Fatigue and the S–N curve, explained

How fatigue happens

A crack starts at a stress concentration, a scratch or an inclusion, usually at the surface. Each load cycle grows it a little. When the remaining section can no longer carry the load, it breaks suddenly. The fracture surface shows it: smooth beach marks where the crack grew, then a rough zone where it finally let go.

Most mechanical failures in service are fatigue failures. Shafts, springs, welds, bolts and aircraft structure are all designed against it first and against static strength second.

Reading the S–N curve

Stress amplitude S is plotted against the number of cycles to failure N on a log scale. Steels show an endurance limit: a stress below which they survive indefinitely, around half the ultimate strength for a polished test specimen. Aluminium has no true endurance limit, so designs quote a strength at a fixed life, often 5 × 10⁸ cycles.

A real part is weaker than the polished specimen. The Marin factors reduce the endurance limit for surface finish, size, load type, temperature and reliability, and a rough machined surface alone can cost a quarter or more.

Mean stress and Goodman

A tensile mean stress shortens fatigue life; a compressive one helps, which is why shot peening works. The Goodman line connects the endurance limit on the alternating axis to the ultimate strength on the mean axis, and is the standard conservative design check.

A worked number

A steel with S_ut = 600 MPa has a specimen endurance limit near 300 MPa. Say the Marin factors together come to 0.6, giving S_e = 180 MPa. Under σ_a = 80 MPa and σ_m = 100 MPa, Goodman gives 80/180 + 100/600 ≈ 0.61, so the factor of safety is about 1.6.

Common mistakes

• Designing to the polished-specimen endurance limit without the Marin factors.
• Ignoring mean stress, which makes a bolt or spring look safer than it is.
• Assuming aluminium has an endurance limit like steel.