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


Geometry
---------

It is a semi-elliptical longitudinal crack opening into a narrow internal cavity plunged into a straight pipe. Symmetry planes are used in order to model only a quarter of the pipe.


.. image:: images/100002010000032800000224DC7A72A08238C33C.png
    :width: 6.3374in
    :height: 4.298in


.. _RefImage_100002010000032800000224DC7A72A08238C33C.png:

Figure 1.1-2: Defect geometry

 The tube has a length of 2,000 mm, a thickness of 60 mm, and an internal radius of 270 mm.

 The defect is 7.5 mm deep and 22.5 mm long.


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

     A rational traction curve of the Ramberg-Osgood type is used.



:math:`\frac{ϵ}{{ϵ}_{0}}=\frac{\sigma }{{\sigma }_{0}}+\alpha \mathrm{.}{(\frac{\sigma }{{\sigma }_{0}})}^{n}`, with :math:`{ϵ}_{0}=\frac{{\sigma }_{0}}{E}`

Where:

:math:`n=6`

:math:`\alpha =1`

:math:`{\sigma }_{0}=163\mathit{MPa}`

:math:`E=174700\mathit{MPa}`

:math:`\nu \mathrm{=}0.3`

     The law of material behavior is elasto-plastic (*' VMIS_ISOT_TRAC ')*

     Below is its traction curve.


.. image:: images/10000201000008EB00000669AB1710074C1F0D82.png
    :width: 6.889in
    :height: 4.9508in


.. _RefImage_10000201000008EB00000669AB1710074C1F0D82.png:

Figure 1.2-1: Ramberg-Osgood traction curve n=6, a=1 provided by [:ref:`bib1 <bib1>`].

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

Symmetry with respect to the 2 main planes:

:math:`{U}_{Y}=0.` in the :math:`Y=0` plan.

:math:`{U}_{Z}=0.` in the :math:`Z=0.` plane out of the crack



     Rigid mode on the :math:`X` axis.

     Average :math:`{U}_{X}` in nodes NODE_CENTRE_FISS_T and NODE_CENTRE_FISS_TM is zero.

The loading conditions are applied in two steps:

1) A pressure load :math:`P` is gradually applied to the inner pipe wall.

 A tensile load :math:`\text{EFFET\_FOND}` at the end of the pipe is also applied to model the background effect.

 :math:`\text{EFFET\_FOND}=P\frac{{R}_{\text{int}}^{2}}{{R}_{\text{ext}}^{2}-{R}_{\text{int}}^{2}}`

 Until maximum pressure :math:`P=36.97\mathit{MPa}`

2) By staying with a maximum pressure applied to the pipe, a next bending moment :math:`M` (:math:`-Y`) applying pressure to open the semi-elliptical defect is applied progressively until :math:`M=-5989.460\mathit{kNm}`.

         It is applied to the end of the tube via a linear distribution following :math:`x` of axial stresses :math:`{\sigma }_{\mathit{zz}}`

:math:`{\sigma }_{\mathit{zz}}=\frac{Mx}{I}` in the :math:`Z=2000\mathit{mm}` plan

where :math:`I` is the squared moment of bending. :math:`I=\pi \frac{({R}_{\text{ext}}^{4}-{R}_{\text{int}}^{4})}{4}`

Below is a figure with the pipe loading path given by [:ref:`bib1 <bib1>`].


.. image:: images/100002010000031000000208AD583E5CF16C276A.png
    :width: 5.9929in
    :height: 3.972in


.. _RefImage_100002010000031000000208AD583E5CF16C276A.png:

Figure 1.3-1: load path provided by [:ref:`bib1 <bib1>`].