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


Geometry
---------

The geometry studied is that of a bi-metal tank slice, limited to 45 degrees in azimuth.


.. image:: images/Cadre1.gif


.. _RefSchema_Cadre1.gif:


Defect considered
~~~~~~~~~~~~~~~~~

In method :math:`K\beta`, the defect is not modeled in the mesh. The mesh makes it possible to calculate the stresses at the nodes. A post-processing is then applied to calculate the stress intensity factor by the :math:`\beta` method from the stresses at the nodes (the method is detailed in [:ref:`R7.02.10 <R7.02.10>`]).

For this test, the defect in question is elliptical and has a longitudinal orientation. Its dimensions are as follows (see figure below):


* depth: :math:`{\mathit{prof}}_{\text{def}}\mathrm{=}\mathrm{6mm}`


* width: :math:`\mathrm{2b}\mathrm{=}\mathrm{60mm}`


* offset in the coating: :math:`\mathit{deca}\mathrm{=}–\mathrm{0,2}\mathit{mm}`


.. image:: images/10011412000069D500004A6A140E8084EC343E1E.svg
    :width: 543
    :height: 386


.. _RefImage_10011412000069D500004A6A140E8084EC343E1E.svg:


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

**For thermal calculation:**

Two properties are specified, they are:


* LAMBDA: isotropic thermal conductivity as a function of temperature, expressed in :math:`{\mathit{W.m}}^{\mathrm{-}1}\mathrm{.}{K}^{\mathrm{-}1}`,


* BETA: volume enthalpy as a function of temperature, expressed in :math:`{\mathit{J.m}}^{\mathrm{-}3}`.

For coating:


.. csv-table::

    "**Temperature (** :math:`°C` **)**", "**LAMBDA**"
    "0", "14.7"
    "20", "14.7"
    "50", "15.2"
    "100", "15.8"
    "150", "16.7"
    "200", "17.2"
    "250", "18"
    "300", "18.6"
    "350", "19.3"



.. csv-table::

    "**Temperature (** :math:`°C` **)**", "**BETA**"
    "0", "0.000000.E+00"
    "50", "1.102100.E+08"
    "100", "3.013300.E+08"
    "150", "5.014300.E+08"
    "200", "7.081300.E+08"
    "250", "9.188800.E+08"
    "300", "1.132910.E+09"
    "350", "1.348980.E+09"


For the base metal:


.. csv-table::

    "**Temperature (** :math:`°C` **)**", "**LAMBDA**"
    "0", "37.7"
    "20", "37.7"
    "50", "38.6"
    "100", "39.9"
    "150", "40.5"
    "200", "40.5"
    "250", "40.2"
    "300", "39.5"
    "350", "38.7"



.. csv-table::

    "**Temperature (** :math:`°C` **)**", "**BETA**"
    "0", "0.000000.E+00"
    "50", "1.061900.E+08"
    "100", "2.903300.E+08"
    "150", "4.829100.E+08"
    "200", "6.832800.E+08"
    "250", "8.921600.E+08"
    "300", "1.109440.E+09"
    "350", "1.335060.E+09"


**For mechanical calculation:**

Four parameters are filled in, they are:


.. csv-table::

    "* :math:`E`:", "Young's modulus, expressed in :math:`\mathit{Pa}`,"
    "* :math:`\mathit{nu}\mathrm{=}0.3` ", "Poisson's ratio,"
    "* ALPHA:", "isotropic thermal expansion coefficient, expressed in :math:`°C`,"
    "* TEMP_DEF_ALPHA = 20:", "value of the temperature at which the values of the thermal expansion coefficient ALPHAont were determined, expressed in :math:`°C`."


For coating:


.. csv-table::

    "**Temperature (** :math:`°C` **)**", ":math:`E`"
    "0", "1.985E+11"
    "20", "1.97E+11"
    "50", "1.95E+11"
    "100", "1.915E+11"
    "150", "1.875E+11"
    "200", "1.84E+11"
    "250", "1.8E+11"
    "300", "1.765E+11"
    "350", "1.72E+11"



+----------------------------------+----------++----------------------------------+---------+
|**Modeling a**                               ||**Modeling b**                              |
+----------------------------------+----------++----------------------------------+---------+
|**Temperature (** :math:`°C` **)**|**ALPHA** ||**Temperature (** :math:`°C` **)**|**ALPHA**|
+----------------------------------+----------++----------------------------------+---------+
|0                                 |1.756E-05 ||                                  |         |
+----------------------------------+----------++----------------------------------+---------+
|20                                |1.764E-05 ||20                                |1.64E-05 |
+----------------------------------+----------++----------------------------------+---------+
|50                                |1.7787E-05||50                                |1.654E-05|
+----------------------------------+----------++----------------------------------+---------+
|100                               |1.8019E-05||100                               |1.68E-05 |
+----------------------------------+----------++----------------------------------+---------+
|150                               |1.8225E-05||150                               |1.704E-05|
+----------------------------------+----------++----------------------------------+---------+
|200                               |1.8575E-05||200                               |1.72E-05 |
+----------------------------------+----------++----------------------------------+---------+
|250                               |1.8568E-05||250                               |1.75E-05 |
+----------------------------------+----------++----------------------------------+---------+
|300                               |1.8768E-05||300                               |1.777E-05|
+----------------------------------+----------++----------------------------------+---------+

For the base metal:


.. csv-table::

    "**Temperature (** :math:`°C` **)**", ":math:`E`"
    "0", "2.05E+11"
    "20", "2.04E+11"
    "50", "2.03E+11"
    "100", "2E+11"
    "150", "1.97E+11"
    "200", "1.93E+11"
    "250", "1.89E+11"
    "300", "1.85E+11"
    "350", "1.8E+11"



+----------------------------------+----------++----------------------------------+---------+
|**Modeling A**                               ||**B Modeling**                              |
+----------------------------------+----------++----------------------------------+---------+
|**Temperature (** :math:`°C` **)**|**ALPHA** ||**Temperature (** :math:`°C` **)**|**ALPHA**|
+----------------------------------+----------++----------------------------------+---------+
|0                                 |1.2878E-05||                                  |         |
+----------------------------------+----------++----------------------------------+---------+
|20                                |1.3002E-05||20                                |1.122E-05|
+----------------------------------+----------++----------------------------------+---------+
|50                                |1.3198E-05||50                                |1.145E-05|
+----------------------------------+----------++----------------------------------+---------+
|100                               |1.3521E-05||100                               |1.179E-05|
+----------------------------------+----------++----------------------------------+---------+
|150                               |1.382E-05 ||150                               |1.214E-05|
+----------------------------------+----------++----------------------------------+---------+
|200                               |1.4102E-05||200                               |1.247E-05|
+----------------------------------+----------++----------------------------------+---------+
|250                               |1.4382E-05||250                               |1.278E-05|
+----------------------------------+----------++----------------------------------+---------+
|300                               |1.4682E-05||300                               |1.308E-05|
+----------------------------------+----------++----------------------------------+---------+


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

The boundary conditions imposed are those of an axisymmetric system.

Two types of loads are applied:


* heat exchange in the inner skin,


* fluid pressure in the inner skin.