Issue 78

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

The large grain size scatter (±203 µm for L d and ±47 µm for TL d ) uncertainty in ξ that propagates into coefficient identification. The relative uncertainty propagates as:

δξ δ d 47

203 395









d 154   



and

30% type A ,

51% type B

(7)

ξ

This intrinsic microstructural uncertainty partly explains the scatter of μ ᵢ around the fitted Chebyshev curve, which is more pronounced for type B (NRMSE = 8.50 %) than for type A (NRMSE = 7.87%). Chebyshev-enriched Paris model The enriched model takes the form of the classical Paris law multiplied by a microstructural modulation factor μ ( ξ ) :

  / · ( ) n da dN C K    

(8)

with the modulation factor expressed as an expansion on the Chebyshev basis:

N

0     1 k

 

  

 

· a T

(9)

k k

Combining these elements, the complete enriched model reads:

 

  

N 

da

a     d





 

  

n

 



with f 

   

· C K

a T

· 1 

1, 1

(10)

k k

dN



k

0

This is the central equation of the proposed model. When all k a = 0, d a /dN = C·( Δ K) ⁿ exactly, confirming that the enriched model is a conservative extension of the classical Paris law. The multiplicative term [1 + Σ a ₖ T ₖ ( ξ )] amplifies or attenuates the Paris prediction according to the crack position in the microstructure, reflecting intragranular accelerations and grain boundary retardations. Physical admissibility further requires μ ( ξ ) to remain strictly positive over the entire domain:

  

  

N 

  









min

a T

1

0

(11)

 



min

k k



1,1



k

0

This positivity condition is verified numerically over a fine grid of 10 000 points uniformly distributed on [ − 1, 1] after each identification. For both specimen types, μ min > 0 is satisfied, guaranteeing that the model never predicts a zero or negative propagation rate: μ min = 0.252 for type A (reached near ξ ≈ − 0.85, corresponding to the initiation region close to the hole edge) and μ min = 0.152 for type B (reached near ξ ≈ +0.72, in the post-GB ₁ deceleration zone). For high-degree polynomials, Runge-type oscillations are mitigated by the Chebyshev basis itself, whose minimax property (Eq 5) minimises the maximum deviation between knots, bounding inter-point excursions to a fraction of the observed μ min margins. Finally, the model relies on a clear scale separation: C and n describe the macroscopic average behaviour identified directly on each holed specimen type, while the coefficients k a describe microstructural fluctuations at the mesoscopic (grain-by-grain) scale. Physical interpretation of the model components The variable ( / ) f a d   is directly linked to the grain size d in the propagation direction. Changing the propagation direction changes d , and therefore modifies the physical scale on which the polynomial oscillations develop. The ratio   / d d   = 2.56 implies that, for the same number of grains crossed, ξ A evolves over a 2.56 times smaller physical scale than ξ B in terms of physical crack length. Combined with the number of grains actually crossed ( N g-A = 3.72 > N g-B = 1.55), this physical scale difference was initially thought to explain a higher optimal order for type A. However, both orientations converge to the same statistically optimal order, N = 21.

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