## 3 Results and Discussion
### Yield Points Defined by the Critical Characteristic Energy \(W^{\prime}\)
The values of \(W^{\prime}_{ch}\) are 20.7 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\), 47.8 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\) for the Kelvin model with relative densities of 0.20 and 0.43, and 19.4 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\), 38.3 \(\times\) 10\({}^{-3}\) mJ/mm\({}^{3}\) for the random model with relative densities of 0.3 and 0.4. The yield point under multiaxial load states is obtained when \(W^{\prime}\) is equal to \(W^{\prime}_{ch}\). The following Figure 13 gives the energy values obtained with the plasticity models in the experiment. It shows that the the energy value obtained by the experiment is bigger than that obtained with the plasticity models. And with the increasing of relative density, the energy values increase. However, the energy values are also dependent on characteristic diameter \(D_{ch}\) and characteristic shape anisotropy \(\alpha_{ch}\). Because of the characteristic diameter \(D_{ch}\) with a small difference, we only discuss the effect of the characteristic shape anisotropy \(\alpha_{ch}\) on the difference energy values. Figure 14 gives the effect of the characteristic shape anisotropy \(\alpha_{ch}\). And this effect can be fitted by the quadratic formula. It shows that the difference between the energy values obtained with the plasticity models and the experiment decreases as characteristic shape anisotropy \(\alpha_{ch}\) increases. Figure 11: The relation of parameter A with k and \(\cos(3\theta)\). Figure 13: The energy values obtained with the plasticity models and the experiment of RVE models in Table 1 under uniaxial loading. To test, we use all models in Table 1 to determine whether these equations apply to models with different inner structures. Figure 15 gives yield points defined by the critical characteristic energy W\({}^{\prime}\) based on macro-scale stresses. Figure 15 compares the yield surfaces defined by the yield method in Formulas (20) and (21) for the Kelvin and random RVE models with the experimental datum. Different from the yield definition methods in the literature, the yield surface defined by this method depends on \(\sigma_{e}\) and \(\sigma_{m}\) only. It also shows that the surfaces determined by this yield definition method are nearly the same as those defined by the stored elastic energy at  
Figure 14: The difference value between the energy values obtained with the plasticity models and the experiment of RVE models with 0.4 relate density under uniaxial loading. Figure 15: The yield surfaces of models with different inner structures determined by the corrected \(W_{ch}^{\prime}\). microscopic scales, and therefore are consistent with the literature experimental data. When \(\rho=1\), Formula (20) becomes  
\[\mathrm{W}^{\prime}=\frac{1}{2\mathrm{E}}\mathrm{e}_{\mathrm{e}}^{2} \tag{22}\]  
Formula (22) describes the yield surface for continuum materials. Therefore, the yield definition in Formulas (20) and (21) can be considered as a general yield definition method. It can be seen from Figures 6-9 and 15 that the yield surface with the random model lies within that of the Kelvin model. This might be due to the regular structure of the Kelvin model. It means that the foam with the regular inner structure could be harder to yield than that with the random inner structure. Finally, it is presented that the yield points of all random models in Table 1 are basic superposition as shown as Figure 13. It means that the yield point defined by \(\mathrm{W}^{\prime}\) gets rid of related density and inner structure influences. Compared with the complicated yield surface in [PERSON] et al. [13; 14], we can obtain a simple yield function to describe the yield. And the yield definition is also used for the continuum materials. In addition, different from yield definition such as critical plastic dissipation energy or the rate of critical plastic dissipation energy and initial energy, the definition of Formulas (3) and (15) can be obtained in the experiments.