PSI - Issue 75

Felix-Christian Reissner et al. / Procedia Structural Integrity 75 (2025) 382–391 Felix-Christian Reissner / Structural Integrity Procedia 00 (2025) 000–000

385

4

density function (PDF) f ( x ):

log 10 ( x ) − g � log 10 ( y )  σ x , log  ,

f ( x ) = ϕ 

log 10 ( x ) − g � log 10 ( y )  σ x , log  .

F ( x ) =Φ 

(4)

Although both log-normal and three-parameter Weibull distributions are widely used in fatigue literature, e.g. Schijve (2005); Brot (2019); Meeker et al. (2024), this paper focuses on the log-normal distribution for consistency and comparability across evaluation strategies as well as simplicity.

3. Evaluation of S-N data with Maximum Likelihood

3.1. Maximum Likelihood

Maximum Likelihood Estimation (MLE) is a widely used method to estimate parameters of statistical S-N models, particularly when dealing with right-censored data (runouts). Unlike simple regression, MLE incorporates runouts using the survival function 1 − F ( x ), which expresses the probability of survival beyond a given load amplitude or number of cycles. In this study, a log-normal distribution is assumed (see Section 2.2). Its mean µ is replaced by the output of the deterministic model, which depends on the estimated parameters ˆ θ = [ ˆ N k , ˆ S a , k , ˆ k 1 ]. Note that in this study, k 2 is kept constant throughout the estimation and is not optimized. The likelihood for a single observation is given by: L i ( ˆ θ ) =   f ( x i | ˆ θ ) , if x i has failed 1 − F ( x i | ˆ θ ) , if x i has not failed (runout) . (5) The overall likelihood function is the product of all the single likelihoods. To ensure computability, the likelihood function is usually log-transformed, which converts the product into a sum and leads to

n  i = 1  δ i log � f ( x i | ˆ θ )  + (1 − δ i ) log � 1 − F ( x i | ˆ θ )   , where δ i =  

1 , if x i has failed 0 , if x i has not failed (runout) .

(6)

L ( ˆ θ ) =

If the standard deviation σ is estimated by the MLE, the estimator is biased. Therefore, the standard deviation is estimated separately with n fail − p degrees of freedom, where n fail represents the number of uncensored observations and p is the number of estimated parameters. Finally, maximum likelihood estimation is conducted by numerically minimizing −L ( ˆ θ ).

3.2. Evaluation Strategies

In the statistical evaluation of S-N curves, the choice of the direction in which stochastic variation is modeled plays a critical role. Depending on the direction, di ff erent formulations for the likelihood function can be constructed. This section introduces four strategies for modeling and evaluating S-N curves using MLE: (1) variation in the fatigue-life direction, (2) variation in the stress direction, and (3) and (4) a hybrid approaches based on transformation between both, fatigue-life and stress direction.