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


Geometry
---------

Axisymmetric cylinder


.. image:: images/Shape1.gif


.. _RefSchema_Shape1.gif:


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

The material properties of the two sub-domains (blue and red) are those of 316L steel, described below. Work hardening restoration parameters are also defined but only on the red subdomain.

**Thermal parameters:**

Volume heat capacity :math:`\rho {C}_{p}=3.9\times {10}^{6}({\mathit{J.m}}^{-3.}°{C}^{-1})`

Conductivity :math:`\lambda ({\mathit{W.m}}^{-1}°{C}^{-1})`:


.. csv-table::

    "T (°C)", ":math:`\lambda`"
    "20. ", "14."
    "100. ", "15.2"
    "200. ", "16.6"
    "300. ", "17.9"
    "400. ", "19.0"
    "500. ", "20.6"
    "600. ", "21.8"
    "700. ", "23.1"
    "800. ", "24.3"
    "900. ", "26."
    "1000. ", "27.3"
    "1200. ", "29.9"
    "1450. ", "35."
    "1500. ", "70."


**Thermo-mechanical parameters:**


* Thermo-elastic parameters:

 Young's module :math:`E(\mathit{Pa})`

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

 Thermal expansion coefficient :math:`\alpha`

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

 Elasticity limit :math:`{\sigma }_{y}(\mathit{Pa})`


.. csv-table::

    "T (°C)", ":math:`E(\times {10}^{6})` ", "", "T (°C)", "T (°C)", ":math:`\alpha (\times {10}^{-6})` ", "", "T (°C)", ":math:`{\sigma }_{y}(\times {10}^{6})`"
    "20. ", "1956". ", "", "20. ", "14.56", "", "20. ", "286."
    "100. ", "191200. ", "", "100. ", "15.39", "", "200. ", "212."
    "200. ", "185700. ", "", "200. ", "16.21", "", "400. ", "180."
    "300. ", "17960. ", "", "300. ", "16.86", "", "600. ", "137."
    "400. ", "17260. ", "", "400. ", "17.37", "", "800. ", "139."
    "500. ", "16450. ", "", "500. ", "17.78", "", "1000. ", "70."
    "600. ", "15000. ", "", "600. ", "18.12", "", "1100. ", "35."
    "700. ", "144100. ", "", "700. ", "18.43", "", "1200. ", "16."
    "800. ", "131400. ", "", "800. ", "18.72", "", "1300. ", "10."
    "900. ", "11680. ", "", "900. ", "18.99", "", "1500. ", "10."
    "1000. ", "100,000. ", "", "1000. ", "19.27", "", "", ""
    "1100. ", "8000. ", "", "1100. ", "19.53", "", "", ""
    "1200. ", "57000. ", "", "1200. ", "19.79", "", "", ""
    "1300. ", "30,000. ", "", "1300. ", "20.02", "", "", ""
    "1400. ", "2000. ", "", "1600. ", "20.02", "", "", ""
    "1500. ", "100. ", "", "", "", "", "", ""



* Thermoplastic parameters (law with linear work hardening)

 Tangent module :math:`{E}_{T}(\mathit{Pa})`


.. csv-table::

    "T (°C)", ":math:`{E}_{T}(\times {10}^{6})`"
    "20. ", "2400."
    "700. ", "2400."
    "800. ", "2350."
    "900. ", "1500."
    "1000. ", "800."
    "1100. ", "725."
    "1200. ", "150."
    "1300. ", "10."



* Work hardening restoration parameter: multiplicative function FONC_MULT

 Restoration start temperature: Td = 600°C

 Temperature at the end of restoration: Tf = 1000°C


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:


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


* Blockages:

uy = 0 in P1, P2, P3 and P4


* Charging:

The following thermal load is applied all over the mesh:


    .. image:: images/Shape3.gif


    .. _RefSchema_Shape3.gif:

with T0 = 20°C and T1 = 1125°C