Reference solution
=====================


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

**Before transformation**, thermo-elastic solution for :math:`t<{\tau }_{1}^{\text{'}}`.

: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}^{\text{'}}=\frac{{\sigma }_{0}^{\mathit{aust}}}{{p}_{0}-{s}^{\mathit{aust}}\mu }=47.06s`

**Before transformation**, thermo-elasto-plastic solution, :math:`{\tau }_{1}^{\text{'}}\le t\le {\tau }_{1}`, :math:`{\tau }_{1}=\mathrm{60 }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)\\ {\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, :math:`{\tau }_{1}<t<{\tau }_{2}`, :math:`{\tau }_{\mathrm{2 }}\mathrm{=}\mathrm{112 }s`.

: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})+{\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 :math:`F({Z}_{\mathit{fbm}})={Z}_{\mathit{fbm}}(1-{Z}_{\mathit{fbm}})`

**After transformation**, thermo-elasto-plastic solution,

.. image:: images/Object_28.svg
    :width: 70
    :height: 19


.. _RefImage_Object_28.svg:

, :math:`{\tau }_{3}=\mathrm{176 }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}}({\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}`


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

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

:math:`{\epsilon }_{\mathit{zz}}^{p}` for :math:`t\mathrm{=}60` and :math:`176` seconds.

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

with:

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

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


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