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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.


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


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