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

Solution field shape
---------------------------

     The solution :math:`\sigma (t)` constraint field is of the form:

:math:`\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:

:math:`{\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 :math:`\sigma (t)` keeps the direction constant, we have:

:math:`{\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 :math:`{\epsilon }^{p}` is the plastic deformation tensor.


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

**Notation:** Hereinafter, we will note :math:`{\epsilon }_{\alpha }^{\mathit{eff}}` (resp. :math:`{\epsilon }_{\gamma }^{\mathit{eff}}`) the effective work-hardening variable for the cold phases (resp. for the hot phase).

**Before transformation**, thermo-elastic solution for :math:`t<{t}_{1}`.

:math:`\{\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 :math:`t={t}_{1}` such that:

:math:`\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, :math:`{t}_{1}\le t\le {\tau }_{1}`.

:math:`\{\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 :math:`{\tau }_{1}<t\le {\tau }_{2}`, we are in an elastic regime, so we have a thermo-elastic solution with phase change.

:math:`\{\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 :math:`{\tau }_{2}<t<{t}_{2}`.

:math:`\{\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 :math:`t={t}_{2}` such that:

:math:`{\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: :math:`R=0` before replastification.

So in :math:`{t}_{2}` we have:

:math:`\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 :math:`t\ge {t}_{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)=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}`


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


.. csv-table::

    "To :math:`t\mathrm{=}60s`:", ":math:`{\sigma }_{\mathit{zz}}` "," :math:`{\epsilon }_{\gamma }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\alpha }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{th}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{meca}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{plas}}`"
    "To :math:`t=89s`:", ":math:`{\sigma }_{\mathit{zz}}` "," :math:`{\epsilon }_{\gamma }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\alpha }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{th}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{meca}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{plas}}`"
    "To :math:`t\mathrm{=}112s`:", ":math:`{\sigma }_{\mathit{zz}}` "," :math:`{\epsilon }_{\gamma }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\alpha }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{th}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{meca}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{plas}}`"
    "To :math:`t\mathrm{=}176s`:", ":math:`{\sigma }_{\mathit{zz}}` "," :math:`{\epsilon }_{\gamma }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\alpha }^{\mathit{eff}}` "," "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{th}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{meca}}` "," :math:`{\epsilon }_{\mathit{xx}}^{\mathit{plas}}`"



Bibliographical references
---------------------------


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


2. DONORE .A.M. - WAECKEL .F. - RAZAKANAIVO .A. - Doc. Aster [:ref:`R4.04.02 <R4.04.02>`].