# Power Dissipation and Wear Modeling in Wheel-Rail Contact
## 2 Formulation of Wheel-Rail Contact Problem
### Wear Evaluation Models  
The main factors generating the wear of wheel or rail are material oxidation, plastic deformation, crack growth and propagation [1; 3; 8; 9; 14; 22]. The estimation of the worm material amount is of primary interest among railway engineers. The accurate estimation of the removed material volume resulting from tribological processes along the contacting surfaces is in general very difficult to perform and requires intensive computational effort. Therefore, nowadays, simplified approaches are used to predict the wear of wheel or rail materials. They consist of the formulation of one simple equation to calculate the amount of worn material. This equation is tuned on experimental data relating wear to the contact patch parameters. However, it is also the weakness of this approach. The wear law has to be usually calibrated in a laboratory experiment. So, it implies its application in operational conditions very similar to the conditions imposed on materials or contact geometry during the laboratory test. The extension of these wear laws beyond the tested condition can lead to incorrect wear evaluation. Since on-track tests are very expensive and the reproduction of the experimental conditions is limited, the required data are obtained in laboratory experiments.  
We shall consider abrasive wear only. In this type of wear, the worn material in the form of wear debris is assumed to be shifted away from the gap between the bodies. The worn and removed material updates the location and geometry of the contacting surfaces. It implies that the position and shape of the contacting surfaces is also updated. The modification of the contact interface leads finally to the updating of tangential and normal forces along the contact patch. In turn, it causes the modification of the wear process and a gap between the bodies. Due to this complex relation between contact geometry and forces, the modeling of the contact problems with wear is recognized as a challenging task [17; 22]. The numerical approach to the model wear process depends on the amount of worn material. When the position and shape of the contacting surfaces do not change significantly and wear debris volume is small, such changes are not considered. The contact displacement and stresses as well as the wear depth are calculated for a given contact geometry. Next, the wear gap and interface location are updated according to the calculated wear depth function. In the second approach, when the volume of the wear debris is large, the impact of wear on deformation has to be considered. So, for a given contact interface position, displacement, stresses, wear depth function as well as worn material volume are calculated. This volume is estimated using the formulas relating it with energy or power dissipated due to the work of friction force [22]. Next, the new position of the contact interface is evaluated using the worn material volume. The removed rail material is distributed among a few finite element layers close to the contact area layer rather than inside the contact interface layer only. It causes an update of the contact node's position as well as the shape of the contacting surfaces.