PSI - Issue 76

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

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(a)

(b)

Fig. 5. (a) Stress intensity factor range, Δ K vs. number of cycles to failure, N f . (b) Stress intensity factor range normalised by the size-dependent threshold, Δ K / Δ K th vs. number of cycles to failure, N f .

The effect of the size-dependency of Δ K th can be considered by plotting the normalised stress intensity factor, Δ K / Δ K th versus N f , as shown in Fig. 5(b). In this graph, the value of Δ K is divided by the respective threshold, Δ K th , calculated with Eq. (2) for each specimen. This means that each specimen features its specific threshold, depending on the size of the most critical inclusion (which is expected to be the largest one in the volume of the gauge section). It is obvious that scatter of the lifetime curves is significantly reduced for all materials by normalising the Δ K / Δ K th -N f curves – which is expressed by higher values of R ² compared with the Δ K-N f curves. However, there are two inconsistencies that are apparent when examining Fig. 5(b) more closely. On the one hand, the VHCF lifetimes of material A are still markedly longer than that of materials B and C, although the difference is less pronounced compared with the conventional S-N curves, see Fig. 2. On the other hand, Eq. (2) does not serve well to calculate the fatigue threshold: no failure should occur at Δ K / Δ K th < 1 in the case of a conservative prediction – which is obviously not the case. Evaluations of experimental results obtained with high-strength steels – as reported, for example, by Murakami et al . (1999), Furuya et al . (2003), Spriestersbach et al . (2017), and Schönbauer et al . (2023) – have shown that the predicted thresholds according to Eq. (2) are often slightly non-conservative in the VHCF regime. Δ K -values approximately 10 – 20 % below the predicted threshold have been reported for interior inclusions when failure occurred beyond 10 8 cycles. In the present investigation, however, interior failures at Δ K -values that are 35 % below the predicted threshold are observed, i.e., failure occurred even at Δ K / Δ K th = 0.65, as shown in Fig. 5(b). This is far away from a conservative prediction. In contrast, no failure from surface inclusions occurred at Δ K / Δ K th -values below 1.3, see open symbols in Fig. 5(b). Comprehensive investigations on the defect sensitivity of high-strength steels revealed that the √ area -parameter model (Eq. (2)) may underestimate the threshold value for surface failure by around 10 – 15 % – see, for example, Schönbauer and Mayer (2019), Sistaninia et al . (2024), and More et al . (2025). Therefore, the underestimation of Δ K th for surface defects by 30 % in the present study has to be considered a deviation from conventionally observed values and must be scrutinised. The above evaluation indicates that the parameters used in Eq. (2) – i.e., the defect size in terms √ area and the Vickers hardness HV – do not cover all of the prevailing factors influencing the fatigue properties of the investigated steel sheets. It is assumed that residual stresses significantly affect the fatigue strength, since no measures – such as stress relief annealing or electropolishing – were taken to remove them from the fatigue test specimens. This is corroborated by the observed location of crack-initiation as described in Section 3.2.