# Fatigue Crack Propagation Analysis of Rail Surface Under Mixed Initial Crack Patterns
## 3 Results
### Stress Intensity Factor and Life Analysis with Different Friction Coefficients  
Figure 12a,b illustrates the assignments of the SIF as a function of the normalized range at the crack tip, considering various friction coefficients. Figure 11: Crack propagation at different initial crack lengths: (**a**) crack propagation length; (**b**) crack propagation rate. Figure 12a,b shows the distribution of the stress intensity factor across various friction coefficients. The analysis of Figure 12a,b reveals that the contribution of \(K_{\text{I}}\) to crack propagation is minimal. As the coefficient of friction increases, the stress intensity factor of maxima for \(K_{\text{II}}\) and \(K_{\text{III}}\) exhibits a decreasing trend. This occurs because a higher coefficient of friction necessitates a greater stress for the propagation of crack surfaces, thereby making the crack propagation more difficult. The \(K_{\text{II}}\) peak occurs at a distance normalized to 1 from the leading edge of the crack, while the maximum \(K_{\text{III}}\) value is noted at a distance normalized to 0.75. Moreover, under identical operating conditions, the value of \(K_{\text{II}}\) consistently remains lower than that of \(K_{\text{III}}\). For varying friction coefficients, the distribution of \(K_{\text{II}}\) exhibits symmetry, while \(K_{\text{III}}\) demonstrates centrosymmetry. Notably, the distribution patterns are largely analogous. Figure 12c shows the profile of equivalent force strength factors in the leading edge node of the crack for different friction coefficients. The stresses observed at both the left and right sides of the crack's leading edge node exhibit negligible variation despite the increasing coefficient of friction. Conversely, stresses within the interval segment of the crack's leading edge node, normalized over a distance range of 0.25 to 0.65, demonstrate a continuous decline as the friction coefficient increases. This finding indicates that an enhancement of the friction coefficient may effectively mitigate the propagation of fatigue cracks. The lengths of fatigue cracks, evaluated across various initial friction coefficients, are calculated based on the most rapid propagation paths of crack growth. Figure 13a illustrates the relation of crack extension length with the number of loading cycles, considering three distinct friction coefficients. When the friction coefficients are set at 0.2, 0.3, and 0.4, the corresponding crack propagation lengths are 1.58 mm, 1.37 mm, and 1.28 mm, respectively, under a load cycle of \(1\times 10^{5}\). It is evident that as the coefficient of friction increases, the crack propagation distances decrease for the same number of cycles. This observation suggests that a higher friction coefficient effectively inhibits crack propagation. Figure 13b illustrates the correlation between the propagation rate and the crack propagation length across varying friction coefficients. An analysis of Figure 13b reveals that the crack propagation rate exhibits a decreasing trend as the friction coefficient increases. Moreover, a larger disparity in the crack propagation rate correlates with a reduced number of cycles necessary to achieve the stable propagation stage. This occurs because higher friction coefficients result in increased stress requirements for crack propagation, thereby rendering the cracks less susceptible to propagation. Through a comparative analysis of the propagation rates of mixed fatigue cracks across varying crack angles, lengths, and friction coefficients, it can be concluded that crack length exerts the most significant influence upon the rate of expansion of fatigue cracks on the track surface. Furthermore, as crack length increases, the propagation rate accelerates. If the rail surface is not polished promptly, surface cracks may rapidly propagate, resulting in the formation of long cracks that penetrate deeply into the rail's interior. This condition can lead to the peeling and detachment of pieces, among other forms of damage.