# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Calculation Model
#### 2.4.6 Solution of the Geometric and Normal Contact Problems  
An isolated wheel transmits its own weight and load to the rail via a common interface: the contact area. This contact area must be greater than zero to avoid infinite normal stress. To compute wear and identify its location, both the contact area and normal stress need to be determined. The first is a geometric problem, while the second pertains to normal contact. As explained in Ref. [23], contact between two bodies can be either non-conformal or conformal. In non-conformal contact, the contact area is relatively small compared to the bodies' characteristic sizes. In contrast, conformal contact occurs when the geometry of one body adapts to that of the other, resulting in a larger contact area--this can happen when the wheel and rail are worn enough that their geometries align. Ref. [23] also notes that if the materials in contact are quasi-identical, the problem simplifies significantly. The quasi-identity requires a relationship between the shear modulus and Poisson's ratio, which is satisfied in wheel--rail contact since both are made of the same material (steel). Figure 10: **(a)** Force diagram when the gradient is positive (left) or negative (right); (b) normal force components at the contact area; and (c) force diagram of a wheelset negotiating a canted curve. Source: own elaboration. Both the geometric and normal contact problems are solved simultaneously, and the theories for addressing them are discussed in Refs. [18; 20; 23]:  
* Hertzian contact theory. This theory was the first satisfactory analysis for the stresses appearing at the contact zone between 2 elastic solid bodies and solves the geometric problem at the same time if a series of hypotheses are fulfilled. According to this theory, the contact area is the intersection of two perfect paraboloids: a perfect ellipse. * Kik-Piotrowski theory. This is a quasi-Hertzian theory and is also based on the virtual interpenetration between surfaces. It assumes the same pressure distribution in the longitudinal direction as Hertz, but not in the lateral direction as the curvature is not always constant in that direction. It is interesting to point out that this theory disregards the real shape of the bodies and replaces them by elastic half-spaces, which allows employing Boussinesq's influence functions. * Ayasse-Chollet. This is also a quasi-Hertzian theory, a variant of the previous one. * Stiff approach. This theory is based on a stiff contact in which there is a theoretical contact point for which a series of constraints is imposed. As stated in Ref. [17], which collects the theories, the Herztian contact theory is the most common due to its high accuracy, low computing effort and because the hypotheses it brings are fulfilled for most of the cases. Here is the list:  
1. The bodies in contact are homogeneous, isotropic, and linearly elastic. 2. Displacements are assumed to be infinitesimal, significantly smaller than the characteristic dimensions of the bodies. 3. The surfaces of the bodies are smooth in the contact zone, meaning they have no roughness. 4. Each body can be modeled as an elastic half-space, necessitating non-conformal contact. 5. The surfaces of the bodies can be approximated by quadratic functions near the point of maximum interpenetration, indicating that the curvatures (the second derivatives of the functions) are constant. 6. The distance between the undeformed profiles of both bodies at the maximum interpenetration point can be modeled as a paraboloid. 7. The contact between the bodies occurs without friction, allowing only normal pressure to be transmitted. In Figure 11, the most representative images of this model are presented. After that, the main formulae are shown, which come from the aforementioned references and also from Refs.