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


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

A notched axisymmetric specimen of type :math:`\mathrm{AE4}` is considered. The cohesive zone represented by interface elements (modeling A) or joint elements (modeling B) is positioned on line :math:`\mathrm{AB}`.


.. image:: images/10000200000001200000024E7BC4E8374B2298D7.png
    :width: 2.9957in
    :height: 6.1311in


.. _RefImage_10000200000001200000024E7BC4E8374B2298D7.png:

**Figure 1**: Geometry of specimen :math:`\mathrm{AE4}`.


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=207\mathit{GPa}` and :math:`\nu =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.

The crack is represented with two different laws for models A and B:

**Modeling A:**

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

:math:`{\sigma }_{c}=1200\mathit{MPa}`, :math:`{G}_{c}=130\mathit{MPa}\cdot \mathit{mm}`, :math:`{\delta }_{e}=0.01\mathit{mm}`, :math:`{\delta }_{p}=0.07\mathit{mm}`, :math:`{\delta }_{c}=0.157\mathit{mm}`,

The resulting law is shown schematically below.


.. image:: images/10000200000001600000010008F7320B67E65E25.png
    :width: 3.6689in
    :height: 2.6689in


.. _RefImage_10000200000001600000010008F7320B67E65E25.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:`{\mathrm{2G}}_{c}`.

**Modeling B:**

For joint elements the following parameters are used in law CZM_LIN_REG:

:math:`{\sigma }_{c}=1200\mathit{MPa}`, :math:`{G}_{c}=130\mathit{MPa}\cdot \mathit{mm}`, :math:`\mathit{pena}\mathit{adherence}=1.E-5`

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


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

With reference to FIG. 1, the boundary conditions are as follows:


* trip to :math:`X` blocked on line :math:`\mathrm{AD}`,


* imposed displacement :math:`l` in the direction :math:`Y` on line :math:`\mathrm{DC}`.

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

.. csv-table::

    "Time :math:`\mathrm{[}s\mathrm{]}` ", "0", "1"
    "Displacement :math:`l` :math:`\mathrm{[}\mathit{mm}\mathrm{]}` ", "0", "0.5"


The cohesive zone is represented by the interface elements on line :math:`\mathrm{AB}`. The upper lip of the interface elements is called :math:`\mathrm{AB}` and the lower lip is called :math:`A\text{'}B\text{'}`. The boundary conditions on the interface elements are:


* move to :math:`X` imposed on the lips :math:`\mathrm{AB}` and :math:`A\text{'}B\text{'}`: :math:`{\mathrm{DX}}_{\mathrm{AB}}={\mathrm{DX}}_{A\text{'}B\text{'}}`


* trip to :math:`Y` blocked on line :math:`A\text{'}B\text{'}`.