## 4 Discussion  
Differences in the stress state appearing at the interface shaft/hub have been quantified from the dimensionless parameter \(\eta_{\rm I}=\sigma_{\rm i,modified}/\sigma_{\rm i,theoretical}\) defined as the ratio between the maximum stress of the modified hub considering a contact ring (\(\sigma_{\rm i,modified}\)) and the stress given by the theory of pressure cylinders for a conventional interference fit (\(\sigma_{\rm i,theoretical}\)), \(i\) being the radial, tangential, and von Mises stress components. The variation of these parameters with the contact ring thickness ratio is shown in Figure 13.  
Figure 12: Axial distributions at the interface shaft-hub of: (**a**) radial stress, (**b**) tangential stress, and (**c**) von Mises stress for diverse materials of the contact ring (\(\gamma=0.125\)). According to the results shown in Figure 13, the stress concentration factor obtained using contact rings is lower than the one obtained in a conventional shrink fit (\(\gamma=0\)). This effect is observed even for low-thickness contact rings. In the case of the parameter radial \(\eta_{r}\), the stress concentration is decreased as the contact ring thickness increases. Such a decrease is more marked for low thickness ratios (\(0<\gamma<0.250\)) and becomes soft for higher thickness ratios. The variation of the von Mises parameter similarly decreases with the contact ring thickness ratio, whereas the tangential parameter is softly increased with \(\gamma\). Thus, according to the obtained results for contact rings with thickness ratios within the range \(0.125<\gamma<0.250\), a reduction in the stress with regard to a conventional fit (without a contact ring) as high as 40% is obtained: radial component (39.1%); tangential (44%); and von Mises (38%). Instead of this, stress reduction increases with the contact ring thickness ratio and, consequently, will be higher for high values of \(\gamma\); from a practical point of view, it is considered that the contact ring thickness should be notably lower than the total hub thickness. Therefore, according to the obtained results, it is considered that the recommended contact ring thickness must be included within the range \(0.125<\gamma<0.250\). From the obtained results, the influence of the contact ring thickness ratio can be derived in terms of the following equations that were obtained by fitting the curves shown in Figure 13 using a quadratic interpolation resulting the following equations for estimating the parameters \(\eta\) radial, tangential, and von Mises, respectively:  
\[\eta_{r}=0.00181\ \gamma^{2}-0.06686\ \gamma+1.92640,\quad R^{2}=0.9700 \tag{3}\]  
\[\eta_{\theta}=-0.00010\ \gamma^{2}+0.0050X\ \gamma+0.57006,\quad R^{2}=0.9983 \tag{4}\]  
\[\eta_{vM}=0.00033\ \gamma^{2}-0.01110\ \gamma+0.83144,\quad R^{2}=0.9519 \tag{5}\]  
According to the results, the correlation coefficient is higher than 0.95, so these equations can be useful for estimating the degree of reduction in terms of the dimensionless contact ring thickness within the interval analyzed. Regarding the variation of the parameter \(\eta\) with the interference radial, it can be seen that it increases linearly with the radial interference for all the cases considered \(\eta\) (radial, tangential, and von Mises). It must be remarked that for the case of the higher contact ring thickness of \(\gamma=0.250\) (blank symbol in Figure 14), the stress concentration is lower. To complete the analysis and quantify the stress reduction, from the stress distributions of Figure 12, the variation with the Young modulus of the material of the contact ring of the parameter \(\eta\) was obtained for the 28 cases of study (Figure 15a). It deserves to highlight that among the selected materials, four of them had a similar Young modulus (around 190 GPa) and different Poisson coefficients varying within a range of 0.2 and 0.33. This fact  
Figure 13: Variation of the parameter \(\eta\) radial, tangential, and von Mises as a function of the contact ring thickness ratio.