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


Geometry
---------

We consider a three-dimensional bar whose dimensions are:


* height: :math:`\mathit{LZ}` = 4 mm,


* side: :math:`\mathit{LX}` = :math:`\mathit{LY}` = 1 mm.

This bar has a plane, semi-elliptical crack. The crack is located in plane :math:`\mathit{Oxy}`. The characteristics of the cracks are as follows:


* semi-major axis: :math:`a` = 119 µm


* semi-minor axis: :math:`b` = 100 µm.


.. image:: images/Cadre11.gif


.. _RefSchema_Cadre11.gif:


.. image:: images/Cadre11-1.gif


.. _RefSchema_Cadre11-1.gif:


.. image:: images/Cadre11-2.gif


.. _RefSchema_Cadre11-2.gif:


.. image:: images/Cadre11-3.gif


.. _RefSchema_Cadre11-3.gif:

● A

● B

● C
Figure 1.1-1: Geometry of the initial crack

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

The material is isotropic elastic whose properties are:

:math:`E\mathrm{=}200000\mathit{MPa}`

:math:`\nu \mathrm{=}\mathrm{0,3}`


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


Cyclic loading for fatigue study
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The structure is subject to a loading of fatigue under constant amplitude: traction :math:`{\sigma }_{\mathit{max}}=220\mathit{MPa}` and a ratio :math:`R\mathrm{=}\mathrm{0,1}`. The temperature is room temperature. The charging frequency is :math:`40\mathit{Hz}`. A load of 4000 cycles is applied.

The traction force is applied to the upper and lower faces.

The blocking of rigid modes is carried out in the following way:


* point :math:`A` is stuck in the :math:`\mathit{Oy}` and :math:`\mathit{Oz}` directions,


* point :math:`B` is stuck in the :math:`\mathit{Oy}` and :math:`\mathit{Oz}` directions,


* point :math:`C` is stuck in the :math:`\mathit{Ox}` and :math:`\mathit{Oz}` directions.


Modeling with cohesive zones: monotonic loading
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For this modeling, the load is monotonic instead of cyclical: the structure is subjected to :math:`{\sigma }_{\mathit{max}}\mathrm{=}220\mathit{MPa}` traction. The traction force is applied to the upper and lower faces.

The blocking of rigid modes is carried out in the following way:


* point :math:`A` is stuck in the :math:`\mathit{Oy}` and :math:`\mathit{Oz}` directions,


* point :math:`B` is stuck in the :math:`\mathit{Oy}` and :math:`\mathit{Oz}` directions,


* point :math:`C` is stuck in the :math:`\mathit{Ox}` and :math:`\mathit{Oz}` directions.