## 2 Methodology
### Multiaxial Loading Conditions  
The multiaxial loading states in this work are axisymmetric compression and tension as they are commonly seen in experiments. Three normal loads (including tension and compression) are applied proportionally to the RVE. Since the RVE models are cubes and the model used is homogenous at the macro-scale, the stresses \(\sigma_{xx}\), \(\sigma_{yy}\), and \(\sigma_{zz}\) herein are normal stresses along the x, y, and z axis of the RVE. Where the shear stresses (\(\tau_{xy}\), \(\tau_{yz}\) and \(\tau_{xz}\)) are equal to zero, so that the normal stresses (\(\sigma_{xx}\), \(\sigma_{yy}\) and \(\sigma_{zz}\)) are the principal stress. Because of the similar structures in the x, y, and z directions, we can choose any of the x, y, and z directions as the principal loading direction in the simulation, and the results would be similar. In the simulation, the loading condition is chosen to let the normal stress \(\sigma_{xx}\geq\sigma_{yy}\geq\sigma_{zz}\). So the x, y, and z direction of RVE is chosen as the direction of principal stress \(\sigma_{1}\), \(\sigma_{2}\) and \(\sigma_{3}\) in all of the following numerical simulations, respectively.  
In the principal-stress space, \(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\) holds. The permissible region should be the intersection between the area of \(\sigma_{1}\geq\sigma_{2}\) and the area of \(\sigma_{2}\geq\sigma_{3}\). The limits for this permissible region are the \(\sigma_{1}=\sigma_{2}\) plane and the \(\sigma_{2}=\sigma_{3}\) plane, respectively. The intersection of the two planes of \(\sigma_{1}=\sigma_{2}\) and \(\sigma_{2}=\sigma_{3}\) is the hydrostatic stress state. The Lode angle \(\theta\) is the included angle between the \(\sigma_{1}=\sigma_{2}\) plane and the plane that passes through the hydrostatic axis. [PERSON] et al. (2019) and [PERSON] (2018) investigated the influence of the triaxial state of stress  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Number**} & \multicolumn{4}{c}{**Number of Element**} \\ \cline{2-5}  & **Size 0.6 mm** & **Size 0.4 mm** & **Size 0.3 mm** & **Size 0.25 mm** \\ \hline
1 & 121,478 & 305,784 & 585,765 & 867,593 \\
3 & 104,871 & 259,658 & 457,528 & 862,369 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The number of elements in a random model with different element sizes.  
Figure 2: The stress-strain curve of random model 1 and 3 in Table 1 with different element sizes.  
on the failure and yield of foam. The range of \(\theta\) is \(0^{\circ}\leq\theta\leq 60^{\circ}\). Formulas (4) and (5) are the expressions of triaxial stress ratio k and Lode angle \(\theta\), respectively,  
\[\mathrm{k}=\frac{\sigma_{e}}{\sigma_{m}}=\frac{\sqrt{\frac{1}{2}\Big{(}( \sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{1}-\sigma_{3} )^{2}\Big{)}}}{(\sigma_{1}+\sigma_{2}+\sigma_{3})/3} \tag{4}\]  
\[\cos(3\theta)=\frac{27(\sigma_{1}-\sigma_{m})(\sigma_{2}-\sigma_{m})(\sigma_{ 3}-\sigma_{m})}{2\sigma_{e}^{3}} \tag{5}\]  
where \(\sigma_{e}\) is Mises equivalent stress, \(\sigma_{m}\) is mean stress. However, due to the lack of tension experimental data, only the yield behavior for \(\sigma_{m}\leq 0\) in the \(\sigma_{e}-\sigma_{m}\) space is investigated. In the study of [PERSON] et al. [29], the position of the yield point in the \(\sigma_{e}-\sigma_{m}\) space is determined by the value of k as well as \(\cos(3\theta)\). To ensure that the yield points are evenly distributed in the \(\sigma_{e}-\sigma_{m}\) space, the triaxial stress ratio k is set as \(-\infty,-3,-1.5,-1,-0.6,-0.3,\ 0\). The value of \(\cos(3\theta)\) is set as \(-1\), \(0\), and \(1\). So the yield surface contains sufficient data, including uniaxial compression, biaxial compression, tri-axial compression, and combined compression-tension load. Table 4 presents the principal stress ratio schemes used in the numerical simulations.