PSI - Issue 71

22 Haru Fujishima et al. / Procedia Structural Integrity 71 (2025) 18–25 3.1. Effects of stress gradient Comparing the results of = 0.2 −1 （ ● ） and = 0.4 −1 （ ▲ ， ▼ ） at a test frequency f = 1 Hz, and = 0.2 −1 （〇） and = 0.4 −1 （△，▽） at f = 67 Hz, for both f , the larger stress gradient ( = 0.4 −1 ) results in longer life. At f = 1 Hz, there is no significant difference between the fatigue lives for = 0.4 −1 and = 0.2 −1 , being in good agreement with the prediction. On the other hand, at f = 67 Hz, the fatigue lives for = 0.4 −1 and = 0.2 −1 are longer than the prediction. Fig. 6 compares the crack growth curves for different stress gradients χ at a test frequency f = 1 Hz, for which the effect of test frequency is minor, as illustrated in section 3.2. Crack initiation occurred before 2% of the fatigue life; thus, most of life is the crack growth life. The crack growth behaviours are similar just after the crack initiation, but the subsequent crack growth rate differs. Since the nominal stress is the same ( σ = 210 MPa), the effect of the stress gradient on the crack initiation and early growth behaviours near the specimen surface should be small. However, as the crack extends, a stress decrease in the inward direction becomes more significant for a larger stress gradient.Accordingly, the fatigue life extends more because the crack growth rate becomes slower in the interior than the surface of a semi elliptical crack originating from the hole. It is known that for many metallic materials, including steels, the effect of test frequency on the tension-compression fatigue limit of the smooth specimen is negligible for a frequency smaller than 500Hz (Frost et al., 1974). Also, the applied stress (210 MPa) was sufficiently smaller than the lower yield point (325 MPa), and the rotating bending fatigue limit of the smooth specimen (240 MPa) at a frequency f = 55 Hz (Endo and Yanase, 2019). Hence, the effect of heat generation at f = 67 Hz may be negligible. The temperature rise on the specimen surface measured by an infrared thermometer was so small as to be unmeasurable. Therefore, the mechanical factor related to the test frequency is supposed to affect the fatigue strength primarily. In Fig. 5, a comparison of fatigue life is drawn between two different test frequencies using the results at = 1 （ ● ） and = 67 （〇） for a stress gradient of = 0.2 −1 and those at = 1 （ ▲ ， ▼ ） and = 67 （△，▽） for = 0.4 −1 . For both χ , the fatigue life became significantly longer at a higher test frequency of f = 67 Hz. In particular, when the specimen with a stress gradient of = 0.4 −1 was tested at 67 Hz, the specimen was not broken at σ = 210 MPa. This stress is about 10% higher than the fatigue limit σ w = 188 MPa predicted by Eq. (2). Figs. 6 and 7 compare the crack growth curves. Cracks were mostly observed before cycles reached 2% of the total life. Therefore, the cause of the difference in fatigue strength (fatigue life and fatigue limit) between = 1 and = 67 is not due to the crack initiation condition but is mainly related to the crack growth rate and threshold condition. It is thought that the reason why the test frequency affected the experimental results is associated with the testing machine’s structure. The rotating shaft of the machine to which a specimen is attached has a much larger mass than the specimen. Therefore, when it rotates at high speed, the resultant inertia force will work to keep the specimen deflection constant. The drilled hole exists in an asymmetric position concerning the axis of rotation. Therefore, the positional relationship between the defect and the neutral plane changes by rotation, and the moment of inertia of area changes, resulting in a change in compliance. The more the crack emanating from a drilled hole expands, the more the difference in compliance during one rotation becomes. When the test speed is low enough, the inertia is so minor that the deflection will vary with the position of the defect during rotation, causing the machine’s rotating shaft to oscillate. During the fatigue test performed at a frequency f = 1 Hz, the shaft fluctuation became large enough to be observed by the naked eye just before failure. As demonstrated by ● In Fig. 5, the experimental results of fatigue life N f are very close to the prediction at f = 1 Hz for a small stress gradient effect, i.e., χ = 0.2 mm -1 . This result means that the cyclic stress amplitude applied to the specimen is an accurate value σ = 210 MPa, calculated by the bending stress formula, assuming static load application. Also, as shown in Fig. 6, the difference in the crack growth curves between χ = 0.2 and 0.4 mm -1 is comparatively small at a frequency 3.2. Effects of test frequency f