## 2 Micromechanical Modeling Using the Generalized Method of Cells
### Continuity Conditions  
The interfaces between adjoining subcells along with the boundaries between neighboring RUCs are required to be imposed the displacement and traction continuity conditions in an average sense. [PERSON]'s strain concentration matrix is constructed using these continuity conditions to make the unknown average subcell strains be related to the applied global average strains. Hence, the subcell strains are utilized to provide the global average strains of RUCs by defining the equation as follows:  
\[\bar{\epsilon}_{\rm ij}=\frac{1}{hl}\sum\limits_{\beta=1}^{N_{\beta}}\sum \limits_{\gamma=1}^{N_{\gamma}}h_{\beta}l_{\gamma}\bar{\epsilon}_{\rm ij}^{( \beta\gamma)} \tag{5}\]  
where \(h\), \(l\) denote the sizes of RUC, as well as \(h_{\beta}\) and \(l_{\gamma}\) represent the length of subcell \(\beta\gamma\), respectively.  
Next, imposing interfacial displacement continuity between adjacent subcells produces the following equations:  
\[\begin{array}{l}\bar{\epsilon}_{11}^{(\beta\gamma)}=\bar{\epsilon}_{11},\ \beta=1,\ldots,N_{\beta};\ \gamma=1,\ldots,N_{\gamma}\\ \sum\limits_{\beta=1}^{N_{\beta}}h_{\beta}\bar{\epsilon}_{22}^{(\beta\gamma) }=h\bar{\epsilon}_{22},\ \gamma=1,\ldots,N_{\gamma}\\ \sum\limits_{\gamma=1}^{N_{\gamma}}l_{\gamma}\bar{\epsilon}_{33}^{(\beta \gamma)}=l\bar{\epsilon}_{33},\beta=1,\ldots,N_{\beta}\\ \sum\limits_{\beta=1}^{N_{\beta}}h_{\beta}\bar{\epsilon}_{12}^{(\beta\gamma) }=h\bar{\epsilon}_{12},\ \gamma=1,\ldots,N_{\gamma}\\ \sum\limits_{\gamma=1}^{N_{\gamma}}l_{\gamma}\bar{\epsilon}_{13}^{(\beta \gamma)}=l\bar{\epsilon}_{33},\ \beta=1,\ldots,N_{\beta}\\ \sum\limits_{\beta=1}^{N_{\beta}}\sum\limits_{\beta=1}^{N_{\beta}}h_{\beta}l_ {\gamma}\bar{\epsilon}_{23}^{(\beta\gamma)}=hl\bar{\epsilon}_{23}\end{array} \tag{6}\]  
Equations of (6) are available to be represented in the matrix form,  
\[A_{\rm G}\epsilon_{\rm s}=J\bar{\epsilon} \tag{7}\]  
where  
\[\bar{\epsilon}=[\bar{\epsilon}_{11},\bar{\epsilon}_{22},\bar{\epsilon}_{33}, 2\bar{\epsilon}_{23},2\bar{\epsilon}_{13},2\bar{\epsilon}_{12}],\epsilon_{ \rm s}=\left[\bar{\epsilon}^{(11)},\bar{\epsilon}^{(12)},\ldots,\bar{ \epsilon}^{(N_{\beta}N_{\gamma})}\right],J=[1,h,l,h,l,hl] \tag{8}\]  
and  
\[A_{\rm G}=\left[1,\sum\limits_{\beta=1}^{N_{\beta}}h_{\beta},\ \sum\limits_{\gamma=1}^{N_{\gamma}}l_{\gamma},\ \sum\limits_{\beta=1}^{N_{\beta}}h_{\beta},\ \sum\limits_{\gamma=1}^{N_{\gamma}}l_{\gamma},\ \sum\limits_{\beta=1}^{N_{\beta}}\sum\limits_{\beta=1}^{N_{\beta}}h_{\beta}l_ {\gamma}\right] \tag{9}\]  
where \(\bar{\epsilon}^{(\beta\gamma)}\) are components of the average subcell strains, and the matrix \(\rm A_{\rm G}\) contains geometrical parameters of each subcell.  
In order to determine the unknown subcell strains successfully, the interfacial traction continuity is necessary to be imposed to ensure enough equations, which can be written as follows,  
\[\bar{\sigma}_{2j}^{(\beta\gamma)}=\bar{\sigma}_{2j}^{(\beta\gamma)},\ \beta=1,\ldots,N_{\beta}-1,\gamma=1,\ldots,N_{\gamma} \tag{10}\]  
\[\bar{\sigma}_{3j}^{(\beta\gamma)}=\bar{\sigma}_{3j}^{(\beta\gamma)},\ \beta=1,\ldots,N_{\beta},\gamma=1,\ldots,N_{\gamma}-1 \tag{11}\]where \(\hat{\beta}\) and \(\hat{\gamma}\) are given by  
\[\begin{split}\widehat{\beta}&=\left\{\begin{array}{cc} \beta+1,&\beta<N_{\beta}\\ 1,&\beta<N_{\beta}\\ \gamma&=\left\{\begin{array}{cc}\gamma+1,&\gamma<N_{\gamma}\\ 1,&\gamma<N_{\gamma}\end{array}\right.\end{split} \tag{12}\]  
These traction conditions can be reformulated in matrix form in terms of the average subcell strains utilizing the subcell constitutive relationships,  
\[A_{M}\varepsilon_{\text{s}}=0 \tag{13}\]  
where the matrix \(A_{M}\) possesses the material properties in each subcell.