## 3 Micromechanical Analysis
### Micromechanical Methods  
#### 3.2.1 Model  
In this section, the properties of the various components, such as the fiber and matrix, as well as their interactions, are used to derive the overall composite properties in a micromechanical analysis. In fact, the fibers in a unidirectional composite are dispersed at random throughout the matrix. Nonetheless, it makes sense to take into account some periodicity in the fiber distribution. There are many models for different fiber arrangements. For our purposes, we present a standard model, from which it is possible to develop more complicated arrangements. This model is made simpler by periodicity and symmetry. The square array in Figure 1 illustrates this circumstance. For composites reinforced with fibers, the following assumptions are considered valuable:The \(X_{2}\)-\(X_{3}\) plane has a rectangular array of fibers that are expanded in the \(X_{1}\) direction. According to our presentation, the polymeric matrix is nonlinearly viscoelastic but isotropic, and the fibers are linearly elastic and anisotropic. Indeed, these presumptions may be changed in the case of the nonlinear material. Under load, no cracks or holes may form since the interaction between the fibers and matrix is only mechanical. The problem's size and complexity are decreased by the aforementioned regular and periodic packing, which enables the study of a representative cell. Note that one of the recurring units of the periodical pattern depicted in Figure 1 is displayed in Figure 2a. The fiber is thought to have a transversal section that is circular. As seen in Figure 2b, the representative unit cell (RUC) (or representative volume element--RVE) can therefore be made up of two subcells, each representing a quarter of a fiber and the corresponding matrix material. In addition, the unit cells are thought to be tiny in relation to the composite's overall dimensions. For each subcell in the representative cell, a local reference frame (\(X_{1}\), \(x_{2}^{(\lambda)}\), \(x_{3}^{(\lambda)}\)), with its origin located as shown in Figure 2 is introduced. \"\(\lambda\)\" is used to symbolically represent both fiber (\(\lambda=f\)) and matrix (\(\lambda=m\)) constituents. Furthermore, the displacement in each of the two subcells is expressed in a first-order linear expansion as a function of the distance from the origin to the local coordinate axes [27,28]. This means that the displacement at any point in the subcell is as follows:  
\[u_{i}^{(\lambda)}=u_{i}^{o(\lambda)}+x_{2}^{(\lambda)}\xi_{i}^{(\lambda)}+x_{ 3}^{(\lambda)}\xi_{i}^{(\lambda)}\ ;\ i=1,2,3\ . \tag{16}\]  
Figure 1: Square fiber packing geometry of a unidirectional composite. Figure 2: The repeating cell of a square fiber array (**a**) and the representative unit cell (RUC) with the local coordinates (**b**). With \(u_{i}^{o}\), the component \(j\) of the displacement of the origin of the local reference frame is noted along the xi axis. \(\xi_{i}^{(\lambda)}\) and \(\zeta_{i}^{(\lambda)}\) are coefficients representing the linear dependence on \(x_{2}^{(\lambda)}\) and \(x_{3}^{(\lambda)}\), respectively. The aim of this presentation is to establish the relationship between the average stresses and average strains in the composite. These relationships will offer us the coefficient of the overall constitutive law. The strain-displacement relationships of anisotropic bodies yield the following:  
\[\epsilon_{ij}^{(\lambda)}=\frac{1}{2}\left(\frac{\partial u_{i}^{(\lambda)}}{ \partial x_{j}}+\frac{\partial u_{j}^{(\lambda)}}{\partial x_{i}}\right)\,;\, i,j=1,2,3 \tag{17}\]  
when \(i\
eq j\), it can be used in the notation \(\gamma_{ij}^{(\lambda)}=2\epsilon_{ij}^{(\lambda)}\); \(i,j=1,2,3\) ; \(i\
eq j\); for the engineering shear strain. Substituting Equation (16) into Equation (17), we obtain  
\[