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


Geometry
---------

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

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

.. _RefImage_Object_1.svg:


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

Behaviour model of CAM_CLAY
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The parameters specific to elastoplastic law CAM_CLAY [:ref:`r7.01.14 <R7.01.14>`] are:

* In C modeling: :math:`\mu=3.846154` MPa, :math:`PORO=0.5`, :math:`\lambda=0.2`,, :math:`\kappa=0.05`,,,, :math:`M=1.02`, :math:`P_{cr}^0=10` MPa, :math:`{K}_{cam}=6.5` MPa,, sq. kPa; :math:`P_{trac}=-100`

* In modeling D: :math:`\mu=610` MPa, :math:`PORO=0.66`, :math:`\lambda=0.25`,, :math:`\kappa=0.05`,,, :math:`M=0.9`, :math:`P_{cr}^0=300` kPa, :math:`{K}_{cam}=0` Pa, :math:`P_{trac}=0` Pa.

Behaviour model MCC
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The parameters specific to the elastoplastic law MCC [:ref:`r7.01.48 <R7.01.48>`] are given in the :numref:`v6.04.160-table_parametres_MCC` concerning the G modeling.

.. _v6.04.160-table_parametres_MCC:
 
.. list-table:: Paramètres du modèle MCC utilisés dans la modélisation G.

 *-**Effect**
   - **Appellation**
   - **Definition**
   - **Symbol**
   - **Value**
 * - Isotropic nonlinear elasticity
   - | *BulkModulus*
     | *ShearModulus*
     | *SwellingIndex*
   - | Compressibility module
     | Shear module
     | Elastic nonlinearity index
   - |:math: `K`
     |:math: `\ mu`
     |:math: `\ kappa`
   - | 160 MPa
     | 100 MPa
     | 50
 * - Initial elasticity domain
   - | *initCritPress*
     | *critstateSlope*
     | *TensileYieldStress*
   - | Initial critical pressure
     | Critical state slope
     | Isotropic tensile strength
   - |:math: `p_ {c0} `
     |:math: `M`
     |:math: `\ sigma_0`
   - | 1 MPa
     | 1
     | 10 kPa
 * - Kinematic-isotropic work hardening
   - | *IncoplastIndex*
   - | Plastic incompressibility index
   - |:math: `\ beta`
   - | 50

Behaviour model CSSM
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The parameters specific to the model CSSM [:ref:`r7.01.44 <R7.01.49>`] are grouped together in the :numref:`v6.04.160-table_parametres_CSSM` concerning the H modeling.

.. _v6.04.160-table_parametres_CSSM:
 
.. list-table:: Paramètres du modèle CSSM utilisés dans la modélisation H.

 *-**Intervention**
   - **Appellation**
   - **Definition**
   - **Symbol**
   - **Value**
 * - Elasticity
   - | *BulkModulus*
     | *ShearModulus*
     | *ShearModulusRatio*
   - | Total compressibility module
     | Total shear modulus
     | Ratio of the shear modulus of component 1 to the total shear modulus
   - |:math: `K`
     |:math: `\ mu`
     |:math: `\ rho`
   - | 516 MPa
     | 238 MPa
     | 0.1
 * - Component 1
   - | *critstateSlope*
     | *initCritPress*
     | *IncoplastIndex*
     | *IsoHardRatio*
     | *isoHardIndex*
   - | Critical state slope
     | Initial critical pressure
     | Plastic incompressibility index
     | Homothetic reduction ratio of the initial elasticity domain
     | Hardening index by homothetic enlargement of the initial elasticity domain 
   - |:math: `M`
     |:math: `p_ {c0} `
     |:math: `\ beta`
     |:math: `\ eta`
     |:math: `\ omega`
   - |:math: `1.38`
     |:math: `100` kPa
     |:math: `30`
     |:math: `0.99`
     |:math: `32`
 * - Component 2
   - | *HypDistortion*
     | *HyperExponent*
     | *miNcritPress*
   - | Reference distortion of the "modified hyperbolic" relationship
     | Curvature parameter for the "modified hyperbolic" relationship
     | Minimum pressure at which the critical state is attainable
   - |:math: `\ gamma_ {\ mathrm {hyp}}`
     |:math: `n_ {\ mathrm {hyp}} `
     |:math: `C`
   - |:math: `2.10^ {-4} `
     |:math: `0.78`
     |:math: `448` kPa

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

In modeling C, the hydrostatic test is performed with a stress state initiated at :math:`\boldsymbol{\sigma}=-P_{trac}\boldsymbol{I}`. We then increase :math:`p` to :math:`p_1=9` MPa, before discharging to :math:`P_{trac}`.

In modeling D, a first elastic calculation is performed up to :math:`p_1=100` kPa. :math:`p` is then increased to :math:`p_2=800` Pa by finally discharging to :math:`p_1`.

In the G model, we increase :math:`p` to :math:`10` MPa before discharging up to :math:`10` kPa.

In the H modeling, :math:`p` is increased to :math:`100` kPa before discharging to :math:`0.1` kPa.

 **Note:**

 In C modeling, an initial compressibility is given as a material parameter, :math:`K_{cam}=6.5` MPa, 
 it is therefore not necessary to initialize the constraints field (cf. [:ref:`r7.01.14 <R7.01.14>`]). 
 However, in D modeling, the initial compressibility value is zero (:math:`K_{cam}=0` Pa). 
 It is therefore necessary to initialize the stress state, because in the expression of the hydrostatic stress in law CAM_CLAY, 
 for zero volume deformation, the stress is non-zero. 
 To initialize this constraint, a purely elastic calculation is initially carried out by changing the pressure from :math:`0` to :math:`p_1=100` kPa. 
 From this calculation only the stress field at Gauss points is extracted.
 This stress field resulting from the elastic calculation is considered to be the initial state of the hydrostatic stress necessary for law CAM_CLAY of the following calculation.