4281_-17_Fatigue

Aerospace Structural Design

MAE 4281

Fatigue

David FlemingAssociate Professor

Aerospace Engineering

Historical Perspective

de Havilland Comet• First jet passenger transport, first

operational flight 1952

http://www.rafmuseum.org.uk/de-havilland-comet-1a.htm

http://www.bamuseum.com/50-60.html

de Havilland Comet 1 suffered several crashes

• March 1953 crash on take-off• May 1953 broke-up in flight

– initially blamed on severe weather

• 1954: two more losses– “disintegrated in flight” over water– grounded

Major effort to investigate Comet 1 incidents became a landmark in fracture mechanics

• recovery of remains from underseas• accelerated fatigue testing of surviving

airframe

Marriott, Stewart, and Sharpe, Air Disasters, Barnes & Noble Books, 1999.

de Havilland Comet 1 testing: Failure of pressurized cabin after only about 9000 flight hours:

Metal fatigue, initiation from square “window” corner (severe stress concentration)

• Incomplete understanding of fracture mechanics• Material performance not well characterized


Stress concentrations related to accelerated crack initiation and growth

Marriott, Stewart, and Sharpe, Air Disasters, Barnes & Noble Books, 1999.


A Contemporary Look at Fatigue

Number of Accidents Fixed Wing Rotary Wing

Bolt, stud or screw 108 32 Fastener hole or other hole 72 12 Fillet, radius or sharp notch 57 22 Weld 53 3 Corrosion 43 19 Thread (other than bolt or stud) 32 4 Manufacturing defect or tool mark 27 9 Scratch, nick or dent 26 2 Fretting 13 10 Surface of subsurface flaw 6 3 Improper heat treatment 4 2 Maintenance-induced crack 4 -- Work-hardened area 2 -- Wear 2 7

Percentage of Failures Engineering Components Aircraft Components

Corrosion 29 16 Fatigue 25 55 Brittle Fracture 16 -- Overload 11 14 High Temperature Corrosion 7 2 SCC/Corrosion Fatigue/HE 6 7 Creep 3 -- Wear/Abrasion/Erosion 3 6

S. J. Findlay and N. D. Harrison, “Why Aircraft Fail,” Materials Today, Vol. 5 (11), November 2002.

Characterization of constant amplitude cyclic loading based on “mean stress” and “stress amplitude”

Constant amplitude loading

Various equivalent ways of representing the loading:• Smax, Smin Note that fatigue people use the notation “S”• Sm (mean stress), Sa (stress amplitude note, as with wave

this is half of the difference between Smax,and Smin)• “R-ratio” R = Smin/ Smax.

– R = −1: fully reversed– R = 0: “zero-to-max”

Constant amplitude fatigue testing: a common testing method is rotating beam test

Rotating beam fatigue (R= −1) is most common.

May use hydraulic testing apparatus with appropriate controller for other R ratios.

N.E. Dowling, Mechanical Behavior of Materials, 2/e, Prentice-Hall, 1999.

S-N diagram describes relationship between stress amplitude and fatigue life

Count number of cycles to failure, Nf (or simply N), for a various constant amplitude load states

Results are conventionally plotted for tests with a single R-ratio, in the form of the Stress Amplitude Sa of the vertical axis, and the corresponding Nf on the horizontal axis– Conventional to use logarithmic scale on the N axis.

Collins, J.A., Failure of Materials in Mechanical Design, 2/e, Wiley, 1993.

Low-Cycle and High-Cycle fatigue define different regimes of fatigue loading

Low-cycle FatigueFor large stress amplitudes, plastic deformations occur, resulting in low fatigue life (typically below 104 or 105 cycles)

High-cycle FatigueStrain remains primarily in the elastic range during loading.

Behavior is quite different in these two regimes. For most engineering design, high-cycle behavior is required, and we will therefore emphasize this behavior in MAE 4284. Most S-N diagrams show only the high-cycle behavior and many mathematical characterizations of constant-amplitude fatigue are also valid only for high-cycle fatigue

S-N Curves may be represented by fitted equations

Curve fit representations of S-N data are sometimes used for analytical convenience.

Validity of the different curve fits depends on the nature of the physical response.

For example:

A, B are curve-fitting constants.[In this case, only high cycle fatigue is considered.]

Bfa ANS

Modified from:Collins, J.A., Failure of Materials in Mechanical Design, 2/e, Wiley, 1993.

There is typically a lot of scatter in fatigue test data

– response is strongly influenced by internal defect states, particularly as regards the initiation of fatigue crack initiation

– Ideally, confidence limits are reported along with the fatigue data, resulting in S-N-P (P for probability) curves.


Endurance Limit describes a stress amplitude below which fatigue life appears to be infinite

Some (not all!) materials exhibit a minimum stress amplitude below which fatigue damage does not appear to occur (within the limits of patience in testing 108 cycles at 10 Hz 116 days!).

Boresi, A. P. et al, Advanced Mechanics of Materials, 5/e, Wiley, 1993.

No endurance limit evident for this aluminum alloy

Endurance Limit can be used as part of simple design process for fatigue safety

–design part such that stress remains below the endurance limit.

–also called “Fatigue Limit”

Easy to implement, but not so useful for us:• Aluminum alloys do not exhibit endurance

limit• Tends to produce very conservative (i.e.

heavy) designs.

Applying S-N data requires consideration of how the mean stress influences behavior

So, we have looked at test data for fatigue life for initially undamaged specimens subject to constant-amplitude loading. A couple of problems with using this…

• Test data is likely to be available for only a limited number of test conditions (typically R = −1 only), but in service a different R ratio may be used. Do we have to retest for each different R?

• In reality, loading is not uniform amplitude. How do we handle variable amplitude cases?

“R-ratio” R = Smin/ Smax.

Constant Life curve illustrates combinations of mean stress and stress amplitude that produce a given “fatigue life,” Nf

Constant Life Diagram showing curves for various Nf illustrates

Mean Stress (R-Ratio) Effects

• Simply takes data from S-N curves for various R ratios and plots them in a different format.


Define σa,0 as the “fully-reversed” stress amplitude: stress amplitude to produce a given Nf when the mean stress is zero

(Note that σa,0 is a function of Nf)

Now normalize the stress amplitude axis on the constant life diagram by σa,0 produces an interesting result:

Data for different fatigue life, Nf, tend to collapse onto a single curve. N.E. Dowling, Mechanical Behavior of Materials, 2/e, Prentice-

Hall, 1999.

Normalized Amplitude Constant-Life Diagram

Normalize the curve further by normalizing the mean stress by the ultimate strength.

Fitting a line to the resulting curve gives a means to estimate Nf for any constant amplitude loading

Megson

m

ult

m

a

a

S

S

S

S1

0,

Goodman Relation assumes a linear relationship between and

If we make a curve fit to the data on the normalized constant life diagram, then we get a relationship between the stress amplitude to produce failure at a given life in zero-mean-stress and nonzero-mean-stress cases.

Important:Here Sa,0 is the fully-reversed stress amplitude giving same fatigue life as some other (Sm Sa) constant amplitude state

1

1

0,

0,

ult

m

a

a

ult

m

a

a

S

S

S

S

S

S

S

S

or

0,a

a

S

S

ult

m

S

S


Goodman Relation

So what good is this?• We can use the Goodman relation to

determine a stress amplitude Sa,0 that results in an equivalent life to a given non-zero mean stress cases (Sa, Sm) that we are studying.

• Existing experimental data (in the form of a S-N diagram or an equivalent curve fit) may be available for the fully-reversed (R = −1, zero mean stress case). Thus for a given Sa,0 the fatigue life Nf is known.

Thus we can predict the fatigue life for our nonzero mean stress case using only empirical data from the fully-reversed case.

ult

m

aa

ult

m

a

a

S

S

SS

S

S

S

S

1

1

0,

0,


Sa,0

Nf

R= −1

Is the Goodman relation a good fit?

Look at experimental data.• Relatively low ductility metals (e.g.

high-strength steel) make good match with Goodman relation

• For high ductility metals (see curve), Goodman may be overly conservative.

• For brittle metals (e.g. cast iron), Goodman may be nonconservative and should not be used.

• Also note that Goodman is typically nonconservative for compressive mean stresses. A common approach when using Goodman is to use the following :


0;10,

ma

a SS

S if

Alternate Approaches

• Modified Goodman relation: replace ultimate strength Sult with a corrected value.

• Alternate curve fits to the normalized constant life diagram. (Must verify that the relation used is appropriate for the material in question.)– Gerber:

12

0,

ult

m

a

a

S

S

S

SN.E. Dowling, Mechanical Behavior of Materials, 2/e,

Prentice-Hall, 1999.

Modified Goodman

Gerber

Goodman

Variable Amplitude Loading

Consider hypothetical case of load experienced by a lower wing skin during flight.

How many times could we repeat this operational cycle before fatigue failure of that part?

Palmgren-Miner rule provides a simple approach to variable amplitude loading

So-called “Miner’s Rule”

Not necessarily the most accurate (fracture mechanics approaches may yield better results.)

Palmgren-Miner Rule is based on an assumption of linear damage accumulation

Assumes each load cycle of a certain type uses up the life of the structure at the same rate as any other load cycle of that type (regardless of the prior loading history, the sequence of load types, etc.)

These assumptions aren’t quite true, of course.

Palmgren-Miner Rule

Say a cycle with a mean stress and stress amplitude that S-N data show has a life Nf = 106 occurs. We will assume that that one cycle uses up 1/106 of the total life of the part. Miner’s rule then says that if we count up the life of the part used up by each cycle as it comes, then failure will come when the result adds up to one.

For convenience, we may do our counting based on grouping together cycles of a given size. Then, Miner’s rule may be expressed as:

jNN

jn

N

n

fj

j

j

j

typecycles of life fatigue amplitudeconstant

typeof cycles ofnumber

1 :if failure Fatigue

Cycle countingDetermine typical loading profile for the part in question

– previous operational history– engineering judgment

For miner’s rule, the order of cycles doesn’t matter, so computer algorithms may be used to count the cycles and lump them together.

Palmgren-Miner Rule

If cycles are counted for one operational cycle, then an alternate form of Miner’s rule can be used to express the number of operational cycles before failure, Bf:

jNN

jn

N

nB

fj

j

j

jf

type cycle for , life, fatigue

cycle op. one in type of cycles of number

cycle loperationa

1

1

Example

Dowling

Palmgren-Miner Rule

Comments:• Not highly accurate (but fatigue life

prediction is filled with uncertainty, generally)

• Fracture mechanics approach give better results

Fracture Mechanics Approach to Fatigue

Consider briefly a more physical look at how fatigue cracks develop and propagate through a material.

Three stages of fatigue crack growth

Stage 1: Initiation

Motion of internal defects during cyclic loading results in the initiation of a crack like flaw.

For example, Cottrell-Hull Mechanism based on dislocation motion

Comments:• Typically a very slow process• Stress concentrations, even

highly localized defects such as surface nicks or internal inclusions can significantly accelerate initiation

• Substantial scatter in the behavior due to sensitivity to internal defect state

N.E. Dowling, Mechanical Behavior of Materials

D.K. Felbeck and A.G. Atkins, Strength and Fracture of Engineering Solids, Prentice-Hall, 1996.

Stage 2: Fatigue Crack GrowthOnce a sufficiently large, crack-

like flaw is produced, the physics of crack growth is dominated by the stress field around the crack tip.

Various mechanisms exist for different materials

• Faster than Stage 1 crack growth• Process results in characteristic

striations on cross-section

D.K. Felbeck and A.G. Atkins, Strength and Fracture of Engineering Solids, Prentice-Hall, 1996.

D. Broek, Elementary Engineering Fracture Mechanics, Kluwer, 1986

J.M. Barsom, S.T. Rolfe, Fracture and Fatigue Control in Structures, 3/e, ASTM, 1999.

During Stage 2 crack growth, cyclic loading produces “beach marks” that are very helpful for failure analysis

Closer view of the fracture surface at the inboard end of the lower spar cap of the right wing rear spar. Unlabeled arrows indicate the location of two offset drilled holes.

Closer view of the fatigue region in the horizontal leg of the lower spar cap of the rear spar. Unlabeled brackets indicate fatigue origin areas at the surfaces of the fastener hole, and dashed lines indicate the extent of the fatigue region visible on the

fracture surface.

Figures, captions from NTSB (www.ntsb.gov) AP Photo

Chalk’s Ocean Chalk’s Ocean Airways G-73T Airways G-73T Crash, Dec. Crash, Dec. 20052005

Stage 3: Unstable Crack Growth (Final Fracture)

Stage 3 is not really a fatigue process.

Eventually the crack grows large enough such that KI = Kc. (Call the crack length when this condition is reached the critical crack length ac).

Then unstable crack growth occurs leading to (essentially) instantaneous fracture.

Sample Fracture Surfaces illustrate the three stages of crack growth


Crack growth analysis: predict growth a crack from a given initial configuration

Find crack length for a given defect as a function of time (or equivalently number of loading cycles).

Because of the difficulties with initiation (large scatter, difficulty in quantifying initial damage state) some starting crack size a0 is assumed. This may be based on the limits of the sensitivity of inspection equipment used to search for initial flaws – Typically, assume an initial flaw size, a0, equal to the largest undetectable crack size.

Crack growth analysis requires data on crack growth rates as a function of loading

Analysis is focused on Stage II crack growth

S-N data will not be useful for this, as crack growth is not monitored during such testing.

Crack growth test using a specimen with a preexisting crack loaded cyclically while monitoring crack growth

Conduct constant amplitude fatigue test (e.g. CT specimen): measure crack length as a function of number of cycles da/dN, Crack Growth Rate.

Data are usually expressed as a function of ΔK, the range of stress intensities (note ΔK is a function of crack length for constant stress amplitude)

Typically plotted on log-log axes

Sarafin

Crack Growth Data

Typical data are shown.• Many (not all) materials

have a lower threshold value of ΔK

• At very high ΔK crack growth rates increase dramatically (unstable crack growth)

• For intermediate ΔK many materials have a linear crack growth curve (on log-log axes)


Paris Law: linear curve fit of crack growth data during Region 2.

loading)geometry,,(

Constants Material :,d

d

afK

nC

KCN

a n

Crack growth may be predicted based on Paris Law

Using the Paris Law, we are in a position to predict crack growth for a given starting crack size, load history.

Can give a prediction of fatigue life.

ac : critical crack size corresponding to ultimate collapse of the part

a0 : initial crack size

Safe-Life AnalysisFrom Megson (p. 257)“In the [safe-life] approach, the structure is designed to have a

minimum life during which it is known that no catastrophic damage will occur. At the end of this life, the structure must be replaced even though there may be no detectable signs of fatigue.”

From Sarafin (p. 391)“A part satisfies safe-life criteria when we do NDE to screen out

cracks above a particular size and then show by analysis or test that an assumed crack of that size will not grow to failure when subjected to the cyclic and sustained loads encountered during four complete service lifetimes.”

“One complete service lifetime includes all significant loading events… that occur after the NDE. The scatter factor of four is to account for variability…”

In aircraft, typically this is applied to parts with critical single-point failures, such as landing gear, or rotating engine components. It is difficult to get necessary data and many tests are required to get a statistical understanding of failure characteristics, and even so there is strong sensitivity to details such as tool marks, etc…

Fail-Safe Design• Requires residual strength after failure of

certain primary components sufficient to allow safe flight– either damage part can tolerate a partial fracture or

there is a secondary load path that can do the job

Typical example: fuselage in transport aircraft design to support 40inch long fatigue crack.

Fail-Safe Design Features


Damage-Tolerant Design

• Accepts that cracks are inevitable, but establish procedures to make sure that they can be found and repaired before the integrity of the structure is compromised.

Fracture Control

Fracture control options (modified from Broek, Practical use of Fracture Mechanics, Kluwer, 1989):

• Safe-life design, replacement or retirement after computed life. [May lead to a heavier part than the following]

• Fail-safe design, structure can tolerate certain large defects, repair upon occurrence of partial failure

• Periodic inspection, repair upon crack detection• Periodic proof testing, repair upon failure of proof test

Inspection

• Inspect part before entering service• Conduct crack growth analysis based on sensitivity

of inspection• Set inspection schedule according to the life divided

by some (typically large) factor of safety• Upon reaching the inspection time, reinspect the

part:– No crack detected: reenter service, using same inspection

interval– Crack detected: repair or replace part

Some Complications

• R-Ratio effects on Crack Growth Rates (can be accounted for with methods analogous to those used with the Goodman relation)

• Paris Law may be replaced with more accurate curve fits.• Details of load history can significantly influence analysis

– retardation• Other effect should be considered such as corrosion (stress

corrosion cracking), creep, other environmental effects.One should expect a lot of uncertainty in fatigue analysis and act

accordingly.

Retardation


4281_-17_Fatigue

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