## 2 Model  
### FE Model  
A schematic diagram of the 3D FE vehicle-track model is shown in Figure 2. The model is based on a symmetrical vehicle-track system assuming a straight track in which lateral movement is negligible. The wheel, rail and sleepers are modeled with solid elements. The nominal radius of the wheel is 0.46 m, and the tread coning is 1/40. The rail is a standard International Union of Railway (UIC) 54 profile with an inclination of 1/40. The contact surface of the wheel is smooth, whereas a length of corrugation is prescribed along the rail surface. The sprung mass (the weight of the car body and the bogie) is lumped into a mass element \(M_{\mathrm{c}}\) supported by the primary suspension, which is represented by spring-damper elements. The sprung mass above a half wheelset is 1/8 of the sprung dynamic load of a whole vehicle, which is approximately a quarter of the sprung mass carried by a bogie. The fastening system and the ballast are also modeled as spring-damper elements. The track parameters are taken from [29] as shown in Table 1 and represent the typical Dutch railway system. In the FE model, the wheel and rail are meshed with 8-node solid elements. To achieve a solution of sufficient accuracy and an acceptable computation time, only the size of the elements in the solution zone is refined (0.8 mm \(\times\) 0.8 mm in the longitudinal and lateral directions). The elements far from the solution zone are meshed at an element size up to 7.5 cm. These choices are based on [24], which concluded that the contact mechanics solution with an element size of 1.3 mm \(\times\) 1.3 mm is sufficiently accurate when the FE approach used here is implemented for engineering applications. The total number of elements in the model is 1,135,384; the number of nodes is 1,297,900; and the model length is 18 m. In [30], a track length of 10 m was sufficient for problems of similar frequency and wavelength. The damage mechanism studied in this paper is wear; thus, the wheel and rail materials are assumed to be elastic. A Coulomb friction law is employed with a friction coefficient \(f_{\mathrm{C}}\) of 0.6 as in [31]. In the literature, the friction coefficient of dry wheel-rail contact is reported to be between 0.4 and 0.65 [32]. The solution process of the simulation includes two steps: an implicit analysis (using Ansys) and an explicit analysis (using Ls-dyna). The implicit analysis is performed to identify the initial deformation in the equilibrium position of the vehicle-track system. Then, the wheel is set to roll along the rail with a constant speed \(v=38.9\) m/s (corresponding to the typical Dutch passenger train speed of 140 km/h). An explicit integration scheme with a central difference method is then implemented to solve the wheel-rail frictional rolling contact problems. The displacements obtained from the implicit process are used as the initial state of the explicit integration process. If the time step (4.67 \(\times\) 10\({}^{-8}\) s in this model) is smaller than the critical time step (5 \(\times\) 10\({}^{-8}\) s) determined by the Courant criterion [33], convergence is guaranteed. By keeping the time step sufficiently small, the model can include all necessary vibration modes. In the explicit analysis, the frictional rolling is modeled using a surface-to-surface algorithm with the penalty method described in [34]. Because of the nature of explicit integration, the effect of transient rolling and the high-frequency dynamic behavior of the vehicle-track system excited by the moving wheel are automatically included in the solution. As shown in Figure 2a, a distance \(L_{1}\) is used during the explicit process to diminish the effect of vibration excited by imperfect initial equilibrium because of numerical errors from the implicit solution. A total of 10 waves of corrugation with length \(L_{2}\) are introduced after \(L_{1}\).