PSI - Issue 76

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

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may govern failure initiation across di ff erent stress regimes. In contrast, the single-mode model provides a single median curve that only approximately represents the overall trend. The most significant di ff erence between the two approaches is observed in terms of survival probability, as highlighted in the right panel. The 5–95% survival bands predicted by the bimodal model align more closely with the experimental data than those of the single-mode model, particularly in the high-cycle regime, without yielding unjustifiably wide confidence intervals. These plots confirm the ability of the bimodal Weibull formulation to capture the stress-dependent evolution of fatigue life scatter and to describe multiple initiation mechanisms within a unified probabilistic framework.

Fig. 2. Comparison between di ff erent modeling approaches for the Z-machined dataset. Left: Median curves obtained from a single-mode Weibull model (dashed red), and from defect-sensitive models based on porosity (dash-dotted brown) and lack-of-fusion (solid brown) dominant popula tions. Right: 5–95% survival bands computed using the single-mode (red dash-dotted) and bimodal (green dashed) log-Weibull models. The calibration of the bimodal log-Weibull model for the two datasets yields the parameters summarized in Table 2. For the Z-as-built specimens, the application of the bimodal model appears to be unnecessary, as the estimated values of S eL and S eP are very close, indicating no significant separation between the defect-related fatigue regimes.

Table 2. Bimodal log-Weibull model parameters Parameters

Z-as-built

Z-machined

S eL (MPa) S eP (MPa) β L (Cycles) β P (Cycles)

77.7 84.1 4366 5680 49.88

116.2 188.1 3806 2777 22.90

γ

Figure 3 illustrates the probabilistic structure of the proposed bimodal fatigue life model. The left panel shows the 3D surface of the probability density function ( p df ), identified for Z-machined samples, as a function of the number of cycles to failure and the maximum applied stress, capturing the evolution of the life distribution across the full range of stress levels. The surface highlights the emergence of bimodal behavior at intermediate stress values, consistent with the presence of multiple defect-driven fatigue mechanisms. The right graph shows two cross-sections of the p df evaluated at constant stress levels ( S = 200MPa and S = 300 MPa). At S = 300 MPa, the p df is sharply peaked, indicating a narrow life distribution, whereas at S = 200 MPa. Furthermore, a set of 10 4 synthetic data points was randomly sampled from the bimodal p df distribution. The resulting histogram was compared to the analytical p df for consistency verification. The distribution is clearly bimodal, reflecting the coexistence of early failures and long-life specimens. Fractographic analysis confirmed that failures predicted by the model at survival probabilities below 30%