Reference problem
=====================

Geometry
---------


.. image:: images/1000069C000023DD0000170C8DA33F3F4F4E49B0.svg
    :width: 462
    :height: 297


.. _RefImage_1000069C000023DD0000170C8DA33F3F4F4E49B0.svg:


Material properties
------------------------

     The following convention is adopted in order to distinguish the parameters of the hot phase (austenitic) from the parameters of the cold phases (ferrito-pearlitic, bainitic and martensitic):


+-------------------------------------------------------------------------------------+-+----------------------------------------------------------------------------------+
|                                                                                     |=|Characteristics relating to the austenitic phase                                  |
+ .. image:: images/Object_30.svg                                                     + +                                                                                  +
|    :width: 462                                                                      | |                                                                                  |
+    :height: 297                                                                     + +                                                                                  +
|                                                                                     | |                                                                                  |
+                                                                                     + +                                                                                  +
|                                                                                     | |                                                                                  |
+-------------------------------------------------------------------------------------+-+----------------------------------------------------------------------------------+
|                                                                                     |=|characteristics relating to the ferrito-pearlitic, bainitic and martensitic phases|
+ .. image:: images/Object_133.svg                                                    + +                                                                                  +
|    :width: 462                                                                      | |                                                                                  |
+    :height: 297                                                                     + +                                                                                  +
|                                                                                     | |                                                                                  |
+                                                                                     + +                                                                                  +
|                                                                                     | |                                                                                  |
+-------------------------------------------------------------------------------------+-+----------------------------------------------------------------------------------+


 **Metallurgical parameters:**

TRC to model a bainitic metallurgical evolution, over the entire structure, of the form:

:math:`{Z}_{\mathit{fbm}}=\{\begin{array}{ccc}0.& \mathit{si}t\le {\tau }_{1}& {\tau }_{1}=60s\\ \frac{t-{\tau }_{1}}{{\tau }_{2}-{\tau }_{1}}& \mathit{si}{\tau }_{1}\le t<{\tau }_{2}& {\tau }_{2}=112s\\ 1.& \mathit{si}t\ge {\tau }_{2}& \end{array}`

**Thermal parameters:**

Calorific capacity: :math:`\rho {C}_{p}=2.{10}^{6}{\mathit{J.m}}^{-3}\mathrm{.}°{C}^{-1}`

Conductivity: :math:`\lambda =9999.9{\mathit{W.m}}^{-1}\mathrm{.}°{C}^{-1}`

**Thermo-mechanical parameters:**


* Thermo-elastic parameters:

Young's module :math:`E=200000{10}^{6}\mathit{Pa}`

Poisson's ratio :math:`\nu =0.3`

Thermal expansion coefficients :math:`{\alpha }_{\mathit{fbm}}={\alpha }_{\mathit{aust}}=20.{10}^{-6}°{C}^{-1}`

Expansion coefficient definition temperature: :math:`{T}_{\mathit{ref}}=900°C`

Reference thermal deformation state: :math:`\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}=2.52{10}^{-3}`

Elasticity limit:

:math:`{\sigma }_{y}^{\mathit{fbm}}=1200.{10}^{6}\mathit{Pa}`

:math:`{\sigma }_{y}^{\mathit{aust}}={400.10}^{6}\mathit{Pa}`


* Thermoplastic parameters (law with linear work hardening)

Tangent modules: :math:`{E}_{T}^{\mathit{fbm}}={E}_{T}^{\mathit{aust}}={2000.10}^{6}\mathit{Pa}`

So we have: :math:`{H}^{\mathit{fbm}}={H}^{\mathit{aust}}=\frac{{\mathit{EE}}_{T}}{(E-{E}_{T})}=\mathrm{2,04}{.10}^{9}\mathit{Pa}`


* Settings for work hardening restoration (full restore): :math:`{\theta }_{\gamma \mathrm{,3}}=0`

:math:`{\theta }_{\gamma \mathrm{,3}}` is the rate of work hardening transmitted from austenite to the ferritic phase 3 (bainite).


Boundary conditions and loads
-------------------------------------


* :math:`{u}_{Y}=0` on the :math:`\mathit{AB}` side; :math:`{u}_{X}=0` in :math:`A`.


* :math:`T={T}^{0}+\mu t`, :math:`\mu =-5°{\mathit{C.s}}^{-1}` throughout the structure.


* The load on the structure is due to thermal and metallurgical expansion phenomena constrained in the :math:`z` direction by the condition of plane deformations.


Initial conditions
--------------------

:math:`{T}^{0}=900°C={T}^{\mathit{ref}}`