PSI - Issue 76

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

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Constraint 2 [ − ] Axisymmetric 0.333 -0.230 0.879 0.138 0.895 0.996 Plane stress 0.104 0.259 2.222 0.518 0.637 0.999 Plane strain 0.322 -0.258 1.105 0.163 0.937 0.997 Q 1 [ − ] Q 2 [ − ] b 1 [ − ] b 2 [ − ] C [ − ] R

Table 1. Parameters for fitting Eq. 9 to the simulation results. Quality of fit is assessed with the coe ffi cient of determination, R 2 .

Now, assuming U ( ∆ a ) corresponds to near-threshold crack closure (being locally independent of σ ), the cyclic R-curve can be estimated via Eq. 8, rewritten as

∆ K th ( ∆ a ) =∆ K th , e ff / U ( ∆ a ) ,

(10)

where the intrinsic threshold is assumed to be ∆ K th , e ff = 2 . 5MPa √ m. The cyclic R-curves corresponding to the fitted crack closure results are presented in Fig. 3(b), combined with the nominal crack driving force ∆ K ( ∆ a ) = 2 σ Y π ( a 0 +∆ a ) , (11) where the shape factor, Y = 2 /π for the internal crack and Y = 1 for the through-plate crack. A condition for the fatigue limit in terms of the cyclic R-curve analysis is the tangency of ∆ K and cyclic R-curve, ∆ K th :

∆ K ( ∆ a ) =∆ K th ( ∆ a ) ∆ K ′ ( ∆ a ) =∆ K ′ th ( ∆ a ) ,

(12)

where the prime symbol is used to indicate a derivative with respect to ∆ a . While the combination of the exponential and power law terms prohibits an analytical solution, the numerical solution to Eq. 12 is straightforward. Table 2 presents the predicted and reference fatigue limits, σ w and σ w , ref , respectively, along with the corresponding crack extensions before arrest, ∆ ˆ a . The long crack thresholds, ∆ K th , lc , inferred from the saturation values of the cyclic R curves of Fig. 3(b), are also included.

∆ K th , lc [MPa √ m]

Constraint

∆ ˆ a [ µ m]

σ w [MPa]

σ w , ref [MPa]

Axisymmetric 2.540 220.943 269.694

2.793 3.925

Plane stress

5.664 160.466 171.693

Plane strain 1.984 138.388 171.693 2.668 Table 2. Predicted values of crack extension and fatigue limit corresponding to the tangency condition of Eq. 12 illustrated in Fig. 3(b). The reference values obtained from Eq. 1 are denoted by σ w , ref . The final column includes the inferred long crack thresholds.

5. Discussion

This section briefly discusses the most relevant aspects of the present work. The long crack thresholds inferred from the simulated cyclic R-curves in Fig. 3 are considerably lower—save for pure plane stress conditions—than the values reported in literature for similar steels (Scho¨nbauer and Stanzl-Tschegg, 2013; Scho¨nbauer et al., 2017; More et al., 2025), which are often closer to ∆ K th , lc = 5MPa √ mat R = 0. This is expected as the present analysis considers only plasticity-induced crack closure, whereas the cyclic R-curve consists of the total closure influence, including also roughness and oxide-induced crack closure, for example (see Fig. 1). It can be inferred that these other mechanisms should often account for the majority of the total crack closure influence of long cracks.