Reference problem
=====================


Geometry
---------


.. image:: images/10000000000004950000020912BE22F2090EFF97.png
    :width: 5.9055in
    :height: 2.6138in


.. _RefImage_10000000000004950000020912BE22F2090EFF97.png:

**Figure 1**: Representation of the system of two elements in plane :math:`(X,Y)`. We choose :math:`L\mathrm{=}1\mathit{mm}`.


Geometry: case X- FEM
~~~~~~~~~~~~~~~~~~~~~~~

In the models in formulation XFEM, there is no longer a cohesive joint or interface element in the model, but the cohesive law is defined on the interface using the DEFI_CONTACT command, as one would do for a contact law. Consequently, the square is meshed with a few elements and the line of discontinuity is introduced in the middle of the square, in an irregular manner: it intersects elements that are elastic in this case.


.. image:: images/10000000000001EC00000104390F31D39D165C18.png
    :width: 4.478in
    :height: 2.3665in


.. _RefImage_10000000000001EC00000104390F31D39D165C18.png:

**Figure 2**: Modeling X- FEM in plane :math:`(X,Y)`. We choose :math:`L=1\mathrm{mm}`.


Material properties
----------------------


Cohesive laws
~~~~~~~~~~~~~~~


Material RUPT_FRAG
^^^^^^^^^^^^^^^^^^^

**Cube:* elastic

Young's module: :math:`E=0.5\mathrm{MPa}` (except for J, K and L models where :math:`E=100\mathrm{MPa}` is taken, this choice is purely practical)

Poisson's Ratio :math:`\nu =0`

**Joint element**: laws CZM_EXP_REG, CZM_LIN_REG

**Interface element**: laws CZM_OUV_MIX, CZM_EXP_MIX,,, CZM_TAC_MIX, CZM_FAT_MIX

**Mixed law for linear finite elements**: law CZM_LIN_MIX


+------------------------------------------------+---------------------------------------------+-------------------------+
|Critical surface energy density:                |:math:`{G}_{c}=0.9N/\mathrm{mm}`             |(keyword: GC)            |
+------------------------------------------------+---------------------------------------------+-------------------------+
|                                                |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|critical stress:                                |:math:`{\sigma }_{c}=1.1\mathrm{MPa}`        |(keyword: SIGM_C)        |
+------------------------------------------------+---------------------------------------------+-------------------------+
|                                                |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|**Joint element:**                              |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|penalization of adherence                       |PENA_ADHERENCE = :math:`{10}^{-3}\mathrm{mm}`|(keyword: PENA_ADHERENCE)|
+------------------------------------------------+---------------------------------------------+-------------------------+
|(small energy regulation parameter at 0, see [])                                                                        |
+------------------------------------------------+---------------------------------------------+-------------------------+
|criminalization of contact                      |PENA_CONTACT = 1 (by default)                |(keyword: PENA_CONTACT)  |
+------------------------------------------------+---------------------------------------------+-------------------------+
|                                                |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|**Interface element:**                          |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|Penalization of Lagrangian                      |PENA_LAGR =100 (by default)                  |(keyword: PENA_LAGR)     |
+------------------------------------------------+---------------------------------------------+-------------------------+
|Slip stiffness                                  |RIGI_GLIS =10 (by default)                   |(keyword: RIGI_GLIS)     |
+------------------------------------------------+---------------------------------------------+-------------------------+
|                                                |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|**Mixed law for linear finite elements:**       |                                             |                         |
+------------------------------------------------+---------------------------------------------+-------------------------+
|Penalization of Lagrangian                      |PENA_LAGR =10                                |(keyword: PENA_LAGR)     |
+------------------------------------------------+---------------------------------------------+-------------------------+

NB: Material data is of course not intended to represent a particular material. They are only intended for numerical validation tests.


Material RUPT_DUCT
^^^^^^^^^^^^^^^^^^^

**Cube:* elastic

Young's module: :math:`E={10}^{6}\mathrm{MPa}` (this choice is purely practical)

Poisson's Ratio :math:`\nu =0`

**Interface element**: law CZM_TRA_MIX


.. csv-table::

    "Critical surface energy density:", ":math:`{G}_{c}=0.9N/\mathrm{mm}` ", "(keyword: GC)"
    "critical constraint:", ":math:`{\sigma }_{c}=9\mathrm{MPa}` ", "(keyword: SIGM_C)"
    "Extrinsic form coefficient", ":math:`0.0625` ", "(keyword: COEF_EXTR)"
    "Plastic tray form factor", ":math:`0.3125` ", "(key word: COEF_PLAS)"
    "Lagrangian penalty", "PENA_LAGR =100 (by default)", "(keyword: PENA_LAGR)"
    "Slip stiffness", "RIGI_GLIS =10 (by default)", "(keyword: RIGI_GLIS)"


NB: Material data is of course not intended to represent a particular material. They are only intended for numerical validation tests.


.. _RefHeading__31154616:


Boundary conditions and loads
-------------------------------------

**Embedment**: The imposed displacements are zero on the face of the cohesive element opposite to the elastic element.

**In mode** :math:`I`: An imposed displacement

.. image:: images/Object_4.svg
    :width: 17
    :height: 18


.. _RefImage_Object_4.svg:

is applied to the face of the elastic element opposite the joint (see FIG. 1).

.. csv-table::

    "DX= 2.16506351", "DY= 1.250", "DZ= 0"


**In mode** :math:`\mathrm{II}`: Imposed displacement

.. image:: images/Object_5.svg
    :width: 17
    :height: 18


.. _RefImage_Object_5.svg:

is applied to all the nodes of the solid element.


.. csv-table::

    "DX= -1.250", "DY= 2.16506350946110", "DZ= 0"


**In mode** :math:`\mathrm{III}`: Imposed displacement

.. image:: images/Object_6.svg
    :width: 17
    :height: 18


.. _RefImage_Object_6.svg:

is applied to all the nodes of the solid element.


.. csv-table::

    "DX= 0.0", "DY= 0.0", "DZ= 2.5"


For laws CZM_OUV_MIX, CZM_EXP_MIX and CZM_TAC_MIX, the same vectors normalized to 1 are used.

For fatigue law CZM_FAT_MIX (in :math:`I` mode only) the same vectors normalized to :math:`\mathrm{0,094}` are used. This value corresponds to the magnitude of the load because this one is multiplied by a cyclic function, zero at zero, which is equal to 1 at odd times and 0 at even times.

For the ductile law CZM_TRA_MIX (in :math:`I` mode only) the same vectors normalized to 1 are used. Cyclic loading is applied to test all the states of the law but also monotonic loading. The tests are carried out with the first load.


Loading: case X- FEM
~~~~~~~~~~~~~~~~~~~~~~~~~

**Embed**: The movements imposed are zero on the left side of the square.

**In mode** :math:`I`: An imposed displacement

.. image:: images/Object_316.svg
    :width: 17
    :height: 18


.. _RefImage_Object_316.svg:

is applied to the right side of the square (see figure 2).

.. csv-table::

    "DX= 2.16506351", "DY= 1.250", "DZ= 0"