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


Geometry
---------


.. image:: images/1000052400001B980000129BB7A905A498F51911.svg
    :width: 355
    :height: 240


.. _RefImage_1000052400001B980000129BB7A905A498F51911.svg:

Figure 1.1-1: Reference geometry


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

The material obeys a law of behavior in large plastic deformations with linear isotropic work hardening.

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


+-----------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|:math:`\sigma \mathrm{=}\frac{F}{S}\mathrm{=}\frac{F}{{S}_{0}}\mathrm{.}\frac{l}{{l}_{0}}`                                                     |                                                                                                                                                                           |
+-----------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                                               |:math:`\begin{array}{c}\nu \mathrm{=}\mathrm{0,3}\\ E\mathrm{=}\mathrm{200000MPa}\\ {E}_{T}\mathrm{=}\mathrm{2000MPa}\\ {\sigma }_{y}\mathrm{=}\mathrm{1000MPa}\end{array}`|
+ .. image:: images/10000BA4000069D500004B3E0227F64FC3AA1029.svg                                                                                +                                                                                                                                                                           +
|    :width: 355                                                                                                                                |                                                                                                                                                                           |
+    :height: 240                                                                                                                               +                                                                                                                                                                           +
|                                                                                                                                               |                                                                                                                                                                           |
+ Figure 1.2-1: Tensile Curve                                                                                                                   +                                                                                                                                                                           +
|                                                                                                                                               |                                                                                                                                                                           |
+-----------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


:math:`{l}_{0}` and :math:`l` are, respectively, the initial length and the current length of the useful part of the test piece.

:math:`{S}_{0}` and :math:`S` are, respectively, the initial and current surfaces.


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

The bar, of initial length :math:`{l}_{0}`, locked in the direction :math:`Ox` on the face [:ref:`1,2 <1,2>`] has a mechanical traction movement :math:`{u}^{\mathit{meca}}` varying linearly in time on the face :math:`\mathrm{[}\mathrm{3,}4\mathrm{]}`:


.. image:: images/10000DDE00000E2900000B440A7CDEAB66C949B6.svg
    :width: 355
    :height: 240


.. _RefImage_10000DDE00000E2900000B440A7CDEAB66C949B6.svg:

Figure 1.3-1: Limit conditions and loading