2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Analytical
\(\begin{array}{ccc}{\sigma }_{\mathrm{zz}}& =& 2\nu P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\\ {\sigma }_{\mathrm{rr}}& =& P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\left[1-\frac{{b}^{2}}{{r}^{2}}\right]\\ {\sigma }_{\theta \theta }& =& P\frac{{a}^{2}}{{b}^{2}-{a}^{2}}\left[1+\frac{{b}^{2}}{{r}^{2}}\right]\\ {\sigma }_{r\theta }& =& 0\\ {u}_{r}& =& \frac{P}{E}\frac{{a}^{2}}{{b}^{2}-{a}^{2}}(1+\nu )\left[(1-2\nu )+\frac{{b}^{2}}{{r}^{2}}\right]r\end{array}\)
We get:
For \(r=0.1\) |
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For \(r=0.2\) |
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\({\sigma }_{\mathrm{rr}}=-60.\) |
\({\sigma }_{\mathrm{rr}}=0.\) |
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\({\sigma }_{\theta \theta }=100.\) |
\({\sigma }_{\theta \theta }=40.\) |
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\({\sigma }_{\mathrm{zz}}=12.\) |
\({\sigma }_{\mathrm{zz}}=12.\) |
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\({\sigma }_{r\theta }=0.\) |
\({\sigma }_{r\theta }=0.\) |
Transition to the Cartesian axis system:
\(\begin{array}{}{\sigma }_{\mathrm{xx}}={\sigma }_{\mathrm{rr}}{\mathrm{cos}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{sin}}^{2}\theta -2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{yy}}={\sigma }_{\mathrm{rr}}{\mathrm{sin}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{cos}}^{2}\theta +2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{xy}}={\sigma }_{\mathrm{rr}}\mathrm{sin}\theta \mathrm{cos}\theta -{\sigma }_{\theta \theta }\mathrm{sin}\theta \mathrm{cos}\theta -2{\sigma }_{r\theta }({\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta )\end{array}\)
with:
\(\theta =0°\) at points \(A\) and \(B\),
\(\theta =22.5°\) at points \(C\) and \(D\),
\(\theta =45°\) at points \(E\) and \(F\).
2.2. Benchmark results#
Displacements \((u,v)\) and constraints \(({\sigma }_{\mathrm{xx}},{\sigma }_{\mathrm{yy}},{\sigma }_{\mathrm{zz}},{\sigma }_{\mathrm{xy}})\) at points \(A,B,C,D,E,F\).
2.3. Bibliographical references#
Y.C. FUNG. Foundations of solid mechanics. Prentice-hall, Inc. Englewood Cliffs. NJ. 1965 p.243-245.