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


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

The various models are identical except for the slope of the crack.


.. image:: images/100000000000010C00000210CE12A53FF4DD8C32.png
    :width: 3.7217in
    :height: 7.3327in


.. _RefImage_100000000000010C00000210CE12A53FF4DD8C32.png:

**Full mesh for one angle** :math:`\beta =60°`


.. image:: images/100014E0000019BC000013D814D399948C887790.svg
    :width: 331
    :height: 238


.. _RefImage_100014E0000019BC000013D814D399948C887790.svg:

**Zoom on the crack point**

The radius is :math:`7.5E-5m`.

There are four crowns defined by command CALC_G:


.. csv-table::

    "crown 1: :math:`\mathrm{Rinf}=0.` "," :math:`R\text{sup}=1.875E-\mathrm{5m}`"
    "crown 2: :math:`\mathrm{Rinf}=1.875E-\mathrm{5m}` "," :math:`R\text{sup}=3.750E-\mathrm{5m}`"
    "crown 3: :math:`\mathrm{Rinf}=3.750E-\mathrm{5m}` "," :math:`R\text{sup}=5.625E-\mathrm{5m}`"
    "crown 4: :math:`\mathrm{Rinf}=5.625E-\mathrm{5m}` "," :math:`R\text{sup}=7.500E-\mathrm{5m}`"


The direction of propagation is defined by: :math:`\mathrm{cos}\theta ,\mathrm{sin}\theta`


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

The mesh consists of 10676 nodes and 4584 elements, including 1392 QUA8 elements and 3168 TRI6 elements.


Features tested
------------------------

The calculation of :math:`{K}_{I}` and :math:`{K}_{\mathrm{II}}` is not valid for a bimaterial: option K cannot be used and only the calculation of the energy return rate is possible.


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

Tested values
~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Aster**", "**% difference**"
    "Left end, :math:`\theta =15°` ", "", "", ""
    ":math:`G`, crown 1", "9.67362E+1", "9.2428E+1", "4.45"
    ":math:`G`, crown 2", "9.67362E+1", "9.6392E+1", "0.356"
    ":math:`G`, crown 3", "9.67362E+1", "9.6417E+1", "0.330"
    ":math:`G`, crown 4", "9.67362E+1", "9.6421E+1", "0.326"
    ":math:`{K}_{I}` ", "5,6694E+6", "-", "-"
    ":math:`{K}_{\mathrm{II}}` ", "2,4852E+6", "-", "-"
    "Right end, :math:`\theta =15°` ", "", "", ""
    ":math:`G`, crown 1", "1.0125E+2", "9.6763E+1", "4.33"
    ":math:`G`, crown 2", "1.0125E+2", "1.0093E+2", "0.315"
    ":math:`G`, crown 3", "1.0125E+2", "1.0095E+2", "0.295"
    ":math:`G`, crown 4", "1.0125E+2", "1.0095E+2", "0.291"
    ":math:`{K}_{I}` ", "6,3145E+6", "-", "-"
    ":math:`{K}_{\mathrm{II}}` ", "4,8309E+5", "-", "-"



notes
~~~~~~~~

To obtain the :math:`G` on the bottom of the crack, we calculate the energy release rate using the relationship between :math:`G` and the :math:`{K}_{j}` [:ref:`bib3 <bib3>`]:

:math:`\begin{array}{c}{\kappa }_{1}\mathrm{=}{\kappa }_{2}\mathrm{=}2.076923\\ {\mu }_{1}\mathrm{=}7.6923E+11\\ {\mu }_{2}\mathrm{=}7.6923E+10\\ \alpha \mathrm{=}\mathrm{-}9.37742E\mathrm{-}2\\ \beta \mathrm{=}2.524488E\mathrm{-}12\\ G\mathrm{=}\beta ({K}_{I}^{2}+{K}_{\mathit{II}}^{2})\end{array}`