## 2 Models and the Theoretical Methods
### Theoretical Method  
In order to study the band gap characteristics of the metamaterial structures, according to the theory of elastic dynamics, the elastic wave equation with displacement as the unknown quantity in the Cartesian coordinate system is established. With the assumptions of material continuity, complete elasticity, homogeneity, isotropy, and small deformations [30], the elastic wave equation is expressed as:  
\[\rho\frac{\partial^{2}u_{i}}{\partial t^{2}}=\sum_{j=1}^{3}\left\{\frac{ \partial}{\partial x_{i}}\left(\lambda\frac{\partial u_{j}}{\partial u_{j}} \right)+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\right]\right\}+ \rho f_{i} \tag{1}\]  
where \(u_{i}\) and \(u_{j}\) denote the displacement components, \(f_{i}\) denotes the external force per unit volume, \(\rho\) denotes the density, \(t\) denotes the time, and \(\lambda\) and \(\mu\) denote the Lame elastic constants of the material. For the particularity of the structure, we establish the elastic wave equation in the cylindrical coordinate system to calculate the dispersion relation curve in the infinite system based on the equations of motion, geometry, and physics. Here, \(u\), \(v\), and \(\omega\) denote the displacement components of the cylindrical coordinate system and \(r\), \(\theta\), and \(z\) denote the coordinate components in the cylindrical coordinate system. \[\left\{ \begin{array}{l}\rho\frac{\partial^{2}u}{\partial t^{2}}=(\lambda+2 \mu)\frac{\partial\theta_{i}}{\partial t}-\frac{2\mu}{r}\frac{\partial u^{ \prime}}{\partial\theta}+2\mu\frac{\partial u^{\prime}}{\partial z}+\rho f_{ i}\\ \\ \rho\frac{\partial^{2}v}{\partial t^{2}}=(\lambda+2\mu)\frac{\partial\theta_{ i}}{\partial z}-2\mu\frac{\partial u^{\prime}}{\partial z}+2\mu\frac{ \partial u^{\prime}}{\partial r}+\rho f_{i}\\ \\ \rho\frac{\partial^{2}w}{\partial t^{2}}=(\lambda+2\mu)\frac{\partial\theta_{i} }{\partial z}-\frac{2\mu}{r}\frac{\partial}{\partial r}\left(rw^{\prime}_{ \theta}\right)+\frac{2\mu}{r}\frac{\partial w^{\prime}_{r}}{\partial\theta}+ \rho f_{i}\end{array} \right. \tag{2}\]  
where the volume strain \(\theta_{l}\) and rotational components (\(w^{\prime}_{r}w^{\prime}_{\theta},w^{\prime}_{z}\)) are defined as:  
\[\left\{ \begin{array}{l}\theta_{l}=\frac{1}{r}\frac{\partial(ru)}{\partial r }+\frac{1}{r}\frac{\partial w}{\partial\theta}\frac{\partial w}{\partial z} \\ \\ w^{\prime}_{r}=\frac{1}{2}\left(\frac{1}{r}\frac{\partial w}{\partial\theta}+ \frac{1}{r}\frac{\partial w}{\partial z}\right)\\ \\ w^{\prime}_{\theta}=\frac{1}{2}\left(\frac{1}{r}\frac{\partial u}{\partial \theta}+\frac{\partial w}{\partial r}\right)\\ \\ w^{\prime}_{z}=\frac{1}{2}\left(\frac{1}{r}\frac{\partial(rv)}{\partial\theta}+ \frac{\partial u}{\partial\theta}\right)\end{array} \right. \tag{3}\]  
According to [PERSON]'s theorem, only a single lattice cell needs to be considered since the periodic structure of lattice cells is infinite along the radial direction. Its lattice boundary condition equation is:  
\[u(r+r_{a},z)=u(r,z)\mathrm{e}^{\mathrm{i}k_{r}r_{a}} \tag{4}\]  
where \(r\) denotes the radial position, \(a\) denotes the lattice constant, and \(k_{r}\) denotes the components of the Bloch wave vector, \(K_{r}\), in the radial direction. Figure 1: REM: (**a**) Schematic diagram of REM lattice structure and BZ wave vector (Gamma-R) in reciprocal space. (**b**) Formation of REM. (**c**) Quasi-static supermaterial. Using Equation (4), by establishing the model as shown in Figure 1a under the 2D axisymmetric of COMSOL Multiphysics, applying a periodic boundary condition on r, sweeping the Bloch wave vector over the boundary of the first irreducible (\(\Gamma(0,0)\) to \(R(1,0)\)) Brillouin zone (IBZ), the dispersion relation curve of the radial metamaterial structure is obtained. It is worth noting here that the quasi-static limit of the dispersion curves can be inferred from the slope or curvature of the dispersion curve around the point \(\Gamma\) when \(K_{r}\to 0\)[30; 32; 33].