2. Benchmark solution

2.1. Calculation method used for the reference solution

The constraint is fixed by the loading path (stress control), i.e.:

\(\sigma =\left[\begin{array}{ccc}{\sigma }^{D}& {\tau }^{D}& 0\\ {\tau }^{D}& 0& 0\\ 0& 0& 0\end{array}\right]\)

The elastic part of the deformation is deduced from this:

\({\varepsilon }^{e}=\frac{1}{E}\left[\begin{array}{ccc}{\sigma }^{D}& (1+\nu ){\tau }^{D}& 0\\ (1+\nu ){\tau }^{D}& -\nu {\sigma }^{D}& 0\\ 0& 0& -\nu {\sigma }^{D}\end{array}\right]\)

If we now assume that we know the total deformation \(\varepsilon\), then we can to derive/to establish the plastic deformation: \({\varepsilon }^{p}=\varepsilon -{\varepsilon }^{e}\)

Note:

\({\varepsilon }_{\mathrm{xx}}^{p}+{\varepsilon }_{\mathrm{yy}}^{p}+{\varepsilon }_{\mathrm{zz}}^{p}=0\) and \({\varepsilon }_{\mathrm{yy}}^{p}={\varepsilon }_{\mathrm{zz}}^{p}\) so \({\varepsilon }_{\mathrm{yy}}^{p}={\varepsilon }_{\mathrm{zz}}^{p}=\frac{-{\varepsilon }_{\mathrm{xx}}^{p}}{2}\)

then the reminder constraint:

\(\chi =C{\varepsilon }^{p}\) with \(\frac{2}{3C}=\frac{1}{{E}^{T}}-\frac{1}{E}\) with \(C\): Prager constant

In addition, to obtain correct precision, it is necessary to use a fairly large number of increments for path \(\mathit{AB}\), in this case, at least 30 in this case. The same goes for trip \(\mathrm{BC}\).

2.2. Benchmark results

The total deformation data \(\varepsilon\) is required for the previous calculations. It is obtained as an average of the results of several codes.

2.3. Bibliographical references

1. Structural Analysis Software Validation Guide - SFM. Afnor technique