## 4 Microstructure Mechanical Models of Cellular Medium
### Multi-Cell Model  
Many scholars have conducted research on the finite element (FE) method. For cellular media, modeling is important. Only when a model more closely describes real conditions, can the calculation agree with experimental values. Some researchers [119; 120; 121; 122; 123; 124] established 2D multi-cell models to investigate the mechanical properties of cellular media, as shown in Figure 7. [PERSON] [120] thought that entrapped gas hardly affects plateau stress, but it did affect the deformation modes. In the densification region, the entrapped gas significantly enhanced crushing stress, as shown in Figure 8a,b. Generally, the effect of entrapped gas on the hexagonal cells is more important than that on the circular cells, indicating that it is dependent on the cell morphology. Figure 6: Improved single cell model (**a**) tetradecahedron model (Kelvin model) [115] and (**b**) octahedron model [116]. [PERSON] [119] established a circle arc (c-a) model, as shown in Figure 7c. Under quasi-static conditions, the cellular medium deformed into an X-shape, and the broken zone spread until it was completely densified. At moderate impact velocities, V-shaped deformation can be found. At high velocities, the crushing zone is positioned high on the impact side, which is an I-shaped deformation, and then spreads along the crushing direction, as shown in Figure 8c. [PERSON] [122; 123; 124] investigated the mechanical properties (Young's modulus, Poisson's ratio, and shear modulus) of 2D honeycomb material based on a regular hexagon-cell model and irregular hexagon-cell model. They considered the viscoelasticity and the irregularity of the model, and found that the [PERSON]'s modulus and shear modulus depend on the viscoelastic parameters, while the Poisson's ratio was independent on the viscoelastic parameters and frequency. They put forward the expressions of elastic modulus under static loading and steady-state vibration conditions. Figure 7: 2D multi-cell models: (**a**) hexagonal-cell model [120]; (**b**) circular-cell model [120]; (**c**) circle arc (c-a) model [119]; (**d**) irregular hexagonal-cell model [122]. Some researchers established 3D multi-cell models to investigate mechanical properties of cellular medium, as shown in Figure 9. [PERSON] [125] used the finite element software ABAQUS to simulate the dynamic mechanical behavior of closed-cell foam with the Voronoi models [126, 127, 128, 129, 130] and tetradecahedron models [131, 132, 133]. At low impact velocities, during the initial compression process, plastic deformation first begins to appear in the middle of the tetradecahedron model. At high impact velocities, cell crushing propagates downwards from the impacted surface, which exhibits an \"I\" shaped deformation mode, as shown in Figure 10a. Figure 8: (**a**) Stress-strain curves of hexagonal-cell model at 100 s\({}^{-1}\)[120], (**b**) stress-strain curves of circular-cell model at 100 s\({}^{-1}\)[120], and (**c**) deformation mode of c-a model at different impact speeds [119]. [PERSON] [135] and [PERSON] [33] concluded that cellular media showed different deformation modes under different impact velocities. They observed that at low impact velocities, the deformation zones were randomly distributed, while at medium velocities there were transitional deformation modes. When the impact velocity was high enough, the broken zones were locally concentrated in the impacted area and then spread along the direction of propagation. Figure 10e compares the strain rate enhancement and inertia enhancement at different impact velocities. In the random mode, the stain rate enhancement decides the strength enhancement. In the band front mode, the inertia enhancement decides the strength enhancement. [PERSON] [134] established an ad-hoc model to generate closed-cell foam microstructures, as shown in Figure 9f. Figure 10c showed that the ad-hoc model was the closest to the real microstructure and compressive behavior of the cellular medium material.