## 3 Probabilistic Model to Explain the Statistical Fatigue Property in Interior-Induced Fracture
### Distribution Characteristics of Inclusion Size and Inclusion Depth  
The inclusion size is not constant, and therefore, the size has its own distribution characteristics peculiar to the fabrication process of the metallic material [11; 12; 13]. Of course, since the inclusion size \(\rho\) is always positive, we have \(\rho>0\). In other words, the lower bound of the inclusion size is assumed \"0\". In addition, a size larger than 40 \(\upmu\)m is seldom found in the bearing steel. Distribution pattern of the inclusion size with the  
Figure 3: Relationship between fatigue strength \(\sigma_{\text{\tiny{uv9}}}\) and inclusion radius \(\rho\). above feature is well-represented by a two-parameter Weibull distribution [27], without a location parameter. The probability density function and the cumulative distribution function are given as follows:  
\[f_{0}(\rho)=\frac{a}{b}\left(\frac{\rho}{b}\right)^{a-1}exp\left\{-\left(\frac{ \rho}{b}\right)^{a}\right\}, \tag{6}\]  
and  
\[F_{0}(\rho)=1-exp\left\{-\left(\frac{\rho}{b}\right)^{a}\right\}, \tag{7}\]  
where \(a\) is the shape parameter and \(b\) is the scale parameter, respectively. When the number of inclusions included within the critical volume of a specimen is denoted by \(n\), the inclusion with the maximum size would be the crack starter under the cyclic loadings. Accordingly, the distribution characteristics of the maximum size among \(n\) of inclusions in the critical volume become the most important factor to control the fatigue strength and fatigue life of the metallic material. Based on the concept of extremes' distribution [28], the probability density function of the maximum size among \(n\) inclusions, \(f(\rho_{n})\), is provided by  
\[f(\rho_{n})=n\{F_{0}(\rho_{n})\}^{n-1}\frac{a}{b}\left(\frac{\rho_{n}}{b} \right)^{a-1}exp\left\{-\left(\frac{\rho_{n}}{b}\right)^{a}\right\} \tag{8}\]  
Although we have to know the original distribution of the inclusion size \(\rho\) itself, it is difficult to directly observe such a distribution pattern. In such a circumstance, the authors determined the respective parameters of the original size distribution, so as to satisfy the following conditions:  
\[P[1\ \upmu\text{m}<\rho<15\ \upmu\text{m}]=0.8,\ F_{0}=0.1\ \text{at}\ \rho=1\ \upmu\text{m}\text{ and}\ F_{0}=0.9\ \text{at}\ \rho=15\ \upmu\text{m}.\]  
This condition for the inclusion size was introduced here as a possible assumption supposed from the observation results by [PERSON] et al. [12]. Thus, we obtain the results for the Weibull parameters as \(a=1.139\) and \(b=7.214\), respectively. In addition, since the number of inclusions inside the critical volume, \(n\), is also unknown, the authors analyzed the distribution of the extremes by supposing various values as the number of inclusions, like \(n=2,3,5,6,10,20\), and \(40\), respectively. Probability density functions of the maximum inclusion size thus calculated by Equation (8) are depicted in Figure 4, where \(f(\rho_{1})\) plotted by a fine dotted curve in the most left hand side indicates the original probability density function of the inclusion size. The mode of the largest inclusion size, i.e., the inclusion size at each peak of the probability density function, tends to shift into the right hand side, with an increase in the number of inclusions in the critical volume of the specimen. As described in Section 3.1, the inclusion depth at the crack initiation site, \(\xi\), plays an important role to control both the fatigue strength and fatigue life of the metallic material, in the case of the rotating bending, due to the steep slope of the stress distribution across the section. Figure 5 indicates the distribution feature of the inclusions projected to the specimen section, assuming random distribution (uniform distribution) [23]. Here, if the inclusion depth from the surface is denoted as \(\xi_{1},\xi_{2},\ldots,\xi_{n}\), from the most shallow inclusion, one can obtain the one-dimensional distribution of the inclusion depth.