Benchmark solution
=====================


Calculation method used for the reference solution
--------------------------------------------------------

For an axisymmetric crack in a cylinder of infinite length, the method of Singular Integral Equations and Asymptotic Developments [:ref:`bib1 <bib1>`] makes it possible to calculate the values of the stress intensity factors.


1. Case 1: Traction and Torsion

Traction induces an opening in mode 1. :math:`\mathrm{K1}` is given by the following formula:

:math:`{K}_{I}=\frac{P}{\pi {a}^{2}}\sqrt{\pi a}{F}_{1}(a/b)`

.. code-block:: text

    where :math:`P` is the force applied to the upper and lower faces and :math:`{F}_{1}` is a given function [:ref:`Figure 2.1-a <Figure 2.1-a>`].


.. code-block:: text

    Torsion induces an opening in mode 3. :math:`\mathrm{K3}` is given by the following formula:

:math:`{K}_{\mathrm{III}}=\frac{\mathrm{2T}}{\pi {a}^{3}}\sqrt{\pi a}{F}_{3}(a/b)`

.. code-block:: text

    where T is the moment applied to the upper and lower faces and :math:`{F}_{3}` is a given function [:ref:`Figure2.1‑a <Figure2.1‑a>`].


1. Case 2: Non-contact bending

Flexion induces an opening in :math:`1` mode. The value of :math:`\mathrm{K1}` at the maximum opening point :math:`A` is given by the following formula:

:math:`{K}_{{I}_{A}}=\frac{\mathrm{4M}}{\pi {a}^{3}}\sqrt{\pi a}{F}_{2}(a/b)`

.. code-block:: text

    where :math:`M` is the moment applied to the upper and lower faces and :math:`{F}_{2}` is a given function [:ref:`Figure2.1‑a <Figure2.1‑a>`].


1. Case 3: Flexion with contact

There is no analytical solution to this problem. On the one hand, it is expected that :math:`\mathrm{K1}` will be close to the case without contact on the part of the crack that is being opened, and on the other hand that :math:`\mathrm{K1}` will be zero on the part of the crack that is being closed.


.. image:: images/100000000000040000000300F8F4D0D03F8E367A.png
    :width: 5.6535in
    :height: 3.1575in


.. _RefImage_100000000000040000000300F8F4D0D03F8E367A.png:


.. _Ref59871376:

Figure 2.1-a: Functions :math:`\mathrm{F1}`, :math:`\mathrm{F2}`, and :math:`\mathrm{F3}`

These three functions come from [:ref:`bib1 <bib1>`].


Benchmark results
----------------------

Digital application:

Unless otherwise stated, in the rest of this document, the parameters used for :math:`a` and :math:`b` are:

:math:`a=0.4m`

:math:`b=0.5m`


.. csv-table::

    "Case 1: Traction and Torsion", "Case 2: Flexion"
    ":math:`\mathrm{K1}=5.35{\mathrm{MPa.m}}^{1/2}`
    :math:`\mathrm{K3}=11.22{\mathrm{MPa.m}}^{1/2}` "," :math:`{\mathrm{K1}}_{A}=11.71{\mathrm{MPa.m}}^{1/2}`"


Table 2.2-1: Reference values


Bibliographical references
---------------------------


* TADA, PARIS, IRWIN: The Stress Analysis Of Cracks Handbook, Del Research Corporation, Hellertoxn, Pennsylvania (1973).


*Calculation of stress intensity factors by extrapolation of the displacement field,*Code_Aster* Reference Manual, R7.02.08


* CORNELIU: Quarter-point elements for curved crack fronts, Computers & Structures Vol. 17, No. 2, pp. 227-231, 1983