Kirsch equations

Result

Loading an infinite plate with circular hole of radius a with stress σ, the resulting stress field is:

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

σ θ θ = σ 2 ( 1 + a 2 r 2 ) − σ 2 ( 1 + 3 a 4 r 4 ) cos ⁡ 2 θ

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