# Fatigue Crack and Residual Life Prediction Based on an Adaptive Dynamic Bayesian Network
## 3 Fatigue Crack Propagation Experiment Verification
### Multiple Adaptive Variables DBN Model  
To provide additional evidence regarding the effectiveness of the DBN particle filter, experiments from [PERSON] [38] were utilized to continue the validation process. The single-edge crack growth test material is domestically sourced commercial galvanized high-strength steel wire, with a diameter of 7 mm. For finite-width, equal-thickness strip specimens with unilateral cracks, the calculation formula for the crack form factor \(F\) can be obtained from the Stress Intensity Factor Manual, as shown below:  
\[F=1.12-0.234\Big{(}\frac{a}{b}\Big{)}+10.55\Big{(}\frac{a}{b}\Big{)}^{2}-21.72 \Big{(}\frac{a}{b}\Big{)}^{3}+30.39\Big{(}\frac{a}{b}\Big{)}^{4} \tag{22}\]  
where \(a\) is the crack length and \(b\) is the specimen width. [PERSON] adopts the Walker law as the prediction formula for crack propagation [27]. \[\frac{da}{dN}=C\big{[}(1-R)^{a}\Delta K\big{]}^{m} \tag{23}\]  
The experiment collected separate data on crack propagation at stress ratios \(R=0.1\), \(R=0.5\), and \(R=0.7\). The parameter \(m\) of Paris' law is 3.3. The crack propagation data under a 220 MPa load were utilized as the validation dataset for the DBN model, as depicted in Figure 13. The author performed a parameter fitting process to determine the values of parameters \(C\) and \(\alpha\); the obtained values were approximately \(7\times 10^{-13}\) and \(-0.3\), respectively, with a survival rate of 95%. Figure 12: Prediction results of residual lives for the specimens with dimensions of (**a**) \(120\times 30\times 15\) mm\({}^{3}\), (**b**) \(160\times 40\times 15\) mm\({}^{3}\) and (**c**) \(240\times 60\times 15\) mm\({}^{3}\). Figure 13: Single-edge crack growth test specimen. The parameters in Table 2 are utilized as sources of uncertainty for the DBN particle filter. The parameters \(C\) and \(m\) are adaptively updated in the DBN model update. The mean and variance of sampling for other uncertain variables remain unchanged. The prediction results of the DBN particle filter are shown in Figure 14.  
Figure 15 demonstrates that the adaptive DBN model is equally capable of accurately forecasting the actual progression of crack expansion across varying stress ratios. In comparison to the three-point bend model examined in Section 3.1, there are less observational data available for this section's single-edge crack model. Additionally, the uncertain variable parameter \(\Theta\) within the Walker law is not linear but includes a power exponent variable, which magnifies the errors. When utilizing the initial model to predict crack expansion reliability, the overlapping uncertainties of various model parameters \(\Theta\) lead to a wider range of predicted crack intervals. This range only narrows, adopting a \"trumpet\" shape, after incorporating the next batch of observational data, which prompt an update in the model's state. Following this update, the precision of crack predictions improves, and the uncertainty interval diminishes, demonstrating that the update enables the particles to more closely mimic the true probability distribution between two observational points. As predictions advance, the estimates increasingly diverge from the actual conditions, and the uncertainty interval expands accordingly. This expansion results from the model's uncertainty propagation, which continually escalates the prediction errors, highlighting the importance of ongoing crack monitoring and frequent updates to the model's state. Figures 16 and 17 showcase the variations in the probability distribution of [PERSON] model parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Initial Value** & **Standard Deviation** & **Meaning** \\ \hline Log(C) & \(-27\) & 1 & The logarithm of parameter \(C\) \\ \(\alpha\) & \(-0.3\) & \(0.01\) & Stress ratio coefficient \\ \(a_{0}\) & \(0.001\) & \(0.0001\) & Prefabricated crack error (mm) \\ \(a_{\text{obs}}\) & \(0.001\) & \(0.0001\) & Measurement error \(\varepsilon_{a}\) \\ \hline \hline \end{tabular}
\end{table}