Issue 78

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

Complete identification algorithm Tab. 6 summarises the complete identification procedure, implemented using Python (NumPy, SciPy, Matplotlib).

Step

Level

Operation

Loading of da /dN vs N and a (N) data – 286 digitization using WebPlotDigitizer from experimental curves. Digitisation uncertainty estimated at ±2% on crack length a (corresponding to ±0.5 pixel at the axis scale used), propagating to ±1% on Δ K via Eq. (1) and ±4–8% on da /dN via finite differences. These uncertainties remain smaller than the physical scatter of μ ᵢ (NRMSE = 7.87% for type A, 8.50 % for type B) and do not significantly affect the Chebyshev coefficient identification Type B only: Savitzky-Golay smoothing of a (N) (window 11 pts, degree 3), followed by computation of da /dN = Δ a / Δ N using central finite differences Log-log regression da /dN vs Δ K on each holed specimen → identification of C A , n A (type A) and C B , n B (type B) Computation of Δ K ᵢ = 1.12·Kt· Δσ · √ ( π a ᵢ ) for each experimental point Computation of μ ᵢ = ( d a /dN) ᵢ / [C·( Δ K ᵢ ) ⁿ ] — experimental modulation factor Normalisation: computation of ξ ᵢ = ( / ) i f a d ∈ [ − 1,1] – d = 154 µm (type A) or 395 µm (type B) Physical validity filter: 0.01 < μ ᵢ < 15 and Δ K ᵢ > 0 Order sweep over N ∈ [2,26]: Chebyshev fit by chebfit, computation of AICc(N) and R²(N) Selection of N* = argmin AICc(N) — N* A = 21, N* B = 21 Computation of final metrics: R² , NRMSE, μ min (positivity check), μ mean , μ max Verification of the positivity condition μ ( ξ ) > 0 on grid of 10 000 points

1

Pre-processing

2

Pre-processing

3

Level 1

4

Level 2

5

Level 2

6

Level 2

7

Level 2

8

N* selection

9

N* selection

10 11

Validation Validation

Table 6. Identification algorithm for the enriched model parameters.

Identification results Tab. 7 summarises all identified parameters for both specimen types and the long-crack reference.

Type A (  ∥ R D ) 35.4

Type B (  ⊥ R D ) 31.5

Long crack reference

Parameter

0 a (µm) d (µm)

— — —

154

395

C (µm·cycle ⁻ ¹·(MPa √ m) ⁻ ⁿ )

3.55 × 10 ⁻⁵

1.07 × 10 ⁻ ³

n

2.382 0.665

1.071 0.147

2.53

R² Paris

0.984

N* (AICc) R² enriched NRMSE (%)

21

21

— — — — — — —

0.907

0.835 8.50% 1.477 2.965 0.152

7.87

μ mean

1.126 3.140 0.312

μ max μ min

N g (number of grains crossed)

3.72

1.55

Table 7: Identified parameters of the enriched model - AA2024-T351.

286