2. Benchmark solution

2.1. Calculation method used for the reference solution

Analytical solution.

2.2. Benchmark results

The mechanical deformation is equal to:

\(\begin{array}{ccc}{\varepsilon }^{\mathrm{mec}}& \text{=}& \varepsilon -{\varepsilon }^{\mathrm{th}}\\ & \text{=}& \varepsilon -\alpha T\end{array}\)

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

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

The constraint in the test will be equal to:

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

2.3. note

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