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


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

BROWN & STRAWLEY [:ref:`bib1 <bib1>`] reference solution:


.. csv-table::

    ":math:`J\mathrm{=}{F}^{2}\pi a{\sigma }^{2}\mathrm{/}E` with :math:`F=1.98`"
    ":math:`a` in :math:`\mathit{mm}`"
    ":math:`\sigma` and :math:`E` in :math:`N/{\mathrm{mm}}^{2}`"



Reference results for :math:`G`
-------------------------------------

Benchmark results :math:`G={1.98}^{2}\times \pi \times 37.5\times 0.5{10}^{-5}=2.3093{10}^{-3}\mathrm{Mpa.mm}`

The formula

.. image:: images/Object_3.svg
    :width: 17
    :height: 18


.. _RefImage_Object_3.svg:

(IRWIN) = :math:`\frac{1}{E}({K}_{1}^{2}+{K}_{2}^{2})` conduit, like

.. image:: images/Object_5.svg
    :width: 17
    :height: 18


.. _RefImage_Object_5.svg:

, at :math:`{K}_{1}=21.491{\mathrm{MPa.mm}}^{1/2}`


Reference results for derivatives of :math:`G`
-----------------------------------------------------

By varying the Young's modulus and the load :math:`\mathrm{Fy}`, we see that:


:math:`G\mathrm{=}\alpha {F}_{Y}^{2}` with :math:`\alpha \mathrm{=}2.3{10}^{\mathrm{-}3}` being :math:`\frac{\mathrm{\partial }G}{\mathrm{\partial }{F}_{Y}}\mathrm{=}2\alpha {F}_{Y}`
--------------------------------------- ---- ----

:math:`G\mathrm{=}\frac{\beta }{E}` with :math:`\beta \mathrm{=}460.` being :math:`\frac{\mathrm{\partial }G}{\mathrm{\partial }E}\mathrm{=}\mathrm{-}\frac{G}{E}`


Bibliographical reference
-------------------------


1. BROWN - STAWLEY ASTM Special Technical Publication No. 410 (1966)