Modeling A
==============


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

The cell is homogenized in MASSIF in order to obtain the parameters for a 3D model.


Tested values
---------------

The values of the homogenized coefficients (elastic and thermal) obtained by the calculation are compared to those of the note [:ref:`VOLDOIRE93 <VOLDOIRE93>`].


+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|**Homogenized coefficients**|[]      |Calculation ASTER|Relative error (%) :math: `\ frac {\ left|\ text {approximate value} -\ text {theoretical value}\ right|} {\ left|\ text {theoretical value}\ right|}\ ast 100`|
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{1111}}`   |0.48437 |0.484961         |0.12%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{2222}}`   |--      |0.484782         |--                                                                                                                                                             |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{3333}}`   |0.73037 |0.730671         |0.04%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{1122}}`   |0.12246 |0.122638         |0.14%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{1133}}`   |0.18205 |0.182280         |0.13%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{2233}}`   |--      |0.182226         |--                                                                                                                                                             |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{1212}}`   |0.16161 |0.163213         |0.98%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{2323}}`   |--      |0.344923         |--                                                                                                                                                             |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{A}_{\text{3131}}`   |0.344589|0.344814         |0.06%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{K}_{\text{11}}`     |0.447967|0.448258         |0.06%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{K}_{\text{22}}`     |--      |0.448399         |--                                                                                                                                                             |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`{K}_{\text{33}}`     |0.62115 |0.621319         |0.03%                                                                                                                                                          |
+----------------------------+--------+-----------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------+

The equalities between the homogenized coefficients described in the preceding paragraph are also verified.


+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|Relationship between homogenized coefficients                                                                                                                                                                                                                                                                                                                                                                                                    |Relative error (%)|
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math:`\begin{array}{c}{A}_{\text{1111}}^{\text{hom}}-{A}_{\text{2222}}^{\text{hom}}=0\end{array}`                                                                                                                                                                                                                                                                                                                                               |0.04%             |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math:`\begin{array}{c}{A}_{\text{1133}}^{\text{hom}}-{A}_{\text{2233}}^{\text{hom}}=0\end{array}`                                                                                                                                                                                                                                                                                                                                               |0.03%             |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math:`\begin{array}{c}{A}_{\text{3131}}^{\text{hom}}-{A}_{\text{2323}}^{\text{hom}}=0\end{array}`                                                                                                                                                                                                                                                                                                                                               |0.03%             |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math: `{A} _ {\ text {3333}}} ^ {\ text {3333}}} ^ {e}} ^ {e} |} {\ int} _ {Z} _ {e} _ {e}}} E\ phantom {\ rule {3333}}} ^ {e}}}\ phantom {\ rule {0.5em} {0ex}}}\ mathit {dz} +2 {\nu} ^ {2} _ {2} _ {e}} _ {e}}} E\ phantom {\ rule {0.5em} {0ex}}}\ mathit {dz} +2 {\nu} ^ {2}}\ {Z} _ {z} _ {e} _ {e}}} E\ phantom {\ rule {0.5em} {0ex}}} \ left ({A} _ {\ text {1111}}} ^ {\ text {hom}}} + {\ text {1122}}} ^ {\ text {hom}}}\ right) =0`|0.34%             |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math:`{A}_{\text{1133}}^{\text{hom}}-\nu \text{.}\left({A}_{\text{1111}}^{\text{hom}}+{A}_{\text{1122}}^{\text{hom}}\right)=0`                                                                                                                                                                                                                                                                                                                  |1.6e -04%         |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+
|:math:`{A}_{\text{3}\mathrm{\alpha }\text{3}\mathrm{\alpha }}^{\text{hom}}-\frac{2\mathrm{\mu }}{k}{K}_{\text{11}}^{\text{hom}}=0`                                                                                                                                                                                                                                                                                                               |2.7th -05%        |
+-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+------------------+