PSI - Issue 38

Robin Hauteville et al. / Procedia Structural Integrity 38 (2022) 507–518 Robin Hauteville, Xavier Hermite, Fabien Lefèbvre / Structural Integrity Procedia 00 (2021) 000 – 000

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Figure 4: Bastenaire model

4. Parameter estimation method 4.1. Least square regression method

The least squares method makes it possible to adjust an S-N curve when all the parts have been brought to failure. Its theoretical basis is linear regression, which aims to produce a model in the form of a regression line that minimises the gap between the experimental points and the model. The model thus obtained has the status of the median line only in the case where the sample tested comes from a normal population. In this sense, this method is well applicable to the field of limited endurance: • The lifetimes follow a lognormal distribution (i.e. regression is performed on logarithms of lifetimes). Example with the Basquin model: A log-linear model is presupposed for the fatigue behaviour of the material/process couple in limited endurance: = − ↔ log( ) = log( ) − log( ) (7) To identify the parameters of the mean curve C and m, it is performed a linear regression on log(N) function of the log(σ). The least squares adjustment informs the following formulations for the point estim ate of the slope parameter m, denoted ̂ , and the intercept ordinate, denoted ̂ . ̂ = − ∑ (log − ̅lo̅̅g̅̅ ̅̅ )(log − ̅lo̅̅g̅̅ ̅ ) =1 ∑ (log − ̅lo̅̅g̅̅ ̅ ) 2 =1 (8) l̂og = ̅lo̅̅g̅̅ ̅̅ + ̂ l̅o̅̅g̅̅ ̅ With l̅o̅̅g̅̅ ̅̅ = ∑ log =1 : mean of log of the lifetimes. • All tests are conducted to failure.