The relation between stress amplitude, σa, and life, N, is one of the oldest methods for fatigue evaluation. It is a very popular method since the relation can be easily obtained from fatigue tests. The relation takes two characteristic shapes, shown in Figure 3-8. The upper shape, which decreases until a point where it becomes a horizontal line, represents materials that have an endurance limit. If the load amplitude in a load cycle is below the endurance limit, the component does not fail in fatigue. If the endurance limit is exceeded, the component does fail in fatigue if the number of load cycles is large enough. The number of cycles where the S–N curve becomes horizontal varies for different materials but is usually in the range of 106 to 108 cycles. Several materials do not have an endurance limit and experience fatigue even at low stress amplitudes. This fatigue behavior is demonstrated by the dashed line in Figure 3-8.

where Δσ is the stress range evaluated as the difference between the largest and the smallest stresses experienced during a fatigue cycle.

where σ1 is the largest principal stress, σ3 is the smallest principal stress, σvM is the equivalent stress according to von Mises, and σh is the hydrostatic (or mean) stress.

One of the oldest models for fatigue prediction is the Wöhler curve, also called the S–N curve. It relates the number of cycles, N, to the stress amplitude, σa, when stress cycle is constant. On the equation form it is defined as

where fSN denotes the S–N function. At low stresses the fatigue life is limited by a Cycle Cutoff. At high stresses, when the stress amplitude exceeds the highest stress as defined by the S–N curve, the fatigue life cannot be determined. No results are computed in such regions. This is demonstrated in Figure 3-9.

Often an S–N curve for the actual operating condition of the component is not available. One way to handle this is to use an S–N curve obtained in the nominal condition and modify it by applying a stress modification factor, k. On the equation form the stress life relation then becomes

where σf and b are material constants and σa is the stress amplitude. Nf is the number of load cycles to failure, and 2Nf is the number of load reversals. This relation becomes a linear in a log-log diagram and is often utilized in engineering applications. At low stresses the fatigue life is limited by a Cycle Cutoff.

In cases where accurate stress life data is missing, an approximate method can be applied. Since the Basquin equation is a linear relation in the log-log space and the endurance limit is a straight line, a bilinear approximate of an S–N curve can be constructed. This requires knowledge of the endurance limit and the transition between the high-cycle fatigue (HCF) and the low-cycle fatigue. In a fatigue life (N) versus stress amplitude (σa) graph, the bilinear relation can be constructed as shown Figure 3-10.

The transition stress between LCF and HCF is denoted by St, the transition life is denoted by Nt, and the endurance limit or the endurance stress is denoted by Se. The point that defines a knee on the S–N curve is denoted Ne and is called the endurance life. Often the key parameters of the approximate curve are expressed using the ultimate tensile stress, Su. In steel, for example, the values are roughly St = 0.9 ⋅ Su, Nt = 103 cycles, and Se = 0.5 ⋅ Su, Ne = 106 cycles.