PSI - Issue 76

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

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3. Analysis

This section provides the analysis tools required in this work and explicates the connections between necessary quantities such as the cyclic R-curve, ∆ K th , crack closure ratio, U , intrinsic threshold, ∆ K th , e ff , as well as nominal and e ff ective stress intensity factor ranges, ∆ K and ∆ K e ff , respectively. The level of plasticity-induced crack closure is measured with the crack closure ratio, U , which is directly the relative portion of the load cycle where the crack is open. Crack closure is defined by the first node contact criterion; the crack is considered open when the displacement of the first node behind the crack tip is positive. Note that in this work U is derived from the opening level, σ op /σ max , of the loading step, according to Eq. 2:

1 − σ op σ max 1 − R

σ max − σ op σ max − σ min

(2)

U =

.

=

There exist two schools of thought in describing the relationship between crack driving force and crack growth threshold. One compares the e ff ective stress intensity factor range, ∆ K e ff , to the intrinsic threshold, ∆ K th , e ff . The other compares the nominal stress intensity factor range, ∆ K , to the cyclic R-curve, ∆ K th . The only di ff erence lies in which term, external loading or internal resistance, is the development of crack closure included in. As both methods describe the same events, there exists a unique mapping between the two, which is briefly demonstrated next. Given a loading that is at the threshold condition for crack growth, the nominal and e ff ective stress intensity factor ranges must equal their corresponding threshold values, ∆ K =∆ K th ∆ K e ff =∆ K th , e ff . (3)

Now, as noted before, the cyclic R-curve can be divided into two parts: the intrinsic threshold, ∆ K th , e ff , and the addition from crack closure, ∆ K th , op ,

∆ K th =∆ K th , e ff +∆ K th , op ,

(4)

where the latter term is, given that Eq. 3 applies, equivalent to the di ff erence between the nominal and e ff ective stress intensity factor ranges,

∆ K th , op =∆ K − ∆ K e ff ,

(5)

Considering that

∆ K = K max − K min = (1 − R ) K max ∆ K e ff = K max − K op = 1 − σ op σ max

(6)

,

K max = U ∆ K