2. Benchmark solution

2.1. Calculation method used for the reference solution

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

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

The elastic limit is reached for:

\({\tau }_{1}=\frac{{\sigma }_{0}^{\mathit{aust}}}{{p}_{0}-{s}^{\mathit{aust}}\mu }\)

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

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

After transformation, thermo-elasto-plastic solution, for \(t\ge {\tau }_{2}\)

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

2.2. Benchmark results

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

with:

\({\epsilon }^{\mathit{meca}}\): mechanical deformations

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