Kirsch equations

The Kirsch equations describe the stress tensor around a traction-free circular hole in an infinite linear elastic plate under uniform remote tension. For remote stress ${\sigma }_{\infty }$ along $x$, hole radius $a$ and polar coordinates $(r,\theta )$ measured from that direction, put $q=a^2/r^2$. In the classical isotropic plane stress model,The fields satisfy equilibrium and the prescribed boundary tractions. They can be obtained using an Airy stress function satisfying the biharmonic compatibility equation.^[ref]

At $r=a$, $q=1$, so ${\sigma }_{rr}={\sigma }_{r\theta }=0$ whileThe tangential stress is $3{\sigma }_{\infty }$ at the top and bottom of the hole and $-{\sigma }_{\infty }$ at its sides. A traction-free boundary does not require every stress component to vanish: the outward boundary normal is radial, so radial normal and shear tractions must vanish.

For an original distance comparison along $\theta =\pi /2$, the tangential stress is ${\sigma }_{\infty }(1+q/2+3q^2/2)$. At $r=2a$ it is $1.21875{\sigma }_{\infty }$, already far below the edge value three. At large distance the polar components approach those of the original uniform Cartesian tension.

A finite-width plate, internal hole pressure, plasticity or out-of-plane bending changes the boundary-value problem. The stress concentration factor three is therefore specific to the stated remote-tension model. The MIT laboratory uses local strain measurements and photoelasticity to examine how actual geometries compare with ideal concentration fields.^[ref]