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


.. _Ref482175678:


Geometry
---------

Consider a notched cylindrical test piece:


* diameter of the test piece: :math:`18\mathrm{mm}`,


* notch radius: :math:`5\mathrm{mm}`.


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

We adopt a Von Mises elasto-plastic behavior law with isotropic work hardening TRACTION whose traction curve is given point by point:


.. csv-table::

    ":math:`\varepsilon` ", "0.0027", "0.005", "0.005", "0.01", "0.01", "0.015", "0.02", "0.03", "0.04", "0.05", "0.05", "0.05", "0.075", "0.075", "0.1"
    ":math:`\sigma` (:math:`\mathit{MPa}`)", "555", "589", "589", "589", "631", "631", "657", "704", "725", "741", "741", "772", "794"



.. csv-table::

    "0.125", "0.15", "0.2", "0.2", "0.3", "0.3", "0.4", "0.5", "0.6", "0.7", "0.8", "0.8", "0.9"
    "812", "827", "851", "887", "887", "97", "912", "933", "950", "965", "978", "990"


The deformations used in the behavior relationship are linearized deformations. Young's modulus :math:`E` is :math:`200\mathit{GPa}` while the Poisson's ratio :math:`\nu` is equal to :math:`\mathrm{0,3}`.

The coefficients of the Weibull and Bordet models used are as follows:

:math:`m=8`,

:math:`{V}_{0}\mathrm{=}100\mu m`,

:math:`{\sigma }_{u}\mathrm{=}2630\mathit{MPa}`,

:math:`{\sigma }_{\mathit{ys}\mathrm{,0}}\mathrm{=}{\sigma }_{\mathit{ys}}\mathrm{555MPa}`,

:math:`{\sigma }_{\mathit{th}}\text{=}\mathrm{600MPa}`.


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

With reference to the figure in [:ref:`§3.1 <§3.1>`] the boundary conditions are as follows:


* :math:`\mathrm{BC}`: displacement imposed according to :math:`(Y)`,


* :math:`\mathrm{OA}`: movements blocked according to :math:`(Y)`,


* :math:`\mathrm{OB}`: movements blocked following :math:`(X)`.


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

Zero stresses and deformations.