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


Geometry
---------

:math:`y`


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

:math:`{u}^{\mathit{meca}}`

:math:`x`

**Figure 1.1-a: Geometry and reference problem load**

It is a cylinder with height :math:`H=1000\mathit{mm}`, and radius :math:`R=1000\mathit{mm}`.

The square in bold corresponds to the axisymmetric modeling used in §3.


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

Material properties are described by the following parameters:

**For thermo-metallic calculation**


* (:math:`\mathrm{16MND5}` steel)


* :math:`\rho {C}_{p}={5.26E}^{-3}{\mathit{J.mm}}^{-3.}°{C}^{-1}`


* :math:`\lambda ={33.5E}^{-3}{\mathit{W.mm}}^{-1.}°{C}^{-1}`

**Coefficient for metallurgy:**


* :math:`\mathrm{TRC}` "standard"


* :math:`\mathrm{AR3}=830°C`, :math:`\mathrm{alpha}=-0.0306`


* :math:`\mathrm{MS0}=400°C`, :math:`\mathrm{AC1}=724°C`, :math:`\mathrm{AC3}=846°C`


* :math:`{\tau }_{1}=0.034`, :math:`{\tau }_{3}=0.034`

**For thermo-metallo-mechanical calculation**


* Young's module: :math:`E=200000\mathit{MPa}`


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

**Definition of elastic characteristics, expansion and elastic limits for modeling a material undergoing metallurgical transformations:**


* :math:`{T}_{\mathrm{ref}}=20°C`


* Average thermal expansion coefficient for cold phases: :math:`{\alpha }_{f}(T)=10{E}^{-6}`


* Average thermal expansion coefficient of the hot phase: :math:`{\alpha }_{\gamma }(T)=10{E}^{-5}`


* Expansion coefficient definition temperature: :math:`{T}_{\gamma }=20°C`


* Choice of the reference metallurgical phase: cold


* Deformation of the non-reference phase with respect to the reference phase at temperature :math:`{T}_{\mathrm{ref}}`: :math:`\Delta \varepsilon =1{E}^{-2}`


* Elastic limit of the cold phase 1 for plastic behavior:

:math:`\text{F\_sigm\_f}(T)=100\mathit{MPa}`


* Elastic limit of the cold phase 2 for plastic behavior:

:math:`\text{F\_sigm\_f}(T)=100\mathit{MPa}`


* Elastic limit of the cold phase 3 for plastic behavior:

:math:`\text{F\_sigm\_f}(T)=100\mathit{MPa}`


* Elastic limit of the cold phase 4 for plastic behavior:

:math:`\text{F\_sigm\_f}(T)=100\mathit{MPa}`


* Elastic limit of the hot phase for plastic behavior:

:math:`\text{F\_sigm\_f}(T)=100\mathit{MPa}`


* Function used for the mixing law on the elastic limit of multiphase material for plastic behavior:


.. image:: images/10000201000001B90000018DC845C39D2B792D23.png
    :width: 3.2854in
    :height: 2.9528in


.. _RefImage_10000201000001B90000018DC845C39D2B792D23.png:

**Figure 1.2-a: Law of Mixing**

**Definition of the 5 traction curves used in the modeling of non-linear isotropic work hardening of a material undergoing metallurgical phase changes:**

Isotropic work hardening curve :math:`R` as a function of the cumulative plastic deformation :math:`p` for the cold phase 1


*

to :math:`20°C`:

.. csv-table::

    ":math:`p` "," :math:`\sigma (\mathit{Mpa})`"
    "0.99", "250"


To :math:`120°C`:

.. csv-table::

    ":math:`p` "," :math:`\sigma (\mathit{MPa})`"
    "0.0105", "90"
    "0.032", "160"
    "0.064", "220"
    "0.1125", "250"
    "0.1815", "270"


Isotropic work hardening curve :math:`R` as a function of the cumulative plastic deformation :math:`p` for cold phase 2:

Same as previous

Isotropic work hardening curve :math:`R` as a function of the cumulative plastic deformation :math:`p` for cold phase 3:

Same as previous

Isotropic work hardening curve :math:`R` as a function of the cumulative plastic deformation :math:`p` for cold phase 4:

Same as previous

Isotropic work hardening curve :math:`R` as a function of the cumulative plastic deformation :math:`p` for the hot phase:

Same as previous


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

The base of the cylinder is stuck following :math:`y`:

:math:`\mathrm{Uy}=0` on the base of the cylinder.

A :math:`{u}^{\mathit{meca}}=30\mathit{mm}` displacement is imposed on the top of the cylinder.

The temperature is imposed all over the cylinder, such that:


.. image:: images/Object_167.png
    :width: 4.5102in
    :height: 2.2366in


.. _RefSchema_Object_167.png:


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

Initial temperature: :math:`T(x,y\mathrm{,0})=20°C`

The following variables are initialized:

:math:`{Z}_{f}(x,y\mathrm{,0})=0.7`

:math:`{Z}_{p}(x,y\mathrm{,0})=0.0`

:math:`{Z}_{b}(x,y\mathrm{,0})=0.3`

:math:`{Z}_{m}(x,y\mathrm{,0})=0.0`

:math:`d(x,y\mathrm{,0})=0.0`