Issue 78

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

K EYWORDS . Short cracks, Enriched Paris law, Chebyshev polynomials, AA2024-T351, Microstructural anisotropy, Grain boundaries.

I NTRODUCTION

F

atigue failure remains one of the main damage mechanisms affecting aeronautical structures, making the accurate prediction of fatigue crack propagation essential for damage-tolerant design approaches [1,2]. The AA2024-T351 aluminium alloy, a precipitation-hardened Al–Cu–Mg alloy, is widely used in primary aircraft structures due to its high specific strength and good fatigue resistance [3,4]. However, its partially recrystallized rolled microstructure exhibits strong grain anisotropy, characterized by elongated grains along the rolling direction and finer grains in the transverse directions. This microstructural heterogeneity strongly influences short fatigue crack propagation, particularly through successive interactions between the crack front and grain boundaries [5,6]. The classical Paris–Erdogan law [7],   / K n da dN C    , remains the reference model for describing stable long-crack propagation in fatigue regime II. Its formulation relies on the assumption of an effectively homogeneous material. However, several studies have shown that this assumption becomes inadequate for short cracks [8–10]. In particular, short cracks may propagate at Δ K levels below the long-crack threshold, while their propagation rates exhibit non-monotonic oscillations strongly influenced by the local microstructure [11,12]. The influence of grain boundaries on short-crack propagation in aluminium alloys has been extensively investigated. Zhai et al. [13] demonstrated that crack transmission across grain boundaries strongly depends on the crystallographic misorientation between adjacent slip systems. More recently, Chen et al. [14] showed through EBSD observations that crack deflection and local retardation preferentially occur at high-angle grain boundaries. In situ SEM observations performed by Chen et al. [15] further revealed that crack–grain-boundary interactions generally proceed through three successive stages: crack deceleration near the boundary, temporary arrest at the boundary, and subsequent re-acceleration after crossing. In a related aluminium alloy, Jin et al. [16] recently showed that fatigue crack initiation in 7075 alloy occurs preferentially at Fe rich particulates located at grain boundaries or within grains, and that crack deflection at grain boundaries strongly depends on crystallographic misorientation between adjacent grains, consistent with the mechanism reported above. More recently still, Shang et al. [17] demonstrated pronounced anisotropy in fatigue crack initiation and propagation in an AA2024-T7 plate depending on specimen orientation relative to the rolling direction (LT, LS, TS planes), with predominantly transgranular propagation in the L-T and LS directions versus intergranular propagation in the T-S direction - a directional dependence of the propagation mechanism directly relevant to the rolling-induced anisotropy investigated in the present study. Several extensions of the Paris law have been proposed to account for specific mechanical effects. The Forman equation [19], the Walker model [20], and the NASGRO formulation [21] extend the description toward threshold, load ratio effects, and final fracture regimes. Other physically motivated approaches, such as the Navarro and de los Rios model [22] or multiscale formulations coupling crystal plasticity and fracture mechanics [23], explicitly incorporate microstructural effects. More recently, MacLachlan et al. [24] developed a crystal plasticity (CPFE) framework coupled with the extended finite element method (XFEM) for polycrystalline alloys, incorporating an explicit grain-boundary hardening model that reduces the local stored energy available for crack growth near boundaries, qualitatively reproducing experimentally observed growth-retardation periods at grain boundaries. Data-driven alternatives have also emerged: Zhou et al. [25] proposed an interpretable machine-learning method integrating physical domain knowledge to model short-crack growth, while Babu Rao and Spear [26] developed a hybrid physics-based/deep-learning framework to accelerate microstructurally small fatigue crack growth simulations. However, these approaches are often computationally expensive, require large training datasets, or are difficult to implement within a simple analytical framework suitable for structural design applications. To capture the intrinsic variability of short-crack growth induced by discrete microstructural barriers, an appropriate mathematical representation must be both flexible and physically consistent over a bounded domain. In this context, orthogonal polynomial bases offer a rigorous framework for approximating spatially varying modulation effects while preserving numerical stability and minimizing parameter correlation. In particular, Chebyshev polynomials of the first kind are well suited due to their near-minimax approximation property and their orthogonality over the interval [ − 1, 1], which ensures an efficient and non-redundant representation of oscillatory crack–microstructure interactions. This makes them especially relevant for describing the non-monotonic evolution of short-crack propagation driven by successive grain boundary encounters in heterogeneous metallic microstructures.

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