2. Benchmark solution#

2.1. Calculation method used for the reference solution#

1 The reference solution is an analytical solution resulting from, for a circular crack with radius \(a\) in an infinite medium, subject to a uniform surface force \(\sigma\) inclined at an angle \(\alpha\) to the plane of the crack, the stress intensity factors for a point \(A\) placed on the crack front are equal to: —-

\({K}_{I}\mathrm{=}\frac{2}{\pi }\sigma ({\mathrm{sin}}^{2}\alpha )\sqrt{\pi a}\)

\({K}_{\mathit{II}}\mathrm{=}\frac{4}{\pi (2\mathrm{-}\nu )}\sigma (\mathrm{sin}\alpha \mathrm{cos}\alpha )\mathrm{cos}\omega \sqrt{\pi a}\)

\({K}_{\mathit{III}}\mathrm{=}\frac{4(1\mathrm{-}\nu )}{\pi (2\mathrm{-}\nu )}\sigma (\mathrm{sin}\alpha \mathrm{cos}\alpha )\mathrm{sin}\omega \sqrt{\pi a}\)

\(\omega\) being the angle characterizing the position of the point \(A\) on the circular background (see).

2.2. Benchmark results#

For the load under consideration and \(a=\mathrm{0,2}m\), the values are shown in the figure and the table.

_images/1000020100000780000003F152354364A7F87BC5.png

Figure 2.2-1 : Reference values for SIFs

Angle \(\mathrm{\omega }\) (rad)

\({K}_{I}\) \((\mathit{MPa}\mathrm{.}\sqrt{M})\)

\({K}_{\mathit{II}}\) \((\mathit{MPa}\mathrm{.}\sqrt{M})\)

\({K}_{\mathit{III}}\) \((\mathit{MPa}\mathrm{.}\sqrt{M})\)

0

0, 252

0, 297

0

\(\mathrm{\pi }/4\)

0, 252

0, 2099

0, 147

\(\mathrm{\pi }/2\)

0, 252

0

0

208 »

\(3\mathrm{\pi }/4\)

0, 252

-0, 2099

0, 147

\(\mathrm{\pi }\)

0, 252

-0, 297

0

2.3. Bibliographical references#

  1. TADA H., PARIS P., IRWIND G.: The stress analysis of cracks handbook, 3rd ed., 2000