PSI - Issue 2_A

Akiyoshi Nakagawa et al. / Procedia Structural Integrity 2 (2016) 1199–1206 Author name / Structural Integrity Procedia 00 (2016) 000–000

1205

7

inclusions included in the critical volume is not clear at the present stage, the above probability of c P was calculated providing several numbers as n . Thus, we have 0.1616 = c P for 5 = n , 0.0261 = c P for 10 = n and 0.0007 = c P for 20 = n , respectively. Based on this calculation, the probability that all the inclusions are located in the core portion deeper than c ξ is supposed to be sufficiently small under the condition of 10 > n . This fact means that at least one inclusion is almost necessarily located within the surface layer having the depth of µ ξ 250 = c m. The size of the interior inclusion at the crack initiation site is supposed to be 0.005 = ρ mm as a tentative value in this paper. In such small defects, fatigue strength is usually discussed by means of the concept of area , where area is the area of the defect projected to the section perpendicular to the loading direction. In the case of interior defect, the stress intensity factor K is given by Murakami et al. (1988) as follows; area K π σ 0 0.5 = , (6) where 0 σ is the applied stress. In the present situation, the local stress at the inclusion site should be substituted into 0 σ in Eq.(6). The lower bound of the fatigue strength at 9 10 = N , 9 w σ , is interpreted as 900MPa from the

experimental results in Fig.1. Accordingly, we can calculate the stress intensity factor at the inclusion edge by Eq.(6) giving the condition of 2 πρ = area and 900 0 = σ MPa. Thus, we have the result of 2.37 = K MPa m . If we assume that this value gives the threshold stress intensity factor range of th K ∆ , the distribution characteristics of the fatigue strength, 9 w σ , can be analytically derived from the distribution of the inclusion site as follows; if the inclusion is sited at the depth of ξ , the fatigue strength as a nominal stress at the surface is provided by 9 w σ on the ordinate. Based on the geometry in Fig.10, the fatigue strength 9 w σ i s given by (7) Based on this concept, we can also calculate the upper bound of the fatigue strength at 9 10 = N by substituting c ξ ξ = into Eq.(7) as 1080 9 = w σ MPa. This value is supposed to be reasonable from the statistical scatter for the experimental data points in the very high cycle regime in Fig.1. At the stress level of 1100 = σ MPa, all the specimens had failed, but some specimens had runout until 9 10 = N at 1000 = σ MPa. Therefore, the upper bound of the fatigue strength at 9 10 = N should be within the range of 1000MPa-1100MPa. Thus, one can successfully explain the lower and upper bounds of the fatigue strength distribution based on the distribution characteristics of the inclusions. Since the pdf of the inclusion depth ξ is given by Eq.(1), one can derive the pdf of the fatigue strength at 9 10 = N transforming the random variable ξ into another random variable 9 w σ through Eq.(7). Thus, we obtain ξ σ − = r r w 900 9 .

9 w σ

Fig.10 Relationship between fatigue strength and inclusion depth

2

2 900 c w σ

(

)

f

=

σ

w

9

F

3

9

Fig.11 Probability density function of fatigue strength at 9 10 = N cycles