## 4 Results and Discussions
### Expansion of the Probabilistic Model to Analyze P-S-N Characteristics in Interior-Induced Fracture  
Target of the probabilistic model described in Section 3 is the distribution characteristics of the fatigue strength only at stress cycles of \(N=10^{9}\). Accordingly, the analytical model should be conceptionally expanded to interpret the whole statistical aspect of the \(S\)-\(N\) property, in the interior inclusion-induced fracture mode. From this point of view, denoting the fatigue strength at any stress cycle, \(N_{fix}\), by \(\sigma_{w}\), Equation (3) is rewritten as follows:  
\[\sigma_{w}=1273\cdot\rho^{-1/6}+\beta, \tag{14}\]  
\[\beta=\lambda\left\{log\left(N_{fix}\right)\right\}+\gamma. \tag{15}\]  
Based on the dotted \(S\)-\(N\) curve in Figure 1, parameters \(\lambda\) and \(\gamma\) in Equation (15) were determined from two points \(\left(N=10^{6},\ \sigma_{w\theta}^{*}=1424\text{ MPa }\right)\) and \(\left(N=10^{9},\ \sigma_{w\theta}^{*}=920\text{ MPa }\right)\) as \(\lambda=-168\) and \(\gamma=1512\), respectively. Substituting these values into Equations (14) and (15), Equation (14) is rewritten as  
\[\sigma_{w}=1273\cdot\rho^{-1/6}-168\cdot\left\{log\left(N_{fix}\right)\right\} +1512. \tag{16}\]  
Here, \(\sigma_{w}-\rho\) relationships under the fixed numbers of the fatigue life of \(N=10^{6}\), \(N=10^{7}\), \(N=10^{8}\), and \(N=10^{9}\) are depicted in Figure 10. As suggested from Equations (14)-(16), analytical curves in Figure 10 tend to shift in parallel along the ordinate. Of course, the curve tends to shift downwards with an increase in the fixed number of stress cycles. Applying the concept of Equation (4), one could calculate the fatigue strength \(\sigma_{w}^{*}\) by  
\[\sigma_{w}^{*}=\frac{1.5}{1.5-\xi}\left[1273\cdot\rho^{-1/6}-168\cdot\left\{ log\left(N_{fix}\right)\right\}+1512\right]. \tag{17}\]  
Equation (17) includes arbitrarily given number of stress cycles \(N_{fix}\), together with the inclusion size \(\rho\) and the inclusion depth \(\xi\). Therefore, combining the joint probability func  
Figure 10: \(\sigma_{w}\)- \(\rho\) relationships at several given numbers of stress cycles.  
tion in Figure 7 with Equation (17), one could analyze the distribution characteristics of the fatigue strength at any number of stress cycles \(N_{fix}\) by repeating the numerical calculations.  
Probability density functions of the fatigue strength thus obtained at \(N=10^{6}\), \(10^{7}\), \(10^{8}\), and \(10^{9}\) are indicated at the respective numbers of the stress cycles in Figure 11. These analytical results are roughly in agreement with the overall feature of the experimental aspect on the statistical fatigue characteristics. The scale of the axis for the probability density is adjusted to correspond to each value listed in the table attached in the right-hand side in Figure 11. The percentile points of \(F=1\%\), \(10\%\), \(50\%\), \(90\%\), and \(99\%\) for the fatigue strength are indicated by marks of \(\Diamond\)'s along the vertical axis of the probability density functions. The respective thin dotted lines passing through the percentile points corresponding to the same probability give the \(P\)-\(S\)-\(N\) curves representing the statistical fatigue characteristics of this steel. All data points yield within the range of 1-99% in Figure 11. This fact suggests that the probabilistic model developed in this study has an availability to interpret the physical meaning of the statistical fatigue property in the very high cycle regime.