# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Analytical Solution of the Contact Problem
The coefficients \(m\) and \(n\) re related to the calculation of elliptical integrals and are determined by interpolation from tables [25], depending on the value of the parameter \(\beta\), which is equal to  
\[\beta=arccos\left(\frac{B-A}{A-B}\right)=arccos\left(\frac{\frac{1}{K_{w}}- \frac{1}{r_{w}}-\frac{1}{r_{r}}}{\frac{1}{K_{w}}+\frac{1}{r_{w}}+\frac{1}{r_{r} }}\right). \tag{5}\]  
At the center of the contact zone, the compressive normal stresses are as follows:  
\[ \begin{array}{c}\sigma_{x}=-p_{max}\left[2\
u+(1-2\
u)\frac{b}{a+b}\right], \ \sigma_{y}=-p_{max}\left[2\
u+(1-2\
u)\frac{a}{a+b}\right],\\ \sigma_{z}=-p_{max}.\end{array}  \tag{6}\]  
At the extreme points of the major axes of the ellipse (on the boundaries of the contact area), equal in magnitude positive radial and negative tangential stresses act:  
\[ \begin{array}{c}\sigma_{x}=-\sigma_{y}=-p_{max}(1-2\
u)\frac{b}{a^{2}} \left[\frac{1}{e}Arth(e)-1\right],\ x=\pm a\,\ y=0;\\ \sigma_{y}=-\sigma_{x}=-p_{max}(1-2\
u)\frac{b}{a^{2}}\left(1-\frac{b}{a^{2}} arctg\frac{ac}{b}\right),\ y=\pm b\,\ x=0,\end{array}  \tag{7}\]  
where \(e=\sqrt{1-\frac{b^{2}}{a^{2}}}\)--is the eccentricity of the ellipse. The maximum shear stresses \(\tau_{max}\) will act at a point on the vertical axis \(z\), passing through the center of the ellipse, at a certain distance from the surface of the bodies, depending on the value of the eccentricity \(e\) and the elastic properties of the material. The variation of \(\tau_{max}\) for \(\
icefrac{{b}}{{a}}=0.6\) and \(v=0.25\) is illustrated by the data in Table 1 [26]. In strength calculations, the value of the stress intensity (Mises equivalent stress) is usually used:  
\[\sigma_{q\
u}=\frac{1}{\sqrt{2}}\sqrt{\left(\sigma_{x}-\sigma_{y}\right)^{2}+ \left(\sigma_{y}-\sigma_{z}\right)^{2}+\left(\sigma_{z}-\sigma_{x}\right)^{2}} \tag{8}\]  
Figure 3 [26] illustrates how, under the above conditions, the relative equivalent stress \(\
icefrac{{e_{q\
u}}}{{p_{max}}}\) varies at points along the semi-axes of the ellipse. From the presented data, it can be concluded that the highest stresses occur at the center of the ellipse and at a point on the ellipse's boundary that is farthest from its center. At the center, all three principal stresses are compressive (although not necessarily equal in magnitude), while at the boundary, a plane stress state is realized with normal stresses of different signs. This aspect is crucial for understanding the formation of a crack network on the running surface (see Figure 1b). During vehicle movement, the position of the wheelset relative to the rails, and hence the location of the contact area, constantly changes due to transverse oscillations of wagons and locomotives, wheelset imperfections, varying degrees of wear of wheels and rails, etc. Similarly, the directions of principal stresses and maximum shear stresses on the contact surfaces also change. For the surface material layers, which have exhausted their plastic properties, the level of these stresses may be sufficient to cause the formation of microcracks that will further develop through fatigue mechanisms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\
icefrac{{b}}{{a}}\) & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \(\
icefrac{{z}}{{b}}\) & 0.785 & 0.745 & 0.665 & 0.590 & 0.530 & 0.480 \\ \(\
icefrac{{\tau_{max}}}{{p_{max}}}\) & 0.300 & 0.322 & 0.325 & 0.323 & 0.317 & 0.310 \\ \hline \end{tabular}
\end{table} 
Table 1: Dependence \(\frac{\tau_{max}}{p_{max}}\) on the size of the contact patch and the distance from the surface. Mathematical modeling of these processes is a rather complex task, even with the use of powerful numerical methods and modern computers, because many factors influencing the formation of surface microdefects are difficult to formalize and remain insufficiently studied experimentally (e.g., the distribution of residual technological stresses, material anisotropy and inhomogeneity, surface roughness, etc.).