PSI - Issue 79

João Alves et al. / Procedia Structural Integrity 79 (2026) 326–334

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1. Introduction Fatigue is a phenomenon that was first studied by a German engineer called Wohler, in 1840 (Zenner & Hinkelmann, 2019), and can be defined as a process of permanent, progressive and localised structural alteration, which occurs in a material subjected to conditions that produce dynamic stress or deformation at one or several points, and which can culminate in cracks or a complete fracture after a sufficient number of load variations (ASTM E1823 24C). This phenomenon can be summarized by three different phases: Initial Period (Cyclic Slip, Crack Nucleation, Micro Crack Growth), Crack Growth Period (Macro Crack Growth) and Final Failure (Beden et al., 2009). However, the problem arises in knowing the influence of each of these phases on the final fatigue resistance of the material, to properly control this phenomenon. To overcome this challenge, different equations have been proposed in the literature that attempt to model the mechanisms behind the crack propagation in the different fatigue phases, however, this work will be focused on the phases concerning microscope crack growth. The first contribution was done by (Kitagawa & Takahashi, 1976), which concluded that for a determined crack or defect size, l 0 , the fatigue limit was independent of the defect size (Janssen et al., 2024). Switching the boundary conditions of propagation, previously established for long cracks, the stress intensity threshold ΔK th [MPa √m ] , for a fatigue threshold, Δσ th [MPa]. Based on that finding, El Haddad et al. (El-Haddad et al., 1979), proposed a material constant length, l 0 , which relates ΔK th [MPa √m ] with the Δσ th [MPa] and the defect length, a. Modelling the loss of fatigue resistance due to the defects’ increasing size, as can be seen by equation 1. ∆ ℎ = ΔK th √ ( + 0 ) (1) The problem is that the cracks emerging from a defect area (area), according to their characteristics, influences on the final fatigue limit behaviour, [MPa]. Therefore, Murakami proposed equation 2 (Murakami, 2002; Murakami et al., 1989; Murakami & Usuki, 1989). = 3 ((√ + 1 )20) 1 6 ×( − 2 1 ) (2) Where , can be determined by equation 3, C 3 assumes a value according to the localisation of the defect, and Hv, is the Vickers hardness of the material (Murakami, 2002; Murakami et al., 1989; Murakami & Usuki, 1989). = 0.226 × × 10 −4 (3) Moreover, other authors (Alves et al., 2024; Bathias, 2000; Beretta et al., 2009; Morgado et al., 2022; Tajiri et al., 2014; Ueno et al., 2012, 2014) have also been developing different models, with different defects ’ counting methodologies, for different materials. (Bathias, 2000), studied the high cycle fatigue behaviour in high-strength steels, and modified the Murakami proposal by adding a term that quantified both defect localisation and the number of cycles. (Beretta et al., 2009) tested the applicability of (El-Haddad et al., 1979) proposed models, by considering the Murakami considerations, for an A1N steel. (Ueno et al., 2014), analysed the influence of the defects in casting aluminium, using artificial holes to quantify the changes in the fatigue response of the metal, creating two different models dependent on the area of the defect. (Tajiri et al., 2014), studied the influence of the defects distributed along three specimens of an aluminium alloy A356, manufactured by casting, and modified the counting methodology of the model proposed by (Ueno et al., 2012). (Morgado et al., 2022), analysed the fatigue limit behaviour of an extruded aluminium alloy 6060 and heat treated - T1 and T4, and proposed two equations to model the expected fatigue limit response. More recently, (Alves et al., 2024). proposed two different equations, which, unlike the other proposals, used nanotomography as a defect acquisition technique to describe the fatigue limit for a Ti-6Al-4V alloy additively manufactured by SLM. Although the proposed models represented an advance on the understanding of fatigue, especially regarding the small crack propagation from defects, the initial framework proposed by (Murakami, 2002) only offered a proposal to quantify the fatigue limit of the metals, not being descriptive of the evolution of the defects during propagation. To