## 2 Characterization of Cellular Medium Mechanical Properties  
The research on cellular media initially studied porous materials in nature [1; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27], such as grapefruit peels, bamboo, turtle shells, and bone, as shown in Figure 1. After a large number of artificial cellular media appeared, comprehensive and systematic research began in the 1950s. Artificial cellular media materials include foam materials obtained by \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Models** & \multicolumn{2}{c}{**Advantages**} & \multicolumn{2}{c}{**Disadvantages**} \\ \hline \multirow{8}{*}{Phenomenological constitutive model} & \(\bullet\) & Can obtain the mathematical expression between stress and strain by fitting & \(\blacklozenge\) & \(\blacklozenge\) & Cannot consider the effect of the microstructure (cell size, cell shape and cell distribution) on the stress-strain curve of the foam material; \\  & \(\bullet\) & Can study the effect of void porosity and strain rate on the stress-strain curve of foam material; & \(\blacklozenge\) & \(\blacklozenge\) & Cannot describe the stress-strain relationship at higher strain rates (\(10^{5}\) s\({}^{-1}\)-\(\sim\)) or under explosive loading; \\  & & & & & \\  & & & & \\  & & & & \\  & & & & \\  & & & & \\ \hline \multirow{8}{*}{Single cell model} & \(\bullet\) & Considered as a multiphase composite of matrix and pores; & \(\blacklozenge\) & Cannot give the detailed strain and stress field in each phase (matrix or pores); \\  & \(\bullet\) & Can quickly calculate the stress, strain, and stiffness. & \(\blacklozenge\) & \(\blacklozenge\) & Cells are limited to the ellipsoidal topological structure. \\ \cline{1-1} \cline{2-5}  & \(\bullet\) & Use the deformation mechanism of a single cell under load to represent the deformation mechanism of the entire material; & \(\blacklozenge\) & Fit poorly when the relative density is higher than 0.3; \\  & \(\bullet\) & Can obtain relative modulus and relative strength of the foam. & \(\blacklozenge\) & \(\blacklozenge\) & The cell shape is too simple and far from the realistic structure. \\ \hline \multirow{8}{*}{Multi-cell model} & \(\bullet\) & Can observe deformation process of the microstructure, local stress and strain; & \(\blacklozenge\) & Complicated modeling algorithm; \\  & \(\bullet\) & Modeling with algorithms. & \(\blacklozenge\) & Need large-scale commercial software; \\ \cline{1-1} \cline{2-5}  & \(\bullet\) & & & & \\ \cline{1-1} \cline{2-5}  & \(\bullet\) & & & & \\ \cline{1-1} \cline{2-5}  & & & & & \\ \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of various cellular medium mechanical models. foaming, two-dimensional (2D) honeycomb materials, and cellular materials with various structures obtained by three-dimensional (3D) printing, as shown in Figure 2a-c. Since 1980, [PERSON] and [PERSON] (2018) had systematically investigated the mechanical behavior of a variety of porous materials starting from two-dimensional honeycomb materials. They made a unified description and analysis of the structure and properties of a variety of porous materials (i.e., polymer foam, metal foam, ceramic foam, wood, and cancellous bone). Figure 1: Cellular medium materials in nature (**a**) grapefruit peel, (**b**) bamboo, (**c**) turtle shell, and (**d**) bone ([PERSON], 2018; [PERSON] and [PERSON], 2018). Figure 2: Artificial cellular medium (**a**) square honeycomb structure, (**b**) hexagonal honeycomb ([PERSON], 2018; [PERSON] and [PERSON], 2018), (**c**) foaming materials ([PERSON] and [PERSON], 2018), and stress-strain curves of cellular medium under (**d**) quasi-static compression ([PERSON] and [PERSON], 2018) and (**e**) dynamic compression ([PERSON] and [PERSON], 2018). A cellular medium has its own unique characteristics. Compared with thin-walled tube and laminate composite materials, cellular media materials are isotropic and have a relatively large compressive strain, stable stress plateau, and lower density.