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


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

Analytical

:math:`\begin{array}{ccc}{\sigma }_{\mathrm{zz}}& =& 2\nu P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\\ {\sigma }_{\mathrm{rr}}& =& P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\left[1-\frac{{b}^{2}}{{r}^{2}}\right]\\ {\sigma }_{\theta \theta }& =& P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\left[1+\frac{{b}^{2}}{{r}^{2}}\right]\\ {\sigma }_{r\theta }& =& 0\\ {u}_{r}& =& \frac{P}{E}\frac{{a}^{2}}{{b}^{2}-{a}^{2}}(1+\nu )\left[(1-2\nu )+\frac{{b}^{2}}{{r}^{2}}\right]r\end{array}`

We get:

.. csv-table::

    "For :math:`r=0.1` "," :math:`{u}_{r}=\mathrm{5,72}{10}^{-5}` ", "For :math:`r=0.2` "," :math:`{u}_{r}=\mathrm{3,64}{10}^{-5}`"
    "", ":math:`{\sigma }_{\mathrm{rr}}=-60.` ", "", ":math:`{\sigma }_{\mathrm{rr}}=0.`"
    "", ":math:`{\sigma }_{\theta \theta }=100.` ", "", ":math:`{\sigma }_{\theta \theta }=40.`"
    "", ":math:`{\sigma }_{\mathrm{zz}}=12.` ", "", ":math:`{\sigma }_{\mathrm{zz}}=12.`"
    "", ":math:`{\sigma }_{r\theta }=0.` ", "", ":math:`{\sigma }_{r\theta }=0.`"


Transition to the Cartesian axis system:

:math:`\begin{array}{}{\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{rr}}{\mathrm{cos}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{sin}}^{2}\theta -2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{yy}}={\sigma }_{\mathrm{rr}}{\mathrm{sin}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{cos}}^{2}\theta +2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{rr}}\mathrm{sin}\theta \mathrm{cos}\theta -{\sigma }_{\theta \theta }\mathrm{sin}\theta \mathrm{cos}\theta -2{\sigma }_{r\theta }({\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta )\end{array}`

with:


* :math:`\theta =0°` at points :math:`A` and :math:`B`,


* :math:`\theta =22.5°` at points :math:`C` and :math:`D`,


* :math:`\theta =45°` at points :math:`E` and :math:`F`.


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

Displacements :math:`(u,v)` and constraints :math:`({\sigma }_{\mathrm{xx}},{\sigma }_{\mathrm{yy}},{\sigma }_{\mathrm{zz}},{\sigma }_{\mathrm{xy}})` at points :math:`A,B,C,D,E,F`.


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


1. Y.C. FUNG. Foundations of solid mechanics. Prentice-hall, Inc. Englewood Cliffs. NJ. 1965 p.243-245.