Why a small hole can cause a big problem — and why the exact same notch is a footnote in one material and a fatigue crack's origin in another.
Drill a hole through the middle of a loaded plate and the average stress across what's left of the cross-section barely changes — a little less area is carrying the same force, so σ = F/A ticks up modestly. But the stress right at the edge of that hole is nowhere near the average. It spikes to several times the nominal value, because the load lines that used to pass straight through now have to crowd around the obstacle, piling up right at its edge. That local spike is a stress concentration, and it is the single most common starting point for real-world cracks and fractures — not the thick, heavily-loaded section of a part, but the small geometric detail everyone assumed was harmless.
Any geometric discontinuity in a loaded part — a hole, a fillet or corner, a groove, a keyway, a thread root, a sudden change in cross-section — forces the internal load path to bend around it, and that bending concentrates stress right at the feature. The stress concentration factor, Kt, is simply the ratio of that actual peak local stress to the nominal (average, F/A) stress: Kt = σ_max / σ_nom. Kt is pure geometry — it depends only on things like hole diameter relative to plate width or fillet radius relative to shaft diameter, and is exactly the same number whether the part is made of rubber, aluminum, or hardened tool steel. What's not the same across materials is how much of that theoretical spike actually costs the part in terms of fatigue strength — and that is a completely separate quantity, the notch sensitivity, q.
The two factors combine into a single practical number engineers actually design against: the fatigue stress concentration factor, Kf — the real reduction in fatigue strength a given notch causes in a given material — through Kf = 1 + q(Kt − 1). When q = 0, Kf = 1: the material's fatigue strength is completely unaffected, as if the notch weren't there at all. When q = 1, Kf = Kt: the material experiences the notch's full theoretical stress-concentration effect, no relief whatsoever. Ductile metals at room temperature typically fall somewhere in between, because a small zone right at the notch tip can yield plastically under load, spreading the peak out over more material and lowering the effective stress it has to survive cycle after cycle. Brittle materials, hardened/high-strength steels, and materials at low temperature have little or no capacity for that local yielding — dislocations can't move freely enough to redistribute the load — so they end up carrying close to the full Kt, cycle after cycle, right at the notch.
False, or at best dangerously incomplete. Geometric discontinuities can locally multiply the actual stress well above the simple nominal calculation — a Kt of 2 to 4 is routine for a hole or sharp fillet, meaning the real peak stress at that feature can be several times higher than what an F/A check would suggest. For statically loaded parts with generous margins, that local spike may genuinely not matter much, since ductile materials can yield locally and redistribute the load. But for fatigue-loaded parts — anything that sees repeated or vibratory loading — the stress-concentration effect, moderated by the material's notch sensitivity, is a common real-world crack-initiation site. Many actual fatigue failures start at exactly this kind of "minor" geometric feature— a keyway, a sharp fillet, a drilled hole — rather than at the part's thickest or most heavily loaded section. That is precisely why generous fillet radii, hole-edge deburring and treatment, and explicit stress-concentration-factor checks are standard design practice, not an afterthought reserved for critical components.
Explains why any geometric discontinuity — a hole, fillet, groove, keyway, or thread root — causes the actual local stress right at that feature to run well above the simple nominal (F/A) stress, quantified by the stress concentration factor Kt (pure geometry), and why real materials only 'feel' a fraction to all of that effect in fatigue, quantified by the notch sensitivity factor q (0 to 1).
A basic strength check compares nominal stress (F/A) against a material's yield or ultimate strength and calls it done. That check silently assumes stress is uniform across the cross-section, which is only true for a plain, unbroken shape. The instant a hole, fillet, groove, or other discontinuity enters the picture, stress redistributes and spikes locally — often by a factor of two to four or more — even though the average, nominal stress barely moved. Ignoring that local spike is exactly how design margins that looked comfortable on paper turn into real cracks in service.
The stress concentration factor, Kt = σ_max / σ_nom, is a purely geometric quantity — it depends on the ratio of notch dimension to part dimension (hole diameter to plate width, fillet radius to shaft diameter, groove depth to section depth) and is available from standard charts (Peterson's, Roark's) for common geometries. It does not change with material. What changes with material is how much of Kt's theoretical effect actually reduces the part's fatigue strength — that's notch sensitivity, q, ranging from 0 (fatigue strength unaffected by the notch) to 1 (fatigue strength fully reduced by the theoretical Kt). The two combine as Kf = 1 + q(Kt − 1), where Kf is the fatigue stress concentration factor actually used in a fatigue-life calculation.
This distinction governs the design of anything with fillets, holes, keyways, or thread roots that sees cyclic or vibratory loading — shafts, axles, bolted and pinned joints, pressure vessel nozzles, and rotating machinery among them. It's also why the same notch geometry can be a non-issue in a soft, ductile low-strength steel and a serious fatigue liability in a hardened, high-strength version of the same alloy — high-strength materials are generally both more notch-sensitive (higher q) and have proportionally less margin to begin with.
No. Kt is a function of geometry alone — the ratio of the notch feature (hole diameter, fillet radius, groove depth) to the surrounding part dimension. A hole of a given relative size in a plate produces the same Kt whether the plate is aluminum, mild steel, or hardened tool steel. What differs by material is the notch sensitivity, q, which determines how much of that theoretical Kt actually reduces fatigue strength.
q = 0 means the material's fatigue strength is essentially unaffected by the notch — as if the geometric feature weren't there at all, because the material yields locally enough to fully redistribute the peak stress. q = 1 means the material experiences the notch's full theoretical stress-concentration effect on fatigue strength, with no relief from local yielding. Most real materials fall somewhere between these extremes.
Kf = 1 + q(Kt − 1). Kt is the theoretical geometric stress concentration factor, q is the material's notch sensitivity (0 to 1), and Kf is the fatigue stress concentration factor actually applied when calculating a part's fatigue strength or endurance limit at a notch. This formula is why a very sharp notch (high Kt) in a low-notch-sensitivity material (low q) can still be far less damaging in fatigue than the raw Kt value would suggest.
Higher hardness and strength generally come from microstructures with less capacity for the local dislocation motion that allows a material to yield plastically at a notch tip and blunt the stress peak. With less of that redistribution available, more of the theoretical stress concentration translates directly into a real reduction in fatigue strength — hence higher q. This is a common reason a part redesigned in a stronger material can, counterintuitively, perform worse in fatigue at the same notch if the notch itself isn't also improved.
Primarily by softening the geometry itself: generous fillet radii instead of sharp corners, larger relative hole sizes or reinforced hole edges, elliptical rather than circular cutouts where load direction is known, gradual rather than abrupt cross-section changes, and smooth machined or polished surface finishes at critical features. Since Kt is pure geometry, the most reliable fix is almost always geometric — reducing Kt at the source rather than trying to compensate for it with a stronger but more notch-sensitive material.
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