B modeling
==============


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

The entire test piece is meshed into quadrangular elements with 8 knots or triangular elements with 6 knots.

It has 2416 nodes, 625 quadrangles, 185 triangles, and 264 segments.


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


.. image:: images/100000000000015C0000014F50B6C4E10275F342.png
    :width: 4.8327in
    :height: 4.652in


.. _RefImage_100000000000015C0000014F50B6C4E10275F342.png:


.. image:: images/100000000000019F00000133ED4158D091129F49.png
    :width: 5.7634in
    :height: 4.2634in


.. _RefImage_100000000000019F00000133ED4158D091129F49.png:


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}`


.. csv-table::

    "**Identification**", "**Reference**
    :math:`{J}_{\mathrm{ASTM}}` ", "**Aster**", ":math:`\text{\%}` **difference**"
    "G (N/mm) Crown No. 0 UY=0.2 mm", "5.8", "6.2", "6.9"
    "G (N/mm) Crown No. 0 UY=0.4 mm", "22.6", "23.1", "2.1"
    "G (N/mm) Crown No. 0 UY=0.6 mm", "47.2", "47.2", "0.2"
    "G (N/mm) Crown No. 0 UY=0.8 mm", "74.7", "72.8", "2.5"
    "G (N/mm) Crown No. 0 UY=1.0 mm", "103.7", "98.2", "4.3"
    "", "", "", ""
    "G (N/mm) Crown #1 UY=0.2mm", "5.8", "5.3", "8.7"
    "G (N/mm) Crown #1 UY=0.4 mm", "22.6", "20.7", "8.3"
    "G (N/mm) Crown #1 UY=0.6 mm", "47.2", "43.8", "7.3"
    "G (N/mm) Crown #1 UY=0.8mm", "74.7", "68.6", "8.2"
    "G (N/mm) Crown #1 UY=1.0 mm", "103.7", "94.2", "9.2"
    "", "", "", ""
    "G (N/mm) Crown #2 UY=0.2 mm", "5.8", "5.4", "7.4"
    "G (N/mm) Crown #2 UY=0.4 mm", "22.6", "21.1", "6.8"
    "G (N/mm) Crown #2 UY=0.6 mm", "47.2", "44.2", "6.4"
    "G (N/mm) Crown #2 UY=0.8mm", "74.7", "69.1", "7.5"
    "G (N/mm) Crown #2 UY=1.0 mm", "103.7", "94.7", "8.8"



.. csv-table::

    "**Identification**", "**Reference**
    **J ASTM**", "**Aster**", ":math:`\text{\%}` **difference**"
    "G (N/mm) Crown #3 UY=0.2 mm", "5.8", "5.3", "9.5"
    "G (N/mm) Crown #3 UY=0.4 mm", "22.6", "20.2", "10.6"
    "G (N/mm) Crown #3 UY=0.6 mm", "47.2", "42.6", "9.8"
    "G (N/mm) Crown #3 UY=0.8 mm", "74.7", "67.2", "10.1"
    "G (N/mm) Crown #3 UY=1.0 mm", "103.7", "92.6", "10.7"
    "", "", "", ""
    "G (N/mm) Crown No. 4 UY=0.2 mm", "5.8", "5.4", "7.9"
    "G (N/mm) Crown #4 UY=0.4 mm", "22.6", "20.8", "8.0"
    "G (N/mm) Crown No. 4 UY=0.6 mm", "47.2", "43.7", "7.5"
    "G (N/mm) Crown #4 UY=0.8mm", "74.7", "68.4", "8.4"
    "G (N/mm) Crown #4 UY=1.0 mm", "103.7", "93.8", "9.5"


**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", "5.4", "5.4", "5.3", "5.4", "2.2"
    "G (N/mm) UY=0.4mm", "21.1", "21.1", "20.2", "20.8", "4.2"
    "G (N/mm) UY=0.6 mm", "44.2", "42.2", "42.6", "43.7", "3.7"
    "G (N/mm) UY=0.8mm", "69.1", "69.1", "67.2", "68.4", "2.8"
    "G (N/mm) UY=1.0mm", "94.7", "94.7", "92.6", "93.8", "2.1"



notes
---------

The value of :math:`G` in the Aster model is lower than the value in the reference.

The difference is around :math:`\text{10\%}` for all crowns. The stability between the crowns is satisfactory.