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{=}\mathrm{2m}`, the gives the analytical values of SIFs along half of the crack bottom, for :math:`\omega` between :math:`0°` and :math:`180°`. These values are also shown on the.


.. image:: images/100000000000032400000182930E329A37432EB1.png
    :width: 4.189in
    :height: 2.0118in


.. _RefImage_100000000000032400000182930E329A37432EB1.png:

Figure 2.2-1: Reference values for SIFs

.. csv-table::

    "Angle :math:`\omega` (°)", ":math:`{K}_{I}` :math:`(\mathit{Pa}\mathrm{.}\sqrt{M})` "," :math:`{K}_{\mathit{II}}` :math:`(\mathit{Pa}\mathrm{.}\sqrt{M})` "," :math:`{K}_{\mathit{III}}` :math:`(\mathit{Pa}\mathrm{.}\sqrt{M})`"
    "0", "7,978E+05", "9,387E+05", "0.0E+05", "0.0E+05"
    "5", "7,978E+05", "9,351E+05", "5,727E+05", "5,727E+04"
    "10", "7,978E+05", "9,244E+05", "1,14E+05", "1,14E+05"
    "15", "7,978E+05", "9,067E+05", "1,701E+05", "1,701E+05"
    "20", "7,978E+05", "8,821E+05", "2,247E+05", "2,247E+05"
    "25", "7,978E+05", "8,507E+05", "2,777E+05", "2,777E+05"
    "30", "7,978E+05", "8,129E+05", "3,285E+05", "3,285E+05"
    "35", "7,978E+05", "7,689E+05", "7,689E+05", "3,769E+05"
    "40", "7,978E+05", "7,191E+05", "7,191E+05", "4,224E+05"
    "45", "7,978E+05", "6,638E+05", "4,646E+05", "4,646E+05"
    "50", "7,978E+05", "6,034E+05", "5,034E+05", "5,034E+05"
    "55", "7,978E+05", "5,384E+05", "5,382E+05", "5,382E+05"
    "60", "7,978E+05", "4,693E+05", "5,690E+05", "4,693E+05"
    "65", "7,978E+05", "3,967E+05", "5,955E+05", "5,955E+05"
    "70", "7,978E+05", "3,211E+05", "6,175E+05", "6,175E+05"
    "75", "7,978E+05", "2,430E+05", "6,347E+05", "6,347E+05"
    "80", "7,978E+05", "1,630E+05", "6,471E+05", "6,471E+05"
    "85", "7,978E+05", "8,181E+04", "6,546E+05"
    "90", "7,978E+05", "5,750E-11", "6,571E+05"
    "100", "7,978E+05", "-1,630E+05", "6,471E+05", "6,471E+05"
    "110", "7,978E+05", "-3,211E+05", "6,175E+05", "6,175E+05"
    "120", "7,978E+05", "-4,693E+05", "5,690E+05", "5,690E+05"
    "130", "7,978E+05", "-6,034E+05", "5,034E+05", "5,034E+05"
    "140", "7,978E+05", "-7,191E+05", "4,224E+05", "4,224E+05"
    "150", "7,978E+05", "-8,129E+05", "3,285E+05", "3,285E+05"
    "160", "7,978E+05", "-8,821E+05", "2,247E+05", "2,247E+05"
    "170", "7,978E+05", "-9,244E+05", "1,14E+05", "1,14E+05"
    "180", "7,978E+05", "-9,387E+05", "8,050E-11", "8,050E-11"


Table 2.2-1: Reference values


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



.. _RefNumPara__9929_1168901107:

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


2. LORENTZ E., Effect of a free mesh of a cracked three-dimensional structure on the quality of the calculation of the energy restoration rate, CR-I20-2010-11, 2010