# Fatigue (material)

In materials science, fatigue is the weakening of a material caused by repeatedly applied loads. It is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading. The nominal maximum stress values that cause such damage may be much less than the strength of the material typically quoted as the ultimate tensile stress limit, or the yield stress limit.

Fatigue occurs when a material is subjected to repeated loading and unloading. If the loads are above a certain threshold, microscopic cracks will begin to form at the stress concentrators such as the surface, persistent slip bands (PSBs), interfaces of constituents in the case of composites, and grain interfaces in the case of metals.[1] Eventually a crack will reach a critical size, the crack will propagate suddenly, and the structure will fracture. The shape of the structure will significantly affect the fatigue life; square holes or sharp corners will lead to elevated local stresses where fatigue cracks can initiate. Round holes and smooth transitions or fillets will increase the fatigue strength of the structure.

## Fatigue lifeEdit

The American Society for Testing and Materials defines fatigue life, Nf, as the number of stress cycles of a specified character that a specimen sustains before failure of a specified nature occurs.[2] For some materials, notably steel and titanium, there is a theoretical value for stress amplitude below which the material will not fail for any number of cycles, called a fatigue limit, endurance limit, or fatigue strength.[3]

Engineers have used any of three methods to determine the fatigue life of a material: the stress-life method, the strain-life method, and the linear-elastic fracture mechanics method.[4] One method to predict fatigue life of materials is the Uniform Material Law (UML).[5] UML was developed for fatigue life prediction of aluminium and titanium alloys by the end of 20th century and extended to high-strength steels,[6] and cast iron.[7]

## Characteristics of fatigueEdit

Fracture of an aluminium crank arm. Dark area of striations: slow crack growth. Bright granular area: sudden fracture.
• In metal alloys, and for the simplifying case when there are no macroscopic or microscopic discontinuities, the process starts with dislocation movements at the microscopic level, which eventually form persistent slip bands that become the nucleus of short cracks.
• Macroscopic and microscopic discontinuities (at the crystalline grain scale) as well as component design features which cause stress concentrations (holes, keyways, sharp changes of load direction etc.) are common locations at which the fatigue process begins.
• Fatigue is a process that has a degree of randomness (stochastic), often showing considerable scatter even in seemingly identical sample in well controlled environments.
• Fatigue is usually associated with tensile stresses but fatigue cracks have been reported due to compressive loads.[8]
• The greater the applied stress range, the shorter the life.
• Fatigue life scatter tends to increase for longer fatigue lives.
• Damage is cumulative. Materials do not recover when rested.
• Fatigue life is influenced by a variety of factors, such as temperature, surface finish, metallurgical microstructure, presence of oxidizing or inert chemicals, residual stresses, scuffing contact (fretting), etc.
• Some materials (e.g., some steel and titanium alloys) exhibit a theoretical fatigue limit below which continued loading does not lead to fatigue failure.
• High cycle fatigue strength (about 104 to 108 cycles) can be described by stress-based parameters. A load-controlled servo-hydraulic test rig is commonly used in these tests, with frequencies of around 20–50 Hz. Other sorts of machines—like resonant magnetic machines—can also be used, to achieve frequencies up to 250 Hz.
• Low cycle fatigue (loading that typically causes failure in less than 104 cycles) is associated with localized plastic behavior in metals; thus, a strain-based parameter should be used for fatigue life prediction in metals. Testing is conducted with constant strain amplitudes typically at 0.01–5 Hz.

## Timeline of fatigue research historyEdit

Micrographs showing how surface fatigue cracks grow as material is further cycled. From Ewing & Humfrey, 1903
• 1837: Wilhelm Albert publishes the first article on fatigue. He devised a test machine for conveyor chains used in the Clausthal mines.[9]
• 1839: Jean-Victor Poncelet describes metals as being 'tired' in his lectures at the military school at Metz.
• 1842: William John Macquorn Rankine recognises the importance of stress concentrations in his investigation of railroad axle failures. The Versailles train wreck was caused by fatigue failure of a locomotive axle.[10]
• 1843: Joseph Glynn reports on the fatigue of an axle on a locomotive tender. He identifies the keyway as the crack origin.
• 1848: The Railway Inspectorate reports one of the first tyre failures, probably from a rivet hole in tread of railway carriage wheel. It was likely a fatigue failure.
• 1849: Eaton Hodgkinson is granted a "small sum of money" to report to the UK Parliament on his work in "ascertaining by direct experiment, the effects of continued changes of load upon iron structures and to what extent they could be loaded without danger to their ultimate security".
• 1854: Braithwaite reports on common service fatigue failures and coins the term fatigue.[11]
• 1860: Systematic fatigue testing undertaken by Sir William Fairbairn and August Wöhler.
• 1870: Wöhler summarises his work on railroad axles. He concludes that cyclic stress range is more important than peak stress and introduces the concept of endurance limit.[9]
• 1903: Sir James Alfred Ewing demonstrates the origin of fatigue failure in microscopic cracks.
• 1910: O. H. Basquin proposes a log-log relationship for S-N curves, using Wöhler's test data.[12]
• 1940: Sidney M. Cadwell publishes first rigorous study of fatigue in rubber.[13]
• 1945: A. M. Miner popularises Palmgren's (1924) linear damage hypothesis as a practical design tool.
• 1952: W. Weibull An S-N curve model.[14]
• 1954: The world's first commercial jetliner, the de Havilland Comet, suffers disaster as three planes break up in mid-air, causing de Havilland and all other manufacturers to redesign high altitude aircraft and in particular replace square apertures like windows with oval ones.
• 1954: L. F. Coffin and S. S. Manson explain fatigue crack-growth in terms of plastic strain in the tip of cracks.
• 1961: P. C. Paris proposes methods for predicting the rate of growth of individual fatigue cracks in the face of initial scepticism and popular defence of Miner's phenomenological approach.
• 1968: Tatsuo Endo and M. Matsuishi devise the rainflow-counting algorithm and enable the reliable application of Miner's rule to random loadings.[15]
• 1970: W. Elber elucidates the mechanisms and importance of crack closure in slowing the growth of a fatigue crack due to the wedging effect of plastic deformation left behind the tip of the crack.
• 1973: M. W. Brown and K. J. Miller observe that fatigue life under multiaxial conditions is governed by the experience of the plane receiving the most damage, and that both tension and shear loads on the critical plane must be considered.[16]

## High-cycle fatigueEdit

Historically, most attention has focused on situations that require more than 104 cycles to failure where stress is low and primarily elastic.

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

In high-cycle fatigue situations, materials performance is commonly characterized by an S-N 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).

S-N curve for a brittle aluminium with an ultimate tensile strength of 320 MPa

S-N curves are derived from tests on samples of the material to be characterized (often called coupons) where a regular sinusoidal stress is applied by a testing machine which also counts the number of cycles to failure. This process is sometimes known as coupon testing. Each coupon test generates a point on the plot though in some cases there is a runout where the time to failure exceeds that available for the test (see censoring). Analysis of fatigue data requires techniques from statistics, especially survival analysis and linear regression.

The progression of the S-N curve can be influenced by many factors such as stress ratio (mean stress), loading frequency, temperature, corrosion, residual stresses, and the presence of notches. A constant fatigue life (CFL) diagram[17] is useful for the study of stress ratio effect. The Goodman-Line is a method used to estimate the influence of the mean stress on the fatigue strength.

### Probabilistic nature of fatigueEdit

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.

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:

2. A histogram of cyclic stress is created from the rainflow analysis to form a fatigue damage spectrum;
3. For each stress level, the degree of cumulative damage is calculated from the S-N curve; and
4. The effect of the individual contributions are combined using an algorithm such as Miner's rule.

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 RuleEdit

In 1945, M. A. Miner[18] 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, Si (1 ≤ ik), each contributing ni(Si) cycles, then if Ni(Si) is the number of cycles to failure of a constant stress reversal Si (determined by uni-axial fatigue tests), failure occurs when:

${\displaystyle \sum _{i=1}^{k}{\frac {n_{i}}{N_{i}}}=C}$

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.

Although Miner's rule may be a useful approximation in many circumstances, it has several major limitations:

1. 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 Ni(Si).
2. 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.[19] 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.

### Constant Fatigue Life (CFL) diagram and Goodman relationEdit

A CFL diagram is useful for stress ratio effect on S-N curve.[20] Also, in the presence of a steady stress superimposed on the cyclic loading, the Goodman relation can be used to estimate a failure condition. It plots stress amplitude against mean stress with the fatigue limit and the ultimate tensile strength of the material as the two extremes. Alternative failure criteria include Soderberg and Gerber.[21]

### Paris' LawEdit

Typical fatigue crack growth rate graph

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[22]

${\displaystyle {\frac {\mathrm {d} a}{\mathrm {d} N}}=C(\Delta K)^{m}}$

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 in 1967[23] 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.

### Strain-cycle (ε-N) curveEdit

Due to the proportionality between stress and strain, high cycle fatigue can also be expressed as strain amplitude vs. number of cycles. High cycle fatigue can be approximated by equating the total strain to just the elastic strain. Using this approximation, ​12Δεelastic ≡ ​σf'E(2Nf)b

where

Δεelastic is the change in elastic strain per cycle
σf' is a parameter that scales with tensile strength obtained by fitting experimental data
E is the Young's modulus
Nf is the number of cycles to failure
b is the slope of the log-log curve again determined by fitting

The figure below shows high cycle fatigue as the right-most linear portion. Any test performed in the bottom left region (i.e. with a low enough strain amplitude and number of cycles) below the dark line has a high probability to avoid failure.

Graph showing fatigue failure as a function of strain amplitude.

As shown in the figure above (the left-most linear section) and as described in the next section, the total strain is approximated to be equal to just the plastic strain. For regions between high and low cycle fatigue, an unweighted sum of the high cycle and low cycle expressions gives a reasonable approximation with a built-in safety factor.[24]

## Low-cycle fatigueEdit

Where the stress is high enough for plastic deformation to occur, the accounting of the loading in terms of stress is less useful and the strain in the material offers a simpler and more accurate description. This type of fatigue is normally experienced by components which undergo a relatively small number of straining cycles. Low-cycle fatigue is usually characterised by the Coffin-Manson relation (published independently by L. F. Coffin in 1954[25] and S. S. Manson in 1953):[26]

${\displaystyle {\frac {\Delta \varepsilon _{p}}{2}}=\varepsilon _{f}'(2N)^{c}}$

where,

• Δεp /2 is the plastic strain amplitude;
• εf' is an empirical constant known as the fatigue ductility coefficient, the failure strain for a single reversal;
• 2N is the number of reversals to failure (N cycles);
• c is an empirical constant known as the fatigue ductility exponent, commonly ranging from −0.5 to −0.7 for metals in time independent fatigue. Slopes can be considerably steeper in the presence of creep or environmental interactions.

A similar relationship for materials such as Zirconium is used in the nuclear industry.[27]

## Fatigue and Fracture MechanicsEdit

Fatigue failures, both for high and low cycle, all follow the same basic four step process of crack initiation, stage I crack growth, stage II crack growth, and finally ultimate failure. To begin the process cracks must nucleate within a material. This process can occur either at stress risers in metallic samples or at areas with a high void density in polymer samples. These cracks propagate slowly at first during stage I crack growth along crystallographic planes, where shear stresses are highest. Once the cracks reach a critical size they propagate quickly during stage II crack growth in a direction perpendicular to the applied force. These cracks can eventually lead to the ultimate failure of the material, often in a brittle catastrophic fashion.

### Crack initiationEdit

The formation of initial cracks preceding fatigue failure is a separate process consisting of four discrete steps in metallic samples. The material will develop cell structures and harden in response to the applied load. This causes the amplitude of the applied stress to increase given the new restraints on strain. These newly formed cell structures will eventually break down with the formation of Persistent Slip Bands (PSBs). Slip in the material is localized at these PSBs, and the exaggerated slip can now serve as a stress concentrator for a crack to form. Nucleation of a crack to a detectable size accounts for most of the cracking process. It is for this reason that cyclic fatigue failures seem to occur so suddenly- the bulk of the changes in the material are not visible without destructive testing. Even in normally ductile materials, fatigue failures will resemble sudden brittle failures.

PSB-induced slip planes result in intrusions and extrusions along the surface of a material, often occurring in pairs. This slip is not a microstructural change within the material, but rather a propagation of dislocations within the material. Instead of a smooth interface, the intrusions and extrusions will cause the surface of the material to resemble the edge of a deck of cards, where not all cards are perfectly aligned. Slip-induced intrusions and extrusions create extremely fine surface structures on the material. With surface structure size inversely related to stress concentration factors, PSB-induced surface slip quickly becomes a perfect place for fractures to initiate.

It should be noted that these steps can be bypassed entirely if the cracks form at a preexisting stress concentrator either from an inclusion in the material or from a geometric stress concentrator such as a sharp corner or small radius. Forming a PSB before failure requires more energy than crack initiation at a preexisting stress concentrator. It is for that reason that part design and material quality must be scrutinized when producing parts that will be subjected to high cycle loading.

Fracture surface in a glass rod showing beach marks surrounding the initiation site.

Sources:[28][29]

## Design against fatigueEdit

Dependable design against fatigue-failure requires thorough education and supervised experience in structural engineering, mechanical engineering, or materials science. There are four principal approaches to life assurance for mechanical parts that display increasing degrees of sophistication:[30]

1. Design to keep stress below threshold of fatigue limit (infinite lifetime concept);
2. Fail-safe, graceful degradation, and fault-tolerant design: Instruct the user to replace parts when they fail. Design in such a way that there is no single point of failure, and so that when any one part completely fails, it does not lead to catastrophic failure of the entire system.
3. Safe-life design: Design (conservatively) for a fixed life after which the user is instructed to replace the part with a new one (a so-called lifed part, finite lifetime concept, or "safe-life" design practice); planned obsolescence and disposable product are variants that design for a fixed life after which the user is instructed to replace the entire device;
4. Damage tolerant design: Instruct the user to inspect the part periodically for cracks and to replace the part once a crack exceeds a critical length. This approach usually uses the technologies of nondestructive testing and requires an accurate prediction of the rate of crack-growth between inspections. The designer sets some aircraft maintenance checks schedule frequent enough that parts are replaced while the crack is still in the "slow growth" phase. This is often referred to as damage tolerant design or "retirement-for-cause".

### Stopping fatigueEdit

Fatigue cracks that have begun to propagate can sometimes be stopped by drilling holes, called drill stops, in the path of the fatigue crack.[31] This is not recommended as a general practice because the hole represents a stress concentration factor which depends on the size of the hole and geometry, though the hole is typically less of a stress concentration than the removed tip of the crack. The possibility remains of a new crack starting in the side of the hole. It is always far better to replace the cracked part entirely.

### Material changeEdit

Changes in the materials used in parts can also improve fatigue life. For example, parts can be made from better fatigue rated metals. Complete replacement and redesign of parts can also reduce if not eliminate fatigue problems. Thus helicopter rotor blades and propellers in metal are being replaced by composite equivalents. They are not only lighter, but also much more resistant to fatigue. They are more expensive, but the extra cost is amply repaid by their greater integrity, since loss of a rotor blade usually leads to total loss of the aircraft. A similar argument has been made for replacement of metal fuselages, wings and tails of aircraft.[32]

### Peening treatment of welds and metal componentsEdit

Example of a HFMI treated steel highway bridge to avoid fatigue along the weld transition.

Increases in fatigue life and strength are proportionally related to the depth of the compressive residual stresses imparted by surface enhancement processes such as shot peening but particularly by laser peening. Shot peening imparts compressive residual stresses approximately 0.005 inches deep, laser peening imparts compressive residual stresses from 0.040 to 0.100 inches deep, or deeper. Laser peening provide significant fatigue life extension through shock wave mechanics which plastically deform the surface of the metal component changing the material properties.[33] Laser peening can be applied to existing parts without redesign requirements or incorporated into new designs to allow for lighter materials or thinner designs to achieve comparable engineering results.

### High frequency mechanical impact (HFMI) treatment of weldsEdit

The durability and life of dynamically loaded, welded steel structures are determined often by the welds, particular by the weld transitions. By selective treatment of weld transitions with the High Frequency Mechanical Impact (HFMI) treatment method,[34][35] the durability of many designs can be increased significantly. This method is universally applicable, requires only specific equipment and offers high reproducibility and a high degree of quality control.

#### Deep Cryogenic treatmentEdit

The use of Deep Cryogenic treatment has been shown to increase resistance to fatigue failure. Springs used in industry, auto racing and firearms have been shown to last up to six times longer when treated. Heat checking, which is a form of thermal cyclic fatigue has been greatly delayed.[36]

## Notable fatigue failuresEdit

### Versailles train crashEdit

Versailles train disaster

Drawing of a fatigue failure in an axle by Joseph Glynn, 1843

Following the King's celebrations at the Palace of Versailles, a train returning to Paris crashed in May 1842 at Meudon after the leading locomotive broke an axle. The carriages behind piled into the wrecked engines and caught fire. At least 55 passengers were killed trapped in the carriages, including the explorer Jules Dumont d'Urville. This accident is known in France as the "Catastrophe ferroviaire de Meudon". The accident was witnessed by the British locomotive engineer Joseph Locke and widely reported in Britain. It was discussed extensively by engineers, who sought an explanation.

The derailment had been the result of a broken locomotive axle. Rankine's investigation of broken axles in Britain highlighted the importance of stress concentration, and the mechanism of crack growth with repeated loading. His and other papers suggesting a crack growth mechanism through repeated stressing, however, were ignored, and fatigue failures occurred at an ever-increasing rate on the expanding railway system. Other spurious theories seemed to be more acceptable, such as the idea that the metal had somehow "crystallized". The notion was based on the crystalline appearance of the fast fracture region of the crack surface, but ignored the fact that the metal was already highly crystalline.

### de Havilland CometEdit

The recovered (shaded) parts of the wreckage of G-ALYP and the site (arrowed) of the failure

Two de Havilland Comet passenger jets broke up in mid-air and crashed within a few months of each other in 1954. As a result, systematic tests were conducted on a fuselage immersed and pressurised in a water tank. After the equivalent of 3,000 flights, investigators at the Royal Aircraft Establishment (RAE) were able to conclude that the crash had been due to failure of the pressure cabin at the forward Automatic Direction Finder window in the roof. This 'window' was in fact one of two apertures for the aerials of an electronic navigation system in which opaque fibreglass panels took the place of the window 'glass'. The failure was a result of metal fatigue caused by the repeated pressurisation and de-pressurisation of the aircraft cabin. Also, the supports around the windows were riveted, not bonded, as the original specifications for the aircraft had called for. The problem was exacerbated by the punch rivet construction technique employed. Unlike drill riveting, the imperfect nature of the hole created by punch riveting caused manufacturing defect cracks which may have caused the start of fatigue cracks around the rivet.

The fuselage roof fragment of G-ALYP on display in the Science Museum in London, showing the two ADF windows at-which the initial failure occurred.[37]

The Comet's pressure cabin had been designed to a safety factor comfortably in excess of that required by British Civil Airworthiness Requirements (2.5 times the cabin proof test pressure as opposed to the requirement of 1.33 times and an ultimate load of 2.0 times the cabin pressure) and the accident caused a revision in the estimates of the safe loading strength requirements of airliner pressure cabins.

In addition, it was discovered that the stresses around pressure cabin apertures were considerably higher than had been anticipated, especially around sharp-cornered cut-outs, such as windows. As a result, all future jet airliners would feature windows with rounded corners, greatly reducing the stress concentration. This was a noticeable distinguishing feature of all later models of the Comet. Investigators from the RAE told a public inquiry that the sharp corners near the Comets' window openings acted as initiation sites for cracks. The skin of the aircraft was also too thin, and cracks from manufacturing stresses were present at the corners.

### Alexander L. Kielland oil platform capsizingEdit

Fractures on the right side of the Alexander L. Kielland rig

The Alexander L. Kielland was a Norwegian semi-submersible drilling rig that capsized whilst working in the Ekofisk oil field in March 1980 killing 123 people. The capsizing was the worst disaster in Norwegian waters since World War II. The rig, located approximately 320 km east of Dundee, Scotland, was owned by the Stavanger Drilling Company of Norway and was on hire to the United States company Phillips Petroleum at the time of the disaster. In driving rain and mist, early in the evening of 27 March 1980 more than 200 men were off duty in the accommodation on the Alexander L. Kielland. The wind was gusting to 40 knots with waves up to 12 m high. The rig had just been winched away from the Edda production platform. Minutes before 18:30 those on board felt a 'sharp crack' followed by 'some kind of trembling'. Suddenly the rig heeled over 30° and then stabilised. Five of the six anchor cables had broken, with one remaining cable preventing the rig from capsizing. The list continued to increase and at 18.53 the remaining anchor cable snapped and the rig turned upside down.

A year later in March 1981, the investigative report[38] concluded that the rig collapsed owing to a fatigue crack in one of its six bracings (bracing D-6), which connected the collapsed D-leg to the rest of the rig. This was traced to a small 6 mm fillet weld which joined a non-load-bearing flange plate to this D-6 bracing. This flange plate held a sonar device used during drilling operations. The poor profile of the fillet weld contributed to a reduction in its fatigue strength. Further, the investigation found considerable amounts of lamellar tearing in the flange plate and cold cracks in the butt weld. Cold cracks in the welds, increased stress concentrations due to the weakened flange plate, the poor weld profile, and cyclical stresses (which would be common in the North Sea), seemed to collectively play a role in the rig's collapse.

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