## 1 Introduction  
The accurate localization of trains is a paramount subject for ensuring the correct functioning of railway tracks and hence their safety. A \"track circuit\" system is classically employed to provide real-time information about the presence of a train in a specific track section. The short circuit between the two rails, created by its wheels and axle, triggers a signal indicating the occupancy of the zone, and the signaling systems can be activated accordingly. Maintaining the best possible electrical conduction at the wheel-rail contact is therefore a major issue for ensuring the reliability of the track circuit system. A high contact resistance leads to a phenomenon known as \"deshunting\", where the presence of the train is no longer detected for a certain period, resulting in a critical safety issue. The thin oxide layer at the wheel-rail interface is a well-known cause of deshunting, especially on tracks with low frequentation, and despite the implementation of various devices to prevent [1, 2] or reduce the occurrence of deshunting, many instances are still detected every year. Deshunting events can have a detrimental effect on railroad management and give rise to potentially dangerous situations. It is therefore necessary to better understand the behavior of the wheel-rail electrical contact in order to minimize their occurrence and make track circuits more robust to failure. The subject of wheel-rail contact is a complex multi-physical subject involving mechanical and electrical aspects. In [3; 4; 5], the detection and impact of the mechanical deterioration of the wheel rail was studied. The parameters of the surface, such as surface roughness [6; 7], local temperature and physical chemistry [8], have a strong influence on the wheel-rail contact resistance. In some cases it was also reported that arcing between the wheel and the rail could occur, thus leading to microstructural modifications of the surface [6; 9]. The state of the surface and the difficulties inherent in this multi-physical character apply not only to the two solids in contact, but also to a layer called \"third body\" present in their interface. This third body has a wide variety of components that can affect the friction at the interface [10; 11] since it can incorporate wear particles from wheels and rails, external contaminants such as dead leaves, frost, sand, and oil, as well as rust formed naturally on the surfaces [12]. In [13], third body samples were examined by tribological, morphological and microscopic analysis, and their electrical conductivity, heterogeneity and compactness properties were compared between good and bad shunting conditions. Rust can have serious effects on the maintenance of railways, especially in costal or humid areas where corrosion can be more severe. As matter of fact, the rain can deposit salts or contaminants contained in the atmosphere, leading to acid rains, acting as a catalyst to the corrosion process [14]. The rust composition was studied by [PERSON] et al. using in situ X-ray diffraction [15]. On the railhead, the main species detected were FeOOH oxyhydroxides in various forms and Fe\({}_{3}\)O\({}_{4}\) magnetite. From a more global point of view, [PERSON] et al. [16] examined the influence of the type of track environment and of the traffic frequency on the thickness of rust formed on rails and wheel roughness. Using a huge test bench involving real wheel and rail elements, they also studied the influence of several factors, such as normal load, current intensity and oxide layer thickness on contact resistance. A few years later, the understanding of wheel-rail electrical contact was enriched by new studies carried out in parallel on a 4-scale test bench and on site using an instrumented regional train, both stationary and running at low speed [17; 18].