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


Geometry
---------

Here we are considering a cube of dimension :math:`\mathrm{1m}\mathrm{\times }\mathrm{1m}\mathrm{\times }\mathrm{1m}`.

Z

1 m


.. image:: images/Object_1.svg
    :width: 12
    :height: 15


.. _RefImage_Object_1.svg:

D

C

1 m


.. image:: images/Object_2.svg
    :width: 12
    :height: 15


.. _RefImage_Object_2.svg:


.. image:: images/Object_3.svg
    :width: 12
    :height: 15


.. _RefImage_Object_3.svg:


.. image:: images/Object_4.svg
    :width: 12
    :height: 15


.. _RefImage_Object_4.svg:

B

A

Y

1 m

X

.. image:: images/1000010A0000271000002710863D401C4058C571.svg
    :width: 12
    :height: 15


.. _RefImage_1000010A0000271000002710863D401C4058C571.svg:

Coordinates of the points (in :math:`m`):


.. csv-table::

    "", ":math:`A` "," :math:`B` "," :math:`C` "," :math:`D`"
    ":math:`x` ", "0", "0", "0", "0.5", "1"
    ":math:`y` ", "0", "1", "1", "0.5", "1"
    ":math:`z` ", "0", "0", "0", "0.5", "1"



Material properties
----------------------

Parameters of the law of elastic behavior:

.. csv-table::

    ":math:`E\mathrm{=}4500\mathit{MPa}`"
    ":math:`\nu \mathrm{=}0.3`"


Parameters of the modified Hoek-Brown law:

.. csv-table::

    ":math:`{\gamma }^{\mathit{rup}}\mathrm{=}0.005`"
    ":math:`{\gamma }^{\mathit{res}}\mathrm{=}0.017`"
    ":math:`{(S{\sigma }_{c}^{2})}^{\mathit{end}}\mathrm{=}225{\mathit{MPa}}^{2}`"
    ":math:`{(S{\sigma }_{c}^{2})}^{\mathit{rup}}\mathrm{=}482.5675{\mathit{MPa}}^{2}`"
    ":math:`{(m{\sigma }_{c}^{2})}^{\mathit{end}}\mathrm{=}13.5\mathit{MPa}`"
    ":math:`{(m{\sigma }_{c}^{2})}^{\mathit{rup}}\mathrm{=}83.75\mathit{MPa}`"
    ":math:`\beta \mathrm{=}3\mathit{MPa}`"
    ":math:`{\phi }^{\mathit{rup}}\mathrm{=}15°`"
    ":math:`{\phi }^{\mathit{res}}\mathrm{=}30°`"
    ":math:`\alpha \mathrm{=}3.3`"



Initial conditions, limits and loading
------------------------------------------------

The test is divided into two phases:


1. First, the sample is brought to a homogeneous state

    .. image:: images/Object_17.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_17.svg:

. For this, the corresponding confinement pressure is imposed on the front faces (

    .. image:: images/Object_18.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_18.svg:

), right lateral (

    .. image: images/Object_19.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_19.svg:

) and higher (

    .. image: images/Object_20.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_20.svg:

), the water pressures are zero everywhere and the movements are taken zero on the back faces (

    .. image: images/Object_21.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_21.svg:

), left lateral (

    .. image: images/Object_22.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_22.svg:

) and lower (

    .. image: images/Object_23.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_23.svg:

).


2. Once the homogeneous state is obtained, the movements are kept blocked on the rear, left and lower lateral faces. Hydraulic flows are zero on all sides. A displacement is imposed on the upper side (

    .. image: images/Object_24.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_24.svg:

) so as to obtain a deformation

    .. image: images/Object_25.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_25.svg:

equal to — 25% from the start of the second phase, in constant deformation increments

    .. image: images/Object_26.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_26.svg:

. On the front and right lateral faces, limit conditions under total stress are imposed:

    .. image: images/Object_27.svg
        :width: 12
        :height: 15


    .. _RefImage_Object_27.svg:

.