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


Geometry
---------


.. image:: images/1000000000000400000003A99482B2EEC56C2432.png
    :width: 3.4165in
    :height: 3.1252in


.. _RefImage_1000000000000400000003A99482B2EEC56C2432.png:


.. csv-table::

    "Plate width:", ":math:`W=\mathrm{0,6}m`"
    "Plate length:", ":math:`L=\mathrm{0,3}m`"
    "Crack length:", ":math:`\mathrm{2a}=\mathrm{0,3}m`"



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

Notation for thermoelastic properties:


.. image:: images/Object_1.svg
    :width: 327
    :height: 234


.. _RefImage_Object_1.svg:

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

:math:`{E}_{x}={E}_{y}={2.10}^{5}\mathrm{MPa}`

:math:`{\nu }_{x}={\nu }_{y}=\mathrm{0,3}`

:math:`{\alpha }_{x}={\alpha }_{y}=\mathrm{1,2}{10}^{-5}°{C}^{-1}`

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


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

Two models are considered:


* the half-model :math:`x\ge 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` at point :math:`(W/\mathrm{2,0})`


* full model

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

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

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

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

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