Issue 78

T. Nasri et alii, Frattura ed Integrità Strutturale, 78 (2026) 275-297; DOI: 10.3221/IGF-ESIS.78.17

Panel (b) presents the SEM micrograph of the crack path. The dark region on the left corresponds to the hole edge, corresponding to the stress concentration region and crack initiation site. The crack propagates from left to right, showing local deflections and tortuous propagation paths at each grain boundary crossing, consistent with the crack-microstructure interactions reported by Chen et al. [14,15]. The dark region on the right does not correspond to a physical feature of the specimen; it only indicates that the SEM field of view does not extend beyond a ≈ 405 µm. Grain boundary positions GB ₄ and GB ₅ in this region were determined from optical microscopy measurements of a (N). Panel (c) shows the corresponding crack growth rate da /dN as a function of crack length a . The classical Paris law (dashed grey, n = 2.38, R² = 0.665) captures the mean propagation trend but fails to reproduce the local oscillations. The enriched Chebyshev model (red, N* = 21, R² = 0.907, NRMSE = 7.87%) accurately reproduces all five observed peaks, each aligned with a grain boundary position GB k . The direct correspondence between the grain boundaries identified in panels (a) and (b) and the d a /dN peaks observed in panel (c) provides physical support for the model. The oscillations in crack growth rate therefore cannot be attributed to numerical artefacts but reflect actual crack–microstructure interactions at successive grain boundaries.

Figure 7: Crack growth rate da /dN vs. crack length a - type A specimen (  ∥ R D , crack ⊥ R D , d  = 154 µm, max  = 200 MPa, K t = 2.3, R = 0.1, 0 a = 35.4 µm). (a) EBSD orientation map - TL plane (a = 0–270 µm); GB ₄ and GB ₅ beyond EBSD range, identified by SEM. (b) SEM micrograph of the crack path. (c) da/dN vs. crack length: Paris law (dashed grey) and enriched Chebyshev model (red, N = 21 , R² = 0.907 , NRMSE = 7.9 %).

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