B modeling
==============


Characteristics of modeling
-----------------------------------

The modeling is of the "material point" type with linear relationships on the components of the strain tensor simulating incompressibility, in non-linear quasistatics.

We consider here phase 2 of loading, with an initial hydrostatic stress state :math:`{\sigma }_{\mathit{xx}}^{0}\mathrm{=}{\sigma }_{\mathit{yy}}^{0}\mathrm{=}{\sigma }_{\mathit{zz}}^{0}\mathrm{=}{\sigma }^{0}\mathrm{=}\mathrm{-}30\mathit{kPa}`

The data is the same as modeling A.


Characteristics of the mesh
----------------------------

No finite element


Tested sizes and results
------------------------------

The solutions are calculated at point :math:`C` and compared to the results obtained with GEFDYN (same tests as modeling A). They are given in terms of isotropic pressure, plastic volume deformation :math:`{\varepsilon }_{v}^{p}` and mobilization factors, and summarized in the following tables:

:math:`3\mathrm{\cdot }P\text{'}\mathrm{=}{\sigma }_{\mathit{ij}}\mathrm{\cdot }{\delta }_{\mathit{ij}}` :math:`(\mathrm{kPa})`

+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|                                                                                                         |Code_Aster (:math:`\mathit{kPa}`)|GEFDYN (:math:`\mathit{kPa}`)|Relative error|
+ .. image:: images/Object_87.svg                                                                         +                                 +                             +              +
|    :width: 91                                                                                           |                                 |                             |              |
+    :height: 24                                                                                          +                                 +                             +              +
|                                                                                                         |                                 |                             |              |
+ (:math:`\mathit{kPa}`)                                                                                  +                                 +                             +              +
|                                                                                                         |                                 |                             |              |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|-15                                                                                                      |-80.223                          |-80.194                      |0.038%        |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|15                                                                                                       |-73.988                          |-74.078                      |-0.122%       |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|-15                                                                                                      |-66.055                          |-66.250                      |-0.295%       |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|15                                                                                                       |-52.625                          |-52.999                      |-0.707%       |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+
|-15                                                                                                      |-45,539                          |-45.672                      |-0.292%       |
+---------------------------------------------------------------------------------------------------------+---------------------------------+-----------------------------+--------------+


:math:`{\varepsilon }_{v}^{p}\mathrm{=}\mathit{trace}({\varepsilon }^{p})`

+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |Code_Aster|GEFDYN  |Relative error|
+ .. image:: images/Object_89.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+ (:math:`\mathit{kPa}`)                                                                                  +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|-15                                                                                                      |-2.62e-5  |-2.63e-5|-0.225%       |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |-2.80e-5  |-2.78e-5|0.854%        |
+ .. image:: images/Object_90.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+                                                                                                         +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |-4.48e-5  |-4.43e-5|1.076%        |
+ .. image:: images/Object_91.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+                                                                                                         +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |-7.11e-5  |-7.05e-5|0.842%        |
+ .. image:: images/Object_92.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+                                                                                                         +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |-1.13e-4  |-1.11e-4|1.412%        |
+ .. image:: images/Object_93.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+                                                                                                         +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+
|                                                                                                         |-1.85e-4  |-1.85e-4|0.261%        |
+ .. image:: images/Object_94.svg                                                                         +          +        +              +
|    :width: 91                                                                                           |          |        |              |
+    :height: 24                                                                                          +          +        +              +
|                                                                                                         |          |        |              |
+                                                                                                         +          +        +              +
|                                                                                                         |          |        |              |
+---------------------------------------------------------------------------------------------------------+----------+--------+--------------+


:math:`({r}_{\mathit{iso}}^{c}+{r}_{\mathit{ela}}^{s})`

+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |Code_Aster|GEFDYN |Relative error|
+ .. image:: images/Object_96.svg                                                                         +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+ (:math:`\mathit{kPa}`)                                                                                  +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |0.00140   |0.00138|1.329%        |
+ .. image:: images/Object_97.svg                                                                         +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+                                                                                                         +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |0.00220   |0.00217|1.406%        |
+ .. image:: images/Object_98.svg                                                                         +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+                                                                                                         +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |0.00341   |0.00337|1.246%        |
+ .. image:: images/Object_99.svg                                                                         +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+                                                                                                         +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |0.00518   |0.00510|1.663%        |
+ .. image:: images/Object_100.svg                                                                        +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+                                                                                                         +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+
|                                                                                                         |0.00195   |0.00189|2.947%        |
+ .. image:: images/Object_101.svg                                                                        +          +       +              +
|    :width: 91                                                                                           |          |       |              |
+    :height: 24                                                                                          +          +       +              +
|                                                                                                         |          |       |              |
+                                                                                                         +          +       +              +
|                                                                                                         |          |       |              |
+---------------------------------------------------------------------------------------------------------+----------+-------+--------------+


:math:`({r}_{\mathit{dev}}^{m}+{r}_{\mathit{ela}}^{d})`

+----------------------------------------------------------------------------------------------------------+----------+------+--------------+
|                                                                                                          |Code_Aster|GEFDYN|Relative error|
+ .. image:: images/Object_103.svg                                                                         +          +      +              +
|    :width: 91                                                                                            |          |      |              |
+    :height: 24                                                                                           +          +      +              +
|                                                                                                          |          |      |              |
+ (:math:`\mathit{kPa}`)                                                                                   +          +      +              +
|                                                                                                          |          |      |              |
+----------------------------------------------------------------------------------------------------------+----------+------+--------------+
|15                                                                                                        |0.326     |0.326 |-0.088%       |
+----------------------------------------------------------------------------------------------------------+----------+------+--------------+
|-15                                                                                                       |0.395     |0.394 |0.172%        |
+----------------------------------------------------------------------------------------------------------+----------+------+--------------+
|15                                                                                                        |0.395     |0.394 |0.172%        |
+----------------------------------------------------------------------------------------------------------+----------+------+--------------+
|-15                                                                                                       |0.579     |0.578 |0.255%        |
+----------------------------------------------------------------------------------------------------------+----------+------+--------------+