# Power Dissipation and Wear Modeling in Wheel-Rail Contact
## 1 Introduction
The analysis of different parameters influencing the evolution of the wear process as well as the results of the experiments allow the researchers to formulate many different wear prediction models available in the literature [1; 3; 8; 12; 13; 14; 15]. Although these models are different in their details, their numerical solution requires us to perform the same steps. Based on the solution of a dynamic equation coupling the vehicle and track, the forces and creepages along the contact area are calculated. Next, it allows us to evaluate the volume of the worn material as well as the wear rate. Using calculated wear data, the original surfaces of the wheel and profile can be updated. The forces in the contact patch or the wear data can be estimated either locally or globally, i.e., in each finite element inside the contact area or along the whole contact interface. In this paper, we shall use and implement the power dissipation model combined with the Archard model to compute wear rather than the classic Archard model [12] only. Recall from [16; 17; 18] that since the wear distribution updates the contact geometry, it causes that numerical modeling of contact problems is not easy and generates interest in developing new methods. [PERSON]'s wear model relating the update of the contact surface with normal traction and slip velocity is frequently used in the railway area to predict the wear depth of the contacting surfaces. From this law, the wear depth function is calculated. This function is treated as an internal variable and is used in the contact nonpenetration condition. The numerical application of this model is based on the assumption that the volume of the worn material is suitably small. It means that changes in the original wheel or rail profiles are small and they can be practically neglected. This assumption allows us to calculate stresses and wear distributions on a given geometry and next slightly update the finite element mesh close the the contact interface. When the material loss in the contact area due to wear is large, it has to be taken into account. It means that the change in the shape of the wheel and rail profiles may be also significant. Moreover, since contact traction is also altered, it leads to the modification of the wear evolution process. From the literature [19; 20; 21; 22; 23; 24; 25], it follows that in this case, the evaluation of the wear rate based on the power or energy dissipation model is more accurate and stable than based on that Archard model. Energy dissipation based on the wear model relates the worn material volume and energy dissipated by the friction force. This energy is calculated as a product of the friction force and the tangential displacement. The proportional coefficient is called the energy wear coefficient [22] and is evaluated experimentally. In numerical computations, in each iteration, contact surface location as well as its deformation due to wear is updated. Numerically, this update can be realized using two different approaches. The first approach consists of using the general adaptive finite element method. Due to an update of the mesh from iteration to iteration, the degradation of the finite elements is avoided. The second approach combines the wear depth function and worn material volume calculation. When this volume is large, the position of mesh nodes in a wear box [17] close to the contact surface is changed to match this worn material volume and to avoid the degradation of the finite element mesh. The worn material is redistributed among the finite elements belonging either to current as well as other layers outside the contact area. Rolling contact problems with friction and/or heat flow are intensively studied in the literature [26; 27; 28]. Paper [26] is concerned with the numerical simulation of the thermo-elastic wheel-rail contact problem, where the stress reduction on the contact boundary is ensured by functionally graded material covering the rail. This problem has been solved numerically using a quasi-static approach.