## 1 Introduction  
Phononic crystals [1; 2] are periodic structures that can control elastic waves in solids. Due to their excellent band gap properties, phononic crystals are widely investigated. The existing band gap formation mechanisms in phononic crystals are mainly the Bragg scattering mechanism [3; 4; 5] and the local resonance mechanism [6]. For the phononic crystals based on the Bragg scattering mechanism, the generation of the band gap results from the interaction between the periodically changing material properties and elastic waves. The scale of the scatterer and the wavelength corresponding to the band gap frequency are in the same order of magnitude. Therefore, a large space size is required to block the vibration [5]. With the excitation of the elastic waves in the phononic crystal based on the local resonance mechanism [7], the scatterer oscillator resonates and interacts with the elastic wave. Furthermore, the band gap frequency is much lower than that of the Bragg band gap of the same lattice size, enabling a small size to control large wavelengths. Phononic crystals dominated by local resonance mechanisms are also called elastic metamaterials [8; 9; 10]. To facilitate the application of elastic metamaterials in vibration control and mechanical engineering, it is essential to obtain low-frequency and large-bandwidth band gaps. Researchers have put a substantial amount of effort in elastic metamaterials with planar periodic arrangement to explore metamaterial structures with excellent band gap properties [11; 12; 13]. [PERSON] et al. [14] designed an elastic metamaterial with double-sided short cylinders periodically arranged in the X and Y directions and obtained a broadband full band gap calculated by the finite element method. [PERSON]. [PERSON] et al. [15] investigated an elastic metamaterial with a rubber coating. They obtained a low-frequency band gap, with an initial frequency below 200 Hz, by arranging the lattice structure periodically in a plane. [PERSON] et al. [16] proposed a two-dimensional elastic metamaterial with the periodic distribution of dual resonators in the X and Y directions, which generated a 5 Hz wide ultra-low-frequency band gap around 35 Hz. Radial elastic metamaterials (REMs) [17; 18; 19; 20; 21; 22; 23; 24; 25; 26] are annular structures arranged periodically in the radial direction with omnidirectional band gap properties. [PERSON] et al. [23] proposed an REM composed of alternating plexiglass and steel materials to obtain a broadband complete band gap. [PERSON] et al. [25] investigated double-layer REMs with lattice slip properties, and their structures exhibited broadband band gap properties. [PERSON] et al. [20] designed a substrate composed of alternating aluminum and epoxy resin-based structure, added periodic rubber and steel block REMs on the substrate, and obtained a band gap with an initial frequency below 500 Hz and a bandwidth of about 100 Hz. [PERSON] et al. [27] designed a radial elastic metamaterial with steel as the matrix and a combined mass of silicone rubber and steel. The center frequency of the first-order band gap was found to exist around 300 Hz. Although the radial elastic metamaterials aim to achieve low-frequency stop bands, it is still difficult to achieve ultra-low-frequency stop bands. Some researchers [21; 22; 23; 24] found that the quasi-static periodic structure can open a wide-band stop band in the ultra-low-frequency range. [PERSON] et al. [28] designed a special mass-spring system. They showed that the quasi-static system effectively attenuated the elastic waves in the ultra-low-frequency range by fixing the external boundary of the model. [PERSON] et al. [22] investigated the shielding phenomenon of the flexural waves in quasi-static elastic plates by imposing fixed constraints on special points on the elastic plate. They found out that the form of point confinement caused the structure to produce ultra-low stop bands. Through the Fourier series and high frequency homogenization theory, [PERSON] et al.