## 3 Results and Discussion
### Yield Points Defined by the Critical Plastic Dissipation Energy  
For the multiaxial loads, the initial yield of the closed foam could also be judged by the critical plastic dissipation energy [29; 30]. The critical plastic dissipation energy value is given as the plastic dissipation energy per unit volume at point B in Figure 4, which is defined by 0.2% offset of plastic strain on the uniaxial principal compression stress-strain curve. The yield points under multiaxial load states are defined by the condition that the plastic dissipation energy per unit volume reaches the critical plastic dissipation energy value per unit volume. For the Kelvin model with 0.2 relative density, the plastic dissipation energy per unit volume is 22.8 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\); with 0.43 relative density, the plastic dissipation energy per unit volume is 60.6 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\); for the random model 6 (in Table 1) with 0.3 relative density, the plastic dissipation energy per unit volume is 20.1 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\); for model 1 (in Table 1) with 0.4 relative density, the plastic dissipation energy per unit volume is 57.3 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\). Table 5 shows the plastic dissipation energy per unit volume for different models with different related densities and inner structure. It is presented that the plastic dissipation energy per unit volume at yield point is not subject solely to the influence of the related density of the foam material. However, the inner structure also affects the plastic dissipation energy per unit volume. And the plastic dissipation energy per unit volume at the yield point of the regular Kelvin model is bigger that that of the random model.  
Figure 7 gives the normalized yield points (\(\sigma_{m}/\sigma_{u}\), \(\sigma_{e}/\sigma_{u}\)) given by the critical plastic dissipation energy in the \(\sigma_{m}-\sigma_{e}\) space using the Kelvin model with relative densities of 0.2 and 0.43, and models 6 and 1, in Table 1, with relative densities of 0.3 and 0.4, respectively. Additionally, like Figure 5, Figure 6 also implies that parameter \(\cos(3\theta)\) affects the distribution of yield points defined by this method because of the scatter of yield points for the given value of parameter k in the \(\sigma_{m}/\sigma_{u}-\sigma_{e}/\sigma_{u}\) space. For the comparison of the yield surfaces obtained by the numerical simulation and available literature biaxial experimental data [4] and triaxial experimental data [9], the result is similar to that in Figure 6. It is shown that the yield surfaces obtained by the numerical simulation are in good accordance with experimental data close to the \(\sigma_{e}/\sigma_{u}\) axis, but there is a larger difference with the stress state close to the \(\sigma_{m}/\sigma_{u}\) axis.  
Overall, contrasting the data, the results demonstrate that the yield surfaces defined by these two yield definition methods above can not only be described by \(\sigma_{m}\) and \(\sigma_{e}\). The parameter k would also skew the distribution of yield points in the \(\sigma_{m}/\sigma_{u}-\sigma_{e}/\sigma_{u}\) space. Meanwhile, there is a large difference between the numerical simulation datum and experimental datum near \(\sigma_{m}/\sigma_{u}\) axis. Therefore, more suitable definitions of yield points should be explored.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Related Density**} & \multicolumn{6}{c}{**Plastic Dissipation Energy Per Unit Volume (\(\times\)10\({}^{-3}\) mJ/mm\({}^{3}\))**} \\ \cline{2-9}  & **Kelvin** & **Model 1** & **Model 2** & **Model 3** & **Model 4** & **Model 5** & **Model 6** & **Model 7** \\ \hline
0.43 & 60.6 & & & & & & & \\
0.2 & 22.8 & & & & & & & \\
0.4 & & 57.3 & 48.1 & 39.7 & 31.5 & & & \\
0.35 & & & & & & & 21.4 & \\
0.3 & & & & & & & & 20.1 & \\
0.25 & & & & & & & & 15.7 \\ \hline \hline \end{tab