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


Geometry
---------

In the Cartesian coordinate system :math:`(x,y)`, consider an elastic rectangular flat plate denoted by :math:`\Omega =]\mathrm{0,}L[\times ]\mathrm{0,}H[` (see [:ref:`Figure 1.1-a <Figure 1.1-a>`]). Note :math:`{\Gamma }_{0}\mathrm{=}\left\{0\right\}\mathrm{\times }\mathrm{]}\mathrm{0,}H\mathrm{[}` the left side of the domain and :math:`\mathrm{\partial }\Omega \mathrm{\setminus }{\Gamma }_{0}` the complementary part of the edge.


.. image:: images/1000000000000114000001621B239BC3370174F8.png
    :width: 2.1398in
    :height: 2.8091in


.. _RefImage_1000000000000114000001621B239BC3370174F8.png:

**Figure** 1.1-a **.1-a: Plate scheme**

Domain :math:`\Omega` dimensions:

:math:`L=\mathrm{1mm},H=2\pi \text{mm}`


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

The material is elastic with a critical stress and a tenacity chosen arbitrarily:

:math:`E=10\text{MPa},\nu =\mathrm{0,}{\sigma }_{c}=1.1\text{MPa},{G}_{c}=0.9{\text{N.mm}}^{-1}`


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

The boundary conditions are determined by the analytical solution presented in the next part in such a way that they lead to a crack with a non-constant jump along :math:`{\Gamma }_{0}`. The loading corresponds to a displacement imposed on the edges of the plate: (see [:ref:`Figure1.3-a <Figure1.3-a>`]).


.. csv-table::

    ":math:`u=U(x,y)` ", "on :math:`\Omega /{\Gamma }_{0}`"
    ":math:`u={U}_{0}\delta (y)-(y)` ", "on :math:`{\Gamma }_{0}`"



.. image:: images/10000000000001C60000016FA6F563CD4C6800DD.png
    :width: 3.5193in
    :height: 2.9126in


.. _RefImage_10000000000001C60000016FA6F563CD4C6800DD.png:

**Figure 1.3-a: Load Diagram**

The values :math:`U,{U}_{0}` and :math:`\delta` are set when constructing the reference solution in the next part.