A stress-concentration-formula generating equation for arbitrary shallow surfaces

Analytical understanding of how stress concentrates is invaluable. An equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces, for the plane stress conditions and to a first-order approximation. Under shallow conditions, for any first-order Hölder-continuous surface function f(x), the derived equation is: kt(x)=1-2Hf′(x), where H is the Hilbert transform and f′(x) is the spatial derivative of f with respect to the independent variable. It is shown that using this generating equation, well-known traditional results can be easily derived. Also, a number of other stress concentration formulas for various cases are generated. Furthermore, a second-order approximation is introduced, which shows the dependence of kt on not only the slope but also on the concavity of the surface. The approach used herein can be extended to finding closed-form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile (work in progress).