## 1 Introduction  
Rail surface defects are a critical safety concern for railway infrastructure owners and operators all over the world. They undermine the safety and operational reliability of both moderate- and high-speed trains in passenger suburban, metro, urban, mixed-traffic, and freight rail systems. Furthermore, the cost of rail replacements due to such defects has become a significant portion of the whole track maintenance costs, especially in European countries, e.g., Austria, Germany, and France [1]. Traditionally, two different defects are classified: rail studs and rail squats [2]. Rail studs initiate from the white etching layer (WEL) due to wheel slides or excessive traction and grow horizontally 3-6 mm below the rail surface. Rail squats propagate from surface cracks initiated by rolling contact fatigue (RCF), and grow at similar depth of 3-6 mm below the rail surface. Both defects are shown in Figure 1. As a result, the rail surface becomes depressed and passing wheels create excess vibration, noise, and impact loads. This leads to uncomfortable rides for passengers [3], and in cases where impact forces exceed acceptable limits the safety of track components can be compromised [3; 4; 5; 6; 7]. Rail squats and studs have been observed in all arrays of track geometries and gradients, in all types of track structures, and in all operational rail traffics. Squats are often found in tangent tracks, in high rails of moderate-radius curves, and in turnouts with vertical, unground rails. Due to the high potential damage caused by rail squats and studs, several research and development projects have been initiated around the world to investigate the causes of, and feasible solutions to, these defects. Computational rail squat and stud modeling has been the topic of several studies. A finite-element (FE) analysis with a two-dimensional (2D) elastic-plastic model under the assumption of plane strain was used to investigate crack growth from the WEL in [8]. The researchers found that the crack growth direction in the interface between the base material and the WEL is determined by the discontinuity of a material rather than the stress state, and that cracks tend to grow along the interface between the WEL and the rail material, because it is comparatively hard for a crack to propagate into the rail material. Field observations and a numerical analysis in [9] showed that squats initiate as a result of differential wear and differential plastic deformation. Numerical simulations in [10] have also shown that the growth of squats is related to some eigenmodes of the wheel-track interaction system and the high-frequency vibration at wheel-rail contact plays an important role. The probability of rail squat initiation from surface defects based on a transient stress analysis was studied in [11] using an FE model of the vehicle-track interaction. The results showed that when a defect is smaller than 6 mm, its chance to grow into a squat is very small, and when it is larger than 8 mm and in the middle of the running band, the chance is large. RCF occurring on Chinese high-speed rails and wheels was investigated in [12]. Based on field observations and a numerical simulation, it was concluded that indentations seem to be the main cause of RCF. If relatively small but deep indentations exist, then peak von Mises stress can occur both on the surface and at the bottom of the crack, but stress at the bottom is likelier to create RCF cracks [13; 14]. The development of rail squats is most commonly studied using the finite-element method, which is based on the classical continuum mechanics theory. It uses spatial derivatives, which do not exist when the displacement field is discontinuous, i.e., when cracks are present. So, as a remedy, techniques of fracture mechanics must be used; however, their major drawback is that the crack path must be known a priori. Due to such limitations, independent crack branching is difficult to implement.