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


Calculation method
------------------

The reference result was obtained analytically with the following hypotheses:


* The plate is assumed to be of infinite dimension,


* Muskheliskvili and Kolosov method in polar coordinates.

:math:`{\mathrm{\sigma }}_{\mathit{rr}}=\frac{P}{2}[(1-\frac{{a}^{2}}{{r}^{2}})-(1-\frac{4{a}^{2}}{{r}^{2}}+\frac{3{a}^{4}}{{r}^{4}})\mathrm{cos}2\mathrm{\theta }]`

:math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}=\frac{P}{2}[(1+\frac{{a}^{2}}{{r}^{2}})+(1+\frac{3{a}^{4}}{{r}^{4}})\mathrm{cos}2\mathrm{\theta }]`

:math:`{\mathrm{\sigma }}_{r\mathrm{\theta }}=\frac{P}{2}(1+\frac{2{a}^{2}}{{r}^{2}}-\frac{3{a}^{4}}{{r}^{4}})\mathrm{sin}2\mathrm{\theta }`


Reference quantities and results
-----------------------------------

The selected reference results relate to circumferential stress :math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}`.

         :math:`{\mathrm{\sigma }}_{\mathrm{\theta }\mathrm{\theta }}(a,\mathrm{\theta })=P(1+2\mathrm{cos}2\mathrm{\theta })`


.. csv-table::

    "Point", "Size", "Value (N/mm²)"
    ":math:`\text{A}(a\mathrm{,0})` "," :math:`{\mathrm{\sigma }}_{\text{}\mathrm{\theta }\mathrm{\theta }}` "," :math:`7.5`"
    ":math:`\text{F}(a,\frac{\mathrm{\pi }}{4})` "," :math:`{\mathrm{\sigma }}_{\text{}\mathrm{\theta }\mathrm{\theta }}` "," :math:`2.5`"
    ":math:`\text{E}(a,\frac{\mathrm{\pi }}{2})` "," :math:`{\mathrm{\sigma }}_{\text{}\mathrm{\theta }\mathrm{\theta }}` "," :math:`-2.5`"



Uncertainties about the solution
----------------------------

Semi-analytical solution


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


1. Guide VPCS - 1990 edition.