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

It is a cylindrical creep bar.


Geometry
---------


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:

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

It is a cylinder with height :math:`H\mathrm{=}1.0m`, and radius :math:`R\mathrm{=}1.0m`.

The square in bold corresponds to the axisymmetric modeling used in [:ref:`§3 <§3>`].


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

Material properties are described by the following parameters:

**For thermo-metallic calculation**

(Zircaloy)

:math:`\rho {C}_{p}\mathrm{=}2000000{\mathit{J.m}}^{\mathrm{-}3.}°{C}^{\mathrm{-}1}`

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

Coefficient for metallurgy:

:math:`\mathit{teqd}\mathrm{=}809°C`, :math:`K\mathrm{=}1.135{E}^{\mathrm{-}2}`, :math:`n\mathrm{=}2.187`

:math:`\mathit{t1c}\mathrm{=}831°C`, :math:`\mathit{t2c}\mathrm{=}0.`, :math:`\mathit{qsr}\mathrm{=}14614`, :math:`\mathit{Ac}\mathrm{=}1.58E\mathrm{-}4`

:math:`m\mathrm{=}4.7`, :math:`\mathit{t1r}\mathrm{=}\mathrm{949,1}°C`, :math:`\mathit{t2e}\mathrm{=}0.`, :math:`\mathit{Ar}\mathrm{=}\mathrm{-}5.725`, :math:`\mathit{Br}\mathrm{=}0.05`

**For thermo-metallo-mechanical calculation**


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


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

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


* :math:`{T}_{\mathit{ref}}\mathrm{=}800°C`


* Average thermal expansion coefficient for cold phases: :math:`{\alpha }_{f}(T)\mathrm{=}0`


* Average thermal expansion coefficient of the hot phase: :math:`{\alpha }_{\gamma }(T)\mathrm{=}0`


* Expansion coefficient definition temperature: :math:`{T}_{\gamma }\mathrm{=}800°C`


* Choice of the reference metallurgical phase: hot


* Deformation of the non-reference phase with respect to the reference phase at temperature :math:`{T}_{\mathit{ref}}`: :math:`\Delta \varepsilon \mathrm{=}0`


* Cold phase elasticity limit 1 for viscous behavior: :math:`{F}_{{\mathit{sigm}}_{f}}(T)\mathrm{=}0`


* Elastic limit of the cold phase 2 for viscous behavior: :math:`{F}_{{\mathit{sigm}}_{f}}(T)\mathrm{=}0`


* Elastic limit of the hot phase for viscous behavior: see [:ref:`Figure 1.2-a <Figure 1.2-a>`]


.. image:: images/10000201000001B50000018D6EF9DA5F2D5BF677.png
    :width: 3.2102in
    :height: 2.9146in


.. _RefImage_10000201000001B50000018D6EF9DA5F2D5BF677.png:

**Figure 1.2.-a: Elastic limit of the hot phase for viscous behavior**


1. Function used for the mixing law on the elastic limit of multiphase material for viscous behavior: :math:`f`


.. image:: images/10000201000001B90000018DEF50D3C942FF1EAE.png
    :width: 3.1126in
    :height: 2.798in


.. _RefImage_10000201000001B90000018DEF50D3C942FF1EAE.png:

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

Definition of the work hardening modules used in modeling the phenomenon of linear isotropic work hardening of a material undergoing metallurgical phase changes:


1. Slope of the traction curve for cold phase 1


.. image:: images/10000201000002C20000018D39182F1A56C4E9E9.png
    :width: 3.811in
    :height: 2.6035in


.. _RefImage_10000201000002C20000018D39182F1A56C4E9E9.png:

**Figure 1.2-c: Tensile curve for cold phase 1**


* Slope of the traction curve for cold phase 2:

:math:`f(T)\mathrm{=}0`


* Slope of the traction curve for the hot phase:

:math:`f(T)\mathrm{=}0`

**Definition of the viscous parameters of the law of viscoplastic behavior taking into account metallurgy:**


1. Parameter :math:`\eta` of the viscoplastic flow law, for cold phase 1


.. image:: images/10000201000001C50000018D824885AB32BD330D.png
    :width: 3.7272in
    :height: 3.2638in


.. _RefImage_10000201000001C50000018D824885AB32BD330D.png:

**Figure 1.2-d: Parameter** :math:`\eta` **of the viscoplastic flow law, for the cold phase 1**


1. Parameter :math:`\eta` of the viscoplastic flow law, for cold phase 2


.. image:: images/10000201000001CB0000018DC7E823325EFA0793.png
    :width: 3.8252in
    :height: 3.3028in


.. _RefImage_10000201000001CB0000018DC7E823325EFA0793.png:

**Figure 1.2-e: Parameter** :math:`\eta` **of the viscoplastic flow law, for the cold phase 2**


* Parameter :math:`\eta` of the viscoplastic flow law, for the hot phase:

:math:`f(T)\mathrm{=}0`


* Parameter :math:`n` of the viscoplastic flow law, for cold phase 1:

:math:`f(T)\mathrm{=}5.76`


* Parameter :math:`n` of the viscoplastic flow law, for cold phase 2:

:math:`f(T)\mathrm{=}2.94`


* Parameter :math:`n` of the viscoplastic flow law, for the hot phase:

:math:`f(T)\mathrm{=}1.0`


* Parameter :math:`C` relating to the restoration of viscous work hardening, for cold phase 1:

:math:`f(T)\mathrm{=}13.70539827`


* Parameter :math:`C` relating to the restoration of viscous work hardening, for cold phase 2:

:math:`f(T)\mathrm{=}0`


* Parameter :math:`C` relating to the restoration of viscous work hardening, for the hot phase:

:math:`f(T)\mathrm{=}0`


* Parameter :math:`m` relating to the restoration of viscous work hardening, for the cold phase1:

:math:`f(T)\mathrm{=}5.76`


* Parameter :math:`m` relating to the restoration of viscous work hardening, for the cold phase2:

:math:`f(T)\mathrm{=}1.0`


* Parameter :math:`m` relating to the restoration of viscous work hardening, for the hot phase:

:math:`f(T)\mathrm{=}1.0`


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

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

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

A :math:`F\mathrm{=}25N` traction force is imposed on the top of the cylinder

The temperature is imposed all over the cylinder for :math:`t\mathrm{=}\mathrm{120s}`.

:math:`T(x,y\mathrm{,120})\mathrm{=}800°C`


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

The following variables are initialized:


* :math:`T(x,y\mathrm{,0})\mathrm{=}800°C`


* :math:`\mathit{V1}(x,y\mathrm{,0})\mathrm{=}1.0`


* :math:`\mathit{V2}(x,y\mathrm{,0})\mathrm{=}0.0`


* :math:`\mathit{V3}(x,y\mathrm{,0})\mathrm{=}20.`


* :math:`\mathit{V4}(x,y\mathrm{,0})\mathrm{=}0.`

:math:`\mathit{V1}` *: proportion of the cold phase* :math:`\alpha`

:math:`\mathit{V2}` *: proportion of the cold phase* :math:`\alpha` *, mixed with the phase* :math:`\beta`

:math:`\mathit{V3}` *: node temperatures*

:math:`\mathit{V4}` *: time corresponding to the temperature at which the start or end of the equilibrium transformation*