Kirsch equations

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title: "Kirsch equations" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["solid-mechanics"] description: "Equations describing elastic stresses" topic_path: "general/solid-mechanics" source: "https://en.wikipedia.org/wiki/Kirsch_equations" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Equations describing elastic stresses ::

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

Result

Loading an infinite plate with a circular hole of radius a with stress σ, the resulting stress field is (the angle is with respect to the direction of application of the stress):

\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. Zeitschrift des Vereines deutscher Ingenieure, 42, 797–807. ::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::