## 2 Experimental Fatigue Test Data Referred in This Study  
Sufficient number of fatigue test data are required to analyze the statistical fatigue property of metallic materials. However, the fatigue test in the very high cycle regime occupies a long time. For example, at the usual loading frequency of 50-60 Hz, it takes around 200 days to reach the loading cycles of \(N=10^{9}\). In such a circumstance, the authors extracted a series of fatigue test results obtained as round robin tests by the Research Group on Statistical Aspects of Materials Strength [RGSAMS] [3]. These fatigue tests were performed in rotating bending in very high cycle regime over \(N=10^{9}\) cycles, for a bearing steel of JIS-SUJ2, by using the same type of testing machine and common specimens. Since such fatigue test results give useful data for discussion on the very high cycle fatigue property of metallic materials, they are often referred to by many researchers in this area [3; 7; 20; 21].  
All experimental results are plotted as an _S-N_ diagram in Figure 1, in which fatigue test data obtained at different laboratories in seven universities in Japan were indicated altogether. Open symbols represent the _S-N_ data for the surface-induced fracture, whereas solid symbols indicate the results for the interior-induced fracture. Several data points attached arrows around \(N=10^{9}\), which indicate runout specimens without fracture. The solid and dotted curves indicate the _S-N_ curves fitted to the respective fracture modes, using the JSMS Standard of \"Standard Regression Method of _S-N_ Curves\", accepting the semi-logarithmic bilinear model [22]. The fatigue limit for the surface-induced fracture was \(\sigma_{\text{w}}=1278\) MPa, as given by the horizontal portion of the solid _S-N_ curve. This conventional fatigue limit cannot provide the true fatigue limit, since fatigue fractures take place in the very high cycle region, at stress levels lower than the horizontal portion of the solid _S-N_ curve. For the experimental data in the interior-induced fracture mode, the linear regression line was indicated by a dotted line. Although the horizontal line was also indicated from the point of \(N=10^{9}\) along the dotted _S-N_ curve, further experimental reconfirmation would be required in order to make clear the appearance of the horizontal portion in the fracture mode by continuing fatigue tests, until further long life region.  
Figure 1: Rotating bending _S-N_ characteristics of bearing steel in very high cycle regime.  
Typical examples of fracture surfaces in the respective fracture modes are shown in Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-induced fracture mode and (b) is the fracture surface in the interior-induced fracture mode [23]. The fatigue crack occurs at the specimen surface in the short life region at a relatively high stress level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture surface in the very high cycle region, at low stress levels, as shown in Figure 2b. In such a case, an inclusion is also found at the core site of the fish-eye. This fact means that the fatigue crack takes place around the inclusion at the crack initiation site, in a very high cycle regime. The main aim of this study was to analyze the statistical distribution characteristics of the _S-N_ property in the interior inclusion-induced fracture mode, by combining distribution characteristics of the inclusion size and the inclusion depth.  
For the sake of important reference, static mechanical properties such as the tensile strength and the Vickers' hardness were examined, prior to performing fatigue tests of this bearing steel. Experimental results of these properties are indicated in Table 1.