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


.. _Ref482175678:


Geometry
---------

A smooth cylindrical half-test specimen is considered.


.. image:: images/10000000000002AB000001BC1F39A487FF1FBC28.png
    :width: 6.8283in
    :height: 4.4374in


.. _RefImage_10000000000002AB000001BC1F39A487FF1FBC28.png:


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

A Von Mises elasto-plastic behavior law with linear isotropic work hardening 'VMIS_ISOT_LINE' is adopted. The deformations used in the behavior relationship are linearized deformations.


.. image:: images/1000214E000069BB00004D4F16B6534DE5FFFC59.svg
    :width: 219
    :height: 160


.. _RefImage_1000214E000069BB00004D4F16B6534DE5FFFC59.svg:

The Young's modulus :math:`E`, the tangent modulus :math:`{E}_{t}` and the Poisson's ratio do not depend on temperature. We take: :math:`E\mathrm{=}200\mathit{GPa}`, :math:`{E}_{t}\mathrm{=}2000\mathit{MPa}` and :math:`\nu \mathrm{=}\mathrm{0,3}`.

The evolution of the elastic limit with temperature is given in the following table:


.. csv-table::

    "**Temperature** :math:`\mathrm{[}°C\mathrm{]}` ", "— **150**", "— **100**", "— **50**", "— **50**"
    ":math:`{\sigma }_{Y}\mathrm{[}\mathit{MPa}\mathrm{]}` ", "750", "700", "700", "650"


Finally, thermal expansion is neglected (thermal expansion coefficient taken to be equal to 0).

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

With reference to the figure in [:ref:`§1.1 <§1.1>`] the boundary conditions are as follows:


* on the surface :math:`\mathit{SSUP}` :math:`\mathit{BC}` (:math:`Y\mathrm{=}{L}_{0}`) displacement

    .. image:: images/Object_1.svg
        :width: 219
        :height: 160


    .. _RefImage_Object_1.svg:

imposed in the direction :math:`\mathit{OY}`,


* on the surface :math:`\mathit{SINF}` :math:`\mathit{OA}` (:math:`Y\mathrm{=}0`) blocked movements in direction :math:`\mathit{OY}`,


* :math:`A` movements blocked following :math:`X` and :math:`Z`,


* :math:`B` movements blocked following :math:`Z`.

The temporal evolution of temperature (assumed to be homogeneous in the test piece) and of elongation

.. image:: images/Object_2.svg
    :width: 219
    :height: 160


.. _RefImage_Object_2.svg:

are shown in the following table:


+----------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----+-----+-----+------+
|Time :math:`\mathrm{[}s\mathrm{]}`                                                                                                                                    |10   |20   |30   |40    |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----+-----+-----+------+
|Temperature :math:`\mathrm{[}°C\mathrm{]}`                                                                                                                            |—50  |—50  |—150 |—150  |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----+-----+-----+------+
|Displacement                                                                                                                                                          |20,35|20,30|20,30|32,525|
+                                                                                                                                                                      +     +     +     +      +
|                                                                                                                                                                      |     |     |     |      |
+ .. image:: images/Object_3.svg                                                                                                                                       +     +     +     +      +
|    :width: 219                                                                                                                                                       |     |     |     |      |
+    :height: 160                                                                                                                                                      +     +     +     +      +
|                                                                                                                                                                      |     |     |     |      |
+                                                                                                                                                                      +     +     +     +      +
|                                                                                                                                                                      |     |     |     |      |
+                                                                                                                                                                      +     +     +     +      +
|                                                                                                                                                                      |     |     |     |      |
+ :math:`\mathrm{[}\mathit{mm}\mathrm{]}`                                                                                                                              +     +     +     +      +
|                                                                                                                                                                      |     |     |     |      |
+----------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----+-----+-----+------+


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

Zero stresses and deformations.