A-D modeling
================


Objectives
---------

The objective of these models is to verify the behavior of options RIGI_MECA and RIGI_MECA_HYST on an element by testing the real and imaginary parts of the stiffness matrix produced.

In isotropic elasticity, the law of behavior is written as: :math:`\sigma =2G(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-K\mathit{Tr}(ϵ)\mathit{Id}`

If the material has complex modules, the real part/imaginary part separation is simply written:

:math:`{\sigma }_{r}=2\mathrm{\Re }(G)(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-\mathrm{\Re }(K)\mathit{Tr}(ϵ)\mathit{Id},{\sigma }_{i}=2\mathrm{\Im }(G)(ϵ-\frac{1}{3}\mathit{Tr}(ϵ))-\mathrm{\Im }(K)\mathit{Tr}(ϵ)\mathit{Id}`


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
--------------------

Three fictional isotope materials are declared:


* elastic, material 1: :math:`{G}_{1}=15\mathit{Pa},{K}_{1}=20\mathit{Pa}`


* elastic, material 2: :math:`{G}_{2}=10\mathit{Pa},{K}_{2}=25\mathit{Pa}`


* viscoelastic, v material: :math:`G=15+10j\mathit{Pa},K=10+25j\mathit{Pa}`

Thus, the matrix calculated on the material 1 must be the real part of the matrix calculated on the material v and the matrix calculated on the material 2 must be the imaginary part of the matrix calculated on the material v.


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


.. csv-table::

    "Modeling", "Material Type", "Element", "Mesh"
    "A", "Isotropic", "3D", "1 HEXA8"
    "B", "Orthotropic", "3D", "1 HEXA8"
    "C", "Transverse Isotropic", "3D", "1 HEXA8"
    "D", "Isotropic", "D_ PLAN ", "1 QUAD4"


For models of non-isotropic materials, materials 1, 2 and v are used by taking :math:`{G}_{L}={G}_{N}={G}_{T},{E}_{L}={E}_{N}={E}_{T},{\nu }_{LT}={\nu }_{LN}={\nu }_{NT}`


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

For each model, the following stiffness matrices are constructed:


* :math:`{K}_{i}`, the real part of the stiffness matrix on the material i or i=1.2 (built with RIGI_MECA)


*:math:`{K}_{i}\text{*}` the stiffness matrix on the material i or i=1.2 (built with RIGI_MECA_HYST)


* :math:`{K}_{v}`, the complex stiffness matrix on material v (built with RIGI_MECA_HYST)

We then check: :math:`{K}_{v}={K}_{1}+\mathit{jK}2` and :math:`{K}_{1}\text{*}={K}_{1}`