In this paper, ultrasonic (20 kHz) fatigue tests were performed on specimens of a high-strength steel in very high cycle fatigue (VHCF) regime. Experimental results showed that for most tested specimens failed in a VHCF regime, a fatigue crack originated from the interior of specimen with a fish-eye pattern, which contained a fine granular area (FGA) centered by an inclusion as the crack origin. Then, a two-parameter model is proposed to predict the fatigue life of high-strength steels with fish-eye mode failure in a VHCF regime, which takes into account the inclusion size and the FGA size. The model was verified by the data of present experiments and those in the literature. Furthermore, an analytic formula was obtained for estimating the equivalent crack growth rate within the FGA. The results also indicated that the stress intensity factor range at the front of the FGA varies within a small range, which is irrespective of stress amplitude and fatigue life.

(Graphical abstract)

A two-parameter model is proposed to predict the fatigue life of high-strength steels with fish-eye mode failure in a very high cycle fatigue regime, which takes into account the inclusion size and the FGA size. The model was verified by the data of present experiments and those inthe literature. Furthermore, an analytical formula was obtained for estimating the equivalent crack growth rate within FGA. It is also indicated that the stress intensity factor range at the front of the FGA varies within a small range, which is irrespective of stress amplitude and fatigue life. Figure Comparison of predicted fatigue life with experimental data, in which the superscript 4 denotes the specimens by oil-quenching and tempering for 2.5 h in a vacuum at \(300\,^\circ \hbox {C}\) by a conventional frequency test.