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


Geometry
---------


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

**C**

**D**

**B**

**A**

.. image:: images/Object_1.png
    :width: 4.1909in
    :height: 1.7772in


.. _RefImage_Object_1.png:


.. csv-table::

    "Length", ":math:`L=50\mathrm{mm}`"
    "Width", ":math:`W=16\mathrm{mm}`"
    "Crack depth", ":math:`a=6\mathrm{mm}`"



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

The material is elastoplastic of the Von Mises type. For models A to D, there is no work hardening. For the E-modeling, various work-hardening laws are compared (linear or power hardening). The properties of the material are as follows:


.. csv-table::

    "Young's module", ":math:`E=\mathrm{2,0601}{10}^{5}\mathrm{MPa}`"
    "Poisson's Ratio", ":math:`\nu =0.3`"
    "Elastic limit", ":math:`{\sigma }_{y}=\mathrm{808,34}\mathrm{MPa}`"
    "Work hardening module", ":math:`H=0` (modeling A to D) or :math:`H=\mathrm{6,867}{10}^{4}\mathrm{MPa}` (modeling E)"



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

For models A, B, and E, the model is limited to half of the structure, the plane of the vertical crack being a plane of symmetry. For the C and D models, the entire structure is represented.

**Boundary conditions**

Vertical displacement :math:`\mathrm{UX}=0` at point :math:`B`

Horizontal displacement :math:`\mathrm{UY}=0` in ligament :math:`\mathrm{AB}` (symmetry condition for models A, B and E)

**Loading**

Horizontal displacement imposed on segment :math:`\mathrm{CD}`: :math:`\mathrm{UY}=\delta`