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


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

The modeling is 3D with a hydro-mechanical coupling in a semi-static non-linear fashion.

In loading phase :math:`1`, the sample is brought to consolidation pressure :math:`{\sigma }_{\mathrm{xx}}^{0}={\sigma }_{\mathrm{yy}}^{0}={\sigma }_{\mathrm{zz}}^{0}={\sigma }_{0}=-200\mathrm{kPa}`. This state of confinement makes it possible to consider the sample as *moderately dense* sand.

We use Hujeux's law *cyclic*.


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

Number of knots: :math:`20`

Number of meshes and type: 1 HEXA20 and 6 QUAD8.


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

The solutions are calculated at point :math:`C` and compared to references GEFDYN. 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:`Q\mathrm{=}\sqrt{\frac{1}{2}{\sigma }_{\text{ij}}^{d}\mathrm{:}{\sigma }_{\text{ij}}^{d}}(\mathit{kPa})`

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference type", "GEFDYN (:math:`\mathit{kPa}`)", "tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "1.015E+2", "3.0"
    "-2.E-3"," SOURCE_EXTERNE ", "1.343E+2", "2.0"
    "-5.E-3"," SOURCE_EXTERNE ", "1.808E+2", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "2.139E+2", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "2.495E+2", "1.0"


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

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference type", "GEFDYN (:math:`\mathit{kPa}`)", "tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "-5.889E+2", "1.0"
    "-2.E-3"," SOURCE_EXTERNE ", "-5.823E+2", "1.0"
    "-5.E-3"," SOURCE_EXTERNE ", "-5.638E+2", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "-5.439E+2", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "-5.442E+2", "1.0"


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

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference Type", "GEFDYN ", "Tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "-1.37E-5", "8.0"
    "-2.E-3"," SOURCE_EXTERNE ", "-2.19E-5", "6.0"
    "-5.E-3"," SOURCE_EXTERNE ", "-4.51E-5", "3.0"
    "-1.E-2"," SOURCE_EXTERNE ", "-7.03E-5", "2.0"
    "-2.E-2"," SOURCE_EXTERNE ", "-7.00E-5", "2.0"


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

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference Type", "GEFDYN ", "Tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "1.51E-3", "1.0"
    "-2.E-3"," SOURCE_EXTERNE ", "2.40E-3", "1.0"
    "-5.E-3"," SOURCE_EXTERNE ", "4.91E-3", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "7.60E-3", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "1.16E-3", "2.0"


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

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference Type", "GEFDYN ", "Tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "0.334", "3.0"
    "-2.E-3"," SOURCE_EXTERNE ", "0.436", "2.0"
    "-5.E-3"," SOURCE_EXTERNE ", "0.583", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "0.693", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "0.790", "1.0"



notes
---------

The comparison between *Code_Aster* and GEFDYN solutions is relatively good, with generally less than :math:`1\text{\%}` errors. Relative errors greater than :math:`1\text{\%}` appear for levels of test values that are relatively low and close to the numerical precision applied during the calculation.