## 4 Wear and Corrugation Simulation
### Prediction of Major Field Observations  
In reality, the rolling of the wheel along the rail excites a range of vibrations of the structure and continua in the system. However, only a few of the vibration components are directly related to the corrugation. The corrugation wavelength observed in the field and introduced in the model is 30 mm, and the wavelength of short pitch corrugation is typically 20-80 mm, while the calculated wear is high-pass filtered to remove wear patterns shorter than 20 mm.  
The wear along the longitudinal axis \(y\) = 0 mm is studied. Figure 8a shows the distribution of the simulated wear along the rail with different corrugation amplitudes for the P4 corrugation position of 0 = 0, with 1.2 m as the corrugation peak. The results for the other corrugation positions show similar trends and are not presented. The wear is not uniform along the rail; thus, corrugation development will not be uniform under the evaluated conditions. Comparing the field observations in Figure 1 with the wear simulation results in Figure 8a reveals that the model closely reproduces the conditions in Figure 1b,c in terms of predicting the periodicity of the wear and the number of corrugation waves in the periodicity. The periodicity in Figure 1c is nine and 11 waves. The periodicity in Figure 8a is nine waves between the largest peak-to-trough wear magnitudes at 0.96 m and 1.23 m. In Figure 1c, the periodicity is counted between the squats because the squats should have been caused by the most severe corrugation, i.e., the largest wear.  
Figure 8: Distribution of wear at different corrugation amplitudes obtained for the corrugation in Figure 3b with \(\theta\) = 0 and P4 at 1.2 m. The middle of a sleeper is at 1.2 m, and the midpoint between two sleepers is at 0.9 m. (**a**) High-pass filtered at \(\lambda\) = 20 mm. (**b**) High-pass filtered at \(\lambda\) = 30 mm.  
Furthermore, the results agree well with the field observations (Figure 1). Specifically, the model predicts that the largest wear occurs at 1.23 m above a sleeper centered at 1.2 m (Figure 8a). This is in agreement with Figure 1c, in which one of the squats was above a sleeper, and with those in Figure 1b, where the corrugation was always the most severe above the sleepers. Rails above sleepers have been reported to be more prone to corrugation [22]. The inset in Figure 8a shows that the magnitude of wear decreases as the corrugation amplitude increases. This pattern can be understood in relation to Figures 6 and 7b because the wear is proportional to the product of the shear stress and the slip. The slip at A = 20 \(\upmu\)m in Figure 7b is generally equal to or less than that at A = 0 \(\upmu\)m, except at P2. The shear stress at A = 20 \(\upmu\)m in Figure 6 is always less than that at A = 0 \(\upmu\)m, and the difference is large for P2 and P3.  
The wear in Figure 8a is of the same wavelength and phase as the modeled corrugation. Thus, the existing corrugation will be erased by wear and will not grow, and if the rail is smooth, no corrugation will initiate. This finding is reasonable because the simulation is based on nominal track parameters; in reality, many tracks are indeed free of corrugation. Thus, the results indicate that corrugation is sensitive to the system parameters, e.g., rail fastening [41].