## 4 Microstructure Mechanical Models of Cellular Medium
### Single Cell Model  
The single cell model is based on analyzing the deformation mechanism of a tiny element (a cell) under load, as shown in Figure 5. [PERSON] and [PERSON] [28] first proposed a GA model suitable for open-cell and closed-cell foam, which simplified the cellular medium into countless cells composed of cell edges and cell walls. Under pressure, the cell edges bend, and the cell walls stretch. The relationship between the relative modulus E\({}_{r}\), and relative strength \(\sigma_{r}\), and relative density \(\rho_{r}\) can be obtained by:  
\[E_{r}=\phi^{2}\rho_{r}^{2}+(1-\phi)\rho_{r} \tag{28}\]  
\[\sigma_{r}=\phi^{1.5}\rho_  
[PERSON] [109; 112] investigated the mechanical strength of open-cell PMMA foam. They assumed that the open-cell foam was composed of many equal cubic frames, and the simplified structural unit is shown in Figure 5c. [PERSON] obtained the relationship between relative elastic modulus and relative density:  
\[\mathrm{E_{r}}=\mathrm{C_{0}}(0.1459+0.7082\mathrm{\rho_{r}})^{4} \tag{30}\]  
where \(\mathrm{C_{0}}\) is a constant fitted by experimental data. [PERSON] [110] explored the constitutive model of closed-cell foam. The basic unit is similar to the open-cell structure unit, as shown in Figure 5d. The cell wall is added to the open-cell model. The thickness of cell wall \(\mathrm{t_{f}}\) is equivalent to the thickness of cell edge \(\mathrm{t_{e}}\) both is \(\mathrm{t}/2\), and the length of cell edge is also \(l\).  
According to the Gibson-Ashby model, \(\mathrm{E_{r}}\) of closed-cell constitutive model can be calculated as:  
\[\mathrm{E_{r}}=\left(\frac{3\mathrm{A_{r}}^{2}-2\mathrm{A_{r}}^{3}}{1-\left( 1-\mathrm{A_{r}}\right)^{3}}\right)\mathrm{A_{r}}^{4}+\beta\left(1-\frac{3 \mathrm{A_{r}}^{2}-2\mathrm{A_{r}}^{3}}{1-\left(1-\mathrm{A_{r}}\right)^{3}} \right)\frac{\mathrm{A_{r}}}{2} \tag{31}\]  
\(\mathrm{A_{r}}\) can be calculated as:  
\[\mathrm{A_{r}}=1-\left(1-\mathrm{\rho_{r}}\right)^{\frac{1}{3}} \tag{32}\]  
When the relative density is high, the closed-cell constitutive model prediction is slightly lower than the measured result. They pointed out that when the thickness of cell wall is different from the thickness of cell edge, the phenomenon of \"too low\" prediction will appear, and the cell structure tends to be round rather than square. [PERSON] believed that when the relative density was lower than 0.3, the relative strength of cellular medium could be predicted well. [PERSON] [111] proposed an improved GA model and derived elastic modulus empirical equations under different uniaxial loadings according to experimental results.  
Figure 5: Deformation mechanism (**a**) elastic collapse and (**b**) plastic collapse [28], and single cell model (**c**) open-cell unit [109], (**d**) closed-cell unit [110], and (**e**) cell model under different uniaxial loadings [111].  
On this basis, researchers developed tetradecahedron models (Kelvin model) [113; 114; 115] and octahedron models [116; 117; 118], as shown in Figure 6. These single cell models provide an easy method to analyze the mechanical properties, but the simple shape and regular permutation of them are too idealistic and far from the realistic structure. They cannot represent the mechanical properties of the entire cellular medium.