2. Benchmark solution

2.1. Method used for the reference solution

The reference solution for the stress intensity factor for mode I \({K}_{I}\) is expressed as follows [1]:

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

with \(F\left(\frac{a}{W}\right)=\mathrm{1,122}-\mathrm{0,231}\left(\frac{a}{W}\right)+\mathrm{10,550}{\left(\frac{a}{W}\right)}^{2}-\mathrm{21,710}{\left(\frac{a}{W}\right)}^{3}+\mathrm{30,382}{\left(\frac{a}{W}\right)}^{4}\)

This expression for \(F\) comes from a least-squares regression (Gross 1964; Brown 1966). The announced accuracy of this formula is 0.5% for \(\frac{a}{W}\mathrm{\le }\mathrm{0,6}\).

Note: in [1] other formulas for \(F\left(a/W\right)\) are given, for example a formula due to Tada (1973) that has an accuracy of less than 0.5% for any \(a/W\) ratio.

The \(G\) energy return rate is obtained thanks to Irwin’s formula: \(G=\frac{\left(1-{\nu }^{2}\right)}{E}{K}_{I}^{2}\). The expected accuracy for the reference value of \(G\) will be less than 1%.

2.2. Benchmark results

With the numerical values of the statement, we find:

\(F\left(a/W\right)\approx \mathrm{1,186}\)

\({K}_{I}\approx \mathrm{6,646}\mathit{MPa}\mathrm{.}\sqrt{m}\)

\(G\approx \mathrm{191,4}J\mathrm{.}{m}^{-2}\)

Note: using the alternative Tada 1973 formula, we obtain \(F\left(a/W\right)\approx \mathrm{1,196}\), i.e. a difference between the 2 references of approximately 0.8%. Taking into account the details announced for each reference, it is deduced that the theoretical value is located between these 2 references. It would therefore certainly be more appropriate to choose the average value \(F\left(a/W\right)\approx \mathrm{1,191}\) as a reference value. However, in the following, the reference solution is considered to be the one derived from least squares regression (Gross 1964; Brown 1966).

2.3. Bibliographical references

• 8. Tada, P. Paris, G. Irwin, G. Irwin, The Stress Analysis of Cracks Handbook, 3rd edition, 2000