2. Benchmark solution

2.1. Calculation method

No calculation is required to obtain the reference solution. The values \(m\) and \({\sigma }_{u}\) (M and SIGM_REFE in option WEIBULL of DEFI_MATERIAU) that one seeks to identify are known and make it possible to generate the experimental data base. Thus, the elongations at break are determined in the following way:

For each pair \(m\) and \({\sigma }_{u}\) associated with a test temperature, a sample of 15 Weibull stress values at break were determined by random drawing taking into account the following statistical law:

\({P}_{f}({\mathrm{\sigma }}_{w})=1-\mathrm{exp}[-{(\frac{{\mathrm{\sigma }}_{w}}{{\mathrm{\sigma }}_{u}})}^{m}]\)

The Weibull constraint is defined by:

\({\sigma }_{w}=\sqrt[m]{\sum _{i}{({\sigma }_{I}^{i})}^{m}\frac{{V}_{i}}{{V}_{0}}}\)

The summation relates to the volumes of plasticized \({V}_{i}\) material, \({\sigma }_{I}^{i}\) designating the maximum main stress in each of these volumes (the volume \({V}_{0}\) (VOLU_REFE in the option WEIBULL of DEFI_MATERIAU) is equal to \((50\mu {m}^{3})\)).

In the case of a simple tensile load with the hypothesis of small deformations, the Weibull stress, \({\sigma }_{W}\), is expressed as a function of the elongation at break \((l-{l}_{0})/{l}_{0}\),, according to:

\({\mathrm{\sigma }}_{w}=[{E}_{t}(\frac{l-{l}_{0}}{{l}_{0}})+(1-\frac{{E}_{t}}{E}){\mathrm{\sigma }}_{Y}]\sqrt[m]{\frac{V}{{V}_{0}}}\)

We therefore deduce from this expression and from the previous random draw the values of the elongations at break reported in the table in [§1.3].

2.2. Reference quantities and results

The reference quantities of \(m\) and \({\sigma }_{u}\) used to create the experimental test bases are as follows:

Temperature \([°C]\)	—50	—100	—150	—150
\(m\)	24	24	24
\({\sigma }_{u}\) \([\mathrm{MPa}]\)	2800	2700	2600

2.3. Uncertainties about the solution

The uncertainty in the solution cannot be determined precisely. It can be quite high. In fact, reference values can only be found when considering experimental populations composed of an infinite number of samples.