Benchmark solution
====


Calculation method used for the reference solution
----

:ref:`1 <RefNumPara__9929_1168901107>`
The reference solution is an analytical solution resulting from, for a circular crack with radius :math:`a` in an infinite medium, subject to a uniform surface force :math:`\sigma` inclined at an angle :math:`\alpha` to the plane of the crack, the stress intensity factors for a point :math:`A` placed on the crack front are equal to:
----

:math:`{K}_{I}\mathrm{=}\frac{2}{\pi }\sigma ({\mathrm{sin}}^{2}\alpha )\sqrt{\pi a}`

:math:`{K}_{\mathit{II}}\mathrm{=}\frac{4}{\pi (2\mathrm{-}\nu )}\sigma (\mathrm{sin}\alpha \mathrm{cos}\alpha )\mathrm{cos}\omega \sqrt{\pi a}`

:math:`{K}_{\mathit{III}}\mathrm{=}\frac{4(1\mathrm{-}\nu )}{\pi (2\mathrm{-}\nu )}\sigma (\mathrm{sin}\alpha \mathrm{cos}\alpha )\mathrm{sin}\omega \sqrt{\pi a}`

:math:`\omega` being the angle characterizing the position of the point :math:`A` on the circular background (see).


Benchmark results
----

For the load under consideration and :math:`a=\mathrm{0,2}m`, the values are shown in the figure and the table.


.. image:: images/1000020100000780000003F152354364A7F87BC5.png
    :width: 7.0992in
    :height: 4.111in


.. _RefImage_1000020100000780000003F152354364A7F87BC5.png:

**Figure** 2.2-1 **: Reference values for SIFs**


.. csv-table::

    "Angle :math:`\mathrm{\omega }` (rad)", ":math:`{K}_{I}` :math:`(\mathit{MPa}\mathrm{.}\sqrt{M})` "," :math:`{K}_{\mathit{II}}` :math:`(\mathit{MPa}\mathrm{.}\sqrt{M})` "," :math:`{K}_{\mathit{III}}` :math:`(\mathit{MPa}\mathrm{.}\sqrt{M})`"
    "0", "0, 252", "0, 297", "0"
    ":math:`\mathrm{\pi }/4` ", "0, 252", "0, 2099", "0, 147"
    ":math:`\mathrm{\pi }/2` ", "0, 252", "0", "0", 208"
    ":math:`3\mathrm{\pi }/4` ", "0, 252", "-0, 2099", "0, 147"
    ":math:`\mathrm{\pi }` ", "0, 252", "-0, 297", "0"



Bibliographical references
----



.. _RefNumPara__9929_1168901107:

1.  TADA H., PARIS P., IRWIND G.: The stress analysis of cracks handbook, 3rd ed., 2000