# Fatigue Crack Propagation Analysis of Rail Surface Under Mixed Initial Crack Patterns
## 1 Introduction  
Long-term wheel-rail rolling contact during train service can lead to various types of fatigue damage on the rail, including cracks, abrasion, and stripping [1; 2; 3]. If not addressed promptly, these damages may cause rail failures that affect train safety. Fatigue cracks on the rail surface (shown in Figure 1) can expand under sustained wheel-rail loading, forming deep cracks inside the rail and potentially leading to block stripping, ultimately resulting in rail failure and serious safety accidents [4]. Therefore, analyzing the extension characteristics of rail surface fatigue cracks under complex working conditions provides a theoretical foundation and methodological support for accurately predicting the remaining life of cracked rails and enabling intelligent rail operation and maintenance [5]. The generation and expansion of wheel-rail rolling contact fatigue cracks are influenced by numerous factors, with a complex interaction force mechanism [6; 7; 8]. In response to this, researchers have conducted experimental studies investigating various factors, including the third medium, material properties, contact conditions, and their effects on both surface and subsurface crack propagation behavior, encompassing both short and long cracks. For example, [PERSON] et al. [9] studied the influence of slip rate and contact stress on the evolution of rolling contact fatigue cracks near rail surface defects through double-disk pair-rolling experiments. Their findings indicated that the slip rate is the primary factor influencing the residual crack angle. [PERSON] et al. [10] employed laser cladding technology to repair rails experiencing side wear, investigating the wear behavior at varying cladding depths and evaluating the anti-wear performance of the cladding on the rails. [PERSON] al. [11] examined the microstructure of various weld regions via SEM and assessed crack propagation in rail samples subjected to cyclic loading. [PERSON] et al. [12] investigated wheel-rail interaction forces using a double-disk test rig with varying sliding ratios for dry and wet testing. They proposed a model to assess the interaction between wear and contact fatigue in order to study the propagation behavior of surface cracks. Alternatively, numerical simulations offer an effective approach for investigating rolling contact fatigue cracks in rails. [PERSON] et al. [13] simulated rolling contact by developing a two-dimensional finite element model to investigate the extension path of fatigue cracks and performed a comparative analysis using physical experiments. [PERSON] et al. [14] developed a mathematical model using a 2D finite element simulation platform to predict crack propagation under rolling contact between the wheel and rail. They investigated crack propagation in rails under mixed loading conditions and found that the crack propagation paths vary with the contact load. [PERSON] et al. [15] conducted a numerical study of diagonal cracks on rail surfaces by developing a two-dimensional finite element model to analyze the effect of driving force on crack propagation behavior under various operating conditions. The above research reduces the three-dimensional crack propagation problem in steel rails to a two-dimensional model. While many valuable insights have been gained, there remains a gap between the model and engineering reality, such as the neglect of the crack leading edge geometry's influence on the propagation path [16]. Consequently, recent research has progressively focused on three-dimensional fatigue crack propagation in rails. For instance, [PERSON] et al. [17] conducted simulations of fatigue tests across four mixed modes to examine the direction of crack propagation in both shear and tensile modes under varying working conditions characterized by different crack lengths. [PERSON] et al. [18] examine the impact of varying elastic mixing levels on fatigue crack propagation behavior through both experimental and numerical investigations. [PERSON] et al.