PSI - Issue 76

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 76 (2026) 11–18

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Fig. 1. Schematic illustration of the composition and evolution of the cyclic resistance curve as a function of crack extension.

1994; More et al., 2025). In such an event, the threshold condition for the fatigue limit is crack arrest rather than the lack of crack initiation. In fact, analyzing the condition for crack arrest provides essentially a lower bound for the fatigue limit, referred to here as the fracture-mechanical fatigue limit . Crack initiation resistance is not considered— rather, crack initiation is assumed. As crack initiation resistance can only increase the fatigue limit from that defined by crack arrest, the term lower bound can be attributed to the fracture-mechanical fatigue limit. Besides, short crack arrest is often the fatigue limit defining mechanism as crack-like defects are an exceedingly common source of failure in real components with considerable volumes (Murakami and Endo, 1994). The authors hold the opinion that, in fatigue testing, discerning between the fatigue limit mechanism—initiation or propagation—is crucial metadata (see More et al., 2025). A paradigm shift in the explanatory physics can cause a manifold deviation in the observed fatigue limit value—prone to causing confusion and uncertainty in data reuse. Many sources qualitatively assert that the development of crack closure is the primary factor in systematic crack arrest (Tanaka and Akiniwa, 1988; Murakami, 2019a)—the event governing the fracture-mechanical fatigue limit. One enhanced method to express the development of crack closure is the cyclic resistance curve. Originally proposed by Tanaka and Akiniwa (1988), the cyclic resistance curve, or R-curve, depicts the development of crack growth re sistance as a function of crack extension from the baseline value of the intrinsic threshold stress intensity factor range, ∆ K th , e ff , to the long crack threshold stress intensity factor range, ∆ K th , lc (Tanaka and Akiniwa, 1988; Maierhofer et al., 2018a). For a given crack extension, a comparison with the nominal driving force, ∆ K , provides a binary prediction for crack growth. Furthermore, the o ff set between ∆ K and ∆ K th correlates with the crack growth rate, d a / d N , and their tangency condition defines the fracture-mechanical fatigue limit. A schematic depiction of the cyclic R-curve is shown in Fig. 1, partitioning the resistance against crack growth into di ff erent mechanisms akin to Maierhofer et al. (2014) and Maierhofer et al. (2018a). The cyclic R-curve can be thought to consist of two distinct parts; a material-inherent resistance, often defined as the intrinsic threshold, ∆ K th , e ff , and the additional, extrinsic resistance, ∆ K th , op . For metallic materials, the intrinsic ∆ K th , e ff is a function of Burger’s vector and Young’s modulus, and thus a material constant (Hertzberg, 1995). For steels, values of ∆ K th , e ff ∈ [2 , 3] MPa √ m are usually reported (Maierhofer et al., 2018a; Scho¨nbauer and Stanzl Tschegg, 2013). Maierhofer et al. (2014), for example, use a value of ∆ K th , e ff = 2 . 5MPa √ m, which is also adopted in this article. In turn, the extrinsic ∆ K th , op is a multivariate function describing the development of crack closure. With the recent rise in numerical methods to model crack closure (Newman, 1981; McClung and Sehitoglu, 1989; Antunes et al., 2004; Camas et al., 2011; Ka¨rkka¨inen et al., 2023), it is possible to derive the cyclic R-curve by simu lating the development of crack closure. While the measured cyclic R-curve includes all active resistance mechanisms as shown in Fig. 1, this article takes the necessary first step by considering only plasticity-induced crack closure. For the fatigue limit analysis, the high saturation rate of plasticity-induced crack closure is expected to emphasize its significance over other, slower closure mechanisms. The article is structured as follows. First, details on the finite element modeling used to obtain the plasticity-induced crack closure are briefly given. Second, important aspects related to the analysis framework are described. Finally, the results from the analysis are presented, along with relevant discussion, followed by the conclusions of the study.