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


Geometry
---------

The crack is circular (penny shaped crack) with radius :math:`a`, in the :math:`\mathrm{Oxy}` plane. For the medium to be considered infinite, the characteristic quantities of the massif are of the order of 5 times greater than radius :math:`a`.


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

Young's module: :math:`E={2.10}^{5}\mathrm{MPa}`

Poisson's ratio: :math:`\nu =0.3`

Density: :math:`\rho =7850\mathrm{kg}/\mathrm{m³}`


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


.. csv-table::

    "Underside", ": uniform tensile stress :math:`{\sigma }_{z}\mathrm{=}1.\mathit{MPa}`"
    "Upper face", ": uniform tensile stress :math:`{\sigma }_{z}=1.\mathrm{MPa}`"


According to the modeling, we also have conditions at the limits of symmetry and the blocking of rigid body movements.

In modeling D, where only a quarter of the parallelepiped is represented, antisymmetry boundary conditions are used for torsional loading: they amount to imposing zero tangential displacements on one face. Torsional loading is introduced in the form of a tangential surface force (distributed shear) applied to the lips of the crack.


* Upper lip: :math:`{F}_{X}\mathrm{=}\mathrm{-}\tau \frac{Y}{a}` and :math:`{F}_{Y}\mathrm{=}+\tau \frac{X}{a}`


* Lower lip: :math:`{F}_{X}\mathrm{=}+\tau \frac{Y}{a}` and :math:`{F}_{Y}\mathrm{=}\mathrm{-}\tau \frac{X}{a}`