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


Geometry
---------

A torsional test is reproduced on a hollow cylinder subjected to internal :math:`{P}_{i}` and external confinement pressures :math:`{P}_{e}`, to a vertical force :math:`F` and to a torsional moment :math:`M`. This test is schematized by a cubic sample (a material point) subjected to confinement stresses :math:`{\sigma }_{z}`, :math:`{\sigma }_{r}` and :math:`{\sigma }_{\theta }`, and to a shear deformation :math:`{\epsilon }_{z\theta }=M(t)` (**Figure**).

If :math:`{P}_{i}={P}_{e}={\sigma }_{0}` and :math:`F=\frac{2{\sigma }_{0}}{\pi }`, the following confinement values are obtained:


* :math:`{\sigma }_{\theta }={\sigma }_{r}={\sigma }_{0}`


* :math:`{\sigma }_{z}=3{\sigma }_{0}`

We then have the following values of the main constraints:


* :math:`{\sigma }_{1}=2{\sigma }_{0}+\mid {\sigma }_{z\theta }\mid`


* :math:`{\sigma }_{2}={\sigma }_{0}`


* :math:`{\sigma }_{3}=2{\sigma }_{0}-\mid {\sigma }_{z\theta }\mid`

But above all, the main axes rotate by an angle :math:`\alpha` equal to [:ref:` **1**  < **1** >`]:

:math:`\alpha (t)=\frac{1}{2}\mathrm{arctan}\left(\frac{2{\sigma }_{z\theta }(t)}{{\sigma }_{z}-{\sigma }_{\theta }}\right)=\frac{1}{2}\mathrm{arctan}\left(\frac{{\sigma }_{z\theta }(t)}{{\sigma }_{0}}\right)`


.. image:: images/100000000000019B000000C842D8CF86D4113104.png
    :width: 4.2811in
    :height: 2.0835in


.. _RefImage_100000000000019B000000C842D8CF86D4113104.png:

Figure 1.1-a: Schematic diagram of the torsional test on a hollow cylinder


Properties of the sample
---------------------------

The elastic parameters are:


* the isotropic compressibility module: :math:`K=\mathrm{516,2}\mathit{MPa}`


* the shear modulus: :math:`\mu =\mathrm{238,2}\mathit{MPa}`

The parameters of the Mohr-Coulomb law are:


* the angle of friction: :math:`\phi =33°`


* the characteristic angle: :math:`\Psi =27°`


* cohesion: :math:`{c}_{0}=1\mathit{kPa}`


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

The torsional test presented here is carried out on a hardware point with the SIMU_POINT_MAT command.

We are taking a :math:`{\sigma }_{0}=-50\mathit{kPa}` lockdown pressure. The constraints imposed are therefore as follows:


* :math:`{\sigma }_{\mathit{xx}}={\sigma }_{\mathit{yy}}=-50\mathit{kPa}`


* :math:`{\sigma }_{\mathit{zz}}=-150\mathit{kPa}`

The imposed shear deformation :math:`{\epsilon }_{\mathit{yz}}` varies linearly between :math:`t=0` and :math:`100\mathit{sec}` from :math:`0` to :math:`\mathrm{0,01}\text{\%}` in :math:`N=10` steps of time.