## 3 Results  
### Influence of the Contact Ring Thickness  
To achieve a better understanding of the effect of the contact ring on the stress distribution, the chromatic maps corresponding to the radial, tangential, and von Mises stress distributions are shown in Figures 5-7 for a conventional fit (without a contact ring) compared with the ones obtained in a shrink fit with a contact ring of 5% of the total hub thickness, \(\gamma\) = 0.050. These distributions allow one to obtain in a qualitative way the effect of the contact ring on the stress field. Figure 4: Mesh convergence validation for contact pressure \(p\) and relative error \(\varepsilon\) in terms of mesh factor \(K\), defined as the ratio between the number of elements of each mesh and the number of elements of the coarsest mesh, corresponding to: (**a**) a case considering the hub and shaft composed of the same material and (**b**) a case considering the hub and shaft composed of different materials. Figure 5: Distributions of the radial stress in: (**a**) a conventional shrink fit (without a contact ring) and (**b**), a modified shrink fit with a contact ring (\(\gamma\) = 0.050). Figure 6: Distributions of the tangential stress in: (**a**) a conventional shrink fit (without a contact ring), and (**b**) a modified shrink fit with a contact ring (\(\gamma\) = 0.050). As can be observed, at the interface of the shaft/hub, a high-stress concentration appears (right side of the hub). This concentration is more intense in the radial component at the hub (Figure 5). In a similar way, it is observed that in the tangential stress (Figure 6) and von Mises stress (Figure 7) fields, the maximum values are moved towards the interface of the hub/contact ring. However, the values of the stress undergone in the case of the contact ring are lower than the ones obtained in the case of a fit with a conventional hub (Figures 5, 6 and 7). To go deep into the analysis and quantify the effect previously described, in Figure 8, the axial distributions are shown for the radial stress and in a similar way, Figure 9 for the tangential stress and Figure 10 for the von Mises stress for each case of study. According to the results shown in Figure 8, the radial stress is uniformly distributed throughout the interface up to the vicinity of the hub edge (\(z>40\) mm). There, a high increment of the radial stress is observed reaching the maximum value at the edge of the hub (\(z=50\) mm), revealing the well-known huge stress concentration. Regarding the stress distribution in shrink fits using contact rings, the same shape with a stress concentration at the hub edge is obtained, but key changes can be easily observed in those distributions. Firstly, the radial stress at the edge of the hub decreases as the contact ring thickness is increased, reaching a stress reduction of up to half. In addition, it can be observed that the zone of high stress (right side of the plot) is reduced as the contact ring thickness is increased. In a similar way, the radial stress at the central section of the hub (where such stress is uniformly distributed) is reduced with the contact ring thickness, but in a less marked way, being only noticeable for contact ring thicknesses higher than 12.5% of the total thickness. Taking into account the relationship between the maximum transmitted  
Figure 8: Axial distributions of the radial stress at the interface contact ring/shaft for diverse values of the thickness ratio. Figure 7: Distributions of the von Mises stress in: (**a**) a conventional shrink fit (without a contact ring), and (**b**) a modified shrink fit with a contact ring (\(\gamma=0.050\)). torque and the contact pressure (equal to the radial stress but with the opposite sign), it can be concluded that the contact ring let to reduce, in both magnitude and size, the stress concentration appearing at the edge of the hub without causing significant changes in the maximum transmitted torque (always lower than 10%) since the contact pressure distribution is similar far away to the stress concentrator.