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


Calculation method used for the reference solution
--------------------------------------------------------

**Before transformation**, thermo-elastic solution for :math:`t<{\tau }_{1}` (no metallurgical transformation :math:`\dot{Z}=0`).

:math:`\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)\\ {\sigma }_{\mathit{zz}}(t)={p}_{0}t\\ {\epsilon }_{\mathit{zz}}^{e}(t)=\frac{{\sigma }_{\mathit{zz}}(t)}{E}\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{aust}}(T-{T}^{0})\end{array}`

The elastic limit is reached for:

:math:`{\tau }_{1}=\frac{{\sigma }_{0}^{\mathit{aust}}}{{p}_{0}-{s}^{\mathit{aust}}\mu }`

**During transformation**, thermo-metallo-elasto-plastic solution, for :math:`{\tau }_{1}\le t\le {\tau }_{2}`, :math:`{\tau }_{\mathrm{2 }}=\mathrm{40 }s`.

:math:`\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{pt}}(t)\\ {\sigma }_{\mathit{zz}}(t)={p}_{0}t\\ {\epsilon }_{\mathit{zz}}^{e}(t)=\frac{{\sigma }_{\mathit{zz}}(t)}{E}\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={Z}_{\mathit{aust}}{\alpha }_{\mathit{aust}}(T-{T}^{0})+{Z}_{\mathit{fbm}}\left({\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\right)\\ {\epsilon }_{\mathit{zz}}^{p}=\frac{{\sigma }_{\mathit{zz}}(t)-({Z}_{\mathit{aust}}{\sigma }_{y}^{\mathit{aust}}(T)+{Z}_{\mathit{fbm}}{\sigma }_{y}^{\mathit{fbm}}(T))}{{Z}_{\mathit{aust}}{H}^{\mathit{aust}}(T)+{Z}_{\mathit{fbm}}{H}^{\mathit{fbm}}(T)}\\ {\epsilon }_{\mathit{zz}}^{\mathit{pt}}(t)={k}^{\mathit{fbm}}\left({\sigma }_{\mathit{zz}}({\tau }_{1})-\frac{{p}_{0}}{2\lambda \varphi }\right)-{k}^{\mathit{fbm}}\left({\sigma }_{\mathit{zz}}(t)-\frac{{p}_{0}}{2\lambda \varphi }\right){(1-{Z}_{\mathit{fbm}})}^{2}\end{array}`

**After transformation**, thermo-elasto-plastic solution, for :math:`t\ge {\tau }_{2}`

:math:`\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{pt}}({\tau }_{2})\\ {\sigma }_{\mathit{zz}}(t)={p}_{0}t\\ {\epsilon }_{\mathit{zz}}^{e}(t)=\frac{{\sigma }_{\mathit{zz}}(t)}{E}\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\\ {\epsilon }_{\mathit{zz}}^{p}(t)=\frac{{\sigma }_{\mathit{zz}}(t)-({\sigma }_{0}^{\mathit{fbm}}+{s}^{\mathit{fbm}}\mu t)}{{H}_{0}^{\mathit{fbm}}+{\lambda }^{\mathit{fbm}}\mu t}\end{array}`


Benchmark results
----------------------

:math:`{\sigma }_{\mathit{zz}}^{}`, :math:`{\epsilon }_{\mathit{zz}}^{}`, :math:`{\epsilon }_{\mathit{zz}}^{p}`, :math:`{\epsilon }_{\mathit{zz}}^{\mathit{th}}`, :math:`{\epsilon }_{\mathit{zz}}^{\mathit{meca}}`, :math:`{\epsilon }_{\mathit{zz}}^{\mathit{plas}}`, and :math:`\chi` to :math:`\mathrm{24 }s`,,,,,,, and to, :math:`\mathrm{26 }s`, :math:`\mathrm{40 }s`, and purpose. :math:`\mathrm{90 }s`

with:

:math:`{\epsilon }^{\mathit{meca}}`: mechanical deformations

:math:`{\epsilon }^{\mathit{plas}}`: plastic deformations (including transformation plasticity)