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


Geometry
---------


.. image:: images/100000000000039700000253F0A9E0A15525FFD5.png
    :width: 5.3154in
    :height: 3.4409in


.. _RefImage_100000000000039700000253F0A9E0A15525FFD5.png:

**Figure 1.1 Problem geometry**

Length: :math:`L=20\mathit{cm}`

Width: :math:`l=20\mathit{cm}`

:math:`\mathit{AC}=\mathit{CD}`


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

Material for half plane 1:


.. csv-table::

    "Young's module", ":math:`E=2\times {10}^{12}\mathit{Pa}`"
    "Poisson's Ratio", ":math:`\nu =0.3`"


Material for half plane 2:


.. csv-table::

    "Young's Module", "A, D Modeling: :math:`E=8\times {10}^{12}\mathit{Pa}`
    B, E modeling: :math:`E=2\times {10}^{12}\mathit{Pa}`
    C, F modeling: :math:`E=5\times {10}^{11}\mathit{Pa}`"
    "Poisson's Ratio", ":math:`\nu =0.3`"



Boundary conditions and loads
-------------------------------------

For models A, B and C:

Imposed displacement:

.. csv-table::

    "Embedding sides :math:`\mathit{AB}`, :math:`\mathit{BC}` "," :math:`\mathit{DX}=0`"
    "Embedding point B:", ":math:`\mathit{DY}=0`"
    "Uniform connection on the side: :math:`\mathit{DE}` "," :math:`{U}_{X}({N}_{i})={U}_{X}({N}_{1})`"
    "Uniform connection on the side: :math:`\mathit{EF}` "," :math:`{U}_{X}({N}_{i})={U}_{X}({N}_{1})`"


For D, E, and F models:

Imposed displacement:

.. csv-table::

    "Embedding point B:", ":math:`\mathit{DX}=0`, :math:`\mathit{DY}=0`"
    "Embedding point E:", ":math:`\mathit{DY}=0`"


For A, B, C, D, E, and F models:

Imposed loading:

.. csv-table::

    "Force contour on the side :math:`\mathit{CD}` "," :math:`{F}_{X}=-75\times {10}^{6}N`"
    "Force contour on the side :math:`\mathit{FA}` "," :math:`{F}_{X}=75\times {10}^{6}N`"



Crack size
-------------------

:math:`c`: characteristic length of the horizontal crack.

:math:`a`: characteristic length of the deflected crack a :math:`45°`.


.. csv-table::

    "", ":math:`c` "," :math:`a`"
    "A, B, and C modeling", ":math:`\mathrm{0,02}` "," :math:`\mathrm{0,02}`"
    "D, E, and F modeling", ":math:`\mathrm{0,02}` "," :math:`\mathrm{0,002}`"