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

Validation of joint :math:`\mathrm{2D}` with the JOINT_MECA_RUPT law


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

Modeling in plane deformations D_ PLAN for the elastic element.

Plane modeling for the joint element (keyword PLAN_JOINT).


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

Number of knots: 6

The elastic element is a QUAD4.

The joint element is a degenerate QUAD4 (knots combined).


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

**Fashion** :math:`I`

The joint is opened until partial damage [1] _
. :math:`U\mathrm{=}\beta \mathrm{\times }{U}_{\mathit{el}}+(1\mathrm{-}\beta )\mathrm{\times }{U}_{\mathit{max}}` Where :math:`\beta \mathrm{=}0.2`. Then it goes into compression zone :math:`U\mathrm{=}\mathrm{-}{U}_{\mathit{el}}`. Finally, the joint is put into tension until complete damage :math:`U\mathrm{=}{U}_{\mathit{max}}`. We test the global response (the resultant of the nodal force, FN) of the system (joint and cube) in the local coordinate system. The reference values are analytical (see page :ref:`6 <Solution analytique JOINT_MECA_RUPT>`).

**Fashion** :math:`\mathrm{II}`

The joint is opened in :math:`I` mode until partial damage occurs, then it is stressed in :math:`\mathit{II}` mode. Tangential stiffness slopes are tested at two values of normal apertures. The incremental nature of the tangential evolution causes the tangential stress to change during the partial discharge in :math:`I` mode, which makes its analytical estimation complex. So we test it after the discharge by regression.

We note by:

:math:`{\sigma }_{\mathit{pena}}\mathrm{=}{\sigma }_{\mathit{max}}{P}_{\mathit{cont}}\mathrm{\times }(E+{L}_{\mathit{cube}}\mathrm{\times }{K}_{N})\mathrm{/}(E+{L}_{\mathit{cube}}\mathrm{\times }{K}_{N}\mathrm{\times }{P}_{\mathit{cont}})`

:math:`{\sigma }_{t}({\delta }_{n})\mathrm{=}{K}_{T}\mathrm{\times }(1\mathrm{-}{\delta }_{n}\mathrm{/}{L}_{\mathit{CT}}){U}_{t}`


+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|**Size tested**                            |**Reference**                                                               |**Tolerance (** :math:`\text{\%}` **)**|
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|**fashion** :math:`I`                                                                                                                                           |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|FN, peak value                             |:math:`{\sigma }_{\mathit{max}}`, which is :math:`\mathrm{1.D05}`           |0.10                                   |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|FN, value damaged                          |:math:`\beta {\sigma }_{\mathit{max}}`, which is :math:`\mathrm{2.D04}`     |0.10                                   |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|FN, compression value                      |:math:`{\sigma }_{\mathrm{pena}}`, which is :math:`\mathrm{-}\mathrm{2.D05}`|0.10                                   |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|**fashion** :math:`\mathit{II}`                                                                                                                                 |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+
|FT, for two values of :math:`{\delta }_{n}`|:math:`{\sigma }_{t}({\delta }_{n})`                                        |0.10                                   |
+-------------------------------------------+----------------------------------------------------------------------------+---------------------------------------+


.. [1] see the analytical solution for ratings