PSI - Issue 75

240 Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 234–244 Author name / Structural Integrity Procedia (2025) 7 For constant amplitude loading, the retardation stress intensity factor, , can be calculated as the sum of a triangular and a rectangular stress block (Fig. 3), using Eq. 13 as kernel function. In the more complex but more realistic variable amplitude loading scenario, instead, the creation of the residual stress zones is governed by the cycles with the largest amplitudes. In other words, when a cycle with a significant amplitude creates a large residual compressive stress ahead of the crack tip, the crack tip itself must travel through this residual wake until it reaches its end or continue through a new residual compressive stress zone created by a more recent large-amplitude cycle (Fig. 4b). Considering the example in Fig. 4b, where each block is caused by an overload cycle, the overall retardation stress intensity factor, , is given by the sum of all the trapezoidal blocks, whose effect is estimated through Eq. 13. Mikheevskiy (2009) proposed four memory rules to deal with the overloads (Fig. 5): • Rule 1 : when the residual stress rises by more than 5% of the original overload, the first block ends and a new block starts. Both blocks are used to calculate the retardation stress intensity. This rule applies when the crack propagates into the climbing region of the original overload cycle and avoids overestimating the retardation effect. • Rule 2 : when an overload is followed by smaller cycles producing significantly lower residual stresses, these lower residual stresses do not contribute to the crack retardation and are neglected. • Rule 3 : when a new overload is 5% (or more) hihger than the current overload, the current block is ended and stored in the summation stack and a new block is started. • Rule 4 : when an overload is followed by cycles with approximately the same residual stress (within ±5%) as the original overload, the residual stress profile of the original overload is simply extended. This improves computational efficiency by avoiding that the number of blocks to add together becomes unnecessarily too large.

Fig. 5. Memory rules.

2.4. Estimation of material parameters The parameters required by the crack tip plasticity model used by the ‘Total - Life’ method can be obtained through either tensile tests or traditional stain-life (EN) fatigue testing. In particular, the parameters to determine are the modulus of elasticity, , the Ramberg- Osgood’s cyclic strength coefficient, ′ , and cyclic strain hardening exponent, ′ , and the Poisson’s ratios for elastic and plastic strain, and , respectively. The crack growth model instead requires the estimation of the following parameters: • The intercept, , and the slope, , of the ℎ segment of the crack growth curve. • Walker’s mean stress correction coefficient, (typically ≈ 1+ 1 ′ ). • Naroozi’s crack -closure, notch-correction coefficient, (typically 2≤ ≤3 ). • The effective crack-tip radius ∗ (typically of the order of the grain-size). The crack growth rate parameters can be estimated through traditional fatigue crack growth testing over a range of stress ratios. The effective crack-tip radius, ∗ , instead, according to Mikheevskiy (2009), can be estimated