## 4 Analytical Methods
### Mori-Tanaka Approach
This results in  
\[E_{11}=\frac{\bar{\varepsilon}_{11}^{0}}{\bar{\varepsilon}_{11}}E_{m}=\frac{E _{m}}{1+v_{f}[a_{11}-v_{m}(a_{12}+a_{13})]}\enspace. \tag{90}\]  
For the other directions, similarly, we obtain the following results for [PERSON]'s moduli:  
\[E_{22}=\frac{\bar{\varepsilon}_{22}^{0}}{\bar{\varepsilon}_{22}}E_{m}=\frac{E _{m}}{1+v_{f}[a_{22}-v_{m}(a_{21}+a_{23})]}\enspace. \tag{91}\]  
and  
\[E_{33}=\frac{\bar{\varepsilon}_{33}^{0}}{\bar{\varepsilon}_{33}}E_{m}=\frac{E _{m}}{1+v_{f}[a_{33}-v_{m}(a_{31}+a_{32})]}\enspace. \tag{92}\]  
The shear moduli can be obtained using the following relations:  
\[\overline{\sigma}_{12}=2G_{12}\bar{\varepsilon}_{12}\qquad;\quad\overline{ \sigma}_{12}=2G_{m}\bar{\varepsilon}_{12}^{0}\quad. \tag{93}\]Recall that  
\[\bar{\varepsilon}_{12}=\bar{v}_{12}^{0}+v_{f}\bar{\varepsilon}_{12}^{*}=\bar{ \varepsilon}_{12}^{*}-v_{f}\frac{G_{12,f}-G_{m}}{\Big{(}G_{12,f}-G_{m}\Big{)} \Big{(}2v_{m}P_{1212}+v_{f}\Big{)}+G_{m}}\bar{\varepsilon}_{12}^{0}\ . \tag{94}\]  
From Equation (93), \(G_{12}\) is obtained as follows:  
\[G_{12}=G_{m}\left(1+\frac{v_{f}}{\frac{G_{m}}{G_{12,f}-G_{m}}+2v_{m}P_{1212}} \right). \tag{95}\]  
In the same way, we have  
\[G_{23}=G_{m}\left(1+\frac{v_{f}}{\frac{G_{m}}{G_{23,f}-G_{m}}+2v_{m}P_{2323}} \right). \tag{96}\]  
and  
\[G_{31}=G_{m}\left(1+\frac{v_{f}}{\frac{G_{m}}{G_{31,f}-G_{m}}+2v_{m}P_{3131}} \right). \tag{97}\]  
For [PERSON]'s ratio, classic relations are used:  
\[\bar{\varepsilon}_{22}=-v_{m}\bar{v}_{11}\ \ \ \ \ \ \ \ \ ;\ \ \ \ \ \ \ \ \bar{v}_{22}^{0}=\bar{v}_{33}^{0}=-v_{m}\bar{v}_{11}^{0}\ . \tag{98}\]  
where  
\[ \begin{array}{l}\bar{\varepsilon}_{11}=\bar{\varepsilon}_{11}^{0}+v_{f} \bar{\varepsilon}_{11}^{*}=\bar{\varepsilon}_{11}^{0}+v_{f}a_{11}\bar{v}_{11}^ {0}+v_{f}a_{12}\bar{\varepsilon}_{22}^{0}+v_{f}a_{13}\bar{\varepsilon}_{33}^ {0}\\ =\bar{v}_{11}^{0}\Big{(}1+v_{f}a_{11}\Big{)}+v_{f}a_{12}\bar{\varepsilon}_{22}^ {0}+v_{f}a_{13}\bar{v}_{33}^{0}\end{array} \ . \tag{99}\]  
We obtain  
\[\bar{\varepsilon}_{22}=\bar{\varepsilon}_{22}^{0}+v_{f}\bar{\varepsilon}_{22 }^{*}=v_{f}a_{21}\bar{\varepsilon}_{11}^{0}+\bar{\varepsilon}_{22}^{0}\Big{(} 1+v_{f}a_{22}\Big{)}+v_{f}a_{23}\bar{v}_{33}^{0}\ . \tag{100}\]  
and this results in  
\[ \begin{array}{l}\bar{\varepsilon}_{11}=\Big{[}\Big{(}1+v_{f}a_{11}\Big{)}-v _{f}a_{12}v_{m}-v_{f}a_{13}v_{m}\Big{]}\bar{\varepsilon}_{11}^{0}\\ =\Big{[}v_{f}a_{21}-v_{m}\Big{(}1+v_{f}a_{22}\Big{)}-v_{m}v_{f}a_{23}\Big{]} \bar{v}_{11}^{0}\end{array} \ . \tag{101}\]  
Now, using Equation (87), we have  
\[v_{12}=-\frac{\bar{\varepsilon}_{22}}{\bar{\varepsilon}_{11}}=-\frac{v_{f}a_{2 1}-v_{m}\Big{(}1+v_{f}a_{22}\Big{)}-v_{m}v_{f}a_{23}}{1+v_{f}a_{11}-v_{f}a_{12 }v_{m}-v_{f}a_{13}v_{m}}\ . \tag{102}\]  
or  
\[v_{12}=\frac{v_{m}-v_{f}[a_{22}-v_{m}(a_{21}+a_{23})]}{1+v_{f}[a_{11}-v_{m}(a_ {12}+a_{13})]}. \tag{103}\]  
In a similar way, the following is obtained:  
\[v_{23}=\frac{v_{m}-v_{f}[a_{22}-v_{m}(a_{21}+a_{23})]}{1+v_{f}[a_{33}-v_{m}(a_ {7}+a_{8})]}\ . \tag{104}\]  
and  
\[v_{31}=\frac{v_{m}-v_{f}[a_{33}-v_{m}(a_{31}+a_{32})]}{1+v_{f}[a_{11}-v_{m}(a_ {12}+a_{13})]}. \tag{105}\]This study presents a method that uses uniquely determined values to provide the elastic/viscoelastic constants of the composite. The bounded methods that are often employed to determine these values provide the lower and upper bounds. The method's uniqueness depends on the fibres' anisotropy or transverse isotropy behavior. [PERSON]'s constitutive equation [80] is used to assess the material's temporal response.