Issue 78

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

  1, 1 ,   

  n T x cos n arccos x x    · ,

n N

(2)

They satisfy the three-term recurrence relation:       1 1 2 · n n n T x x T x T x    

(3) with initial conditions T ₀ (x) = 1 and T ₁ (x) = x. The first six polynomials are: T ₀ = 1, T ₁ = x, T ₂ = 2x² − 1, T ₃ = 4x³ − 3x, T ₄ = 8x ⁴ − 8x²+1, T ₅ = 16x ⁵ − 20x³+5x. Three mathematical properties justify their choice as a functional basis for the enrichment of the Paris law. First, Chebyshev polynomials are orthogonal on [ − 1, 1] with respect to the weight function w(x) = (1 − x²) ⁻ ½ expressed as:       1 1 2 · 0 ; 0 ; 0 2 1 m n T x T x dx ifm n ifm n if m n x            (4) This orthogonality has a key practical consequence: the normal equations for least-squares identification of the coefficients k a are decoupled. Adding or removing a term does not affect the values of the other coefficients. This property guarantees numerical stability and the consistency of order selection by the AICc criterion. Second, the minimax property states that the normalised Chebyshev polynomial T ₙ (x)/2 ⁿ ⁻ ¹ minimises the maximum error among all monic polynomials of degree n on [ − 1, 1]: This makes the Chebyshev basis optimal for approximating oscillatory functions on a bounded interval in the sense of the uniform error. This property is particularly relevant here since the microstructural modulation factor μ ( ξ ) presents quasi periodic oscillations associated with successive grain boundary crossings. Third, when all coefficients k a are zero, the enriched model reduces exactly to the classical Paris law. This conservation property ensures that the enriched model is a conservative extension that preserves the reference behaviour in the absence of microstructural information. Normalised variable ξ = f(a/d): the microstructure–propagation constitutive link The polynomial argument is defined as the crack length normalised by the grain size in the propagation direction. For a grain size d , the dimensionless ratio ( / ) a d represents the number of grains crossed from the crack origin. This ratio provides a natural scale for describing crack–grain boundary interactions: when ( / a d )= k (integer), the crack reaches the k-th grain boundary GB ₖ from the initiation point. The normalised variable ξ is then constructed as:      1,1 p x 1 1 2 p x n min max   (5)

a              a d d





min



1   



2 ·

1, 1

(6)

max            a d a d  

min

where ( / ) min a d and ( / ) max a d correspond to the bounds of the experimental domain. For type A specimens: TL d d d     154 µm; for type B: L d d d     395 µm. The polynomial T ₙ ( ξ ) exhibits n oscillations on [ − 1, 1]. If the crack crosses N g grains in total, dominant contributions in the decomposition are expected at orders n close to g N , whose oscillation frequency is associated with the intergranular spacing. The normalisation by d thus directly links the expected oscillation frequency to the microstructure. This constitutive link is illustrated as follows: for type A ( d  154 µm), g A N  = 0 )/ max a a d    (610 − 35.4)/154 ≈ 3.7 grains crossed, consistent with the identified value 3.72 (Tab. 7); for type B ( d  = 395 µm), g B N   (645 − 31.5)/395 ≈ 1.55 grains crossed, consistent with the identified value 1.55 (Tab. 7).

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