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


Geometry
---------

The crack is a circular crown in a plane orthogonal to the cylinder axis [Figure 1.1-a :ref:`None <fig1>` :ref:`None <fig1>`]. Parameters :math:`a` and :math:`b` determine the radius of the cylinder and the depth of the crack. The [:ref:`Figure 1.1-b <Figure 1.1-b>`] is a section of the cylinder in the crack plane (plane :math:`\mathrm{Oyz}`). For the medium to be considered infinite, the height of the cylinder is :math:`h=10b`.


.. image:: images/10000000000001AE000002055669D92139E5B977.png
    :width: 2.1453in
    :height: 3.7154in


.. _RefImage_10000000000001AE000002055669D92139E5B977.png:


.. _Ref113932890:

**Figure 1.1-** a **: Geometry of the cracked cylinder**


.. image:: images/Object_1.svg
    :width: 259
    :height: 249


.. _RefImage_Object_1.svg:


.. _Ref54005267:


.. _Ref54004673:

**Figure 1.1-** b **: Crack plan**


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

Young's module: :math:`E=205000\mathrm{MPa}`

Poisson's ratio: :math:`\nu =0.3`


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

Three loads will be applied in order to calculate the stress intensity factors :math:`\mathrm{K1}` and :math:`\mathrm{K3}` in :math:`\mathrm{3D}` using the POST_K1_K2_K3 operator.

Load 1 tests :math:`\mathrm{K1}` and :math:`\mathrm{K3}`.

Charging 2 tests :math:`\mathrm{K2}` without taking contact into account.

Charging 3 tests :math:`\mathrm{K2}` with contact taken into account.

:math:`\mathrm{K1}` and :math:`\mathrm{K3}` are expected to be constant along the crack bottom and for :math:`\mathrm{K2}` to vary.

Note: cases of traction and twisting can be treated equally with or without contact (here, without contact) because there is never a closure of the crack.


.. csv-table::

    "", "Case 1: traction and twist", "Case 2: non-contact bending", "Case 3: contact bending"
    "Top side", ":math:`{N}_{x}=6\mathrm{MN}`
    :math:`{T}_{x}=3\mathrm{MN}` "," :math:`{M}_{y}=1.5\mathrm{MN}` "," :math:`{M}_{y}=1.5\mathrm{MN}`"


Table 1.3-1: Load cases

The above efforts are applied to the structure through discrete :math:`\mathrm{3D}` elements located at the center of the upper face. Note that the maximum opening point due to the imposed bending (next moment :math:`\mathrm{Oy}`) will be the :math:`A` node (see [:ref:`Figure 1.1‑b <Figure 1.1‑b>`].

Rigid body movements are blocked by the same method by embedding the center of the underside.