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


2D geometry
------------



The structure is a rectangle made up of two plates of the same material, separated by an interface.

The dimensions of the plate, on which the pressures are applied, are:

:math:`{L}_{X}\mathrm{=}\mathrm{2m}`, :math:`{L}_{Y}\mathrm{=}\mathrm{1,8}m`

The second plate has the following dimensions:

:math:`{L}_{X}\mathrm{=}\mathrm{2m}`, :math:`{L}_{Y}’\mathrm{=}\mathrm{1,2}m`

The position of the reference points is:


.. csv-table::

    "", ":math:`x` "," :math:`y`"
    ":math:`A` ", "-1", "0"
    ":math:`B` ", "1", "0"
    ":math:`C` ", "1", "1.8"
    ":math:`D` ", "-1", "1.8"
    ":math:`O` ", "0", "0"



3D geometry
------------

The structure is a rectangular parallelepiped made up of two plates of the same material, separated by an interface.

The dimensions of the top plate, on which the pressures are applied, are:

:math:`{L}_{X}\mathrm{=}\mathrm{2m}`, :math:`{L}_{Y}\mathrm{=}\mathrm{1,8}m`, :math:`{L}_{Z}\mathrm{=}\mathrm{0,01}m`

The second plate underneath has the following dimensions:

:math:`{L}_{X}\mathrm{=}\mathrm{2m}`, :math:`{L}_{Y}’\mathrm{=}\mathrm{1,2}m`, :math:`{L}_{Z}\mathrm{=}\mathrm{0,01}m`

The position of the reference points is:


.. csv-table::

    "", ":math:`x` "," :math:`y` "," :math:`z`"
    ":math:`A` ", "-1", "0", "0"
    ":math:`B` ", "1", "0", "0"
    ":math:`C` ", "1", "1.8", "0"
    ":math:`D` ", "-1", "1.8", "0"
    ":math:`O` ", "0", "0", "0"
    ":math:`A\text{'}` ", "-1", "0", "-0.01"
    ":math:`B\text{'}` ", "1", "0", "-0.01"
    ":math:`C\text{'}` ", "1", "1.8", "-0.01"
    ":math:`D\text{'}` ", "-1", "1.8", "-0.01"
    ":math:`O\text{'}` ", "0", "0", "0", "-0.01"



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

Poisson's ratio: :math:`0.2`

Young's module: :math:`1.{10}^{10}N/{m}^{2}`


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

The lower plate :math:`(y<0)` is blocked by an embedding of its lower face.

Plan :math:`\mathit{ABCD}` is stuck in the :math:`{e}_{z}` direction.

In the 3D case, the :math:`(x\mathrm{=}0)` plane is blocked in the :math:`{e}_{x}` direction, and in the 2D case, the line :math:`(x\mathrm{=}0)` is blocked in the :math:`{e}_{x}` direction.

The upper plate :math:`(y>0)` is subjected to a distributed horizontal pressure acting on the lateral faces :math:`P\mathrm{=}\mathrm{\pm }(\mathrm{-}{5.10}^{4}y+{1.10}^{5})N\mathrm{/}{m}^{2}` (according to the principle of compression). A quadratic tensile displacement is applied to it on the upper face :math:`d\mathrm{=}\mathrm{-}\mathrm{2,5}{.10}^{\mathrm{-}6}{x}^{2}+{1.10}^{\mathrm{-}5}m`.


.. image:: images/10000000000001FA00000240A603EA54F10F55F9.jpg
    :width: 4.4929in
    :height: 4.4256in


.. _RefImage_10000000000001FA00000240A603EA54F10F55F9.jpg:


.. _Ref15420311621:

Figure 1: Structure geometry and interface positioning and 3D loads