## 2 Materials and Methods  
The bulk material employed in the experiments was the quenching and tempered 32 CrMoV 13 steel, whose composition is (wt %): 0.32% C; 0.35% Si; 0.52% Mn; 0.14% Ni; 3.0% Cr; 0.83% Mo; 0.28% V. Austenitizing was performed at 950 \({}^{\circ}\)C for 30 min followed by quenching in oil. In the following, the specimens were subjected to tempering, at 635 \({}^{\circ}\)C for a holding time of 1 h, followed by cooling in air. Finally, the specimens were subjected to plasma nitriding with parameters chosen to obtain a nitrided layer of about 0.3 mm depth. The thickness, morphology, composition, and structure of the nitrided layers were observed by optical microscopy, scanning electron microscopy (SEM) and X-ray diffraction (XRD). Vickers microhardness profiles were carried out along the treated layers on the cross section of the specimens, using a load of 25 g for 15 s, enabling us to analyse how mechanical properties vary throughout the treated layers. A two-disc type RCF testing machine with lubrication was used to simulate rolling contact conditions [27]. Specimens of 70 mm diameter, 7 mm width and 40 mm curvature radius were used. In the RCF tests, both discs were subjected to 11.7 m/s linear velocity and the oil temperature was maintained at 56 \({}^{\circ}\)C. In RSCF the linear velocity of the discs was U\({}_{1}\) = 11.7 m/s and U\({}_{2}\) = 10.3 m/s, which means that the faster (driver) specimen has the same rolling and sliding directions and the slower (follower) specimen has opposite rolling and sliding directions. As lubricant, a hydraulic oil with 46 mm\({}^{2}\)/s kinematic viscosity at 40 \({}^{\circ}\)C (ISO VG 46) and a viscosity pressure coefficient equal to 13.39 GPa\({}^{-1}\) was used [28]. The initial contact conditions and the results obtained for the minimum oil film thickness, \(h\), superficial roughness, \(Ra\), lambda ratio, \(\Lambda\), and sliding rate are shown in Table 1. The oil film thickness, \(h\), was determined by the formula of [PERSON] and [PERSON] [29]. The lambda ratio, \(\Lambda\), is the relationship between the oil film thickness, \(h\), and the composite surface roughness, \(\sigma\)[30]:  
\[\Lambda=\text{h}/\sigma \tag{1}\]  
In the RSCF tests the sliding rate, \(\lambda\) was determined by the following equation,  
\[\lambda=(\text{U}_{1}-\text{U}_{2})/(\text{U}_{1}+\text{U}_{2}) \tag{2}\]  
where U\({}_{1}=11.7\) m/s and U\({}_{2}=10.3\) m/s are the circumferential linear velocity of the specimens. To identify the failure mechanisms, the specimens were analysed after testing with magnifying lenses, by optical microscopy and SEM. A cut-off of 0.8 mm was used for measuring the arithmetic average value of filtered roughness (Ra) before and after the RCF tests, following the standard DIN 4768. Residual stress determination was performed by XRD using the sin\({}^{2}\)\(\psi\) method [31]. Lattice deformations of the [211] diffraction planes of the \(\alpha\)-Fe phase were determined for six \(\phi\) directions using Cr K\({}_{\alpha}\) wavelength with vanadium filter in the diffraction beam. Surface and in-depth residual stresses were determined before and after RFC tests, in three zones of the cylindrical specimens, A, B and C, equally spaced by 120 degrees (Figure 1a). Before the RFC tests, the residual stress measurements were also carried out, for some specimens, in three points equal spaced along the width of the specimens, as shown in Figure 1b, to understand the homogeneity of the nitriding treatment. After RCF tests, due to the width of the rolling track, the measures are taken only in the centre of the specimens. The subsurface residual stress was determined at varying depths below the surface, using the electropolishing removal technique. Stress directions were defined, as shown in Figure 1b: \(\sigma_{11}\) is the circumferential or hoop stress tangent to the rolling track in the rolling direction, and \(\sigma_{22}\) is the axial stress, perpendicular to the rolling direction (Figure 1b).