## 3 Asymptotic Homogenization Method: Homogeneous Problem, Local Problems and Effective Coefficients
### Solution of Local Problems through the Finite Element Method
 \tag{42}\]where \(\xi_{1},\ \xi_{2},\ \xi_{3}\) and \(\xi_{4}=1-\xi_{1}-\xi_{2}-\xi_{3}\) represent the element's natural coordinates. The element node vector \(p_{q}\mathbf{w}\) is defined as  
\[p_{q}\mathbf{w}_{30\times 1}=\Big{[}\ _{pq}w_{1(1)\ pq}w_{2(1)\ pq}w_{3(1)}\ \cdots\ _{pq}w_{1(10)\ pq}w_{2(10)\ pq}w_{3(10)}\Big{]}^{T}. \tag{43}\]  
Taking into account Equations (41) and (42), the relationship between deformations and pseudo-displacements as shown in Equation (39) can be written in matrix form as functions of the natural coordinates and the element's shape functions, as follows:  
\[p_{q}\varepsilon=\mathbf{B}\ _{pq}\mathbf{w}, \tag{44}\]  
where \(\mathbf{B}\) is the element strain-displacement matrix:  
\[\mathbf{B}_{6\times 30}= \begin{bmatrix}\Im_{11}&\Im_{12}&\Im_{13}&0&0&0&0&0 &0\\ 0&0&0&\Im_{21}&\Im_{22}&\Im_{23}&0&0&0\\ 0&0&0&0&0&\Im_{31}&\Im_{32}&\Im_{33}\\ 0&0&0&\Im_{31}&\Im_{32}&\Im_{33}&\Im_{21}&\Im_{22}&\Im_{23}\\ \Im_{31}&\Im_{32}&\Im_{33}&0&0&0&\Im_{11}&\Im_{12}&\Im_{13}\\ \Im_{32}&\Im_{32}&\Im_{31}&\Im_{12}&\Im_{13}&0&0&0\end{bmatrix} _{\mathbf{H}_{9 \times 30}}, \tag{45}\]  
\(\Im_{ij}\) represent the inverse matrix coefficients of the Jacobian transformation \(\mathbf{J}\) between the Cartesian system \((y_{1},y_{2},y_{3})\) and the natural coordinate system \((\xi_{1},\ \xi_{2},\ \xi_{3})\), and the matrix \(\mathbf{H}_{9\times 30}\) is defined in Appendix C.  
The stress values are determined as functions of the natural coordinates and the element's shape functions by means of the relations shown in Equations (37)-(42), such that  
\[p_{q}\boldsymbol{\sigma}=\mathbf{D}\mathbf{B}\ _{pq}\mathbf{w}. \tag{46}\]  
Replacing Equations (44) and (46) into Equation (36), we obtain the deformation energy \(\Pi_{e}\) associated with an element \(e\) in the following form:  
\[\Pi_{e}=\frac{1}{2}\ _{pq}\mathbf{w}^{T}\ \mathbf{K}_{e}\ _{pq}\mathbf{w}, \tag{47}\]  
where \(\mathbf{K}_{e}\) is the stiffness matrix of the element:  
\[\mathbf{K}_{e}=\int_{0}^{1}\int_{0}^{1-\xi_{1}}\int_{0}^{1-\xi_{1}-\xi_{2}} \mathbf{B}^{T}\mathbf{D}\ \mathbf{B}\ \det(\mathbf{J})\ d\zeta_{1}\ d\zeta_{2}\ d\zeta_{3}. \tag{48}\]  
Therefore, taking into account the connectivity of the elements, the total energy is equal to  
\[\Pi=\sum_{e}\Pi_{e}=\sum_{e}\frac{1}{2}\ _{pq}\mathbf{w}^{T}\ \mathbf{K}_{e}\ _{pq} \mathbf{w}=\frac{1}{2}\ _{pq}\mathbf{W}^{T}\ \mathbf{K}\ _{pq}\mathbf{W}, \tag{49}\]  
where \(\mathbf{K}\) and \({}_{pq}\mathbf{W}\) represent the global rigid matrix and the global pseudo-displacement vector on \(S\), respectively. The minimum potential energy is found by means of an algebraic system of equations obtained through  
\[\frac{\partial\Pi}{\partial_{pq}\mathbf{W}_{i}}=0, \tag{50}\]  
and by applying the corresponding boundary conditions of each \({}_{pq}L\) local problem; see Equations (A15)-(A20). constituents are defined in Table 2 and taken from different works, in which the effective engineering and the Hill's constants in the \(y_{1}\)-direction are related by  
\[ \begin{split} E_{11}^{*}&=n-\frac{l^{2}}{k},\\ E_{22}^{*}&=E_{33}^{*}=\frac{4m\left(kn-l^{2}\right) }{n(k+m)-l^{2}},\\ \
u_{12}^{*}&=\frac{l}{2k},\\ \
u_{23}^{*}&=\frac{n(k-m)-l^{2}}{n(k+m)-l^{2}}, \end{split}  \tag{61}\]  
where \(m=\mathrm{C}_{2323}^{*}=(\mathrm{C}_{3333}^{*}-\mathrm{C}_{2233}^{*})/2\) and \(p=\mathrm{C}_{1313}^{*}=\mathrm{C}_{1212}^{*}\) are the transverse and longitudinal shear moduli, \(k=(\mathrm{C}_{3333}^{*}+\mathrm{C}_{2233}^{*})/2\) is the transverse bulk modulus, \(l=\mathrm{C}_{1122}^{*}=\mathrm{C}_{1133}^{*}\) is the associated cross modulus and \(n=\mathrm{C}_{1111}^{*}\) is related to uniaxial straining in the \(y_{1}\)-direction [49].