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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.

Additional recommended knowledge

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.
 
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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