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


Geometry
---------

A

B

.. image:: images/10000000000003400000038EB39691399E2A024F.png
    :width: 2.5in
    :height: 2.7252in


.. _RefImage_10000000000003400000038EB39691399E2A024F.png:


.. csv-table::

    "Inner radius:", ":math:`{R}_{1}=\mathrm{0,1}m`"
    "External radius:", ":math:`{R}_{2}=\mathrm{0,6}m`"
    "Thickness:", ":math:`t=\mathrm{0,2}m`"
    "Semi major axis:", ":math:`a=\mathrm{0,05}m`"
    "Half minor axis:", ":math:`b=\mathrm{0,0125}m`"



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


.. csv-table::

    "Young's module", ":math:`E\mathrm{=}2{10}^{5}\mathit{MPa}`"
    "Poisson's Ratio", ":math:`\nu \mathrm{=}0.3`"
    "Density", ":math:`\rho \mathrm{=}7800\mathit{kg}\mathrm{/}{m}^{3}`"



Boundary conditions and loading
------------------------------------

The model will be limited to the part of the thick disk located in half-space :math:`Y\ge 0`, the plane of the vertical crack being a plane of symmetry.

In the absence of knots on the axis of revolution, a rigid mode will be blocked by a linear relationship between degrees of freedom.

.. csv-table::

    "Let's say the points:", ":math:`A({R}_{1}\mathrm{,0},t)` "," :math:`B(\mathrm{-}{R}_{1}\mathrm{,0},t)`"


Blocking translation in :math:`X`: :math:`\mathit{UX}(A)+\mathit{UX}(B)\mathrm{=}0`

Blocking the translation in :math:`Y`: :math:`\mathrm{UY}=0` in plane :math:`\mathrm{XOZ}`, except for the lips of the crack.

Blocking translation in :math:`Z`: :math:`\mathrm{UZ}(A)=0`

Blocking the rotation around :math:`\mathrm{OX}`: ensured by the condition at the symmetry limits in the :math:`\mathrm{XOZ}` plane

Blocking rotation around :math:`\mathrm{OY}`: :math:`\mathrm{UZ}(B)=0`

Blocking the rotation around :math:`\mathrm{OZ}`: ensured by the condition at the symmetry limits in the :math:`\mathrm{XOZ}` plane

Loading: stationary angular speed :math:`\omega =500\mathrm{rad}/s`