Kirsch equations

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):

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

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

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

References

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