D modeling
==============

We recall that modeling D concerns the use of model MCC [:ref:`r7.01.48 <R7.01.48>`]. It is characterized by:

* A hydrostatic compression path in drained condition up to a pressure equal to :math:`2` MPa.
* An undrained path by maintaining lateral stresses on the sample while imposing a vertical compression displacement.

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

Same as modeling A.

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

At moment :math:`t=2`, at the end of an undrained load path, 
the components :math:`\sigma'_{xx}` and :math:`\sigma'_{zz}` of the effective stress tensor, 
and water pressure :math:`PRE1`, are tested at node NO8. The reference values are non-regression values.

.. list-table:: Valeurs de :math:`\sigma'_{xx}=\sigma'_{yy}`.

 *-**Size**
   - **Moment**
   - **Reference**
   - **Aster code**
 * - :math:`\sigma'_{xx}`
   - :math:`2`
   - No regression
   - :math:`-804134` Not
 * - :math:`\sigma'_{zz}`
   - :math:`2`
   - No regression
   - :math:`-2017717` Not
 * - :math:`PRE1`
   - :math:`2`
   - No regression
   - :math:`1195865` Not

:numref:`v7.31.122-modelisation_D` shows the path of the effective stresses in their meridian plane, where:

.. math::
\ sigma'_m=\ frac {\ mathrm {tr} (\ boldsymbol {\ sigma} ')} {3},\ quad\ sigma'_ {eq} =\ sqrt {\ frac {3} {tr} {tr} (\ boldsymbol {\ sigma} '-\ sigma'_m\ boldsymbol {I}) :(\ boldsymbol {\ sigma}) :(\ boldsymbol {\ sigma}) :(\ boldsymbol {\ sigma}) :(\ boldsymbol {\ sigma} '-\ sigma'_m\ boldsymbol {I})}

The load is hydrostatic up to stress :math:`-\sigma'_m=2` MPa (drained isotropic compression phase). Then, during undrained loading, the diverter becomes non-zero and plastic flow occurs very early. The stress path joins the critical state line of equation :math:`\sigma'_{eq}+M(\sigma'_m-\sigma_0)=0` [:ref:`r7.01.48 <R7.01.48>`] _ during which the mean effective pressure :math:`-\sigma'_m` decreases. This path is similar to that obtained with modeling A.

.. _v7.31.122-modelisation_D:

.. figure:: images/figure_modelisation_D.svg
 :align: center
 :width: 960
 :height: 360

 Equivalent stress response and effective stress path in the meridian plane (D modeling).