## 3 Results and Discussion
### Yield Points Defined by the Critical Plastic Dissipation Energy at the Microscopic Scale  
Since the yield points determined by the yield definition methods given above are not consistent with the experimental datum, other kinds of yield definition methods should be explored. It is found that the ability of volumetric deformation of closed-cell foam materials must be considered because the great difference between simulation results and experimental results occurred mainly near the \(\sigma_{m}/\sigma_{u}\) axis. Therefore, the yield definition based on plastic dissipation energy could be improved by including the volumetric deformation energy. At the microscopic scale, the internal deformation of the wall material is commonly divided into two parts: volumetric deformation and distortional deformation. Theoretically, the total energy, plastic dissipation energy, and volumetric deformation energy of the RVE at any time \(t\) can be calculated with the following equations,  
\[E_{t}=\int_{0}^{t}(\int_{V}\left(\sigma_{xx}\dot{\varepsilon}_{xx}+\sigma_{yy} \dot{\varepsilon}_{yy}+\sigma_{zz}\dot{\varepsilon}_{zz}+2*\left(\sigma_{xy} \dot{\varepsilon}_{xy}+\sigma_{yz}\dot{\varepsilon}_{yz}+\sigma_{xz}\dot{ \varepsilon}_{xz}\right)\right)dV)dt \tag{13}\]  
\[E_{p}=\int_{0}^{t}(\int_{V}\left(\sigma_{xx}\dot{\varepsilon}_{xx}^{p}+\sigma_ {yy}\dot{\varepsilon}_{yy}^{p}+\sigma_{zz}\dot{\varepsilon}_{zz}^{p}+2*\left( \sigma_{xy}\dot{\varepsilon}_{xy}^{p}+\sigma_{yz}\dot{\varepsilon}_{yz}^{p}+ \sigma_{xz}\dot{\varepsilon}_{xz}^{p}\right)\right)dV)dt \tag{14}\]  
\[E_{V}=\int_{0}^{t}(\int_{V}\left(\left(\sigma_{xx}+\sigma_{yy}+\sigma_{zz} \right)/3*\left(\dot{\varepsilon}_{xx}+\dot{\varepsilon}_{yy}+\dot{\varepsilon }_{zz}\right)\right)dV)dt \tag{15}\]  
where \(E_{t}\), \(E_{p}\), and \(E_{V}\) indicates total energy, plastic dissipation energy, and volumetric deformation energy, respectively, \(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{yz},\sigma_{xz}\) are components of the stress tensor in the cell wall material, \(\dot{\varepsilon}_{xx},\dot{\varepsilon}_{yy},\dot{\varepsilon}_{zz},\dot{ \varepsilon}_{xy},\dot{\varepsilon}_{yz},\dot{\varepsilon}_{xz}\) are components of the strain rate tensor, \(\dot{\varepsilon}_{xx}^{p},\dot{\varepsilon}_{yy}^{p},\dot{\varepsilon}_{zz}^{p},\dot{\varepsilon}_{xy}^{p},\dot{\varepsilon}_{yz}^{p},\dot{\varepsilon}_{xz}^ {p}\) are plastic strain rates, and V is the total volume of the cell wall material. \(E_{t}\), \(E_{p}\), and \(E_{V}\) (Formulas (13)-(15)) can be numerically evaluated during the finite element analysis. With the energy obtained with Formulas (13)-(15), a critical parameter \(E_{cri}\) at yield points can be assigned to \(E_{p}+M*E_{V}\) per unit RVE volume at point B in Figure 4. Similarly, for the case of the multiaxial stress state, the yield point is also thought to be reached when \(E_{p}+M*E_{V}\) per unit RVE volume equals the critical parameter, where M is an empirical weight coefficient  
Figure 7: The yield surfaces are given with the critical plastic dissipation energy method. By comparison with biaxial and triaxial experiments given in [4; 11] respectively, it can be found that the value of M = 4 is a suitable weight coefficient of the volumetric deformation energy in the yield point definition. For the Kelvin model RVE with a relative density of 0.2, the critical parameter is 28.9 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\); with a relative density of 0.43, the critical parameter is 79.8 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\); for the random model 6 (Table 1) RVE with a relative density of 0.4, the critical parameter is 76.3 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\). Figure 8 shows that the yield surface for the Kelvin model with a relative density of 0.2 is on the outside of that for the Kelvin model with a relative density of 0.43. The yield surface of the random model with a relative density of 0.4 is on the inside of the yield surface for the Kelvin model.