## 4 Analysis of Fatigue Crack Propagation Test Results
### Effect of Notch Shape on Crack Propagation Rate  
In order to investigate the influence of shape of machine-cut notches on the fatigue crack propagation of steel wire, notches with different shapes are used for the test in this paper. S-1, S-2 and S-3 represent the specimens with semi-ellipsoidal notch, trapezoidal notch and v-shaped notch, respectively. The schematic diagram of steel wire and corresponding wire specimen with notches of different shapes are shown in Figure 24.  
Figure 25 shows the section morphologies of the three specimens without the hooking, the section morphologies with the hooking and the schematic diagrams of the hooking. It can be seen that with the increase of loading numbers, the three specimens all have different beach-like load shedding lines at the cross section. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Numbering**} & **Fitting** & **Coefficient** & **Coefficient** & **Fitting** & **Coefficient** & **Coefficient** \\  & **Curve** & **A (10\({}^{-4}\))** & **B** & **Curve** & **A\({}^{\prime}\) (10\({}^{-4}\))** & **B\({}^{\prime}\)** \\ \hline L-1 & _a-N_ & 0.519 & \(-\)0.753 & _c-N_ & 1.187 & \(-\)1.018 \\ L-2 & _a-N_ & 0.506 & \(-\)0.715 & _c-N_ & 0.931 & \(-\)0.527 \\ L-3 & _a-N_ & 0.431 & \(-\)0.434 & _c-N_ & 1.065 & \(-\)0.492 \\ L-4 & _a-N_ & 0.383 & \(-\)0.392 & _c-N_ & 0.777 & 2.910 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Coefficient table. Figure 23: Variation of crack width with cyclic numbers. Figure 24: _Cont._Figure 24: Wire specimens with machine-cut notches of different shapes. (**a**) No. S-1 steel wire specimen; (**b**) dimension of No. S-1 steel wire; (**c**) No. S-2 steel wire specimen; (**d**) dimension of No. S-2 steel wire; (**e**) No. S-3 steel wire specimen; and (**f**) dimension of No. S-3 steel wire. Figure 25: Steel wire sections with notches at different shapes. (**a**) S-1 without “bench-like pattern”; (**b**) S-1 with “bench-like pattern”; (**c**) schematic diagram of S-1 wire specimen; (**d**) S-2 without “bench-like pattern”; (**e**) S-2 with “bench-like pattern”; (**f**) schematic diagram of S-2 wire specimen; (**g**) S-3 without “bench-like pattern”; (**h**) S-3 with “bench-like pattern”; and (**i**) schematic diagram of S-3 wire specimen. Table 7 shows the change of crack depth and width of three groups of steel wires with the cyclic loading numbers. Here, \(a\) is the depth of crack; \(c\) is the width of crack, and \(N\) is cyclic loading numbers. Figure 26 shows the variation of crack depth with cyclic numbers according to the data in Table 7. It can be seen that the rate of crack propagation presents a uniform trend. The sharper the shape of the notch, the faster the rate of crack propagation. It can also be seen from Figure 26 that the crack depth shows a linear correlation with the cyclic loading numbers. In order to distinguish the influence of notch shape on crack propagation, linear fit is used for the a-\(N\) and c-N curves. \(a=A\times N+B\), \(c=A^{\prime}\times N+B^{\prime}\). The fitting coefficients are shown in Table 8. It can be seen that the sharper the notch shape, the larger the coefficient A, and the higher the crack propagation rate.