Benchmark solution
=====================


Calculation method used for the reference solution
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Analytical solution.


Benchmark results
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The mechanical deformation is equal to:

:math:`\begin{array}{ccc}{\varepsilon }^{\mathrm{mec}}& \text{=}& \varepsilon -{\varepsilon }^{\mathrm{th}}\\ & \text{=}& \varepsilon -\alpha T\end{array}`

With an element at degree one and an integration diagram :math:`2\mathrm{\times }2` we will have:

:math:`\begin{array}{ccc}{\varepsilon }^{\mathrm{mec}}& \text{=}& \frac{{u}_{\mathrm{xB}}-{u}_{\mathrm{xA}}}{a}-\alpha \left[\frac{1+\xi }{2}{T}_{\mathrm{max}}\right]\\ & \text{=}& \frac{{\sigma }_{d}}{E}+\frac{1}{2}\alpha {T}_{\mathrm{max}}-\alpha \left[\frac{1+\xi }{2}{T}_{\mathrm{max}}\right]\end{array}`

The constraint in the test will be equal to:

:math:`\sigma \mathrm{=}E{\varepsilon }^{\mathit{mec}}` with :math:`{\varepsilon }^{\mathit{mec}}\mathrm{=}{10}^{\mathrm{-}3}\mathrm{-}\alpha \left[\frac{1+\xi }{2}{T}_{\mathit{max}}\right]`


note
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Since the thermal component of the stress depends on the intrinsic coordinate, the solution is to consider an average temperature per element.