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


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

The entire test piece is meshed into quadrangular elements with 4 knots or triangular elements with 3 knots.

It has 799 nodes, 624 quadrangles, 185 triangles, and 261 segments.


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

Very small elements (:math:`\mathrm{0,02}\mathrm{mm}`) at the tip of the crack.

The first ring is located in a single material, the 4 other rings cross the interface between the two materials.


.. image:: images/100041CA00002E6800002BED034B05B8F3BDF5A4.svg
    :width: 598
    :height: 566


.. _RefImage_100041CA00002E6800002BED034B05B8F3BDF5A4.svg:


.. image:: images/10000000000001C90000013DAF6A1894698B8CC3.png
    :width: 625
    :height: 414


.. _RefImage_10000000000001C90000013DAF6A1894698B8CC3.png:


.. image:: images/1000410C00003393000022361F217C689A3DD090.svg
    :width: 598
    :height: 566


.. _RefImage_1000410C00003393000022361F217C689A3DD090.svg:


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

The energy return rate :math:`G` is calculated by the :math:`\mathrm{THETA}` method for the following 5 crowns:


* Crown 0: :math:`{R}_{\mathrm{inf}}=\mathrm{0,02}\mathrm{mm}` :math:`{R}_{\text{sup }}=\mathrm{0,18}\mathrm{mm}`


* Crown 1: :math:`{R}_{\mathrm{inf}}=\mathrm{0,2}\mathrm{mm}` :math:`{R}_{\text{sup}}=1\mathrm{mm}`


* Crown 2: :math:`{R}_{\mathrm{inf}}=1\mathrm{mm}` :math:`{R}_{\text{sup}}=2\mathrm{mm}`


* Crown 3: :math:`{R}_{\mathrm{inf}}=2\mathrm{mm}` :math:`{R}_{\text{sup}}=3\mathrm{mm}`


* Crown 4: :math:`{R}_{\mathrm{inf}}=3\mathrm{mm}` :math:`{R}_{\text{sup}}=5\mathrm{mm}`

Tested values


.. csv-table::

    "**Identification**", "**Reference**
    **J ASTM**", "**Aster**", ":math:`\text{\%}` **difference**"
    "G (N/mm) Crown No. 0 UY=0.2 mm", "5.8", "6.0", "3.1"
    "G (N/mm) Crown No. 0 UY=0.4 mm", "22.6", "23.2", "2.8"
    "G (N/mm) Crown No. 0 UY=0.6 mm", "47.2", "48.2", "2.0"
    "G (N/mm) Crown No. 0 UY=0.8 mm", "74.7", "75.0", "0.5"
    "G (N/mm) Crown No. 0 UY=1.0 mm", "103.7", "102.8", "-0.8"
    "", "", "", ""
    "G (N/mm) Crown #1 UY=0.2 mm", "5.8", "6.10", "4.6"
    "G (N/mm) Crown #1 UY=0.4 mm", "22.6", "23.2", "2.8"
    "G (N/mm) Crown #1 UY=0.6 mm", "47.2", "47.3", "0.1"
    "G (N/mm) Crown #1 UY=0.8mm", "74.7", "73.1", "2.2"
    "G (N/mm) Crown #1 UY=1.0mm", "103.7", "99.8", "3.8"
    "", "", "", ""
    "G (N/mm) Crown #2 UY=0.2 mm", "5.8", "6.1", "4.3"
    "G (N/mm) Crown #2 UY=0.4 mm", "22.6", "23.3", "3.2"
    "G (N/mm) Crown #2 UY=0.6 mm", "47.2", "48.0", "1.6"
    "G (N/mm) Crown #2 UY=0.8mm", "74.7", "74.9", "0.3"
    "G (N/mm) Crown #2 UY=1.0 mm", "103.7", "103.7", "103.1", "0.3"



.. csv-table::

    "**Identification**", "**Reference**
    **J ASTM**", "**Aster**", ":math:`\text{\%}` **difference**"
    "G (N/mm) Crown #3 UY=0.2 mm", "5.8", "6.1", "4.4"
    "G (N/mm) Crown #3 UY=0.4 mm", "22.6", "23.3", "3.3"
    "G (N/mm) Crown #3 UY=0.6 mm", "47.2", "48.2", "1.9"
    "G (N/mm) Crown #3 UY=0.8mm", "74.7", "75.5", "1.0"
    "G (N/mm) Crown #3 UY=1.0 mm", "103.7", "104.2", "0.4"
    "", "", "", ""
    "G (N/mm) Crown No. 4 UY=0.2 mm", "5.8", "6.1", "4.5"
    "G (N/mm) Crown No. 4 UY=0.4 mm", "22.6", "23.4", "3.4"
    "G (N/mm) Crown No. 4 UY=0.6 mm", "47.2", "48.2", "2.1"
    "G (N/mm) Crown #4 UY=0.8mm", "74.7", "75.5", "1.1"
    "G (N/mm) Crown #4 UY=1.0 mm", "103.7", "104.4", "0.7"


**Stability of** :math:`G` **with the choice of crowns**


.. csv-table::

    "**Identification**", "**Crown 2**", "**Crown 3**", "**Crown 4**", ":math:`\text{\%}` **Maximum variance**"
    "G (N/mm) UY=0.2mm", "6.1", "6.1", "6.1", "6.1", "0.2"
    "G (N/mm) UY=0.4mm", "23.3", "23.3", "23.4", "0.1"
    "G (N/mm) UY=0.6mm", "48.0", "48.2", "48.2", "0.4"
    "G (N/mm) UY=0.8mm", "74.9", "75.5", "75.5", "0.5"
    "G (N/mm) UY=1.0mm", "103.1", "103.1", "104.2", "104.4", "0.4"



notes
---------

In all cases, the absolute value of the difference in the calculation of :math:`G` is less than :math:`\text{5\%}`. For rings 3 to 5, the difference decreases as a function of the displacement to reach an almost zero value.

The stability on rings 2, 3 and 4 is very good, the gap between rings is always less than :math:`\text{0,5 \%}`.