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


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


Geometry
----------



Cube geometry:

 Center :math:`O(0.\mathrm{,0}\mathrm{.}\mathrm{,0}\mathrm{.})`

Point :math:`A(0.5\mathrm{,0}.5\mathrm{,0}.5)`

 Side cube :math:`1m`


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


*
    *
        * Elastic


            * :math:`E=5800.0\mathrm{E6}\mathrm{Pa}` Young's module


            * :math:`\rho =2500{\mathrm{kg.m}}^{-3}` Density


            * :math:`\nu =0.3` Poisson's ratio


*
    *
        * DRUCK_PRAGER with linear negative work hardening


            * :math:`\alpha =0.33` Pressure Dependence Coefficient


            * :math:`{p}_{\mathrm{ultm}}=0.01` Ultimate cumulative plastic deformation


            * :math:`{\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}` Plastic constraint


            * :math:`h=-2.00\mathrm{E8}\mathrm{Pa}` Work hardening module


*
    *
        * DRUCK_PRAGER with parabolic negative work hardening


            * :math:`\mathrm{\alpha }=0.33` Pressure Dependence Coefficient


            * :math:`{p}_{\mathit{ultm}}=0.01` Ultimate cumulative plastic deformation


            * :math:`{\mathrm{\sigma }}^{Y}=2.57E6\mathit{Pa}` Plastic constraint


*
    *
        *
            * :math:`{\mathrm{\sigma }}_{\mathit{ultm}}^{Y}=0.57E6\mathit{Pa}` Ultimate constraint


        * DRUCK_PRAG_N_A with parabolic negative work hardening


            * :math:`\alpha =0.33` Pressure Dependence Coefficient


            * :math:`{p}_{\mathrm{ultm}}=0.01` Ultimate cumulative plastic deformation


            * :math:`{\sigma }^{Y}=2.57\mathrm{E6}\mathrm{Pa}` Plastic constraint


            * :math:`{\sigma }_{\mathrm{ultm}}^{Y}=0.57\mathrm{E6}\mathrm{Pa}` Ultimate constraint


            * :math:`\mathrm{\beta }=0.33` Initial expansion coefficient


*
    *
        * Hydraulic behavior: saturated liquid




*
    *
        *
            * :math:`\mathrm{Pre1}=1\mathrm{Pa}` Reference liquid pressure


            * :math:`{\rho }_{\mathrm{pre1}}=1000{\mathrm{kg.m}}^{-3}` Density of water


            * :math:`\mathrm{Poro}=0.14` Initial porosity


            * :math:`{\rho }_{\mathrm{vh}}=2400{\mathrm{kg.m}}^{-3}` Homogenized density


            * :math:`\mathrm{bio}=1` Biot coefficient


            * :math:`{K}_{\mathrm{intrinsèque}}=1E-18{m}^{2}` Intrinsic permeability


            * :math:`\frac{1}{{K}_{l}}=0` Incompressible liquid


            * :math:`\mathrm{vi}=1.0E-3\mathrm{Pa.s}` Viscosity




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

Boundary conditions and loads are applied in two steps:


* Step :math:`A`: :math:`t\in [\mathrm{0,1}\mathrm{.}]`

**Boundary conditions**


*
    * Knot pressure :math:`\mathrm{PRE1}=0.`


    * Imposed movements, of symmetry, on the faces of the cube belonging to the planes

:math:`X=-0.5` :math:`\mathrm{DX}=0`

:math:`Y=-0.5` :math:`\mathrm{DY}=0`

:math:`Z=-0.5` :math:`\mathrm{DZ}=0`

**Loading**


*
    * We progressively apply :math:`p={2.10}^{6}\mathrm{Pa}` compression to the faces of the cube belonging to the planes: :math:`X=0.5`, :math:`Y=0.5` and :math:`Z=0.5`


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:


*
    *
        * Step B: :math:`t\in \text{]}\mathrm{1,2}\mathrm{.}\text{]}`

From the state of constraints obtained at instant :math:`t=1.s`, the following conditions are applied to the faces of the cube:

**Movations**


*
    * For the face belonging to plane :math:`Z=0.5`, the :math:`\mathrm{DZ}` displacement is gradually applied, following a ramp:


.. image:: images/Shape3.gif


.. _RefSchema_Shape3.gif:


*
    * For faces belonging to planes :math:`X=-0.5,Y=-0.5,Z=-0.5`, symmetry conditions are applied.

Loads: the loads applied are constant:


*
    *
        *
            * Face belonging to plane :math:`X=0.5` :math:`p={2.10}^{6}\mathrm{Pa}`


            * Face belonging to plane :math:`Z=0.5` :math:`p={2.10}^{6}\mathrm{Pa}`