Issue 78

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

P ARAMETER IDENTIFICATION

T

his section describes the two-level strategy used to identify the enriched model parameters from the experimental data, together with the resulting values obtained for both specimen orientations.

Hierarchical identification strategy Parameter identification follows a two-level hierarchical strategy that exploits the multiplicative structure of Eqn. (10). This approach decouples the identification of the Paris constants C and n — performed at level 1 by direct regression on experimental data — from the identification of the microstructural coefficients a ₖ , performed at level 2 using the residual modulation factor. The optimal order N* is selected at level 2 by the corrected Akaike information criterion (AICc). At level 1, the Paris constants C and n are obtained by log-log regression applied directly to the experimental / da dN vs Δ K data of each holed specimen: (13) This regression provides, for each orientation, the macroscopic reference constants used at level 2. For type A: C A = 3.55 × 10 ⁻⁵ µm/cycle·(MPa √ m) ⁻ ⁿ , n A = 2.382, R² (Paris) = 0.665, over Δ K ∈ [4.9; 20.2] MPa √ m. For type B: C B = 1.07 × 10 ⁻ ³ µm/cycle·(MPa √ m) ⁻ ⁿ , n B = 1.071, R² (Paris) = 0.147, over Δ K ∈ [4.6; 20.8] MPa √ m. The low R² values - particularly for type B -confirm that the classical Paris law does not adequately describe short-crack propagation, thereby motivating the proposed enrichment. At level 2, the experimental microstructural modulation factor is evaluated at each data point i as:   C n log K  log   · da dN       log

· da dN C K      

i exp

  , 



i 

(14)

n

i

The values μ ᵢ are then fitted with a Chebyshev series of order N, by solving:

2



  

 

  

N 

1       i 

  

 

  

1  



 

a T

minimise

a T

)

(15)

k k

k k i





k

i

k

0

Owing to the orthogonality of the Chebyshev basis (Eqn. (4)), the least-squares problem is well-conditioned and yields a numerically stable solution. The coefficients k a are then obtained by linear regression through the numpy. polynomial. chebyshev.chebfit function in Python. Computation of Δ K The nominal stress intensity factor range, already introduced in Eq. (1), is used throughout the identification procedure described below. with t K = 2.3,   = 180 MPa, and where a is the measured crack length. For type B, a (N) is smoothed using a Savitzky– Golay filter prior to Δ K computation. The normalisation of ξ , already introduced in Eq. (6), is applied at each experimental point i . For type B, a ( N ) is smoothed using a Savitzky–Golay filter prior to Δ K computation, for the reasons discussed below. Optimal order selection N* by AICc criterion The polynomial order N reflects a bias–variance trade-off. Orders N = 0 and N = 1 are excluded from the sweep: N = 0 is equivalent to the classical Paris law ( μ = 1 uniformly, no enrichment), and N = 1 yields a strictly monotonic modulation with no oscillatory capacity, which is physically inconsistent for a grain-boundary interaction model. The sweep therefore starts at N = 2. Order selection is guided by the corrected Akaike information criterion (AICc) [29], the uncorrected form of which is given by:

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