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


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

**C** modeling is performed on a hardware point, using SIMU_POINT_MAT.

The axial deformation imposed during the isotropic compression phase is equal to :math:`-{\mathrm{\sigma }}_{0}/3K` (axial deformation making it possible to reach :math:`{\mathrm{\sigma }}_{\mathit{zz}}={\mathrm{\sigma }}_{0}`, applied in :math:`t=1s`). During the same time interval, lateral constraints change from :math:`{\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{\mathit{yy}}=0` to :math:`{\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{\mathit{yy}}={\mathrm{\sigma }}_{0}`. The maximum axial deformation applied during the deviatoric phase is :math:`{\mathrm{ϵ}}_{\mathit{zz}}=-0.03\text{\%}` (final instant :math:`t=30s`). During the deviatoric phase, the lateral stresses are kept constant, equal to :math:`{\mathrm{\sigma }}_{\mathit{xx}}={\mathrm{\sigma }}_{\mathit{yy}}={\mathrm{\sigma }}_{0}`.


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


Tested values
~~~~~~~~~~~~~~~~

Five non-regression tests are carried out at the end of the test at :math:`t=30s`, on:


* SIXX — lateral constraint that must be equal to the imposed lockdown, i.e. :math:`{\mathrm{\sigma }}_{0}`;


* SIZZ — constraint resulting from the law of behavior;


* V7— plastic multiplier (equal to the equivalent plastic deformation of Von-Mises);


* V8— plastic volume deformation (:math:`{\mathrm{ϵ}}_{\mathit{kk}}-{\mathrm{ϵ}}_{\mathit{kk}}^{e}`);


* V9— dissipation (:math:`\int {\mathrm{\sigma }}_{\mathit{ij}}\left(\mathrm{\delta }{\mathrm{ϵ}}_{\mathit{ji}}-\mathrm{\delta }{\mathrm{ϵ}}_{\mathit{ji}}^{e}\right)`).


.. csv-table::

    ":math:`t=30\mathit{sec}` ", "Analytical solution from modeling A", "Calculated solution from modeling C"
    "SIZZ (Pa)", ":math:`-173289.5416041` "," :math:`-173077.28029053428`"
    "SIXX (Pa)", ":math:`-50000.` "," :math:`-50000.00000615134`"
    "V8"," :math:`0.00016784502547224` "," :math:`0.00011457113677334`"
    "V7"," :math:`0.00033099447556585` "," :math:`0.0002523646130019748`"
    "V9 (Pa)", "-", ":math:`2.7623234303334216`"


Table 4.2.1-1: Validation of results for modeling C

It is normal to notice some differences with the analytical solution for components SIZZ, V7, and V8 since the smoothed criterion used by the Mohr-Coulombas law deviates somewhat from the original Mohr-Coulomb criterion.