# Fatigue Crack Propagation Analysis of Rail Surface Under Mixed Initial Crack Patterns
## 3 Results
### Stress Intensity Factor and Lifetime Analysis at Different Initial Crack Angles  
To facilitate clarity in description, the positions of the crack leading edge nodes are normalized, establishing a reference point at a normalized distance of zero and extending to a normalized distance of one, which delineates the boundary surrounding the crack leading edge node. Figure 8a,b illustrates the assignments of the SIF as a function of the normalized range at the crack tip, considering various initial cracking angles. As illustrated in Figure 8a, the stress intensity factor \(K_{\text{I}}\) at the cracked tips consistently approaches as well as remains below zero, suggesting that the crack surface is perpetually in a closed state throughout the crack extension processes, while the mode-I of cracking has a relatively minor impact on crack extension. The \(K_{\text{II}}\) values at the crack edge are negative at the center of the crack's leading edge, attaining a positive maximum at the crack edge. This distribution is predominantly symmetrical, with \(K_{\text{II}}\) values approaching zero as the initial crack angle increases. As indicated in Figure 8b, the crack tip \(K_{\text{III}}\) exhibits a centrosymmetric distribution, characterized by a consistent increase in magnitude from the left to the right sides of the leading edge of the crack. Additionally, the \(K_{\text{III}}\) value of the crack leading edge decreases with an increase in the initial crack angle. The largest \(K_{\text{II}}\) value consistently remains smaller than the largest \(K_{\text{III}}\) value under identical operating conditions. When the wheel is positioned directly above the crack, crack propagation primarily occurs as a composite propagation of the mode II and III induced by shear stresses. Figure 8c illustrates the equivalent stress strength factors assignments along the crack leading edge node for different initial crack angles. Figure 8c reveals an equivalent stress intensity factor \(K_{eq}\), where maximum values at various initial crack angles are observed to occur on both sides along the leading edge of the crack. Furthermore, it is evident that \(K_{eq}\) progressively diminishes as the position at the leading edge of the crack approaches the center. This implies that the proximity of the crack node to the surface increases the likelihood of crack propagation. The propagation lengths of fatigue cracks, determined for various initial crack angles, are calculated utilizing the most efficient pathway of crack propagation. The correlation is between the crack extension length and the counts of load cycling for four various initially cracked angles. The measured crack extension lengths were 1.64 mm, 1.47 mm, 1.31 mm, and 1.23 mm for initial crack angles of \(0^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), and \(45^{\circ}\), respectively, at a cycle number of 1.5 \(\times\)\(10^{5}\). This observation underscores the significant impact of the initial crack angles upon the fatigue crack propagation life. Within the angular range of \(0^{\circ}\) to \(45^{\circ}\), an increasing amount of load cycling and accelerated crack expansion rate leads to a transition from a stable propagation phase to an unstable one. Furthermore, at a consistent number of cycles, a decrease in the initial crack angle correlates with a heightened susceptibility to crack propagation behavior. Based on Figure 9a, an analysis was conducted to elucidate the correlation that exists between the crack extension length and the corresponding count of fatigue cycles to derive the rate of crack extension along the extension path. Figure 9b  
Figure 8: Crack tip stress intensity factor at various initial cracked angles: (**a**) \(K_{\text{I}}\), \(K_{\text{II}}\); (**b**) \(K_{\text{III}}\); (**c**) \(K_{eq}\). illustrates the rate of crack extension versus crack extension length, specifically for cracks with varying initial propagation angles along the optimal propagation path. Upon entering the propagation stage, the crack exhibits a rapid increase in its propagation rate.