C modeling
==============

In this modeling, the extended finite element method (:math:`\text{X-FEM}`) is used. A radius of geometric enrichment is defined with a number of element layers equal to 3.


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

The domain is meshed with linear triangles (mesh TRIA3). We maintain the refinement of the previous models, namely 100 quadrangles (divided into 2 triangles) along the :math:`X` axis and 100 quadrangles (divided into 2 triangles) along the :math:`Y` axis. The crack is not meshed.


.. image:: images/10000000000004590000040C9E5F3D85B588CA85.png
    :width: 6.889in
    :height: 6.4126in


.. _RefImage_10000000000004590000040C9E5F3D85B588CA85.png:

**Figure** 4.1-1 **: mesh with triangles**


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

The crack is inclined according to 3 angular values: :math:`\theta \mathrm{=}0°,30°,60°`

For each angle of inclination, the stress intensity factors are tested as in models :math:`A` and :math:`B`, by the :math:`G\mathrm{-}\mathit{thêta}` method and by the method of extrapolation of displacement jumps.

For method :math:`G-\mathrm{thêta}` (command CALC_G), the following theta field crowns are chosen: :math:`{R}_{\mathrm{inf}}=\mathrm{0,1}a` and :math:`{R}_{\text{sup}}=\mathrm{0,3}a`.


Results for :math:`\theta \mathrm{=}0°`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "CALC_G ", "", ""
    "C1 + background1: K1", "2.5725 105"," 2.0%"
    "C1 + background2: K1", "2.5725 105"," 2.0%"
    "C1 + background1: K2", "0", "257"
    "C1 + background2: K2", "0", "257"
    "C1 + background1: G", "0.29"," 2.0%"
    "C1 + background2: G", "0.29"," 2.0%"
    "C2 + background1: K1", "2.5725 105"," 2.0%"
    "C2 + background2: K1", "2.5725 105"," 2.0%"
    "C2 + background1: K2", "0", "257"
    "C2 + background2: K2", "0", "257"
    "C2 + background1: G", "0.29"," 2.0%"
    "C2 + background2: G", "0.29"," 2.0%"
    "POST_K1_K2_K3 ", "", ""
    "C1 + background1: K1", "2.5725 105"," 2.0%"
    "C1 + background2: K1", "2.5725 105"," 2.0%"
    "C1 + background1: K2", "0", "514.5"
    "C1 + background2: K2", "0", "514.5"
    "C2 + background1: K1", "2.5725 105"," 2.0%"
    "C2 + background2: K1", "2.5725 105"," 2.0%"
    "C2 + background1: K2", "0", "514.5"
    "C2 + background2: K2", "0", "514.5"


The zero values of :math:`{K}_{2}` are tested in absolute terms with a tolerance equal to :math:`{K}_{1}^{\mathit{ref}}\mathrm{/}1000` for CALC_G and a tolerance equal to :math:`{K}_{1}^{\mathit{ref}}\mathrm{/}500` for POST_K1_K2_K3.


Results for :math:`\theta \mathrm{=}30°`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "CALC_G ", "", ""
    "C1 + background1: K1", "1.9294 105"," 2.0%"
    "C1 + background2: K1", "1.9294 105"," 2.0%"
    "C1 + background1: K2", "1.1139.105"," 3.0%"
    "C1 + background2: K2", "1.1139.105"," 3.0%"
    "C1 + background1: G", "0.215"," 2.0%"
    "C1 + background2: G", "0.215"," 2.0%"
    "C2 + background1: K1", "1.9294 105"," 2.0%"
    "C2 + background2: K1", "1.9294 105"," 2.0%"
    "C2 + background1: K2", "1.1139.105"," 3.0%"
    "C2 + background2: K2", "1.1139.105"," 3.0%"
    "C2 + background1: G", "0.215"," 2.0%"
    "C2 + background2: G", "0.215"," 2.0%"
    "POST_K1_K2_K3 ", "", ""
    "C1 + background1: K1", "1.9294 105"," 2.0%"
    "C1 + background2: K1", "1.9294 105"," 2.0%"
    "C1 + background1: K2", "1.1139.105"," 3.0%"
    "C1 + background2: K2", "1.1139.105"," 3.0%"
    "C2 + background1: K1", "1.9294 105"," 2.0%"
    "C2 + background2: K1", "1.9294 105"," 2.0%"
    "C2 + background1: K2", "1.1139.105"," 3.0%"
    "C2 + background2: K2", "1.1139.105"," 3.0%"



Results for :math:`\theta \mathrm{=}60°`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Tolerance**"
    "CALC_G ", "", ""
    "C1 + background1: K1", "6.4312105"," 2.0%"
    "C1 + background2: K1", "6.4312105"," 2.0%"
    "C1 + background1: K2", "1.1139.105"," 3.0%"
    "C1 + background2: K2", "1.1139.105"," 3.0%"
    "C1 + background1: G", "7,1692 10-2"," 2.0%"
    "C1 + background2: G", "7,1692 10-2"," 2.0%"
    "C2 + background1: K1", "6.4312105"," 2.0%"
    "C2 + background2: K1", "6.4312105"," 2.0%"
    "C2 + background1: K2", "1.1139.105"," 3.0%"
    "C2 + background2: K2", "1.1139.105"," 3.0%"
    "C2 + background1: G", "7,1692 10-2"," 2.0%"
    "C2 + background2: G", "7,1692 10-2"," 2.0%"
    "POST_K1_K2_K3 ", "", ""
    "C1 + background1: K1", "6.4312105"," 2.0%"
    "C1 + background2: K1", "6.4312105"," 2.0%"
    "C1 + background1: K2", "1.1139.105"," 3.0%"
    "C1 + background2: K2", "1.1139.105"," 3.0%"
    "C2 + background1: K1", "6.4312105"," 2.0%"
    "C2 + background2: K1", "6.4312105"," 2.0%"
    "C2 + background1: K2", "1.1139.105"," 3.0%"
    "C2 + background2: K2", "1.1139.105"," 3.0%"