## 2 Materials and Methods
### Fiber Structure Analysis and Generating Mechanical Properties Method
#### 2.4.2 Mechanical Properties Method
 \tag{4}\]  
The mechanical properties can be represented by \(C_{ij}\), and in finite element analysis, if the values of \(E_{1}\), \(E_{2}\), \(E_{3}\), \(G_{12}\), \(G_{23}\), \(G_{31}\), and \(v_{ij}\) (\(i,j=1,\,2,\,3\)) are known for each element, the relationship between the components of the general anisotropic elastic tensor and the anisotropic engineering constant is as follows:  
\[C_{11}=\frac{1-v_{23}v_{32}}{E_{2}E_{3}\Delta},\;C_{22}=\frac{1-v_{13}v_{31}} {E_{1}E_{3}\Delta},\;C_{33}=\frac{1-v_{12}v_{21}}{E_{1}E_{2}\Delta},\]  
\[C_{12}=\frac{v_{12}+v_{31}v_{23}}{E_{2}E_{3}\Delta}=\frac{v_{12}+v_{32}v_{13}} {E_{1}E_{3}\Delta}\]  
\[C_{23}=\frac{v_{12}+v_{31}v_{23}}{E_{2}E_{3}\Delta}=\frac{v_{12}+v_{32}v_{13}} {E_{1}E_{3}\Delta}\]  
\[C_{13}=\frac{v_{12}+v_{31}v_{23}}{E_{2}E_{3}\Delta}=\frac{v_{12}+v_{32}v_{13}} {E_{1}E_{3}\Delta}\]  
\[C_{44}=G_{23}\;,\;C_{55}=G_{31},\;C_{66}=G_{12}\]  
where \(\Delta=\frac{1-v_{12}v_{12}-v_{12}v_{23}-v_{11}v_{12}-v_{21}v_{23}v_{13}}{E_{1 }E_{2}E_{3}}\). To define the properties of an orthotropic material, nine elasticity tensors are required, namely \(C_{11}\), \(C_{22}\), \(C_{33}\), \(C_{44}\), \(C_{55}\), \(C_{66}\), \(C_{12}\), \(C_{13}\), \(C_{23}\)[19; 20]. In finite element analysis, mechanical property characteristic values (\(E_{1}\), \(E_{2}\), \(E_{3}\), \(G_{12}\), \(G_{23}\), \(G_{31}\), \(v_{ij}\) (\(i,j=1,\,2,\,3\))) were predicted using the orientation distribution of glass fibers obtained from injection molding process simulation results at positions corresponding to each element, utilizing GEODICT. These predicted mechanical property values were then used in structural analysis. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Tensor notationindex(ij)** & 11 & 22 & 33 & 23, 32 & 31, 13 & 12, 21 \\ \hline \multirow{2}{*}{**Voigt notationindex**} & xx & yy & zz & yz & zx & xy \\ \cline{2-6}  & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Index assignment. RVE modeling was conducted using the FiberGeo module of the GeoDict software. The key parameters include fiber diameter, fiber length, fiber volume fraction, and the fiber orientation tensor values. In the injection molding simulation of the motor mount, fiber orientation tensor results were obtained for each mesh element using SIGMAsoft software. The orientation ratio in the primary flow direction varied across the model, ranging from approximately 35% to 90%. However, due to the computational limitations of generating a unique RVE model for each element, a representative approach was adopted. Elements with fiber orientation ratios between 35% and 40% were grouped and modeled using an RVE corresponding to 40% orientation. Similarly, elements with orientation ratios of 81% or higher were modeled using the 90% orientation RVE. Intermediate ranges were assigned to the nearest 10% increment (e.g., 50%, 60%, 70%, etc.). In the RVE modeling, the X-axis value of the fiber orientation tensor was used as the primary reference for flow direction alignment. The Y-axis and Z-axis components were held constant across all RVE models to maintain consistency in transverse dispersion. This approach allowed for a practical yet representative mapping of fiber orientation effects into the multi-scale simulation framework. Table 7 summarizes the fiber orientation tensor values used for each RVE model. The table presents the representative orientation ratio ranges, the corresponding X-axis tensor values applied in the RVE generation, and the fixed Y and Z components. This classification enabled efficient yet accurate integration of orientation-dependent mechanical behavior into the multi-scale analysis.