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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stresses at the edges of deep inclusions (No.2, No.3, No.4) are corresponding to the dotted line giving 2.0 ≅ α , peak stresses at the inclusion edges (No.1) exceed the above dotted line. Based on this aspect, inclusions within such a thin surface layer are apt to cause the fatigue cracks easily. Denoting the critical depth having such an effect by δ , we have peek stresses higher than the dotted line in the region of δ ξ ≤ in Fig.6. If we denote the fatigue limit for specimen without inclusion by 0 w σ , peek stresses around inclusions of No.1 and No.2 can exceed the fatigue limit of 0 w σ . In such a circumstance, the fatigue crack can take place at these two inclusions, while the crack cannot occur at other inclusions of No.3 and No.4. In addition, the fatigue crack can easily occur at the inclusion of No.1 comparing with inclusion No.2, since the peek stress of No.1 inclusion is much higher than that of No.2 inclusion. If No.1 inclusion is not included in the material, the fatigue crack must occur at No.2 inclusion and the depth at the core of the fish-eye becomes also deep on the fracture surface.

0.45 = ν

0.30 = ν

0.25 = ν

ξ

B α

A α

0.45 = ν

0.30 = ν

0.25 = ν

Stress concentration factor

Location of cavity

ρ ξ /

(a) Definition of parameters

(b) Variation of stress concentration factor

Fig.7 Effect of inclusion depth on stress concentration factor around cavity in rotating bending

Fig.7 indicates the effect of the inclusion depth on the stress concentration factor for a semi-infinite body with a spherical cavity in tension. Top three curves in Fig.7(b) provide the analytical results at the site of B nearest to the surface as shown in Fig.7(a), whereas the bottom results give the analytical results at the deepest site of A. Analytical results obtained under three different values of Poisson’s ratio ( = ν 0.25, 0.30, 0.45) are indicated in Fig.7(b). The stress concentration factor at the site A, A α , tends to keep the value around 2.0 ≅ α regardless of the depth ( ρ ξ / ). But, the factor at the site B, B α , tends to increase significantly with a decrease of the inclusion depth. The region giving the significantly high stress is restricted within / 2.0 < ρ ξ so that the stress increase is limited within a vicinity of ρ ξ 2 < . Since the stress concentration is distinctly limited within a vicinity around the inclusion, the peek stress at the interior inclusion is assumed to be given by ξ σ α σ = ⋅ , where ξ σ denotes the local stress at the interior inclusion. 3.2. Distribution characteristics of non-metallic inclusions As a simple model, let us suppose that interior inclusions having the same size are distributed at random inside the material and they are projected to a common section of a bar specimen as depicted in Fig.8. If we denote the inclusion depth as 1 ξ , 2 ξ , ⋅ ⋅ ⋅ , n ξ from the smallest depth for the entire inclusions, one can obtain the one dimensional distribution of the inclusion depth of ξ . When the inclusions are distributed at random over the section, the probability density function (pdf) of the depth ξ is given by the following expression; ( ) f ξ ξ 2 1 , (1)

  

  =  − r

r