## 5 Fatigue Life Assessment of the Key Fatigue Detail  
Numerical calculations of mixed-mode fatigue-crack growth are carried out for the fatigue life when the initial crack size \(a_{0}\) at the location of the stress concentration of the load-bearing structure is 1 mm, 2 mm, and 3 mm. First, the ratios of the stress-intensity factor ranges \(\Delta K_{II}\) and \(\Delta K_{III}\) to \(\Delta K_{I}\) for the fatigue crack tips in the process of fatigue-crack growth are calculated, respectively, as shown in Figure 9. It can be seen from Figure 9 that the ratios of \(\Delta K_{II}\) and of \(\Delta K_{III}\) to \(\Delta K_{I}\) fluctuate within 3%; therefore, the fatigue crack of this particular fatigue detail is a mixed-mode fatigue crack, which is in line with the assumption; however, the ratios are rather insignificant as the crack is mainly subjected to unidirectional axial force, and hence the fatigue-crack growth is still dominated by mode I opening-mode cracks. The effective stress-intensity factor range for the tip of the fatigue crack in the process of fatigue-crack growth can be obtained by Equation (8) [24]:  
\[\Delta K_{eff}=\sqrt{K_{I}^{2}+K_{II}^{2}+\frac{K_{III}^{2}}{(1-v)}} \tag{8}\]  
where \(K_{I}\), \(K_{II}\), and \(K_{III}\) are the effective stress-intensity factors corresponding to mode I, mode II, and mode III cracks at the crack tip, respectively, and \(v\) is Poisson's ratio. Figure 8: Curve of geometric correction coefficient \(Y\) with respect to \(a/t\). Figure 9: _Cont._The effective stress-intensity factor ranges of the fatigue crack tip during the fatigue-crack growth process for the three initial crack sizes are calculated. The relationship between the effective stress-intensity factor range and the fatigue crack length is shown in Figure 10. It can be seen from the figure that the effective stress-intensity factor range increases approximately linearly with the increase of the fatigue crack length, and the trends are almost the same for the three cases. Figure 10: Relationship between \(\Delta K_{\textit{eff}}\) and crack length \(a\). After obtaining the relationship between the effective stress-intensity factor range for the tip of the fatigue crack and the length of the fatigue crack, the number of loadings required for each step of the fatigue-crack growth and the fatigue life of the particular fatigue detail can be obtained by Equation (9) [24]:  
\[N=\sum_{1}^{n}N_{i}=\sum_{1}^{n}\int_{a_{0}}^{a_{i}}\frac{da}{C(\Delta K)^{m}}= \int_{a_{0}}^{a_{cr}}\frac{da}{C(\Delta K)^{m}} \tag{9}\]  
where \(N\) is the fatigue life; \(a_{0}\) is the initial crack size; \(a_{i}\) is the length of the crack growth; \(a_{cr}\) is the critical crack size; and \(C\) and \(m\) are the fatigue-crack growth rate parameters measured by the test. The number of loadings required for each step of fatigue-crack growth and the fatigue life of the fatigue detail are calculated, as shown in Figure 11. When the initial crack size is 1 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure for the fatigue details is 5.03 \(\times\) 10\({}^{11}\) times; when the initial crack size is 2 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure of the fatigue details is 3.62 \(\times\) 10\({}^{11}\) times; and when the initial crack size is 3 mm, the number of loadings for the fatigue-load calculation model II two-car model corresponding to the fatigue failure of the fatigue details is 2.84 \(\times\) 10\({}^{11}\) times. As is known from Section 2 of this paper, the fatigue damage caused by the fatigue-load calculation model II two-car model through a single pass of the uncorroded fatigue detail is 4.04 \(\times\) 10\({}^{-13}\), according to the Palmgren-Miner linear cumulative damage principle, the fatigue life of the detail corresponds to 2.48 \(\times\) 10\({}^{12}\) passes.