## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients  
The homogenized local problems over \(S\), the homogeneous static problem and effective coefficients linked to the periodic elastic FRC are derived from Equations (1)-(5) using the AHM [17; 22; 23] by means of the asymptotic expansion of the displacement \(u_{i}\) in powers of the parameter \(\epsilon\), resulting in the following:  
\[u_{i}=u_{i}^{(0)}(\mathbf{x})+\epsilon u_{i}^{(1)}(\mathbf{x},\mathbf{y})+ \epsilon^{2}u_{i}^{(2)}(\mathbf{x},\mathbf{y})+\cdots, \tag{6}\]  
where \(u_{i}^{(0)}\) is a function only of the slow variable \(\mathbf{x}\), \(u_{i}^{(k)}(\mathbf{x},\mathbf{y})\)\((k=1,2,\cdots)\) are S-periodic functions with respect to the fast variable \(\mathbf{y}\) and the superscripts denote the order of terms in the expansions. Here, the two-scales, denoted by \(\mathbf{x}\) (global or slow) and \(\mathbf{y}\) (local or fast), characterize the global behavior of the composite and the heterogeneities at a local level over \(S\), respectively. The scales are related by \(\mathbf{y}=\mathbf{x}/\epsilon\), where \(\epsilon\) is a small enough parameter which represents the ratio between the dimensions of the periodic cell \(S\) and the characteristic dimensions of the nanocomposite. The mathematical statement of the \(p_{q}L\) local problems over the unit cell \(S\) are stated as  
\[\frac{\partial\ _{pq}\tau_{ij}(\mathbf{y})}{\partial y_{j}}=-\frac{\partial C _{ijpq}(\mathbf{y})}{\partial y_{j}}, \tag{7}\]  
subject to the perfect interface conditions, as shown in Equation (5), which in this case are  
\[{}_{pq}N_{k}^{(1)}(\mathbf{y})\ =\ _{pq}N_{k}^{(2)}(\mathbf{y})\ \ \ \text{on}\ \ \Gamma \tag{8}\]  
\[C_{ijpq}^{(1)}(\mathbf{y})+\ _{pq}\tau_{ij}^{(1)}(\mathbf{y})n_{j}^{(1)} =\ C_{ijpq}^{(2)}(\mathbf{y})+\ _{pq}\tau_{ij}^{(2)}(\mathbf{y})n_{j}^{(2)},\ \ \ \text{on}\ \ \Gamma \tag{9}\]where the components of the Y-periodic local stress tensor are  
\[p_{\mathit{q}}\tau_{ij}(\mathbf{y})=C_{ijkl}(\mathbf{y})\frac{\partial_{pq}N_{k}( \mathbf{y})}{\partial y_{l}}. \tag{10}\]  
and \(p_{\mathit{q}}N_{k}\) (\(k,p,q=1,2,3\)) are the components of the \(S\)-periodic local pseudo-displacements vector, providing the solution of the \(p_{\mathit{q}}L\) local problems. Additionally, the following condition, over \(S\), is needed to guarantee a unique solution of the local problems: \(\langle p_{\mathit{q}}N_{k}\rangle=0\). Here, \(\langle\cdot\rangle=(1/|S|)\int(\cdot)d\mathbf{y}\) is the average operator over the periodic cell \(S\) and \(|S|\) is the volume of the cell. The homogeneous static problem equivalent to Equation (1) is stated as follows:  
\[\frac{\partial\overline{\sigma}_{ij}^{(0)}(\mathbf{x})}{\partial x_{j}}=0, \tag{11}\]  
where the constitutive equations of the homogeneous problem are  
\[\overline{\sigma}_{ij}^{(0)}(\mathbf{x})=\langle\sigma_{ij}^{(0)}(\mathbf{x} )\rangle=C_{ijpq}^{*}\frac{\partial u_{p}^{(0)}(\mathbf{x})}{\partial x_{q}}, \tag{12}\]  
where \(u_{p}^{(0)}(\mathbf{x})\) is the solution of the homogenized problem in Equation (11) and \(C_{ijpq}^{*}\) are the effective coefficients, which are defined as follows:  
\[C_{ijpq}^{*}=\langle C_{ijpq}(\mathbf{y})+\ p_{\mathit{q}}\tau_{ij}(\mathbf{y })\rangle. \tag{13}\]  
The boundary conditions expressed in Equation (2) are transformed into  
\[ \begin{split} u_{i}^{(0)}(\mathbf{x})&=0,\ \text{on}\ \partial_{1}\Omega,\\ \overline{\sigma}_{ij}^{(0)}(\mathbf{x})n_{j}&=0,\ \text{on}\ \partial_{2}\Omega.\end{split}  \tag{14}\]  
The mathematical procedures to obtain the \(p_{\mathit{q}}L\) local problems (Equations (7)-(10)) and the homogeneous static problem (Equations (11)-(12)) can be seen in Appendix A. More details explaining he AHM procedure and the rigorous theoretical mathematical background can be found in the classical works [17; 20; 44; 45]. Six local problems \(p_{\mathit{q}}L\) (\(p,q=1,2,3\)) in terms of the local variable \(\mathbf{y}\) are defined in the unit cell \(S\).