S–N Curve

The S–N curve — named from the stress amplitude S on the vertical axis and the number of cycles to failure N on the horizontal axis — is the foundational diagram of classical fatigue design. It was introduced by August Wöhler in the 1860s through systematic fatigue testing of railway axles, which is why it is also called the Wöhler curve. The diagram maps the experimental relationship between cyclic stress level and fatigue life, providing the data needed to design components against failure by accumulated cyclic damage.

Construction of the S–N curve

To construct an S–N curve, a series of nominally identical specimens are tested under sinusoidal (or other controlled) cyclic loading at different stress amplitudes. Each specimen is cycled until it fractures, and the number of cycles to failure N_f is recorded. The results are plotted on a semi-logarithmic scale (linear S, logarithmic N) or a log-log scale, and a curve — usually a straight line on the chosen scale — is fitted through the data.

Because fatigue data are inherently scattered (reflecting variability in surface finish, material microstructure, internal defects, and test conditions), multiple specimens are tested at each stress level, and statistical methods are used to construct curves corresponding to specified survival probabilities (typically 50%, 90%, or 99%). Design codes specify which probability level to use.

Regions of the S–N curve

The S–N curve for a typical steel shows three characteristic regions. At high stress amplitudes — in the low-cycle fatigue (LCF) regime, roughly N < 10^4 cycles — plastic deformation occurs in each cycle and life is short. This regime is better handled by strain-life methods rather than stress-life analysis. In the intermediate high-cycle fatigue (HCF) regime (10^4 < N < 10^6–10^7 cycles), the curve slopes downward on a log scale: higher stress means shorter life. For steels and cast irons, the curve flattens out below a characteristic stress level at approximately 10^6–10^7 cycles — this horizontal asymptote is the endurance limit S_e.

For aluminium alloys, titanium alloys, and most non-ferrous metals, no true horizontal asymptote exists. The curve continues to slope downward at high cycle counts, and fatigue strength is reported at a specified number of cycles — typically 10^7 or 10^8 — rather than as an endurance limit.

Fatigue strength at finite life

The S–N curve can be represented analytically. A common form for the finite-life region is the Basquin equation:

or equivalently:

where b and C are material constants. In this form, the S–N curve is a straight line on a log-log plot with slope -1/b. Given the Basquin constants, the fatigue strength at any number of cycles N can be computed directly.

Specimen versus component

S–N data are almost always generated on small, polished, unnotched specimens under completely reversed bending or axial loading. Real engineering components differ from test specimens in surface finish, size, loading mode, mean stress, and the presence of stress concentrations. These differences are accounted for by applying correction factors to the specimen endurance limit:

where k_a is the surface finish factor, k_b the size factor, k_c the loading factor, k_d the temperature factor, and k_e the reliability factor. The corrected endurance limit S_e' is the appropriate design threshold for the actual component.

Limitations

The S–N approach is a stress-based method best suited to high-cycle fatigue where stresses remain predominantly elastic. It does not directly account for plastic strain accumulation, crack initiation mechanisms, crack growth, or variable amplitude loading (for which the Miner rule is used as an approximation). Where fracture mechanics data are available, damage tolerance analysis using the Paris–Erdogan law provides a more physically transparent alternative.

Common mistakes

A common mistake is applying a polished-specimen S–N curve directly to a real component with notches, welds, corrosion, residual stress, surface roughness, size effects, or mean stress. Another is treating the endurance limit as universal when many non-ferrous alloys have no true fatigue limit. A strong S–N review states material condition, test method, stress ratio, survival probability, surface condition, size correction, mean-stress correction, environment, loading spectrum, and whether crack-growth assessment is required.