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


Geometry
---------

The global frame of reference is frame :math:`(A,X,Y,Z)`. In this coordinate system the coordinates of the nodes are:

:math:`A(0.,0.,0.)`

:math:`B(3.,1.,0.)`

:math:`C(2.,3.,0.)`

:math:`D(3.,1.,-1)`

We will study the behavior of the tetrahedron :math:`\mathrm{ABCD}` whose material properties are defined in a local coordinate system :math:`(A,x,y,z)` obtained by rotating the global coordinate system according to nautical angles :math:`(\alpha =30°,\beta =20°,\gamma =10°)`.


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

The materials used are orthotropic and transverse isotropic. In order to validate orthotropic deformations of thermal origin, a thermomechanical calculation is also carried out.

We adopt the terminology convention used in Code_Aster. Let the suffixes :math:`L`, :math:`T` and :math:`N` mean Longitudinal, Transverse, and Normal.

The units are not specified.


.. image:: images/Object_1.svg
    :width: 259
    :height: 98


.. _RefImage_Object_1.svg:

(We know that

.. image:: images/Object_2.svg
    :width: 259
    :height: 98


.. _RefImage_Object_2.svg:

,

either

.. image:: images/Object_3.svg
    :width: 259
    :height: 98


.. _RefImage_Object_3.svg:

For transverse isotropy, we keep the same values knowing that:


.. image:: images/Object_4.svg
    :width: 259
    :height: 98


.. _RefImage_Object_4.svg:

It should be noted that these coefficients are defined in a local coordinate system :math:`(A,L,T,N)` rotated with the nautical angles :math:`(30°,20°,10°)` with respect to the global coordinate system.


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

The boundary conditions are of the Dirichlet type. A linear displacement field is assumed in :math:`x` and :math:`y` so that the deformation field is constant.


.. csv-table::

    "Dirichlet conditions", ":math:`\mathrm{dX}=\mathrm{2x}+\mathrm{3y}+\mathrm{4z}`"
    "", ":math:`\mathrm{dY}=\mathrm{3x}+\mathrm{5y}+\mathrm{6z}`"
    "", ":math:`\mathrm{dZ}=\mathrm{4x}+\mathrm{6y}+\mathrm{7z}`"
    "Thermal conditions", "Temperature imposed on the entire structure of 100"


We will therefore impose:


.. csv-table::

    "* for node :math:`A` "," :math:`\mathrm{dX}=\mathrm{0,}\mathrm{dY}=\mathrm{0,}\mathrm{dZ}=0`"
    "* for node :math:`B` "," :math:`\mathrm{dX}=\mathrm{9,}\mathrm{dY}=\mathrm{14,}\mathrm{dZ}=18`"
    "* for node :math:`C` "," :math:`\mathrm{dX}=\mathrm{13,}\mathrm{dY}=\mathrm{21,}\mathrm{dZ}=26`"
    "* for node :math:`D` "," :math:`\mathrm{dX}=\mathrm{5,}\mathrm{dy}=\mathrm{8,}\mathrm{dZ}=11`"