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


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

For this modeling, the 3 topological parameters of the crack block are:


* :math:`\mathrm{NS}`: number of sectors on :math:`90°`


* :math:`\mathrm{NC}`: number of crowns


* :math:`\mathrm{rt}`: the radius of the largest crown (with a: half length of the crack)

:math:`\mathit{NS}\mathrm{=}8`

:math:`\mathit{NC}\mathrm{=}4`

:math:`\mathit{rt}\mathrm{=}\mathrm{0,001}\mathrm{\times }a`

The values of the upper and lower radii, to be specified in the CALC_G command, are:


.. csv-table::

    "", "Crown 1", "Crown 2", "Crown 3", "Crown 3", "Crown 4", "Crown 5", "Crown 6"
    ":math:`\text{Rinf}` ", "3,75E—5", "7,500E—5", "7,500E—5", "1,500E—4", "1,875E—4", "2,250E—4", "2,250E—4"
    ":math:`\text{Rsup}` ", "7,50E—5", "1,125E—4", "1,500E—4", "1,875E—4", "2,250E—4", "3,000E—4", "3,000E—4"



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

Half-mesh; mesh radiating at the right end of the crack.

3831 knots,

1516 elements,

884 TRI6,

632 QUA8.


Quantities tested and results of modeling A
----------------------------------------


.. csv-table::

    "**Identification**", "**Reference**", "**Aster**", "**% difference**"
    ":math:`{K}_{\mathit{II}}`, crown #1", "2.2347E+7", "2.2814E+7", "2.09"
    ":math:`{K}_{\mathit{II}}`, crown #2", "2.2347E+7", "2.2813E+7", "2.08"
    ":math:`{K}_{\mathit{II}}`, crown #3", "2,2347E+7", "2,2814E+7", "2,09"
    ":math:`{K}_{\mathit{II}}`, crown no. 4", "2.2347E+7", "2.2814E+7", "2.09"
    ":math:`{K}_{\mathit{II}}`, crown no. 5", "2,2347E+7", "2,2817E+7", "2,10"
    ":math:`{K}_{\mathit{II}}`, crown #6", "2.2347E+7", "2.2818E+7", "2.11"
    "", "", "", ""
    ":math:`G`, crown No. 1", "2.4969E+3", "2.5984E+3", "4.07"
    ":math:`G`, crown #2", "2.4969E+3", "2.5990E+3", "4.09"
    ":math:`G`, crown #3", "2,4969E+3", "2,5992E+3", "4,10"
    ":math:`G`, crown no. 4", "2.4969E+3", "2,5993E+3", "4,10"
    ":math:`G`, crown no. 5", "2.4969E+3", "2.6013E+3", "4.18"
    ":math:`G`, crown #6", "2.4969E+3", "2.5985E+3", "4.07"



notes
---------

In the reference, the author assumes :math:`{K}_{I}\mathrm{=}0`, but he does not check it after the fact. In view of the deformations resulting from Code_Aster, the coefficient :math:`{K}_{I}` is different from zero, but it remains very low compared to :math:`{K}_{\mathit{II}}` (the crack slides more than it opens).

With regard to the energy return rate :math:`G`, if we assume that :math:`{K}_{I}\mathrm{=}0`, we derive the reference value from the formula of IRWIN in plane constraints:

:math:`{G}_{\mathit{ref}}\mathrm{=}(1\mathrm{/}E)\mathrm{\times }{K}_{\mathit{II}}^{2}`