G modeling
==============


Reference problem description
------------------------------------


Geometry
~~~~~~~~~~

We consider a slender sandwich structure with a square section composed of two skins and a heart. This is the same calculation case as that presented by Lucie Rouleau in chapter 4 of her thesis [:ref:`bib1 <bib1>`]


.. image:: images/10000201000002C30000024B43968A4756A7B573.png
    :width: 5.6083in
    :height: 4.6555in


.. _RefImage_10000201000002C30000024B43968A4756A7B573.png:

x=1m


Material properties
~~~~~~~~~~~~~~~~~~~~~~~~

The skins are composed of an isotropic elastic material whose properties are:


* Young's module :math:`E=2.1{10}^{11}\mathit{Pa}`


* Poisson's ratio :math:`\mathrm{\nu }=0.3`


* Density :math:`\mathrm{\rho }=7800\mathit{kg}/m\mathrm{³}`

The core is composed of a linear viscoelastic material whose shear coefficient is modelled by a fractional Zener model, i.e.: :math:`G(\omega )=\frac{{G}_{0}+{G}_{\mathrm{\infty }}{(j\omega )}^{\alpha }}{1+{(j\omega )}^{\alpha }}`, the compressibility coefficient is taken to be constant. Numeric values are:


* :math:`{G}_{0}=\mathrm{1,4e6}\mathit{Pa}`


* :math:`{G}_{\mathrm{\infty }}=\mathrm{0,54e9}\mathit{Pa}`


* :math:`\tau =\mathrm{0,52e-6}s`


* :math:`\alpha =\mathrm{0,59}`


* Density :math:`\rho =1460\mathit{kg}/m\mathrm{³}`


* Compressibility coefficient :math:`K=\mathrm{2,2e9}\mathit{Pa}`


Boundary conditions and loads
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This structure is excited punctually by a unitary harmonic effort at point :math:`(1m,\mathrm{0,5}m,\mathrm{0,5}m)`, we will be interested in the displacement along :math:`z` at this same point. The structure is embedded in x=0.


Benchmark solution
---------------------


Reference quantities and results
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We compare the modulus values of the displacement along z at given frequencies calculated by the new Code_Aster methods and the calculation code of [:ref:`bib1 <bib1>`]


Uncertainties about the solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

It is considered that there is a regression if the results obtained are not identical.


Tested sizes and results
------------------------------


Characteristics of the mesh
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The mesh contains 1750 elements of type HEXA20 and 35 elements of type QUAD8.


Results
~~~~~~~~

The following 4 frequencies corresponding to the first resonances and anti-resonances of the structure are tested.


.. csv-table::

    "**Frequency**", "**Calculated value**", "**Reference value**"
    "45 Hz", "-2.19729959e-06-5.31250496e-06j", "-2.52885263e-06-5.26288448e-06j"
    "179 Hz", "1.15296519e-09-3.39972232e-09j", "1.11783628e-09-3.30841894e-09j"
    "255", "1.04664929e-07-2.81759671e-07j", "1.01911277e-07-2.96162398e-07j"
    "696", "1.57582867e-08-6.87887584e-08j", "1.48442697e-08-7.13562822e-08j"