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.^[1] Most materials, such as composites, plastics and ceramics, seem to experience some sort of fatigue-related failure.^[2]

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.^[2]

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.^[3] The stress values that cause fatigue damage are typically much less than the yield strength of the material.

Stages of fatigue

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.^[4]

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.^[5] 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:^[2]

• Mean stress effect: Higher mean stress increases the rate of crack growth.
• Environment: Increased moisture increases the rate of crack growth. 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 hydrogen embrittlement. 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.^[6]
• Short crack effect: In 1975, Pearson observed that short cracks grow faster than expected.^[7] Possible reasons for the short crack effect include the presence of the T-stress, the tri-axial stress state at the crack tip, the lack of crack closure associated with short cracks and the large plastic zone in comparison to the crack length. In addition, long cracks typically experience a threshold which short cracks do not have.^[8] There are a number of criteria for short cracks:^[9]
  • cracks are typically smaller than 1 mm,
  • cracks are smaller than the material microstructure size such as the grain size, or
  • crack length is small compared to the plastic zone.
• Underloads: Small numbers of underloads increase the rate of growth and may counteract the effect of overloads.
• Overloads: Initially overloads (> 1.5 the maximum load in a sequence) lead to a small increase in the rate of growth followed by a long reduction in the rate of growth.

Characteristics of fatigue

• 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 samples in well controlled environments.
• Fatigue is usually associated with tensile stresses but fatigue cracks have been reported due to compressive loads.^[10]
• 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 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 research history

• 1837: Wilhelm Albert publishes the first article on fatigue. He devised a test machine for conveyor chains used in the Clausthal mines.^[11]
• 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.^[12]
• 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.^[13]
• 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.^[11]
• 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.^[14]
• 1940: Sidney M. Cadwell publishes first rigorous study of fatigue in rubber.^[15]
• 1945: A. M. Miner popularises Palmgren's (1924) linear damage hypothesis as a practical design tool.^[16][17]
• 1952: W. Weibull An S-N curve model.^[18]
• 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.^[19]
• 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.^[20]
• 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.^[21][22]
• 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.^[23]

Predicting fatigue life

The American Society for Testing and Materials defines fatigue life, N_f, as the number of stress cycles of a specified character that a specimen sustains before failure of a specified nature occurs.^[24] 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.^[25] However, in practice, several bodies of work done at greater numbers of cycles suggest that fatigue limits do not exist for any metals.^[26][27][28]

Engineers have used a number of methods to determine the fatigue life of a material:^[29]

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.

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:

1. Complex loading is reduced to a series of simple cyclic loadings using a technique such as rainflow analysis;
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 non-proportional loading, critical plane analysis must be applied.

Miner's rule

In 1945, Milton A. Miner popularised a rule that had first been proposed by Arvid Palmgren in 1924.^[16] 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_i (1 ≤ i ≤ k), each contributing n_i(S_i) cycles, then if N_i(S_i) is the number of cycles to failure of a constant stress reversal S_i (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 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 N_i(S_i).
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.^[30] 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).

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.^[31] 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.^[32] 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),^[33] loading frequency, temperature, corrosion, residual stresses, and the presence of notches. A constant fatigue life (CFL) diagram^[34] 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.^[35] 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.^[36]

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.^[37]

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 ${\displaystyle \Delta \varepsilon /2}$ is the sum of the elastic strain amplitude ${\displaystyle \Delta \varepsilon _{\text{e}}/2}$ and the plastic strain amplitude ${\displaystyle \Delta \varepsilon _{\text{p}}/2}$ and is given by^[2][38]

${\displaystyle {\Delta \varepsilon \over 2}={\Delta \varepsilon _{\text{e}} \over 2}+{\Delta \varepsilon _{\text{p}} \over 2}}$.

Basquin's equation for the elastic strain amplitude is

${\displaystyle {\Delta \varepsilon _{\text{e}} \over 2}={\frac {\Delta \sigma }{2E}}={\frac {\sigma _{\text{a}}}{E}}}$

where ${\displaystyle E}$ is Young's modulus.

The relation for high cycle fatigue can be expressed using the elastic strain amplitude

${\displaystyle {\Delta \varepsilon _{\text{e}} \over 2}={\frac {\sigma _{\text{f}}^{\prime }}{E}}(2N_{\text{f}})^{b}}$

where ${\displaystyle \sigma _{\text{f}}^{\prime }}$ is a parameter that scales with tensile strength obtained by fitting experimental data, ${\displaystyle N_{\text{f}}}$ is the number of cycles to failure and ${\displaystyle b}$ 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

${\displaystyle {\Delta \varepsilon _{\text{p}} \over 2}=\varepsilon _{\text{f}}^{\prime }(2N_{\text{f}})^{c}}$.

Combining the elastic and plastic portions gives the total strain amplitude accounting for both low and high cycle fatigue

${\displaystyle \frac{\mathrm{\Delta <!--\; \Delta \; -->}\mathrm{\epsilon <!--\; \epsilon \; -->}}{2}=\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_{\text{f}}^{\mathrm{\prime <!--\; \prime \; -->}}}{E}(2N_{}}$Zdroj:https://en.wikipedia.org?pojem=Material_fatigue
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