Fatigue

In, fatigue is the weakening of a material caused by cyclic loading that results in progressive and localized structural damage and the growth of cracks. The nominal maximum values that cause such damage may be much less than the strength of the material, typically quoted as the, or the.

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 initiate at such as holes, persistent slip bands (PSBs),  interfaces or  in metals.

Once a crack has initiated, each loading cycle will grow the crack a small amount, typically producing striations on some parts of the fracture surface. The crack will continue to grow until it reaches a critical size which occurs when the of the crack exceeds the  of the material producing rapid propagation and typically complete fracture of the structure.

Stages of fatigue
Fatigue failures, both for high and low cycle, all follow the same basic steps 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 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 initiation
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 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.

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.

Crack growth
Fatigue cracks grow from material or manufacturing defects from as small as 10 µm. In the case of aluminium, cracks generally grow from the surface, where water vapour from the atmosphere is able to reach the tip of the crack and dissociate into atomic hydrogen which causes. Cracks growing internally are isolated from the atmosphere and grow in a vacuum where the rate of growth is typically an order of magnitude slower than a surface crack.

When the rate of growth becomes large enough, can be seen on the fracture surface. Striations mark the position of the crack tip and the width of each striation represents the growth from one loading cycle. Striations are a result of plasticity at the crack tip.

An estimate of the fatigue life of a component can be made using a by summing up the width of each increment of crack growth for each loading cycle. Most of the fatigue life is generally consumed in the crack growth phase. Safety or scatter factors are applied to the calculated life to account for any uncertainty and variability associated with fatigue. When the stress intensity exceeds a critical value known as the fracture toughness, unsustainable fracture will occur, usually by a process of micro-void coalescence. Prior to final fracture, the fracture surface may contain a mixture of fatigue and fast fracture.

The rate of growth used in crack growth predictions is typically measured by applying thousands of constant amplitude cycles to a coupon and measuring the rate of growth from the change in compliance of the coupon or by measuring the growth of the crack on the surface of the coupon. Standard methods for measuring the rate of growth have been developed by ASTM International.

The growth rate $$da/dN $$, is measured over a range of $$\Delta K$$, although additional variables such as mean stress, environment, overloads and underloads can also affect the rate of growth. Crack growth may stop below a certain threshold.

Characteristics of fatigue

 * In metal alloys, and for the simplifying case when there are no macroscopic or microscopic discontinuities, the process starts with 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, often showing considerable scatter even in seemingly identical samples in well controlled environments.
 * Fatigue is usually associated with tensile stresses but fatigue cracks have been reported due to compressive loads.
 * The greater the applied stress range, the shorter the life.
 * Fatigue life scatter tends to increase for longer fatigue lives.
 * Damage is irreversible. Materials do not recover when rested.
 * Fatigue life is influenced by a variety of factors, such as, , metallurgical microstructure, presence of or  chemicals, es, scuffing contact , etc.
 * Some materials (e.g., some and  alloys) exhibit a theoretical  below which continued loading does not lead to fatigue failure.
 * High cycle (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&mdash;like resonant magnetic machines&mdash;can also be used, to achieve frequencies up to 250 Hz.
 * (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 history

 * 1837: publishes the first article on fatigue. He devised a test machine for  chains used in the.
 * 1839: describes metals as being 'tired' in his lectures at the military school at.
 * 1842: recognises the importance of s in his investigation of   failures. The  was caused by fatigue failure of a locomotive axle.
 * 1843: reports on the fatigue of an axle on a locomotive tender. He identifies the  as the crack origin.
 * 1848: The 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: is granted a "small sum of money" to report to the  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: F. Braithwaite reports on common service fatigue failures and coins the term fatigue.
 * 1860: Systematic fatigue testing undertaken by Sir and.
 * 1870: 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.
 * 1903: Sir 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.
 * 1940: publishes first rigorous study of fatigue in rubber.
 * 1945: A. M. Miner popularises Palmgren's (1924) linear damage hypothesis as a practical design tool.
 * 1952: An S-N curve model.
 * 1954: The world's first commercial jetliner, the, 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  in the tip of cracks.
 * 1961: 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: and M. Matsuishi devise the  and enable the reliable application of Miner's rule to  loadings.
 * 1970: W. Elber elucidates the mechanisms and importance of 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 must be considered.

Predicting fatigue life
The defines fatigue life, Nf, as the number of stress cycles of a specified character that a specimen sustains before  of a specified nature occurs. For some materials, notably and, there is a theoretical value for stress amplitude below which the material will not fail for any number of cycles, called a.

Engineers have used a number of methods to determine the fatigue life of a material:
 * 1) the stress-life method,
 * 2) the strain-life method,
 * 3) the crack growth method and
 * 4) probabilistic methods, which can be based on either life or crack growth methods.

Historically, fatigue has been separated in two regions of high cycle fatigue that require more than 104 cycles to failure where stress is low and primarily and  where there is significant plasticity. Experiments have shown that low cycle fatigue is also crack growth.

Whether using stress/strain-life approach or using crack growth approach, complex or variable amplitude loading is reduced to a series of fatigue equivalent simple cyclic loadings using a technique such as.

Stress/Strain-life methods
A mechanical part is often exposed to a complex, often, sequence of loads, large and small. In order to assess the safe life of such a part using the fatigue damage or stress/strain-life methods the following series of steps is usually performed:
 * 1) Complex loading is reduced to a series of simple cyclic loadings using a technique such as ;
 * 2) A  of cyclic stress is created from the rainflow analysis to form a ;
 * 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 and therefore some equivalence rule is needed whenever the loading is multiaxial. For simple, proportional loading histories (lateral load in a constant ratio with the axial), may be applied. For more complex situations, such as non-proportional loading, must be applied.

Miner's rule
In 1945, M. A. Miner popularised a rule that had first been proposed by 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 = i = k), 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:
 * $$\sum_{i=1}^k \frac {n_i} {N_i}=C $$

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 recognize 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. 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.

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

S-N curves are derived from tests on samples of the material to be characterized (often called ) where a regular al stress is applied by a testing machine which also counts the number of cycles to failure. This process is sometimes known as coupon testing. For greater accuracy but lower generality component testing is used. Each coupon or component 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 ). Analysis of fatigue data requires techniques from, especially survival analysis and.

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

A Constant Fatigue Life (CFL) diagram is useful for stress ratio effect on S-N curve. Also, in the presence of a steady stress superimposed on the cyclic loading, the can be used to estimate a failure condition. It plots stress amplitude against mean stress with the fatigue limit and the of the material as the two extremes. Alternative failure criteria include Soderberg and Gerber.

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.

Strain-life (e-N) method
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,
 * $1/2$?eelastic = $s_{f}^{'}/E$(2Nf)-b

where
 * ?eelastic is the change in elastic strain per cycle
 * sf' 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.

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.

Crack growth methods
such as the are used to predict the life of a component. They can be used to predict the growth of a crack from 10 um to failure. For normal manufacturing finishes this may cover the most of the fatigue life of a component where growth can start from the first cycle.. The conditions at the crack tip of a component are usually related to the conditions of test coupon using a characterising parameter such as the stress intensity, or. All these techniques aim to match the crack tip conditions on the component to that of test coupons which give the rate of crack growth.

Additional models may be necessary to include retardation and acceleration effects associated with overloads or underloads in the loading sequence. In addition, small crack growth data may be needed to match the increased rate of growth seen with small cracks.

Typically, a cycle counting technique such as is used to extract the cycles from a complex sequence. This technique, along with others, has been shown to work with crack growth methods.

Crack growth methods have the advantage that they can predict the intermediate size of cracks. This information can be used to schedule inspections on a structure to ensure safety whereas strain/life methods only give a life until failure.

Sources:

Design
Dependable design against fatigue-failure requires thorough education and supervised experience in, , or. There are at least five principal approaches to life assurance for mechanical parts that display increasing degrees of sophistication:
 * 1) Design to keep stress below threshold of  (infinite lifetime concept);
 * ,, and : Instruct the user to replace parts when they fail. Design in such a way that there is no , and so that when any one part completely fails, it does not lead to catastrophic failure of the entire system.
 * 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); and  are variants that design for a fixed life after which the user is instructed to replace the entire device;
 * : Is an approach that ensures aircraft safety by assuming the presence of cracks or defects even in new aircraft. Crack growth calculations, periodic inspections and component repair or replacement can be used to ensure critical components that may contain cracks, remain safe. Inspections usually use to limit or monitor the size of possible cracks and require an  prediction of the rate of crack-growth between inspections. The designer sets some  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".
 * : Ensures the probability of failure remains below an acceptable level. This approach is typically used for aircraft where acceptable levels may be based of probability of failure during a single flight or taken over the lifetime of an aircraft. A component is assumed to have a crack with a probability distribution of crack sizes. This approach can consider variability in values such as crack growth rates, usage and critical crack size. It is also useful for considering damage at multiple locations that may interact to produce multi-site or . Probability distributions that are common in data analysis and in design against fatigue include the, , , and.

Testing
can be used for components such as a coupon or a full-scale test article to determine:
 * 1) the rate of crack growth and fatigue life of components such as a coupon or a full-scale test article.
 * 2) location of critical regions
 * 3) degree of  when part of the structure fails
 * 4) the origin and cause of the crack initiating defect.

These tests may form part of the certification process such as for.

Repair

 * 1) Stop drill Fatigue cracks that have begun to propagate can sometimes be stopped by  holes, called drill stops, in the path of the fatigue crack. This is not recommended as a general practice because the hole represents a  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.
 * 2) Blend. Small cracks can be blended away and the surface cold worked or shot peened.
 * 3) Oversize holes. Holes with cracks growing from can be drilled out to a larger hole to remove cracking and bushed to restore the original hole. Bushes can be cold shrink interference bushes to induce compressive residual stress. The oversized hole can also be cold worked by drawing an oversized mandrel through the hole.
 * 4) Patch. Cracks may be repaired by installing a patch or repair fitting. Composite patches have been used to restore the strength of aircraft wings after cracks have been detected or to lower the stress prior to cracking in order to improve the fatigue life. Patches may restrict the ability to monitor fatigue cracks and may need to be removed and replaced for inspections.

Life improvement

 * 1) Change material. 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  blades and s in metal are being replaced by  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.
 * 2) Peening  a surface can reduce such tensile stresses and create compressive, which prevents crack initiation.  Forms of peening include: , using high-speed projectiles,  (also called high-frequency mechanical impact) using a mechanical hammer, and  which uses high-energy laser pulses. Increases in fatigue life and strength are proportionally related to the depth of the compressive residual stresses imparted.  Shot peening imparts compressive residual stresses approximately 0.005 inches (0.1 mm) deep, while laser peening can go 0.040 to 0.100 inches (1 to 2.5 mm) deep, or deeper.
 * 3) Deep Cryogenic treatment. 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.
 * 4) Re-profiling. Changing the shape of a stress concentration such as a hole or cutout may be used to extend the life of a component.  using numerical optimisation algorithms have been used to lower the stress concentration in wings and increase their life.

Versailles train crash
Following the 's celebrations at the, a train returning to Paris crashed in May 1842 at 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. This accident is known in France as the "Catastrophe ferroviaire de Meudon". The accident was witnessed by the British locomotive engineer and widely reported in Britain. It was discussed extensively by engineers, who sought an explanation.

The derailment had been the result of a broken axle. 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 Comet
Two 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 immersed and pressurised in a water tank. After the equivalent of 3,000 flights, investigators at the (RAE) were able to conclude that the crash had been due to failure of the pressure cabin at the forward  window in the roof. This 'window' was in fact one of two apertures for the of an electronic navigation system in which opaque  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 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 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 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 s 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 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 capsizing
The was a Norwegian   that capsized whilst working in the  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, Scotland, was owned by the Stavanger Drilling Company of Norway and was on hire to the United States company 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 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 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 ), seemed to collectively play a role in the rig's collapse.

Others

 * The 1862 was caused by the fracture of a steam engine beam and killed 220 people.
 * The 1919 Boston has been attributed to a fatigue failure.
 * The 1948 crash due to fatigue failure in a wing spar root
 * The, presidential plane of , crashed due to engine failure caused by metal fatigue.
 * The 1965 capsize of the UK's first offshore oil platform, the, was due to fatigue in part of the suspension system linking the hull to the legs.
 * The 1968 lost one of its main rotor blades due to fatigue failure.
 * The 1968 lost a wing due to improper maintenance leading to fatigue failure.
 * The 1969 crash due to a fatigue failure of the wing pivot fitting from a material defect resulted in the development of the  approach for fatigue design.
 * The caused by fatigue failure resulting in the loss of the right horizontal stabilizer.
 * The 1979 crashed after engine separation attributed to fatigue damage in the pylon structure holding the engine to the wing, caused by improper maintenance procedures.
 * The 1980 crashed due to fatigue in an engine turbine shaft resulting in engine disintegration leading to loss of control.
 * The 1985 crashed after the aircraft lost its vertical stabilizer due to faulty repairs on the rear bulkhead.
 * The 1988 suffered an explosive decompression at 24000 ft after a fatigue failure.
 * The 1989 lost its tail engine due to fatigue failure in a fan disk hub.
 * The 1992 lost both engines on its right-wing due to fatigue failure in the pylon mounting of the #3 Engine.
 * The 1998 was caused by fatigue failure of a single composite wheel.
 * The 2000 was likely caused by rolling contact fatigue.
 * The 2000 on Ford Explorers originated from fatigue crack growth leading to separation of the tread from the tire.
 * The 2002 disintegrated in-flight due to fatigue failure.
 * The 2005 lost its right wing due to fatigue failure brought about by inadequate maintenance practices.
 * The 2009 due to fatigue failure.
 * The due to metal fatigue of turbine mountings.