2. Reference solution

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

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

Where:

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

And the energy release rate for mode I pure reads:

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

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

[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.