PSI - Issue 75

Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 219–233 Author name / Structural Integrity Procedia (2025) 9 where = − max is the compressive stress intensity factor due to the confinement induced by the elastic body on the plastic zone, with representing the stress intensity factor required to extend the current plastic zone to the edge of the plastic zone created by the tensile overload, . This condition can be expressed as: + = + (27) where, as defined in Fig. 2b, is the current crack length, is the size of the current plastic zone, is the crack length when the tensile overload occurred and is the size of the plastic zone generated by the tensile overload. Using Irwin’s formula ( Eq. 23) to estimate the plastic zone size, Eq. 27 becomes: + 1∙ ∙ ( ) 2 = + 1∙ ∙ ( ) 2 (28) Solving for yields the expression: = ∙√1− − (29) Therefore, can be calculated as: = ∙√1− − − max (30) The Willenborg model, despite its popularity, does not include delayed retardation, overshooting and crack acceleration (compressive underload). However, it can be argued that is reasonable to neglect these phenomena as they are often second order effects. Generalised Willenborg Model Gallagher (1974) generalised the Willenborg model to improve its convergence with test results in the threshold region and near crack arrest. The generalised Willenborg model proposed by Gallagher is documented in NASA (1999) and is expressed as: ′ = ∙ (31) where = 1 − ∆ ℎ max −1 (32) with indicating the shut-off value of the stress ratio max , varied to optimise the fit between life prediction and test results. Reasonable values of are 3.0 for aluminium, 2.7 for titanium and 2.0 for steel (Harter (2000)). However, Harter stated that these values are not truly material dependent and can vary. This retardation model does not consider compressive underload, delayed retardation or overshoot. 227 3.3.