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


Geometry
---------

We consider a cube with side :math:`\mathrm{2L}` and a lens-shaped crack with radius :math:`R` such as :math:`\frac{L}{R}\mathrm{=}5` and central angle :math:`\alpha \mathrm{=}\frac{\pi }{4}` (see).

The equation characteristic of the shape of the crack surface is:

:math:`{x}^{2}+{y}^{2}+{(z\mathrm{-}R)}^{2}\mathrm{=}{R}^{2}` with :math:`0\mathrm{\le }z\mathrm{\le }(1\mathrm{-}\mathrm{cos}\alpha )R`.

We pose :math:`a\mathrm{=}R\mathrm{sin}\alpha`.


The characteristic equation of the crack background is:
----------------------------------------

:math:`{x}^{2}+{y}^{2}\mathrm{=}{a}^{2}` with :math:`z\mathrm{=}(1\mathrm{-}\mathrm{cos}\alpha )R`


.. image:: images/1000000000000445000002EBAB849099C5E163DF.png
    :width: 5.6929in
    :height: 3.8898in


.. _RefImage_1000000000000445000002EBAB849099C5E163DF.png:

Figure 1: geometry of the cracked cube


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

The material is elastic isotropic in properties:

:math:`E\mathrm{=}210000\mathit{MPa}`

:math:`\mathrm{\nu }=\mathrm{0,22}`


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

The cube is subjected to hydrostatic tension :math:`\sigma`.