Kirsch’s Infinite Plate¶
Consider a 2D plate containing a circular hole and being subjected to remote tensile load \(\sigma_{\inf}\).
Taking under consideration the boundary conditions, we require our stress to be zero on the hole’s free surface:
We will start by considering a circular portion of the plate concentric with the hole and having a radius large enough with resect to a. The stresses will effectively be the same as in the plate without the hole and thus:
The stress distribution in the plate can be regarded as having two parts, the first arising from the normal applied traction and is \(\frac{\sigma_{\inf}}{2}\) and the second part having an angular dependence.
A good guess for \(\Phi\), seems to have the general form :
Let us ignore \(g(r)\) for now.
Substituting \(\Phi\) into the bi-harmonic equation we readily obtain:
A general solution to this ODE on \(f(r)\) will have the form :
and so the stress components are
From the conditions prescribed above we obtain:
The uniform tension \(\frac{1}{2}\sigma_{\inf}\) is accounted for by \(g(r)\) which can be written as :
which is a general solution for a stress distribution symmetrical about an axis1.
Adding the two parts of the solution we finally obtain:
For \(r=a\) we can see that $\( \sigma_{rr}=\tau_{r \theta} =0 \quad ; \quad \sigma_{\theta \theta} = \sigma_{\inf} \left(1-2cos2\theta \right) \)$
the maximum stress experienced appear at \(\theta = \pi /2 ; 3\pi/2\) where \(\sigma_{\theta \theta} = 3\sigma_{inf}\)
Note that for \(\theta=0,\pi\) we obtain \(\sigma_{\theta \theta}=-\sigma_{\inf}\).
From the figure below, it is evident that the stress magnitude rapidly decreases to the remote load as \(r\) increases.
- 1
Theory of Elasticity; J.N. Goodier & S. Timoshenko 1951, New York.