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


Geometry
---------

The homogeneous solution is obtained on a plane volume element (square with side :math:`1`) or 3D (cube with side :math:`1`) volume element. The dimensions have no influence on the solution.


.. image:: images/1000000000000109000000BAD6A0788DBB6F42AA.png
    :width: 1.8937in
    :height: 1.4335in


.. _RefImage_1000000000000109000000BAD6A0788DBB6F42AA.png:


.. image:: images/10000000000000DF00000097125EB0AC85421A99.png
    :width: 1.9264in
    :height: 1.2626in


.. _RefImage_10000000000000DF00000097125EB0AC85421A99.png:


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

:math:`\begin{array}{c}E\mathrm{=}2.{10}^{7}\mathit{Pa}\\ \nu \mathrm{=}0.3\\ {\sigma }_{y}\mathrm{=}2.{10}^{8}\mathit{Pa}\\ {E}_{T}\mathrm{=}2.{10}^{9}\mathit{Pa}\end{array}`

For diffusion, the material coefficients are, for a temperature of :math:`T\mathrm{=}293K`:

:math:`\begin{array}{c}{D}_{L}\mathrm{=}1.27{10}^{\text{-8}}\\ {N}_{L}\mathrm{=}5.1{10}^{29}{m}^{\text{-3}}\\ {K}_{T}\mathrm{=}{e}^{60000.\mathrm{/}\mathit{RT}}\\ {a}_{1}\mathrm{=}23.26{a}_{2}\mathrm{=}\mathrm{-}2.33{a}_{3}\mathrm{=}\mathrm{-}5.5\\ {V}_{h}\mathrm{=}2.e\mathrm{-}6R\mathrm{=}8.3144J\mathrm{/}\mathit{mol}\mathrm{/}K\end{array}`


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


.. csv-table::

    "Side :math:`1\mathrm{-}3` "," :math:`{F}_{x}` imposed"
    "Side :math:`2\mathrm{-}4`:", ":math:`{u}_{x}\mathrm{=}0.`"
    "Point :math:`2`:", ":math:`{u}_{y}\mathrm{=}0.`"


Loading by a distributed force :math:`{F}_{x}` increasing over time:


.. csv-table::

    "Instant (:math:`s`)", "0"," :math:`{10}^{7}` "," :math:`{2.10}^{7}`"
    "Force :math:`{F}_{x}` ", "0"," :math:`{\sigma }_{y}` "," :math:`3{\sigma }_{y}`"


Temporal discretization:


* 1 step until t= :math:`{10}^{7}`


* 100 steps until t= :math:`{2.10}^{7}`: for each step, a mechanical calculation then a diffusion calculation.

The initial concentrations in :math:`\mathit{H2}`:


* In the crystal lattice :math:`{C}_{L}(0)\mathrm{=}2.08{10}^{21}{m}^{\text{-3}}`


* In the traps :math:`{C}_{T}(0)\mathrm{=}8.42{10}^{20}{m}^{\text{-3}}`