Reference solution
==================


.. image:: images/10000000000005D000000002FCD2737BD3670FFA.jpg
    :width: 6.4819in
    :height: 0.0161in


.. _RefImage_10000000000005D000000002FCD2737BD3670FFA.jpg:

The analytical SIF solution for this crack geometry (Figure
`1) <#page4>`_is given by Eshraghi *et al.* [:ref:`1 <1>`]:

:math:`{K}_{I}=\frac{2}{\pi }\sigma \sqrt{\mathit{\pi a}}F\left(\frac{a}{b}\right)`

Where:

:math:`F\left(\frac{a}{b}\right)=\frac{1-0.5\frac{a}{b}+0.148{\left(\frac{a}{b}\right)}^{3}}{\sqrt{1-\frac{a}{b}}}`

And the energy release rate for mode I pure reads:

:math:`J=\frac{{K}_{I}^{2}\times \left(1-{v}^{2}\right)}{E}`

In this calculation, the input parameters are: a = 2 mm, b = 10 mm, which gives the values of J = 0.118 N/mm.

[:ref:`1 <1>`] I. Eshraghi and N. Soltani. Stress intensity factor calculation for internal circumferential cracks in functionally graded cylinders using the weight function approach. *Engineering Fracture Mechanics*, *134*: 1-19, 2015.