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

Geometry
---------


.. image:: images/1000097C000069D5000031320E42FBA4DB364EED.svg
    :width: 475
    :height: 220


.. _RefImage_1000097C000069D5000031320E42FBA4DB364EED.svg:


.. csv-table::

    "Bar length", ": :math:`1m`"
    "Bar section", ": :math:`5{\mathrm{cm}}^{2}`"



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

Isotropic hardening and linear kinematics
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. image:: images/10000A0A000069BB00003A9A478AF30F2329E070.svg
    :width: 475
    :height: 220


.. _RefImage_10000A0A000069BB00003A9A478AF30F2329E070.svg:


.. csv-table::

    "Young's module:", ":math:`E=2.{10}^{11}\mathrm{Pa}`"
    "Work hardening slope:", ":math:`{E}_{t}={2.10}^{9}\mathrm{Pa}`"
    "Elastic limit:", ":math:`\sigma ={2.10}^{8}\mathrm{Pa}`"
    "Poisson's ratio:", ":math:`\nu =\mathrm{0,3}`"
    "Coefficient of thermal expansion:", ":math:`\alpha ={1.10}^{-5}{K}^{-1}`"



Pinto-Menegotto model
~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. image:: images/10001994000069BB0000437D9A4C85602318F7A0.svg
    :width: 475
    :height: 220


.. _RefImage_10001994000069BB0000437D9A4C85602318F7A0.svg:


.. csv-table::

    "Young's modulus:", "E", "", "=", "2." :math:`{10}^{11}\mathrm{Pa}`"
    "Elastic limit:", ":math:`{\sigma }_{y}^{0}` ", "", "", "=", ":math:`{2.10}^{8}\mathrm{Pa}`"
    "Poisson's ratio:", ":math:`\nu` ", "", "=", "0.3"
    "Coefficient of thermal expansion:", ":math:`\alpha` ", "", "=", ":math:`{1.10}^{-5}{K}^{-1}`"
    "Work hardening deformation:", ":math:`{\varepsilon }_{h}` ", "", "=", ":math:`2.3{10}^{-3}`"
    "Ultimate constraint:", ":math:`{\sigma }_{u}` ", "", "", "=", ":math:`2.58{10}^{8}\mathrm{Pa}`"
    "Ultimate deformation:", ":math:`{\varepsilon }_{u}` ", "", "", "=", ":math:`{3.10}^{-2}`"
    "Coefficient defining curve :math:`\zeta`:", ":math:`{R}_{0}` ", "", "=", "", "20"
    "Coefficient defining the curve :math:`\zeta` "," :math:`{A}_{1}` ", "", "=", "18.5"
    "Coefficient defining the curve :math:`\zeta` "," :math:`{A}_{2}` ", "", "=", "0.15"
    "Buckle coefficient:", ":math:`C` ", "", "=", "0.5"
    "Buckling coefficient:", ":math:`A` ", "", "=", "0.008"



Boundary conditions and loading
------------------------------------

**Boundary conditions:**

The bar is recessed. Travel is therefore blocked in all three directions.

In :math:`\mathrm{N1}` and :math:`\mathrm{N2}`: :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=0`

**Charging:**

The loading path is described by the evolution of the temperature, which is uniform across the bar:

.. csv-table::

    ":math:`t` ", "0", "1", "2", "2", "2", "3", "4", "5", "6", "7"
    ":math:`T(°C)` ", "50", "—50", "—50", "—300", "—300", "—100", "—150", "—350", "—200"


The reference temperature is :math:`0°C`.


.. image:: images/Object_133.png
    :width: 334
    :height: 185


.. _RefSchema_Object_133.png: