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


Geometry
---------

Consider a square with side :math:`1m` crossed by a horizontal crack:


.. image:: images/1000000000000208000001B68A88A403A1F8B567.png
    :width: 3.9465in
    :height: 3.3244in


.. _RefImage_1000000000000208000001B68A88A403A1F8B567.png:

The crack is defined by the normal "level set" :math:`{l}_{n}(Y)=Y-0.5` and the tangent :math:`{l}_{t}(X)=-X+0.5`.


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

The material is isotropic elastic whose properties are:


* :math:`E=100000\mathrm{MPa}`


* :math:`\nu =0.0`


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


Dirichlet boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

*For the A* modeling, Dirichlet boundary conditions are imposed along the entire contour of the square corresponding to the asymptotic opening mode in I :ref:`[1] <RefNumPara__1250_178636250>` mode. We then have:

:math:`{u}_{X}=\frac{1}{2\mu }\sqrt{\frac{r}{2\pi }}{K}_{I}\mathrm{cos}\frac{\theta }{2}(\kappa -\mathrm{cos}\theta )` and :math:`{u}_{Y}=\frac{1}{2\mu }\sqrt{\frac{r}{2\pi }}{K}_{I}\mathrm{sin}\frac{\theta }{2}(\kappa -\mathrm{cos}\theta )`,

with:

:math:`\mu =\frac{E}{2(1+\nu )}` and :math:`\kappa =3-4\nu` in plane constraints,

:math:`r` and :math:`\theta` are the polar coordinates as defined in the following figure:


.. image:: images/10000000000001B90000019B565D5099C07B41FD.png
    :width: 2.5083in
    :height: 2.3374in


.. _RefImage_10000000000001B90000019B565D5099C07B41FD.png:

With :math:`\nu =0.0` and :math:`{K}_{I}=1`, we finally impose:

:math:`{u}_{X}=\frac{1}{E}\sqrt{\frac{r}{2\pi }}\mathrm{cos}\frac{\theta }{2}(3-\mathrm{cos}\theta )` and :math:`{u}_{Y}=\frac{1}{E}\sqrt{\frac{r}{2\pi }}{K}_{I}\mathrm{sin}\frac{\theta }{2}(3-\mathrm{cos}\theta )`.


Neumann boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

*For modelling* *B*, Neumann boundary conditions are imposed. A uniform compression :math:`P=-20\mathit{MPa}` is imposed on the right edge while the left edge is embedded.


.. image:: images/10000000000001F0000001AC6D1621B47E7B64E9.png
    :width: 2.7492in
    :height: 2.372in


.. _RefImage_10000000000001F0000001AC6D1621B47E7B64E9.png: