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


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

The general solution on the go is as follows:

:math:`\mathrm{\{}\begin{array}{c}{u}_{r}\mathrm{=}\frac{{\mathit{Pa}}^{2}}{E({b}^{2}\mathrm{-}{a}^{2})}(1+\nu )\left[(1\mathrm{-}2\nu )r+\frac{{b}^{2}}{r}\right]\\ {u}_{\theta }\mathrm{=}{u}_{z}\mathrm{=}0\end{array}`

In deformations:

:math:`\mathrm{\{}\begin{array}{c}{\varepsilon }_{\mathit{rr}}\mathrm{=}\frac{{\mathit{Pa}}^{2}}{E({b}^{2}\mathrm{-}{a}^{2})}(1+\nu )\left[(1\mathrm{-}2\nu )\mathrm{-}\frac{{b}^{2}}{r}\right]\\ {\varepsilon }_{\theta \theta }\mathrm{=}\frac{{\mathit{Pa}}^{2}}{E({b}^{2}\mathrm{-}{a}^{2})}(1+\nu )\left[(1\mathrm{-}2\nu )+\frac{{b}^{2}}{r}\right]\\ {\varepsilon }_{r\theta }\mathrm{=}{\varepsilon }_{\mathit{zz}}\mathrm{=}0\end{array}`

In constraints:

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

For a perfectly incompressible cylinder :math:`(\nu =0.5)`, we obtain:


.. csv-table::

    "", ":math:`r=\mathrm{0,1}` "," :math:`r=\mathrm{0,2}`"
    ":math:`{u}_{r}` "," :math:`6.{10}^{\text{-}5}\mathrm{mm}` "," :math:`3.{10}^{\text{-}5}\mathrm{mm}`"
    ":math:`{\varepsilon }_{\mathrm{rr}}` "," :math:`-6.{10}^{\text{-}4}` "," :math:`-\mathrm{1,5}{10}^{\text{-}4}`"
    ":math:`{\varepsilon }_{\theta \theta }` "," :math:`-6.{10}^{\text{-}4}` "," :math:`\mathrm{1,5}{10}^{\text{-}4}`"
    ":math:`{\sigma }_{\mathrm{rr}}` "," :math:`-60\mathrm{MPa}` "," :math:`0\mathrm{MPa}`"
    ":math:`{\sigma }_{\theta \theta }` "," :math:`100\mathrm{MPa}` "," :math:`40\mathrm{MPa}`"
    ":math:`{\sigma }_{\mathrm{zz}}` "," :math:`20\mathrm{MPa}` "," :math:`20\mathrm{MPa}`"


The transition through the Cartesian system is made using the following relationships:

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

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

The following are compared to the reference values:


* trips :math:`(u,v)` to points :math:`A` and :math:`F`,


* the deformations (:math:`{\varepsilon }_{\mathrm{xx}}`, :math:`{\varepsilon }_{\mathrm{yy}}`, :math:`{\varepsilon }_{\mathrm{xy}}`) and the constraints (:math:`{\sigma }_{\mathrm{xx}}`, :math:`{\sigma }_{\mathrm{yy}}`, :math:`{\sigma }_{\mathrm{zz}}`,) and the constraints (,,, :math:`{\sigma }_{\mathrm{xy}}`) at the points :math:`A` and :math:`F`,


* equivalent deformations and stresses equivalent to point :math:`A`.

Finally, to test the passage of the quantities from the Gauss points to the nodes for the middle nodes, we also test the non-zero deformations and stresses at a middle node of the structure.


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


1. Y.C. FUNG: Foundations of Solid Mechanics. Prenctice-Hall, Inc. Englewood Cliffs. NJ.1965, pp. 243-245


2. [:ref:`V3.04.100 <V3.04.100>`] Hollow cylinder in plane deformations