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					Fatigue of Metals
• mechanical components can fail at stresses well below the tensile strength of the material if subjected to alternating loads • failure of ductile materials unter alternating loads occurs in a quasi brittle manner, i.e. by crack propagation • failure is preceded by characteristic changes in the material microstucture • this phenomenon is called ‘metals fatigue’

I. Case Studies (the Disasters Catalogue) A. The Comet Disasters

A famous series of fatigue failures led to the de Havilland Comet crashes of the early 50s The Comet was designed and built in the UK. It was the world’s first commercial jet airliner.

• May 2, 1953: G-ALYV disintegrated in a thunderstorm at 10000 ft during its initial climb on a flight from Calcutta to Delhi • January 10, 1954: G-ALYP crashed from at 27000 ft in good weather on a Rome to London flight

• April 8, 1954: G-ALYY disappeared on a flight from Rome to Cairo. All comet aircraft were grounded.

By investigation of recovered wreckage from G-ALYP and pressure cycle testing of the fuselage of G-ALYU in a water tank, fatigue failure of the fuselage was identified as the cause of the Comet accidents

Cause of the Comet disasters 1) The economic backdrop: In order to provide an economically satisfactory payload and range at the high cruising speed which the turbo-jet engines offered, it was essential that the cruising height should be upwards of 35,000 ft. double that of the then current airliners and that the weight of the structure and equipment should be as low as possible (Official accident report) This forced engineers to go at (and beyond) the limits of then current engineering practice in designing an aircraft operating under (for the times) extreme conditions.

2) This led to engineering design flaws: Engineers were aware of potential problems due to the novel and (for the time) extreme service conditions and tried to ensure stability of the pressure cabin, also against fatigue, but: • • • They failed to properly evaluate stress concentrations near the corners of windows Instead of doing calculations, design engineers relied on practical tests (very British...) ... which, though in line with ‘good engineering practice’ of the time, were substantially flawed (use of static tests to evaluate fatigue resilience)

( But – it can’t happen these days?!

B The B737-200 disasters • On August 22, 1981 the fuselage of Far Eastern Airlines Flight 103 disintegrated in mid-flight. • On April 28, 1988, the B737-200 of Aloha Airlines Flight 243 mutated into a convertible at an altitude of 27000ft. By some kind of miracle, only one person was killed.

These accidents are interesting in view of the safety philosophy employed (‘fly-till-it-breaks’) which puts emphasis on ability to contain failures once occurred (‘safe decompression’) scenario rather than on detection and replacement of fatigued parts

But: detection programs may not help either... (Enschede 1998)

II. Fatigue Lifetime Evaluation A. Characterization of alternating loads A1. Reversed/repeated stress cycles: Periodic stress vs time signals
σmax : maximal stress σmin : minimal stress σm = (σmax+σmin)/2: mean stress σr = σmax-σmin: stress range σa = σr/2: stress amplitude

σ σmax σr



Maximum and minimum stress equal in magnitude: mean stress = 0, reversed stress signal Otherwise: repeated stress signal But: Real stress vs time signals are hardly periodic. What to if the stress vs time signal is irregular?

A2: Characterization of irregular load vs time curves 1) Characterize load curve in terms of maxima and minima only 2) Introduce ‘classes’ (stress intervals) such that maxima/minima belonging to the same class are counted similarly (effectively: round the values of σmax / σmin values
800 600 400
1 2 3 4 5 6 7

Stress [MPa]

200 0 -200 -400 -600 -800 0.0 0.2 0.4 0.6 0.8


Time [s]

Signal may now be characterized in terms of • • • • peak counting: Histogram of maxima/minima range counting: Histogram of ranges (differences between adjacent maxima/minima) level crossing counting: Histogram of level crossings rainflow counting: Conversion of irregular time series into a sequence of ‘cycles’

The rainflow algorithm
• • • • Reduce the time history to a sequence of maxima and minima Turn the graph by 90° (earliest time to the top) and imagine it depicts a pagoda roof. Each minimum is imagined as a source of water that "drips" down the pagoda. Count the number of half-cycles by looking for terminations in the flow occurring when either: a) It reaches the end of the time history; b) It merges with a flow that started at an earlier minimum c) It flows above a minimum of greater depth than its origin. Repeat this for the maxima Assign a magnitude to each half-cycle equal to the stress difference between its start and termination. Pair up half-cycles of identical magnitude (but opposite sense) to count the number of complete cycles. Typically, there are some residual half-cycles. Sometimes, they can be paired up to close the loop.

• • •

After counting: Characterize each cycle / residual half cycle in terms of its mean stress and amplitude Rainflow matrix: Characterize each cycle in terms of its start, maximum and minimum stresses (start stress => first index) Rainflow matrix for load signal above:
1 1 2 3 4 5 I 6 7 2 I I 3 4 5 6 I 7 1 2 1 2 I I
1 2



1 2

6 I



3 4 5 6 7

1 2

B. Fatigue lifetime evaluation B1. Characterization of fatigue properties Fatigue testing is commonly done assuming reversed / repeated loads • Determine number N of cycles to failure as a function of stress amplitude σa (‘S’) • Lifetime depends on mode of testing (eg. bending vs. torsion) and possibly on specimen geometry (notch effects) • Representation: S-N curve (Woehler curve)


Tensile stress amplitude [MPa]


σm=-400MPa σm=400 MPa




0 1 10 100 1000 10000 100000 1000000 1E7

Number of cycles, N

Typical S-N curve for a high strength steel • Lifetime decreases with increasing stress amplitude • Existence of an endurance limit (‘infinite’ lifetime below EL)

• Little or no influence of cycling rate • Influence of mean stress: > tensile mean stress reduces lifetime (crack opening) > compressive stress increases lifetime (crack closure)

B2: Lifetime evaluation for loads with nonzero mean stress: 1) If lifetime data with mean stress are available: Read lifetime off S-N curve (interpolate if necessary) 2) If only lifetime data without mean stress are available: Use some empirical estimate to convert true amplitude and mean stress into ‘effective’ stress amplitude and use the corresponding lifetime from S-N curve

B3: Lifetime evaluation for irregular load vs time signals

• Perform rainflow classification of load vs time signal • Determine fatigue lifetimes Ni for all occurring types of cycles (for all elements of the rainflow matrix) • Let ni be the number of cycles of type i Then failure occurs if




(Palmgren-Miner rule)

Example: For previously discussed stress vs time signal: When will the material of the example S-=N curve fail if subjected to this periodically repeated signal? 4 classes of cycles. Two (1-2-1 and 2-3-2) below fatigue limit: N=∞ 1-6-1: N ~ 800 1-5-1: N~3000
1   1 T [s] ×  +  =1  3000 800  T = 631s

C. Statistics of fatigue Measured lifetimes exhibit huge statistical scatter. Consequence: need for statistical description to predict failure safety margins Some probabilistic notations: • probability density function p (t )dt probability that a part fails (under given conditions) between times t, t+dt • cumulative probability (distribution function)
F (t ) = ∫ p(t ')dt ' probability of failure before time t
0 t

Mean lifetime:
t = ∫ t p(t )dt

Variance of lifetimes:
st 2 = t 2 − t

= ∫ t p(t )dt −  ∫ t p (t )dt   



Fatigue lifetimes cannot be described by a Gaussian distribution (Why?) Instead one usually uses a log-normal distribution, ie., the logarithm of the lifetime (in cycles) is assumed to be Gaussian distributed

Log-normal distribution: p(log N ) = 1
2 2π slog N

 ( log N − log N exp  − 2 2 slog N  



   

This can be used to evaluate time-dependent failure/survival probabilities (see tutorials)

If fatigue lifetime is governed by ‘weakest link’ in a chain of identical elements (eg. in cables, Forth Road Bridge): Lifetimes obey Weibull distribution Probability for failure before N cycles:
  N β  F ( N ) = 1 − exp  −      N0    

(N0, β) parameters of the Weibull distribution NB: What is the variance of this distribution?

II. Microstructural Aspects of Fatigue

Fatigue leads to characteristic microstructure changes on the level of the dislocation arrangement.

Dislocation microstructure in a fatigued Ni polycrystal (SEM, channeling contrast)

Microstructure depends on stress amplitude: • At stress amplitudes below endurance limit: Patchy ‘matrix’ pattern with very little plastic activity (plastic strain amplitude ∆γ<10-4 • At the endurance limit: Formation of lamellar ‘persistent slip bands’ (PSB) parallel to slip plane with highest resolved shear stress. Local plastic activity in PSBs is about 100 times higher. • Fatigue cracks develop where the PSB hits the surface and grow along the PSB’s

There seems to be a one-to-one correspondence between PSB occurrence and fatigue failure (but we don’t know why PSB occur...) Important for practice: Fatigue cracks nucleate at surfaces
⇒Possibility to deal with fatigue problems by surface treatment (Grain refinement, large plastic deformation, hard coatings)

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