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


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

There is a known analytical solution to this problem, if we consider :math:`{R}_{1}` the inner radius and :math:`{R}_{2}` the outer radius, then the radial stress expressed in polar coordinates is written as:

:math:`{\mathrm{\sigma }}_{\mathit{rr}}(r)=\frac{{R}_{1}^{3}}{{R}_{2}^{3}-{R}_{1}^{3}}\cdot \frac{{R}_{2}^{3}-{r}^{3}}{{r}^{3}}\cdot P`.



So we definitely find :math:`{\mathrm{\sigma }}_{\mathit{rr}}({R}_{1})=P` and :math:`{\mathrm{\sigma }}_{\mathit{rr}}({R}_{2})=0`.


Reference quantities and results
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We test the contact pressure on the interfaces, in :math:`r=30\mathit{mm}`, on both sides of the discontinuity. With :math:`{R}_{1}=20\mathit{mm}` and :math:`{R}_{2}=40\mathit{mm}`, we then have:

:math:`{\mathrm{\sigma }}_{\mathit{rr}}(r)=4.894179894\mathit{MPa}`.


Uncertainty about the solution
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Analytical solution.