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


Geometry
---------


.. image:: images/1000000000000400000003A99482B2EEC56C2432.png
    :width: 3.4134in
    :height: 3.1236in


.. _RefImage_1000000000000400000003A99482B2EEC56C2432.png:


.. csv-table::

    "Plate width:", ":math:`W\mathrm{=}0.6m`"
    "Plate length:", ":math:`L=0.3m`"
    "Crack length:", ":math:`\mathrm{2a}=0.3m`"



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

Rating for thermoelastic properties:

:math:`\left\{\begin{array}{c}{\varepsilon }_{x}\\ {\varepsilon }_{y}\\ {\gamma }_{\mathit{xy}}\end{array}\right\}\mathrm{=}\left[\begin{array}{ccc}{S}_{11}& {S}_{12}& 0\\ {S}_{12}& {S}_{22}& 0\\ 0& 0& {S}_{66}\end{array}\right]\left\{\begin{array}{c}{\sigma }_{x}\\ {\sigma }_{y}\\ {\tau }_{\mathit{xy}}\end{array}\right\}+\left\{\begin{array}{c}{\alpha }_{11}\\ {\alpha }_{22}\\ 0\end{array}\right\}\mathrm{\cdot }(T\mathrm{-}{T}_{\mathit{ref}})`

:math:`\begin{array}{c}{S}_{11}\mathrm{=}1\mathrm{/}{E}_{x}\\ {S}_{22}\mathrm{=}1\mathrm{/}{E}_{y}\\ {S}_{12}\mathrm{=}\mathrm{-}{\nu }_{x}\mathrm{/}{E}_{x}\mathrm{=}\mathrm{-}{\nu }_{y}\mathrm{/}{E}_{y}\\ {S}_{66}\mathrm{=}1\mathrm{/}{G}_{\mathit{xy}}\\ {\alpha }_{11}\mathrm{=}{\alpha }_{x}\\ {\alpha }_{22}\mathrm{=}{\alpha }_{y}\end{array}`

We are limited to isotropic material, both from a thermal and mechanical point of view:

:math:`{E}_{x}\mathrm{=}{E}_{y}\mathrm{=}2.{10}^{5}\mathit{MPa}`

:math:`{\nu }_{x}\mathrm{=}{\nu }_{y}\mathrm{=}0.3`

:math:`{\alpha }_{x}\mathrm{=}{\alpha }_{y}\mathrm{=}1.2{10}^{\mathrm{-}5}°{C}^{\mathrm{-}1}`

:math:`{\lambda }_{x}\mathrm{=}{\lambda }_{y}\mathrm{=}54.W\mathrm{/}m°C`


Boundary conditions and loading
------------------------------------

Two models are considered:


* the half-model :math:`x=0`


* the complete model

**Mechanical boundary conditions:**


* half-model

:math:`\mathrm{UX}=0` along the axis of symmetry :math:`X=0`

:math:`\mathrm{UY}=0` in focus (:math:`W/2.`)


* full model

:math:`\mathrm{UX}=0` at point :math:`(\mathrm{0,}L/2.)`

:math:`\mathrm{UY}=0` at points :math:`(-L/2.)` and :math:`(L/2.)`

**Thermal boundary conditions:**


* half-model

:math:`T=100°C` on the top edge :math:`Y=L/2`.

:math:`T=-100°C` on the bottom edge :math:`Y=-L/2`.

zero flow on the axis of symmetry, on the free edge :math:`X=W/2.` and on the edge of the crack


* full model

:math:`T\mathrm{=}100°C` on the top edge :math:`Y=L/2`.

:math:`T\mathrm{=}\mathrm{-}100°C` on the bottom edge :math:`Y=-L/2`.

zero flux on free edges :math:`X\mathrm{=}\mathrm{\pm }W\mathrm{/}2.` and on the edge of the crack