Issue 78

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

boundary interactions that cannot be captured by a single power-law slope. Threshold-based extensions such as Forman [19] or NASGRO [21] are also not strictly applicable as they are designed for long-crack behaviour near threshold and instability. The proposed Chebyshev-enriched formulation overcomes these limitations by separating the macroscopic Paris contribution (level 1) from the microstructural modulation (level 2), enabling a consistent description of crack growth in the short-crack regime. Compared with existing approaches, the proposed model occupies an intermediate position. Extensions such as Walker [20] and Forman [19] introduce load ratio and toughness effects but remain based on homogeneous material assumptions and cannot capture grain-scale oscillations. The Navarro and de los Rios model [22] explicitly incorporates grain boundary shielding mechanisms but requires incremental grain-by-grain treatment, limiting its practical use. Crystal plasticity-based frameworks [23] provide a more complete physical description but remain computationally intensive for engineering applications. In contrast, the present approach requires only the mean grain size d , readily obtained from EBSD, and remains computationally inexpensive. Its main advantage is therefore the combination of physical interpretability and simplicity.

Metric

Type A - Paris

Type A - Enriched

Type B - Paris

Type B - Enriched

R²

0.665

0.907 (+36%)

0.147

0.835 (+468%)

NRMSE (%)

—

7.87

—

8.50

μ mean μ max

1.00 (ref.)

1.126 3.140

1.00 (ref.)

1.477 2.965

— —

— —

N* (AICc)

21

21

Table 9. Results summary - classical Paris vs. enriched model.

C ONCLUSION

T

his study demonstrates the effectiveness of the Chebyshev-enriched Paris law in describing short fatigue crack growth in rolled AA2024-T351 aluminum alloy. The key findings are structured as follows: - The enriched model substantially outperforms the classical Paris law for both crack orientations relative to the rolling direction ( R D ), yielding a marked increase in the coefficient of determination:  Type A (Crack ⊥ R D ): R² increases from 0.665 to 0.907 ( N* = 21), with the five modulation peaks matching the five grain boundaries GB ₁ -GB ₅ identified experimentally by EBSD and SEM.  Type B (crack ∥ R D ): R² increases from 0.147 to 0.835 ( N = 21). the model captures the dominant peak at the single grain boundary GB ₁ identified at the specimen surface, together with additional oscillatory content plausibly arising from sub-surface crack-microstructure interactions, given the marked difference between the in-plane and through-thickness grain sizes for this orientation ( d L = 395 µm vs. d TC = 133 µm). - The proposed formulation, ⁄ = C·( Δ K) ⁿ · μ ( ξ ), ξ = ሺ ሻ ⁄ ∈ [ − 1, 1] explicitly incorporates the grain size measured by EBSD into the normalised variable ξ , providing a direct measure of the number of grains crossed by the crack front and introducing microstructural effects without resorting to grain-resolved or multiscale simulations. - Rolling-induced anisotropy is established through the EBSD-measured grain-size ratio ( ∥ ୄ ⁄ = 2.56) and the number of grains crossed ( N g-A = 3.72 vs N g-B = 1.55). A reproducibility check on a second specimen per orientation further confirms the robustness of the Level-1 Paris constants, with type B showing remarkable inter-specimen consistency and type A showing somewhat greater variability, both within the physically plausible long-crack range. - The proposed formulation provides a bridge between classical Paris modelling and multiscale approaches, retaining a closed-form, physically interpretable structure while capturing local propagation fluctuations relevant to anisotropic aluminium alloys. - Perspectives for future work include establishing a quantitative link between | ௞ | and grain boundary misorientation θ ₖ through full crack-path EBSD mapping, assessing the representativeness of 2D surface EBSD characterization through 3D or serial-sectioning approaches — particularly relevant where the through-thickness and in-plane grain sizes differ markedly, as observed here for type B — extending the approach to other rolled aluminium alloys and to variable load ratios, and integrating the model into fatigue life prediction frameworks.

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