Issue 78

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

The correspondence between the model and the underlying microstructure operates at two distinct levels. First, the positions of the μ ( ξ ) oscillation peaks are directly validated against experimentally identified grain boundaries in the present study: Fig. 7 shows that all five peaks predicted for type A align with the five grain boundaries GB ₁ -GB ₅ located by EBSD and SEM along the crack path. For type B, the dominant peak aligns with the single grain boundary GB ₁ identified experimentally at the specimen surface. The model also captures secondary oscillations beyond this single feature. A plausible explanation lies in the three-dimensional nature of the crack front. EBSD characterization was performed on the specimen surface only, whereas ( / ) da dN reflects propagation through the full specimen thickness. For type B, the grain size in the propagation direction ( d L = 395 µm) is nearly three times larger than the through-thickness grain size ( d TC = 133 µm, Tab. 3). Sub-surface grain boundaries, invisible to the surface scan, could therefore contribute to the observed signal. For type A, this concern does not arise: the TL and TC grain sizes are nearly equal ( d TL / d TC = 1.16), so the surface characterization remains representative of the through-thickness microstructure. This asymmetry suggests that the representativeness of 2D surface EBSD depends on how closely the in-plane and through-thickness grain sizes match — a distinction not previously considered in the interpretation of short-crack modulation models. Second, the amplitudes | k a | reflect the degree of grain boundary misorientation encountered at each crossing, consistent with the mechanistic observations of Chen et al. [14,15], who reported that crack deflection and local retardation-then-acceleration behaviour occur systematically at high-angle twist boundaries ( α > 20°) in a related aluminium alloy. A full quantitative mapping between | k a | and the misorientation angle θ ₖ at each individual grain boundary represents a natural extension of this work, building on the EBSD-based approach already established here. The mean modulation factor μ mean = ∫ μ ( ξ ) d ξ / 2 integrates the effect of all crossed boundaries over the entire crack path and quantifies the global anisotropy. The fact that μ mean-B = 1.477 > μ mean-A = 1.126 reflects that type B propagates on average faster than its own Paris reference law predicts, an effect amplified relative to the originally reported value following re identification with the statistically optimal order N = 21. This result is consistent with the experimental observations and with Chen et al. [14,15]. Finally, the optimal order N* was initially connected to the number of grains crossed N g via the following empirical relationship:     g * · N , 5.7 type A , 5.2 type B N T with T A and T B       (12) This relationship should be understood as an empirical scaling factor specific to the present dataset, rather than a general physical law. With the statistically optimal order for type B now established at N = 21, the corresponding coefficients become α T,A ≈ 5.65 and α T,B ≈ 13.55. This substantial difference, combined with the inter-specimen variability quantified in Tab. 8, indicates that α T does not scale simply with grain size across orientations. (Eq. 12) is therefore best regarded as a dataset-specific observation rather than evidence of a general grain-scale modulation mechanism. Summary table of model components Tab. 5 summarises the correspondence between the mathematical components of the model and their microstructural interpretation. This correspondence constitutes the foundation of the microstructurally explicit nature of the model - that can be identified and measured by EBSD.

Component

Mathematical meaning

Physical interpretation Mean behaviour in regime II Normalised grains crossed - oscillation scale

EBSD source

C , n

Macroscopic Paris constants Normalised variable ∈ [ − 1, 1] Orthogonal minimax basis k-th coefficient amplitude

Not required

d  =154µm, d  =395µm

ξ = ( / ) f a d

T ₖ ( ξ )

Microstructural oscillations Mathematical property

Misorientation at k th crossing

| k a |

k  [14,15]

Significant modes ≈ α T ×N N* _B g (dataset-specific)

N*

Optimal order (AICc)

N g from d and a (N)

μ ( ξ )=1+ Σ a ₖ T ₖ SEM and EBSD observation [14,15] Table 5: Correspondence between mathematical components and microstructural interpretation. Modulation factor Ratio d a /dN exp /d a /dN Paris

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