It has been recognized since 1830 that a metal subjected to a repetitive or fluctuating stress will fail at a stress much lower than that required to cause fracture on a single application of load. Failures occurring under conditions of dynamic loading are called fatigue failures, presumably because it is generally observed that these failures occur only after a considerable period of service. Fatigue has become progressively more prevalent as technology has developed a greater amount of equipment, such as automobiles, aircraft, compressors, pumps, turbines, etc., subject to repeated loading and vibration. Today it is often stated that fatigue accounts for al least 90 percent of all service failures due to mechanical causes.

A fatigue failure is particularly insidious because it occurs without any obvious warning. Fatigue results in a brittle-appearing fracture, with no gross deformation at the fracture. On a macroscopic scale the fracture surface is usually normal to the direction of the principal tensile stress. A fatigue failure can usually be recognized from the appearance of the fracture surface, which shows a smooth region, due to the rubbing action as the crack propagated through the section, and a rough region, where the member has failed in a ductile manner when the cross section was no longer able to carry the load. Frequently the progress of the fracture is indicated by a series of rings, or "beach marks", progressing inward from the point of initiation of the failure.

Three basic factors are necessary to cause fatigue failure. These are:

• maximum tensile stress of sufficiently high value,
• large enough variation or fluctuation in the applied stress, and
• sufficiently large number of cycles of the applied stress.

In addition, there are a host of other variables, such as stress concentration, corrosion, temperature, overload, metallurgical structure, residual stresses, and combined stresses, which tend to alter the conditions for fatigue. Since we have not yet gained a complete understanding of what causes fatigue in metals, it will be necessary to discuss each of these factors from an essentially empirical standpoint. Because of the mass of data of this type, it will be possible to describe only the highlights of the relationship between these factors and fatigue.

Stress Cycles

At the outset it will be advantageous to define briefly the general types of fluctuating stresses which can cause fatigue. Figure 1 serves to illustrate typical fatigue stress cycles.

Figure 1a illustrates a completely reversed cycle of stress of sinusoidal form. For this type of stress cycle the maximum and minimum stresses are equal. Tensile stress is considered positive, and compressive stress is negative.

Figure 1b illustrates a repeated stress cycle in which the maximum stress σ_max (R_max) and minimum stress σ_min (R_min) are not equal. In this illustration they are both tension, but a repeated stress cycle could just as well contain maximum and minimum stresses of opposite signs or both in compression.

Figure 1c illustrates a complicated stress cycle which might be encountered in a part such as an aircraft wing which is subjected to periodic unpredictable overloads due to gusts.

Figure 1. Typical fatigue stress cycles. (a) Reversed stress; (b) repeated stress; (c) irregular or random stress cycle.

A fluctuating stress cycle can be considered to be made up of two components, a mean, or steady, stress σ_m (R_m), and an alternating, or variable, stress σ_a. We must also consider the range of stress σ_r. As can be seen from Fig. 1b, the range of stress is the algebratic difference between the maximum and minimum stress in a cycle.

The S-N Curve

The basic method of presenting engineering fatigue data is by means of the S-N curve, a plot of stress S against the number of cycles to failure N. A log scale is almost always used for N. The value of stress that is plotted can be σ_a, σ_max, or σ_min. The stress values are usually nominal stresses, i.e., there is no adjustment for stress concentration. The S-N relationship is determined for a specified value of σ_m, R (R=σ_min/σ_max), or A (A=σ_a/σ_m). Most determinations of the fatigue properties of materials have been made in completed reversed bending, where the mean stress is zero.

It will be noted that this S-N curve is concerned chiefly with fatigue failure at high numbers of cycles (N > 10⁵ cycles). Under these conditions the stress, on a gross scale, is elastic, but as we shall see shortly the metal deforms plastically in a highly localized way. At higher stresses the fatigue life is progressively decreased, but the gross plastic deformation makes interpretation difficult in terms of stress. For the low-cycle fatigue region (N <>4 or 10⁵ cycles) tests are conducted with controlled cycles of elastic plus plastic strain instead of controlled load or stress cycles.

The usual procedure for determining an S-N curve is to test the first specimen at a high stress where failure is expected in a fairly short number of cycles, e.g., at about two-thirds the static tensile strength of the material. The test stress is decreased for each succeeding specimen until one or two specimens do not fail in the specified numbers of cycles, which is usually at least 10⁷ cycles.

The highest stress at which a runout (non-failure) is obtained is taken as the fatigue limit. For materials without a fatigue limit the test is usually terminated for practical considerations at a low stress where the life is about 10⁸ or 5x10⁸ cycles. The S-N curve is usually determined with about 8 to 12 specimens.

Statistical Nature of Fatigue

A considerable amount of interest has been shown in the statistical analysis of fatigue data and in reasons for the variability in fatigue-test results. Since fatigue life and fatigue limit are statistical quantities, it must be realized that considerable deviation from an average curve determined with only a few specimens is to be expected.

It is necessary to think in terms of the probability of a specimen attaining a certain life at a given stress or the probability of failure at a given stress in the vicinity of the fatigue limit. To do this requires the testing of considerably more specimens than in the past so that the statistical parameters for estimating these probabilities can be determined.

The basic method for expressing fatigue data should then be a three-dimensional surface representing the relationship between stress, number of cycles to failure, and probability of failure.

In determining the fatigue limit of a material, it should be recognized that each specimen has its own fatigue limit, a stress above which it will fail but below which it will not fail, and that this critical stress varies from specimen to specimen for very obscure reasons. It is known that inclusions in steel have an important effect on the fatigue limit and its variability, but even vacuum-melted steel shows appreciable scatter in fatigue limit.

The statistical problem of accurately determining the fatigue limit is complicated by the fact that we cannot measure the individual value of the fatigue limit for any given specimen. We can only test a specimen at a particular stress, and if the specimen fails, then the stress was somewhere above the fatigue limit of the specimen. The two statistical methods which are used for making a statistical estimate of the fatigue limit are called probit analysis and the staircase method. The procedures for applying these methods of analysis to the determination of the fatigue limit have been well established.

Effect of Mean Stress on Fatigue

Much of the fatigue data in the literature have been determined for conditions of completely reversed cycles of stress, σ_m = 0. However, conditions are frequently met in engineering practice where the stress situation consists of an alternating stress and a superimposed mean, or steady, stress. There are several possible methods of determining an S-N diagram for a situation where the mean stress is not equal to zero.

Cyclic Stress-Strain Curve

Cyclic strain controlled fatigue, as opposed to our previous discussion of cyclic stress controlled fatigue, occurs when the strain amplitude is held constant during cycling. Strain controlled cyclic loading is found in thermal cycling, where a component expands and contracts in response to fluctuations in the operating temperature. In a more general view, the localized plastic strains at a notch subjected to either cyclic stress or strain conditions result in strain controlled conditions near the root of the notch due to the constraint effect of the larger surrounding mass of essentially elastically deformed material.

Since plastic deformation is not completely reversible, modifications to the structure occur during cyclic straining and these can result in changes in the stress-strain response. Depending on the initial state a metal may undergo cyclic hardening, cyclic softening, or remain cyclically stable. It is not uncommon for all three behaviors to occur in a given material depending on the initial state of the material and the test conditions.

Generally the hysteresis loop stabilizes after about 100 cycles and the material arrives at an equilibrium condition for the imposed strain amplitude. The cyclically stabilized stress-strain curve may be quite different from the stress-strain curve obtained on monotonic static loading. The cyclic stress-strain curve is usually determined by connecting the tips of stable hysteresis loops from constant-strain-amplitude fatigue tests of specimens cycled at different strain amplitudes. Under conditions where saturation of the hysteresis loop is not obtained, the maximum stress amplitude for hardening or the minimum stress amplitude for softening is used. Sometimes the stress is taken at 50 percent of the life to failure. Several shortcut procedures have been developed.

Low-Cycle Fatigue

Although historically fatigue studies have been concerned with conditions of service in which failure occurred at more than 10⁴ cycles of stress, there is growing recognition of engineering failures which occur at relatively high stress and low numbers of cycles to failure. This type of fatigue failure must be considered in the design of nuclear pressure vessels, steam turbines, and most other types of power machinery. Low-cycle fatigue conditions frequently are created where the repeated stresses are of thermal origin. Since thermal stresses arise from the thermal expansion of the material, it is easy to see that in this case fatigue results from cyclic strain rather than from cyclic stress.