B modeling
==============


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

The simulation is carried out axisymmetrically. The interface elements allow the crack to be represented along :math:`{\Gamma }_{i}`. Their modeling is AXIS_JOINT and their cohesive behavior is CZM_LIN_REG. The other elements of the mesh are QUAD4 with ELAS elastic behavior in AXIS modeling.


Material parameters
-------------------

The values of the Young's modulus, the Poisson's ratio, the critical stress and the toughness of the material are taken as follows:


.. image:: images/Object_38.svg
    :width: 360
    :height: 25


.. _RefImage_Object_38.svg:

Moreover, the parameter for regularizing the cohesive law is taken to be equal to PENA_ADHERENCE = 0.00001. (NB: these are "test" values that do not correspond to any particular material.)


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

The mesh is identical to the previous one except that the layer of cohesive elements is composed of elements with a small thickness, oriented with the command ORIE_FISSURE.


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

To test the numerical solution, we use the equation [:ref:`éq 2-1 <éq 2-1>`]. We note

.. image:: images/Object_39.svg
    :width: 360
    :height: 25


.. _RefImage_Object_39.svg:

the resultant of the force along :math:`{\Gamma }_{i}` multiplied by :math:`2\pi`.


+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|**Size tested**                                                                                 |**Theory**  |**Aster_code**     |**Difference (** :math:`\text{\%}` **)**|
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |2.298338E-01|2.2983379490657E-01|-2.22E-06                               |
+ .. image:: images/Object_42.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 3                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |1.049985E+01|1.0499850000001E+01|1.05E-11                                |
+ .. image:: images/Object_43.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 3                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |3.884798E-01|3.8847977822122E-01|-5.61E-06                               |
+ .. image:: images/Object_44.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 6                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |5.99982E+00 |5.9998200000014E+00|2.27E-11                                |
+ .. image:: images/Object_45.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 6                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |5.2809E-01  |5.2809010497189E-01|1.99E-05                                |
+ .. image:: images/Object_46.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 8                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+
|                                                                                                |2.03974E+00 |2.0397408000020E+00|3.92E-05                                |
+ .. image:: images/Object_47.svg                                                                +            +                   +                                        +
|    :width: 360                                                                                 |            |                   |                                        |
+    :height: 25                                                                                 +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+ right now: 8                                                                                   +            +                   +                                        +
|                                                                                                |            |                   |                                        |
+------------------------------------------------------------------------------------------------+------------+-------------------+----------------------------------------+