## 4 Analytical Methods
### Mori-Tanaka Approach  
The Mori-Tanaka theory is applied to determine the elastic/viscoelastic coefficients of the biphasic composite reinforced with long, anisotropic fibers [147]. The RUC(RVE) is used for this. It consists of a viscoelastic epoxy matrix reinforced with uniformly distributed, monotonically aligned, parallel fibers. In light of the aforementioned theory, the resulting composite is considered orthotropic. However, the composite body in question is considered transversely isotropic if the fibers of the elliptical cylinders are randomly oriented. The elliptical fibers are defined by the ratio (see Figure 6), which is 1 if the fibers are circular. [PERSON] and [PERSON] [148] combine [PERSON]'s [147] mean-field theory with [PERSON]'s [149] method for a cylindrical fiber with an elliptical section. The two phases are regarded as isotropic materials in [150; 151]. The fibers are thought to exhibit anisotropic behavior in our investigation. The current investigation's contribution to the research of these materials for determining the elastic/viscoelastic constants for this kind of composite is represented by this method. In practical applications, it is especially crucial to obtain engineering constants that are as close to reality as possible. All of the elastic constants required to characterize an orthotropic material, especially those that are  
Figure 6: Composite with aligned elliptic fibers. transversely isotropic, may be computed using the Mori-Tanaka method. By obtaining the creep curves, the compliance matrix may be constructed, enabling an experimental verification of the results. Now, let us look at a two-phase composite reinforced with fibers evenly distributed throughout the matrix. We use (CM) to indicate a comparative material. Examine both the CM and the RVE, the actual composite, under the same boundary traction \(\overline{\sigma}\). Boundary traction is a type of traction that is applied at the RVE's end, or boundary surface. There is a traction \(\overline{\sigma}\) observed since the loading condition for the RVE and the actual composite is the same. The matrix of elastic coefficients \(C_{m}\) and the fiber's elastic coefficients \(C_{f}\) is used. It is clear that the average strain field in the matrix and the average strain in (CM), where the mean stress is \(\overline{\sigma}\), are different under traction. Consider \(\widetilde{\epsilon}\) the difference between the average values of the strain. Between the two kinds of material, there is a different mean stress \(\widetilde{\sigma}\). In CM, the relationship between the mean strain field \(\varepsilon^{0}\) and the mean stress field \(\overline{\sigma}\) is as follows:  
\[\overline{\sigma}=C_{m}\varepsilon^{0}\enspace. \tag{76}\]  
The mean strain field in the composite is \(\varepsilon^{m}=\varepsilon^{0}+\overline{\epsilon}\) and the mean stress field is \(\sigma^{m}=\overline{\sigma}+\widetilde{\sigma}\). This results in  
\[\sigma^{m}=\overline{\sigma}+\widetilde{\sigma}=C_{m}(\varepsilon^{0}+ \overline{\epsilon})\enspace. \tag{77}\]  
The mean strain field in the fiber differs from that in the matrix through an additional term \(\varepsilon^{pt}\), so \(\varepsilon^{f}=\varepsilon^{m}+\varepsilon^{pt}=\varepsilon^{0}+\widetilde{ \epsilon}+\varepsilon^{pt}\). A similar mean stress field differs by the term \(\sigma^{pt}\): \(\sigma^{f}=\overline{\sigma}+\widetilde{\sigma}+\sigma^{pt}\). The stress-strain is as follows:  
\[\sigma^{f}=\overline{\sigma}+\widetilde{\sigma}+\sigma^{pt}=C_{f}(\varepsilon ^{0}+\widetilde{\epsilon}+\varepsilon^{pt})\enspace, \tag{78}\]  
or  
\[\sigma^{f}=\overline{\sigma}+\widetilde{\sigma}+\sigma^{pt}=C_{f}(\varepsilon ^{0}+\widetilde{\epsilon}+\varepsilon^{pt})=C_{m}(\varepsilon^{0}+\widetilde{ \epsilon}+\varepsilon^{pt}-\varepsilon^{*})\enspace, \tag{79}\]  
with  
\[\varepsilon^{pt}=P\varepsilon^{*}\enspace. \tag{80}\]  
\(P\) is the [PERSON]'s transformation tensor, where the symmetry property \(P_{ikjl}=P_{jikl}=P_{jikl}\) is valid [139].