## 3 The Constitutive Relationship of Cellular Medium
### Homogenization Algorithm Model  
A cellular medium can be considered as a two-phase composite with matrix and pores. The homogenization algorithm model [97; 98; 99; 100; 101; 102; 103; 104] combines a macro-scale and a micro-scale to predict the effect of microstructure on the overall properties of the material. The two scales are connected by a representative volume element (RVE). In a micro-macro-approach, at each macro-point \(\mathbb{X}\), we know the macro-strain \(\overline{z}\) and need to compute the macro-stress \(\overline{\sigma}\). At micro-level, the macro-point is viewed as the center of a RVE with domain \(\omega\) and boundary \(\partial\omega\), as shown in Figure 4b. The Hill-Mandell condition, expressing the equality between energies at both scales, transforms the relation between macro-strains \(\overline{z}\) and macro-stresses \(\overline{\sigma}\) into a relation between average strains \(\langle\varepsilon\rangle\) and average stresses \(\langle\sigma\rangle\) over the RVE, with \(\overline{z}=\langle\varepsilon\rangle\) and \(\overline{\sigma}=\langle\sigma\rangle\). The matrix (phase 0) has volume \(V_{0}\) and volume fraction \(v_{0}=V_{0}/V\), where \(V\) is the volume of the RVE. The pores (phase 1) have a total volume  
Figure 4: (**a**) Simplified one-dimensional shock models for cellular medium [46; 87]; (**b**) basic model of homogenization algorithm model and progressive damage model [30; 97]. and a volume fraction \(v_{1}=V_{1}/V=1-v_{0}\) (subscript 0 refers to the matrix and 1 to the pores). The volume averages are defined as:  
\[\langle f\rangle=\frac{1}{V}\int_{\omega}f(\overline{x},x)dV,\ \ \langle f\rangle_{\omega_{i}}=\frac{1}{V_{i}}\int_{\omega_{j}}f( \overline{x},x)dV_{i},\ \ i=0,1 \tag{13}\]  
where \(\overline{x}\) is the macro-point, and \(x\) is the micro-point. The averages over \(\omega\) (the entire RVE), \(\omega_{0}\) (the matrix phase) and \(\omega_{1}\) (the pores phase) are related by \(\langle f\rangle=(1-v_{1})\langle f\rangle_{\omega 0}+v_{1}\langle\epsilon \rangle_{\omega 1}\). The strain averages and stress averages can be written as [97; 101]:  
\[\langle\epsilon\rangle_{\omega}=(1-v_{1})\langle\epsilon\rangle_{\omega_{0}}+ v_{1}\langle\epsilon\rangle_{\omega_{1}},\langle\sigma\rangle_{\omega}=(1-v_{1}) \langle\sigma\rangle_{\omega_{0}}+v_{1}\langle\sigma\rangle_{\omega_{1}} \tag{14}\]  
Under linear boundary conditions, the strain averages per phase are related by a strain concentration tensor \(\mathrm{B}^{\epsilon}\) as follow:  
\[\langle\epsilon\rangle_{\omega_{1}}=\mathrm{B}^{\epsilon}:\ \langle\epsilon \rangle_{\omega_{0}} \tag{15}\]  
To express this strain concentration tensor \(\mathrm{B}^{\epsilon}\), assumptions should be made on the micro-mechanics. This model is based on the Eshelby tensor \(S(I,C_{0})\) and assumes that the strain field inside the inclusion (pore) is uniform and related to the macro-strain [103]. \[\mathrm{B}^{\epsilon}=\left\{I+S:\left[(C_{0})^{-1}:C_{1}-I\right]\right\}^{-1} \tag{16}\]  
where Eshelby tensor \(S(I,C_{0})\) depends on the geometry of the pore (\(I\)) and the stiffness tensor of the matrix \(C_{0}\). The per-phase average strains are related to the macro-strain \(\langle\epsilon\rangle\) by:  
\[\langle\epsilon\rangle_{\omega_{0}}=\left[v_{1}\mathrm{B}^{\epsilon}+(1-v_{1} )I\right]^{-1}:\ \langle\epsilon\rangle,\ \ \langle\epsilon\rangle_{\omega_{1}}=\mathrm{B}^{\epsilon}:\ [v_{1} \mathrm{B}^{\epsilon}+(1-v_{1})I]^{-1}:\ \langle\epsilon\rangle \tag{17}\]  
We consider two-phase composites where the pores (phase 1) have the same shape, orientation and stiffness tensor \(C_{1}\). Thus the macroscopic material behavior can be written in the form [30; 97; 101; 102; 103]  
\[\overline{\sigma}=\overline{C}:\bar{\epsilon},\ \ with\ \ \ \ \overline{C}=\left[v_{1}C_{1}:\mathrm{B}^{\epsilon}+(1-v_{1})C_{0}\right]: \left[v_{1}\mathrm{B}^{\epsilon}+(1-v_{1})I\right]^{-1} \tag{18}\]  
where \(\overline{C}\) is the homogenized macro-stiffness tensor.