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Kirsch equations

The Kirsch equations describe the elastic stresses around the hole in an infinite plate in one directional tension. They are named after Ernst Gustav Kirsch.

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Result

Loading an infinite plate with circular hole of radius a with stress σ, the resulting stress field is:

$\sigma_{rr} = \frac{\sigma}{2}\left(1 - \frac{a^2}{r^2}\right) + \frac{\sigma}{2}\left(1 + 3\frac{a^4}{r^4} - 4\frac{a^2}{r^2}\right)\cos 2\theta$

$\sigma_{\theta\theta} = \frac{\sigma}{2}\left(1 + \frac{a^2}{r^2}\right) - \frac{\sigma}{2}\left(1 + 3\frac{a^4}{r^4}\right)\cos 2\theta$

$\sigma_{r\theta} = - \frac{\sigma}{2}\left(1 - 3\frac{a^4}{r^4} + 2\frac{a^2}{r^2}\right)\sin 2\theta$

References

Kirsch, 1898, Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre. Zeitshrift des Vereines deutscher Ingenieure, 42, 797–807.

Category: Solid mechanics

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kirsch_equations". A list of authors is available in Wikipedia.

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