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


Geometry and loading
-----------------------

We consider a Compact Tension specimen (:math:`\mathit{CT}`) with a thickness of :math:`25\mathit{mm}`. The geometry includes a rigid pin to which the load is applied.


.. image:: images/10000200000002110000014980051032472A5BCD.png
    :width: 4.4098in
    :height: 2.7402in


.. _RefImage_10000200000002110000014980051032472A5BCD.png:

**Figure 1**: Geometry


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

To describe the behavior of the axisymmetric specimen material (bulk material), an elastoplastic behavior law with isotropic work hardening is used (law VMIS_ISOT_TRAC).

We take: :math:`E\mathrm{=}207\mathit{GPa}` and :math:`\nu \mathrm{=}0.3` and the work hardening curve used is given below:


.. image:: images/10000200000001C100000134E1017A238A1EAC7A.png
    :width: 4.678in
    :height: 3.2083in


.. _RefImage_10000200000001C100000134E1017A238A1EAC7A.png:

**Figure 2**: Isotropic work hardening curve of solid material.

For interface elements the following parameters are used in law CZM_TRA_MIX:

:math:`{\sigma }_{c}\mathrm{=}1800\mathit{MPa}`, :math:`{G}_{c}\mathrm{=}150\mathit{MPa}\mathrm{\cdot }\mathit{mm}`, :math:`{\delta }_{e}\mathrm{=}0.01\mathit{mm}`, :math:`{\delta }_{p}\mathrm{=}0.06\mathit{mm}`, :math:`{\delta }_{c}\mathrm{=}0.117\mathit{mm}`

The resulting law is shown schematically below.


.. image:: images/100002000000016900000108F387E7EDCA4FFB9E.png
    :width: 3.7563in
    :height: 2.7472in


.. _RefImage_100002000000016900000108F387E7EDCA4FFB9E.png:

**Figure 3**: Law of behavior of interface elements.

NB: Only half of the crack is modelled thanks to the symmetry of the problem, the toughness of the material is :math:`2{G}_{c}`.

Finally, the rigid pin has elastic behavior (law ELAS) with: :math:`E\mathrm{=}1\mathrm{\times }{10}^{9}\mathit{MPa}`, :math:`\nu \mathrm{=}0.3`


Boundary conditions and loading
------------------------------------

The limit conditions imposed on the pin are as follows:


* move to :math:`X` blocked,


* imposed displacement :math:`l` in the :math:`Y` direction.

The evolution of displacement :math:`l` over time is given in the following table:

.. csv-table::

    "Time :math:`\mathrm{[}s\mathrm{]}` ", "0", "0.4"
    "Displacement :math:`l` :math:`\mathrm{[}\mathit{mm}\mathrm{]}` ", "0", "1.6"


The cohesive zone is represented by the interface elements on the test piece ligament. The boundary conditions on the interface elements are:


* displacement in :math:`X` imposed identical on both lips of the cohesive zone,


* :math:`Y` movement stuck on the lower lip.