G modeling
==============


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

The geometry and the load are identical to the F model: a quarter of the structure is modeled, pressure is applied on the upper side. In this modeling, the initial crack is meshed. Compared to the F model, cohesive zones are introduced in the extension of the crack. This extension is represented by level-sets, so that the discontinuity is taken into account by modeling XFEM, as for modeling F. The cohesive law CZM_LIN_MIX is introduced into this model XFEM by the command DEFI_CONTACT.

**Loading:** Unit pressure distributed on the face of the block opposite the plane of the lip:

:math:`P=1\mathrm{MPa}` in the :math:`Z=1250\mathrm{mm}` plan.

The cohesive parameters are chosen so that this has the effect of opening a few cohesive elements in the vicinity of the initial crack bottom:


*
    * To not have a complete rupture, but simply a de-cohesion close to the initial crack point, we take :math:`{G}_{c}>{G}_{\mathit{max}}`, with :math:`{G}_{\mathit{max}}` the maximum local :math:`G` for the long front, while maintaining the same order of magnitude for both values. In our case, :math:`{G}_{c}=2.5\times {10}^{-4}{\mathit{N.mm}}^{-1}` versus :math:`{G}_{\mathit{max}}=7.2\times {10}^{-5}{\mathit{N.mm}}^{-1}`.


    * To still observe de-cohesion in the vicinity of the tip, the characteristic size of the cohesive zone :math:`{l}_{c}=\frac{E{G}_{c}}{(1-{\nu }^{2}){\sigma }_{c}^{2}}` is chosen so as to cover a few elements while remaining small compared to the size of the structure :math:`h\le {l}_{c}\le a`. In this test case, to reduce the calculation time, we took :math:`{l}_{c}=14\mathit{mm}`, which leads to :math:`{\sigma }_{c}=2\mathit{MPa}`, to be compared with typical element sizes :math:`h=1\mathit{mm}` on the short side of the ellipse, and :math:`h=2\mathit{mm}` on the long side.


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

Number of knots: 4522

Number of meshes and types: 22300 TETRA4


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

The values tested are the :math:`\mathrm{K1}` equivalent stress intensity factors along the crack bottom, calculated by CALC_G. In order to obtain a regular result, only 5 points distributed uniformly along the crack base are post-treated (NB_POINT_FOND =5). We test the values at the end points of the front :math:`A` (:math:`s\mathrm{=}0`) and :math:`B` (:math:`s\mathrm{=}\mathrm{26,7}`).

The smoothing 'LAGRANGE' is used for :math:`\theta` and 'LAGRANGE_NO_NO' for :math:`G`.


.. csv-table::

    "**Identification**", "**Reference type**", "**Reference value**", "**Tolerance**"
    "**CALC_G_** XFEM: :math:`\mathit{K1}(A)` ", "'ANALYTIQUE'", "0.000"," 4.0%"
    "**CALC_G_** **XFEM** ****: **:** :math:`\mathrm{K1}(B)` ", "'ANALYTIQUE'", "4.068"," 8.0%"



note
--------

For this test, there is a discrepancy of a few percent. Precision can be improved by choosing a smaller cohesive zone and further refining the mesh. This step was not done here so that the modeling could be completed in less than a minute.