# Fretting Fatigue Behavior under Tension-Bending Mixed-Mode Loading
## 4 Results and Discussion
### Stress Analysis  
The maximum Von Mises equivalent stress of the specimen \(\sigma_{\text{eff,max}}\), the distribution of normal stress \(p(x)\), the horizontal component of stress \(\sigma_{\text{xx}}\), and the tangential stress \(q(x)\) on the contact surface are examined in this section to understand the fretting fatigue behavior of the material under tension-bending mixed-mode loading. The \(\sigma_{\text{eff,max}}\) produced in any loading case is less than the yield stress, \(\sigma_{\text{y}}\), for the material (\(\sigma_{\text{y}}=\)310 Mpa [13]), as shown in Figure 7, and it demonstrates that the linear elastic constructive relation is reasonable. Figure 7 also shows that the \(\sigma_{\text{eff,max}}\) increases with the increase in the dimensionless stiffness ratio, \(\gamma\), and oblique load, \(\lambda\), but the slope of the curve related to the mode angle \(\theta\) decreases rapidly. The curve of \(\sigma_{\text{eff,max}}\) vs. \(\theta\) grows rapidly when \(\theta<45^{\circ}\) but slowly increases when \(\theta>45^{\circ}\), as shown in Figure 7c, which indicates that the risk of material yielding is the highest for the bending-dominated (\(45^{\circ}<\theta<90^{\circ}\)) mixed-mode loading and it can be conservative and simple to estimate the bending-dominated fretting fatigue by the bending fretting fatigue strength. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\lambda\) (\(10^{-4}\)) & **1.063** & **1.328** & **1.594** & **1.860** \\ \hline Contact width (\(x/t\)) & 0.259 & 0.259 & 0.260 & 0.260 \\ Stick zone width (\(x/t\)) & 0.162 & 0.143 & 0.132 & 0.132 \\ Ratio of stick–slip width & 1.680 & 1.231 & 1.038 & 1.039 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contact sizes for different oblique loads (\(\theta=45^{\circ}\), \(\gamma=1\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\gamma=0.4\) & \(\gamma=0.5\) & \(\gamma=0.67\) & \(\gamma=1\) & \(\gamma=1.5\) & \(\gamma=2\) & \(\gamma=2.5\) \\ \hline Contact width (\(x/t\)) & 0.341 & 0.316 & 0.289 & 0.259 & 0.236 & 0.225 & 0.217 \\ Stick zone width (\(x/t\)) & 0.232 & 0.211 & 0.188 & 0.162 & 0.142 & 0.131 & 0.123 \\ Ratio of stick–slip width & 2.126 & 2.022 & 1.867 & 1.680 & 1.515 & 1.394 & 1.312 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Contact sizes for different stiffness ratios (\(\theta=45^{\circ}\), \(\lambda=1.063\times 10^{-4}\)). Figure 8 shows the dimensionless normal stress, \(p(x)/E_{1}\), distribution on the contact surface. As shown in Figure 8a,c, the peak value and the curve profile of the normal stress are almost the same for different oblique loads and mode angles, but the location of the peak normal stress (i.e., contact center) moves towards the trailing edge of the contact as the two variables increase. These phenomena indicate that the peaks of the contact stress and the contact width are dominated by the normal load, \(P\), but the position of the contact center and the slip regime of the contact zone will be affected by the mode angle and oblique load. Unlike \(\lambda\) and \(\theta\), the peak value of contact pressure increases with the increase in stiffness ratio \(\gamma\), but the position of the contact center remains constant (shown in Figure 8b), which indicates that the variable, \(\gamma\), can function in adjusting the preload between two contact bodies (such as bolts and other fasteners). Figure 9 demonstrates the effects of \(\theta\), \(\lambda\), and \(\gamma\) on the dimensionless surface stress component \(\sigma_{xx}/E_{1}\). As shown in Figure 9, \(\sigma_{xx}/E_{1}\) is compressive at most of the contact surface but transforms into tensile stress at the leading edge, and the fitting fatigue crack propagation is dominantly by the high gradient of tensile stress [39]. As expected, when a larger oblique load is applied, the peak of tensile stress is higher, and its position moves toward the contact center, as shown in Figure 9a.