Kirsch equations
= Kirsch equations
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{wiki}
The \b[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,
$$
\begin{aligned}
\sigma_{rr}&=\frac{\sigma_\infty}2(1-q)+\frac{\sigma_\infty}2(1-4q+3q^2)\cos2\theta,\\
\sigma_{\theta\theta}&=\frac{\sigma_\infty}2(1+q)-\frac{\sigma_\infty}2(1+3q^2)\cos2\theta,\\
\sigma_{r\theta}&=-\frac{\sigma_\infty}2(1+2q-3q^2)\sin2\theta.
\end{aligned}
$$
The fields satisfy equilibrium and the prescribed boundary tractions. They can be obtained using an <Airy stress function> satisfying the biharmonic compatibility equation.<Stresses Around a Circular Hole by MIT Materials Science and Engineering>[one-page handout, stress function, component equations and boundary conditions]{ref}
At $r=a$, $q=1$, so $\sigma_{rr}=\sigma_{r\theta}=0$ while
$$
\sigma_{\theta\theta}(a,\theta)=\sigma_\infty(1-2\cos2\theta).
$$
The 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.<Isotropic Linear Elastic Stress Concentration by MIT Mechanical Engineering>[§4.1 and experimental §§6–7]{ref}