PSI - Issue 76

Afshin Khatammanesh et al. / Procedia Structural Integrity 76 (2026) 115–122

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Fig. 4. Distribution of inclusion sizes.

3.3. Fracture-mechanics evaluation The larger sizes of inclusions observed at the crack-initiation sites of specimens extracted from material B, as shown in Fig.4, provide a reasonable explanation for the lower fatigue strength. It is generally accepted that non metallic inclusions can be consider as crack-like defects. Hence, their damaging effect can be evaluated using fracture mechanics principles. Murakami (2019) provided a simple equation to calculate the stress intensity factor range, Δ K , of 3D defects or cracks using √ area as the size parameter: ∆ K � f ∙ ∆ � √ area , (1) with f = 0.65 and 0.5 for surface and interior defects or cracks, respectively. The value of Δ K has been calculated for each crack-initiating inclusion according to Eq. (1), and the results are plotted vs. the number of cycles to failure, N f , in Fig. 5(a). The Δ K - N f curves provide a better correlation between the different steel sheets than the S-N curves shown in Fig. 2. However, there is a significant increase in scatter, particularly for interior failure, as reflected by the low coefficients of determination, R ², of the power fit curves in Fig. 5(a). It is noted that the R ² values indicated in Figs. 2 and 5 refer to interior failure only. Due to the limited number of specimens that failed from the surface, only datapoints from material B (blue, open diamond symbols) are fitted. The size dependence of Δ K th for small defects and cracks can be estimated using the following equation, as provided by Murakami and Endo (1986): ∆ K th � g ∙ � HV � 120 � ∙ �√ area � 1 / 3 , (2) where g = 3.3×10 − 3 and 2.77×10 − 3 for surface and interior defects/cracks, respectively, and HV is the Vickers hardness in kgf/mm². Note that the units of √ area in Eqs. (1) and (2) must be m and µm, respectively, to obtain Δ K and Δ K th in MPa √ m .