## 5 Relationship between Contact Forces and Wear as well as New Insights
### Additional Discussions  
In this section, 2D and 3D analysis results are briefly compared. In [16], a 2D non-Hertzian contact model is developed in which the stress distribution is not symmetric at the falling and rising edges of the corrugation. The same phenomenon is observed in the present 3D numerical analysis. A clear difference between the 2D and 3D approaches is that in the 2D analysis, the contact patches at the falling and rising edges are symmetric in the longitudinal direction about the contact patch center. In the case of the 3D analysis, the contact patches are not symmetric about the contact patch center.  
Wavelength fixing is also observed. In the Dutch railway network, the wavelength of the short pitch corrugation is 25-50 mm at a corresponding speed of 30-140 km/h. If the corrugation is purely determined by frequency fixing, then the wavelength should have varied in a broader range. Therefore, the numerical analyses and the field observations indicate that frequency fixing and wavelength fixing may have occurred simultaneously. One such possibility is that for a certain traffic speed, if the preferred frequency is \(\mathit{nf}_{0}\), with \(f_{0}\) representing the fundamental frequency, the frequency will jump to harmonics of a higher frequency \((n+\mathit{if})_{0}\), \(i\geq 1\) when the speed is increased so that the wavelength remains at the fixed wavelength.  
As shown in Figure 11a, approximately two wavelengths are required for the normal contact force to transit from the steady state before the corrugation, which starts at 0.87 m, to the new steady state controlled by the corrugation. For the tangential force, approximately 3.5 wavelengths are required. Thus, the longitudinal force lags approximately 1.5 wavelengths behind the normal force, i.e., approximately 45 mm. Because the length of the contact patch is approximately 15 mm (see Figure 6), the lag is approximately three-times the length of a contact patch. In [14], the lag for static rolling contact is one length of a contact patch, which reveals another difference between static and dynamic rolling contact. Moreover, Figure 11 shows that the tangential contact force contains considerably more harmonics than the normal force.  
Metallurgical investigations on corrugated rail material show that the corrugation crest experiences high dynamic contact forces, high deformation, work hardening and, consequently, high wear resistance. The corrugation trough, instead, is reported to be developed from wear [11; 52]. This is in agreement with the numerical results in the present paper, i.e., high contact pressure is found at the corrugation crest and lower pressure and high micro-slip at the corrugation trough. Additionally, the delay in corrugation growth with reduction of wear by friction modifier [66] also supports wear as a mechanism of the corrugation.  
Although more research using more sophisticated models for the longitudinal support is needed, the findings of this paper have relevance for the numerical modeling of longitudinal modes. The inclusion of longitudinal support stiffness, such as in [62], results in a longitudinal mode at approximately 1298 Hz, which represents a difference of 7 Hz with respect to the model without longitudinal rail support. Hence, the more complex model does not result in a significant difference in the longitudinal dynamics. This situation should be further explored by investigating the actual parameters experimentally under Dutch conditions and verifying the dynamics.