# Fatigue Characteristics Analysis of Carbon Fiber Laminates with Multiple Initial Cracks
## 5 Fatigue Reliability Modeling of Multi-Cracked Laminates
### Sensitivity Analysis  
To ascertain the effects of parameter uncertainties on the fatigue life of carbon fiber laminates with multiple cracks, this section utilizes the Sobol method to conduct a sensitivity analysis on the fatigue life. The necessary matrix, derived from Latin hypercube sampling, is fed into the fatigue proxy model described earlier to obtain the fatigue life response. For each parameter, the first-order and cumulative sensitivities are evaluated, as indicated in Table 11 and visually represented in Figure 17.  
The sensitivity analysis, as evidenced in Table 10 and Figure 17, reveals that the first-order Sobol sensitivity indices suggest the edge crack length L\({}_{1}\), internal crack length L\({}_{2}\), and angle \(\theta\) have a pronounced influence on fatigue resistance of carbon fiber plates, with the edge crack length L\({}_{1}\) being the most influential. This finding aligns with the crack propagation trends depicted in Figure 11 for carbon fiber laminates. Additionally, the elastic modulus E\({}_{1}\), shear modulus G\({}_{1}\), and Poisson's ratio \(\
u_{1}\) in the direction of the fibers is significantly more impactful than in other directions, which is related to the orientation of the applied loads and confirms the validity of the sensitivity analysis.  
In the context of the Sobol total sensitivity analysis, the edge crack length L\({}_{1}\), internal crack length L\({}_{2}\), angle \(\theta\), and the distance between cracks S are identified as having substantial global influence, with a noticeable increase compared to the first-order analysis. In  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Uncertainty Parameter** & **S\({}_{1}\)** & **S\({}_{1}\)** \\ \hline Edge crack length L\({}_{1}\) & 0.44073 & 0.46249 \\ Internal crack length L\({}_{2}\) & 0.20386 & 0.23427 \\ Internal crack angle \(\theta\) & 0.18209 & 0.18644 \\ Distance between cracks & 0.03724 & 0.04179 \\ Elastic modulus E\({}_{1}\) & 0.03387 & 0.03984 \\ Elastic modulus E\({}_{2}\) & 0.01586 & 0.01429 \\ Elastic modulus E\({}_{3}\) & 0.01064 & 0.0024 \\ Shear modulus G\({}_{1}\) & 0.01376 & 0.00941 \\ Shear modulus G\({}_{2}\) & 0.01127 & 0.00197 \\ Shear modulus G\({}_{3}\) & 0.01121 & 0.00227 \\ Poisson’s ratio \(\
u_{1}\) & 0.01454 & 0.00368 \\ Poisson’s ratio \(\
u_{2}\) & 0.01379 & \(5.20072\times 10^{-4}\) \\ [PERSON]’s ratio \(\
u_{3}\) & 0.01113 & \(6.22547\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 11: Sensitivity analysis results.  
Figure 17: Sobol sensitivity analysis.  
contrast, there is a decline in the impact of the elastic modulus E, shear modulus G, and Poisson's ratio \(\
u\), with the shear moduli G\({}_{2}\) and G\({}_{3}\) and Poisson's ratios \(\
u_{2}\) and \(\
u_{3}\) exerting a negligible effect on the life cycle in fatigue of the carbon fiber laminates.  
In summary, the parameters exerting the most considerable influence on the fatigue life of carbon fiber plates are the edge crack length L\({}_{1}\), internal crack length L\({}_{2}\), and angle \(\theta\), which are identified as key factors contributing to material failure and fracture.