Benchmark solution
====


Method used for the reference solution
----

The reference solution for the stress intensity factor for mode I :math:`{K}_{I}` is expressed as follows [:ref:`1 <1>`]:

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

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

Note: in [:ref:`1 <1>`] other formulas for :math:`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 :math:`a/W` ratio.

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


Benchmark results
----

With the numerical values of the statement, we find:

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

:math:`{K}_{I}\approx \mathrm{6,646}\mathit{MPa}\mathrm{.}\sqrt{m}`

:math:`G\approx \mathrm{191,4}J\mathrm{.}{m}^{-2}`

*Note*: using the alternative Tada 1973 formula, we obtain :math:`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 :math:`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).


Bibliographical references
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*   H. Tada, P. Paris, G. Irwin, G. Irwin, The Stress Analysis of Cracks Handbook, 3rd edition, 2000