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

Geometry
---------

.. _v7.31.126-cube:

.. figure:: images/Object_1.svg
 :width: 480
 :height: 360

 height: :math:`h=1` m, width: :math:`l=1` m, thickness: :math:`e=1` m

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

The values used for the sixteen parameters of the Barcelona model are given in :numref:`v7.31.126-parametres_barcelone`.

.. _v7.31.126-parametres_barcelone:

.. list-table:: Paramètres du modèle de Barcelone.

 *-**Category**
   - **Appellation**
   - **Definition**
   - **Symbol**
   - **Value**
 * - Elasticity
   - | *BulkModulus*
     | *ShearModulus*
     | *SwellingIndex*
     | *SuccionIndex*
     | *SuccionModulus*
   - | Compressibility module
     | Shear module
     | Elastic compressibility coefficient at:math: `p_c=\ mathrm {Cte} `
     | Elastic compressibility coefficient at:math: `p"=\ mathrm {Cte} `
     | Initial compressibility module in capillary pressure
   - |:math: `K`
     |:math: `\ mu`
     |:math: `\ kappa`
     |:math: `\ kappa_s`
     |:math: `k_S`
   - |:math: `15` MPa
     |:math: `10` MPa
     |:math: `0.02`
     |:math: `0.008`
     |:math: `30` MPa
 * - Initial elasticity domain
   - | *critstateSlope*
     | *initConspress*
     | *YieldSuccion*
   - | Critical state slope
     | Initial consolidation pressure at:math: `p_c=0`
     | Initial irreversibility threshold under capillary pressure
   - |:math: `M`
     |:math: `p_ {con} ^0`
     |:math: `s_0^0`
   - |:math: `1`
     |:math: `100` kPa
     |:math: `200` kPa
 * - | Pressure work hardening
     | capillary
   - | *YieldTensSlope*
     | *PlastSuccionIndex*
   - | Slope of the isotropic tensile limit
     | Plastic compressibility coefficient at:math: `p"=\ mathrm {Cte} `
   - |:math: `k_s`
     |:math: `\ lambda_s`
   - |:math: `0.6`
     |:math: `0.08`
 * - | Work hardening by deformation 
     | volume plastic
   - | *PlastStrainIndex*
     | *PlastStrainPara1*
     | *PlastStrainPara2*
     | *RefeConsPress*
   - | Total compressibility coefficient at:math: `p_c=0`
     |:math: `1^ {er} `total compressibility coefficient parameter at:math: `p_c=\ mathrm {Cte}`
     |:math: `2^ {nd} `total compressibility coefficient parameter at:math: `p_c=\ mathrm {Cte}`
     | Reference consolidation pressure to:math: `p_c=0`
   - |:math: `\ lambda_0`
     |:math: `\ beta`
     |:math: `r`
     |:math: `p_r`
   - |:math: `0.2`
     |:math: `12.5` mpa:Math: `^ {-1} `
     |:math: `0.75`
     |:math: `100` kPa
 * - Plastic flow
   - | *FlowCorrection*
   - | Plastic flow correction coefficient
   - |:math: `\ alpha`
   - |:math: `1`
 * - Initial state 
   - | *initVoidsRatio*
   - | Initial void index
   - |:math: `e_0`
   - |:math: `0.9`

The properties of fluids (dry gas and liquid water) and of diffusion within the solid skeleton, necessary to solve the coupled calculation LIQU_GAZ, are presented in :numref:`v7.31.126-parametres_fluides`.

.. _v7.31.126-parametres_fluides:

.. list-table:: Paramètres des fluides et de diffusion.

 *-**Keyword**
   - **Definition**
   - **Symbol**
   - **Value**
 * - THM_LIQU
   - | Initial density of the liquid
     | Dynamic liquid viscosity
     | Inverse of liquid compressibility
   - |:math: `\ rho_ {q} ^0`
     |:math: `\ mu_ {q}`
     |:math: `1/K_ {q} `
   - |:math: `1000` kg.m:math: `^ {-3} `
     |:math: `1\ times 10^ {-3} `Not.
     |:math: `1/ (2\ times 10^9) `pa:Math: `^ {-1}`
 * - THM_GAZ
   - | Molar mass of dry gas
     | Dynamic viscosity of dry gas
   - |:math: `M_ {as} ^ {ol} `
     |:math: `\ mu_ {as}`
   - |:math: `0.03` kg.mole:math: `^ {-1} `
     |:math: `9\ times10^ {-6} `Not.
 * - THM_DIFFU
   - | Isotropic Biot coefficient
     | Intrinsic permeability
     | Initial homogenized density
     | Isothermal saturation function
     | capillary pressure
   - |:math: `b`
     |:math: `K^ {int} `
     |:math: `\ rho_ {home} ^ {0}`
     | VG_N, VG_SR, VG_SMAX,
     | VG_SATUR, VG_PR (Van Genuchten)
   - |:math: `1`
     |:math: `1\ times 10^ {-8} `m:math: `^ {2}`
     |:math: `2400` kg.m:math: `^ {-3} `
     |:math: `1.6, 0, 0.99, `
     |:math: `1, 16` MPa

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

The gas pressure remains constant equal to :math:`\mathrm{PRE2}={1}` Pa. In the first loading phase, desaturation is carried out, during which the capillary pressure decreases to zero since :math:`\mathrm{PRE1}={100}` kPa. In the second phase, isotropic compression is carried out, until the average stress :math:`\sigma_m=-400` kPa is reached. In the third and final phase, while maintaining the mean stress, the capillary pressure is increased to :math:`\mathrm{PRE1}={300}` kPa.