Endurance Limit

The endurance limit S_e is the stress amplitude at which the S–N curve becomes horizontal — the stress level below which fatigue failure does not occur, regardless of how many cycles are applied. It is a material property that emerges from the physical mechanisms of fatigue damage: below a certain cyclic stress amplitude, the plastic strain per cycle is insufficient to propagate dislocations, nucleate persistent slip bands, or advance any pre-existing micro-cracks. Fatigue damage simply does not accumulate.

Materials with and without an endurance limit

The endurance limit is a well-defined property for steels, cast irons, and some titanium alloys. For these materials, the S–N curve flattens at approximately 10^6–10^7 cycles, and components designed to operate below S_e are considered to have infinite fatigue life. A rough correlation for steels relates the endurance limit to the ultimate tensile strength \sigma_\text{UTS}:

Above \sigma_\text{UTS} \approx 1400 \, \text{MPa}, the endurance limit saturates at approximately 700 \, \text{MPa} and does not continue to increase with strength — a consequence of increasing notch sensitivity and susceptibility to hydrogen embrittlement in very high-strength steels.

Aluminium alloys, copper alloys, magnesium alloys, and most non-ferrous metals do not exhibit a true endurance limit. Their S–N curves continue to slope downward at high cycle counts with no horizontal asymptote. For these materials, a fatigue strength at a specified life — most commonly 10^7 or 10^8 cycles — is reported instead. This is sometimes loosely referred to as an “endurance limit” in engineering practice, but it is more precisely called a fatigue strength at N cycles.

Corrected endurance limit for real components

The endurance limit S_e determined from laboratory specimens on small, polished, unnotched samples under fully reversed bending does not directly apply to real engineering components. Several factors reduce the effective endurance limit, and a set of correction factors is applied to obtain the modified endurance limit S_e':

The surface factor k_a accounts for the effect of surface finish: ground, machined, hot-rolled, and forged surfaces all have progressively lower k_a than a polished specimen, because rougher surfaces have more initiation sites. The size factor k_b reflects that larger components have a greater volume of material at high stress and a higher probability of containing a critical defect; k_b < 1 for diameters above approximately 8 mm. The load factor k_c corrects for loading mode: axial loading produces a more uniform stress distribution than bending, resulting in a lower effective endurance limit. The temperature factor k_d reduces S_e' at elevated temperatures. The reliability factor k_e applies a statistical correction to achieve a specified survival probability.

Endurance limit and stress concentration

Geometric discontinuities — notches, holes, fillets, keyways — reduce the effective endurance limit through the fatigue strength reduction factor K_f:

The fatigue strength reduction factor K_f = 1 + q(K_t - 1) depends on the theoretical stress concentration factor K_t and the notch sensitivity index q of the material. High-strength steels are more notch-sensitive than low-strength steels, meaning that geometric discontinuities cause a proportionally greater reduction in endurance limit as material strength increases.

Design use

In infinite-life design — the traditional approach for components expected to operate for an indefinitely large number of cycles without scheduled replacement — the design criterion is that the alternating stress at the critical location, corrected for mean stress using the Goodman criterion or a similar model, must not exceed S_e'. This approach is standard for rotating shafts, springs, gear teeth, and other components subjected to regular cyclic loading throughout their service life.