## 2 Micromechanical Modeling Using the Generalized Method of Cells  
### Constitutive Models  
It is deemed that fiber-reinforced composites can be justified as a collection of RUCs which are divided into \(N_{\beta}\times N_{\gamma}\) (\(\beta\), \(\gamma\) = 1, 2,..., \(N_{\beta}\), \(N_{\gamma}\)) subcells as displayed in Figure 1. The number of subcells is arbitrary, and each of them can contain a specified material representing the constituents in the composite; the constituents are matrices and fibers in general, and the interfaces between the subcells can be ignored in the perfect bonding conditions [26].  
A local coordinate system \((x_{1},\overline{x}_{2}^{(\beta)},\overline{x}_{3}^{(\gamma)})\) is set for each subcell of the RUCs, where the \(x_{1}\) axis shown in Figure 1 represents the fiber direction, and the size of each subcell is defined as \(h_{\beta}\) and \(l_{\gamma}\) along the direction of the \(x_{2}\) and the \(x_{3}\). It is assumed employing a linear displacement approximation is suitable to obtain the average behavior of the RUCs,  
\[\mu_{i}^{(\beta\gamma)}=\omega_{i}^{(\beta\gamma)}(x)+\overline{x}_{2}^{( \beta)}\phi_{i}^{(\beta\gamma)}+\overline{x}_{3}^{(\gamma)}\psi_{i}^{(\beta \gamma)}\ i=1,2,3 \tag{1}\]  
where \(\omega_{i}^{(\beta\gamma)}\) denotes the displacement located at the center of subcell \(\beta\gamma\). \(\phi_{i}^{(\beta\gamma)}\) and \(\psi_{i}^{(\beta\gamma)}\) are microvariables which can be easily related to the subcell strains \(\varepsilon_{ij}^{(\beta\gamma)}\), the local strain tensors can be represented as follows,  
\[\varepsilon_{ij}^{(\beta\gamma)}=\frac{1}{2}\Big{(}\partial_{i}\mu_{j}^{( \beta\gamma)}+\partial_{j}\mu_{i}^{(\beta\gamma)}\Big{)} \tag{2}\]  
where \(\partial_{1}=\partial/\partial x_{1}\), \(\partial_{2}=\partial/\partial\overline{x}_{2}^{(\beta)}\), \(\partial_{3}=\partial/\partial\overline{x}_{3}^{(\gamma)}\).  
Substituting Equation (1) into Equation (2) generates the average strain equations for each subcell,  
\[\begin{array}{c}\bar{\varepsilon}_{11}^{(\beta\gamma)}=\partial_{1}\omega_ {1}^{(\beta\gamma)}\\ \bar{\varepsilon}_{22}^{(\beta\gamma)}=\phi_{2}^{(\beta\gamma)}\\ \bar{\varepsilon}_{33}^{(\beta\gamma)}=\psi_{3}^{(\beta\gamma)}\\ 2\bar{\varepsilon}_{23}^{(\beta\gamma)}=\phi_{3}^{(\beta\gamma)}+\psi_{2}^{( \beta\gamma)}\\ 2\bar{\varepsilon}_{13}^{(\beta\gamma)}=\psi_{1}^{(\beta\gamma)}+\partial_{1} \omega_{3}^{(\beta\gamma)}\\ 2\bar{\varepsilon}_{12}^{(\beta\gamma)}=\psi_{1}^{(\beta\gamma)}+\partial_{1} \omega_{2}^{(\beta\gamma)}\end{array} \tag{3}\]  
Herein, only elastic subcell strains, \(\bar{\varepsilon}_{ij}(\beta\gamma)\), will be taken into account; [PERSON]'s law is utilized to express subcell stresses in terms of its corresponding strains is written as,  
\[\left[\begin{array}{c}\overline{\sigma}_{11}^{(\beta\gamma)}\\ \overline{\sigma}_{22}^{(\beta\gamma)}\\ \overline{\sigma}_{33}^{(\beta\gamma)}\\ \overline{\sigma}_{12}^{(\beta\gamma)}\\ \overline{\sigma}_{33}^{(\beta\gamma)}\\ \overline{\sigma}_{33}^{(\beta\gamma)}\end{array}\right]=(1-D_{\max})\left[ \begin{array}{cccccc}C_{11}^{(\beta\gamma)}&C_{12}^{(\beta\gamma)}&C_{13}^ {(\beta\gamma)}&0&0&0\\ C_{21}^{(\beta\gamma)}&C_{22}^{(\beta\gamma)}&C_{23}^{(\beta\gamma)}&0&0&0\\ C_{31}^{(\beta\gamma)}&C_{32}^{(\beta\gamma)}&C_{33}^{(\beta\gamma)}&0&0&0\\ 0&0&0&C_{44}^{(\beta\gamma)}&0&0\\ 0&0&0&0&C_{55}^{(\beta\gamma)}&0\\ 0&0&0&0&0&C_{66}^{(\beta\gamma)}\end{array}\right]=\left[\begin{array}{c} \bar{\varepsilon}_{11}^{(\beta\gamma)}\\ \bar{\varepsilon}_{22}^{(\beta\gamma)}\\ \bar{\varepsilon}_{33}^{(\beta\gamma)}\\ \bar{\varepsilon}_{12}^{(\beta\gamma)}\\ \bar{\varepsilon}_{13}^{(\beta\gamma)}\\ \bar{\varepsilon}_{23}^{(\beta\gamma)}\end{array}\right] \tag{4}\]  
where \(C_{ij}^{(\beta\gamma)}\) denotes the elastic stiffness matrix of the subcell \(\beta\gamma\), and \(D_{max}\) represents the stiffness deduction coefficient of each failed subcell.  
Figure 1: Discretization of the repeat unit cell.