Issue 78

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

Figure 13: Comparison μ ( ξ ) - type A (red) vs. type B (blue). Both orientations converge to the same statistically optimal order ( N = 21); this convergence is independently validated by the physical peak-counting criterion for type A only, consistent with N g-A = 3.72 > N g-B = 1.55. The modulation factor oscillates around μ = 1, corresponding to the local Paris reference. Type A exhibits five well-defined oscillations associated with the five grain boundaries crossed. Type B, following re-identification at the statistically optimal order N = 21, exhibits a more complex oscillatory pattern than initially reported, dominated by a pronounced peak associated with GB ₁ but accompanied by several secondary oscillations without a direct experimental counterpart Unlike the earlier interpretation of this contrast as a direct microstructural signature of rolling-induced anisotropy, the comparable number of significant Chebyshev coefficients now identified for both orientations (18 for type A, 16 for type B) shows that spectral richness alone does not track grain-boundary crossing frequency. Rolling-induced anisotropy remains established independently through the EBSD-measured grain-size ratio ( / d d   = 2.56) and the number of grains crossed, rather than through the Chebyshev spectral content. Synthesis and general discussion Tab. 9 summarises the main results. For type A, R² increases from 0.665 to 0.907, corresponding to a gain of approximately +36%. For type B, R ² increases from 0.147 to 0.835, a gain of approximately +468%. Type A crosses 3.72 grains over the tested range, producing five well-defined peaks in / da dN that coincide precisely with the grain boundary positions observed in the microstructural maps; the modulation factor reaches μ max = 3.140 at GB ₅ , indicating that the local propagation rate exceeds the Paris prediction by a factor greater than three, consistent with the high misorientation angles reported by Chen et al. [14] at this boundary. Type B, in contrast, is dominated experimentally by a single grain boundary identified at the specimen surface (GB ₁ ) yet requires the same statistically optimal order as type A ( N = 21) to achieve a physically admissible fit. As discussed in "Physical interpretation of the model components" above, this additional oscillatory content plausibly reflects sub-surface crack-microstructure interactions not captured by the 2D EBSD characterization, given the pronounced difference between the in-plane and through-thickness grain sizes for this orientation ( d L = 395 µm vs. d TC = 133 µm). The higher mean value μ mean-B = 1.477 confirms that type B propagates faster on average than predicted by its own Paris baseline, consistent with the lower grain boundary density in the ∥ R D direction ( f  = 2.5 vs. f  = 6.5 boundaries/mm). For both orientations, the enriched model captures the dominant experimentally identified grain boundaries, with type A additionally validated by the full peak-count correspondence to all five GB positions. A key feature of the present dataset is the limited Δ K range covered by short-crack propagation. Both specimen types lie within Δ K ∈ [4.6; 22] MPa √ m, which does not overlap with the long-crack regime ( Δ K ∈ [10; 35] MPa √ m). This confirms established observations in short-crack mechanics [5,6,9]: cracks can propagate at stress intensity levels below those required for long-crack growth, due to incomplete crack closure and dominant microstructural barriers. In this regime, the classical Paris law loses predictive capability. The scatter in the d a /dN- Δ K relation is significantly higher than for long cracks, and the low exponent obtained for type B ( n B = 1.071) reflects the combined effect of weak Δ K dependence and strong grain

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