Kirsch equations explained

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} = (\sigma)/(2) \left( 1 - (a ^{2})/(r ^{2}) \right) + (\sigma)/(2) \left( 1 +3 (a ^{4})/(r ^{4}) -4 (a ^{2})/(r ^{2}) \right)\cos 2 \theta \sigma _{\theta\theta} = (\sigma)/(2) \left( 1 + (a ^{2})/(r ^{2}) \right) - (\sigma)/(2) \left( 1 +3 (a ^{4})/(r ^{4}) \right)\cos 2 \theta \sigma _{r\theta} = - (\sigma)/(2) \left( 1 -3 (a ^{4})/(r ^{4}) +2 (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.