## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
### Solution of Local Problems through the Finite Element Method  
For many cases of FRCs with complex geometrical structures, the determination of analytical solutions for the local problems and the exact expressions of the effective coefficients through the analytic AHM are difficult or even impossible to solve. Only for specific inclusion geometries, the exact analytical solutions have been reported; see for instance [23]. In this sense, an alternative approach is the finite element method. In the present work, the SAFEM-3D approach is developed to find the solution for the local problems and the effective elastic properties. The SAFEM-3D is implemented by combining the AHM and the FEM; i.e., the local problems obtained by the AHM are solved using the FEM through the minimum potential energy principle, in a similar manner to the methodology developed by [PERSON] et al. [29; 46] applied to a bi-dimensional elastic two-phase FRC. The total corresponding potential energy \(\Pi\) for an elastic solid medium \(\Omega\)--see for instance [47]--is defined as  
\[\Pi=\frac{1}{2}\int_{V}\mathbf{\sigma}^{T}\mathbf{\epsilon}\ dV-\int_{V}\mathbf{u}^{T}\bm {f}\ dV-\int_{S}\mathbf{u}^{T}\mathbf{T}\ dS-\sum_{i}u_{i}P_{i}, \tag{36}\]  
where \(\Pi\) involves the deformation energy per unit volume of the body and the potential energies associated with the body forces \(\mathbf{f}\), tensile forces \(\mathbf{T}\) and point loads \(P_{i}\) respectively. In the present work, the forces \(\mathbf{f}\) and \(\mathbf{T}\) and loads \(P_{i}\) are assumed to be zero. The relationships given by Equations (26)-(29) are related as follows:  
\[{}_{pq}\mathbf{\sigma}=\mathbf{D}\ _{pq}\mathbf{\varepsilon}, \tag{37}\]  
where  
\[{}_{pq}\mathbf{\sigma}=\left[ \begin{array}{cccc}{}_{pq}\sigma_{11}&{}_{pq}\sigma _{22}&{}_{pq}\sigma_{33}&{}_{pq}\sigma_{23}&{}_{pq}\sigma_{13}&{}_{pq}\sigma_ {21}\end{array} \right]^{T}, \tag{38}\]  
\[{}_{pq}\mathbf{\varepsilon}=\left[ \begin{array}{cccc}{}_{pq}\epsilon_{11}&{}_{pq }\epsilon_{22}&{}_{pq}\epsilon_{33}&{}_{pq}\epsilon_{23}&{}_{pq}\epsilon_{13} &{}_{pq}\epsilon_{21}\end{array} \right]^{T} \tag{39}\]  
\[=\left[ \begin{array}{cccc}\frac{\partial_{pq}M_{1}}{\partial y_{1}}&\frac{ \partial_{pq}M_{2}}{\partial y_{2}}&\frac{\partial_{pq}M_{3}}{\partial y_{3}}& \frac{\partial_{pq}M_{2}}{\partial y_{3}}+\frac{\partial_{pq}M_{3}}{\partial y _{2}}&\frac{\partial_{pq}M_{1}}{\partial y_{3}}+\frac{\partial_{pq}M_{3}}{ \partial y_{1}}&\frac{\partial_{pq}M_{1}}{\partial y_{2}}+\frac{\partial_{pq}M _{2}}{\partial y_{1}}\end{array} \right]^{T},\]  
and  
\[\mathbf{D}= \begin{bmatrix}C_{1111}&C_{1122}&C_{1133}&0&0&0\\ C_{1122}&C_{2222}&C_{2233}&0&0&0\\ C_{1133}&C_{2233}&C_{3333}&0&0&0\\ 0&0&0&C_{2323}&0&0\\ 0&0&0&0&C_{1313}&0\\ 0&0&0&0&0&C_{1212}\end{bmatrix} . \tag{40}\]  
The region of study (one-eighth of the unit cell \(S\)) is divided into a finite number of tetrahedral elements. Each element is characterized by 10 nodes located at the vertices, and the remaining ones are at the midpoints of the edges. Additionally, each node is associated with a pseudo-displacement vector connected with the \({}_{pq}L\) local problem to be solved. The pseudo-displacements \({}_{pq}M_{k}\) are defined as a linear combination of the shape functions \(\Psi_{i}\) whose coefficients are the nodal values of the unknown pseudo-displacement \({}_{pq}w_{k(s)}\), i.e.,  
\[{}_{pq}M_{k}=\Psi_{s}\ {}_{pq}w_{k(s)}, \tag{41}\]  
where \({}_{pq}w_{k(s)}\) represents the \(k\)-th component of the pseudo-displacement at the \(s\)-th node \((s=1,\cdots,10)\) for the local problem \({}_{pq}L\). The shape functions \(\Psi_{i}\) are defined through natural coordinates [48], as follows:  
\[ \begin{array}{l}\Psi_{1}=\xi_{1}(2\xi_{1}-1),\ \Psi_{2}=\xi_{2}(2\xi_{2}-1),\ \Psi_{3}=\xi_{3}(2\xi_{3}-1),\ \Psi_{4}=\xi_{4}(2\xi_{4}-1),\\ \Psi_{5}=4\xi_{1}^{*}\xi_{2}^{*},\ \Psi_{6}=4\xi_{2}\xi_{3},\ \Psi_{7}=4\xi_{3}\xi_{1},\ \Psi_{8}=4\xi_{1}^{*}\xi_{4},\ \Psi_{9}=4\xi_{2}\xi_{4},\ \Psi_{10}=4\xi_{3}\xi_{4},\\ \end{array}