Duplex S-N fatigue curves: statistical distribution of the transition fatigue life

In recent years, very-high-cycle fatigue (VHCF) behavior of metallic materials has become a major point of interest for researchers and industries. The needs of specific industrial fields (aerospace, mechanical and energy industry) for structural components with increasingly large fatigue lives, up to 1010 cycles (gigacycle
fatigue), requested for a more detailed investigation on the experimental properties of materials in the VHCF
regime. Gigacycle fatigue tests are commonly performed using resonance fatigue testing machines with a loading frequency of 20 kHz (ultrasonic tests). Experimental results showed that failure is due to cracks which nucleate at the specimen surface if the stress amplitude is above the conventional fatigue limit (surface nucleation) and that failure is generally due to cracks which nucleate from inclusions or internal defects (internal nucleation) when specimens are subjected to stress amplitudes below the conventional fatigue limit. Following the experimental evidence, the Authors recently proposed a new statistical model for the complete description of SN curves both in the high-cycle-fatigue (HCF) and in the VHCF fatigue regions (Duplex S-N curves). The model differentiates between the two failure modes (surface and internal nucleation), according to the estimated distribution of the random transition stress (corresponding to the conventional fatigue limit). No assumption is made about the statistical distribution of the number of cycles at which the transition between surface and internal nucleation occurs (i.e., the transition fatigue life).
In the present paper, the statistical distribution of the transition fatigue life is obtained, according to the
statistical model proposed. The resulting distribution depends on the distance between the HCF and the VHCF
regions and on the distribution of the random transition stress. The estimated distribution can be effectively
used to predict, with a specified confidence level, the number of cycles for which an internal nucleation may
probabilistically occur in a VHCF test and it is also informative for properly choosing the end of HCF tests in terms of number of cycles. A numerical example, based on experimental datasets taken from the literature, is provided.