# Effect of the Depth of Decarburized Layer in SKL15 Tension Clamp on Fatigue Strength
## 3 \(\Delta\)K\({}_{l}\) According to the Depth of the Decarburized Layer  
Of the section forces acting on the fatigue fracture surface, the tensile force and the moment about the \(y\) and \(z\) axis affect the opening mode of cracking. Thus, assuming the decarburized layer as the initial crack, \(\Delta K_{l}\) at the crack tip is defined as \(\Delta K_{l}\) generated by each force. Because the tension clamp was made of a circular steel bar, the \(\Delta K_{l}\) equation for tensile force and bending moment was used when there was a crack around the circular steel bar [9]. When the depth of the crack is significantly shallower than the diameter of the round steel bar, \(\Delta K_{l}\) by tensile force is defined as Equation (4). \[\Delta K_{l,Tension}=\frac{P}{\pi b^{2}}\times 1.12\times\sqrt{\pi a} \tag{4}\]  
where \(P\) is the tensile force acting on the cross-section and \(b\) is the radius of the steel bar. \(\Delta K_{l}\) by bending moment is defined as Equation (5). \[\Delta K_{l,Moment}=\frac{4M}{\pi b^{3}}\times 1.12\times\sqrt{\pi a}\cos\theta \tag{5}\]  
where \(M\) is the bending moment acting on the cross-section and \(\theta\) is the angle from the maximum \(\Delta K_{l}\). Under a tensile force, a constant \(\Delta K_{l}\) occurs in the crack notch around the round steel bar, but under a bending moment, the \(\Delta K_{l}\) value changes according to the direction of the moment and the position of the notch. A schematic illustration showing the stress range of the fatigue fracture surface and the position of the notch in angles from the \(y\)-axis is shown in Figure 8a. The \(\Delta K_{l}\) value at the notch considering the tensile force and the bending moment in \(y\) and \(z\) directions is defined as Equation (6). \[\Delta K_{l}=\ \left(\frac{P}{\pi b^{2}}+\frac{4M_{y}}{\pi b^{3}}\cos(\theta+90^{ \circ})+\frac{4M_{z}}{\pi b^{3}}\cos(\theta+180^{\circ})\right)\times\ 1.12\times\sqrt{\pi a} \tag{6}\]  
Substituting the section force calculated in the previous section into Equation (6) and solving for \(\theta\), the position and the maximum value at which \(\Delta K_{l}\) becomes the maximum can be obtained. Figure 8b shows the distribution of \(\Delta K_{l}\) at the notch when the depth of the  
Figure 8: Results of the case of decarburized layer depth of 0.2 mm: (**a**) schematic illustration of the notch location; (**b**) distribution of \(\Delta K_{l}\) along the notch. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Maximum Stress** & **Minimum Stress** & **Stress Range** & **Stress Ratio** \\ \hline
1262 MPa & 1116 MPa & 146 MPa & 0.884 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Stress condition at the cracking location. decarburized layer is 0.2 mm. The maximum value of \(\Delta K_{l}\) was 3.96 MPa\(\surd\)m at the position where \(\theta\) is 243\({}^{\circ}\). Compared with the principal stress distribution in Figure 8a, it can be seen that the maximum stress occurred at the position where \(\theta\) is 235\({}^{\circ}\), which is consistent with the calculation result of \(\Delta K_{l}\). Prior to determining \(\Delta K_{lh}\), the critical crack size for short and long cracks should be checked [10]. If the depth of the decarburized layer is too small, the fatigue crack may not be propagated when the applied stress range is smaller than the fatigue limit. Thus, stress range governs the crack initiation. According to [PERSON] et al. [10], the minimum crack size of the long crack for which the stress intensity factor governs the crack initiation, was defined as Equation (7). \[a_{D}=\frac{1}{\pi}\bigg{(}\frac{\Delta K_{lh}}{F_{c}\Delta\sigma_{o}}\bigg{)} ^{2} \tag{7}\]  
where \(\Delta\sigma_{o}\) is the fatigue limit of unnotched steel. The correction factor \(F_{c}\) is 1.12 in Equation (7).