## Appendix A Mathematical Derivation of Local Problems and Homogenized Equation  
In this appendix, the mathematical derivation of the \({}_{pq}L\) local problems (Equations (7)-(10)) and the homogeneous static problem (Equations (11) and (12)) is developed, as follows:  
Substituting the asymptotic expression in Equation (6) into Equations (1)-(5), where the differentiation is applied according to the relation  
\[\frac{\partial\ f_{i}(\mathbf{x},\mathbf{y})}{\partial x_{j}}=\frac{\partial \ f_{i}(\mathbf{x},\mathbf{y})}{\partial x_{j}}+\epsilon^{-1}\frac{\partial \ f_{i}(\mathbf{x},\mathbf{y})}{\partial y_{j}}, \tag{11}\]  
and the resulting terms are grouped according to power of \(\epsilon\), we obtain the following:  
\[\sigma_{ij}=\sigma_{ij}^{(0)}(\mathbf{x},\mathbf{y})+\epsilon\sigma_{ij}^{(1 )}(\mathbf{x},\mathbf{y})+\epsilon^{2}\sigma_{ij}^{(2)}(\mathbf{x},\mathbf{y} )+\cdots, \tag{12}\]  
\[\epsilon_{ij}=\epsilon_{ij}^{(0)}(\mathbf{x},\mathbf{y})+\epsilon\epsilon_{ij }^{(1)}(\mathbf{x},\mathbf{y})+\epsilon^{2}\epsilon_{ij}^{(2)}(\mathbf{x}, \mathbf{y})+\cdots, \tag{13}\]  
where  
\[\sigma_{ij}^{(0)}(\mathbf{x},\mathbf{y})=C_{ijkl}(\mathbf{y})\frac{\partial u _{k}^{(0)}(\mathbf{x})}{\partial x_{l}}+C_{ijkl}(\mathbf{y})\frac{\partial u _{k}^{(1)}(\mathbf{x},\mathbf{y})}{\partial y_{l}}, \tag{14}\]  
\[\sigma_{ij}^{(1)}(\mathbf{x},\mathbf{y})=C_{ijkl}(\mathbf{y})\frac{\partial u _{k}^{(1)}(\mathbf{x},\mathbf{y})}{\partial x_{l}}+C_{ijkl}(\mathbf{y})\frac{ \partial u_{k}^{(2)}(\mathbf{x},\mathbf{y})}{\partial y_{l}}, \tag{15}\]  
\[\epsilon_{ij}^{(0)}(\mathbf{x},\mathbf{y})=\frac{1}{2}\left(\frac{\partial u _{i}^{(0)}(\mathbf{x})}{\partial x_{j}}+\frac{\partial u_{j}^{(1)}(\mathbf{x} )}{\partial x_{i}}\right)+\frac{1}{2}\left(\frac{\partial u_{i}^{(1)}( \mathbf{x},\mathbf{y})}{\partial y_{j}}+\frac{\partial u_{j}^{(1)}(\mathbf{x },\mathbf{y})}{\partial y_{i}}\right), \tag{16}\]  
\[\epsilon_{ij}^{(1)}(\mathbf{x},\mathbf{y})=\frac{1}{2}\left(\frac{\partial u _{i}^{(1)}(\mathbf{x},\mathbf{y})}{\partial x_{j}}+\frac{\partial u_{j}^{(1)} (\mathbf{x},\mathbf{y})}{\partial x_{i}}\right)+\frac{1}{2}\left(\frac{ \partial u_{i}^{(2)}(\mathbf{x},\mathbf{y})}{\partial y_{j}}+\frac{\partial u _{j}^{(2)}(\mathbf{x},\mathbf{y})}{\partial y_{i}}\right). \tag{17}\]  
Replacing Equations (6) and (12) into Equation (1) and rearranging terms with the same power in \(\epsilon\), we obtain for \(\epsilon^{-1}\):  
\[\frac{\partial\sigma_{ij}^{(0)}(\mathbf{x},\mathbf{y})}{\partial y_{j}}=0, \tag{18}\]  
and for \(\epsilon^{0}\):  
\[\frac{\partial\sigma_{ij}^{(0)}(\mathbf{x},\mathbf{y})}{\partial x_{j}}+\frac {\partial\sigma_{ij}^{(1)}(\mathbf{x},\mathbf{y})}{\partial y_{j}}=0. \tag{19}\]  
Substituting Equation (14) into Equation (18):  
\[\frac{\partial}{\partial y_{j}}\left(C_{ijkl}(\mathbf{y})\frac{\partial u_{k} ^{(1)}(\mathbf{x},\mathbf{y})}{\partial y_{l}}\right)=-\frac{\partial u_{k} ^{(0)}(\mathbf{x})}{\partial x_{l}}\frac{\partial C_{ijkl}(\mathbf{y})}{ \partial y_{j}}, \tag{20}\]  
Thus, it is possible to obtain from Equation (20) the expression for \(u_{k}^{(1)}(\mathbf{x},\mathbf{y})\) as follows:  
\[\sigma_{ij}=\sigma_{ij}^{(0)}(\mathbf{x},\mathbf{y})+\epsilon\sigma_{ij}^{(1 )}(\mathbf{x},\mathbf{y})+\epsilon^{2}\sigma_{ij}^{(2)}(\mathbf{x},\mathbf{y} )+\cdots, \tag{12}\]0  
where \({}_{pq}N_{k}\)\((k,p,q=1,2,3)\) are \(Y\)-periodic local pseudo-displacements vector components. Substituting Equation (A11) into Equation (A10), the \({}_{pq}L\) local problems statement in Equation (7) is obtained.