## 3 Micromechanical Analysis
### Micromechanical Methods
#### 3.2.4 Average Strains in RUC  
The average of the internal strain over volume is expressed as follows:  
\[\bar{\varepsilon}_{ij}=\frac{1}{V}\int_{V}\varepsilon_{ij}dV, \tag{46}\]  
and, for a representative cell,  
\[\bar{\varepsilon}_{ij}=\frac{1}{A}\underset{\lambda=f,m}{\sum}\bar{\varepsilon }_{ij}^{(\lambda)}A_{\lambda}\bar{\varepsilon}_{ij}=\frac{1}{A}\underset{ \lambda=f,m}{\sum}\bar{\varepsilon}_{ij}^{(\lambda)}A_{\lambda}, \tag{47}\]  
where notations \(\bar{\varepsilon}_{ij}^{(\lambda)}\) and \(A_{\lambda}\) are the same as before. Using Equations (16) and (18) for \(i=j=1\), the following is obtained:  
\[\bar{\varepsilon}_{ij}^{(\lambda)}=\frac{\partial u_{1}^{o}}{\partial X_{1}}. \tag{48}\]  
Considering (24), we have  
\[\bar{\varepsilon}_{11}=\frac{1}{\left(R+\frac{h}{2}\right)^{2}}\Bigg{\{}\frac {\pi R^{2}}{4}\frac{\partial u_{1}^{o}}{\partial X_{1}}+\left[\left(R+\frac{h }{2}\right)^{2}-\frac{\pi R^{2}}{4}\right]\!\frac{\partial u_{1}^{o}}{ \partial X_{1}}\Bigg{\}}, \tag{49}\]  
or, after some calculus,  
\[\bar{\varepsilon}_{11}=\frac{\partial u_{1}^{o}}{\partial X_{1}}. \tag{50}\]  
Now that all the equations required to anticipate the composite's overall behavior have been established, it is reasonable to draw out instances to assess the analysis's applicability. As was previously indicated, the specimen undergoes a complex three-axial condition of stress even in a uniaxial test. In order to allow the stress-dependent material properties\(g_{\infty},g_{1},g_{2}\) and \(a_{\sigma}\) to rely on a single invariant instead of the applied stress provided by Equations (25) and (26), it is appropriate to have an octahedral shear stress in the matrix. The octahedral shear stress in the matrix is introduced as this so-called equivalent stress. Since the analysis is performed at the microlevel, the same analogy may be applied here to let the material properties be a function of \(S_{oct}^{(m)}\), an equivalent stress in the subcell \"m\" defined below. \[S_{oct}^{(m)}=\left\{\frac{1}{2}\bigg{[}\left(\overline{S}_{11}^{(m)}- \overline{S}_{22}^{(m)}\right)^{2}+\left(\overline{S}_{22}^{(m)}-\overline{S} _{33}^{(m)}\right)^{2}+\left(\overline{S}_{33}^{(m)}-\overline{S}_{11}^{(m)} \right)^{2}\bigg{]}+3\bigg{[}\left(\overline{S}_{12}^{(m)}\right)^{2}+\left( \overline{S}_{23}^{(m)}\right)^{2}+\left(\overline{S}_{31}^{(m)}\right)^{2} \bigg{]}\bigg{\}}\right\}^{\frac{1}{2}} \tag{51}\]  
It should be mentioned that the process used to solve the unknowns is gradual in nature. This implies that the process is carried out again until the variables of interest are identified at the desired time, and that the unknown variables are assessed gradually over time. However, a set of thirteen algebraic equations must be solved at each time step. The equations contain unknown variables: the stresses \(\overline{S}_{11}^{(\lambda)}\), \(\overline{S}_{22}^{(\lambda)}\), and \(\overline{S}_{33}^{(\lambda)}\) (six unknowns); the microvariables \(\xi_{2}^{(\lambda)}\) and \(\zeta_{3}^{(\lambda)}\) (four unknowns); the average strains \(\overline{\varepsilon}_{11}\), \(\overline{\varepsilon}_{22}\), and \(\overline{\varepsilon}_{33}\) (three unknowns). As previously stated, a relationship between the fiber's stress in the representative unit cell and the average stress in the matrix is sought. It should be noted that [PERSON]'s model [98, 99] may oversimplify the actual situation by assuming that the neighbouring subcells of the representative unit cell (RUC) have equal average stresses. In practice, the unit cell's two phases may experience differing average stresses and strains [100, 101, 102, 103, 104].