PSI - Issue 2_A

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

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where r is the radius of the bar specimen. As illustrated by a sloped line in Fig.9, the density becomes maximum at the surface, whereas it becomes zero at the center of the section. It should be noted that the probability density function of ξ is not given by the uniform distribution indicated by a horizontal dashed line.

ξ 2 1

ξ

  

( )

c ξ

f

  =  − r

r 2 •

r

Uniform distribution

1 =

( ) ξ

f

r

Probability density function

r

0

ξ

Depth of inclusion

Fig.8 Definition of inclusion depth on the common section

Fig.9 Distribution pattern of the inclusion depth

In the case of rotating bending, the crack initiation sites are usually limited within the surface layer having a critical depth of c ξ . As discussed in Section 2.2, the value of c ξ is supposed to be µ 250 m approximately. If the inclusion depth is less than the radius of the inclusion at the crack initiation site, i.e. ρ ξ < , the crack initiation mode is assumed as the surface-initiated fracture. Therefore, the probability occurring the surface-initiated fracture, s P , can be given as follows; ( ) { } ( ) { } ( ) ( ) c c c s s r r r r r r A P A ξ ξ ρ ρ ξ π ρ π − − = − − − − = = 2 2 2 2 2 2 0 , (2) c ξ ξ < . In the case of fatigue test results in Fig.1, we have only two specimens failed in the surface-initiated fracture among a lot of specimens failed in the very high cycle regime. Total number of specimens failed along the second S-N curve is 89, and the number of specimens failed in the surface-initiated fracture is only two as described in Section 2.1. Thus, the probability the surface-initiated fracture occurring is calculated as 2 / 89 0.0225 = = s P (2.25%). On the other hand, the corresponding probability of s P can be also calculated by Eq.(2) under the practical data of r , ρ and c ξ , respectively. Making reference to the experimental data by Sakai et al. (1999), if we put 1.5 = r mm, 0.005 = ρ mm and 0.250 = c ξ mm, this probability of s P is calculated as follows; ( ) ( ) 0.0218 0.250 2 1.5 0.250 0.005 2 1.5 0.005 = × − × − = s P . (3) 2 / 89 0.0225 = = s P (2.25%) described above is very close to the analytical value of 0.0218 = s P (2.18%) calculated by Eq.(3). Thus, the appearing frequency of the surface-initiated fracture in the very high cycle regime can be well explained from the distribution characteristics of the inclusions inside the material. 3.3. Probabilistic model for distribution pattern of fatigue strength in very high cycle regime Since the pdf of the inclusion depth ξ is given by Eq.(1), the cumulative distribution function (cdf) of ξ is provided as follows; ( ) 2 2 2 1 2 ξ ξ ξ ξ ξ r r r r F  = −   =  − . (4) Denoting the probability that an inclusion is located within the surface layer having the thickness of c ξ by c F , the probability that all inclusions are located in the core portion deeper than c ξ is provided by ( ) n c c P F = − 1 , (5) where n means the total number of inclusions included inside the critical volume of the specimen. As mentioned previously, the value of c ξ is given as µ ξ 250 = c m from the experimental results. Since the total number of where s A is the area of ρ ξ < , and 0 A the area of The experimental result of  