E and F models
====================


Objectives
---------

The objective of these models is to verify on an element that the modeling of a viscoelastic material whose Poisson's ratio is constant/real and whose complex Young's modulus is equivalent to the modeling of an elastic material with a hysteretic damping coefficient.

In isotropic elasticity, the law of behavior is written as:

:math:`\sigma =\frac{E}{1+\nu }(ϵ+\frac{\nu }{1-2\nu }\mathit{Tr}(ϵ)\mathit{Id})`

So if :math:`E={E}_{r}+j{E}_{i}` then

:math:`\sigma =(1+j\frac{{E}_{i}}{{E}_{r}})\frac{{E}_{r}}{1+\nu }(ϵ+\frac{\nu }{1-2\nu }\mathit{Tr}(ϵ)\mathit{Id})`

This corresponds to an isotropic elastic material with Young's modulus :math:`{E}_{r}`, Poisson :math:`\nu` and hysteretic damping :math:`\eta =\frac{{E}_{i}}{{E}_{r}}`


Geometry
---------

As a stiffness matrix construction test, the geometry corresponds to a cube for 3D elements and a square plane for 2D elements.


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

The two declared materials are:


* viscoelastic, material 1: :math:`G=\mathrm{0,53e9}+\mathrm{9,3e6}j\mathit{Pa},\nu =\mathrm{0,3}`


* elastic, material 2: :math:`E=\mathrm{1,3e9}\mathit{Pa},\nu =\mathrm{0,3,}\eta =\mathrm{1,8e-2}`


Characteristics of the models
----------------------------------


.. csv-table::

    "Modeling", "Element", "Mesh"
    "E", "3D", "1 HEXA8"
    "F", "C_ PLAN ", "1 QUAD4"



Tested sizes
-----------------

For each model, complex stiffness matrices are constructed:


*:math:`{K}_{1}\text{*}` of material 1


*:math:`{K}_{2}\text{*}` of material 2

We then check :math:`{K}_{1}\text{*}={K}_{2}\text{*}`