PSI - Issue 82

Xiangnan Pan et al. / Procedia Structural Integrity 82 (2026) 125–130 X. Pan / Structural Integrity Procedia 00 (2026) 000–000

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1. Introduction For most engineering materials, strength is the most important mechanical property index, which serves as a measure of a material’s ability to resist failure under external loads and has the dimension of stress (Ashby, 2016; Meyers and Chawla, 2009). Fatigue strength σ w specifically refers to the maximum value of applied stress amplitude σ a or the corresponding maximum stress σ max that a material can withstand under a specified number N f of loading cycles (Schijve, 2009; Suresh, 1998). When N f = 1 cycle, fatigue strength is equal to the ultimate tensile strength (UTS, σ u ) under axial loading (Pan and Hong, 2024; Pan et al., 2021). In this case, fatigue and tensile tests are equivalent, with another key strength parameter involved: yield strength σ y , or more specifically the 0.2 % proof yield strength σ 0.2 for materials without a distinct yield point. When N f = 10 7 cycles, the fatigue strength has traditionally been regarded as the “fatigue limit” (Schijve, 2009; Suresh, 1998) for carbon steels and other similar materials with relatively low UTS. Below this limit, the material is generally considered to have “infinite life” — i.e., no fatigue failure will occur regardless of how many additional cycles are applied. When the loading cycles beyond 10 7 , extensive investigations (Bathias, 1999; Bathias and Paris, 2005; Sakai, 2023) have shown that fatigue failure still occurs in many metallic materials, such as: high-strength steels (Atrens et al., 1983; Naito et al., 1983), titanium alloys (Atrens et al., 1983; Furuya and Takeuchi, 2014; Gao et al., 2024; Pan and Hong, 2019; Pan et al., 2018, 2020, 2024a), and aluminium alloys (Pan et al., 2024b, 2025). This phenomenon is known as very-high-cycle fatigue (VHCF). Although Furuya (2021), Furuya et al. (2022), Pan and Hong (2024), and Pan et al. (2024b) continually persist in emphasizing the potential existence of a “true” fatigue limit for an “infinite” fatigue life, engineers consistently use the fatigue strength at a specified number of cycles (e.g., 10 8 , 10 9 , 10 10 and 10 11 ) as a substitute in VHCF research. Among these specified cycle counts for fatigue strength evaluation; the most used one is 10 9 cycles. Thus, this study labels the regime where fatigue life exceeds 10 9 cycles as the “runout”, i.e., the safe regime with no fatigue failure observed. Accordingly, the VHCF regime is defined herein as ranging from 10 7 to 10 9 cycles. To ensure equal lengths on a logarithmic coordinate axis, this study defines the regime from 10 5 to 10 7 cycles as high-cycle fatigue (HCF), with low-cycle fatigue (LCF) defined as the regime below 10 5 cycles.

Nomenclature σ a

applied stress amplitude

minimum stress amplitude applied to runout specimens maximum stress amplitude applied to runout specimens

σ a-0 σ a-1 σ a-2 σ a-3 σ a-4 σ a-5 σ max

minimum stress amplitude applied to specimens failed in very-high-cycle fatigue regime maximum stress amplitude applied to specimens failed in very-high-cycle fatigue regime minimum stress amplitude applied to specimens failed in high-cycle fatigue regime maximum stress amplitude applied to specimens failed in high-cycle fatigue regime

applied maximum stress UTS, ultimate tensile strength

σ u σ w

fatigue strength

σ w-5% fatigue strength with failure probability of 5 % σ w-50% fatigue strength with failure probability of 50 % σ w-95% fatigue strength with failure probability of 95 % σ y yield strength σ 0.2

yield strength corresponding to 0.2 % residual tensile strain

number of cycles to failure

N f

stress ratio, the ratio of minimum stress to maximum stress

R