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


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

It is a three-dimensional model. The mesh was made using procedure GIBI of cracked block :math:`\mathrm{3D}` [:ref:`bib1 <bib1>`]. Only an eighth of the structure (and therefore a quarter of the crack front) has been represented, with a quarter of this front being discretized into 16 sectors.


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

The mesh is composed of quadratic elements

Number of meshes and types: 624 PENTA15, 5600 HEXA20


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

The values tested are those of the local energy restoration rate at points :math:`A` and :math:`B` using the various integration crowns and the two methods for defining the fields :math:`\theta`:


+---------------------------------------------------------------+-------------+---------------------------+
|**Identification**                                             |**Reference**|:math:`\text{\%}` Tolerance|
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` **local** discretization LEGENDRE **(degree** 5 **)**              |                           |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi 1                                          |1.13210-1    |2%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi2                                           |1.13210-1    |2%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi3                                           |1.13210-1    |3%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi4                                           |1.13210-1    |2%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi 5                                          |1.13210-1    |2%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` **local** discretization LINEAIRE                                  |                           |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi1                                           |1.13210-1    |3%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi2                                           |1.13210-1    |3%/1%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi3                                           |1.13210-1    |6%/4%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi4                                           |1.13210-1    |4%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`G` Mini/Maxi 5                                          |1.13210-1    |4%/2%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` discretization LEGENDRE **(degree** 5 **)**                        |                           |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi1                                           |1.5773105    |3%/4%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi2                                           |1.5773105    |6%/7%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi3                                           |1.5773105    |4%/9%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi 5                                          |1.5773105    |4%/9%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` discretization LINEAIRE                                            |                           |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi1                                           |1.5773105    |1%/8%                      |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi2                                           |1.5773105    |3%/25%                     |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi3                                           |1.5773105    |7%/34%                     |
+---------------------------------------------------------------+-------------+---------------------------+
|:math:`K` Mini/Maxi 5                                          |1.5773105    |7%/34%                     |
+---------------------------------------------------------------+-------------+---------------------------+


.. csv-table::

    "Crown 1:", ":math:`\mathrm{Rinf}=0.07` "," :math:`\text{Rsup}=0.2`"
    "Crown 2:", ":math:`\mathrm{Rinf}=0.2` "," :math:`\text{Rsup}=0.4`"
    "Crown 3:", ":math:`\mathrm{Rinf}=\mathrm{0.4.}` "," :math:`\text{Rsup}=\mathrm{0.6.}`"
    "Crown 4:", ":math:`\mathrm{Rinf}=0.07` "," :math:`\text{Rsup}=0.6`"


Crown 5: Rinf and Rsup are automatically determined by CALC_G.

Where

The supports of the local :math:`\theta` field correspond to the first three crowns of the global field.


notes
---------


* The reference value is the value of the local energy return rate: :math:`{G}_{\mathrm{réf}}=5.66{10}^{-8}J/{m}^{2}`.


* The results of local :math:`G` are only given for points :math:`A` and :math:`B` respectively located on a plane of symmetry and in the middle of the crack front. The results for point :math:`B` (middle of the forehead) show a difference of around :math:`\text{3\%}` compared to the reference result. The results for point :math:`A` are less good (the difference is between :math:`\text{3\%}` and :math:`\text{13.5\%}`), which is a usual observation for estimating the local :math:`G` for points located on a plane of symmetry.