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


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

:ref:`(1) <RefNumPara__9929_1168901107>`
The reference solution for a lens-shaped crack with radius :math:`R` in an infinite medium, subjected to a uniform pressure :math:`\sigma` quite far from the crack, shows that the stress intensity factors are constant along the crack background and equal:
--------------------------------------------------------- ----

:math:`{K}_{I}\mathrm{=}\mathrm{0,877}(\frac{2}{\pi })\sigma \sqrt{\pi a}`

:math:`{K}_{\mathit{II}}\mathrm{=}\mathrm{0,235}(\frac{2}{\pi })\sigma \sqrt{\pi a}`

:math:`{K}_{\mathit{III}}\mathrm{=}0`

with :math:`a\mathrm{=}R\mathrm{sin}\alpha`.


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

For the load under consideration :math:`\sigma \mathrm{=}1\mathit{MPa}` and the following geometric characteristics:

:math:`L\mathrm{=}10m`, :math:`R\mathrm{=}2m`, :math:`a=\sqrt{2}`, we find:

:math:`{K}_{I}\mathrm{=}\mathrm{1,177}\mathit{MPa}\mathrm{.}\sqrt{m}`

:math:`{K}_{\mathit{II}}\mathrm{=}\mathrm{0,3153}\mathit{MPa}\mathrm{.}\sqrt{m}`

:math:`{K}_{\mathit{III}}\mathrm{=}0`


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



.. _RefNumPara__9929_1168901107:

1. J.P. Pereira, C. A. Duarte, D. Duarte, D. Guoy, and X. Jiao: HP-Generalized FEM and Crack Surface Representation for Non-Planar 3-D Cracks, Int. Mr Numer. Engng, 77:601-633, 2009.