## 3 Results and Discussion
### Yield Points Defined by the Critical Characteristic Energy \(W^{\prime}\)  
To derive a more convenient method to define the yield state macroscopically, we use the stress tensor of RVE in Formula (2) to define the yield of closed foam material. It is reported that the elastic energy stored per unit RVE volume \(E_{e}/V_{RVE}\) can be approximately represented with W given by Formula (18) macroscopically in [39],  
\[\text{W}=\frac{1}{2\overline{\text{E}}}\Big{(}\sigma_{\text{e}}^{2}+\beta^{2} \sigma_{\text{m}}^{2}\Big{)} \tag{18}\]  
where \(\beta\) is the material constant defined by Formula (19), and \(\overline{E}\) is defined by,  
\[\overline{E}=\frac{3\text{E}}{2*(1+v)} \tag{19}\]  
Figure 9: The yield surfaces defined by the \(E_{ch}\) based on the micro-scale. Table 6 gives the \(E_{ch}\) and W for the random model 1 with 0.4 relative density under different multiaxial stress states. It is shown that the W is not equal to \(E_{ch}\) and varies with the stress state. So we give yield surfaces defined by the critical characteristic energy \(E_{ch}\) and W in Figure 10, respectively. It shows that the yield surface defined by the critical characteristic energy W is bigger than that defined by the critical characteristic energy \(E_{ch}\), especially near \(\sigma_{m}/\sigma_{u}\) axis. So we cannot use the value of W at the yield point B in Figure 4 to define the yield of the RVE. To propose a more proper engineering yield definition method, we can reformulate Formula (18). \begin{table}
\begin{tabular}{l l l l l} \hline \(\sigma_{1}\) & \(\sigma_{2}\) & \(\sigma_{3}\) & \(E_{ch}\) (J/mm\({}^{3}\)) & W (J/mm\({}^{3}\)) \\ \hline
0 & 0 & \(-1\) & 0.03833 & 0.03373 \\
1 & \(-1.366\) & \(-3.732\) & 0.03833 & 0.03537 \\
1 & \(-2\) & \(-2\) & 0.03833 & 0.03742 \\ \(-1\) & \(-1\) & \(-4\) & 0.03833 & 0.03381 \\ \(-0.134\) & \(-1\) & \(-1.8661\) & 0.03833 & 0.03714 \\
0 & \(-1\) & \(-1\) & 0.03833 & 0.0355 \\ \(-1\) & \(-1\) & \(-2.5\) & 0.03833 & 0.03517 \\ \(-1\) & \(-2.367\) & \(-3.733\) & 0.03833 & 0.03734 \\ \(-1\) & \(-4\) & \(-4\) & 0.03833 & 0.0363 \\ \(-4\) & \(-4\) & \(-7\) & 0.03833 & 0.03643 \\ \(-1.9364\) & \(-3\) & \(-4\) & 0.03833 & 0.03726 \\ \(-1\) & \(-2\) & \(-2\) & 0.03833 & 0.03689 \\ \(-3\) & \(-3\) & \(-4\) & 0.03833 & 0.03742 \\ \(-1\) & \(-1\) & \(-1\) & 0.03833 & 0.03776 \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of \(E_{ch}\) and W for the random model with a relative density of 0.4. Figure 10: The yield surfaces defined by the \(E_{ch}\) based on the micro-scale. Comparison of yield surfaces defined by the \(E_{ch}\) and W of random model 1 with a relative density of 0.4. To do this, we propose Formula (20), which is modified for Formula (18), to represent the elastic energy per unit volume of RVE based on macro-scale stresses as follows,  
\[\mathrm{W}^{\prime}=\frac{1}{2\mathrm{E}}\Big{(}\sigma_{\mathrm{e}}^{2}+(1- \rho)\Big{(}\mathrm{A}+\beta^{2}\Big{)}*\sigma_{\mathrm{m}}^{2}\Big{)} \tag{20}\]  
Let \(\mathrm{W}^{\prime}=E_{ch}\), and we can get the value of coefficient A for different stress states. Figure 11 shows that the coefficient A changes as a function of the stress-proportional parameter k and \(\cos(3\theta)\). As shown in Figure 9, the effect of \(\cos(3\theta)\) on the yield surface is small, so only the effect of the parameter k is considered in the following discussion. Figure 12 shows the relationship between parameter A and \(\rho(1-2\
u)\) for different values of k. It is shown that the coefficient A scales linearly with the \(\rho(1-2\
u)\) at several fixed values of parameter k. Therefore, parameter A can be written as Formula (21),  
\[\mathrm{A}=-0.2+0.2\mathrm{k}+0.3k^{2}+(7.4-0.3\mathrm{k})\rho(1-2\
u) \tag{21}\]  
With Formulas (20) and (21), the value of \(W^{\prime}\) at yield point B in Figure 9 can be obtained and is defined as the critical value \(W^{\prime}_{ch}\).