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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 297
    :height: 297


.. _RefImage_Object_1.svg:

**Figure 1.1-a: Geometry of the reference problem**

It is a cylinder with height :math:`H\mathrm{=}\mathrm{20mm}`, inner radius :math:`\mathit{Rint}\mathrm{=}\mathrm{4.118mm}`, and outer radius :math:`\mathit{Rext}\mathrm{=}\mathrm{4.746mm}`.


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

Material properties are described by the following parameters:

Material properties are described by the following parameters:

**Thermal properties:**


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


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

**Metallurgical properties:**


* :math:`\mathit{TDEQ}\mathrm{=}809°C`


* :math:`K\mathrm{=}{\mathrm{1.135.10}}^{\mathrm{-}2}`


* :math:`N\mathrm{=}2.187`


* :math:`\mathit{T1C}\mathrm{=}831°C`


* :math:`\mathit{T2C}\mathrm{=}0.°C`


* :math:`{\mathit{QSR}}_{K}\mathrm{=}14614`


* :math:`\mathit{AC}\mathrm{=}{\mathrm{1.58.10}}^{\mathrm{-}4}`


* :math:`M\mathrm{=}4.7`


* :math:`\mathit{T1R}\mathrm{=}\mathrm{949,1}°C`


* :math:`\mathit{T2R}\mathrm{=}0.°C`


* :math:`\mathit{AR}\mathrm{=}\mathrm{-}5.725`


* :math:`\mathit{BR}\mathrm{=}0.05`

**Thermo-elastic mechanical properties:**


* Young's modulus: :math:`E\mathrm{=}80000\mathit{MPa}`


* Poisson's ratio: :math:`\mathit{NU}\mathrm{=}0.35`


* Same expansion coefficient for hot and cold phases :math:`{F}_{\mathit{ALPHA}}=\mathrm{8,0}\times {10}^{-6}°{C}^{-1}` and :math:`{C}_{\mathit{ALPHA}}=\mathrm{8,0}\times {10}^{-6}°{C}^{-1}`

**Mechanical Properties of the Law** META_LEMA_ANI


* Parameters related to viscosity, phase :math:`\alpha` pure

F1_A = 2.39

F1_M = 0.

F1_N = 4.39

F1_Q = 19922.8


* Viscosity parameters, blend :math:`\alpha +\beta`

F2_A = 0.22

F2_M = 0.77 E-4

F2_N = 2.96

F2_Q = 21023.7


* Parameters related to viscosity, phase :math:`\beta` pure

C_A = 9.36

C_M = 0.99 E-4

C_N = 6.11

C_Q = 6219


* Coefficient of the anisotropy matrix in plane :math:`(r,\theta ,z)`, phase a

F_ MRR_RR = 0.4414

F_ MTT_TT = 0.714

F_ MZZ_ZZ = 1

F_ MRT_RT = 0.75

F_ MRZ_RZ = 0.75

F_ MTZ_TZ = 0.75


* Coefficient of the anisotropy matrix in plane :math:`(r,\theta ,z)`, phase b

C_ MRR_RR = 1

C_ MTT_TT = 1

C_ MZZ_ZZ = 1

C_ MRT_RT = 0.75

C_ MRZ_RZ = 0.75

C_ MTZ_TZ = 0.75


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

**Thermal part:**

A uniform temperature is imposed all over the tube:


.. csv-table::

    "Time (:math:`s`)", "Temperature (:math:`°C`)"
    "-1. ", "20."
    "0. ", "609."
    "36.1", "609."
    "44. ", "799.7"
    "46. ", "838.67"
    "48. ", "876.52"
    "49.2", "894.5"


**Mechanical part:**

The lower part of the cylinder (:math:`\text{FACE\_INF}`) is blocked when moving as follows :math:`z`: :math:`\mathit{UZ}(x,y\mathrm{,0})\mathrm{=}0`

The entire upper part of the cylinder (:math:`\text{FACE\_SUP}`) has a uniform :math:`z` displacement

Pressure is imposed on the inside face of the tube (:math:`\text{FACE\_INT}`):


.. csv-table::

    "Time (:math:`s`)", "Pressure (:math:`\mathit{MPa}`)"
    "-1.0", "0."
    "0. ", "0."
    "36.1", "6.74"
    "49.2", "6.74"


We take into account the background effect on the upper part of the tube (FACE_SUP):


.. csv-table::

    "Time (:math:`s`)", "Pressure (:math:`\mathit{MPa}`)"
    "-1.0", "0."
    "0. ", "0."
    "36.1", "6.74*coeff"
    "49.2", "6.74*coeff"


With :math:`\mathit{coef}\mathrm{=}(\mathit{Rint}\mathrm{\times }\mathit{Rint})\mathrm{/}\mathrm{[}(\mathit{Rext}\mathrm{\times }\mathit{Rext})\mathrm{-}(\mathit{Rint}\mathrm{\times }\mathit{Rint})\mathrm{]}`