## 2 Micro- and Macro-Mechanical Analysis Models for Fiber-Reinforced Composites  
As fiber-reinforced composite laminates are widely applied in the fields of aerospace and aviation, their mechanical performance prediction is very important and meaningful. In this section, several kinds of micro-mechanical mathematical models will be introduced to model elastic mechanical properties of fiber-reinforced composites. As is known to all, a composite laminate is a stack of layers of unidirectional fiber-reinforced laminae, which are generally made of fibers and a matrix, which are two different constituent materials. Properties of composite laminates depend on the properties of the constituent materials, distributions, py orientations and other material parameters. As the basic unit of composite material mechanics, consider a typical unidirectional fiber-reinforced composite lamina as shown in Figure 1, where a coordinate system is used with the 1-axis aligned parallel to the axis of the fibers and the 2-3 plane is transversely isotropic. The 3-axis denotes the thickness direction. Figure 1: Sketch map of a typical fiber-reinforced composite lamina [39]. According to the theory of composite materials mechanics, one can make good use of the micro-mechanical analysis method to predict macro elastic mechanical parameters of composite laminates, where the fibers can be transversely isotropic and the matrix material is considered to be isotropic. As a result, the fiber-reinforced composite lamina can be viewed as an orthotropic anisotropic material having three material principal axes coinciding with those of the fibers, which have four independent elastic mechanical parameters including \(E_{1},E_{2},v_{12}\) and \(G_{12}\). Here, \(E_{1}\) is the longitudinal tension modulus and \(E_{2}\) is the transverse tension modulus. \(v_{12}\) is the Poisson's ratio and \(G_{12}\) is the shear modulus. The micro-scale quantification analysis models can mainly consist of the series model and the parallel model [40], whose sketch maps are as shown in Figure 2. As the series model says, the following quantification formulas can be obtained as  
\[ \begin{split} E_{1}=V_{f}E_{11}^{f}+V_{m}E_{m},E_{2}=\frac{E_{2}^ {f}E_{m}}{V_{m}E_{12}^{f}+V_{f}E_{m}}\\ v_{12}=V_{f}v_{f}+V_{m}v_{m},G_{12}=\frac{G_{f}G_{m}}{V_{m}G_{f} +V_{f}G_{m}}\end{split}  \tag{1}\]  
where the analytical relationships between material properties of composite micro constitutes and macro elastic mechanical parameters can be established. The symbols \(f\) and \(m\) separately represent the fibers and matrix. \(E_{11}^{f}\) and \(E_{22}^{f}\) are [PERSON]'s tension moduli in the longitudinal and in-plane transverse orientations of fibers, respectively. \(V_{f}\) and \(V_{m}\) are the volume fractions of the fibers and matrix, respectively. \(v_{f}\) and \(v_{m}\) are [PERSON]'s ratios of the fibers and matrix, respectively. \(G_{f}\) and \(G_{m}\) are shear moduli of the fibers and matrix, respectively. When the void volume fraction is taken into account [41; 42], one can also obtain the following formula as:  
\[V_{f}+V_{m}+V_{V}=1 \tag{2}\]  
in which \(V_{V}\) is the void volume fraction. On the other hand, as the parallel model says, the following quantification formulas can be obtained as:  
\[ \begin{split} E_{1}=V_{f}E_{11}^{f}+V_{m}E_{m},E_{2}=E_{1}\\ v_{12}=\frac{v_{f}V_{f}E_{11}^{f}+v_{m}V_{m}E_{m}}{V_{f}E_{11}^{f}+V _{m}E_{m}},G_{12}=V_{m}G_{m}+V_{f}G_{f}\end{split}  \tag{3}\]  
For the sake of predicting macro elastic mechanical properties of composite laminates more precisely, some researchers suggest combining the series model with the parallel model [27]. The corresponding formulas can be achieved as  
\[ \begin{split} E_{1}=V_{f}E_{11}^{f}+V_{m}E_{m},E_{2}=(1-C)E_{2}^ {I}+CE_{2}^{II}\\ v_{12}=(1-C)v_{12}^{I}+Cv_{12}^{II},G_{12}=(1-C)G_{12}^{I}+CG_{12}^{ II}\end{split}  \tag{4}\]  
Figure 2: Sketch maps of the series model and parallel model for the lamina. Here, the superscripts I and II represent the series model and parallel model, respectively.