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Metal fatigue

This article is part of the Mechanical failure modes series.

Buckling

Corrosion

Creep

Fatigue

Fracture

Melting

Wear

In materials science, fatigue is a process by which a component is weakened by repeated bending or other cyclic stress, often to the point of fracture. The stress can be small enough to be below the ultimate tensile stress, or even the yield stress of the material. However, over a large number of cycles the effect can be catastrophic.

Characteristics of fatigue failures

The following characteristics are common to fatigue in all materials:

• The process starts with a microscopic crack, called the initiation site, which then widens with each subsequent movement, a phenomenon analysed in the topic of fracture mechanics.
• Failure is essentially probabilistic. The number of cycles required for failure varies between homogeneous material samples. Analysis demands the techniques of survival analysis.
• The greater the applied stress, the shorter the life.
• Damage is cumulative. Materials do not recover when rested.
• Fatigue life is influenced by a variety of factors, such as temperature and surface finish, in complicated ways.
• Some materials, for example steel and titanium, exhibit a fatigue limit, a limit below which repeated stress has no effect. Most others, for example aluminium, exhibit no such limit and even infinitesimally small stresses will eventually cause failure.

Timeline of fatigue history

• 1829: Wilhelm Albert first discusses the phenomenon on observing the failure of iron mine-hoist chains in Clausthal mines.
• 1839: The term fatigue becomes current when Jean-Victor Poncelet describes metals as being tired in his lectures at the military school at Metz.

• 1843: William John Macquorn Rankine recognises the importance of stress concentration in his investigation of railroad axle failures following the Versailles accident.
• 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.
• 1860: The first systematic investigations of fatigue life by Sir William Fairbairn and August Wöhler. Wöhler's study of railroad axles leads him to the idea of a fatigue limit and to propose the use of S-N curves in mechanical design.

• 1903: Sir James Alfred Ewing demonstrates the origin of fatigue failure in microscopic cracks.
• 1910: O. H. Basquin clarifies the shape of a typical S-N curve.
• 1939: Invention of the strain gauge at Baldwin-Lima-Hamilton catalyses fatigue research.
• 1945: A. M. Miner popularises A. Palmgren's (1924) linear damage hypothesis as a practical design tool.
• 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. Matsuiski devise the rainflow-counting algorithm and enable the reliable application of Miner's rule to random loadings.
• 1970: W. Elber elucidates the mechanisms and importance of crack closure.
• 1975: S. Pearson observes that propagation of small cracks is sometimes surprisingly arrested in the early stages of growth.

High-cycle fatigue

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

The S-N curve

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

figure to be done

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

Probabilistic nature of fatigue

As coupons sampled from a homogeneous frame will manifest variation in their number of cycles to failure, the S-N curve should more properly be an S-N-P curve capturing the probability of failure after a given number of cycles of a certain stress. Probability distributions that are common in data analysis and in design against fatigue include the lognormal distribution, extreme value distribution and Weibull distribution.

Complex loadings

In practice, a mechanical part is exposed to a complex, often random, sequence of loads, large and small. In order to assess the safe life of such a part:

1. Reduce the complex loading to a series of simple cyclic loadings using a technique such as rainflow analysis;
2. Create an histogram of cyclic stress from the rainflow analysis;
3. For each stress level, calculate the degree of cummulative damage incurred from the S-N curve; and
4. Combine the individual contributions using an algorithm such as Miner's rule.

Miner's rule

In 1945, M. A. Miner popularised a rule that had first been proposed by A. Palmgren in 1924. The rule, variously called Miner's rule or the Palmgren-Miner linear damage hypothesis, states that where there are k different stress magnitudes in a spectrum, 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, failure occurs when:

<math>\sum_{i=1}^k \frac {n_i} {N_i} \ge 1 <math>

This can be thought of as assessing what percentage of life is consumed by stress reversal at each magnitude then forming a linear combination of their aggregate.

Though Miner's rule is a useful approximation in many circumstances, it has two major limitations:

1. It fails to recognise the probabilistic nature of fatigue and there is no simple way to relate life predicted by the rule with the charateristics of a probability distribution.
2. There is sometimes an effect in the order in which the reversals occur. In some circumstances, cycles of high stress followed by low stress cause more damage than would be predicted by the rule.

Low-cycle fatigue

Where the stress is high enough for plastic deformation to occur, the account in terms of stress is less useful and the strain in the material offers a simpler description. Low-cycle fatigue is usually characterised by the Coffin-Manson relation (popularised by L. F. Coffin in 1979 based on S. S. Manson's 1960 work):

<math>\frac {\Delta \epsilon_p} {2} = \epsilon_f '(2N)^c<math>

-where:

• Δε_p /2 is the plastic strain amplitude;
• ε_f' is an empirical constant known as the fatigue ductility coefficient, the failure strain for a single reversal;
• 2N is the number of reversals to failure (N cycles);
• c is an empirical constant known as the fatigue ductility exponent, commonly ranging from -0.5 to -0.7 for metals.

Fatigue and fracture mechanics

The account above is purely phenomenological and, though it allows life prediction and design assurance, it does not enable life improvement or design optimisation. For the latter purposes, an exposition of the causes and processes of fatigue is necessary. Such an explanation is given by fracture mechanics in four stages.

1. Crack initiation;
2. Stage I crack-growth;
3. Stage II crack-growth; and
4. Ultimate ductile failure.

Factors that affect fatigue-life

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Design against fatigue

Dependable design against fatigue-failure requires thorough education and supervised experience in mechanical engineering or materials science. There are three principle approaches to life assurance for mechanical parts that display increasing degrees of sophistication:

1. Design to keep stress below threshold of fatigue limit (infinite lifetime concept);
2. 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);
3. Instruct the user to inspect the part periodically for cracks and to replace the part once a crack exceeds a critical length. This approach usually uses the technologies of nondestructive testing and requires an accurate prediction of the rate of crack-growth between inspections.

Famous fatigue failures

Versailles accident

On May 8, 1842 one of the trains carrying revellers on their return from Versailles to Paris, having witnessed the celebrations of the birthday of Louis Philippe, derailed and caught fire. Though the resulting conflagration mutilated the dead beyond recognition or enumeration, it is estimated that 53 perished and around 40 were seriously injured.

The derailment had been the result of a broken locomotive axle and Rankine's investigation highlighted the importance of stress concentration for the first time.

De Havilland Comet

Metal fatigue came strongly to the notice of aircraft engineers in 1954 after three de Havilland Comet passenger jets had broken up in mid-air and crashed within a single year. Investigators from the Royal Aircraft Establishment at Farnborough in England told a public enquiry that the sharp corners around the plane's window openings acted as initiation sites for cracks. All aircraft windows were immediately redesigned with rounded corners.

Others

• Capsize of the oil platform Alexander Kielland
• Loss of United Airlines Flight 232

See also

• Air safety

Bibliography

• Andrew, W. (1995) Fatigue and Tribological Properties of Plastics and Elastomers, ISBN 1884207154
• Dieter, G. E. (1988) Mechanical Metallurgy, ISBN 0071004068
• Little, R. E. & Jebe, E. H. (1975) Statistical design of fatigue experiments ISBN 0470541156