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


Geometry
---------


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


.. _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
----------------------

:math:`E=31\mathrm{GPa}`

:math:`\nu =0.2`

Here, we also enter the sorption-desorption curve that relates the water content

.. image:: images/Object_4.svg
    :width: 480
    :height: 360


.. _RefImage_Object_4.svg:

To hygrometry

.. image:: images/Object_5.svg
    :width: 480
    :height: 360


.. _RefImage_Object_5.svg:

.

In this case it was assumed that the numerical values of

.. image:: images/Object_6.svg
    :width: 480
    :height: 360


.. _RefImage_Object_6.svg:

And of

.. image:: images/Object_7.svg
    :width: 480
    :height: 360


.. _RefImage_Object_7.svg:

are the same.

Parameters specific to the natural creep of BETON_UMLV:


+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent stiffness associated with the skeleton formed by hydrate blocks at the mesoscopic scale    |
+ .. image:: images/Object_8.svg                                                                              +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent stiffness intrinsically associated with hydrates at the microscopic scale                  |
+ .. image:: images/Object_9.svg                                                                              +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: stiffness associated with the capacity of adsorbed water to transmit charges (*load bearing water*)|
+ .. image:: images/Object_10.svg                                                                             +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa}`]                                                                                      +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the diffusion mechanism within the capillary porosity            |
+ .. image:: images/Object_11.svg                                                                             +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |spherical part: apparent viscosity associated with the mechanism of interlamellar diffusion                         |
+ .. image:: images/Object_12.svg                                                                             +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatory part: viscosity associated with water adsorbed by hydrate sheets                                          |
+ .. image:: images/Object_13.svg                                                                             +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+
|                                                                                                             |deviatoric part: viscosity of free water.                                                                           |
+ .. image:: images/Object_14.svg                                                                             +                                                                                                                    +
|    :width: 480                                                                                              |                                                                                                                    |
+    :height: 360                                                                                             +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+ [:math:`\mathrm{MPa.s}`]                                                                                    +                                                                                                                    +
|                                                                                                             |                                                                                                                    |
+-------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------+


Parameters specific to the natural creep of BETON_BURGER:


+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |spherical part: apparent stiffness associated with the reversible domain of delayed deformations          |
+ .. image:: images/Object_205.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa}`]                                                                                       +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |deviatoric part: associated stiffness associated with the reversible domain of delayed deformations       |
+ .. image:: images/Object_209.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa}`]                                                                                       +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |spherical part: apparent viscosity associated with the reversible range of delayed deformations           |
+ .. image:: images/Object_211.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                     +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |spherical part: apparent viscosity associated with the irreversible diffusion mechanism                   |
+ .. image:: images/Object_213.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                     +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |deviatory part: viscosity associated with the reversible domain of delayed deformations                   |
+ .. image:: images/Object_215.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                     +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|                                                                                                              |deviatoric part: apparent viscosity associated with the irreversible diffusion mechanism                  |
+ .. image:: images/Object_217.svg                                                                             +                                                                                                          +
|    :width: 480                                                                                               |                                                                                                          |
+    :height: 360                                                                                              +                                                                                                          +
|                                                                                                              |                                                                                                          |
+ [:math:`\mathrm{MPa.s}`]                                                                                     +                                                                                                          +
|                                                                                                              |                                                                                                          |
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+
|:math:`\kappa =3.0\times {10}^{-3}`                                                                           |Irreversible deformation standard controlling the nonlinearity applied to the long-term deformation module|
+--------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------+



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

In this test, a homogeneous drying field that is invariant in the structure is created, the humidity being :math:`\text{100\%}` (condition of a sealed test piece). Mechanical loading corresponds to unidirectional compression in the vertical direction (:math:`z` in :math:`\mathrm{3D}` or :math:`y` in :math:`\mathrm{2D}`); its intensity is :math:`1[\mathrm{MPa}]`. The load is applied in :math:`\mathrm{1s}` and is kept constant for 100 days.


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

The start of the calculation is assumed to be instant :math:`–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.