Statistical distributions of Transition Fatigue Strength and Transition Fatigue Life in duplex S-N fatigue curves

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 probabilistic model for the complete description of S–N 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 TFL). In the present paper, the TFL distribution is obtained. The resulting distribution depends on the distance between the HCF and the VHCF regions and on the distribution of the random transition stress. It is also shown that the statistical distribution of the fatigue strength at the median TFL (i.e., the Transition Fatigue Strength TFS) has median which corresponds to the mean transition stress. Finally, a procedure for computing Likelihood Ratio Confidence Intervals (LRCIs) for both the median TFL and the median TFS is given in the paper. The estimated TFL and TFS distributions can be effectively used for properly choosing the duration of HCF tests in terms of number of cycles and the stress amplitude below which VHCF failures more probably occur. LRCIs for the median TFL and TFS can be usefully computed for assessing uncertainty in the estimation procedure when a limited number of experimental data is available. A numerical example based on an experimental dataset taken from the literature is provided.