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

In this D_ PLAN modeling, the plate is cracked over half a length. The crack is described by method XFEM. The crack is enriched geometrically, on a radius :math:`{R}_{\mathit{ENRI}}\mathrm{=}\mathrm{0,1}`.

The elements are linear of type TRIA3.


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

The unit square is meshed regularly []. To build the mesh, we rely on a regular grid :math:`100\mathrm{\times }100`.

*NOMBRE OF NOEUDS: 10201*

*NOMBRE OF MAILLES: 20400*

 *TRIA3: 20000*


.. image:: images/100000000000023F00000241640D3D6365C7CCA9.png
    :width: 4.2937in
    :height: 4.6728in


.. _RefImage_100000000000023F00000241640D3D6365C7CCA9.png:

Figure 2.1-1: Meshing with triangle-elements

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


Tested sizes:
~~~~~~~~~~~~~~~~~~~

For this horizontal crack, we test the value of the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` as well as the value of the energy restoration rate :math:`G` given by CALC_G.

For method :math:`G-\mathrm{thêta}` (command CALC_G), the following theta field crown is chosen:

:math:`{R}_{\mathit{inf}}\mathrm{=}\mathrm{0,1}a` and :math:`{R}_{\text{sup}}\mathrm{=}\mathrm{0,3}a` where :math:`a` is the length of the crack.

On the other hand, we test the displacement field calculated by Code_Aster. Instead of performing a local test on a few cells by TEST_RESU, we test the displacement field on a large number of cells. An arbitrary test area has been delimited in the field [].

In practice, we compare: :math:`{∥{U}^{\mathit{calc}}\mathrm{-}{U}^{\mathit{ana}}∥}_{{L}_{2}}<\mathit{tolerance}\mathrm{\times }{∥{U}^{\mathit{ana}}∥}_{{L}_{2}}`.


.. image:: images/100002010000012300000110D9A38DDE904A9EF8.png
    :width: 3.0311in
    :height: 2.8335in


.. _RefImage_100002010000012300000110D9A38DDE904A9EF8.png:

Figure 2.2.1-1: Definition of test GROUP_MA

      We are finally testing the energy of the structure, the :math:`{L}^{2}` travel norm, throughout the field.


Results:
~~~~~~~~~~~

Stress Intensity Factors Test:


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "CALC_G ", "", ""
    "K1", "1.00"," 1.0%"
    "K2", "0.00"," 1.0%"
    "G", "1.0 10-5"," 1.0%"


Testing the standard_L2 of the error on the displacement field: :math:`{\Vert {U}^{\mathit{calc}}-{U}^{\mathit{ana}}\Vert }_{{L}_{2}}<\mathit{tolerance}\times {\Vert {U}^{\mathit{ana}}\Vert }_{{L}_{2}}`


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "POST_ELEM ", "", ""
    "NORME ", "0.00"," 0.1%"


Structure energy test:


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "POST_ERREUR ", "", ""
    "REFERENCE ", "3,50687407712 10-6"," 0.1%"


Testing the :math:`{L}^{2}` travel standard across the field:


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "POST_ERREUR ", "", ""
    "REFERENCE ", "7,6057690825 10-6"," 0.1%"



Additional results:
---------------------------

On the [], the displacement field is represented with amplification of the movement jump at the interface. We can see that the crack opens rigorously in :math:`\mathit{mode}I`, as expected.


.. image:: images/1000000000000233000001E0D23F553334A666B1.png
    :width: 5.8646in
    :height: 5in


.. _RefImage_1000000000000233000001E0D23F553334A666B1.png:

Figure 2.3-1: Field of movement (with offset)