## 3 Contact Solutions at Corrugation
### Tangential Contact  
Figure 6b,d show the surface shear stress distributions when the rail is smooth and corrugated, respectively. In the presence of corrugation, especially when the amplitude is large, e.g., A = 20 \(\upmu\)m, the distributions of the shear stress differ from those of the smooth rail case. Similar to the pressure distributions, the maximum shear stress, which occurs in the central part and at the adhesion-slip boundary within the contact patch, is cut in the middle and protrudes on the gauge and field sides, especially at P2 and P3. The development of the surface shear stress with the increase in corrugation amplitude follows a trend similar to that of the normal pressure in the slip zone of the contact patch.  
When a tangential force is applied, slip arises in the contact area. In the adhesion zone, there should not be relative slip between the contacting particles of the wheel and rail surface the wheel and rail surfaces stick together. There should, however, be relative slip between the contacting particles of the wheel and rail surfaces in the slip zone. The tangential contact force \(\mathrm{F_{n\_T}}\) at a node in the slip zone equals the bound of the tangential nodal force, which is the product of the nodal force \(\mathrm{F_{n\_N}}\) in the normal direction and the friction coefficient \(f_{\mathrm{C}}\) according to [PERSON]'s law. In the adhesion zone, the tangential nodal contact force is lower than the bound. Therefore, a node is in a slip zone if:  
\[f_{\mathrm{C}}|\mathrm{F_{n\_N}}|-|\mathrm{F_{n\_T}}|<\varepsilon_{\mathrm{T}} \tag{4}\]  
where \(\varepsilon_{\mathrm{T}}\) is the tolerance. In this paper, \(\varepsilon_{\mathrm{T}}\) is 5% of the maximal tangential nodal force in the contact patch.  
The adhesion-slip distribution in the presence of corrugation is shown in Figure 6d. The boundary between the adhesion and the slip zones at P2 and P3 is far from elliptic [14]. As assumed, differential wear proportional to the frictional work [9; 18; 22] causes the development of corrugation. Frictional  
Figure 6: Contact pressure, shear stress, contact patch, adhesion-slip distributions and vector graphs of micro-slips (vectors in Figure 6b,d) when A = 20 \(\upmu\)m along one corrugation wavelength (P1–P4) (projection onto the \(xOy\) plane; Border 1: contact patch border; Border 2: adhesion-slip distribution border).  
work only occurs in the slip zone. Therefore, the size of the slip zone in the contact patch, as well as the magnitude of the slip might be indicators of corrugation development.  
In Figure 7a, changes in the size of the slip zone with increases in corrugation amplitude at P1, P2, P3 and P4 are shown. The size fluctuates slightly at A = 0-10 \(\upmu\)m. To study the trend of change, the corrugation amplitude is further extended to 30 \(\upmu\)m. When the rail is smooth, the size of the slip zone is approximately 69 mm\({}^{2}\), which is 40% of the whole contact patch (170 mm\({}^{2}\)). After the initial fluctuations at A = 2.5 \(\upmu\)m and 5 \(\upmu\)m, the value at P2 presents the largest change and increases up to 98 mm\({}^{2}\) at A = 30 \(\upmu\)m, which is 57% of the total smooth contact zone. At the other three positions, the change in the slip zone size is small. Specifically, at P3, this value remains almost at the same level, whereas at P1 and P4, a 10% decrease is observed compared with that of the smooth rail (A = 30 \(\upmu\)m).  
Figure 7b shows the change in the slip distribution along the longitudinal axis \(y\) = 0 mm. The maximum slip at P2 is much larger than that of the other cases, which is in agreement with the trend shown in Figure 7a. At P3 and P4, the magnitudes of the slip decrease compared with that of the smooth rail. The increase in the size of the slip zone and the maximum magnitude of the slip at P2 may indicate that large micro-slips in the trough could be a major driver of corrugation development. The distribution of the slip within the contact patch is shown in Figure 6b,d.