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

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3.2. Forward physics model (Paris’ law) The fatigue crack growth in metallic structures is commonly described using the Paris–Erdogan law (Paris and Erdogan 1963), which relates the crack growth rate to the applied stress intensity factor range: ' ' 1 0 = ( ) 2 (1) where is the crack length, is the load cycle count, delta is the stress intensity factor range, and and are material-dependent constants experimentally. The stress intensity factor is expressed as: ∆ = ∆ √ (2) where ∆ is the applied stress range and is the geometry correction factor. As reported in (Fierro and Meo 2015), for cracks initiating from a hole, the geometry factor stabilises once the crack length exceeds approximately one-fifth of the hole radius, with typically ranging between 1.00 and 1.05. In this work, is assumed constant during the stable crack growth phase (Lim and Sohn 2019), since its variation within this range has a negligible influence on Δ and thus on the Paris’ law predictions compared to other sources of modelling uncertainty. In the prognosis module, the Paris’ law serves as the forward model, propagating the crack length from the inferred to the predicted RUL under specified loading conditions. The model provides a foundation for targeted corrections via the Bayesian update framework described in the following section. The crack growth inference framework is formulated as a three-level hierarchical Bayesian model (Fig. 4). At global level, hyperpriors encode population-wide knowledge of model parameters based on engineering expertise including , , 3/45 and 3/45 . At first local level, the hyperparameters 3/45 and 3/45 define a prior distribution for the . At second level beliefs, trial datasets ( ∗ , ∗ ) are generated by the forward crack growth model, with uncertainty propagated via Monte Carlo sampling. Here, and are sampled as input material parameters with sampled as the initial condition. The observed data ( 7,8% , 7,8% ) are incorporated through the likelihood function to update posterior distributions. The hierarchical update can be expressed as: ( | ) ∝ ( | ) ( | ) ( ) (3) where are local parameters, are hyperparameters, and is the observed data. The structure forms the basis for sequential Bayesian updating, allowing the model to incorporate new information while maintaining physical constraints. 3.3. Dynamic Bayesian network formulation 3.3.1. Hierarchical Bayesian foundation