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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symmetrically from both sides of the central hole. Crack growth was monitored using a high-resolution camera equipped with a 21-megapixel full-frame CMOS sensor, with measurements calibrated against a scale placed on the specimen surface. Four PZT sensors were bonded to each specimen in two configurations. Two sensors were vertically aligned with the hole centre at coordinates (50, 210) and (50, 90), while the other two were positioned farther from the central axis at (70, 180) and (20, 70) – the origin is at the corner of the sample. This arrangement was designed to evaluate the robustness of the proposed detection algorithm under varying sensor layouts. Loading was applied at 6 Hz, and the resulting dynamic responses were captured using a digital oscilloscope. Data were sampled at 10 kHz for a duration of 10 seconds every 1000 cycles, yielding 100,000 samples per measurement without interrupting the test. The fatigue loading was terminated when the crack reached approximately 15 mm in length. 2.2. Automatic crack detection procedure The automatic crack detection workflow (Fig. 2) is based on passive sensing through higher harmonic analysis combined with the dynamic piecewise linear (DPL) method and consists of four sequential stages: (1) fatigue loading, (2) signal acquisition, (3) third harmonic calculation and fatigue life classification, and (4) crack length characterisation. In the first stage, fixed-frequency fatigue loading is applied to the specimen, which inherently serves as the excitation source, and the response is analysed in both time and frequency domains. During the data acquisition stage, surface-mounted PZT sensors record the structural response induced by the loading. In the third harmonic calculation stage, the DPL method is employed to extract slope values from the linear fits of segmented windows along the curve representing the relative third harmonic parameter as a function of loading cycles. This enables identification of transition points corresponding to distinct phases of fatigue progression by applying predefined thresholds. Using the extracted features, the fatigue process is classified and modelled. Subsequently, a physics informed LSTM model is applied to estimate the crack length and the corresponding number of cycles. The resulting crack length estimates and cycle counts then feed into the prognosis module for life prediction. Fig. 2. Automated framework for crack detection and length prediction using harmonic analysis, piecewise linear fitting, and LSTM-based modelling. 3. Probabilistic prognosis of crack growth While the diagnosis and prognosis modules share a common goal of characterising crack growth, their functional requirements and modelling demands are distinct. In the diagnosis module, crack size estimation was achieved using a physics-informed LSTM model trained on third harmonic parameters, where Paris’ law was embedded as a regularisation mechanism to improve physical consistency. However, the prediction remains fundamentally deterministic, limiting its ability to represent uncertainty and to quantify confidence in long-term fatigue forecasts. More critically, the LSTM model does not support inference of latent quantities such as the . To address these limitations, the current study adopts a probabilistic framework that integrates a Paris’ law-based crack growth model with sequential Bayesian inference. The approach enables continuous assimilation of online measurements, while supporting joint estimation of crack length, material parameters, and under uncertainty. Importantly, by decoupling the prognosis module from the preceding diagnosis pipeline, the framework preserves modularity and scalability, allowing integration with alternative sensing or damage detection schemes through standardised interfaces. This design philosophy aligns with the principles of federated digital twin architectures, promoting flexibility and generalisability across different structural configurations.