High cycle fatigue (HCF) occurs at relatively large numbers of cycles. The arbitrary classification between high cycle fatigue and low cycle fatigue (LCF) is considered to be roughly 10,000 cycles, where stress is relatively low and deformation is primarily elastic. In general high cycle fatigue is caused by alternating stresses in the elastic or slightly inelastic range, while low cycle fatigue is normally in the inelastic or plastic range which exhibits large strains; the exact number of cycles depends on the properties of the metal.

A study in 1992 conducted by the Scientiﬁc Advisory Board of the US Air Force targeted high cycle fatigue as the single biggest cause of turbine engine failures in military aircraft. High cycle fatigue results in rapid, essentially unpredictable failures due to fatigue-crack propagation under ultrahigh frequency loading. Furthermore, cracking often initiates from small defects which are associated with damage caused by a variety of drivers, including fretting and impacts by foreign objects.

### Stress-cycle (S-N) curve

In high cycle fatigue situations, materials performance is commonly characterized by an *S-N curve*, or stress-cycle curve, or fatigue curve, also known as a *Wöhler curve*. This is a graph of the magnitude of a cyclic stress (*S*) against the logarithmic scale of cycles to failure (*N*).

The S-N curve in the high-cycle region is sometimes described by the Basquin equation:

σ_{a} = σ’_{f}(2N_{f})^{b}

S-N curves of ASTM A-36 and AISI 4340 based on HCF data

The progression of the *S-N curve* can be influenced by many factors such as corrosion, temperature, residual stresses, and the presence of notches. The Goodman-Line is a method used to estimate the influence of the mean stress on the fatigue strength.

### Probabilistic nature of fatigue

As coupons sampled from a homogeneous frame will display a variation in their number of cycles to failure, the S-N curve should more properly be a Stress-Cycle-Probability (S-N-P) curve to capture the probability of failure after a given number of cycles of a certain stress. Probability distributions that are common in data analysis and in design against fatigue include the log-normal distribution, extreme value distribution, Birnbaum–Saunders distribution, and Weibull distribution.

### Complex loadings

In practice, a mechanical part is exposed to a complex, often random, sequence of loads, large and small. In order to assess the safe life of such a part:

- Complex loading is reduced to a series of simple cyclic loadings using a technique such as rainflow analysis;
- A histogram of cyclic stress is created from the rainflow analysis to form a fatigue damage spectrum;
- For each stress level, the degree of cumulative damage is calculated from the S-N curve; and
- The effect of the individual contributions are combined using an algorithm such as
*Miner’s rule*.

#### For multiaxial loading

Since S-N curves are typically generated for uniaxial loading, some equivalence rule is needed whenever the loading is multiaxial. For simple, proportional loading histories (lateral load in a constant ratio with the axial), Sines rule may be applied. For more complex situations, such as nonproportional loading, Critical plane analysis must be applied.

### Miner’s Rule

In 1945, M A Miner popularised a rule that had first been proposed by A. Palmgren in 1924. The rule, variously called *Miner’s rule* or the *Palmgren-Miner linear damage hypothesis*, states that where there are *k* different stress magnitudes in a spectrum, *S _{i}* (1 ≤

*i*≤

*k*), each contributing

*n*(

_{i}*S*) cycles, then if

_{i}*N*(

_{i}*S*) is the number of cycles to failure of a constant stress reversal

_{i}*S*(determined by uni-axial fatigue tests), failure occurs when:

_{i}C is experimentally found to be between 0.7 and 2.2. Usually for design purposes, C is assumed to be 1. This can be thought of as assessing what proportion of life is consumed by a linear combination of stress reversals at varying magnitudes.

Though Miner’s rule is a useful approximation in many circumstances, it has several major limitations:

- It fails to recognise the probabilistic nature of fatigue and there is no simple way to relate life predicted by the rule with the characteristics of a probability distribution. Industry analysts often use design curves, adjusted to account for scatter, to calculate
*N*(_{i}*S*)._{i} - The sequence in which high vs. low stress cycles are applied to a sample in fact affect the fatigue life, for which Miner’s Rule does not account. In some circumstances, cycles of low stress followed by high stress cause more damage than would be predicted by the rule. It does not consider the effect of an overload or high stress which may result in a compressive residual stress that may retard crack growth. High stress followed by low stress may have less damage due to the presence of compressive residual stress.

### Paris’ Law

In Fracture mechanics, Anderson, Gomez, and Paris derived relationships for the stage II crack growth with cycles N, in terms of the cyclical component ΔK of the Stress Intensity Factor K

where a is the crack length and m is typically in the range 3 to 5 (for metals), which states that the rate of crack growth with respect to the cycles of load applied is a function of the stress intensity factor; this is named Paris’ law.

This relationship was later modified (by Forman, 1967) to make better allowance for the mean stress, by introducing a factor depending on (1-R) where R= min stress/max stress, in the denominator.

### Mean Stress Correction

A number of models exist to compensate for mean stresses:

- Goodman (England 1899)
- Soderberg (USA 1930)
- Gerber (Germany 1870)

These models work in the same manner. The stress amplitude that is to be used in Basquin’s law is corrected according to the mean stress σ_{m} and yield stress σ_{y }or tenstil stress σ_{u.}