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

The objective of this test case is to validate the 1D modeling of a cracked rotor (option ROTOR_FISS of DYNA_VIBRA) which simulates the behavior of a crack in a shaft line by an equivalent law.

The law is established thanks to 3D calculations carried out in a semi-static manner (see modeling D).

The results obtained by the "beam" and the crack element modeling are compared with the 3D calculation for extreme positions: closed crack and crack completely at most.


Geometry
---------

For the cracked rotor, we consider a simple cylindrical straight beam with a length :math:`\mathrm{2L}\mathrm{=}4m` and a diameter :math:`D\mathrm{=}\mathrm{0,8}m`. The crack is in the middle of the beam and has a straight bottom. The depth of the crack is 65%.


.. image:: images/Cadre1.gif


.. _RefSchema_Cadre1.gif:

Figure 1: Cracked rotor geometry


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

The rotor has a density of :math:`\rho =7800\mathrm{kg}/{m}^{3}`.

The Young's modulus is :math:`E\mathrm{=}210{10}^{9}N{m}^{\mathrm{-}2}` and the Poisson's ratio is :math:`\nu \mathrm{=}\mathrm{0,3}`.


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

For models A, B and C, the beam is embedded on the left and is subjected to a bending moment of unit amplitude according to :math:`Y` on its right end. It is considered that the crack rotates relatively slowly at the speed :math:`5` revolutions per second (:math:`300\text{rpm}`).

For the D modeling, the boundary conditions imposed are on the one hand, an embedment in the sense of the theory of beams of one of the ends of the cylinder by means of a 3D- POU connection, and on the other hand, a unilateral contact without friction between the lips of the cracks. The imposed load is a unit bending moment of components :math:`({M}_{x},{M}_{y})` applied at the free end. The orientation of this moment changes according to the moment of calculation.


Initial conditions
--------------------

In the initial state, :math:`t=0`, the crack is closed. It is gradually opened by a linear ramp spread over :math:`0.2s` leading the moment along :math:`Y` from :math:`0` to :math:`1\mathrm{Nm}`.