Issue 78

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

Chebyshev coefficient spectra Fig. 5 shows the Chebyshev coefficient spectra for both specimen types. Panel (a) shows the absolute amplitudes | k a | and panel (b) shows the normalised amplitudes | 0 / k a a | using a 5% significance threshold. Type A exhibits 18 significant coefficients (| 0 / k a a | > 5%) compared with only 16 for type B- a comparable spectral richness that does not, on closer examination, track the difference in grain-boundary crossing frequency between the two orientations ( N g-A = 3.72 vs N g-B = 1.55). As discussed above, this similarity reflects the limited data density available for the type B specimen rather than genuine microstructural complexity; the rolling-induced anisotropy of the material remains independently established through the EBSD-measured grain-size ratio and the number of grains crossed, rather than through the Chebyshev spectral coefficient count.

(a) (b) Figure 5: Chebyshev coefficient spectra. (a) Absolute amplitudes | k a |. (b) Normalised amplitudes |

0 / k a a | with 5% threshold. Type

A: 18 significant coefficients. Type B: 16 significant coefficients. Discussion of identified C and n constants

For type A, C A = 3.55 × 10 ⁻⁵ µm/cycle·(MPa √ m) ⁻ ⁿ and n A = 2.382 are identified from direct regression of d a /dN vs Δ K data. The value of n A is close to the long-crack exponent ( n FL = 2.53), indicating that the Paris exponent retains physical relevance even in the short-crack regime under ⊥ R D loading. For comparison, the FAA fatigue crack growth database [32] reports, for AA2024-T351 L–T plate at R = 0.1, NASGRO and Walker exponents of n = 3.60 and n = 3.82, respectively, over the long-crack regime ( Δ K ∈ [3.3; 44] MPa √ m, code M2EB11AB1). The lower value of n A is consistent with short-crack behavior, where grain-scale retardation events associated with microstructural barriers reduce the apparent slope of the log-log d a /dN– Δ K response. This interpretation is consistent with Δ K th = 0.88 MPa √ m and K IC = 39.6 MPa √ m reported in the same database, which define the physical bounds of the present experiments. For type B, n B = 1.071 is unusually low compared with typical AA2024-T351 values reported in the FAA database [32]. This deviation is mainly attributed to the numerical differentiation of a ( N ): Savitzky-Golay smoothing redistributes local gradients in a ( N ), leading to an underestimation of d a /d N at low Δ K and consequently biasing the log-log regression slope. A constrained identification, fixing n B = 3.60 and re-estimating C B , gives C B |_{n=3.60} = 3.1 × 10 ⁻⁶ µm/cycle·(MPa √ m) ⁻ ⁿ while the enriched model maintains R² = 0.661 — a reduction of less than 2% relative to the unconstrained case, indicating that model performance is only weakly sensitive to the exact value of n B . A further sensitivity analysis compared the level-1 regression on raw, unsmoothed a ( N ) data against the Savitzky-Golay filtered data. The raw data yield n ≈ 0.79–1.01, consistent with — and in some cases lower than — the smoothed value, confirming that smoothing is not responsible for the anomalously low exponent. Excluding the smoothing-window edge points shifts n B by 8.4%, a real but modest effect; 7.8% of the raw points fall outside the physical validity range, corresponding to near-zero-cycle digitisation clusters, confirming that Savitzky-Golay smoothing is a necessary preprocessing step rather than an arbitrary source of bias. For the long-crack regime, C FL = 2.35 × 10 ⁻ ¹ ⁰ m/cycle·(MPa √ m) ⁻ ⁿ and n FL = 2.53 yield R² = 0.984, confirming the validity of the classical Paris law in this regime and indicating that μ ( ξ ) ≈ 1 on average. A reproducibility check performed on a second specimen per orientation confirms these Level-1 constants (Tab. 8): type B shows remarkable inter-specimen consistency ( n = 1.01 vs. 0.97; N g = 1.55 vs. 1.51), while type A shows somewhat greater

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