# The Vibration Isolation Design of a Re-Entrant Negative Poisson's Ratio Metamaterial
## 3 Mechanical Properties of the Proposed NPRM  
The proposed NPRM is manufactured by 3D-printing technology, and made of a polymer plastic material named WEILAI 8000, which is from a company named WeNext (Shenzhen, China). The pieces at both ends are made of aluminum, the assembly pins are made of metal, and the particle damping comprises tungsten particles. The properties of the material are listed in Table 2.  
A simulation and experiment were conducted to study the mechanical properties of the NPRM. The testing method is shown in Figure 2. Specifically, for the equivalent setup of particle damping in the simulation, the loose particles were considered to have no impact on the structural stiffness and to only affect the mass distribution at the locations of the connecting circular rings. Therefore, in the finite element analysis, the circular rings were separately segmented, and their material properties, except for density, were the same as that of the plastic (WL8000). The equivalent density \(\rho_{eq}\) can be obtained using the following formula:  
\[\pi\Big{(}(R+T)^{2}-R^{2}\Big{)}\cdot\rho_{1}\cdot h+\pi R^{2}\cdot\rho_{2} \cdot h=\pi\Big{(}(R+T)^{2}-R^{2}\Big{)}\cdot h\cdot\rho_{eq} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Density (kg/m\({}^{3}\))** &
\begin{tabular}{c} **Young’s** \\ **Modulus (MPa)** \\ \end{tabular} & **Poisson’s Ratio** & **Size (mm)** \\ \hline WL8000 & 1120 & 2850 & 0.42 & / \\ Aluminum & 2700 & \(7.1\times 10^{4}\) & 0.33 & / \\ Steel & 7850 & \(2.1\times 10^{5}\) & 0.3 & / \\ Copper & 8900 & \(1.17\times 10^{5}\) & 0.33 & / \\ Tungsten & 19,350 & / & / & 0.6\(\sim\)0.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The material properties. Figure 1: Schematic diagrams of (**a**) T-NPRM, (**b**) N-NPRM, (**c**) N-NPRM with pins, and (**d**) N-NPRM with particle damping. (**e**) Geometrical parameters of N-NPRM. By solving Equation (1), we obtained the following result:  
\[\rho_{eq}=\frac{\left(2R+T\right)\cdot T\cdot\rho_{1}+R^{2}\cdot\rho_{2}}{\left( R+T\right)^{2}-R^{2}} \tag{2}\]  
where \(R\) represents the radius of the pin, \(T\) denotes the wall thickness of the NPRM, \(\rho_{1}\) is the density of WL8000, \(\rho_{2}\) is the density of the pin material, and \(h\) represents the thickness of the NPRM plate. The simulation of the static properties was carried out using the software ABAQUS 2018. For improving the computational efficiency, a 2D model was established, as shown in Figure 2. The element type was CPS4R, and as the thickness of the thin wall was 1.2 mm, in order to ensure the accuracy of the simulation, the maximum size of the element was 0.2 mm. The properties of the materials were set according to Table 2. The bottom aluminum block was fixed, and the force F was loaded on the middle point of the top surface of the top aluminum block. The compressive testing of the metamaterial structure was performed using a universal testing machine in the configuration shown in Figure 2. The same loading conditions were applied in finite element simulations. The length of the simulated compression curve was set to 20 mm, starting at 0 and gradually increasing in the testing machine. The loading was stopped when the specimen exhibited failure. During the experiments, it was observed that the specimen did not reach the full compression of 20 mm before failure occurred. This behavior can be attributed to the material properties of the metamaterial and the quality of the 3D printing. A comparison was made between the stress-strain curve obtained from the experiment, where failure did not occur, and the simulated curve, as shown in Figure 3a. The stress-strain curve corresponds to the conventional T-NPRM structure. In Figure 3a, it can be observed that there is good agreement between the experimental and simulated data, confirming the reliability of the simulation method. And in Figure 3a, the figures A-D are the deformation shapes of the metamaterial by simulation and experiment at different strain points.