# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 1 Introduction  
Recently, there have been many publications of the results of research on modeling the life cycle of railway rails [1, 2, 3]. The relevance of this topic is associated with the need to plan certain types of repair work on the track, since these models make it possible to predict the durability of the track depending on its operating conditions [4]. A railway track is a complex spatial structure consisting of many elements, including rails, sleepers, spacers, fasteners, as well as ballast. All these components of the upper structure of the track differ significantly in shape, dimensions, materials, and mechanical and thermophysical properties. Rails often fail due to direct interaction with the wheels of rolling stock and require repair and replacement and therefore are the most valuable component of the railway track. One of the main causes of rail damage is the formation of specific defects that are of a contact-fatigue nature [5, 6, 7, 8]. The main types of such defects on therail surface include cracks on the lateral side of the rail head (head checking), surface metal delamination (spalling), localized metal chipping on the running surface (flaking), horizontal delamination of the rail head due to the appearance and growth of internal cracks under the running surface (squats) or propagation of cracks originating from the rail's lateral edges (shells) [7; 9; 10]. All these defects are related to the formation of external fatigue cracks (on running surfaces) and internal (at some depth below the running surface) and their subsequent interaction during propagation. External cracks occur as a result of the exhaustion of the plastic properties of the thin surface layers of the rail material due to the cyclic action of high contact stresses, combined with shear stresses caused by wheel slippage in longitudinal and transverse directions, compressive stresses from rail bending, shear stresses from compressed rail torsion, and residual technological stresses. Thus, microcracks (head checking) on the side face of the rail head in curved sections of the track appear in places with increased material hardness, which is the result of deformation hardening of the surface layers due to the action of high tangential stresses during micro-slipping of the wheels. Areas with increased hardness are very narrow (less than 0.5 mm) and are located periodically at a distance from 2 to 5 mm from each other (Figure 1a). Cracks propagate downward from the lateral edge at an angle of \(\sim\)45\({}^{\circ}\) to the direction of movement and penetrate the metal at an angle of \(\sim\)15\({}^{\circ}\) to the face surface. At the same time, in the middle part of the rail head, such surface cracks are located chaotically (Figure 1b), which indicates that the wheels slip relative to the rail which occurs in various directions and the entire rolling surface is subjected to work hardening. At the same time, in the central part of the rail head, such surface cracks are distributed chaotically (Figure 1b), indicating that wheel slippage relative to the rail occurs in various directions and the entire running surface is subjected to work hardening. The thickness of the work-hardened layer depends on the properties of the rail and wheel materials, their relative hardness, technological stresses, temperature, etc., and can reach several millimeters. For internal cracks, the mechanism of formation is different. It is associated with the cyclic action of contact stresses (shear stresses from wheel micro-slippage have a negligible effect on their magnitude), which can cause plastic deformation of the rail material at a certain depth from the running surface (starting from 2 mm and beyond, depending on the axial load). Given the complexity of loading processes (stress trajectories  
Figure 1: Surface microcracks on rails of the Kyiv Metro in curved track sections: (**a**) on the lateral side of the rail; (**b**) in the central part of the rail head.