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


Geometry
---------

Width: :math:`a\mathrm{=}0.05m`

Height: :math:`h\mathrm{=}0.2m`


.. image:: images/10000000000001D600000206F8B564E4A5CA893F.png
    :width: 2.2264in
    :height: 2.6547in


.. _RefImage_10000000000001D600000206F8B564E4A5CA893F.png:


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_6.svg                                                  + +                                                                                  +
|    :width: 32                                                                   | |                                                                                  |
+    :height: 18                                                                  + +                                                                                  +
|                                                                                 | |                                                                                  |
+                                                                                 + +                                                                                  +
|                                                                                 | |                                                                                  |
+---------------------------------------------------------------------------------+-+----------------------------------------------------------------------------------+
|                                                                                 |=|characteristics relating to the ferrito-pearlitic, bainitic and martensitic phases|
+ .. image:: images/Object_7.svg                                                  + +                                                                                  +
|    :width: 32                                                                   | |                                                                                  |
+    :height: 18                                                                  + +                                                                                  +
|                                                                                 | |                                                                                  |
+                                                                                 + +                                                                                  +
|                                                                                 | |                                                                                  |
+---------------------------------------------------------------------------------+-+----------------------------------------------------------------------------------+


 **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}}=15.{10}^{-6}°{C}^{-1}` :math:`{\alpha }_{\mathit{aust}}=23.5{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}`

Elastic limit:

:math:`{\sigma }_{y}^{\mathit{fbm}}={\sigma }_{0}^{\mathit{fbm}}+{s}^{\mathit{fbm}}(T-{T}^{0})` with :math:`{\sigma }_{0}^{\mathit{fbm}}=530.{10}^{6}\mathit{Pa}` and :math:`{s}^{\mathit{fbm}}=0.5{10}^{6}\mathit{Pa.}°{C}^{-1}`

:math:`{\sigma }_{y}^{\mathit{aust}}={\sigma }_{0}^{\mathit{aust}}+{s}^{\mathit{aust}}(T-{T}^{0})` with :math:`{\sigma }_{0}^{\mathit{aust}}=400.{10}^{6}\mathit{Pa}` and :math:`{s}^{\mathit{aust}}=0.5{10}^{6}\mathit{Pa.}°{C}^{-1}`


* Thermoplastic parameters (law with linear work hardening)

Tangent modules: :math:`{E}_{T}^{\mathit{fbm}}` and :math:`{E}_{T}^{\mathit{aust}}` are chosen such that:

:math:`{H}^{\mathit{fbm}}={H}_{0}^{\mathit{fbm}}+{\lambda }^{\mathit{fbm}}(T-{T}^{0})` with :math:`{H}_{0}^{\mathit{fbm}}=-50.{10}^{6}\mathit{Pa}` and :math:`{\lambda }^{\mathit{fbm}}=-5.{10}^{6}\mathit{Pa.}°{C}^{-1}`

:math:`{H}^{\mathit{aust}}={H}_{0}^{\mathit{aust}}+{\lambda }^{\mathit{aust}}(T-{T}^{0})` with :math:`{H}_{0}^{\mathit{aust}}=1250.{10}^{6}\mathit{Pa}` and :math:`{\lambda }^{\mathit{aust}}=-5.{10}^{6}\mathit{Pa.}°{C}^{-1}`

We remind you that :math:`H=\frac{{\mathit{EE}}_{T}}{E-{E}_{T}}`


* Parameters for transformation plasticity:

Reminder:


.. csv-table::

    "In the case of a metallurgical evolution of the bainitic type, the model of transformation plasticity is as follows:
    :math:`{\dot{\epsilon }}^{\mathit{pt}}=\frac{3}{2}\stackrel{̃}{\sigma }{k}^{\mathit{fbm}}{F}^{\text{'}}({Z}_{\mathit{fbm}}){\dot{Z}}_{\mathit{fbm}}`"


             Model parameters: :math:`{k}^{\mathit{fbm}}=1.{10}^{-10}{\mathit{Pa}}^{-1}` and :math:`{F}^{\text{'}}({Z}_{\mathit{fbm}})=2(1-{Z}_{\mathit{fbm}})`


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


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


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

Ratings: :math:`\begin{array}{c}T({\tau }_{1})={T}_{1}\\ T({\tau }_{2})={T}_{2}\end{array}`


* 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}}`