## 2 Methodology  
### The Finite Element Model  
The representative volume element (RVE) approach is often used in the literature. By using this method, classical homogenization over the micro-scale domain can be used to analyze the mechanical property at the macroscopic scale. RVE is a micro-scale model of the material used to determine the corresponding homogenized/effective properties at the macroscopic scale. For foam material, the cubic RVE model contains cell wall material and pores. A random modeling procedure that builds closed-cell foam models with various cell structures was proposed by [PERSON] et al. [32; 33; 34]. The mechanical properties of foam material are influenced by bulk material and relative density. However, the research on inner structure is not confined to the Voronoi model. The Voronoi model is put to wide use in simulation, and the disturbance factor can describe the randomness of the inner structure, but there is no method to measure the disturbance factor for the experiment in practice. Different from the Voronoi model, [PERSON] uses characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\) defined based on the two-dimensional geometric parameters to reflect the inhomogeneity in cell size distribution and irregularity in cell shape for the 3D foam materials. When the area fraction of the pores whose diameters are larger than d in the total area of pores is 0.2, this diameter is the characteristic diameter \(D_{ch}\). As the characteristic diameter \(D_{ch}\) increases, the bigger pores in random model are more vulnerable generated. Taking the facts into account, the range of characteristic diameter \(D_{ch}\) is set as (3.11-3.79), as shown in Table 1. For characteristic shape anisotropy \(\alpha_{ch}\), \(\alpha_{ch}=0\) represents that all pores in the random model are nearly spherical pores. As characteristic shape anisotropy \(\alpha_{ch}\) increases, the shape of the big pores is more promiscuous. Figure 1 shows the random models with different characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\). And the characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\) can be measured in experiments. The characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\) are calculated cross section of the random model cubic. There are six cross sections, which are perpendicular to the x, y, and z axis. The Model 1 and Model 2 with \(\alpha_{ch}=0\) only have sphere cells as shown in Figure 1a,b. It shows that there are more large pores with the increasing of the characteristic diameter \(D_{ch}\). The Models 3-7 with \(\alpha_{ch}>0\) have polyhedron cells as shown in Figure 1c-g. It shows that the large pores are more irregular as the characteristic shape anisotropy \(\alpha_{ch}\) increases. And the characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\) can be measured in experiments. The characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\) are calculated cross section of the random model cubic. There are six the cross sections, which are perpendicular to the x, y, and z axis. Meanwhile, the researchers in [35; 36] proposed that a rectangular cube with at least five cellular pores on each edge and more than 125 cellular pores overall can be utilized to investigate the mechanical properties. And the simulation results corroborate the experimental findings well. Therefore, the RVE thus has dimensions of 14 mm in height, breadth, and depth, which are sufficient to meet RVE's requirements for material size.  
Compared with the random model above, we also use the Kelvin model of RVE, which regularly has a body centered cubic (BCC) lattice of cells to explore the mechanical property. So we can use a few cells as RVE because of the periodic and symmetric properties of the Kelvin model. The RVE is a cuboid with a height, width, and depth of \(2\sqrt{2}\) mm, as shown in Figure 1h.