## 4 Microstructure Mechanical Models of Cellular Medium
### Multi-Cell Model
The ad-hoc model adopted a dynamic method that considered the effect of temperature on the material characteristics. It can simultaneously consider cell nucleation, growth through gas expansion, and the interactions between growing cells. [PERSON] [113; 136] developed a randomly distributed ellipsoidal cell model based on the Kelvin cell model, as shown in Figure 9c. [PERSON] [137; 138] presented a three-dimensional (3D) mesoscopic model of closed-cell foam material [139]. These models assume equal thickness of cell wall, which is inconsistent with reality. And they are affected by random factors, and the algorithm stability is not high. There is unsatisfactory practical application of these models, and they can only provide a simple analytical method of cellular media. To truly describe the microstructure of cellular medium, some researchers use computed tomography (CT) or X-ray scanning technology to characterize the cellular medium, as shown in Figure 11. This technology can display the material microstructure layer by  
Figure 9: 3D multi-cell models: (**a**) Voronoi model [33]; (**b**) tetradecahedron model [33]; (**c**) ellipsoidal-cell model [113]; (**d**) body-centered cubic (BCC) model [134]; (**e**) face-centered cubic (FCC) model [134]; (**f**) “Ad-hoc” model [134]. layer intuitively and nondestructively. The 3D microstructure of cellular medium can be further acquired by image reconstruction technology, which restores the topological structure to the greatest extent [60; 130; 131; 132; 133; 134; 135; 136; 137; 138; 139; 140; 141; 142; 143; 144; 145; 146; 147; 148; 149]. It has authenticity unmatched by other methods. Figure 10: (**a**) Deformation process of tetradecahedron model at 8.5 m/s and 136 m/s [125], (**b**) deformation process of Voronoi model at different impact velocities [33], (**c**) stress-strain curves of Ad-hoc model, BCC model, FCC model and Kelvin model [134], (**d**) stress-strain curves of metal foam at 90 m/s [33], and (**e**) crushing strength under various impact velocities [33]. [PERSON] [141] established the finite element model of cenosphere epoxy syntactic foams in ABAQUS/Explicit software using X-ray scanning technology. As shown in Figure 12a, the stress is concentrated near the top and bottom of any single cenosphere as well as the junction between adjacent cenospheres. Microcracks tend to initiate in the cenosphere connection area of the matrix. Then, the microcracks spread and merge with other microcracks and voids left by the broken cenospheres to form macroracks. [PERSON] [150] used micro-CT technology to numerically model the yield characteristics of aluminum (Al) foam [151; 152; 153]. Figure 12b presents deformation modes under compression and tension. During the compressive process, the plastic hinges are formed first, and then the plastic collapse arises in the weak area [143]. During the tension process, crack first occurs at the cell walls in the weak area, then spreads the load, and then completely fractured. [PERSON] [34] developed a FE model based on micro-CT to study the elastoplastic behavior of shock wave propagation and the cell collapse mechanisms of aluminum foams. As shown in Figure 12c, a sudden rise of free surface speed was discovered at 845 m/s, indicating the development of the shock wave. It describes elastic and plastic wave propagation characteristics of the impact load in the loading direction. The elastic wave front tagged with a red line is migrating in the direction of shock. The result shows that the cell collapses in an uneven manner and continue to collapse as the waves move. However, the modeling algorithm based on X-ray and CT is complicated, with many intermediate processing steps, and often need large-scale commercial software. The cost of using X-ray or CT scanning technology is high, which limits its wide application. Figure 11: Flow chart of foam geometry development based on micro-CT: a flat panel detector and micro-focus X-ray source of 100 kV were used for imaging.