2. Reference solution

2.1. Calculation method used for the reference solution

Before transformation, thermo-elastic solution for \(t<{\tau }_{1}^{\text{'}}\).

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

The elastic limit is reached for:

\({\tau }_{1}^{\text{'}}=\frac{{\sigma }_{0}^{\mathit{aust}}}{{p}_{0}-{s}^{\mathit{aust}}\mu }=47.06s\)

Before transformation, thermo-elasto-plastic solution, \({\tau }_{1}^{\text{'}}\le t\le {\tau }_{1}\), \({\tau }_{1}=\mathrm{60 }s\).

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

During transformation, thermo-elasto-metallurgical solution, \({\tau }_{1}<t<{\tau }_{2}\), \({\tau }_{\mathrm{2 }}\mathrm{=}\mathrm{112 }s\).

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

with \(F({Z}_{\mathit{fbm}})={Z}_{\mathit{fbm}}(1-{Z}_{\mathit{fbm}})\)

After transformation, thermo-elasto-plastic solution,

, \({\tau }_{3}=\mathrm{176 }s\).

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

2.2. Benchmark results

\({\epsilon }_{\mathit{zz}}^{p}\), \(\chi\), \({\sigma }_{\mathit{zz}}\), and \({\epsilon }_{\mathit{zz}}\) for \(t=\mathrm{47,}\mathrm{48,}64\) and \(114\) seconds.

\({\epsilon }_{\mathit{zz}}^{p}\) for \(t\mathrm{=}60\) and \(176\) seconds.

\({\epsilon }_{\mathit{zz}}^{\mathit{th}},{\epsilon }_{\mathit{zz}}^{\mathit{meca}}\) and \({\epsilon }_{\mathit{zz}}^{\mathit{plas}}\) in the case of B and D models, for \(t\mathrm{=}\mathrm{47,}\mathrm{48,}64\) and \(114\) seconds.

with:

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

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

2.3. Bibliographical references

1. DONORE A.M. - WAECKEL F.: Influence of structural transformations in elasto-plastic behavior laws Note HI-74/93/024.