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


Geometry
---------

We consider a square plate with side :math:`20m` in plane deformations, centered in the :math:`(X,Y)` coordinate system. The cracks are defined by the points :math:`A` to :math:`J` and :math:`A\text{'}` to :math:`J\text{'}`, whose coordinates are given in table 1. The cracks are shown in FIG. 1.


+----------+---------+---------++-----------------+---------+---------+
|**Points**|:math:`X`|:math:`Y`||**Points**       |:math:`X`|:math:`Y`|
+----------+---------+---------++-----------------+---------+---------+
|:math:`A` |−3.0851  |0.75     ||:math:`A\text{'}`|3,08512  |−0.75    |
+----------+---------+---------++-----------------+---------+---------+
|:math:`B` |0.50000  |1,13327  ||:math:`B\text{'}`|−0.5000  |−0.3667  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`C` |0.20309  |1,55730  ||:math:`C\text{'}`|−0.2031  |0.05730  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`D` |−2,1454  |1.09202  ||:math:`D\text{'}`|2,14543  |−0.4080  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`E` |−1.3794  |0.44923  ||:math:`E\text{'}`|1,37939  |−1.0508  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`F` |−3.0851  |−0.25    ||:math:`F\text{'}`|3,08512  |−1.75    |
+----------+---------+---------++-----------------+---------+---------+
|:math:`G` |−2.3780  |0.45711  ||:math:`G\text{'}`|2,37802  |−1.0429  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`H` |−1.3794  |1.09202  ||:math:`H\text{'}`|1,37939  |−0.4080  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`I` |−0.5134  |1,59202  ||:math:`I\text{'}`|0.51336  |0.09202  |
+----------+---------+---------++-----------------+---------+---------+
|:math:`J` |−0.4397  |0.79125  ||:math:`J\text{'}`|0.43969  |−0.7087  |
+----------+---------+---------++-----------------+---------+---------+

**Table 1: coordinates of the points defining the cracks.**


.. image:: images/100002010000037700000372D14D9EF2D3DCEBE7.png
    :width: 4.9807in
    :height: 4.9217in


.. _RefImage_100002010000037700000372D14D9EF2D3DCEBE7.png:

**Figure 1: position of points and cracks.**


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

The material is isotropic elastic with the following properties:


* :math:`E\mathrm{=}\mathrm{0,1}\mathit{MPa}`


* :math:`\nu =0.3`


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

Constraint :math:`\sigma \mathrm{=}(\begin{array}{cc}1& 1\\ 1& 1\end{array})` is imposed on the entire outline of the structure. This corresponds to a loading in bi-traction and unit shear. Rigid fashions are blocked.


.. image:: images/10000201000005550000054EA7B69E68B642AD45.png
    :width: 4.748in
    :height: 4.7244in


.. _RefImage_10000201000005550000054EA7B69E68B642AD45.png:

**Figure 2: loading the structure.**