2. Benchmark solution#
2.1. Solution field shape#
The solution \(\sigma (t)\) constraint field is of the form:
\(\sigma (t)={\sigma }_{o}(t)\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)\)
The following form of the elastic deformation tensor is deduced from this:
\({\epsilon }^{e}(t)=\frac{{\sigma }_{o}(t)}{E}\left(\begin{array}{ccc}-\nu & 0& 0\\ 0& -\nu & 0\\ 0& 0& 1\end{array}\right)\)
In addition, since \(\sigma (t)\) keeps the direction constant, we have:
\({\epsilon }^{p}(t)={\epsilon }_{o}^{p}(t)\left(\begin{array}{ccc}\frac{-1}{2}& 0& 0\\ 0& \frac{-1}{2}& 0\\ 0& 0& 1\end{array}\right)\)
where \({\epsilon }^{p}\) is the plastic deformation tensor.
2.2. Calculation method used for the reference solution#
Notation: Hereinafter, we will note \({\epsilon }_{\alpha }^{\mathit{eff}}\) (resp. \({\epsilon }_{\gamma }^{\mathit{eff}}\)) the effective work-hardening variable for the cold phases (resp. for the hot phase).
Before transformation, thermo-elastic solution for \(t<{t}_{1}\).
\(\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)=0\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{aust}}(T-{T}^{0})\\ {\sigma }_{\mathit{zz}}(t)=-E{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)\end{array}\)
The elastic limit is reached for \(t={t}_{1}\) such that:
\(\begin{array}{ccc}{\sigma }_{\mathit{zz}}({t}_{1})=-E{\epsilon }_{\mathit{zz}}^{\mathit{th}}({t}_{1})={\sigma }_{y}^{\mathit{aust}}& \iff & T({t}_{1})-{T}^{0}=\frac{-{\sigma }_{y}^{\mathit{aust}}}{E{\alpha }_{\mathit{aust}}}=-100.°C\\ & \iff & {t}_{1}=\frac{T({t}_{1})-{T}^{0}}{\mu }=20s\end{array}\)
Before transformation, thermo-elasto-plastic solution, \({t}_{1}\le t\le {\tau }_{1}\).
\(\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}(t)=0\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{aust}}(T-{T}^{0})\\ {\epsilon }_{\mathit{zz}}^{p}(t)=\frac{-{\sigma }_{y}^{\mathit{aust}}-E{\alpha }_{\mathit{aust}}(T-{T}^{0})}{E+{H}^{\mathit{aust}}}\\ {\sigma }_{\mathit{zz}}(t)=-E({\epsilon }_{\mathit{zz}}^{p}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t))\\ {\epsilon }_{\gamma }^{\mathit{eff}}(t)=p(t)={\epsilon }_{\mathit{zz}}^{p}(t)\\ {\epsilon }_{\alpha }^{\mathit{eff}}(t)=0\end{array}\)
During the transformation, for \({\tau }_{1}<t\le {\tau }_{2}\), we are in an elastic regime, so we have a thermo-elastic solution with phase change.
\(\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})=0\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={Z}_{\mathit{aust}}{\alpha }_{\mathit{aust}}(T-{T}^{0})+{Z}_{\mathit{fbm}}\left({\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\right)\\ {\sigma }_{\mathit{zz}}(t)=-E({\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t))\\ {\epsilon }_{\gamma }^{\mathit{eff}}(t)={\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})\\ {\epsilon }_{\alpha }^{\mathit{eff}}(t)=0\end{array}\)
After transformation, thermo-elastic solution for \({\tau }_{2}<t<{t}_{2}\).
\(\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})=0\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\\ {\sigma }_{\mathit{zz}}(t)=-E({\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1}))\\ {\epsilon }_{\gamma }^{\mathit{eff}}(t)={\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})\\ {\epsilon }_{\alpha }^{\mathit{eff}}(t)=0\end{array}\)
The elastic limit is reached for \(t={t}_{2}\) such that:
\({\sigma }_{\mathit{zz}}({t}_{2})=R(T,Z,{\epsilon }^{\mathit{eff}})+{\sigma }_{y}(T,Z)\)
Because of the restoration of work hardening and the fact that we were in elastic regime throughout the transformation: \(R=0\) before replastification.
So in \({t}_{2}\) we have:
\(\begin{array}{ccc}{\sigma }_{\mathit{zz}}({t}_{2})=-E({\epsilon }_{\mathit{zz}}^{\mathit{th}}({t}_{2})+{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1}))={\sigma }_{y}^{\mathit{fbm}}& \iff & T({t}_{2})-{T}^{0}=-\frac{{\sigma }_{y}^{\mathit{fbm}}+E(\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}+{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1}))}{E{\alpha }_{\mathit{fbm}}}\simeq -624°C\\ & \Rightarrow & {t}_{2}=\frac{T({t}_{2})-{T}^{0}}{\mu }\simeq 125s\end{array}\)
After transformation, thermo-elasto-plastic solution for \(t\ge {t}_{2}\).
\(\{\begin{array}{c}{\epsilon }_{\mathit{zz}}(t)={\epsilon }_{\mathit{zz}}^{e}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)+{\epsilon }_{\mathit{zz}}^{p}(t)=0\\ {\epsilon }_{\mathit{zz}}^{\mathit{th}}(t)={\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}\\ {\epsilon }_{\mathit{zz}}^{p}(t)=\frac{-{\sigma }_{y}^{\mathit{fbm}}-E({\alpha }_{\mathit{fbm}}(T-{T}^{0})+\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}})+{H}^{\mathit{fbm}}{\epsilon }^{p}({\tau }_{1})}{E+{H}^{\mathit{fbm}}}\\ {\sigma }_{\mathit{zz}}(t)=-E({\epsilon }_{\mathit{zz}}^{p}(t)+{\epsilon }_{\mathit{zz}}^{\mathit{th}}(t))\\ {\epsilon }_{\gamma }^{\mathit{eff}}(t)=0\\ {\epsilon }_{\alpha }^{\mathit{eff}}(t)={\epsilon }_{\mathit{zz}}^{p}(t)-{\epsilon }_{\mathit{zz}}^{p}({\tau }_{1})\end{array}\)
2.3. Benchmark results#
To \(t\mathrm{=}60s\): |
\({\sigma }_{\mathit{zz}}\) |
|
|
|
|
|
||
To \(t=89s\): |
\({\sigma }_{\mathit{zz}}\) |
|
|
|
|
|
||
To \(t\mathrm{=}112s\): |
\({\sigma }_{\mathit{zz}}\) |
|
|
|
|
|
||
To \(t\mathrm{=}176s\): |
\({\sigma }_{\mathit{zz}}\) |
|
|
|
|
|
2.4. Bibliographical references#
DONORE A.M. - WAECKEL F. - Influence of structural transformations in elasto-plastic behavior laws Note HI-74/93/024.
DONORE .A.M. - WAECKEL .F. - RAZAKANAIVO .A. - Doc. Aster [R4.04.02].