Циклдық жүктеме кесірінен материалдағы жарықтардың пайда болуы мен таралуы

Initiation and propagation of cracks in a material due to cyclic loading

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. The stress values that cause fatigue damage are typically much less than the yield strength of the material.

Historically, fatigue has been separated into regions of high cycle fatigue that require more than 104 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.

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.

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.

1837: Wilhelm Albert publishes the first article on fatigue. He devised a test machine for conveyor chains used in the Clausthal mines. 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. 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: F. Braithwaite reports on common service fatigue failures and coins the term fatigue. 1860: Systematic fatigue testing undertaken by Sir William Fairbairn and August Wöhler. 1870: A. 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. 1940: Sidney M. Cadwell 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: W. Weibull An S N curve model. 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. 1970: Smith, Watson, and Topper developed a mean stress correction model, where the fatigue damage in a cycle is determined by the product of the maximum stress and strain amplitude. 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.

In 1945, Milton A. Miner popularised a rule that had first been proposed by Arvid Palmgren in 1924. 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 (or localized plastic damages around crack tip).

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.

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. 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.

Stop drill Fatigue cracks that have begun to propagate can sometimes be stopped by drilling holes, called drill stops, at the tip of the crack. The possibility remains of a new crack starting in the side of the hole. Blend. Small cracks can be blended away and the surface cold worked or shot peened. Oversize holes. Holes with cracks growing from them can be drilled out to a larger hole to remove cracking and bushed to restore the original hole. Bushes can be cold shrink Interference fit bushes to induce beneficial compressive residual stresses. The oversized hole can also be cold worked by drawing an oversized mandrel through the hole. 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.

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 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. Induce residual stresses Peening a surface can reduce such tensile stresses and create compressive residual stress, which prevents crack initiation. Forms of peening include: shot peening, using high speed projectiles, high frequency impact treatment (also called high frequency mechanical impact) using a mechanical hammer, and laser peening which uses high energy laser pulses. Low plasticity burnishing can also be used to induce compresses stress in fillets and cold work mandrels can be used for holes. 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. 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. 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. Shape optimisation using numerical optimisation algorithms have been used to lower the stress concentration in wings and increase their life.

Composite materials can offer excellent resistance to fatigue loading. In general, composites exhibit good fracture toughness and, unlike metals, increase fracture toughness with increasing strength. The critical damage size in composites is also greater than that for metals. The primary mode of damage in a metal structure is cracking. For metal, cracks propagate in a relatively well defined manner with respect to the applied stress, and the critical crack size and rate of crack propagation can be related to specimen data through analytical fracture mechanics. However, with composite structures, there is no single damage mode which dominates. Matrix cracking, delamination, debonding, voids, fiber fracture, and composite cracking can all occur separately and in combination, and the predominance of one or more is highly dependent on the laminate orientations and loading conditions. In addition, the unique joints and attachments used for composite structures often introduce modes of failure different from those typified by the laminate itself. The composite damage propagates in a less regular manner and damage modes can change. Experience with composites indicates that the rate of damage propagation in does not exhibit the two distinct regions of initiation and propagation like metals. The crack initiation range in metals is propagation, and there is a significant quantitative difference in rate while the difference appears to be less apparent with composites.

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.

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 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 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.