## 3 The Constitutive Relationship of Cellular Medium
### Homogenization Algorithm Model
The homogenization model can predict heterogeneity by considering the volume composition of composite materials [97; 100; 101; 102; 103]. Compared with the finite element (FE) calculation which takes up much CPU computing time, the homogenization model can quickly calculate the stress, strain, and stiffness of composite materials. However, it cannot give the detailed strain and stress field in each phase, and the inclusions are limited to the ellipsoidal surface. The homogenization model is only available when the matrix phase and the inclusion phase are both isothermal linear elastic materials. However, a cellular medium is not just a linear elastic material. To study the plasticity of material, it is necessary to adopt a progressive damage model. The progressive damage model of composite material belongs to the category of continuum damage mechanism. Different from the damage model that studies the evolution of a single macroscopic crack inside a composite material, the progressive damage model introduces damage variables to express the microdefects in the entire material [105; 106; 107; 108]. These micro-defects can be considered to be continuously distributed inside the material. They will cause the actual internal stress of the material to be higher than the characteristic stress obtained by external measurement. It can be expressed that the stiffness of the material with defects is lower than the stiffness of the non-damaged material, as shown in Figure 4b. Based on the Kachanov-Rabotnov theory, the damage variable D is introduced to describe the constitutive relationship of the damaged material. The one-dimensional characterization stress \(\sigma\) and the real stress (macro-stress) \(\overline{\sigma}\) can be simplified as:\[\sigma=(1-\mathrm{D})\times\overline{\sigma} \tag{19}\]  
Based on the equivalent strain, the constitutive relationship of the three-dimensional damaged material can be obtained. The three-dimensional characterization stress \(\sigma\) is:  
\[\overline{\sigma}=\overline{C}:\epsilon,\ \sigma=\mathrm{C}(\mathrm{D}):\epsilon \tag{20}\]  
\[\sigma=(1-\mathrm{D}):\overline{\sigma}=\left(\mathrm{S}^{-1}:\overline{S} \right):\overline{\sigma} \tag{21}\]  
where \(\overline{S}\) is the stiffness with no damage, and \(\mathrm{S}\) is the characteristic damage stiffness related to the damage variable. \(\mathrm{D}\) is a second-order tensor, and \(\mathrm{S}\) is a fourth-order tensor. The stiffness matrix of the damaged material \(\mathrm{S}(\mathrm{D})\) is  
\[\mathrm{S}(\mathrm{D})=\left[ \begin{array}{cccccc}\frac{\overline{S}_{111 1}}{\mathrm{I}-\mathrm{D}_{11}}&\overline{S}_{1122}&\overline{S}_{1133}&0&0&0 \\ \frac{\overline{S}_{2222}}{\mathrm{I}-\mathrm{D}_{22}}&\overline{S}_{2233}&0&0 &0\\ \frac{\overline{S}_{3333}}{\mathrm{I}-\mathrm{D}_{33}}&0&0&0\\ \mathrm{sym.}&&\frac{\overline{S}_{3222}}{\mathrm{I}-\mathrm{D}_{22}}&0\\ \mathrm{sym.}&&\frac{\overline{S}_{1313}}{\mathrm{I}-\mathrm{D}_{13}}\\ \end{array} \right] \tag{22}\]  
where \(D_{ij}\) is the element of the damage variable matrix \(\mathrm{D}\). \(\overline{S}_{ijkl}\) is the element of the non-damaged material matrix \(\overline{S}\).The progressive damage model describes the generation and evolution of damage variable \(\mathrm{D}\). Damage evolution is a function of plasticity: damage initiates at the onset of plasticity. The damage variable \(\mathrm{D}\) in damage evolution is the key issue and foundation. It directly reflects the damage mechanism of the material. In damage mechanics, the damage variable \(\mathrm{D}\) is used to describe the degradation process of material. \[\mathrm{D}=\varphi(\mathrm{f}) \tag{23}\]  
where \(\varphi(\mathrm{f})\) is the damage evolution function, which reflects the law of the internal damage of the material. Generally, \(\mathrm{D}=0\) in the non-destructive state, and \(\mathrm{D}=1\) in the complete damage state. The Hashin damage model describes damage factor \(\mathrm{D}\) as a function of the Hashin failure criterion coefficient \(\mathrm{f}\).