PSI - Issue 76

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

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the cyclic R-curve may be written as

R  K max =∆ K th , e ff + (1 − U ) ∆ K ,

∆ K th =∆ K th , e ff + 

σ op σ max −

(7)

which reduces further to a form already apparent from the combination of Eqs. 3 and 6,

∆ K th =   

σ max  

1 − R 1 − σ op

 ∆ K th , e ff =∆ K th , e ff / U .

(8)

Eq. 8 can be used to estimate the cyclic R-curve if the intrinsic threshold and near-threshold crack closure devel opment are known, which is demonstrated in the next section.

4. Results

The analysis results concerning the prediction of the fracture-mechanical fatigue limit are provided in this section. The development of the crack closure ratio, U , during crack propagation is presented in Fig. 3(a).

Fig. 3. Plasticity-induced crack closure results as a function of crack extension for axisymmetric (AX), plane stress (PS), and plane strain (PE) constraint conditions. (a) Simulated closure ratios along with the corresponding analytical fits (see Eq. 9 and Table 1). (b) Cyclic R-curves estimated via Eq. 10 using the fitted results of (a) together with nominal crack driving forces obtained by Eq. 11 at the predicted fatigue limit.

As evident from Fig. 3(a), the evolution of plasticity-induced crack closure can be accurately captured by fitting a double exponential decay function,

b 1 ∆ a

b 2 ∆ a

U ( ∆ a ) = Q 1 e −

+ Q 2 e −

+ C ,

(9)

augmented by a constant term, C , corresponding to the saturation value, i.e., stable-state crack closure. Table 1 presents the curve fitting parameters determined for the closure ratio. Quality of fit is assessed with the coe ffi cient of determi nation, R 2 .