Reference solution
=====================


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


.. image:: images/10000CA8000016A200000EE22445279D7C236AB8.svg
    :width: 292
    :height: 192


.. _RefImage_10000CA8000016A200000EE22445279D7C236AB8.svg:

In plane deformations or in plane stresses, the distribution of displacements is given in this coordinate system :math:`(\mathrm{0,}{x}_{\mathrm{1,}}{x}_{2})` by:

:math:`\mathrm{\{}\begin{array}{c}{u}_{1}\mathrm{=}\frac{1+\nu }{E}\sqrt{\frac{r}{2\pi }}({K}_{I}\mathrm{cos}\frac{\theta }{2}(k\mathrm{-}\mathrm{cos}\theta )+{K}_{\mathit{II}}\mathrm{sin}(\frac{\theta }{2})(k\mathrm{-}\mathrm{cos}\theta +2))\\ {u}_{2}\mathrm{=}\frac{1+\nu }{E}\sqrt{\frac{r}{2\pi }}({K}_{I}\mathrm{sin}\frac{\theta }{2}(k\mathrm{-}\mathrm{cos}\theta )\mathrm{-}{K}_{\mathit{II}}\mathrm{cos}(\frac{\theta }{2})(k+\mathrm{cos}\theta \mathrm{-}2))\end{array}`

with :math:`k=3-4\nu` in plane deformations

:math:`k=\frac{3-\nu }{1+\nu }` in plane constraints

or in the :math:`(O,X,Y)` frame by: :math:`\{\begin{array}{}{u}_{x}=\mathrm{cos}\alpha {u}_{1}-\mathrm{sin}\alpha {u}_{2}\\ {u}_{y}=\mathrm{sin}\alpha {u}_{1}+\mathrm{cos}\alpha {u}_{2}\end{array}`

On the outline of the plate, we have: :math:`r=\mathrm{OA}=100\mathrm{mm}`.

We choose to take :math:`{K}_{I}=2.` and :math:`{K}_{\mathrm{II}}=1.` and impose the movements on the outline of the circular plate.


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


.. csv-table::

    ":math:`{K}_{I}=2.` ", ""
    ":math:`{K}_{\mathrm{II}}=1.` ", ""
    ":math:`G=2.275{10}^{-5}` ", "in plane deformations"
    ":math:`G=2.5{10}^{-5}` ", "in plane constraints"



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


1. H.D. BUI Mechanics of Fragile Fracture - Ed. Masson 1978