## 3 Contact Solutions at Corrugation  
### Normal Contact  
The FE approach defines whether a node is within the contact patch through the node force. A node is in contact if:  
\[|F_{\text{n\_N}}|>0 \tag{3}\]  
where \(F_{\text{n\_N}}\) is the nodal force in the direction normal to the local surface. Because of the presence of corrugation, the stress and slip distributions will vary along a corrugation wavelength. Those differential distributions of the normal pressure, shear stress and slip are the direct factors in corrugation development. Figure 4a shows the changes in the contact patch size at P1, P2, P3 and P4 with increasing amplitude A. Compared with A = 0 \(\upmu\)m, an increase in size of up to 16% at P2 is observed in the case of A = 20 \(\upmu\)m, and the increase is approximately 4% at P3. At the other two positions, the changes are minor. The maximum contact patch size is approximately 200 mm\({}^{2}\) (A = 20 \(\upmu\)m) at P2, i.e., the corrugation trough, whereas it is approximately 170 mm\({}^{2}\) when the rail is smooth. Figure 4b displays the maximum contact pressure along the longitudinal axis (\(y\) = 0 mm) for different corrugation amplitudes A. As the corrugation amplitude increases from 0-20 \(\upmu\)m, the contact pressure at P2 drops significantly to 56% of the original level, i.e., from 1320-745 MPa. This decrease is partly because of the 15% increase in the contact patch size (Figure 4a). At P1 and P3, the maximum pressure declines slightly to 78% of the original level. An increase in pressure of 6% is observed at P4 for the case of A = 20 \(\upmu\)m.  
Figure 5 shows a typical profile of the variation of the magnitude of normal force, contact size and maximum contact pressure along the longitudinal axis. In the figure, the typical normal force is the lowest at trough P2 and the highest at peak P4, with intermediate values at points P1 and P3. As the corrugation amplitude increases, the normal contact force at P2 decreases and the area of the contact patch increases; thus, the pressure decreases. At P4, the normal contact force increases, and the area of the contact patch decreases; thus, the pressure increases. At P1 and P3, the normal contact force and the area of the contact patch are intermediate between cases P2 and P4. These patterns are further evident in Figure 4b, where the maximum contact pressure of P1 and P3 for the different corrugation amplitudes decreases, although it does not decrease as strongly as in the case of P2. Figure 6 shows the projections of contact pressure and shear stress onto the \(xOy\) plane, the shape of the contact patch under the smooth condition and the evolution within one corrugation wavelength from P1 to P4 when A = 20 \(\upmu\)m. Most shapes are not elliptic and thus differ from the Hertzian solution. Because of the downhill slope of the corrugation at P1, a large drop in pressure is observed at the leading edge of the contact patch and vice versa at P3. Thus, the pressure distributions at P1 and P3 take the shapes of a \"waning crescent moon\" and \"waxing crescent moon\", respectively. At P2, the maximal pressure shifts to the field and gauge sides, thereby decreasing the contact pressure in the contact patch center to 600-800 MPa. The pressure distribution at P4 is close to the solution under the smooth rail except for a slight shrinkage of the contact patch at the leading and trailing edge centers. The maximum pressure is slightly higher than that at the other locations as shown in the pressure scaling color bar. As also shown in Figure 5, the pressure variation within a corrugation wavelength along the longitudinal axis \(y\) = 0 mm is approximately in phase with the corrugation. Furthermore, the shapes of the contact patch differ from those under the smooth rail condition.