PSI - Issue 75

Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 219–233 Author name / Structural Integrity Procedia (2025)

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2.2. Forman Crack Growth Law Forman et al. (1967) proposed an extension of the Paris law to allow also the description of the fast fracture region (Region III). Forman’s crack growth law is described by the following equation: = ∙ ∆ (1 − ) ∙ −∆ (4) where and are the Forman coefficients, different from the Paris coefficients due to the units in the denominator of Eq. 4, is the plane stress fracture toughness and the effective ∆ is expressed as: ∆ = max ∙ (1− ) (5) By substituting Eq. 5 into Eq. 4, the Forman law takes the following expression: = ∙ m ax (1− ) −1 − max (6) The ability of the Forman crack growth law to implicitly model mean stress effects is often criticised because it misses an independent controlling parameter allowing to ‘fine - tune’ the fit to experimental data. Finally, it is worth highlighting that the Forman law is valid for stress ratios ∈ [ min , max ] , where min and max are the minimum and maximum stress ratios used in the tests, respectively. 2.3. Walker Crack Growth Law Walker (1970) also proposed an extension of the Paris law to account for mean stress effects. Walker achieved this by introducing the stress ratio, , into Paris formula (Eq. 2) as follows: = ∙ [∆ ∙ (1 − ) −1 ] (7) where the exponent controls the degree of influence of the mean stress on the crack growth allowing to ‘fine tune’ the fit to experimental data. Since Eq. 7 is a power law, similarly to the Paris law, it is not suitable for describing the threshold (Region I) and the fast fracture (Region III) regions, but only the crack propagation region (Region II). Walker, considering the hypothesis of crack closure, proposed to neglect compressive stresses in the cycle, leading to the following formulation (with max >0 necessary condition for crack propagation): = { ∙ [∆ ∙ (1 − ) −1 ] for > 0 ∙ [ max ∙ (1− ) −1 ] for ≤0 (8) Similarly to the Forman law, the Walker law can be considered valid for stress ratios ∈ [ min , max ] . 2.4. Austen Crack Growth Law Austen proposed another modification of the Paris law to implicitly consider both threshold and onset of fast fracture (nCode (2003)) . Austen’s crack growth law is given by: