The S-N curve, a.k.a., Stress Life Method, is the basic method presenting fatigue failure in high cycles (N > 10⁵) which implies the stress level is relatively low and the deformation is in elastic range.

The S-N curve for a specific material is the curve of nominal stress S (y axis) against the number of cycles to failure N (x axis). A log scale is almost always used for N. The stress is usually nominal stress and is no adjustment for stress concentration. The curve is usually obtained one by reversed bending experiments with zero mean stress.

The S-N curve of 1045 steel and 2014-T6 aluminum alloy is enclosed below to represent two tipical S-N curves of metal materials.

The 1045 steel, as well as some other steels and titanium alloys, exhibit a fatigue limit. When the amplitude of repeat loading is below the fatigue limit, small stresses do not shorten the fatigue life of the material. On the other hand, the 2014-T6 aluminum alloy and most metal materials do not have the fatigue limit, i.e., small stresses will eventually cause failure.

In short, the S-N curve is used to predicts the number of cycles sustained under certain stress before failure. The curve gives designers a quick reference of the allowable stress level for an intended service life.

While S-N curve is clear and straight forward on addressing the service life under fatigue, its accuracy leaves some room to be improved. Partially because of the statistical nature of fatigue and Partially because of the difference between laboratory experiments and the real-life practice.

For example, most S-N curves are constructed based on zero mean stress. However, it is more often the time-varying stresses are oscillating near a non-zero mean stress. Multiple S-N curves are determined by a several sets of fatigue experiments. Each curve represents a specific mean stress of that particular material.

The non-zero mean stress S-N relation requires huge amount of experiments to obtain the required data and form the mesh over a wide range of mean stresses. There are two approaches to present the data. The first is to present it in a diagram format. The second is to resemble the data with a formula based on the zero-mean stress S-N curve.

A more popular diagram for design purposes is called master diagram which accumulates fatigue data under different mean stresses and presents each line as the fatigue life under the net of maximum and minimum stresses in addition to mean stress and alternating stress as the reference axises. An example of master diagram of AISI 4340 steel is enclosed for your reference.

Users may check the maximum and minimum stress directly. Define R is the ratio of minimum stress to the maximum stress. Alternatively, define A is the ratio of alternating stress to mean stress.

An approximation based on the zero-mean stress S-N curve proposed by Goodman and Gerber is written as

where	is the amplitude of allowable stress (alternating stress).
	is the stress at fatigue fracture when the material under zero mean stress cycled loading.
	m is the mean stress of the actual loading.
	u is the tensile strength of the material.
	r = 1 is called Goodman line which is close to the results of notched specimens. r = 2 is the Gerber parabola which better represents ductile metals.

Please note that the S-N curve and its elaborated master diagram require a lot of experiments to accumulate the necessary data. On the other hand, Goodman and Gerber's approximations, although simple, they might not properly represent the specific material. Finally, they are only good for uniform mean and alternating stresses.

With the complexity of a master diagram, not to mention the time and effort to create one, the fatigue prediction is still less that perfect. In practice, a mechanical component is exposed to a complex, often random, sequence of loads, large and small and different mean values.

The procedure to establish the combined effect of a sequence of loads may involve

1. Simplify and divide the complex loading to a series of simple cyclic loadings
2. Create a histogram of cyclic stress
3. For each stress level, calculate the degree of cumulative damage incurred from the S-N curve
4. Combine the individual contributions to the total effect

Palmgren (1924) and Miner (1945) suggested an algorithm to combine individual contributions, known as Palmgren-Miner's linear damage hypothesis or Miner's rule.

where	k is the total number of different stress magnitudes in a spectrum
	Si (1 <= i <= k) is the magnitudes of each different stress in a spectrum
	ni(Si) is the actual number of cycles under the specific stress Si
	Ni(Si) is the total number of cycles to failure under the specific stress Si
	0.7 < c < 2.2 is an material dependent constant obtained by experiments. Set c = 1, if there is no further information available.

Miner's rule assume the fatigue life is consumed by the linear combination of different portion of stress state, both cycles and magnitude. This approximation, which is simple and straight forward, does not take the squences of loading history into account. For example, a serial of high stress loading, which weaken the material, followed by a serial low stress loading may cause more damage than a serial of low stress loading followed by a serial of high stress loading. But Miner's rule can not catch this effect.

Finally, the probabilistic nature of fatigue makes Miner's rule look over simplified.