Reference problem
=====================

In this test [:ref:`Willam et al., 1987 <Willam et al., 1987>`], the specimen is subjected to a specific loading path that creates a continuous rotation of the main stress directions. This determines the ability of the model to converge despite such changes.


Geometry
---------

The test is based on a unit cubic finite element (1 m x 1 m).


Property of the materials
-----------------------

Young's module: :math:`E=32000\mathrm{MPa}`

Poisson's ratio: :math:`\mathrm{\nu }=0.2`

Parameter of fragility of concrete under tension: :math:`\mathit{MT}=\mathrm{1,7}`

Parameter of fragility of concrete under compression: :math:`\mathit{MC}=\mathrm{1,5}`

Equivalent tensile stress of concrete: :math:`{\mathrm{\sigma }}_{\mathit{ft}}=7.3\mathit{MPa}`

Equivalent stress of concrete in compression: :math:`{\mathrm{\sigma }}_{\mathit{fc}}=38.3\mathit{MPa}`

Angle of the Drucker Prager criterion: :math:`\mathrm{\alpha }=\mathrm{0,15}\mathit{rad}`


Boundary conditions and loads
-------------------------------------

This is a cube subjected to a uniform non-proportional loading, consisting of movements imposed in plane :math:`(\mathrm{Ox},\mathrm{Oy})`.

The material is first subjected to uniaxial traction in the direction :math:`\mathrm{xx}` up to the peak of the stress-deformation curve, in a second stage a shear :math:`\mathrm{xy}` and an orthogonal tension :math:`\mathrm{yy}` are superimposed on the uniaxial loading :math:`\mathrm{xx}` (which continues), this loading path results in a rotation of the main directions of the stresses resulting in the appearance of a stress of shear :math:`{\sigma }_{\mathrm{xy}}`.

This results in imposed deformations that evolve finely by pieces as a function of time, with:


* to :math:`t=0.01\mathrm{jour}`, :math:`\epsilon ={10}^{-4}(\begin{array}{cc}0.84& 0\\ 0& -0.105\end{array})`


* to :math:`t=0.05\mathrm{jour}`, :math:`\epsilon ={10}^{-4}(\begin{array}{cc}5.6& 4.76\\ 4.76& 7.035\end{array})`

The following boundary conditions apply:

 - for the nodes in the plane X=0 → DX = 0

 - for node N1 (0, 0, 0) → DX = DY = DZ = 0

 - for node N5 (0, 0, 1) → DY = 0


.. _Ref121124284: