# Crack Initiation and Propagation Fatigue Life of Ultra High-Strength Steel Butt Joints
## 4 Fatigue Assessment
### Crack Initiation and Propagation Life  
The procedure for optical crack detection and tracking presented in Section 3 allows a separation of the crack initiation life from the total fatigue life until burst fracture. However, the method's main result is the surface crack length \(2c\), whereas the crack depth \(a\) is of importance for specimen failure. Therefore, a formula for the crack aspect ratio \(a\)/\(c\) given in [62] is applied (see Equation (5)). This expression was derived from fillet weld toes by fitting of experimental data, but according to the authors, it can be conservatively used for butt welds as well.  
\[2c=-0.27+6.34\cdot a\quad\text{for 0.1 mm}<a<3\text{ mm} \tag{5}\]  
The threshold crack depth defining the line between crack initiation and crack propagation life is set to \(a_{th}=0.5\) mm for all investigated specimens [3; 4; 63]. According to Equation (5), the corresponding surface crack length is \(2c=2.9\) mm, giving a rather low crack aspect ratio of \(a\)/\(c=0.345\). This value is quite comprehensible, as the local stress concentration at the weld toe decreases rapidly in the sheet thickness direction and therefore encourages the crack to grow in the surface direction first [64; 65]. The fracture surface in Figure 14b exhibits a small crack on the left hand side marked by a yellow arrow exhibiting a very low aspect ratio. A detailed measurement reveals an aspect ratio of \(a\)/\(c=0.35\) validating the result of Equation (5) for the present case. The crack detection procedure delivers the surface crack length in quite coarse steps of load cycle numbers. In order to determine the according threshold load cycle number \(N_{init}\) for \(a_{th}\), linear interpolation is applied.  
Figure 18 shows the effective notch S-N curve using \(\rho^{*}=0.13\) mm of specimen failure in blue dye and crack initiation with \(a_{th}=0.5\) mm in green dye. Table 6 summarizes the effective notch stress fatigue test data. Furthermore, the statistical evaluation including lines of 50% and 97.7% probability of survival are depicted. Some specimens exhibit more than one propagating crack leading to a number of fourteen crack initiation points of ten specimens. In such cases, the effective notch stress values are evaluated at the respective position of the crack initiation points of the particular specimen.  
Figure 19 displays the portion of crack initiation life for the ten through cracks with respect to the effective notch stress range. Hereby, the portion of crack propagation life goes along with increasing stress range although large scatter is observed. However, the technical crack initiation phase seems to be the major portion of the fatigue life. In this context, Equation (6) presents a formula for the  
Figure 18: S-N curve including crack initiation life and cycles to final rupture of all investigated specimens.  
trace of the crack propagation fraction based on the statistical evaluation of the test results for crack initiation (index _init_) and burst fracture (index _f_). The consequential equation is a power law with basis \(\Delta\sigma\) and the difference of the slopes of the S-N curves as the exponent are applicable for varying the probabilities of survival. However, the scatter indices in the present case are nearly identical causing the equation to be almost independent of the employed probability of survival \(P_{S}\)-value. The result of Equation (6) is depicted in Figure 19 for \(P_{S}=97.7\%\).  
\[\frac{N_{init}}{N_{f}}(\Delta\sigma)=\frac{N_{\text{FAT}}\Big{(}\frac{FAT_{ init}}{\Delta\sigma}\Big{)}^{k_{init}}}{N_{\text{FAT}}\Big{(}\frac{FAT_{f}}{ \Delta\sigma}\Big{)}^{k_{f}}}=\frac{(FAT_{init})^{k_{init}}}{\big{(}FAT_{f} \big{)}^{k_{f}}}\cdot(\Delta\sigma)^{k_{f}-k_{init}}=\text{const}\cdot(\Delta \sigma)^{k_{f}-k_{init}} \tag{6}\]