Fatigue (material)
In materials science, fatigue is the initiation and propagation of cracks in a material due to cyclic loading. Once a fatigue crack has initiated, it grows a small amount with each loading cycle, 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 stress intensity factor of the crack exceeds the fracture toughness of the material, producing rapid propagation and typically complete fracture of the structure. Fatigue has traditionally been associated with the failure of metal components which led to the term  metal fatigue. In the nineteenth century, the sudden failing of metal railway axles was thought to be caused by the metal "crystallising" because of the brittle appearance of the fracture surface, but this has since been disproved. Most materials, such as composites, plastics and ceramics, seem to experience some sort of fatigue-related failure. To aid in predicting the fatigue life of a component, fatigue tests are carried out using coupons to measure the rate of crack growth by applying constant amplitude cyclic loading and averaging the measured growth of a crack over thousands of cycles. However, there are also  a number of special cases that need to be considered where the rate of crack growth is significantly different compared to that obtained from constant amplitude testing. Such as the reduced rate of growth that occurs for small loads near the "threshold" or after the application of an "overload"; and the increased rate of crack growth associated with "short cracks" or after the application of an "underload". If the loads are above a certain threshold, microscopic cracks will begin to "initiate" at stress concentrations such as holes, "persistent slip bands" (PSBs), composite interfaces or grain boundaries in metals. The stress  values that cause fatigue damage are typically much less than the yield strength strength of the material. Stages of fatigue. Historically, fatigue has been separated into regions of "high cycle fatigue" that require more than 10 cycles to failure where stress is low and primarily elastic and low cycle fatigue where there is significant plasticity. Experiments have shown that low cycle fatigue is also crack growth. Fatigue failures, both for high and low cycles, all follow the same basic steps: crack initiation, crack growth stages I and II, 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 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 and growth 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 where 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 can cause fractures to initiate. These steps can also be bypassed entirely if the cracks form at a pre-existing stress concentrator such as from an inclusion in the material or from a geometric stress concentrator caused by a sharp internal corner or fillet. Crack growth. Most of the fatigue life is generally consumed in the crack growth phase. The rate of growth is primarily driven by  the range of cyclic loading although additional factors such as mean stress, environment, overloads and underloads can also affect the rate of growth. Crack growth may stop if the loads are small enough to fall below a critical threshold. Fatigue cracks can grow from material or manufacturing defects from as small as 10 μm. When the rate of growth becomes large enough, fatigue striations 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. When the stress intensity exceeds a critical value known as the fracture toughness, unsustainable "fast fracture" will occur, usually by a process of microvoid coalescence. Prior to final fracture, the fracture surface may contain a mixture of areas of fatigue and fast fracture. Acceleration and retardation. The following effects change the rate of growth: Predicting fatigue life. The American Society for Testing and Materials defines "fatigue life", "N", as the number of stress cycles of a specified character that a specimen sustains before failure of a specified nature occurs. 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 or endurance limit. However, in practice, several bodies of work done at greater numbers of cycles suggest that fatigue limits do not exist for any metals. Engineers have used a number of methods to determine the fatigue life of a material: 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 the rainflow-counting algorithm. Stress -life and strain-life methods. A mechanical part is often exposed to a complex, often random, 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: 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 non-proportional loading, critical plane  analysis 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, "S" (1 ≤ "i" ≤ "k"), each contributing "n"("S") cycles, then if "N"("S") is the number of cycles to failure of a constant stress reversal "S" (determined by uni-axial fatigue tests), failure occurs when: 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: Stress-life (S-N) method. Materials fatigue performance is commonly characterized by an "S-N curve", also known as a "Wöhler curve". This is often plotted with the cyclic stress ("S") against the cycles to failure ("N") on a logarithmic scale. S-N curves are derived from tests on samples of the material to be characterized (often called coupons or specimens) 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". 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 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 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. 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 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. 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. With body-centered cubic materials (bcc), the Wöhler curve often becomes a horizontal line with decreasing stress amplitude, i.e. there is a "fatigue strength" that can be assigned to these materials. With face-centered cubic metals (fcc), the  Wöhler curve generally drops continuously, so that only a "fatigue limit" can be assigned to these materials. Strain-life (ε-N) method. When strains are no longer elastic, such as in the presence of stress concentrations, the total strain can be used instead of stress as a similitude parameter. This is known as the strain-life method. The total strain amplitude formula_2 is the sum of the elastic strain amplitude formula_3 and the plastic strain amplitude formula_4 and is given by Basquin's equation for the elastic strain amplitude is where formula_7 is Young's modulus. The relation for high  cycle fatigue can be expressed using the elastic strain amplitude where formula_9 is a parameter that scales with tensile strength obtained by fitting experimental data, formula_10 is the number of cycles to failure and formula_11 is the slope of the log-log curve again determined by curve fitting. In 1954, Coffin and Manson proposed that the fatigue life of a component was related to the plastic strain amplitude using Combining the elastic and plastic portions gives the total strain amplitude accounting for both low and high cycle fatigue where formula_14 is the fatigue strength coefficient, formula_11 is the fatigue strength  exponent, formula_16 is the fatigue ductility coefficient, formula_17 is the fatigue ductility exponent, and formula_18 is the number of cycles to failure (formula_19 being the number of reversals to failure). Crack growth methods. An estimate of the fatigue life of a component can be made using a crack growth equation by summing up the width of each increment of crack growth for each loading cycle. Safety or scatter factors are applied to the calculated life to account for any uncertainty and variability associated with fatigue. 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. Crack growth equations such as the Paris–Erdoğan equation 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 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, J-integral or crack tip opening displacement. 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 rainflow-cycle counting 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. Dealing with fatigue. Design. Dependable design against fatigue-failure requires thorough education and supervised experience in structural engineering, mechanical engineering, or materials  science. There are at least five principal approaches to life assurance for mechanical parts that display increasing degrees of sophistication: Testing. Fatigue testing can be used for components such as a coupon or a "full-scale test article" to determine: These tests may form part of the certification process such as for airworthiness certification. Notable fatigue failures. Versailles train crash. Following the King Louis-Philippe I'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 locked 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 Comet. 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 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 capsizing. 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 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.