PSI - Issue 71

Haru Fujishima et al. / Procedia Structural Integrity 71 (2025) 18–25

19

since the initial investment and the running/maintenance costs are small. In addition to the cost efficiency, this machine has been believed to be reliable because it has a simple mechanism that mechanically applies cyclic bending moment to a specimen by a deadweight. However, the machine has a drawback in that its surrounding vibrations, like earthquakes, can affect the test result (Endo et al., 2019). Also, it is well known that the stress gradient, the inward stress variation in a specimen, affects the fatigue limit (Heywood, 1962; Forrest, 1962), whereas its effect on fatigue life has not been discussed well. The reason is that the physical background determining the form of the S-N curve has not been studied well, and a comparison was impossible between the experimental data and the intrinsic form of the S-N curve. Murakami and Endo recently proposed an S-N curve prediction model (Murakami et al., 2023). When a defect on the specimen surface becomes the origin of fatigue fracture, this model’s basic equation under zero -mean stress ( R = – 1) is given as: = ∗ ( −1) ∗ ∗ (1) where is the crack length expressed as the square root of the area of a defect projected onto the plane normal to the tensile direction [ ] ; i.e., a = √ , N is the number of cycles, is the stress amplitude [MPa], is the fatigue limit at a stress ratio of R = – 1 [MPa], and C *, m * and n * are constants. The following equation can predict based on the √ parameter model (Murakami, 2019): = ( √ ( + 1 )20) 1/6 (2) where HV is the Vickers hardness, and C is the coefficient changing according to the position of a defect or crack: C = 1.43 for the surface and C = 1.56 for the interior. Interestingly, it is confirmed that the constants in Eqs. (1) are approximately fixed values as C * = 10 -4 , m * = 2, and n * = 1, regardless of material (Murakami et al., 2023). Fatigue life N f is calculated by the numerical integral from the initial crack size to the final crack size for a given stress amplitude σ . A sufficiently accurate calculation of N f is possible using = 1000 μ m. It should be noted that calculated by Eqs. (2) decreases cycle by cycle with an increase in the crack size = √ according to crack extension. The difference in materials and the small crack growth behavior are considered in Eqs. (1) through Eqs. (2). Murakami and Endo (2023) referred to this model as “the sc S-N prediction model,” emphasizing it is a small crack-based model. Fig. 1 and Fig. 2 show examples of a comparison between the S-N curve predicted based on Eqs. (1) and the results obtained by the tension-compression and RB fatigue tests (Murakami et al., 2023). The material is an annealed 0.37% carbon steel, the same material investigated in this study. Eqs. (1) and Eqs. (2) predicts an S-N curve from the Vickers hardness (HV = 160) and the initial defect size （ a i = √ = 158 ). Therefore, the predicted S-N curve is the same for both figures. Fig. 1 shows that the tension-compression fatigue test results are in good agreement with the prediction despite the highly different shape of three defects (drilled hole, circumferential notch, and pre-crack). This result means that the mechanical factor represented by the stress amplitude σ and initial defect size a i determines the S N curve dominantly, and the statistical effect on the experimental results is relatively small. Fig. 2 shows the RB fatigue test results for circumferential notches are close to the predicted S-N curve. However, the experimental fatigue lives for drilled holes are longer than the prediction. It is considered that one possible reason for this peculiar result is related to the difference between stress gradients at the surface of specimens under tension and bending. However, the difference between the experimental results for the circumferential notch and the drilled hole cannot be explained fully only for this reason. Based on careful observations, recently, the authors found that the test frequency is another factor affecting the RB fatigue test results. Thus, this study performed the RB fatigue tests by setting test conditions in a planned way to quantitatively investigate the effects of the stress gradient and the test frequency on the fatigue life under bending loading.