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


Geometry
---------

We consider a plate with a height :math:`h=\mathrm{0,1}m`, a width :math:`l=\mathrm{0,05}m` and a thickness :math:`e=0.005m`. A crack is positioned in the middle of the height of the beam, with a depth of :math:`\mathrm{0,1}l` *.*


+----------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|Fz y x                                                                                              |                                                                                             |
+ .. image:: images/Object_1.png                                                                     + .. image:: images/Object_2.png                                                              +
|    :width: 1.6772in                                                                                |    :width: 2.8965in                                                                         |
+    :height: 2.3646in                                                                               +    :height: 2.3984in                                                                        +
|                                                                                                    |                                                                                             |
+                                                                                                    +                                                                                             +
|                                                                                                    |                                                                                             |
+----------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------+



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

We consider the classical properties of a steel:


.. csv-table::

    "Young's module:", ":math:`E={2.10}^{5}\mathrm{MPa}`"
    "Poisson's ratio:", ":math:`\nu =0.3`"
    "Density", ":math:`\rho =7800\mathrm{kg}/{m}^{3}`"



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

The plate is:


1. embedded on surface :math:`{S}_{\mathrm{1 }}`;


2. subjected to a :math:`F(t)` force on the :math:`{S}_{2}` surface.

The evolution of the :math:`F(t)` standard is shown in the figure above. We take :math:`\tau =\mathrm{0,001}s`. The direction of force :math:`F(t)` is as follows:


1. :math:`F(t)=f(t)\mathrm{.}{e}_{x}` for modeling A;


2. :math:`F(t)=(a{e}_{x}+b{e}_{y}+c{e}_{z})f(t)` for B modeling, with :math:`b=\mathrm{2a}` and :math:`c=0.4a`.

For modeling A, we block the movements in the :math:`z` direction (plane problem).