# Theoretical Study of the Wear of a Reduced-Diameter Wheel for Freight Wagons, Based on Its Diameter
## 2 Methods and Materials
### Calculation Model
#### 2.4.9 Calculation of Wear  
Wear significantly damages wheels, drastically reducing their lifespan. This wear occurs due to both abrasive and adhesive processes, and there are models based on wear rates that help determine the wear depth and characterize the damage [23]. Abrasive wear results from the relative movement and roughness of the wheel and rail surfaces, leading to friction and the subsequent loss of material from both. In contrast, adhesive wear arises from plastic deformation and the cohesive forces (such as Van der Waals, electrostatic, or chemical) that develop between the surfaces, resulting in material transfer from one surface to the other [40]. For characterizing wheel wear, Ref. [18] proposed the following hypotheses:  
1. The equations are parameterized for abrasive wear rather than adhesive wear for several reasons: (1) Plastic deformation occurs, but modeling it accurately is challenging without finite-element methods, which are computationally intensive. (2) It is  
Figure 13: (**a**) Curve negotiation by a bogie; (**b**) velocity vectors; (**c**) reference frame \(\overline{uv\omega}\); and (**d**) forces involved when curving. Source: own elaboration. reasonable to assume that abrasive wear is the primary contributor. (3) When the mathematical tools are calibrated with experimental data, both wear phenomena are inherently included in the resulting wear law. 2. The various mathematical tools analyze the wear on the wheel profile, where the wear calculated at each moment is cumulative. 3. Wear is considered to be uniform, i.e., the focus is on the variation of the transverse profile rather than on pattern formation in the longitudinal (circumferential) direction. Therefore, the wear measured at a specific position and time is extrapolated to the entire circumference. 4. There are no pollutants present at the contact interface. The impact of pollutants is accounted for by adjusting the friction coefficient or by introducing new wear laws. Regarding these hypotheses, the models examined in Refs. [18; 23] can be applied to characterize wheel wear: * Energy transfer models. These models compute the energy dissipated at the wheel-rail interface and associate it with the wear rate, which can be ultimately associated with the wear depth. There are various models, each with its own wear law: [PERSON]'s model, which is based on the energy flow; the model developed by the British Railway Research (BRR), which is based on a non-continuous wear law depending on the wear regime (mild, transition, severe); and the model developed by the University of Sheffield (USFD), which is based on a continuous wear law divided into several regimes (mild, severe, catastrophic). * Reye-Archand-Khruschchov (RAK) model. This is the simplest model and characterizes the abrasive wear appearing at the slip zone of the contact area. In this model, the volume of material lost is expressed as a function of the slip speed, normal force, hardness of the wheel material (steel), and a coefficient coming from a wear chart divided in wear zones depending on the normal pressure and the slip speed. In Ref. [17], energy transfer models and the RAK model are collected and assessed. The RAK model is hard to implement because it requires a high computational accuracy; if pressure or slip speed is miscalculated, then the RAK coefficient may be wrong and be in another order of magnitude. As for the energy transfer models, the one with the easiest implementation and lowest computational effort is the USFD model since its wear law is continuous, so small errors do not lead to great errors in the end. There are also other models that compute wheel wear by considering dynamic loading, especially in the event of the train running on a turnout/switch [41; 42]. However, this is neither considered nor introduced in the current paper as clarified in the hypotheses list. Figure 14 presents the wear calculation according to the USFD, which can be removed by reprofiling when its depth reaches a determinate threshold [21; 43; 44].