About the Bug-Rivet Paradox
The bug-rivet paradox is a classic illustration from fatigue and fracture mechanics: a small circular hole in a stressed plate concentrates stress by the same theoretical factor, Kt = 3, whether that hole is the size of a rivet or the size of a pit left by a bug in wet paint. Classical elasticity theory (Kirsch's solution for a hole in an infinite plate under uniaxial tension) says the peak stress is always three times the nominal stress, regardless of the hole's absolute size. Yet engineers know from experience that a tiny flaw is far less damaging to fatigue life than a large one — the paradox is resolved by notch sensitivity, a separate, size-dependent correction.
Understanding the formula
Peterson's empirical notch-sensitivity equation bridges the gap: q = 1 / (1 + a/r), where r is the notch (hole) radius and a is a material-dependent characteristic length (Peterson's constant, larger for tougher, lower-strength materials and smaller for harder, higher-strength ones). The notch sensitivity q then converts the size-independent Kt into the size-dependent fatigue stress-concentration factor: Kf = 1 + q(Kt − 1). As r shrinks toward or below a, q falls toward 0 and Kf approaches 1 — the notch stops mattering for fatigue even though Kt never changes. As r grows much larger than a, q approaches 1 and Kf approaches the full Kt.
Working with units
- Enter the hole radius r and the material constant a in the same length unit (this calculator uses millimeters); mixing units silently breaks the a/r ratio.
- Kt = 3 is the standard value for a small circular hole in an infinite plate under uniaxial tension; adjust it if you are modeling a different notch geometry (an elliptical hole or a fillet, for example).
- Peterson's constant a is empirical and material-specific — typical published values for steels range from roughly 0.01 mm (very high strength) to over 0.2 mm (mild, ductile steel), decreasing as ultimate tensile strength increases.
Knowing the limits
This formula assumes a single circular through-hole in a wide, thin plate under uniform far-field tension, with Kt taken from Kirsch's solution. It does not model multiple interacting holes, sharp cracks (which use fracture-mechanics stress-intensity factors instead), or fully three-dimensional stress states. Peterson's a is a curve-fit to test data for a given material and should be sourced from a reference table for that specific alloy and strength level, not assumed universal.