# Radial Gradient Seismic Metamaterials with Ultra-Low Frequency and Ultra-Wide Band Gap
## 1 Introduction  
Earthquakes are one of the most catastrophic natural disasters, and the spread of seismic sources causes significant damage to structures, further leading to consumption and damage to various aspects of social resources [1; 2; 3]. Seismic metamaterial, as an artificial structure that attenuates seismic waves, can generate a band gap corresponding to the seismic frequency, providing a new idea for building protection [4; 5; 6]. Seismic surface waves have the characteristics of large amplitude, long propagation distance, and low frequency, which cause great harm to buildings. For a long time, the research on seismic metamaterial has focused on surface waves [7; 8]. Seismic Lamb waves are ignored as another form of seismic wave propagation. Research has found that in cases of near-surface earthquakes, the propagation of Lamb waves is particularly important for damage to buildings [9], usually dominating the seismic records of local and regional events [10]. Therefore, in the design of seismic metamaterial, not only the isolation design of surface waves but also the shielding characteristics of lamb waves should be studied [11; 12]. [PERSON] et al. [13] studied the shielding effect of periodic concrete structures on seismic Lamb waves, determined the blocking effect of periodic structures on Lamb waves, and confirmed the existence of low-frequency band gaps. However, there is still a narrow range of shock absorption. [PERSON] et al. [14] used triangular and honeycomb periodic arrays to drill holes on marble to form a metamaterial structure, which verified that thisstructure can generate band gaps and can be used to attenuate Lamb waves in seismic waves. [PERSON] et al. [15] developed a new seismic isolation method, proposed a method of applying metamaterials to the field of seismic isolation, and designed a hollow cylinder as an earthquake metamaterial wave barrier. The structure does not contact the protected building, which can effectively convert the energy of the earthquake's Lamb waves into heat energy for dissipation. However, at present, the Lamb band gap of seismic metamaterials is still narrow, and the engineering application is limited. Radial metamaterials are widely used in various fields of vibration isolation and noise reduction because of their excellent band gap characteristics and omnidirectional shielding [16; 17; 18; 19]. [PERSON] et al. [20] introduced radial metamaterials into seismic metamaterials, proposed radial seismic metamaterials, studied the propagation characteristics of seismic Lamb waves and surface waves, and obtained a wide band gap. Gradient design is applied to metamaterial [21; 22; 23], showing diversified characteristics. [PERSON] et al. [24] designed a two-dimensional gradient refractive index phononic crystal, indicating that gradient refractive structures can perform acoustic focusing over a wide operating frequency range. [PERSON] et al. [25] designed a functional gradient material pipeline structure and found that functionally gradient materials can effectively regulate the bandgap characteristics of classical periodic structures. The biggest advantage of gradient arrangement is that it can manipulate the transmission of elastic waves over a wide frequency range [21; 22]. [PERSON] et al. [26] put forward that the arrangement is the seismic metamaterial with gradient burial depth. The transmission spectrum of its finite structure shows that it has better isolation effect on surface waves in a wider band gap. However, the dispersion curve of the graded seismic metamaterial structure has not been calculated and studied, and the mechanism of band gap widening has not been analyzed. Supercells are composed of several cells with different parameters. The method of calculating the electronic band structure based on the periodic structure of supercells is called supercells method [27]. [PERSON] et al. [27] proposed a supercell phononic crystal plate composed of dual oscillator units and single oscillator units.