Reference solution
=====================


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

The results are experimental [:ref:`ref 1 <ref 1>`]. The measured values are the stress amplitudes after stabilization:


.. csv-table::

    "Loading", ":math:`\frac{\Delta {\varepsilon }_{\mathit{xx}}}{2}` "," :math:`{\varepsilon }_{\mathit{moy}}` "," "," :math:`\frac{\Delta {\sigma }_{\mathit{xx}}}{2}` "," "," :math:`{\sigma }_{\mathit{moy}}` "," :math:`\frac{\Delta \gamma }{2}` "," :math:`{\gamma }_{\mathit{moy}}` "," :math:`\frac{\Delta {\sigma }_{\mathit{xy}}}{2}` "," :math:`{{\sigma }_{\mathit{xy}}}_{\mathit{moy}}`"
    "FRI15 ", "0.0034", "0", "0", "413", "413", "-5", "0.0058", "0", "237", "0"
    "FRI15 ", "0.0034", "0", "0", "398", "398", "-5", "0.0058", "0", "231", "1"


So we get the following averages :math:`\frac{\Delta {\sigma }_{\mathit{xx}}}{2}\mathrm{=}405.5\mathit{MPa}` and :math:`\frac{\Delta {\sigma }_{\mathit{xy}}}{2}\mathrm{=}234\mathit{MPa}`,

which, like :math:`{\sigma }_{\mathit{xx}}\mathit{moy}\mathrm{=}\mathrm{-}5\mathit{Mpa}`, and :math:`{\sigma }_{\mathit{xx}}\mathit{moy}\mathrm{=}\mathrm{0,5}\mathit{Mpa}` leads to:

:math:`{\sigma }_{\mathit{xx}}\mathit{max}\mathrm{=}400.5\mathit{MPa}`, :math:`{\sigma }_{\mathit{xx}}\mathit{min}\mathrm{=}\mathrm{-}410.5\mathit{MPa}`

:math:`{\sigma }_{\mathit{xy}}\mathit{max}\mathrm{=}234.5\mathit{MPa}`, :math:`{\sigma }_{\mathit{xy}}\mathit{min}\mathrm{=}\mathrm{-}233.5\mathit{MPa}`


Uncertainty about the solution
---------------------------

The uncertainty resulting from the variability of the experimental results is 2%. That which comes from the identification of material parameters can be estimated at 5% to 10% (cf. [:ref:`ref. 2 <ref. 2>`]).


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


1. "Multiaxial fatigue evaluation using discriminating strain paths" Nima Shamsaei, Ali Fatemi, Darrell F. Socie International Journal of Fatigue 33 (2011) 597—609


2. J.M. PROIX "Viscoplastic behavior taking into account the non-proportionality of the load" EDF R&D-CR- AMA12 -284, 12/12/12