# Fatigue Crack Propagation Analysis of Rail Surface Under Mixed Initial Crack Patterns
## 3 Results
### Stress Intensity Factor and Lifetime Analysis with Different Initial Crack Lengths  
Figure 9(a),b illustrates the assignments of the SIF as a function of the normalized range at the crack tip, considering various initial cracking sizes. As indicated in Figure 9(a), it can thus be seen that \(K_{\text{I}}\) consistently remains near and below zero across various initial crack lengths, suggesting that \(K_{\text{I}}\) contributes negligibly throughout the crack propagation process. For varying initial crack lengths, the \(K_{\text{II}}\) values exhibit a symmetric distribution, characterized by the maximum values located at the termini of the crack's leading edge nodes, while the minimum values are observed at the midpoint (normalized to 0.5). As the length of the initial crack increases, the values of the leading edge nodes at both ends of the crack progressively rise, while those in the middle exhibit a gradual decline. This observation suggests that a longer initial crack length facilitates the propagation of mode-II cracks at the ends of the crack leading edge nodes, concurrently inhibiting center-region mode-II cracks from growing. Figure 10: Crack tip stress intensity factor at various initial crack lengths: (**a**) \(K_{\text{I}},\ K_{\text{II}}\); (**b**) \(K_{\text{III}}\); (**c**) \(K_{eq}\). Figure 9: Expansion of cracks at various initial crack angles: (**a**) extension length of cracks; (**b**) extension rate of cracks. As indicated in Figure 10b, the \(K_{\text{III}}\) values exhibit a centrosymmetric distribution, wherein both the maximum and minimum \(K_{\text{III}}\) values progressively increase with the initial crack length. Furthermore, the values of \(K_{\text{III}}\) at both extremities of the crack leading edge node approach zero. Under identical operating conditions, the largest \(K_{\text{II}}\) values consistently remain smaller than the largest \(K_{\text{III}}\) values. Figure 10c illustrates the distribution for equivalent stress strength factor \(K_{eq}\) in the crack leading edge node for varying initial crack lengths. It is observed that the maximum \(K_{eq}\) values of all initial crack sizes are situated at the ends of the crack leading edge nodes, whereas the \(K_{eq}\) values approach a decrease before rising as one moves toward the center. The maximum value of the equivalent stress strength factor at the crack leading edge increases as the initial crack length increases. This observation suggests that larger initial crack sizes render the crack more susceptible to propagation behavior, particularly at both the terminal and central regions of the leading edge. The evaluation of fatigue cracks at various initial crack lengths was performed using the optimal trajectory of crack extension. Additionally, Figure 11a illustrates three types of crack extension lengths in relation to the number of load cycles with various lengths of initially cracked fissures. For original crack lengths of 1 mm, 1.5 mm, and 2 mm, correspondingly, crack propagation lengths were observed to be 0.1 mm, 0.51 mm, and 0.83 mm, respectively, under a load cycle of \(5\times 10^{4}\). It reveals that original crack size significantly influences the extensional life of fatigue cracks. The stress on the surface of the rail exhibits an initial increase with depth, subsequently reaching a peak, followed by a gradual decrease in this trend. With increasing original crack size, the equivalent force strength factor at the leading edge of the crack progressively escalates. This phenomenon consequently accelerates the crack propagation rate. Figure 11b illustrates the correlation between the propagation rate and the length of crack propagation across varying initial crack sizes. As depicted in Figure 11b, it is evident that the rates of crack extension increase as the original crack length enlarges. The propagation rates associated with varying initial crack sizes exhibit significant differences; however, their propagation trends appear to be relatively consistent.