PSI - Issue 76

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

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where N is the number of cycles to failure, ˜ N is the scale factor, and m is the shape factor (also known as the Weibull modulus). By definition, the scale factor ˜ N corresponds to the life at which, for the considered stress level, the probability of failure reaches 63.2%. This formulation allows a probabilistic description of fatigue life, capturing the observed scatter at each stress level. To better reflect the increasing scatter in fatigue life as the applied stress approaches the fatigue limit, the Weibull modulus m is defined as a rational function of the stress level:

S u S u − S

m ( S ) = γ ·

(4)

where γ is a calibration parameter controlling the magnitude of the scatter. This expression ensures that the Weibull modulus increases at higher stress levels, thereby reducing the predicted dispersion in life, while it decreases near the endurance limit, consistent with experimental observations. To account for the distinct fatigue behaviors associated with the two defect populations, Eq. 3 is extended, leading to a bimodal distribution of N . The resulting expression is a weighted sum of two log-Weibull distributions:

˜ N L

m

+ (1 − ω ) 1 − exp −

˜ N P

m

P f ( N , S ) = ω 1 − exp −

log( N )

log( N )

(5)

where ω is the weighting factor representing the fraction of LoF-induced failures, and ˜ N P and ˜ N L are the scale pa rameters corresponding to porosity and lack-of-fusion defects, respectively. For simplicity, the same shape parameter m is assumed for both defect populations. This last framework is specifically designed to reflect the presence of two dominant defect classes—porosity and lack of fusion or coalesced porosity—that act as distinct fatigue crack initia tion sites. The value of ω can be treated as an additional model parameter to be calibrated or assumed based on defect classification through fracture surface analysis, representing the proportion of failures initiated by lack-of-fusion de fects over the total number of failures. For example, in the present study, fracture surface analysis revealed that only 30% of the fractured specimens failed due to a critical porosity defect. Accordingly, the weighting factor was set to ω = 0 . 7, reflecting the predominance of lack-of-fusion-induced failures in the tested population. The proposed model is calibrated using the MLEM, which identifies the most probable values of model parameters by maximizing the likelihood of the observed experimental data. As shown in Esposito et al. (2019), MLEM o ff ers a robust statistical basis for fatigue life analysis, particularly when incorporating censored data such as runouts or interrupted tests. When a single defect population is assumed without distinguishing the nature of the initiating flaw (Eqs. (1), (3), and (4)), the likelihood function used in the estimation process takes the following form:

n

p df ( N i , S i , S e ,β,γ ) δ i

· 1 − P f ( N i , S i , S e ,β,γ )

i = 1

1 − δ i

L ( S e ,β,γ ) =

(6)

where S e , β , γ are the unknown values to be optimized; p df is the probability density function of the log-Weibull distribution, P f is its cumulative distribution function given in Eq. 3. When two triggering defect populations are assumed, leading to a bimodal distribution of N (Eqs. (2), (5), and (4)), the likelihood function takes the following form: