## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
Each problem is decoupled into an independent set of equations according to the plane and out-of-plane deformation state assumed in linear elasticity. Table 1 shows the correspondence between the effective elastic coefficients and the local problems \(p_{\mathit{q}}L\). The local problems can be presented as follows: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{n}\,\boldsymbol{L}\) & \(\mathbf{z}\,\boldsymbol{L}\) & \(\mathbf{3}\,\boldsymbol{L}\) & \(\mathbf{2}\,\boldsymbol{S}\,\boldsymbol{L}\) & \(\mathbf{13}\,\boldsymbol{L}\) & \(\mathbf{12}\,\boldsymbol{L}\) \\ \hline C\({}_{1111}^{*}\) & C\({}_{1122}^{*}\) & C\({}_{1133}^{*}\) & 0 & 0 & 0 \\ C\({}_{2211}^{*}\) & C\({}_{2222}^{*}\) & C\({}_{2233}^{*}\) & 0 & 0 & 0 \\ C\({}_{3311}^{*}\) & C\({}_{3322}^{*}\) & C\({}_{3333}^{*}\) & 0 & 0 & 0 \\
0 & 0 & 0 & C\({}_{3223}^{*}\) & 0 & 0 \\
0 & 0 & 0 & 0 & C\({}_{1313}^{*}\) & 0 \\
0 & 0 & 0 & 0 & 0 & C\({}_{1212}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Local problems and associated effective coefficients. Problems \({}_{\lambda 3}L\) with \(\lambda,\beta=1,2\) and \(\lambda\
eq\beta:\)  
\[ \begin{split}\frac{\partial_{\lambda 3}\tau_{\lambda 1}}{ \partial y_{1}}+\frac{\partial_{\lambda 3}\tau_{\lambda 2}}{\partial y_{2}}+\frac{ \partial_{\lambda 3}\tau_{\lambda 3}}{\partial y_{3}}&=-\frac{\partial C_{ \lambda 3\lambda 3}}{\partial y_{3}},\\ \frac{\partial_{\lambda 3}\tau_{31}}{\partial y_{1}}+\frac{ \partial_{\lambda 3}\tau_{32}}{\partial y_{2}}+\frac{ \partial_{\lambda 3}\tau_{33}}{\partial y_{3}}&=-\frac{\partial C_{ \lambda 3\lambda 3}}{\partial y_{\lambda}},\\ \frac{\partial_{\lambda 3}\tau_{\beta 1}}{\partial y_{1}}+\frac{ \partial_{\lambda 3}\tau_{\beta 2}}{\partial y_{2}}+\frac{ \partial_{\lambda 3}\tau_{\beta 3}}{\partial y_{3}}&=0.\end{split}  \tag{15}\]  
Problems \({}_{\beta\beta}L\) with \(\beta=1,2,3:\)  
\[ \begin{split}\frac{\partial_{\beta\beta}\tau_{11}}{ \partial y_{1}}+\frac{\partial_{\beta\beta}\tau_{12}}{\partial y_{2}}+\frac{ \partial_{\beta\beta}\tau_{13}}{\partial y_{3}}&=-\frac{ \partial C_{11\beta\beta}}{\partial y_{1}},\\ \frac{\partial_{\beta\beta}\tau_{21}}{\partial y_{1}}+\frac{ \partial_{\beta\beta}\tau_{22}}{\partial y_{2}}+\frac{\partial_{\beta\beta} \tau_{23}}{\partial y_{3}}&=-\frac{\partial C_{22\beta\beta}}{ \partial y_{2}},\\ \frac{\partial_{\beta\beta}\tau_{31}}{\partial y_{1}}+\frac{ \partial_{\beta\beta}\tau_{32}}{\partial y_{2}}+\frac{\partial_{\beta\beta} \tau_{33}}{\partial y_{3}}&=-\frac{\partial C_{33\beta\beta}}{ \partial y_{3}}.\end{split}  \tag{16}\]  
Problem \({}_{12}L:\)  
\[ \begin{split}\frac{\partial_{12}\tau_{11}}{ \partial y_{1}}+\frac{\partial_{12}\tau_{12}}{\partial y_{2}}+\frac{ \partial_{12}\tau_{13}}{\partial y_{3}}&=-\frac{\partial C_{1212 }}{\partial y_{2}},\\ \frac{\partial_{12}\tau_{21}}{\partial y_{1}}+\frac{\partial_{12 }\tau_{22}}{\partial y_{2}}+\frac{\partial_{12}\tau_{23}}{\partial y_{3}}& =-\frac{\partial C_{2112}}{\partial y_{1}},\\ \frac{\partial_{12}\tau_{31}}{\partial y_{1}}+\frac{\partial_{12 }\tau_{32}}{\partial y_{2}}+\frac{\partial_{12}\tau_{33}}{\partial y_{3}}& =0.\end{split}  \tag{17}\]  
In the Equations (15)-(17), the elastic coefficients \(C_{ijkl}(\mathbf{y})=C_{ijkl}(y_{1},y_{2},y_{3})\) are assumed to be even functions with respect to the variables \(y_{1},\)\(y_{2}\) and \(y_{3}\). Then, the following conditions are satisfied:  
\[ \begin{split}{}_{pq}N_{i}\mid_{y_{n}=d_{h}}=0,\text{ if }\delta_{hi}+\delta_{hp}+\delta_{hq}\text{ \ is odd},\end{split}  \tag{18}\]  
\[ \begin{split}{}_{pq}\tau_{i}\mid_{y_{n}=d_{h}}=0,\text{ if }\delta_{hi}+\delta_{hp}+\delta_{hq}+1\text{ \ is even},\end{split}  \tag{19}\]  
where \(d_{h}=\{0,l_{h}\}\)\((h=1,2,3)\), \(l_{h}\) is the semi-length of the cell in the \(y_{h}\) direction and the function \(\delta_{hi}\) is defined as follows:  
\[ \begin{split}\delta_{hi}=\left\{\begin{array}{ll}1& \text{if }h=i,\\ 0&\text{if }h\
eq i.\end{array}\right.\end{split}