## 4 Analytical Methods
### Mori-Tanaka Approach
The average stress in the whole RVE is as follows:  
\[ \begin{array}{l}\overline{\sigma}=v_{f}\sigma_{f}+v_{m}\sigma_{m}=v_{f}( \overline{\sigma}+\widetilde{\sigma}+\sigma^{pt})+v_{m}(\overline{\sigma}+ \widetilde{\sigma})\\ =(v_{f}+v_{m})\overline{\sigma}+(v_{f}+v_{m})\widetilde{\sigma}+v_{f}\sigma^{ pt}=\overline{\sigma}+\widetilde{\sigma}+v_{f}\sigma^{pt}\enspace.\end{array}  \tag{81}\]  
reduced to  
\[\widetilde{\sigma}=-v_{f}\sigma^{pt}\enspace. \tag{82}\]  
For the strain, a similar procedure offers the follows:  
\[\overline{\epsilon}=-v_{f}(\varepsilon^{pt}-\varepsilon^{*})=-v_{f}(P\varepsilon ^{*}-\varepsilon^{*})=-v_{f}(P-I)\varepsilon^{*}\enspace. \tag{83}\]  
Here, \(I\) is denoted as the unit tensor. Substituting Equation (83) into Equation (79), we have  
\[C_{f}\Big{[}\varepsilon^{0}-v_{f}(P-I)\varepsilon^{*}+P\varepsilon^{*}\Big{]} =C_{m}\Big{[}\varepsilon^{0}-v_{f}(P-I)\varepsilon^{*}+P\varepsilon^{*}- \varepsilon^{*}\Big{]}\enspace. \tag{84}\]After some calculus, this is reduced to  
\[\Big{[}C_{f}\Big{(}-v_{f}(P-I)+P\Big{)}+C_{m}\Big{(}v_{f}(P-I)-P+I\Big{)}\Big{]} \varepsilon^{*}+\Big{(}C_{f}-C_{m}\Big{)}\varepsilon^{0}=0\enspace. \tag{85}\]  
and finally, it is reduced to  
\[\Big{[}\Big{(}C_{f}-C_{m}\Big{)}\Big{(}v_{m}P+v_{f}I\Big{)}+C_{m}\Big{]} \varepsilon^{*}+\Big{(}C_{f}-C_{m}\Big{)}\varepsilon^{0}=0\enspace. \tag{86}\]  
Using Equation (80), we have  
\[ \begin{array}{l}\varepsilon_{1}^{*}=\frac{1}{A}\big{(}A_{11}\varepsilon_{11} ^{0}+A_{12}\varepsilon_{22}^{0}+A_{13}\varepsilon_{33}^{0}\big{)}\enspace;\\ \varepsilon_{22}^{*}=\frac{1}{A}\big{(}A_{21}\varepsilon_{11}^{0}+A_{22} \varepsilon_{22}^{0}+A_{23}\varepsilon_{33}^{0}\big{)}\enspace;\\ \varepsilon_{33}^{*}=\frac{1}{A}\big{(}A_{31}\varepsilon_{11}^{0}+A_{32} \varepsilon_{22}^{0}+A_{33}\varepsilon_{33}^{0}\big{)}\enspace.\end{array}  \tag{87}\]  
where \(A_{ij}\) can be found in literature (i.e., [139]). Considering the definitions for the shear strain, the following is obtained [139]:  
\[ \begin{array}{l}\varepsilon_{12}^{*}=\frac{\big{(}C_{12,f}-C_{m}\big{)}}{ \big{(}C_{12,f}-C_{m}\big{)}\big{(}2v_{m}P_{122}+v_{f}\big{)}+C_{m}} \varepsilon_{12}^{0};\\ \varepsilon_{23}^{*}=\frac{\big{(}C_{23,f}-C_{m}\big{)}}{\big{(}C_{23,f}-C_{m }\big{)}\big{(}2v_{m}P_{232}+v_{f}\big{)}+C_{m}}\varepsilon_{23}^{0};\\ \varepsilon_{31}^{*}=\frac{\big{(}C_{31,f}-C_{m}\big{)}}{\big{(}C_{31,f}-C_{m }\big{)}\big{(}2v_{m}P_{331}+v_{f}\big{)}+C_{m}}\varepsilon_{31}^{0}.\end{array}  \tag{88}\]  
We subject the examined composite and the comparison material to a pure traction \(\overline{\sigma}_{11}\) in order to calculate the longitudinal Young's modulus \(E_{m}\) of an orthotropic body. Then, it follows that \(\overline{\sigma}_{11}=E_{11}\bar{\varepsilon}_{11}\) for the composite and \(\overline{\sigma}_{11}=E_{m}\overline{\varepsilon}_{11}^{0}\); \(\bar{\varepsilon}_{22}^{0}=\bar{\varepsilon}_{33}^{0}=-v_{m}\bar{\varepsilon }_{11}^{0}\) if the comparison material is considered. Equation (87) offers th following:  
\[ \begin{array}{l}\bar{\varepsilon}_{11}=\bar{\varepsilon}_{11}^{0}+v_{f} \bar{\varepsilon}_{11}^{*}=\bar{\varepsilon}_{11}^{0}+v_{f}\Big{(}\frac{A_{11} }{A}\bar{\varepsilon}_{11}^{0}+\frac{A_{12}}{A}\bar{\varepsilon}_{22}^{0}+ \frac{A_{13}}{A}\bar{\varepsilon}_{33}^{0}\Big{)}\\ \phantom{\bar{\varepsilon}_{11}^{0}}=\bar{\varepsilon}_{11}^{0}\Big{(}1+v_{f} a_{11}\Big{)}-v_{f}a_{12}v_{m}\bar{\varepsilon}_{11}^{0}-v_{f}a_{13}v_{m}\bar{ \varepsilon}_{11}^{0}\\ \phantom{\bar{\varepsilon}_{11}^{0}}=\bar{\varepsilon}_{11}^{0}\Big{[}1+v_{f}[ a_{11}-v_{m}(a_{12}+a_{13})]\Big{]}\end{array} \enspace. \tag{89}\]  
Here, \(a_{ij}=A_{ij}/A\), with \(A_{ij}\) and \(A\) defined in [139].