T modeling
==============

Validation of the implementation of law CZM_LIN_MIX in formulation X- FEM. We are testing the opening mode. This modeling is an adaptation to X- FEM of the :math:`G` modeling.


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

The line of discontinuity is modelled by an X- FEM interface, which is introduced into the model by the DEFI_FISS_XFEM operator, with TYPE_DISCONTINUITE =' INTERFACE '. This line crosses the block from one end to the other. It is located at a distance of :math:`\mathrm{0,4}\mathit{mm}` from the left edge and crosses the elements (see fig.2). The interface is said to be non-compliant.

The contact elements are introduced by the discretization CONTACT =' MORTAR 'in the operator MODI_MODELE_XFEM.

The cohesive law is then defined in the operator DEFI_CONTACT, by the keywords ALGO_CONT =' CZM 'and RELATION =' CZM_LIN_MIX'.

The surface elements are of type C_ PLAN.


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

The square is discretized at the rate of :math:`4` elements per side. Therefore:

Number of elements, type HEXA4: 16

Number of knots: 65.


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

The remark on control mentioned for modeling A also applies to this modeling. Strictly speaking, no longer having any interface elements in the model, we replace the tests on SIGN and SIGTX by tests on the contact multipliers X- FEM LAGS_C and LAGS_F1 respectively.

**Fashion** :math:`I`


.. csv-table::

    "**Size tested**", "**No time**", "**Reference**", "**Tolerance (** :math:`\text{\%}` **)**"
    "DX on node 2", "4", "1.71003596", "0.10"
    "ETA_PILO ", "4", "7.898168291D-01", "0.10"
    "SIXXsur mesh 32", "2", "6.599934D-01", "0.10"
    "LAGS_Csur node 9", "9", "1.099989D-01", "0.10"
    "LAGS_F1sur node 9", "8", "0.D+00", "0.10"


These values are obtained according to the explanations in part :ref:`2.2 <RefNumPara__4776_1148470081>`.

The move jump at time :math:`t` is: :math:`⟦u⟧=\frac{t}{\text{COEF\_MULT}}\frac{2{G}_{c}}{{\sigma }_{c}}`. So, :math:`⟦u⟧=0.163636t` (in :math:`\mathit{mm}`).

We can then obtain the displacement standard applied by :math:`U=⟦u⟧+L\frac{{\sigma }_{c}}{E}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)`. For :math:`t=4`, we get :math:`U=1.97\mathit{mm}`, from where :math:`\text{DX}=U\mathrm{cos}(\frac{\pi }{6})=1.71\mathit{mm}` and :math:`\text{ETA\_PILO}=\frac{U}{{U}_{0}}=0.78\mathit{mm}`

Starting with :math:`⟦u⟧`, we get the constraint by :math:`\sigma ={\sigma }_{c}\left(1-⟦u⟧\frac{{\sigma }_{c}}{2{G}_{c}}\right)`. For :math:`t=9`, :math:`\sigma =0.11\mathit{Mpa}`, which, considering the loading in pure I mode, can be tested by the lagrange multiplier LAGS_C. For :math:`t=2`, we find :math:`\sigma =0.88\mathit{Mpa}`, and we have :math:`{\sigma }_{\mathit{xx}}=\sigma {\mathrm{cos}}^{2}(\frac{\pi }{6})=0.66\mathit{MPa}`, which is the tested value.