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


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


.. image:: images/1000000000000146000000F98EF9E7A01EAE6B84.png
    :width: 4.5272in
    :height: 3.4575in


.. _RefImage_1000000000000146000000F98EF9E7A01EAE6B84.png:


.. image:: images/10000000000001600000013274B90166B2B1A37F.png
    :width: 4.8882in
    :height: 4.2492in


.. _RefImage_10000000000001600000013274B90166B2B1A37F.png:


.. image:: images/10000000000001CC00000135BA5E13C2E43B9AD6.png
    :width: 6.3882in
    :height: 4.2909in


.. _RefImage_10000000000001CC00000135BA5E13C2E43B9AD6.png:


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

The initial mesh consists of 8890 nodes and 2203 elements, including 1264 :math:`\mathrm{CU20}` elements and 939 :math:`\mathrm{PR15}` elements.


Tested features
------------------------

Calculation of the local stress intensity factors, at all the nodes of the crack bottom, using the THETA method.

The local stress intensity factors are calculated on a crown with a lower radius :math:`\mathrm{Rinf}=\mathrm{0,00075}m` and an upper radius :math:`\text{Rsup}=\mathrm{0,0025}m`.


Tested values and results of modeling A
-------------------------------------------------


.. csv-table::

    "**Identification**", "**Reference** (:math:`\mathit{Pa.}\sqrt{}m`)", "**Aster** (:math:`\mathit{Pa.}\sqrt{}m`)", "**% difference**"
    ":math:`{K}_{I}`, :math:`s` = 0 (point 1)", "6.09E+007", "5.50E+007", "9.7"
    ":math:`{K}_{I}`, :math:`s` = 5,34881e-3 (point 8)", "7.35E+007", "7.44E+007", "1.2"
    ":math:`{K}_{I}`, :math:`s` = 3.44482e-2 (point 25)", "1.02E+008", "1.02E+008", "0.1"
    ":math:`{K}_{I}`, :math:`s` = 5.36143e-2 (point 33)", "1.03E+008", "9.70E+007", "6.1"


The average difference is less than 2%.


.. image:: images/10000000000004280000029EA4BFB69AEF2A9E7F.png
    :width: 6.889in
    :height: 4.3374in


.. _RefImage_10000000000004280000029EA4BFB69AEF2A9E7F.png:


Tested values and results of modeling A with a linear mesh and as a reference the Murakami solution
-----------------------------------


.. csv-table::

    "**Identification**", "**Reference** (:math:`\mathit{Pa.}\sqrt{}m`)", "**Aster** (:math:`\mathit{Pa.}\sqrt{}m`)", "**% difference**"
    ":math:`{K}_{I}`, :math:`\theta` = 0 degrees", "5.657E+07", "5.789E+07", "-2.33"
    ":math:`{K}_{I}`, :math:`\theta` = 1.4 degrees", "5.945E+07", "5.360E+07", "9.84"
    ":math:`{K}_{I}`, :math:`\theta` = 2.8 degrees", "6.292E+07", "6.596E+07", "-4.84"
    ":math:`{K}_{I}`, :math:`\theta` = 4.3 degrees", "6.638E+07", "6.606E+07", "0.48"
    ":math:`{K}_{I}`, :math:`\theta` = 5.9 degrees", "6.984E+07", "6.902E+07", "1.18"
    ":math:`{K}_{I}`, :math:`\theta` = 7.6 degrees", "7.273E+07", "7.289E+07", "-0.22"
    ":math:`{K}_{I}`, :math:`\theta` = 9.5 degrees", "7.562E+07", "7.597E+07", "-0.47"
    ":math:`{K}_{I}`, :math:`\theta` = 11.6 degrees", "7.908E+07", "8.053E+07", "-1.83"
    ":math:`{K}_{I}`, :math:`\theta` = 14.4 degrees", "8.197E+07", "8.261E+07", "-0.78"
    ":math:`{K}_{I}`, :math:`\theta` = 16.9 degrees", "8.543E+07", "8.695E+07", "-1.78"
    ":math:`{K}_{I}`, :math:`\theta` = 20.5 degrees", "8.889E+07", "8.785E+07", "1.17"
    ":math:`{K}_{I}`, :math:`\theta` = 25.1 degrees", "9,178E+07", "9,190E+07", "-0,13"
    ":math:`{K}_{I}`, :math:`\theta` = 31.1 degrees", "9.466E+07", "9.173E+07", "3.09"
    ":math:`{K}_{I}`, :math:`\theta` = 39.5 degrees", "9.640E+07", "9.562E+07", "0.81"
    ":math:`{K}_{I}`, :math:`\theta` = 51.5 degrees", "9.755E+07", "9.510E+07", "2.51"
    ":math:`{K}_{I}`, :math:`\theta` = 68.5 degrees", "9.755E+07", "9.824E+07", "-0.71"
    ":math:`{K}_{I}`, :math:`\theta` = 90 degrees", "9.640E+07", "9.720E+07", "-0.83"


The parametric angles of the values tested correspond to the position of the 17 points at the bottom of the crack. The figure below allows you to compare the result of the calculation with the reference solution. The mean square deviation is very satisfactory:

Mean square deviation = :math:`\varepsilon \mathrm{=}\sqrt{\frac{{\mathrm{\int }}_{\Gamma }{({K}_{I}^{\mathit{ref}}\mathrm{-}{K}_{I}^{\mathit{aster}})}^{2}\mathit{ds}}{{\mathrm{\int }}_{\Gamma }{({K}_{I}^{\mathit{ref}})}^{2}\mathit{ds}}}\mathrm{=}3.11\text{\%}`


.. image:: images/Object_3.svg
    :width: 438
    :height: 257


.. _RefImage_Object_3.svg:

**Note:**

*Volume loading is introduced here using the keyword* *FORCE_INTERNE (command* *AFFE_CHAR_MECA) and* *formula. The results are equivalent if we use the keyword* *ROTATION.*