“…Recently, extensive efforts had been conducted to develop the fatigue damage models for HCF reliability estimation, including HCF probability analysis considering the uncertainty (Li et al ., 2021; Wang et al ., 2022), HCF evaluation under variable load (Gao and Yuan, 2020; Nya et al ., 2019; Abdullah et al ., 2019), stress-life fatigue modeling (Zhang et al ., 2022b; Kebir et al ., 2021; Song et al ., 2021a) and so forth. Based on the above models, HCF reliability analysis, sensitivity assessment and probabilistic estimation based on standard test specimens have been abundantly studied (Yang et al ., 2021; Wu et al ., 2020). However, the researches on practical engineering problems with small failure probability and high nonlinearity are still insufficient, i.e.…”