PSI - Issue 68

Kimmo Kärkkäinen et al. / Procedia Structural Integrity 68 (2025) 646–652 K. Ka¨rkka¨inen et al. / Structural Integrity Procedia 00 (2024) 000–000

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e ff ect of over- and underloads on the fatigue limit can be analyzed in a simple and e ff ective manner using the cyclic resistance curve, or R-curve (Tanaka and Akiniwa, 1988; Maierhofer et al., 2018). It is commonly assumed that the development of crack closure corresponds to the R-curve, allowing for the transients in crack closure to be linked to the changes in the R-curve. The modeling results presented in the preceding section show that both over- and underloads can cause a resetting of crack closure development, but overloads are also able to significantly alter the subsequent closure levels. Thus, a resetting of the R-curve can be assumed to follow either loading spike, but the shape of the new R-curve is altered with overloads. The R-curves are modified by adjusting the ∆ K th , lc and a 0 parameters (see Eq. (1)). Fig. 3 presents the R-curves based on the simulated crack closure response, which are used in the present analysis. Notice that for the fatigue limit R-curve analysis, only the initial part of the R-curve is important.

Fig. 3. (a) Simulated crack closure response to an over- or underloads. (b) Corresponding R-curves. The closure response after an underload corresponds to an unaltered R-curve. With overloads, R-curves are modified by adjusting the ∆ K th , lc and a 0 parameters (see Eq. (1)).

Another issue is determining when during crack propagation loading spikes would likely occur. Present analysis assumes sporadic or relatively rare loading spikes in otherwise constant amplitude base loading below the constant amplitude fatigue limit. In this case, the majority of the service time is spent in the state where the crack has arrested at the intersection point of the nominal driving force and the R-curve. A sporadic over- or underload is most likely to occur at this stage, and consequently assumed so in this analysis. As the loading spikes are assumed to be infrequent and finite, their inherent damaging e ff ect can be neglected in this analysis. The proposed method is schematically illustrated for the underload case in Fig. 4 and encapsulated in the following assumptions, whose justifications were discussed above. Overloads follow the same principle; the only di ff erence is that an altered R-curve (Fig. 3(b)) is used from the first overload onwards. • R-curve is reset by a loading spike. • R-curve shape is altered by an overload but not by an underload. • A loading spike occurs at the point marking crack arrest, i.e., the intersection point of the ∆ K -curve and R-curve. Based on these considerations, the e ff ective fatigue limit, σ w , e ff , reduced by single or numerous over- or underloads, can be computed as a function of the number of over- or underloads, n . A condition for crack arrest in terms of R-curve analysis is the equality of ∆ K and cyclic R-curve ∆ K th (Zerbst and Madia, 2015; Maierhofer et al., 2018), ∆ K =∆ K th ⇔    2 σ Y √ π ( a init +∆ a ) =∆ K th , lc  ∆ a + a ∗ − a s ∆ a + a ∗ + a 0 − a s a ∗ = a 0 ( ∆ K th , e ff / ∆ K th , lc ) 2 1 − ( ∆ K th , e ff / ∆ K th , lc ) 2 , a 0 = 1 π  ∆ K th , lc Y ∆ σ th , 0  2 , (1) where σ is the nominal stress amplitude, Y shape factor, a init initial crack length, ∆ a crack advance, ∆ K th , lc long crack threshold stress intensity factor range, ∆ K th , e ff intrinsic threshold stress intensity factor range, and ∆ σ th , 0 material threshold stress range. A new solution-dependent variable, a s , is introduced, which shifts the R-curve by a length equal to ∆ a of the previous arrested crack. At the e ff ective fatigue limit, ∆ K is a tangent of the final R-curve. As Eq.