## 3 Finite Element Simulation Analysis
### Analysis of Results
#### 3.2.4 Crack Propagation Analysis  
Figure 14 shows the distribution of the cohesive-zone element damage in the brazed weld during the fatigue cycles under a cyclic displacement of \(U\) = 0.5 mm. During the cyclic loading process of the butt-brazed joints, damage first appeared at the edge of the brazed interface layer as the number of cycles N increased. When the damage reached the maximum value of 0.9999, the cohesive element failed and was removed, and a crack initiated and propagated toward the center until the final fracture occurred.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Serial Number** & **Scale Factor** & **Fatigue Life** & **Error/\%** \\ \hline
1 & 600 & 40,200 & 27.20 \\
2 & 700 & 46,200 & 6.33 \\
3 & 800 & 52,800 & 4.38 \\
4 & 900 & 59,400 & 7.57 \\
5 & 1000 & 66,000 & 19.52 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fatigue life and test error corresponding to different scaling factors.  
Figure 13: Comparison of the test value of the load-life curve with the simulated value.  
Figure 13(a) shows the initial stage of the cycle, where SDV3 (solution-dependent state variables 3) represents the total damage value D. When the number of cycles was 8000, the total damage was D = 0.0712, indicating that the cohesive zone element has not yet reached the critical damage value and no cracks formed in the brazed layer.  
Figure 13(b) shows the crack initiation stage. After 35,200 cycles, the first cohesive zone element reached a total damage value of D = 0.9999, leading to its failure and deletion, resulting in crack initiation.  
Figure 13(c) shows the crack propagation stage. As the number of fatigue cycles increased, damage accumulated continuously. When the number of cycles reached 45,000, the number of cohesive zone elements that failed and were deleted increased, and the crack length increased accordingly.  
Figure 13(d) represents the fracture failure stage. When the number of cycles reached 52,800, all the cohesive zone elements failed and were removed, leading to fatigue failure and fracture of the butt-brazed joint.  
To further illustrate the damage status of the cohesive elements, we selected the first cohesive element with fatigue failure, as shown in point A in Figure 13(a), and analyzed the damage variation rate of the cohesive element under different numbers of cycles. When the number of cycles is 10,000, the slope k of the corresponding point on the curve is 1.310 \(\times\) 10\({}^{-5}\), indicating that the damage variation rate is 1.310 \(\times\) 10\({}^{-5}\). When the number of cycles increases to 20,000, the damage variation rate of the corresponding point is 2.275 \(\times\) 10\({}^{-5}\). As the number of cycles further increases to 30,000, the damage variation rate of the corresponding point becomes 4.684 \(\times\) 10\({}^{-4}\). The results indicate that during the initial stages of the cycles, the damage accumulated slowly at a lower rate (Figure 15). As the number of cycles increased, the damage accumulation rate also increased, leading to a rapid increase in damage and a decrease in material stiffness. When the damage reached 0.999, the cohesive element failed and was removed, which resulted in crack initiation.  
Figure 14: Damage distribution of cohesive units at different cycles.