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

Geometry
---------

Cylindrical test tube.


.. image:: images/Object_1.svg
    :width: 411
    :height: 241


.. _RefImage_Object_1.svg:


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

For models :math:`A` and :math:`B`, the material is assumed to be elastic and the material characteristics are constant to be able to validate the calculation with MECA_STATIQUE,

For modeling :math:`C`, we use the law of MAZARS and some parameters depend on hydration and drying.

Modeling :math:`E` makes it possible to test the law ENDO_ISOT_BETON, with the method of NEWTON and IMPLEX, and the modeling :math:`F` the coupling ENDO_ISOT_BETON/BETON_UMLV_FP, knowing that the material parameters of the law BETON_UMLV_FP are chosen so that there is no creep and therefore that we find the behavior of the law ENDO_ISOT_BETON. In both cases, the material characteristics are constant.

Note that given the loading (expansion, hydration and free drying), no damage develops: we therefore find the elastic solution in all cases.


+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**Modeling**                                                                                                                                                                                                                                                                                                        |**Modeling**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
| :math:`A`                                                                                                                                                                                                                                                                                                          | :math:`C`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
| **and**                                                                                                                                                                                                                                                                                                            | ****: **:**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
| :math:`B`                                                                                                                                                                                                                                                                                                          | **MAZARS**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
| **: Isotropic elasticity** BELARY**                                                                                                                                                                                                                                                                                | :math:`E=10000\mathrm{MPa}` :math:`C=\mathrm{100l}/{m}^{3}` :math:`30000\mathrm{MPa}` :math:`C=\mathrm{80l}/{m}^{3}` :math:`\nu =0.25` for :math:`h=0` 0.15 for :math:`h=1` :math:`\kappa =1.66{10}^{-5}{(l/{m}^{3})}^{-1}` :math:`{\beta }_{\mathrm{endo}}=1.5{10}^{-5}` :math:`\alpha =1.0{10}^{-5}°{C}^{-1}`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + :math:`{A}_{c}=1.4` :math:`{A}_{t}\mathrm{=}1.0` for :math:`C=\mathrm{100l}/{m}^{3}` :math:`0.8` :math:`C=\mathrm{80l}/{m}^{3}` :math:`{B}_{c}=2000` :math:`\mathrm{Bt}=10000` for :math:`h=0` :math:`11000` for :math:`h=1` :math:`{\varepsilon }_{\mathrm{d0}}={10}^{-4}` :math:`k\mathrm{=}0.7`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
| :math:`E=30000\mathrm{MPa}` :math:`\nu =0.2` :math:`\kappa =1.66{10}^{-5}{(l/{m}^{3})}^{-1}` :math:`{\beta }_{\mathrm{endo}}=1.5{10}^{-5}` :math:`\alpha =1.0{10}^{-5}°{C}^{-1}`                                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**Modeling**                                                                                                                                                                                                                                                                                                        |**Modeling**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
| :math:`E`                                                                                                                                                                                                                                                                                                          | :math:`F`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|                                                                                                                                                                                                                                                                                                                    | ****: **:**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
+ **ENDO_ISOT_BETON**                                                                                                                                                                                                                                                                                                +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|                                                                                                                                                                                                                                                                                                                    | **ENDO_ISOT_BETON/BETON_UMLV_FP**                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
+                                                                                                                                                                                                                                                                                                                    + See modeling :math:`E+`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_2.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + MPa                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_3.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    + MPa                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_4.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    + MPa                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_5.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    + MPa .j                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_6.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    + MPa .j                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_7.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    + MPa .j.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + .. image:: images/Object_8.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   +
|                                                                                                                                                                                                                                                                                                                    |    :width: 411                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
+                                                                                                                                                                                                                                                                                                                    +    :height: 241                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
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+                                                                                                                                                                                                                                                                                                                    +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+                                                                                                                                                                                                                                                                                                                    + MPa .j                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           +
|                                                                                                                                                                                                                                                                                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



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

On the :math:`\mathrm{AB}` side: :math:`{u}_{z}=0`

The following are varied uniformly over the structure:


1. the temperature from :math:`T=20°C` at the initial time to :math:`T=120°C` at the final time


2. the water content from :math:`100l/{m}^{3}` at the initial time to :math:`80l/{m}^{3}` at the final time


3. hydration varies from 0. at the initial time to 1. at the final time.