Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

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

:math:`\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:

:math:`{\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 :math:`\varepsilon`, then we can to derive/to establish the plastic deformation: :math:`{\varepsilon }^{p}=\varepsilon -{\varepsilon }^{e}`

Note:

:math:`{\varepsilon }_{\mathrm{xx}}^{p}+{\varepsilon }_{\mathrm{yy}}^{p}+{\varepsilon }_{\mathrm{zz}}^{p}=0` *and* :math:`{\varepsilon }_{\mathrm{yy}}^{p}={\varepsilon }_{\mathrm{zz}}^{p}` *so* :math:`{\varepsilon }_{\mathrm{yy}}^{p}={\varepsilon }_{\mathrm{zz}}^{p}=\frac{-{\varepsilon }_{\mathrm{xx}}^{p}}{2}`

then the reminder constraint:

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

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


Benchmark results
----------------------

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


Bibliographical references
---------------------------


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