Examples
========

You can consult the test cases:

.. csv-table::

    "name", "title", "documentation"
    "HPLV105 ", "Homogenized parameters of GV tube plates", "[:ref:`V7.03.105 <V7.03.105>`]"
    "HPLV106 ", "Homogenized parameters of a fibreglass/resin composite", "[:ref:`V7.03.106 <V7.03.106>`]"
    "HPLV107 ", "Homogenized parameters of N4 tube plates on a whole cell", "[:ref:`V7.03.107 <V7.03.107>`]"
    "HPLV108 ", "Homogenized parameters of a pierced plate with unitary materials and comparison", "[:ref:`V7.03.108 <V7.03.108>`]"


We can also perform a trivial check with test case HPLV101a
entitled "*Homogenization of a homogeneous material*" [:ref:`V7.03.101 <V7.03.101>`], in thermal:
isotropic stationary and isotropic in elasticity.


laminated elastic plate
---------------------------

We consider a material laminated in 3 layers of different thicknesses of isotropic materials. On
note in particular the thicknesses of the successive layers: :math:`{e}_{K}={z}_{K}-{z}_{K-1}`,
see :ref:`r5.03.37-fig-stratifie`. The layers are assumed to be symmetric with respect to the sheet.
medium, so as to cancel the flexion-membrane coupling.

In this case, the base cell has any periodicity on the directions of the tangential plane
:math:`(X,Y)`. Therefore, the fields of membrane correctors
:math:`{\chi }_{11}^{m}`, :math:`{\chi }_{22}^{m}` are finely piecewise dependent on the
position :math:`Z` in the thickness of the plate, while the concealer in
Membrane :math:`{\chi }_{12}^{m}` associated with distortion sucks everywhere. And the fields of
flexure correctors :math:`{\chi }_{11}^{f}`, :math:`{\chi }_{22}^{f}` are parabolic by
pieces in :math:`Z` in the thickness of the plate, while the concealer in
Flexion :math:`{\chi }_{12}^{f}` sucks everywhere.


.. figure:: images/100088E8000069D500003780FC28E5379CFA9C1E.svg
 :name: r5.03.37-fig-stratified
    :align: center
    :width: 197
    :height: 104

    Diagram of a laminated plate with three symmetric layers


The analytical resolution of elementary problems in membrane and then in flexure provides the
analytical expressions of the following homogenized linear elastic characteristics:


.. math::
\ begin {array} {c}
        {A} _ {\ text {name} 1111}} ^ {\ text {memb}}
        =
        {A} _ {\ text {name} 2222}} ^ {\ text {memb}}
        =
        \ sum _ {K} {e} _ {K}\\,\ frac {{E} _ {K}} {1- {\nu} _ {K} ^ {2}}\\
        {A} _ {\ text {name} 1122}} ^ {\ text {memb}}
        =
        \ sum _ {K} {e} _ {K}\\,\ frac {{\nu} _ {K} {E} _ {K}} {1- {\nu} _ {K} _ {K}} ^ {2}}\\
        {A} _ {\ text {name} 1212}} ^ {\ text {memb}}
        =
        \ sum _ {K} {e} _ {K}\\,\ frac {{E} _ {K}} {2 (1+ {\nu}} _ {K})}
    \ end {array}

.. math::
\ begin {array} {c}
        {A} _ {\ text {name} 1111}} ^ {\ text {flex}}
        =
        {A} _ {\ text {home} 2222}} ^ {\ text {flex}}
        =
        \ frac {1} {3}\ sum_ {K} ({z} {K} _ {K} _ {K} ^ {3} - {z} _ {K-1} ^ {3})\,\ frac {{E} _ {K} _ {K} _ {K} ^ {2}}\\
        {A} _ {\ text {home} 1122}} ^ {\ text {flex}}
        =
        \ frac {1} {3}\ sum _ {K} ({z}} ({z} _ {K} _ {1}} ^ {3})\,\ frac {{\nu} _ {K} {K} {E} {E} _ {E} _ {E} _ {K}} {K} {2}} _ {K} {2}}\\
        {A} _ {\ text {home} 1212}} ^ {\ text {flex}}
        =
        \ frac {1} {3}\ sum _ {K} ({z}} ({z} _ {K} _ {K} {3})\,\ frac {{E} _ {K}} {K}} {2 (1+ {\nu} _ {3})})}
    \ end {array}

This analysis can also be carried out using transverse shear.