P modeling
==============

Validation of joint :math:`\mathrm{2D}` in quadratic with the JOINT_MECA_FROT 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: 13

The elastic element is a QUAD8.

The joint element is a degenerate QUAD8 (knots combined).


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

**Mode I**

The joint is opened up to its tensile strength threshold [8] _
. :math:`U\mathrm{=}{U}_{\mathit{el}}` Then we take advantage to get to :math:`U\mathrm{=}2{U}_{\mathit{el}}`. It passes into compression zone :math:`U\mathrm{=}\mathrm{-}{U}_{\mathit{el}}\mathrm{/}3`. Finally, the joint is recharged up to its equilibrium point :math:`U\mathrm{=}0`. 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 obtained analytically (see page :ref:`6 <Solution analytique JOINT_MECA_RUPT>`).

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

The joint is solicited by sliding in :math:`\mathit{II}` mode. We then test the value of the :math:`c` membership. Then we change the normal aperture to :math:`I` mode and we slide back and forth in :math:`\mathit{II}` mode for :math:`{U}_{t}\mathrm{=}\mathrm{\pm }{U}_{\mathit{el}}\mathrm{/}3`.


+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|**Size tested**                   |**Reference**                                                    |**Tolerance (** :math:`\text{\%}` **)**|
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|**fashion** :math:`I`                                                                                                                       |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FN, peak value                    |:math:`{\sigma }_{\mathrm{max}}` is 2.D05                        |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FN, no change in peak value       |:math:`{\sigma }_{\mathrm{max}}` is 2.D05                        |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FN, compression value             |:math:`{\sigma }_{\mathit{max}}\mathrm{/}3` is -6.66666667D04    |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FN, value at the equilibrium point|0.0                                                              |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|**fashion** :math:`\mathrm{II}`                                                                                                             |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FT, membership value              |:math:`C` is 1.D05                                               |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FT, sliding value and membership  |:math:`\mathrm{-}{K}_{N}{U}_{t}\mu \mathrm{-}c` is -1.2445666D+06|0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+
|FT, sliding value and membership  |:math:`{K}_{N}{U}_{t}\mu \mathrm{-}c` is 1.244647D+06            |0.10                                   |
+----------------------------------+-----------------------------------------------------------------+---------------------------------------+

.. [1] see the analytical solution for ratings