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


Geometry
---------


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    :height: 2.5in


.. _RefImage_1000020000000164000000F04B4E75C57E08C7B1.png:


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

The material obeys a law of behavior in large plastic deformations with isotropic work hardening defined by a traction curve (point by point or power law).

The tensile curve is given in the plane logarithmic deformation - rational stress.


+-----------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\sigma =\frac{F}{S}=\frac{F}{{S}_{o}}\cdot \frac{l}{{l}_{o}}`                                                         |:math:`R(p)={\sigma }_{y}+{\sigma }_{y}{(\frac{E}{a{\sigma }_{y}}p)}^{1/n}`                                                                             |
+-----------------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                             |:math:`\begin{array}{cc}\nu & =0.3\\ \alpha & ={10}^{-4}{K}^{-1}\\ {\sigma }_{y}& =1000\mathrm{MPa}\\ E& =200000\mathrm{MPa}\\ n& =7\\ a& =1\end{array}`|
+ .. image:: images/100000000000012A0000013200CD6E22C2675915.png                                                              +                                                                                                                                                        +
|    :width: 3.0575in                                                                                                         |                                                                                                                                                        |
+    :height: 3.1398in                                                                                                        +                                                                                                                                                        +
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+                                                                                                                             +                                                                                                                                                        +
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:math:`{l}_{o}` and :math:`l` are, respectively, the initial length and the current length of the useful part of the test piece.

:math:`{S}_{o}` and :math:`S` are, respectively, the initial and current surfaces. Between temperatures :math:`20°C` and :math:`120°C`, the characteristics are interpolated linearly.


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

The bar, of initial length :math:`{l}_{o}`, locked in the direction :math:`\mathrm{Ox}` on the face [:ref:`1,2 <1,2>`] is subjected to a uniform temperature :math:`T` and to a mechanical tensile movement :math:`{u}^{\text{meca}}` on the face [:ref:`3, 4 <3, 4>`]. The loading sequences are as follows:

Y



X



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    :width: 180
    :height: 144


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    :width: 180
    :height: 144


.. _RefImage_10000E8A00002D7A00000D3BC61E35BAFDB27AA4.svg:

Reference temperature: :math:`{T}_{\text{réf}}\mathrm{=}20°C`.

**Note:**

*The mechanical displacement is measured from the configuration deformed by thermal loading (* :math:`t\mathrm{=}\mathrm{1s}` *). To get the total displacement, you must therefore add the thermal displacement obtained to the time* :math:`t\mathrm{=}\mathrm{1s}` *.*