PSI - Issue 38

512 Robin Hauteville et al. / Procedia Structural Integrity 38 (2022) 507–518 Robin Hauteville, Xavier Hermite, Fabien Lefebvre / Structural Integrity Procedia 00 (2021) 000 – 000 l̅o̅̅g̅̅ ̅ = ∑ log =1 : mean of log of the solicitations. This provides a point estimate of the regression line of the population produced from the sample, denoted. log( ̂ ) log( ̂ ) = ̂ l̅o̅̅g̅̅ ̅ + l̂og (9) It is in most cases this line that is retained as the S-N curve at 50% probability of rupture. The experimental standard deviation associated with lifetimes (called the “ residual standard deviation ”) is given by the expression: l og = √ ∑ [log − (log( ̂ ) − ̂ log )] 2 =1 − 2 (10) The experimental standard deviation and coefficient of variation associated with resistance are given by the expressions: l og = l og (11) The associated degree of freedom number is: = − 2 . 4.2. Maximum likelihood estimates The maximum likelihood method is a fit method based on the probability of an occurring event. In the analysis of fatigue test results, these events are failures or censored data. Its main interest is its ability to consider censored lifetimes, regardless of their distribution in the scatterplot and without the shape of an imposed S-N curve. As the least squares method, it is then a question of fitting a S-N curve supposed to best represent the test results, not by minimising the difference between the latter and the model, but by maximising the probability of occurrence of all these events. Unlike the least squares method, which implies that the residuals are normally distributed around zero, the maximum likelihood method applies to any statistical distribution. The occurrence of a failure can be estimated by the probability density function of lifetime distribution for each stress level: ( , , ) (12) where N is the lifetime to failure , α and β are the parameters of the distribution (for example, the median μ and the standard deviation σ for a normal or log -normal distribution, or the scale parameter η and the form parameter β for a two-parameter Weibull distribution). When a batch of specimens survives the test (censored data), the probability of this event is the reliability at this stress level associated to the number of cycles reached at the end of the test: ( , , ) = 1 − ( , , ) (13) where F is the cumulative distribution function. We call "likelihood" the probability of existence of all events failure or survival through the model expected to represent that behaviour: 6

• A distribution law of lifetimes for each stress level. • An acceleration law to link the lifetimes to the stress levels: