A static strength check compares stress to yield and calls it safe. That check says nothing about a part that sees the same load thousands of times a day — and a stress level that's perfectly safe once can still break the part on the ten-thousandth cycle.
Fatigue is damage that accumulates, not damage that happens all at once. A microscopic crack starts at a surface defect, a machining mark, or — most commonly — a sharp corner, and grows a little further with every load cycle until the remaining cross-section can't carry the load and fails suddenly, often with no visible warning beforehand. The number that matters for a part under repeated load isn't yield strength — it's how many cycles the part sees before a crack that started small enough to miss has grown large enough to matter.
Plot stress against cycles-to-failure on a log-log scale and you get an S-N curve. For most steels, that curve flattens out at a large-but-finite number of cycles (conventionally around 10⁶–10⁷) into something close to a floor: the endurance limit, roughly 40–50% of the material's tensile strength, below which the material is treated as good for effectively infinite cycles. Aluminium's S-N curve never flattens — there is no stress level low enough to guarantee infinite life, which is why aluminium fatigue strength is always quoted at a specific cycle count instead of as a limit.
Stress doesn't distribute evenly around a sharp change in cross-section — it concentrates. A stepped shaft with a sharp corner (r/d = 0.01) can see local stress roughly 3.7× the nominal average; open that same corner to a generous fillet (r/d = 0.20) and the same geometry drops to roughly 1.6× — more than halving the peak stress with no change to the part's function, only its corner radius. Because the S-N curve is steep, that difference in peak stress translates to an outsized difference in cycles to failure: a part that looks safe on nominal (average) stress can still fail early because the crack starts at the concentrated stress, not the average one.
A part cycling between 0 and 200 MPa and a part cycling between 300 and 500 MPa can have the same 200 MPa stress range and very different fatigue lives — the second one is also carrying a steady 400 MPa mean stress, which uses up margin the first part doesn't have to spend. A tensile mean stress reduces how much cyclic swing a part can survive, which is why preloaded or continuously-loaded parts need a different check than a part that only sees load occasionally.
Most fatigue failures I've seen traced back weren't a material problem — they were a stress-below-yield check standing in for a fatigue check that was never run, on a part with a sharp internal corner that nobody flagged. A generous fillet costs nothing at the drawing stage and can be the difference between a part that lasts and one that doesn't. When a part sees repeated load, I ask for the cycle count and the corner radii before I ask for the material.