Issue 78

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

RSS

  

 

AIC M 

k

·ln

2

 

(16)

M

 k k





2

1

 

AICc AIC

(17)

 

M k

1

where RSS = Σ ᵢ [ log ( μ ᵢ ) − log ( μ̂ ᵢ )]², k = N+ 1 parameters, and M the number of experimental points ( M = 116 for type A, M = 98 for type B). The optimal order is:

 * 2,26 n N argmin AICc N     

(18)

Fig. 4 shows the convergence curves R² (N) and NRMSE(N) for type A, together with the physical peak-counting validation criterion. Panel (a) shows the evolution of R² and NRMSE as a function of polynomial order N . The AICc minimum is reached at N* A = 21, where R² = 0.907 and NRMSE = 7.87%. The Paris law reference ( R² = 0.665, dashed line) is shown for comparison. Panel (b) shows the number of peaks in μ ( ξ ) as a function of N . The physical criterion requires the model to reproduce the 5 experimentally identified peaks — corresponding to the 5 grain boundaries GB ₁ –GB ₅ identified experimentally. N* = 21 is the first order satisfying this physical constraint (green point), providing an independent validation of the AICc selection. Orders above N* generate spurious peaks (red points), confirming that N* = 21 is not merely an overfitting artefact but a physically meaningful selection. The same dual-criterion approach was applied to type B , where the statistically optimal order does not converge as cleanly with the physical peak-counting criterion. Two complementary metrics are used to evaluate fit quality:



ˆ ) log       ) i i log

2

log

(

2

i

1  

R

(19)



2

log

(

i

i

RSS M

  100 %





NRMSE

(20)











log

log

max

min

Figure 4: Convergence analysis for type A. (a) R² and NRMSE vs. polynomial order N . Red dashed: N* = 21 (AICc minimum). (b) Number of model peaks vs. N - N* = 21 is the first order reproducing the 5 experimental grain boundaries. For type A: R² = 0.907, NRMSE = 7.87%. For type B: R² = 0.835, NRMSE = 8.50 %. The lower performance for type B reflects the lower number of grains crossed ( N g-B = 1.55) and the higher relative uncertainty on ∥ (51% vs. 30%). Prior to identification, a physical validity filter is applied. Points are retained if μ ᵢ > 0.01 (eliminates noise near zero), μ ᵢ < 15 (removes unstable propagation outliers), and Δ K ᵢ > 0. This filtering step removes less than 5% of the experimental points.

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