2. Benchmark solution#

2.1. Calculation method#

The general solution on the go is as follows:

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

In deformations:

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

In constraints:

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

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

\(r=\mathrm{0,1}\)

\(r=\mathrm{0,2}\)

\({u}_{r}\)

\(6.{10}^{\text{-}5}\mathrm{mm}\)

\(3.{10}^{\text{-}5}\mathrm{mm}\)

\({\varepsilon }_{\mathrm{rr}}\)

\(-6.{10}^{\text{-}4}\)

\(-\mathrm{1,5}{10}^{\text{-}4}\)

\({\varepsilon }_{\theta \theta }\)

\(-6.{10}^{\text{-}4}\)

\(\mathrm{1,5}{10}^{\text{-}4}\)

\({\sigma }_{\mathrm{rr}}\)

\(-60\mathrm{MPa}\)

\(0\mathrm{MPa}\)

\({\sigma }_{\theta \theta }\)

\(100\mathrm{MPa}\)

\(40\mathrm{MPa}\)

\({\sigma }_{\mathrm{zz}}\)

\(20\mathrm{MPa}\)

\(20\mathrm{MPa}\)

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

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

2.2. Reference quantities and results#

The following are compared to the reference values:

  • trips \((u,v)\) to points \(A\) and \(F\),

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

  • equivalent deformations and stresses equivalent to point \(A\).

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.

2.3. Bibliographical references#

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

  2. [V3.04.100] Hollow cylinder in plane deformations