Where Cracks Begin
Fatigue cracks almost always initiate at stress concentrations — holes, notches, fillets, keyways, threads, and surface defects where the local stress is amplified far beyond the nominal value. Understanding and quantifying these stress concentrations is the first step in fatigue design. The stress concentration factor Kt, a purely geometric quantity, tells you how much the local stress exceeds the nominal stress and where the critical point for crack initiation lies.
The Kirsch Solution
In 1898, Ernst Gustav Kirsch derived the exact stress field around a circular hole in an infinite plate under uniaxial tension. His solution showed that the tangential stress at the hole equator is exactly three times the applied stress — the famous Kt = 3.0. This elegant closed-form result remains one of the most important solutions in elasticity theory and serves as the benchmark for all stress concentration analyses.
Real Geometries
Real components have finite width, multiple notches, and complex loading. Finite-width corrections increase Kt above 3.0 as the hole diameter approaches the plate width. Elliptical holes have Kt = 1 + 2(a/b), reaching extreme values for sharp cracks. Shoulder fillets, keyways, and screw threads each have characteristic Kt values tabulated in Peterson's handbook. This simulation computes Kt for common geometries and visualizes the stress field to build intuition about stress flow.
From Kt to Fatigue Life
In fatigue, the full elastic Kt may overestimate the severity of a notch. Materials have a characteristic length scale below which they are insensitive to stress gradients — small, sharp notches are less damaging than Kt suggests. Peterson's notch sensitivity factor q converts Kt to the fatigue notch factor Kf = 1 + q(Kt - 1). For ductile metals with large grain sizes, q can be as low as 0.5, meaning only half the geometric concentration translates to fatigue damage.