurs inspirés par les cristaux phononiques et les métamatériaux acoustiques ont proposé des métamatériaux sismiques. La protection des bâtiments est réalisée d'une nouvelle manière : Les métamatériaux sismiques sont placés autour des bâtiments pour réfléchir et atténuer les ondes sismiques à des fins de protection. Après un séisme, les vibrations mécaniques générées à la source se propagent à l'intérieur et à la surface de la Terre, formant ainsi des ondes sismiques. Les ondes sismiques peuvent être simplement distinguées en deux catégories principales: les ondes de masse et les ondes de surface. Dans les régions de champ moyen et lointain d'un tremblement de terre, les ondes de masse sont couplées sur la surface du demiespace pour former des ondes de surface (également appelées ondes de Rayleigh). L'amplitude de la vibration décroît de manière exponentielle dans le sens de la profondeur du sol, de sorte que l'énergie des ondes de Rayleigh est presque entièrement concentrée près du sol. Les ondes de Rayleigh se trouvent généralement loin de la source du tremblement de terre. Les ondes de Rayleigh ont une grande amplitude, une décroissance lente, et peuvent se propager dans la surface de la terre plusieurs fois après un grand tremblement de terre et avoir encore une grande énergie. Par conséquent, dans le champ lointain d'un tremblement de terre, les ondes de Rayleigh sont plus destructrices pour les bâtiments que les ondes de masse et peuvent menacer les bâtiments à une plus grande distance. La conception et la recherche de métamatériaux sismiques pour les ondes de Rayleigh revêtent donc une grande importance. Les métamatériaux sismiques, généralement basés sur la résonance locale et la diffusion de Bragg, utilisent les propriétés de la bande interdite pour isoler et atténuer les ondes sismiques, voire changer la direction de leur propagation. Les métamatériaux sismiques peuvent être placés autour ou au pied d'un bâtiment pour contrôler les ondes sismiques. La fréquence de résonance de la plupart des bâtiments construits par l'homme est inférieure à 20 Hz, de sorte que la plupart des métamatériaux sismiques actuels sont conçus pour cette gamme. En outre, les chercheurs utilisent souvent des matériaux courants sur les chantiers de construction pour réduire le coût des métamatériaux sismiques, comme l'acier, le caoutchouc industriel et le béton. Afin d'atténuer les ondes sismiques de Rayleigh dans une large gamme de basses fréquences, plusieurs types de SMs sont conçus dans cette thèse. Les caractéristiques de transmission des ondes sismiques de Rayleigh dans les SMs sont étudiées en combinant l'analyse du modèle théorique, la simulation numérique et l'investigation expérimentale. Les principaux contenus et conclusions se déclinent comme suit : 1. En se basant sur des expé riences à l'é chelle du laboratoire et sur la simulation numé rique du SM constitué d'une plaque mé tamaté riau sismique (SMP) à deux composants et d'un demi-espace, une nouvelle mé thodologie de conception pour cré er des SM afin d'atté nuer les ondes sismiques de surface est dé montré e. En outre, les expé riences à l'é chelle du laboratoire constituent é galement une base importante pour l'application du mé tamaté riau sub-longueur d'onde à trois composants. Ensuite, un SM sous-longueur d'onde à trois composants, composé d'un SMP à trois composants et d'un demi-espace, est proposé . En analysant la propagation des ondes de Rayleigh dans les SM unidimensionnels (1D) et 2D dans la gamme de fré quences de la bande interdite à ultra-basse fré quence, on constate que les ondes de Rayleigh sont dé vié es à un certain angle et converties en ondes de masse. En ré sumé , le SM constitué d'une seule couche de SMP enterré e et d'un substrat possè de une bande interdite à ultra-basse fré quence et peut convertir les ondes de Rayleigh en ondes de masse dans la gamme de fré quences de la bande interdite.2. Un SM 1D en forme de T inversé (ITSM) avec une premiè re bande interdite ultralarge (FBG) a é té proposé en amé liorant le SM en forme de pilier. Les effets des paramè tres gé omé triques et maté riels de l'ITSM sur le FBG ont é galement é té discuté s.

3 . 4 . 5 .

 345 La proprié té mé canique de la zone d'atté nuation de la plus basse fré quence (LFAZ) du mé tamaté riau é lastique constitué de piliers sur un substrat a é té é tudié e numé riquement. Il s'avè re que la ZAF est immobile lorsque les distances entre les piliers changent. L'"effet de pié geage de l'arc-en-ciel" dans la deuxiè me zone d'atté nuation (SAZ) a ensuite é té dé montré par simulation numé rique et par des expé riences. Par consé quent, pour un mé tamaté riau é lastique correctement conç u, il est possible de relier la limite infé rieure de la SAZ et la limite supé rieure de la LFAZ. Une large zone d'atté nuation à partir d'une fré quence ultra-basse pour les ondes de surface peut ê tre obtenue en utilisant simplement un type de structures artificielles sur un substrat. Dans le cas des ondes sismiques de Rayleigh, ces deux caracté ristiques de la LFAZ et de la SAZ sont utilisé es pour concevoir un type de mé tamaté riau sismique en utilisant simplement un type de structure artificielle en acier sur un substrat de sol. Le mé tamaté riau sismique proposé peut atté nuer les ondes de Rayleigh de 5 à 22 Hz. Une plaque mé tamaté rielle conç ue (MMP) avec des structures à bras de levier a é té é tudié e expé rimentalement et discuté e. Il est dé montré qu'une large bande interdite pour les ondes de flexion apparaî t dans une ré gion à basse fré quence. L'analyse de propagation mesuré e sur la face arriè re de la MMP a dé montré que les structures à bras de levier atté nuent fortement les ondes de flexion. Enfin, les structures à bras de levier similaires sont introduites dans un demi-espace de sol pour obtenir les bandes interdites induites par l'amplification inertielle pour les ondes sismiques de surface dans une gamme de basses fré quences. L'avantage significatif est que les mé canismes d'amplification inertielle avec une petite masse peuvent atteindre une forte atté nuation des vibrations à basse fré quence, une proprié té qui devrait ê tre ré alisé e pour les SMs. Un mé tamaté riau unidimensionnel à amplification inertielle est proposé , dont la cellule unitaire est composé e d'une poutre de base, de deux bras et de trois articulations. Les ré sultats expé rimentaux et simulé s des transmissions des ondes de flexion dans la poutre nue, le mé tamaté riau traditionnel et le mé tamaté riau à amplification inertielle ont dé montré que le mé tamaté riau à amplification inertielle est capable d'isoler les ondes de flexion dans la bande interdite induite par l'amplification inertielle. Enfin, le mé canisme d'amplification inertielle inté gré a é té introduit dans un demi-espace de sol pour concevoir le SM. La structure de bande et les ré sultats de transmission ont montré 144 que le SM peut isoler les ondes sismiques de Rayleigh à des fré quences ultra-basses.En résumé, plusieurs métamatériaux sismiques ont été conçus dans cette thèse pour isoler les ondes sismiques de Rayleigh. Ces métamatériaux peuvent atténuer les ondes sismiques de Rayleigh dans la gamme des basses fréquences et des ultra-basses fréquences, afin de protéger les bâtiments construits par l'homme et de réduire les dommages causés par les tremblements de terre. Ce travail fournit non seulement de nouvelles approches pour contrôler les ondes sismiques de surface à ultra-basse fréquence, mais aussi de nouvelles idées de conception pour diriger les ondes de surface.Mots clés: Métamatériaux sismiques; ondes de Rayleigh; bande interdite; largebasse fréquence; dispersion inverse; résonance locale; amplification inertielle.

Table 3 -

 3

		1 Material parameters of the unit cell
	Material	Density (kg/m 3 )	Young's modulus (Pa)	Poisson's ratio
	Rubber	1300	1.2 x 10 5	0.47
	Concrete	2500	4 x 10 10	0.3
	Steel	7784	2.07 x 10 11	0.3
	foamed plate	1053	1.6 x 10 5	0.39
	Soil	1800	2 x 10 7	0.3

  -1(d). It is found that the addition of the substrate does not affect the original band structure of SMP. That is to say, the band structure of the SM composed of the two-component SMP and a half space is almost equal to band structure of the two-component SMP plus the sound cone. The sound cone, the boundary of which is formed by the slowest bulk wave propagating in the substrate, can be used to immediately identify real propagating surface waves, bulk modes as well as leaky surface modes and spurious PML modes. Therefore, the wide BG of the SMP itself is preserved. At 110 Hz -300 Hz, we can verify that the SM can attenuate the surface waves well by lab-scale experiments and simulated calculations of the supercell, which corresponds to the BG in Fig.3-1(d). The geometric model, material parameters, and position of the supercell and substrate used in the simulation calculations are consistent with the lab-scale experiments. In addition, the boundary formed by the glass wall is replaced by a low reflection boundary to reduce reflected waves.

Table 3 -

 3 2 The assumed material parameters when V S2 >V R2 >V S1 >V R1

		V P	V S	V R	ρ
	Cover layer	1500	1000	893.1	1800
	Substrate	2000	1400	1229.0	2300

Table 3 -

 3 3 The assumed material parameters when V S1 >V R1 >V S2 >V R2

		V P	V S	V R	ρ
	Cover layer	2000	1400	1229.0	2300
	Substrate	1000	800	636.7	1800

Table 4 -

 4

	a	H	l 1	w 1	l 2 (l)	w 2 (w)
	1.5m	300a	0.95a	0.05a	1.4285a	0.07a

1

The geometric parameter of the SMs

Table 4 -

 4

2

The material parameters

Table 4

 4

			-3 The geometric parameters
	a 1	b	t	h	D	d
	0.25 m	0.2375 m	0.012 m	0.75 m	0.051 m	0.041 m

Table 5 -

 5 2 The geometric parameters of the ITSM

	l 1	w 1	l 2	w 2
	1.35 m	0.075 m	2.25 m	0.15 m

  is the transfer matrix. It is shown in Appendix of this chapter.As for the nth lever-type structure on the beam, the dynamic equilibrium equation

	where 𝐇 2 is
	𝑓 𝑛1 (𝑡) + 𝑓 𝑛2 (𝑡) + 𝑚𝑍 ̈𝑛(𝑡) = 0,	(6-14A)
	𝑙 2 𝑓 𝑛1 (𝑡) + (𝑙 1 -𝑙 2 )𝑚𝑍 ̈𝑛(𝑡) = 0,	(6-14B)
	where 𝑍 𝑛 (𝑡) is the displacement of the nth mass of the lever-type structure; and
	𝑓 𝑛1 (𝑡) and 𝑓 𝑛2 (𝑡) are the forces of the spring and support bar on the lever,
	respectively. Due to the Hooke's Law, we obtain
	𝑓 𝑛1 (𝑡) = 𝑘 [ 𝑍 𝑛 (𝑡) 𝜇-1	+ 𝑧 𝑛,1 (0, 𝑡) -𝑧 𝑛,2 (𝑙 2 , 𝑡)],	(6-15)
	1 (0, 𝑡) = 𝐇 1𝛼 [𝐴 𝑛,1 𝐵 𝑛,1 𝐶 𝑛,1 𝐷 𝑛,1 ] 𝑇 𝑒 𝑖𝜔𝑡 ,	(6-9)
	𝚿 𝑛,1 (𝑙 2 , 𝑡) = 𝐇 1𝛽 [𝐴 𝑛,1 𝐵 𝑛,1 𝐶 𝑛,1 𝐷 𝑛,1 ] 𝑇 𝑒 𝑖𝜔𝑡 ,	(6-10)
	where 𝚿(𝑥, 𝑡) = [𝑧(𝑥, 𝑡) 𝜃(𝑥, 𝑡) 𝑀(𝑥, 𝑡) 𝑄(𝑥, 𝑡)] 𝑇 is the state vector; and
	[𝐴 𝑛,1 𝐵 𝑛,1 𝐶 𝑛,1 𝐷 𝑛,1 ] 𝑇 is the coefficient vector. Substituting Eq. (6-10) into Eq. (6-9),
	we obtain
	𝚿 𝑛,1 (𝑙 2 , 𝑡) = 𝐇 1𝛽 𝐇 1𝛼 -1 𝚿 𝑛,1 (0, 𝑡) ≜ 𝐇 1 𝚿 𝑛,1 (0, 𝑡),	(6-11)
	where 𝐇 1 is the transfer matrix. It is shown in Appendix of this chapter.
	Similarly, the out-plane displacement of section 2 is
	𝑧 𝑛,2 (𝑥 2 , 𝑡) = (𝐴 𝑛,2 𝑒 -𝑖𝜆𝑥 2 + 𝐵 𝑛,2 𝑒 𝑖𝜆𝑥 2 + 𝐶 𝑛,2 𝑒 -𝜆𝑥 2 + 𝐷 𝑛,2 𝑒 𝜆𝑥 2 )𝑒 𝑖𝜔𝑡 , (6-12)
	where 𝑥 2 = 𝑥 -(𝑛 -1)𝑎 and (𝑛 -1)𝑎 + 𝑙 2 ≤ 𝑥 ≤ 𝑛𝑎. So, from the left interface to
	the right one of section 2 of the nth unit cell, we obtain
	𝚿 𝑛,2 (𝑎, 𝑡) = 𝐇 2𝛽 𝐇 2𝛼 -1 𝚿 𝑛,2 (𝑙 2 , 𝑡) ≜ 𝐇 2 𝚿 𝑛,2 (𝑙 2 , 𝑡),	(6-13)

Table 6

 6

		-1 The material parameters used in this chapter
	Material	Density (kg/m 3 )	Young's modulus (Pa)	Poisson's ratio
	Steel	7784	2.07×10 11	0.3
	Al	2700	7×10 10	0.33
	Rubber	1300	1.2×10 5	0.47
	Soil	1800	2×10 7	0.3

Table 7

 7

		-1 The material parameters used in this chapter
	Material	Density (kg/m 3 )	Young's modulus (Pa)	Poisson's ratio
	PLA	1086.3	3.4398 × 10 9	0.35
	Steel	7784	2.07 × 10 11	0.3
	Soil	1800	2 × 10 7	0.3

Chapter 4 Design of the inverted T-shaped seismic metamaterials and implementation of the ultra-wide first bandgap

The 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][START_REF] Zeng | Low-frequency broadband seismic metamaterial using I-shaped pillars in a half-space[END_REF][START_REF] Roux | Toward seismic metamaterials: The metaforet project[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. waves, these two characteristics of the LFAZ and the SAZ are used to design a kind of seismic metamaterial by just using one kind of artificial steel structure on a soil substrate. The proposed seismic metamaterial can attenuation the Rayleigh waves from 5 to 22 Hz. Different with the interplay of the local resonance and Bragg bandgaps [START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF][START_REF] Guo | Interplay of local resonances and Bragg band gaps in acoustic waveguides with periodic detuned resonators[END_REF] , the LFAZ and the SAZ of the SMS hardly interfere with each other and together create a common attenuation zone. In Fig. 6-4(a), the out-of-plane polarization in the band structure of the MMP is given.

Results and Discussions of the MMPs

Band structures

The parameter of the out-of-plane polarization 𝜉 is defined as

where u, v and w are the displacement components in the three directions (x, y and z);

and S is the whole unit cell. A wide bandgap for the flexural waves can be found from 385 to 1200 Hz. It is worth noting that, the end parts of the second and third dispersion curves have high out-of-plane polarization value. However, these two bands almost do not affect the wide bandgap in Fig. 6-4(a). We can easily find these two bands are almost flat, especially when out-of-plane polarization value is large. It means it is just the local resonance at a single frequency, which can be ignored when we determine the bandgap.

The vibration modes at these two bands are illustrated in Fig. 6 As shown in Fig. 6-4(b), there is a significant AZ at these frequencies in the simulated and experimental results. Compared with the simulated results, the attenuation effects in the experimental results are not particularly obvious at low frequencies in the bandgap. In addition, in the whole bandgap, the attenuation effects in the experimental results are weaker than those in the simulated results. This is due to the fact that the hinge joint in the experiments is not completely perfect like how it is considered in the simulation. In the experiments, the friction and backlash in the hinged part are difficult to eliminate. Instead, in the simulations, all boundaries in the hinge joint are rigid and smooth. However, it is easy to find that the MMP in the experiments can obviously attenuate the flexural waves in the bandgap compared with the bare plate.

Chapter 7 Design of enhanced inertial amplification seismic metamaterials with a low frequency bandgap

Inertial amplification mechanism, [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF] which can achieve low-frequencies resonance just by using a small mass, has been introduced to design elastic metamaterials to obtain low-frequencies bandgaps. The embedded amplification mechanism was proposed by Yilmaz et al. [START_REF] Yilmaz | Phononic band gaps induced by inertial amplification in periodic media[END_REF][START_REF] Yilmaz | Theory of phononic gaps induced by inertial amplification in finite structures[END_REF] to amplify the effective inertia of the wave propagation medium. Frandsen et al. [START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[END_REF] introduced the similar structures into a continuous elastic rod to obtain wide bandgap induced by inertial amplification. The three-dimensional metamaterials based on embedded inertial amplification mechanism have been realized. It has been experimentally demonstrated that they are capable of isolating elastic waves in a wide frequency range. [START_REF] Taniker | Design, analysis and experimental investigation of three-dimensional structures with inertial amplification induced vibration stop bands[END_REF] Then, Yuksel et al. [START_REF] Yuksel | Realization of an ultrawide stop band in a 2-d elastic metamaterial with topologically optimized inertial amplification mechanisms[END_REF] have designed a two-dimensional solid structure with embedded inertial amplification mechanisms by using topology optimization to achieve an ultrawide stop band at low frequency. However, the embedded inertial amplification mechanisms have not been introduced into a half space to isolate surface waves.

In where the L is the Lagrange function:

Substituting Eq. (7-8) into Eq. (7-7), we obtain

When the two sides of Eq. (7-9) are equal to zero, the first resonance (ωr) and first anti-resonance (ωar) [START_REF] Li | Generating ultra wide low-frequency gap for transverse wave isolation via inertial amplification effects[END_REF][START_REF] Frandsen | Inertial amplification of continuous structures: Large band gaps from small masses[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] frequencies are obtained:

⁄ , (7-10)

As can be seen from Eqs. (7-10) and (7-11), the ωr is always lower than the ωar.

When the angle θ is less than 16.7°, the function 𝜂 = [5 + cot 2 (𝜃)] 16 ⁄ is larger than 1, which means the amplified motion for the mass is caused by the inertially amplified mechanism and the inertial mass is amplified η times. Of course, according to Eq. (7-10), if the stiffness k is weaker or the mass of two arms is heavier, the lower ωr can be obtained. It is worth noting that, comparing with the metamaterials based on local resonance, [START_REF] Mei | Dark acoustic metamaterials as super absorbers for low-frequency sound[END_REF][START_REF] Mei | Ultrathin metasurface with high absorptance for waterborne sound[END_REF] the inertial amplification mechanism is also based on the local resonance, which can be found from Eq. (7-10), but it can easily obtain lower-frequency bandgap because of the function 𝜂 = [5 + cot 2 (𝜃)] 16 ⁄ appeared in Eq. (7-10). For example, in this work, the angle θ is 5°, so the inertial mass is amplified about 8.5 times.

Therefore, the small mass can achieve strong resonance at low frequencies.

In the simulations, all results are calculated by using finite element software COMSOL 5.4 (Multibody Dynamics Module). The Floquet-Bloch boundary conditions are achieved on the unit cell along x direction by using both boundary similarity and pointwise constraint. In the Fig. 7-1(e), the hinge-connection condition is set on the hinge joints between two bodies. The transmission spectrums of the flexural waves in

Étude et réalisation de métamatériaux séismiques large bande

Résumé

Le tremblement de terre désigne la vibration de la surface de la terre provoquée par la vibration d'un endroit ou d'une certaine zone sous la surface de la terre ou sous la surface de la terre. Il est généralement causé par des phénomènes naturels tels que les mouvements de la croûte terrestre, l'impact d'une météorite et une éruption volcanique. Ces métamatériaux peuvent atténuer les ondes sismiques de Rayleigh dans la gamme des basses fréquences et des ultra-basses fréquences, afin de protéger les bâtiments construits par l'homme et de réduire les dommages causés par les tremblements de terre.

Ce travail fournit non seulement de nouvelles approaches pour contrôler les ondes sismiques de surface à ultra-basse fréquence, mais aussi de nouvelles idées de conception pour la direction des ondes de surface.

Mots clés: Métamatériau sismique; ondes de Rayleigh; bande interdite; largebasse fréquence; dispersion inverse; résonance locale; amplification inertielle