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


Calculation method used for the reference solution
--------------------------------------------------------


.. image:: images/Object_7.svg
    :width: 624
    :height: 313


.. _RefImage_Object_7.svg:

The constants :math:`A` and :math:`B` are calculated by solving the linear system obtained by writing:

:math:`{\sigma }_{\mathrm{rr}}(a)=-p` :math:`{\sigma }_{\mathrm{rr}}(b)=0`

We get:


.. csv-table::

    "For :math:`r=0.1` "," :math:`{u}_{r}=7.3398{10}^{-3}` ", "For :math:`r=0.2` "," :math:`{u}_{r}=4.6816{10}^{-3}`"
    "", ":math:`{\sigma }_{\mathrm{rr}}=-1` ", "", ":math:`{\sigma }_{\mathrm{rr}}=0.`"
    "", ":math:`{\sigma }_{\theta \theta }=1.6685` ", "", ":math:`{\sigma }_{\theta \theta }=0.66738`"
    "", ":math:`{\sigma }_{\mathrm{zz}}=0.20055` ", "", ":math:`{\sigma }_{\mathrm{zz}}=0.20031`"


Transition to the Cartesian axis system:

:math:`\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:

:math:`\theta =0°` at points :math:`A` and :math:`B`

:math:`\theta =22.5°` at points :math:`C` and :math:`D`

:math:`\theta =45°` at points :math:`E` and :math:`F`


Benchmark results
----------------------

Displacements :math:`(u,v)` and constraints

.. image:: images/Object_12.svg
    :width: 624
    :height: 313


.. _RefImage_Object_12.svg:

to points :math:`A,B,C,D,E,F`.


Uncertainty about the solution
---------------------------

Precision of the calculation of Bessel functions.


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


1. M. BONNET: Methods of regularized integral equations in elastodynamics - Bulletin of DER - Series C - No. 1/2 - (1987).


2. ERINGEN - SUHUBI - Elastodynamics, Vol.2: Linear Theory Academic Press (1975).