## 2 Materials and Methods
### Fiber Structure Analysis and Generating Mechanical Properties Method
#### 2.4.2 Mechanical Properties Method  
By averaging the strain in the structure, it is impossible to obtain the [PERSON]'s law for the general anisotropic case. \[\sigma_{ij}=\sum_{r,s=1}^{3}c_{ijrs}\epsilon_{rs}\;i,j\;\in\{1,2,3\} \tag{1}\]  
Here, the tensor \(c\) is a symmetric fourth-order tensor and is referred to as the elasticity tensor. Typically, the elasticity tensor, stresses, and strains are written in Voigt notation. In this method, the elasticity tensor takes the form of a 6 \(\times\) 6 matrix, and the stresses and strains can be expressed as 6 \(\times\) 1 vectors. In Table 6, symbols for representing the stresses of each component are organized. Subscripts x, y, and z signify the geometric x-axis, y-axis, and z-axis, respectively. The stress tensor \(\sigma_{ij}\) and the strain \(\epsilon_{ij}\) can be expressed in Voigt notation. \[\sigma= \begin{pmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{pmatrix} \epsilon= \begin{pmatrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{31}\\ \epsilon_{12}\end{pmatrix}  \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Fiber Orientation Components**} & \multicolumn{4}{c}{**Fiber Orientation**} \\ \cline{2-6} \multicolumn{2}{c}{**by Layer (vol.\%)**} & \multicolumn{2}{c}{**X-Axis (\%)**} & \multicolumn{2}{c}{**Y-Axis (\%)**} & \multicolumn{2}{c}{**Z-Axis (\%)**} & \multicolumn{1}{c}{**V\({}_{\text{f}}\) (\%)**} \\ \hline \multirow{4}{*}{Micro-CT analysis (Core layer)} & 1 st Layer & 63.0477 & 17.6956 & 19.2567 & 31.8003 \\ \cline{2-6}  & 2 nd Layer & 66.1844 & 18.2655 & 15.5502 & 31.9499 \\ \cline{2-6}  & 3 rd Layer & 70.9015 & 15.0393 & 14.0592 & 32.2798 \\ \cline{2-6}  & 4 th Layer & 76.1767 & 11.3675 & 12.4558 & 32.0796 \\ \hline \multirow{4}{*}{Micro-CT analysis (Surface layer)} & 1 st Layer & 81.4451 & 7.7954 & 10.7565 & 32.7301 \\ \cline{2-6}  & 2 nd Layer & 79.8369 & 7.9518 & 12.2113 & 32.4027 \\ \cline{1-1} \cline{2-6}  & 3 rd Layer & 80.0405 & 7.8215 & 12.1378 & 32.3877 \\ \cline{1-1} \cline{2-6}  & 4 th Layer & 81.2374 & 7.4823 & 11.2802 & 32.7593 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fiber orientation tensors and fiber volume fraction of central layer and surface layer. Accordingly, [PERSON]'s law can be expressed as a matrix vector product. \[ \begin{pmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{pmatrix} = \begin{pmatrix}c_{11}&c_{12}&c_{13}&c_{14}&c_{15}&c_{ 16}\\ c_{12}&c_{22}&c_{23}&c_{24}&c_{25}&c_{26}\\ c_{13}&c_{23}&c_{33}&c_{34}&c_{35}&c_{36}\\ c_{14}&c_{24}&c_{34}&c_{44}&c_{45}&c_{46}\\ c_{15}&c_{25}&c_{35}&c_{45}&c_{55}&c_{56}\\ c_{15}&c_{26}&c_{36}&c_{46}&c_{56}&c_{66}\end{pmatrix}\begin{pmatrix}\epsilon _{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{31}\\ \epsilon_{12}\end{pmatrix}  \tag{3}\]  
This is the stress-strain relationship of an anisotropic material where there is no symmetry between the material properties. Considering SFRP composite materials as orthotropic materials, if the properties are symmetric with respect to each symmetrical plane for \(E_{1}\), \(E_{2}\) and \(E_{3}\), the general form of the stress-strain relationship for an orthotropic material can be expressed as follows. \[ \begin{pmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{pmatrix} = \begin{pmatrix}c_{11}&c_{12}&c_{13}&0&0&0\\ c_{12}&c_{22}&c_{23}&0&0&0\\ c_{13}&c_{23}&c_{33}&0&0&0\\ 0&0&0&c_{44}&0&0\\ 0&0&0&0&c_{55}&0\\ 0&0&0&0&c_{66}\end{pmatrix}\begin{pmatrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{23}\\ \epsilon_{31}\\ \epsilon_{12}\end{pmatrix}