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


Geometry
---------

It is a semi-elliptical circumferential crack opening out into a narrow internal hole plunged into a straight pipe. We only model a quarter of a pipe:


.. image:: images/100002010000032800000224C719561A7006B63A.png
    :width: 6.889in
    :height: 4.672in


.. _RefImage_100002010000032800000224C719561A7006B63A.png:

Figure 1.1-2: Defect geometry

The tube has a length of 1500 mm, a thickness of 60 mm, and an internal radius of 270 mm.

The defect is 15 mm deep and 45 mm long.


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

     316L austenitic steel at 350°C- is used for all models.

:math:`E=177000\mathit{MPa}`

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

:math:`{\sigma }_{\gamma }=120\mathit{MPa}`

     The laws of material behavior vary for each model.



     Linear elastic (*' ELAS '*, modeling :math:`A`)

     Nonlinear elastic (*' ELAS_VMIS_TRAC ',* modeling :math:`B`)

     Elasto-plastic (*' VMIS_ISOT_TRAC ',* modeling :math:`C`)

     Below is the traction curve used.


.. image:: images/10000201000008F20000053D8A21AC68115FA9F6.png
    :width: 6.889in
    :height: 4.0335in


.. _RefImage_10000201000008F20000053D8A21AC68115FA9F6.png:

Figure 1.2-1:316L austenitic steel tensile curve at 350°C 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:

A following bending moment (:math:`-Y`) stresses the semi-elliptical defect in the opening in the most penalizing manner.

:math:`M=-5216\mathit{kNm}` where :math:`M` is the moment on the :math:`Y` axis.

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=1500\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}`