PSI - Issue 76

Luca Esposito et al. / Procedia Structural Integrity 76 (2026) 50–58

54

n 

i = 1 

p df ( N i , S i ) 

·  1 − P f ( N i , S i ) 

1 − δ i

δ i

(7)

L ( S eL ,β L , S eP ,β P ,γ ) =

where P f is the cumulative distribution function given in Eq. 5 and p df is its probability density function. This formulation requires a larger number of parameters to identify ( S eL , β L , S eP , β P , γ ), increasing the complexity of the calibration process and potentially requiring a more robust dataset to ensure stable and meaningful parameter estimation. In the likelihood functions of Eqs. 6 and 7, the index i runs over all n tested specimens. The failure indicator δ i takes the value 1 if the i-th specimen failed, and 0 if it was censored (runout). The maximization of the likelihood function was performed using the multi-objective optimization platform modeFRONTIER (2025), which provides global search capabilities and constraint handling. In this work, the Non-dominated Sorting Genetic Algorithm II (NSGA-II) was employed as the optimization criterion due to its robustness in exploring non-convex and multi-modal parameter spaces. Once the parameters of the bimodal model have been identified, the survival probability ( P s ) for a given stress level S and number of cycles N can be computed as: P s ( N , S ) = ω exp  −  log N ˜ N P  m  + (1 − ω )exp  −  log N ˜ N L  m  (8)

where the stress-dependent scale parameters ˜ N P and ˜ N L are derived by inverting the asymptotic S – N relationships:

− 1  

− 1   

˜ N P ( S ) = β P    

   

S u − S eP S − S eP  2

S u − S eL S − S eL  2

 ˜ N L ( S ) = β L

(9)

This formulation is valid for S > S eP and S > S eL .

2.5. Fracture surface analysis for triggering defect characterization

A fractographic analysis was conducted on all fatigue-failed specimens using a V8 Zeiss stereo microscope in order to identify the crack initiation sites and characterize the associated trigger defects. Particular attention was paid to distinguishing between porosity-type and lack-of-fusion features based on morphology, location, and surface appearance. The outcomes of this analysis were used to retrospectively classify each failure and to estimate the relative frequency of failure modes, supporting the defect-based probabilistic modeling approach adopted in this work. Further details of the analysis and classification criteria can be found in Esposito et al. (2025).

3. Results and discussion

The unimodal log-Weibull distribution described in Eq. 3 (referred to as the single-mode log-Weibull model) was initially applied to both the machined and as-built datasets to fit the median S–N curve and estimate the model pa rameters. Figure 1a) compares the fatigue behavior of AlSi10Mg specimens built in the Z-direction in the as-built condition under a load ratio of R = 0 . 1. The experimental data (black dots) are shown along with the median S – N curve estimated according to the ASTM-E739 (2015) procedure (black solid line), and the corresponding 5-95% suvi val band (black dot lines). The results obtained using the proposed single-mode log-Weibull model are also reported: the red solid line represents the median S – N curve, while the red dash-dot lines denote the P s range (5–95%). Data