## 2 Experimental Setup  
The principle of the setup developed was described in [19]. It consists of a reduced-scale wheel (referred to in the following as the \"roller\") mounted in a carriage running on a full-scale rail 1 m long. For the dynamic investigations, a linear motion axis allows the roller carriage to move along the rail at a controlled, scaled-down speed. Mechanical criteria were considered in order for the roller to be representative of the full-scale behavior of the wheel-rail contact. In particular, the aspect ratio of the contact areas as well as the average contact pressure at both scales are maintained. The normal force is applied to the roller axis by an electrical actuator. It is measured using a load cell positioned between the actuator and the carriage and kept constant by a feedback loop. In this paper, the sizing of the roller has been designed to mimic the situation of a train of interest used on the French rail network (FRN). For a clean, smooth and fully conductive contact surface, with a normal force \(F=400\) N, the half-axes of the resulting elliptical contact area were calculated as \(a=0.47\) mm and \(b=0.29\) mm. The measurement of the wheel-rail electrical contact resistance is performed using the four-point method, as shown in Figure 1a, which illustrates the principle of this technique applied to the test bench. A first pair of wires and connections are dedicated to current injection, whereas a second one--as close to the contact elements as possible--is allocated for voltage measuring. This method eliminates resistive contributions from wires and connections. The current intensity is evaluated by measuring the voltage \(\mathrm{V_{s}}\) across a well-known shunt resistance \(\mathrm{R_{s}}\). Figure 1b shows a photograph of the concrete implementation, where connections to roller and rail are ensured using copper thread brushes. In practice, for the direct current (DC) tests reported in this paper, we used a KEITHLEY 2400 source meter (Keithley Instruments, Cleveland, OH, USA) with a resolution of 5\(\mathrm{  
The theoretical value of the contact resistance in the case of a single-spot elliptical contact between two bodies of resistivity \(\rho\) can be expressed as follows [21]:  
\[R_{c}=\left(\frac{\rho}{2a}\right)f(\gamma) \tag{1}\]  
with  
\[\gamma=\left(\frac{a}{b}\right)^{\frac{1}{2}}=1.27 \tag{2}\]  
Here, we have the following:  
\[f(\gamma)\simeq 1 \tag{3}\]  
and  
\[R_{c}\simeq\frac{\rho}{2a} \tag{4}\]  
Figure 3: _U-I_ characteristics for clean contact with \(F\) = 400 N and current I: (O) increasing from 0.1 A to 1 A, and (\(\square\)) decreasing from 1 A to 0.1 A.  
Figure 2: Evolution of the contact current (-left axis) and of the contact voltage (-right axis) for a clean contact between roller and rail (_F_ = 400 N). Parallelepiped coupons, measuring 100 mm \(\times\) 20 mm \(\times\) 10 mm, were made of rail and wheel steel in order to experimentally determine their respective resistivities. We found very similar values, in the order of \(2.3\times 10^{-7}\Omega\cdot\)m; thus  
\[R_{c}\simeq\frac{\rho}{2\pi}=0.24\;\mathrm{m}\Omega \tag{5}\]  
There is a factor of 40 between the measured value and the theoretical value. As the surfaces are a priori considered to be oxide-free, this discrepancy would tend to indicate that the current flow area is much smaller than expected due to the roughness, leading to the hypothesis of a multi-spot contact. Furthermore, it is reasonable to assume that this contact geometry did not change during the test, since the resistance remained constant (the heating effect of the current was not sufficient to soften the material locally). In the case of multi-spot contact, the number of spots (noted \(N\)) is influenced by the degree of surface polish (roughness), material hardness (\(H\)), and applied force (\(F\)). According to [22], \(N\) can be estimated by the following equation:  
\[N=2.5\cdot 10^{-5}H^{0.625}\;F^{0.2} \tag{6}\]  
This equation is deduced empirically from the experimental results obtained in [10].