Modeling A
==============


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

.. figure:: images/Object_8.svg
 :align: center
 :width: 480
 :height: 360

 3D modeling.

.. _RefImage_Object_8.svg:


.. list-table:: Caractéristiques du maillage.

 * - Number of knots
   - 20
 * - Number of meshes
   - | 1 of the HEXA 20 type
     | 6 of the QUAD 8 type

.. list-table:: Mailles définies.

 *-* DROITE *
   - NO3, NO5, NO8, N10, N10, NO12,, NO15, NO17, NO20
 *-* GAUCHE *
   - NO1, NO4, NO6, NO9, NO11,,, NO13, NO16, NO18
 *-* DEVANT *
   - NO6, NO7, NO8, NO11, NO12,,, NO18, NO19, NO20
 *-* DERRIERE *
   - NO1, NO2, NO3, NO9, NO10,,, NO13, NO14, NO15
 *-* BAS *
   - NO13, NO14, NO15, NO16, NO17,,, NO18, NO19, NO20
 *-* HAUT *
   - NO1, NO2, NO3, NO4, NO5,,, NO6, NO7, NO8


Modeling A with law CAM_CLAY of the 3D type is*tridimensional* and*static nonlinear*.

The initial preconsolidation of the sample is equal to :math:`{p}_{\mathrm{co}}={100}` kPa. We check that Young's modulus respects the inequality: :math:`E<1.8` MPa.


In this modeling A, on the group of elements *HAUT*, :math:`DZ` therefore varies between :math:`0` and :math:`-0.1` m in 70 steps of time. During the calculation, the automatic subdivision of the time step is activated to manage situations of non-convergence of local integration.

In the integration of equilibrium equations, an update of the tangent matrix is required, which is provided by the routines of law CAM_CLAY and significantly accelerates convergence. We also require the subdivision of the time step (command DEFI_LIST_INST) to deal with situations where local integration fails due to excessively large load increments.

To represent an oedometric test, the common boundary conditions are (symmetry conditions, because the cubic element represents one eighth of the sample):


*On the stitches* DEVANT *and* DERRIERE *: :math:`DX=0`;

*On the stitches* GAUCHE *and* DROITE *: :math:`DY=0`;

*On the mesh* BAS *: :math:`DZ=0`.


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

The solutions are calculated at node NO8 for four levels of deformation :math:`\epsilon_{zz}` and compared to *FLAC* solutions provided by EDF/CIH. They are given in terms of mean pressure :math:`p` (internal variable V3), equivalent stress :math:`q` (internal variable V4) and vacuum index :math:`e` (internal variable V7). These are summarised by the tables below.

.. list-table::

 * - :math:`{\epsilon}_{zz}`
   - :math:`p` (Pa) **Code_Aster**
   - :math:`p` (Pa) **FLAC**
 * - :math:`-0.1`%
   - :math:`10132.3`
   - :math:`10070.0`
 * - :math:`-0.5`%
   - :math:`10505.9`
   - :math:`10500.0`
 * - :math:`-1`%
   - :math:`11014.5`
   - :math:`11010.0`
 * - :math:`-2`%
   - :math:`12499.3`
   - :math:`12480.0`
 * - :math:`-10`%
   - :math:`41989.0`
   - :math:`41840.0`

.. list-table::

 * - :math:`{\epsilon}_{zz}`
   - :math:`q` (Pa) **Code_Aster**
   - :math:`q` (Pa) **FLAC**
 * - :math:`-0.1`%
   - :math:`514.6`
   - :math:`521.0`
 * - :math:`-0.5`%
   - :math:`1998.8`
   - :math:`2016.0`
 * - :math:`-1`%
   - :math:`3051.1`
   - :math:`3068.0`
 * - :math:`-2`%
   - :math:`4189.7`
   - :math:`4219.0`
 * - :math:`-10`%
   - :math:`12904.9`
   - :math:`13020.0`

.. list-table::

 * - :math:`{\epsilon}_{zz}`
   - :math:`e` **Aster_Code**
   - :math:`e` **FLAC**
 * - :math:`-0.1`%
   - :math:`1.998`
   - :math:`1.997`
 * - :math:`-0.5`%
   - :math:`1.986`
   - :math:`1.985`
 * - :math:`-1`%
   - :math:`1.971`
   - :math:`1.970`
 * - :math:`-2`%
   - :math:`1.941`
   - :math:`1.940`
 * - :math:`-10`%
   - :math:`1.701`
   - :math:`1.700`


The following figures show the various comparisons between Code_Aster and FLAC, in terms of the loading path in the :math:`(p,q)` plane (with :math:`p'=p` in this drained test), and the evolution of the void index. In the latter, two Code_Aster solutions are shown:


* The one-phase solution, which corresponds to modeling A.


* The solution in two phases, where we replaced the assignment of the initial stress field :math:`{P}_{\mathit{cons}}^{0}` (CREA_CHAM) by the calculation of the elastoplastic law CAM_CLAY under isotropic consolidation up to pressure :math:`{P}_{\mathit{cons}}^{0}`.

The procedure adopted in modeling A is the one used for calculation FLAC. In terms of the loading path in plan :math:`(p,q)`, there is an excellent convergence of solutions between FLAC and Code_Aster, both in one phase and in two. With regard, on the other hand, to the void index, while the Code_Aster solution in one phase coincides well with solution FLAC, it is quite different for the Code_Aster solution in two phases. This is easily explained: during isotropic loading with law CAM_CLAY, the void index changed from :math:`e_{0}=2` to :math:`e\approx 1.82`. This result therefore shows how the two approaches (in one or two phases) are not entirely equivalent, and that it is important to be aware of this.

.. figure:: images/Object_166.svg
 :align: center
 :width: 480
 :height: 360

 Plan :math:`(p,q)` loading path: comparison between Code_Aster and FLAC solutions.

.. figure:: images/Object_167.svg
 :align: center
 :width: 480
 :height: 360

 Index of voids as a function of vertical stress: comparison between solutions FLAC and Code_Aster for two calculations: in one phase and in two phases.