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


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

The reference solution is an analytical solution from SIH [:ref:`bib1 <bib1>`] and [:ref:`bib2 <bib2>`].


.. image:: images/Object_1.png
    :width: 5.9161in
    :height: 3.0098in


.. _RefImage_Object_1.png:

Note that the angle :math:`\alpha` here designates the parametric angle of the point :math:`M` (angle with respect to the axis :math:`\mathit{Ox}` projected of :math:`M` onto the circle of radius :math:`b`) and not the polar coordinate of this point.

Here: :math:`a\mathrm{=}25\mathit{mm}` and :math:`b\mathrm{=}6\mathit{mm}`, so :math:`k\mathrm{=}\mathrm{0,9707728}`

The values of the elliptical integral :math:`E(k)` are tabulated in [:ref:`bib3 <bib3>`], according to :math:`\mathit{asin}(k)` which is equal here to :math:`\mathrm{76,11}°`. We then find: :math:`E(k)\mathrm{=}\mathrm{1,0672}`.

Hence the stress intensity factor in :math:`\mathit{MPa.}\sqrt{}\mathit{mm}`: :math:`{K}_{I}(\mathrm{\alpha })=4.0680{\left[{\mathrm{sin}}^{2}\mathrm{\alpha }+\frac{{b}^{2}}{{a}^{2}}{\mathrm{cos}}^{2}\mathrm{\alpha }\right]}^{1/4}`

Then, using Irwin's formula (plane deformation):

:math:`g(\mathrm{\alpha })=\frac{1-{\mathrm{\nu }}^{2}}{E}{K}_{I}{(\mathrm{\alpha })}^{2}`


Bibliography
-------------


1. G.C. SIH: Mathematical Theories of Brittle Fracture - FRACTURE, VOLII‑academic Press - 1968


2. M.K. KASSIN and G.C. SIH: Three-dimensional stress distribution around an elliptical crack under arbitrary loadings J. Appl. Mech., 88, 601-611, 1966.


3. H. TADA, P. PARIS, G. IRWIN: The Stress Analysis of Cracks Handbook - Third Edition - ASM International - 2000