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




Geometry
---------


.. image:: images/Cadre3.gif


.. _RefSchema_Cadre3.gif:

Image 1.1-1: Sandwich plate geometry.

Rectangular sandwich plate composed of three layers of different materials:

         Side: 0.05m x 0.15m

 Thickness: aluminum (top): 0.5 mm

             viscoelastic material (center): 1 mm

             steel (bottom): 1 mm




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

The material of the top layer is aluminum (isotropic elastic); its properties are constant:


* Young's modulus :math:`E=70000\mathit{MPa}`


* Poisson's ratio :math:`\nu =\mathrm{0,3}`


* density :math:`\rho =2700\mathit{kg}/{m}^{3}`


* hysteretic damping :math:`\eta =\mathrm{0,001}`

The material in the bottom layer is steel (isotropic elastic); its properties are constant:


* Young's modulus :math:`E=210000\mathit{MPa}`


* Poisson's ratio :math:`\nu =\mathrm{0,3}`


* density :math:`\rho =7800\mathit{kg}/{m}^{3}`


* hysteretic damping :math:`\eta =\mathrm{0,002}`

The core layer material is viscoelastic (elastomer); some of its properties are frequency-dependent:


.. csv-table::

    "Frequency (Hz)", "Real part of Young's modulus :math:`E` (MPa)", "Loss factor :math:`\eta`"
    "1", "23.2", "1.1"
    "10", "58", "0.85"
    "50", "145", "0.7"
    "100", "203", "0.6"
    "500", "348", "0.4"
    "1000", "435", "0.35"
    "1500", "464", "0.34"


Table 1.2-1: Frequency-dependent properties of the viscoelastic material.

The others are constant:


* Poisson's ratio :math:`\nu =\mathrm{0,45}`


* density :math:`\rho =1200\mathit{kg}/{m}^{3}`




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

Boundary conditions:


* embedding on one edge of the steel layer.

Loading:


* nodal force at point A: FZ=1.




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

Not applicable (harmonic calculation).