is a correspondence between the strength of the material with flaw of diameter $\theta$ and the truncated strength of this same material without flaw under level of stress $s^*$, i.e. we assume that noting $R_{\theta }$ the strength of the specimen with flaw of size $\theta $, it holds that there exists $s^*$ such that $$\mathcal{L}(R_{\theta })\approx \mathcal{L}(R\mid R\leq s^*).$$

Before launching a validation campaign for this procedure, a batch of 27 specimen has been machined including spherical defects whose sizes vary between 0 and 1.8mm (see Figure \[defautseprouvettes\]). These first trials aim at estimating the decreasing relation between mean allowable stress and defects diameter $\theta$. This preliminary study enabled to draw the abatement fatigue curve as a function of $\theta$, as shown in Figure \[abattement\].

![image](RXeprouvettes_defauts_sign.png) \[defautseprouvettes\]

![image](abattement_edit.png) \[abattement\]

Results in Figure \[abattement\] will be used during the splitting procedure to select the diameter $\theta$ to be incorporated in the batch of specimens tested at the current iteration as reflecting the sub-population of material of smaller resistance.

Modeling the distribution of the strength, Pareto model {#GPD_model}
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The events under consideration have small probability under $\mathbb{P}.$ By (\[Prod\]) we are led to consider the limit behavior of conditional distributions 