# Fatigue Crack Propagation Analysis of Rail Surface Under Mixed Initial Crack Patterns
## 2 Materials and Methods
### Finite Element Model  
The wheel-rail rolling contact model is presented in Figure 2a. The rail is made of U71 Mn material, with a length of 200 mm, a height of 192 mm, and the bottom slope set to 1:40. The wheel has an LM-type tread with an outer diameter of 457 mm, an inner diameter of 95 mm, and an axle weight of 25 T. The elastic modulus of the wheel and rail material was set to 2.1 \(\times\) 10\({}^{5}\) MPa, with a Poisson's ratio of 0.3, and fatigue limits of the rail were set to 550 MPa. The wheel-rail contact surface is defined as surface and surface, with a friction coefficient of 0.3 [28; 29]. The boundary conditions were as follows: the bottom surface of the rail were fixed, and symmetric constraints were applied at both ends of the rail. Additionally, the effect of the rail's elastic constraints on crack propagation was neglected. The coupling constraints for all nodes of the wheelset were defined at its center, with a reference point located at this position where a uniform load was applied. The motion of the wheelset on the rail is modeled in two analysis steps: The first step constrains the X and Z directions while relaxing the Y direction to establish surface-to-surface contact between the rail and the wheelset. The second analysis step constrains the X direction, relaxes the Y and Z directions, and applies counterclockwise rotation about the X-axis to simulate rolling contact between the wheel and track. Figure 2b illustrates the network refinement diagram of the contact region, where the grid size near the contact region is set to 1 mm \(\times\) 1 mm; in contrast, the grid size in regions further away from the contact area can be gradually increased to enhance computational efficiency. Figure 2c presents the schematic for crack surface insertion. To ensure calculation accuracy, adaptive mesh technology was employed to remesh the initial crack insertion region [30]; the smallest dimension of an element at a crack leading edge node is 0.1 mm. At the leading edge of the crack, there is an inner ring of 15-node singular wedge cells, consisting of 600 nodes and 400 cells, and an outer ring of 20-node hexahedral cells, consisting of 45,920 nodes and 2296 cells. In the region surrounding the crack, 10-node tetrahedral cells are used, totaling 4,010,380 nodes and 401,038 cells. Additionally, 8-node hexahedral cells, totaling 2,907,848 nodes and 363,481 cells, are used further away from the crack's surface. In accordance with the EN13999-1 [31] standard, the uniform distribution load on a single-wheel rail is defined as half of the axle weight. To investigate the allocation for Von Mises equivalent stress field within the wheel-rail contact area, uniform loads of 100 kN, 125 kN, and 150 kN were applied vertically downward at the center of the wheel. This specific area experiences the highest Von Mises equivalent stress, making it susceptible to crack initiation. Throughout the prolonged service life of railway tracks, the orientation and morphology of microcracks that develop on the surface of the rails exhibit significant variability, attributable to the cyclic forces exerted by the wheels and inherent material defects. In this paper, a consistent load of 125 kN was applied to investigate the effect of varying initial crack patterns on the expansion of fatigue cracks. This was achieved by manipulating the crack angle, size, and the coefficient of friction of wheels to rails. In analyzing the crack angle, the vertical plane aligned with the direction of wheel-rail contact (\(y\)-axis direction), which served as the reference point. Subsequently, the crack was rotated clockwise from this reference plane to incorporate a semi-circular initial crack where the radius was 1 mm and the angles were 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) relative to the vertical plane.