2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The constants \(A\) and \(B\) are calculated by solving the linear system obtained by writing:
\({\sigma }_{\mathrm{rr}}(a)=-p\) \({\sigma }_{\mathrm{rr}}(b)=0\)
We get:
For \(r=0.1\) |
|
For \(r=0.2\) |
|
\({\sigma }_{\mathrm{rr}}=-1\) |
\({\sigma }_{\mathrm{rr}}=0.\) |
||
\({\sigma }_{\theta \theta }=1.6685\) |
\({\sigma }_{\theta \theta }=0.66738\) |
||
\({\sigma }_{\mathrm{zz}}=0.20055\) |
\({\sigma }_{\mathrm{zz}}=0.20031\) |
Transition to the Cartesian axis system:
\(\begin{array}{ccc}{\sigma }_{\mathrm{xx}}& =& {\sigma }_{\mathrm{rr}}{\mathrm{cos}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{sin}}^{2}\theta -2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{yy}}& =& {\sigma }_{\mathrm{rr}}{\mathrm{sin}}^{2}\theta +{\sigma }_{\theta \theta }{\mathrm{cos}}^{2}\theta +2{\sigma }_{r\theta }\mathrm{sin}\theta \mathrm{cos}\theta \\ {\sigma }_{\mathrm{xy}}& =& {\sigma }_{\mathrm{rr}}\mathrm{sin}\theta \mathrm{cos}\theta -{\sigma }_{\theta \theta }\mathrm{sin}\theta \mathrm{cos}\theta -2{\sigma }_{r\theta }({\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta )\end{array}\)
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
to points \(A,B,C,D,E,F\).
2.3. Uncertainty about the solution#
Precision of the calculation of Bessel functions.
2.4. Bibliographical references#
BONNET: Methods of regularized integral equations in elastodynamics - Bulletin of DER - Series C - No. 1/2 - (1987).
ERINGEN - SUHUBI - Elastodynamics, Vol.2: Linear Theory Academic Press (1975).