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


Geometry
---------

The stresses and deformations are homogeneous in the volume element. This can be represented by a plane or a solid element, for example:


.. image:: images/1000058200000FB600000CD1EFC9375AA8655098.svg
    :width: 202
    :height: 165


.. _RefImage_1000058200000FB600000CD1EFC9375AA8655098.svg:


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

Material properties are described by the following parameters:

**Thermo-metallurgical parameters:**


* Zircaloy:


    * :math:`\rho {C}_{p}={2.0E}^{-3}{\mathit{J.mm}}^{-3.}°{C}^{-1}`


    * :math:`\lambda =9.9999{\mathit{W.mm}}^{-1.}°{C}^{-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{t2r}=0.`, :math:`\mathit{Ar}\mathrm{=}\mathrm{-}5.725`, :math:`\mathit{Br}\mathrm{=}0.05`

**Thermo-metallo-mechanical parameters:**


* Thermo-metallo-elastic parameters:


    * Young's module: :math:`E=195000\mathit{MPa}`


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


    * Average thermal expansion coefficient for cold phases: :math:`{\alpha }_{f}=15E-6`


    * Average thermal expansion coefficient of hot phases: :math:`{\alpha }_{\gamma }=23E-6`


    * Expansion coefficient definition temperature: :math:`{T}_{\mathit{ref}}=600°C`


    * Choice of the reference metallurgical phase: :math:`\mathit{Froide}`


    * Deformation of the non-reference phase compared to the reference phase, at :math:`{T}_{\mathit{ref}}`: :math:`\Delta {\epsilon }_{f\gamma }^{{T}_{\mathit{ref}}}={2.52E}^{-3}`


    * Cold phase elasticity limit 1: :math:`{\sigma }_{y\mathrm{,1}}=181\mathit{MPa}`


    * Elastic limit of the cold phase 2: :math:`{\sigma }_{y\mathrm{,2}}=181\mathit{MPa}`


    * Elastic limit of the hot phase: :math:`{\sigma }_{y,\gamma }=0\mathit{MPa}`


    * Mixture function (calculation of the elastic limit of the multiphase material): :math:`\mathit{fonction}\mathit{identité}`


* Thermo-metallo-plastic parameters (law with linear work hardening)


    * Slope of the traction curve for cold phase 1: :math:`{E}_{T\mathrm{,1}}=1930\mathit{MPa}`


    * Slope of the traction curve for cold phase 2: :math:`{E}_{T\mathrm{,2}}=1930\mathit{MPa}`


    * Slope of the hot phase traction curve: :math:`{E}_{T,\gamma }=0\mathit{MPa}`


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


* **Modeling A:**

The volume element is locked along :math:`\mathit{Ox}` along the :math:`[\mathrm{2,4}]` side while being subjected to a pull :math:`{\sigma }^{D}` and a shear force :math:`{\tau }^{D}`.


.. image:: images/10000C960000110E00000B93C8C5FAB7DA58B9F8.svg
    :width: 202
    :height: 165


.. _RefImage_10000C960000110E00000B93C8C5FAB7DA58B9F8.svg:

:math:`{\tau }^{D}`
The loading path is as follows:

:math:`{\sigma }^{D}`

A

B

+------------------------------------------+---------------------------------+-------------------------------+
|                                          |:math:`{\sigma }^{D}\text{[MPa]}`|:math:`{\tau }^{D}\text{[MPa]}`|
+ .. image:: images/Shape1.gif             +                                 +                               +
|                                          |                                 |                               |
+                                          +                                 +                               +
|                                          |                                 |                               |
+------------------------------------------+---------------------------------+-------------------------------+
|:math:`A`                                 |151.2                            |93.1                           |
+------------------------------------------+---------------------------------+-------------------------------+
|O :math:`B`                               |257.2                            |33.1                           |
+------------------------------------------+---------------------------------+-------------------------------+


     OA path from :math:`t=0` to :math:`\mathrm{1s}`.

     AB route from :math:`t=1` to :math:`\mathrm{2s}`.

     BO route from :math:`t=2` to :math:`\mathrm{3s}`.

The temperature is imposed to be constant and equal to 600° C.


* **B modeling:**

B modeling is the exact equivalent of A modeling taking into account large deformations via the keyword DEFORMATION =' SIMO_MIEHE '.