# Fatigue Crack and Residual Life Prediction Based on an Adaptive Dynamic Bayesian Network
## 2 Principle and Methods  
### Fatigue Crack Growth Theory  
In general, the fatigue failure of materials will go through three stages: crack initiation, stable expansion, and rapid unstable failure of cracks. Due to the relatively short time of rapid unstable failure of cracks, the fatigue life is often calculated by selecting the crack initiation life (from the beginning of use to the appearance of engineering detectable cracks) and the crack expansion life (from detectable cracks to critical crack size), as shown in Figure 1.  
Figure 1: The cross-scale propagation of cracks. The mechanism of crack formation life is quite complex. From the beginning of crack nucleation to the point where microscopic cracks expand into macroscopic cracks, it accounts for the largest proportion of fatigue life. Crack nucleation is mainly caused by local stress and strain concentration, and the existence of tiny inclusions or voids in engineering materials has a great influence on the crack nucleation process, resulting in a large degree of discreteness in crack formation life. The length range of microscopic cracks is generally 10-100 \(\upmu\)m, which is at the same order of magnitude as the size of the plastic zone at the crack tip. Currently, there are two commonly used methods for estimating crack formation life without obvious macroscopic cracks. The first method is to use the relationship curve between external constant amplitude cyclic stress amplitude \(S\) and fatigue life \(N\) (referred to as \(S\)-\(N\) curve) as the main design basis for anti-fatigue design methods. This method has a long history and is still adopted by many anti-fatigue design specifications today. The other method is to assume a reasonable microscopic crack size based on experience (such as a microscopic crack length of 0.0588 inches commonly used in flight structures as the minimum observation size) and use the formula for macroscopic crack extension in fracture mechanics to calculate residual life. Therefore, whether it is the fatigue life prediction of cracks that assumes reasonable microcrack sizes or the prediction of macroscopic crack fatigue life, it ultimately comes down to the study of the crack propagation rate. Currently, within the category of online elastic fracture mechanics, the strength of the crack tip singularity is characterized by a single parameter, stress intensity factor \(K\). Therefore, many scholars consider linking the fatigue crack propagation rate with \(K\) and then propose a series of fatigue crack propagation rate formulas, such as the most basic and commonly used [PERSON] law [25], the Forman law [26], and the Walker law [27] that considers the influence of stress ratio \(R\) and strength factor threshold \(K_{th}\), as well as the full crack propagation formula and [PERSON] law [28] that integrates simulation software. According to [PERSON] law [25], the crack growth is described by the following:  
\[\frac{da}{dN}=\text{C}(\Delta K)^{m} \tag{1}\]  
where \(K=F\sigma\sqrt{\pi a}\) is the stress intensity factor. \(F\) is the shape factor. \(\sigma\) is the equivalent stress, which is the stress at the crack location calculated without cracks. \(a\) is the crack length. \(C\) and m are material parameters determined by experiments. \(da/dN\) is the fatigue crack propagation rate. This formula expresses the fatigue crack propagation rate in the form of a power function of the stress intensity factor amplitude \(\Delta K\), which describes the law of crack propagation well and has the advantage of convenient calculation. Therefore, it is still widely used in engineering structural fatigue life prediction. Fatigue crack growth life is the total number of load cycles from the initial crack \(a_{0}\) to the critical crack length \(a_{c}\). First, Equation (2) is used to obtain the size of critical crack size \(a_{c}\). \[a_{c}=\frac{1}{\pi}\left(\frac{K_{C}}{F\sigma_{\text{max}}}\right)^{2} \tag{2}\]  
where \(\sigma_{\text{max}}\) is the maximum cyclic stress. \(K_{C}\) is the fracture toughness of the material.