2. 2 Reminders

2.1. 2.1 The WEIBULL model

We consider a structure with elastoplastic behavior subjected to thermomechanical stress. It is assumed that the cumulative failure probability of this structure follows the following two-parameter law of WEIBULL [bib1]:

\({P}_{f}({\sigma }_{w})=1-\text{exp}\left[-{(\frac{{\sigma }_{w}}{{\sigma }_{u}})}^{m}\right]\) eq 2.1-1

expression in which the module of WEIBULL \(m>1\) describes the tail of the statistical distribution of the sizes of the defects causing the cleavage, \({\sigma }_{u}\) is the cleavage stress and \({\sigma }_{w}\) is the WEIBULL stress that depends on the history of the main stress field in the plasticized zone of the structure. For example, in the case of a monotonous loading trip, it is written as:

\({\sigma }_{w}=\sqrt[m]{\sum _{p}{({\sigma }_{I}^{p})}^{m}\frac{{V}_{p}}{{V}_{0}}}\). Eq 2.1-2

The summation relates to the volumes of plasticized \({V}_{p}\) material, \({\sigma }_{I}^{p}\) designating the maximum principal stress in each of these volumes (\({V}_{0}\) is a characteristic volume of the material).

2.2. 2.2 Parameter identification

Very generally, we consider an experimental base consisting of tests of various types (type 1, 2,…, n), each type of test being carried out \({n}_{j}\) times so that the total number of tests amounts to:

\(N=\sum _{j=1}^{j=n}{n}_{j}\).

This experimental base could for example consist of tests on axisymmetric specimens notched with different notch radii carried out at different temperatures. Given the random nature of the fracture properties of the material in question, this base only constitutes a sample. The greater the number of these samples, the more representative it will be of the behavior of the material in question.

Among the various identification methods proposed in the literature (see for example [bib2]), we select two: the linear regression method, which is often used, as well as the maximum likelihood method recommended by the « European Structural Integrity Society (ESIS) » [bib3].

Note:

A systematic comparative study of the results given by these two methods [bib2] as a function of the number of samples taken randomly from a theoretical distribution showed that the maximum likelihood method leads to a better estimation of the parameters of the WEIBULL model. However, as the linear regression method remains widely used, we have integrated it into our developments.

In the two adjustment methods used, a first calculation of the constraints of WEIBULL is carried out with a given set of parameters (typically, \(m=20\), \({s}_{u}=3000\text{MPa}\)). These N tests are classified using their stress of WEIBULL reached at the time of failure. We therefore have a growing list of constraints from WEIBULL \(({\sigma }_{w}^{1}\text{,...,}{\sigma }_{w}^{i}\text{,...,}{\sigma }_{w}^{N})\), such that for each \((i)\), the number of test pieces broken with a stress of WEIBULL less than or equal to \({\sigma }_{w}^{i}\) is \({n}_{w}^{i}\) (in general \({n}_{w}^{i}=i\)). Among the various possible estimators of the cumulative rupture probability \({P}_{f}^{i}\) corresponding to \({\sigma }_{w}^{i}\) [bib2], we choose the generally recommended one: \({P}_{f}^{i}=\frac{i}{N+1}\).

Note:

In the particular case where the constraint of WEIBULL depends on temperature, the previous ranking must be done temperature by temperature, each temperature corresponding to a different statistical distribution. The estimator of the probability of previous rupture therefore becomes: \({P}_{f}^{i}=\frac{i}{{N}_{T}+1}\) , if the test test \((i)\) was broken at the temperature \(T\) , for which there were \({N}_{T}\) tests.

The two recalibration methods used are valid as long as [éq 2.1-1] remains true. If the identification is carried out on the results of anisothermal tests when the cleavage stress is supposed to depend on temperature, this condition is no longer verified (cf. POST_ELEM [U4.81.22]). In this specific case, it will therefore not be possible to apply the following developments.