PSI - Issue 80

Sadjad Naderi et al. / Procedia Structural Integrity 80 (2026) 77–92 Sadjad Naderi et al. / Structural Integrity Procedia 00 (2025) 000–000

89 13

Fig. 7. Final cycle count distributions at critical crack length for the three schemes.

4.3. Added value of EIFS in the Bayesian process A natural question arises whether the estimation of , representing the early damage state, may be redundant given that crack initiation can already be detected by the diagnostic technique. To examine this, an additional case is performed using Scheme I but excluding from the Bayesian update process. In this case, crack growth curves are sampled only from the posterior distributions of the Paris constants ( +.*&%,-., , +.*&%,-., ) , and the corresponding initial crack sizes are obtained by back-extrapolation of the − curves to =0 . The resulting mean trajectories are shown in Fig. 8, where the dashed lines indicate the extrapolated segments between ( 3E&,0F.;0&%' , =0) and the first detected observation ( $%&%"&%' , $%&%"&%' ) .

Fig. 8. Crack growth predictions without explicit integration in Scheme I. First, the exclusion of forces the calibration process to assume a deterministic initial crack size, which suppresses the natural variability in flaw characteristics. This results in artificially tight uncertainty bounds (0.67% vs. ±1.6% in Scheme I) and removes the adaptive uncertainty quantification that progressive Bayesian updating provides. Second, the mean life prediction (147k cycles) overestimates compared with Scheme I’s final prediction (137k cycles), reflecting a systematic bias introduced when initial flaws are treated only as post-processed values . Third, the predicted trajectories show poorer agreement with the observed data compared with Scheme I, further underscoring the importance of treating as an integrated stochastic variable. Most importantly, integration accelerates convergence of the Bayesian learning process. While the extrapolation-based approach could, in principle, be improved by fine-tuning hyperparameters at additional computational cost, incorporating directly achieves comparable or better performance more efficiently. This highlights the added value of not only as a representation of pre-existing flaws but also as a critical enabler of robust and adaptive uncertainty quantification in fatigue prognosis.