## 3 Results and Discussion
### Yield Points Defined by 0.2% offset of the Characteristic Plastic Strain  
The characteristic stress \(\overline{\sigma}\) and strain \(\overline{\epsilon}\) were given in Formulas (6)-(12),  
\[\overline{\sigma}=\sqrt{\sigma_{e}^{2}+\beta^{2}\sigma_{m}^{2}} \tag{6}\]  
\[\overline{\epsilon}=\sqrt{\varepsilon_{e}^{2}+\varepsilon_{m}^{2}/\beta^{2}} \tag{7}\]  
where \(\sigma_{e}\) and \(\sigma_{m}\) are the Mises effective stress and mean stress, respectively,  
\[\sigma_{e}=\sqrt{\frac{1}{2}[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{ 3})^{2}+(\sigma_{3}-\sigma_{1})^{2}]} \tag{8}\]  
\[\sigma_{m}=\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{3} \tag{9}\]  
where \(\varepsilon_{e}\) and \(\varepsilon_{m}\) are corresponding Mises effective strain and mean strain, respectively,  
\[\varepsilon_{e}=\sqrt{\frac{2}{9}\Big{[}\left(\varepsilon_{1}-\varepsilon_{2} \right)^{2}+\left(\varepsilon_{2}-\varepsilon_{3}\right)^{2}+\left(\varepsilon _{3}-\varepsilon_{1}\right)^{2}\Big{]}} \tag{10}\]  
\[\varepsilon_{m}=\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3} \tag{11}\]  
where \(\beta^{2}\) is a material parameter associated with the Poisson ratio \(\
u\), defined as  
\[\beta^{2}=\frac{9(1-2\
u)}{2(1+\
u)} \tag{12}\]  
With the characteristic stress \(\overline{\sigma}\) and strain \(\overline{\epsilon}\) given above, the characteristic stress-strain curves of different models under uniaxial compression and multiaxial load state can be established. Figure 4 shows the uniaxial principal compression stress-strain curve and corresponding characteristic stress-strain curve for the Kelvin model with a relative  
Figure 5: The characteristic stress-strain curve under multiaxial load (\(-1.963\):\(-3\):\(-4\)). Note: Point A is the yield point determined by 0.2% plastic strain offset in the characteristic stress-strain curve. density of 0.2. Figure 5 shows the corresponding characteristic stress-strain curve of the multiaxial load process. The yield points defined by 0.2% offset of characteristic plastic strain are shown as point A in Figures 4 and 5. In Figure 4, the yield point defined by 0.2% offset of the principal plastic strain is also shown at point B. It is shown that there is little difference between the yield points defined by the characteristic plastic strain method and the principal plastic strain method. The difference is due to the compressibility of the foam material. The yield points defined by the 0.2% characteristic strain offset method for different foam models under multiaxial load states can be obtained. The cell wall materials used in the numerical simulation and experiments are not identical. So the comparison of different aluminum alloys as cell wall materials can have little significance. To solve this problem, we give a comparison of dimensionless yield surfaces. To give a normalized yield surface, the uniaxial compression yield stress \(\sigma_{u}\) is used to nondimensionalize the yield points (\(\sigma_{m}/\sigma_{u}\),\(\sigma_{e}/\sigma_{u}\)) in the \(\sigma_{m}/\sigma_{u}-\sigma_{e}/\sigma_{u}\) space. The uniaxial compression yield stress \(\sigma_{u}\) is obtained with the method shown as point B in Figure 4. The normalized yield points (\(\sigma_{m}/\sigma_{u}\), \(\sigma_{e}/\sigma_{u}\)) for each model are plotted in the following \(\sigma_{m}/\sigma_{u}-\sigma_{e}/\sigma_{u}\) space. Among them, the yield surface defined by 0.2% offset of the characteristic plastic strain is shown in Figure 6. It shows that, for the isotropic Kelvin RVE, the yield surface for the aluminum alloy foam with a relative density of 0.2 is a little larger than the yield surface for the foam with a relative density of 0.43, especially for the loading states close to the \(\sigma_{m}\) axis. For the random RVE model, the difference in the yield surface with different relative densities is small. It can also find that there is a larger scatter on the yield surface for the load ratio with the value of k in the range of \(-0.3\) and \(-1.5\) for both models.