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


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

In this modeling, the mesh method is tested for crack propagation. Level-sets are determined by orthogonal projection onto the segments making up the crack.


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

The structure is modelled by a regular "healthy" mesh composed of :math:`40\times 101` QUAD4, respectively along the :math:`x,y` axes. The crack is represented by a succession of SEG2, regardless of the mesh of the structure.


.. image:: images/1000000000000459000003306C0E7AA868D19B2D.jpg
    :width: 5.9008in
    :height: 4.3256in


.. _RefImage_1000000000000459000003306C0E7AA868D19B2D.jpg:

**Figure** 3.2-a **: mesh of the cracked plate**


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

For each propagation step (:math:`\mathrm{2,5}m`), we test the value of the stress intensity factors :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` given by CALC_G.

We also test the ordinate of the bottom of the crack given by PROPA_FISS.


Results on :math:`{K}_{I}`:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Code_Aster**", "**Reference**", "**difference**"
    "CALC_G ", "", "", ""
    "KI_1 ", "4.17448 106", "4.205998 106", "-0.749%", "-0.749%"
    "KI_2 ", "4.60197 106", "4.63286 106", "-0.667%"
    "KI_3 ", "5.0668 106", "5.09492 106", "-0.552%"
    "KI_4 ", "5.575 106", "5.59908 106", "-0.43%"
    "KI_5 ", "6.1334 106", "6.15349 106", "6.1334 106", "-0.326%"
    "KI_6 ", "6.7499 106", "6.76776 106", "-0.264%"
    "KI_7 ", "7.4338 106", "7.4531 106", "-0.259%", "-0.259%"
    "KI_8 ", "8,1959,106", "8,224,106", "-0.322"", "-0.322""
    "KI_9 ", "9.0497 106", "9.0905 106", "-0.449%"
    "KI_10 ", "1.0011 107", "1.0074 107", "-0.627%"
    "KI_11 ", "1.1099 107", "1.1192 107", "-0.828%", "-0.828%"
    "KI_12 ", "1.2339 107", "1.2465 107", "-1.011%"
    "KI_13 ", "1.37603 107", "1.3916 107", "-1.121%"
    "KI_14 ", "1.54018 107", "1.55716 107", "-1.09%"
    "KI_15 ", "1.7313 107", "1.74586 107", "-0.834%", "-0.834%"



Results on :math:`{K}_{\mathrm{II}}`:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For this test, we want :math:`{K}_{\mathrm{II}}` to be less than :math:`{10}^{-4}{K}_{I}`. So, we make sure that :math:`{K}_{\mathrm{II}}` is fairly close to zero, the reference value.


.. csv-table::

    "**Identification**", "**Code_Aster**", "**Reference**"
    "CALC_G ", "", ""
    "KII_1 ", "-2,7313 102", "0"
    "KII_2 ", "-8,5062 101", "0"
    "KII_3 ", "-2,6061 102", "0"
    "KII_4 ", "1,5995 102", "0"
    "KII_5 ", "-2,7309 102", "0"
    "KII_6 ", "-2,3176 102", "0"
    "KII_7 ", "-3,1276 102", "0"
    "KII_8 ", "3,1327 102", "0"
    "KII_9 ", "-3,8393 102", "0"
    "KII_10 ", "-4,1916 102", "0"
    "KII_11 ", "-4,986 102", "0"
    "KII_12 ", "-5,6998 102", "0"
    "KII_13 ", "-6,7642 102", "0"
    "KII_14 ", "-7,9542 102", "0"
    "KII_15 ", "-9,5344 102", "0"



Results on the ordinate of the bottom of the crack:
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is verified that the ordinate coordinates of the successive crack bottoms are close to the initial value. This check gives the same information as the test on :math:`{K}_{\mathrm{II}}`.


.. csv-table::

    "**Identification**", "**Code_Aster**", "**Reference**", "**Difference**"
    "CALC_G ", "", "", ""
    "y_1", "15", "15", "15", "0%"
    "y_2", "15", "15", "15", "2.18 10-4%"
    "y_3", "15", "15", "15", "2.8 10-4%"
    "y_4", "15", "15", "15", "4.51 10-4%"
    "y_5", "15", "15", "15", "5.47 10-4%"
    "y_6", "15", "15", "15", "6.95 10-4%"
    "y_7", "15", "15", "15", "8.1 10-4%"
    "y_8", "15", "15", "15", "9.5 10-4%"
    "y_9", "15,0002", "15", "0.001%"
    "y_10", "15,0002", "15", "0.001%"
    "y_11", "15,0002", "15", "0.001%"
    "y_12", "15,0002", "15", "0.002%"
    "y_13", "15,0002", "15", "0.002%"
    "y_14", "15,0003", "15", "0.002%"
    "y_15", "15,0003", "15", "0.002%"



Additional results
-------------------------


.. image:: images/10000000000003C2000002560CA3F637DFCDA0C4.gif
    :width: 5.9008in
    :height: 3.6673in


.. _RefImage_10000000000003C2000002560CA3F637DFCDA0C4.gif:

**Figure** 3.4-a **: Influence of the choice of RI and RS crowns on the error on KI**

Here we can see that the most suitable configuration for choosing :math:`\mathrm{RI}` and :math:`\mathrm{RS}` (lower and upper crowns of the theta field) is: :math:`\mathrm{RI}=2\ast {L}_{0}` and :math:`\mathrm{RS}=7\ast {L}_{0}` where :math:`{L}_{0}` is the smallest edge of the mesh.