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


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

The 3D modeling is carried out on a 3D hardware point with STAT_NON_LINE. The difference compared to modeling B is the calculation of the initial state by a thermal loading. To bring the sample to the initial isotropic stress :math:`{P}_{0}=10\mathit{kPa}`, the sample is brought from 20° to 30° Celsius. The movements of the sample are blocked, so that thermal expansion brings the sample into compression. We get:

:math:`{\mathrm{\sigma }}_{0}=\frac{E}{9\left(1-2\mathrm{\nu }\right)}\mathrm{\alpha }\mathrm{\Delta }T={P}_{0}`

Let the following value for the coefficient of thermal expansion be: :math:`\mathrm{\alpha }=\frac{9\left(1-2\mathrm{\nu }\right){P}_{0}}{E\mathrm{\Delta }T}`


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


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

The solutions are post-treated at point :math:`C`, in terms of:


* vertical constraint :math:`{\mathrm{\sigma }}_{\mathit{zz}}`;


* horizontal deformation :math:`{\mathrm{ϵ}}_{\mathit{xx}}`;


* :math:`{e}^{P}=\Vert {e}^{P}\Vert` deviatoric plastic deformation norm

They are compared to an analytical solution (described in the next paragraph) in terms of *maximum difference between* :math:`t=0` *and* :math:`t=20`. The results are summarised in the following tables:

:math:`Q=\sqrt{\frac{1}{2}\underline{\underline{s}}\mathrm{:}\underline{\underline{s}}}[\mathit{Pa}]`


+----------------------------------------+---------------------------------------------+
|Variable                                |Absolute deviation | Code_Aster − Analytics ||
+----------------------------------------+---------------------------------------------+
|:math:`{\mathrm{\sigma }}_{\mathit{zz}}`|0                                            |
+----------------------------------------+---------------------------------------------+
|:math:`{\mathrm{ϵ}}_{\mathit{xx}}`      |3.10-5                                       |
+----------------------------------------+---------------------------------------------+
|:math:`{e}^{P}`                         |1.333 10-5                                   |
+----------------------------------------+---------------------------------------------+


Comments
~~~~~~~~~~~~

The difference with the analytical solution is very small.