## 3 Micromechanical Analysis
### Micromechanical Methods
#### 3.2.2 Average Stresses in RUC  
Having volume \(V\) and a rectangular parallel-piped shape, the composite specimen in question has edges parallel to the coordinate axes (\(X_{1}\), \(X_{2}\), \(X_{3}\)). Calculating the average stress \(\overline{\sigma}_{ij}\) in RUC is the issue that will be covered next [93; 94; 95; 96; 97]. This is a sample of the entire composite that is typical of the bulk material from a structural perspective, while also having a suitable amount of the constituent materials. As a result, the characteristics of the complete composite are defined by the characteristics of the RUC. The average of the stress distribution over the volume is determined using the following equation:  
\[\overline{\sigma}_{ij}=\frac{1}{V}\int\limits_{V}S_{ij}dV. \tag{22}\]  
Considering a RUC as a one-quarter of a fiber in the composite square matrix, we obtain the following for (22):  
\[\overline{\sigma}_{ij}=\frac{1}{A}\left(\overline{S}_{ij}^{(f)}A_{f}+ \overline{S}_{ij}^{(m)}A_{m}\right). \tag{23}\]  
In Equation (23), \(\overline{S}_{ij}^{(\lambda)}\) is the average stresses in RUC, A is the total area of the cell \(A\), and \(A_{\lambda}\) is the area of the subcells. For the sake of simplicity in computation, a unit depth of the composite is considered, i.e., V = \(A\times 1\). For the case presented here, when the fibers are circular, it results in the following:  
\[A=(R+h/2)^{2};\ A_{f}=\pi R^{2}/4\ ;\ A_{m}=(R+h/2)^{2}-\pi R^{2}/4 \tag{24}\]  
We obtain  
\[\overline{\sigma}_{ij}=\frac{1}{\left(R+\frac{h}{2}\right)^{2}}\left\{\frac{ \pi R^{2}}{4}\overline{S}_{ij}^{(f)}+\left[\left(R+\frac{h}{2}\right)^{2}- \frac{\pi R^{2}}{4}\right]\overline{S}_{ij}^{(m)}\right\} \tag{25}\]  
The average stress in fiber and matrix is as follows:  
\[\overline{S}_{ij}^{(\lambda)}=\frac{1}{A_{v}}\int\limits_{A}\sigma_{ij}^{( \lambda)}dA=\frac{1}{A_{v}}\iint\sigma_{ij}^{(\lambda)}dx_{2}dx_{3} \tag{26}\]  
Changing the Cartesian coordinates to polar coordinates, the Jacobian is as follows:  
\[J=\frac{\partial(x_{2},x_{3})}{\partial(r,\theta)}= \begin{vmatrix}\cos\theta &\sin\theta\\ -r\sin\theta&r\cos\theta\end{vmatrix} =r \tag{27}\]  
For subcell \"\(f\)\", Equation (26) has the following form in polar coordinates:  
\[\overline{S}_{ij}^{(f)}=\frac{4}{\pi R^{2}}\int_{0}^{\pi/2}\int_{0}^{R} \sigma_{ij}^{(f)}rdrd\theta \tag{28}\]  
In Equation (28), \(\sigma_{ij}^{(f)}\) represents the classic Hook law of the transversely isotropic material. Using Equation (18) in the Hook law, we have  
\[\overline{S}_{ij}^{(f)}= \begin{cases}C_{11}\varepsilon_{11}^{(f)}+C_{12} \left(\varepsilon_{22}^{(f)}+\varepsilon_{33}^{(f)}\right)\\ C_{12}\varepsilon_{11}^{(f)}+C_{22}\varepsilon_{22}^{(f)}+C_{23}\varepsilon_{33 }^{(f)}\\ C_{12}\varepsilon_{11}^{(f)}+C_{23}\varepsilon_{22}^{(f)}+C_{22}\varepsilon_{3 3}^{(f)}\\ C_{66}\varepsilon_{23}^{(f)}\\ C_{44}\gamma_{31}^{(f)}\\ C_{44}\gamma_{12}^{(f)}\end{cases} = \begin{cases}C_{11}\frac{\partial u_{1}^{ (f)}}{\partial x_{1}}+C_{12}\left(\varepsilon_{2}^{(f)}+\varepsilon_{3}^{(f )}\right)\\ C_{12}\frac{\partial u_{1}^{(f)}}{\partial x_{1}}+C_{22}\varepsilon_{2}^{(f)}+ C_{23}\varepsilon_{3}^{(f)}\\ C_{12}\frac{\partial u_{1}^{(f)}}{\partial x_{1}}+C_{23}\varepsilon_{2}^{(f)}+ C_{22}\varepsilon_{3}^{(f)}\\ C_{66}\left[\varepsilon_{3}^{(f)}+\varepsilon_{2}^{(f)}\right]\\ C_{44}\left[\frac{\partial u_{1}^{(f)}}{\partial x_{1}}+\varepsilon_{1}^{(f )}\right]\\ C_{44}\left[\frac{\partial u_{2}^{(f)}}{\partial x_{1}}+\varepsilon_{1}^{(f )}\right]\end{cases}  \tag{29}\]Now, we apply the same procedure to determine the average stresses for subcell \"\(m\)\".