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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 395
    :height: 291


.. _RefImage_Object_1.svg:


+--------------------------------+--------------------------+
| .. code::                      |:math:`h=\mathrm{1,00}[m]`|
+                                +                          +
|    Hauteur:                    |                          |
+--------------------------------+--------------------------+
|Width:                          |:math:`l=\mathrm{1,00}[m]`|
+--------------------------------+--------------------------+
|Thickness:                      |:math:`e=\mathrm{1,00}[m]`|
+--------------------------------+--------------------------+



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


.. csv-table::

    ":math:`E=31[\mathrm{GPa}]` ", "modulus of elasticity"
    ":math:`\nu =0.2` ", "Poisson's ratio"


Parameters specific to model BETON_UMLV:


+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent stiffness associated with the skeleton formed by hydrate blocks at the mesoscopic scale    |
+ .. image:: images/Object_4.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent stiffness intrinsically associated with hydrates at the microscopic scale                  |
+ .. image:: images/Object_5.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: stiffness associated with the capacity of adsorbed water to transmit charges (*load bearing water*)|
+ .. image:: images/Object_6.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the diffusion mechanism within the capillary porosity            |
+ .. image:: images/Object_7.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the mechanism of interlamellar diffusion                         |
+ .. image:: images/Object_8.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatory part: viscosity associated with water adsorbed by hydrate sheets                                          |
+ .. image:: images/Object_9.svg                                                                              +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: viscosity of free water.                                                                           |
+ .. image:: images/Object_10.svg                                                                             +                                                                                                                    +
|    :width: 395                                                                                              |                                                                                                                    |
+    :height: 291                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+


Parameters specific to model BETON_BURGER:


+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent stiffness associated with the reversible domain of delayed deformations          |
+ .. image:: images/Object_66.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|:math:`\kappa =10.0`                                                                                         |Irreversible deformation standard controlling the nonlinearity applied to the long-term deformation module|
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: associated stiffness associated with the reversible domain of delayed deformations       |
+ .. image:: images/Object_70.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the reversible range of delayed deformations           |
+ .. image:: images/Object_72.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the irreversible diffusion mechanism                   |
+ .. image:: images/Object_74.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatory part: viscosity associated with the reversible domain of delayed deformations                   |
+ .. image:: images/Object_76.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: apparent viscosity associated with the irreversible diffusion mechanism                  |
+ .. image:: images/Object_78.svg                                                                             +                                                                                                          +
|    :width: 395                                                                                              |                                                                                                          |
+    :height: 291                                                                                             +                                                                                                          +
|                                                                                                             |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                          +
|                                                                                                             |                                                                                                          |
+-------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+



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

In this test, a drying field that is homogeneous and constant is created in the structure.

The mechanical load corresponds to shear in plane :math:`\mathrm{xz}`; its intensity is :math:`\mathrm{10 }[\mathrm{MPa}]`. The load is applied in :math:`\mathrm{1s}` and is maintained constant for 750 days.


Initial conditions
--------------------

The start of the calculation is assumed to be the instant —1. At this moment there is no drying field or mechanical stress.

At instant 0, a drying field corresponding to :math:`\text{100 \%}` of humidity is applied.