Issue 78

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

locally dominate crack-tip driving forces [5,6]. These observations provide direct physical motivation for the proposed Chebyshev enrichment. Furthermore, the FAA database reports a long-crack threshold Δ K th ≈ 0.88 MPa √ m at R = 0 (approximately 0.79 MPa √ m at R = 0.1), while all specimens in the present study propagate at Δ K min ≥ 4.6 MPa √ m, i.e. more than five times above the threshold. This excludes any near-threshold propagation mechanism and confirms that the observed oscillations originate primarily from crack–microstructure interactions within the Paris regime II. Second, the linear Paris fits exhibit a steeper slope for type A ( n A = 2.38) than for type B ( n B = 1.07). Third, the long-crack data (purple, R² = 0.984, n FL = 2.53) closely follow the classical Paris law, indicating that μ ( ξ ) ≈ 1 on average in the long crack regime and supporting the need for a dedicated short-crack formulation. Microstructural modulation factor μ ( ξ ) - anisotropy quantification Figs. 11-13 present the modulation factor μ ( ξ ) = d a /dN exp / [C·( Δ K) ⁿ ] for both specimen types as a function of the normalised variable ξ = ( / ) f a d ∈ [ − 1, 1].

Figure 11: Modulation factor μ ( ξ ) -type A. Five peaks corresponding to GB ₁ –GB ₅ ( N* = 21, R² = 0.907, μ mean = 1.126, μ max = 3.140).

Figure 12: Modulation factor μ ( ξ ) - type B ( N* = 21, R² = 0.835, μ mean = 1.477, μ max = 2.965).

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