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

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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):

σ r r = σ 2 ( 1 − a 2 r 2 ) + σ 2 ( 1 + 3 a 4 r 4 − 4 a 2 r 2 ) cos ⁡ 2 θ {\displaystyle \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 }

σ θ θ = σ 2 ( 1 + a 2 r 2 ) − σ 2 ( 1 + 3 a 4 r 4 ) cos ⁡ 2 θ {\displaystyle \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 }

σ r θ = − σ 2 ( 1 − 3 a 4 r 4 + 2 a 2 r 2 ) sin ⁡ 2 θ {\displaystyle \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

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