C modeling
==============


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

The modeling is :math:`\mathrm{3D}` sub-integrated (3D_ HM_SI) with a hydro-mechanical coupling in non-linear quasistatics.

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}=-50\mathrm{kPa}`. This state of confinement makes it possible to consider the sample as *dense* sand.

We use Hujeux's law *cyclic*.


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

Number of knots: :math:`20`

Number of meshes and type: 1 :math:`\mathit{HEXA20}` and 6 :math:`\mathit{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 ", "3.154E+1", "3.0"
    "-2.E-3"," SOURCE_EXTERNE ", "4.013E+1", "2.0"
    "-5.E-3"," SOURCE_EXTERNE ", "5.194E+1", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "6.829E+1", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "1.032E+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 ", "-1.389E+2", "1.0"
    "-2.E-3"," SOURCE_EXTERNE ", "-1.338E+2", "1.0"
    "-5.E-3"," SOURCE_EXTERNE ", "-1.250E+2", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "-1.368E+2", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "-1.860E+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 ", "-2.42E-5", "6.0"
    "-2.E-3"," SOURCE_EXTERNE ", "-3.55E-5", "4.0"
    "-5.E-3"," SOURCE_EXTERNE ", "-5.56E-5", "3.0"
    "-1.E-2"," SOURCE_EXTERNE ", "-2.88E-5", "5.0"
    "-2.E-2"," SOURCE_EXTERNE ", "7.437E-5", "5.0"


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

.. csv-table::

    ":math:`{\varepsilon }_{\text{zz}}` ", "Reference type", "GEFDYN ", "tolerance (%)"
    "-1.E-3"," SOURCE_EXTERNE ", "0.02", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "0.0248", "1.0"


: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 ", "1.49E-3", "2.0"
    "-2.E-3"," SOURCE_EXTERNE ", "2.18E-3", "2.0"
    "-5.E-3"," SOURCE_EXTERNE ", "3.36E-3", "2.0"
    "-1.E-2"," SOURCE_EXTERNE ", "1.68E-3", "3.0"


: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 ", "0.353", "3.0"
    "-2.E-3"," SOURCE_EXTERNE ", "0.451", "2.0"
    "-5.E-3"," SOURCE_EXTERNE ", "0.593", "1.0"
    "-1.E-2"," SOURCE_EXTERNE ", "0.699", "1.0"
    "-2.E-2"," SOURCE_EXTERNE ", "0.794", "1.0"



notes
---------

The comparison between *Code_Aster* and GEFDYN solutions is relatively good, with generally fewer :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.