## 6 Finite Element Method  
### FEM Used to Determine the Field of Strain and Stress in a RUC  
The FEM is a powerful tool to study the response of a composite materials [225; 226; 227; 228; 229; 230; 231]. We do not insist on presenting the basics of the method. FEM is instead presented in conjunction with the study of composites (Figure 8). Consider an FE model where the boundary conditions are taken as \(u_{i}=\alpha_{ij}x_{j}\) (where \(\alpha_{ij}=\alpha_{ji}\)). The average strains computed for such behavior of the material are \(\overline{z}_{ij}=\alpha_{ij}\). This results in the following:  
\[\overline{z}_{ij}=\frac{1}{V}\int_{\Gamma}\epsilon_{ij}dV= \frac{1}{2V}\int_{\Gamma}\left(\frac{\partial u_{j}}{\partial x_{i}}+\frac{ \partial u_{i}}{\partial x_{j}}\right)dV. \tag{111}\]  
Using [PERSON]'s theorem, we have  
\[ \begin{split}&\overline{z}_{ij}=\frac{1}{2V}\int_{\partial \Gamma}\big{(}n_{i}u_{j}+n_{j}u_{i}\big{)}ds=\\ &=\frac{1}{2V}\Big{(}\int_{\partial\Gamma}n_{i}\alpha_{jk}x_{k}ds +\int_{\partial\Gamma}n_{j}\alpha_{li}x_{l}ds\Big{)}=\\ &=\frac{1}{2V}\Big{(}\alpha_{jk}\int_{\partial\Gamma}n_{i}x_{k} ds+\alpha_{il}\int_{\partial\Gamma}n_{j}x_{l}ds\Big{)}.\end{split}  \tag{112}\]  
Figure 8: Composite material with fiber and FEM model or RUC. In our case, it can be written as follows:  
\[ \begin{split}\bar{z}_{ij}=\frac{1}{2V}\left(\alpha_{jk}\int_{\Gamma} \frac{\partial x_{k}}{\partial x_{i}}dV+\alpha_{il}\int_{\Gamma}\frac{\partial x _{l}}{\partial x_{j}}dV\right)=\\ =\frac{1}{2V}\left(\alpha_{jk}\int_{\Gamma}\delta_{kl}dV+\alpha_ {il}\int_{\Gamma}\delta_{lj}dV\right)=\frac{1}{2V}\left(\alpha_{ji}+\alpha_{ ij}\right)=\alpha_{ij}\end{split}  \tag{113}\]  
The FEM can be used to obtain average values of strains and stresses, viz. \(\overline{\sigma}_{22},\overline{\sigma}_{33},\overline{\sigma}_{11}\), \(\overline{\sigma}_{23}=\overline{\tau}_{23},\bar{\varepsilon}_{22},\bar{ \varepsilon}_{33},\bar{\varepsilon}_{11},\bar{\varepsilon}_{23}=1/2\, \gamma_{23}\), considering the case of plane strain. Using these averages, the constants of the unidirectional composite can be evaluated. The rule of mixture is applied. To obtain the longitudinal elastic modulus \(E_{11}\), the rule of the mixture is used:  
\[E_{11}=E_{f}\upsilon_{f}+E_{m}\upsilon_{m} \tag{114}\]  
where  
\[\upsilon_{f}=\frac{A_{f}}{A}\ ;\ \upsilon_{f}=\frac{A_{m}}{A}\, \tag{115}\]  
and \(A=A_{f}+A_{m}\). \(A_{f}\) is denoted as the cross-section of the fiber, and \(A_{m}\) is the cross-section of the matrix. In the plane strain case, the constitutive law is considered:  
\[ \begin{split}\overline{\sigma}_{22}=C_{22}\bar{\varepsilon}_{22} +C_{23}\bar{\varepsilon}_{33};\\ \overline{\sigma}_{33}=C_{23}\bar{\varepsilon}_{22}+C_{22}\bar{ \varepsilon}_{33};\\ \overline{\sigma}_{11}=C_{12}(\bar{\varepsilon}_{22}+\bar{ \varepsilon}_{33});\\ \overline{\tau}_{23}=C_{66}\,\overline{\tau}_{23},\end{split}  \tag{116}\]  
or  
\[ \begin{bmatrix}\bar{\varepsilon}_{22}&\bar{\varepsilon}_{33}\\ \bar{\varepsilon}_{33}&\bar{\varepsilon}_{22}\end{bmatrix}\begin{cases}C_{2 }\\ C_{23}\end{cases} = \begin{cases}\overline{\sigma}_{22}\\ \overline{\sigma}_{33}\end{cases}  \tag{117}\]  
from which we obtain  
\[ \begin{cases}C_{22}\\ C_{23}\end{cases} =\frac{1}{\bar{\varepsilon}_{22}^{2}-\bar{\varepsilon}_{33} ^{2}} \begin{bmatrix}\bar{\varepsilon}_{22}&-\bar{\varepsilon}_{33}\\ -\bar{\varepsilon}_{33}&\bar{\varepsilon}_{22}\end{bmatrix}\begin{cases} \overline{\sigma}_{22}\\ \overline{\sigma}_{33}\end{cases} .