Fatigue• Fatigue is the growth of cracks as a result of

cyclic loading• At one time it was thought that the material got

tired and broke- hence the name• Fatigue is one of the most common causes of

failure of engineering components• There is now sufficient knowledge and

understanding of fatigue to be able to predict failure in most situations- provided the necessary information is available

Fatigue design• Almost all engineering structures are subject to

some kind of cyclic loading and fatigue is one of the most common causes of failure.

• For many parts it is sufficient to build in a substantial safety factor and hope for the best.

• For highly stressed or safety critical components, a more sophisticated approach is needed.

• For structures such as aircraft (highly stressed, safety critical, made of materials vulnerable to fatigue), much of the design work is on fatigue

Stages of fatigue crackingFatigue crack growth is conventionally

divided into three stages:-• Stage I. The crack initiates and grows by

reversed slip on planes at 45° to the maximum principal stress

• Stage II. The crack turns and grows at right angles to the principal stress

• Stage III. The crack grows rapidly leading to final failure by fast fracture

Stage I

• Plastic deformation concentrates on persistent slip bands

Stage I• Although the average stress is below the yield

stress, at small local stress concentrations it is greater and plastic deformation occurs

• Plastic deformation occurs on persistent slip bands leading to step formation at the surface

• Reverse slip on unloading causes opposite movement leading to notches

• Usually extends for 2- 5 grains before the crack makes a sufficient stress concentration to change to stage II.

Influences on stage I• The number, size and shape of stress raisers is

of critical importance in determining the load and number of cycles to start a crack

• Surface finish, notches or other stress concentrators, pores, inclusions, scratches, coatings may all be important

• Length of slip systems determines number of dislocations accumulated at the ends- hence grain size is important

• Stress is loosely related to yield stress

Crack grows by stage I

mechanism at ±45°, then turns

and grows in stage II (arrows)

Stage II

Work hardening and cracking

Stage II• When the crack grows larger a sufficient stress

concentration is generated to cause localised yielding over a volume around the crack tip

• This volume may then crack due to work hardening

• The blunted crack tip may be extruded forward as the crack closes

• Characteristic striations produced on fracture surface

Influences on stage II• Rate of crack propagation in stage II

depends on bulk plastic properties more than surface condition

• The size of the crack and hence the stress intensity at the crack tip is most important

• Stress amplitude and ratio of maximum to minimum stress (R- ratio) are still very important.

S-N approach to fatigue• Pioneered by Wohler in the 19th

century using rotating bend tests• Other test geometries are possible• Plot stress (maximum or

amplitude) against log cycles to failure for multiple specimens

• Produces an S-N curve

S-N curve approach to design• Identify maximum tensile stresses in

structure• Obtain loading history• Obtain S-N curve for specimens• Read off expected life• Use Goodman diagram to account for R ratio• Use Miner’s rule to account for variable

loading

Simple example• An aircraft part is made from 7075-T6

aluminium alloy. The most highly stressed region is smooth and free from stress concentrations. It is loaded in bending with a maximum stress of 250MPa, and the load is fully reversed, so that it bends equally in both directions.

• Estimate the life of the part using the S-N curve given.

Miner’s rule• Assumes fatigue occurs by cumulative

damage, proportional to work input• A given number of cycles at a given load

will cause a certain amount of cumulative damage

• For a number of cycles ni, at a load that would cause failure after Ni cycles, the work input wi is a fraction of the work input Wi to failure i

i

i

i

N

n

W

w

Miner’s rule• Break the loading down into numbers of

cycles at each load level, eg 100 cycles at 1kN, 1000 cycles at 2kN etc

• Loading is assumed to be sequence independent

• Damage caused by each block as a fraction of the total life can be added to predict failure

• Then, at failure,1

i

i

N

n

Goodman diagram• For design purposes, it is best to use a constant

life diagram, which shows the actual life as a function of maximum and minimum stress

• If this data is unavailable, it is possible to construct a diagram from data for one R ratio and the static tensile properties

• Plot stress amplitude vs mean stress• Draw straight lines from fatigue strength R=-1 to

static tensile strength (Goodman) or yield stress (Soderberg). Gerber uses parabola.

Limitations of Miner’s rule• Experimental data shows that predictions

are non- conservative ie overestimates life • Divide predicted life by 10 for safety• For short cracks, cumulative damage

assumption does not necessarily hold and predictions may be non- conservative

• Occasional overloads may break through microstructural barriers, giving much faster crack propagation

Effect of R ratio• S-N curves usually show stress amplitude• Fatigue life depends heavily on R ratio (ratio of

maximum to minimum stress)• High R ratio means maximum stresses are higher

for same amplitude, hence faster crack growth• Can use Gerber, Goodman, Soderberg or

constant life diagrams to account for these effects

• Diagrams show life at various stress amplitudes and R- ratios

Stress amplitudeStress or load

Time

Max

Min

Mean

Mean=(max+min)/2 Amplitude=(max-min)/2 R=min/max

Amplitude

Goodman/Soderberg diagrams

y-UTS UTS-y

Safe

Unsafe

Goodman

Soderberg

Fatigue endurance limit

eg 106 cycles

amplitude

mean

Example• An aircraft part is made from 7075-T6 aluminium

and is smooth with no significant notches or stress concentrations in the region of interest. It is expected that it will experience 2000 loading cycles with maximum stress 500MPa and minimum 100MPa, 100,000 cycles with a mean stress of 250MPa and amplitude 150MPa and 5000 cycles with an amplitude of 120MPa and R ratio of 0.1. Determine whether it will fail or not, using the constant life curves given.

Advantages of S-N approach• Quick and simple to use• Material data widely available• Useful for tests on actual components

under simulated loading• Useful as a quick approximation• Often fatigue strength is sufficiently

great that no further analysis is needed

S-N testing

Limitations of S-N approach• Limited to uniaxial loading• Cannot cope well with stress concentrations

except where these are duplicated exactly in the test specimens

• Specimen surface condition must be identical to finished part

• Miner’s rule for variable amplitude is non- conservative and may be inaccurate

• Cannot predict crack growth from existing or assumed flaws

Effect of notches• S-N curve approach cannot predict failure of

notched components unless the notch is exactly reproduced in the test specimens

• The elastic stress concentration factor kt can be used to predict fatigue life from an S-N curve but will underestimate the life considerably

• At the notch root there is plastic deformation and so the stress concentration is different

• The plastically deformed material is highly constrained by the elastic material around it

Local stress- strain• We can analyse this situation by examining the

plastic strain at the notch root• We can test smooth specimens at the same

cyclic strain, and the number of cycles to start a crack will be similar

• Under cyclic plastic strains the load- displacement curve will form a hysteresis loop

• Work hardening or softening may occur, but the material soon stabilises and exhibits constant cyclic properties

True stress and strain• To examine the local stresses and strains at a

notch we need to use the true stresses and strains, not the engineering ones

• True strains can be added to give total strain

areasectionalcrossousinstantane

loadstresstrue

true

lengthinitial

lengthfinallnstraintrue true

Basquin’s law• We can divide the cyclic strain into elastic

and plastic components e and p which add together to give the total strain t

• The elastic part is related to fatigue life by Basquin’s law:-

• This is an empirical relationship and can be used to predict fatigue life

bffe

a NE

2'2

Manson- Coffin relationship• Similarly, the plastic strain can be related

to the fatigue life by the Manson- Coffin relationship:-

• Again this is an empirical relationship• We can relate the total strain to the fatigue

life by combining both relationships:-

cffp N2'

2

cffb

fft NNE

2'2'

2

Neuber’s rule• We can now predict the fatigue life of the component

if we know the local stresses and strains, for example from a finite element model

• Alternatively, we can estimate the local stress and strain using Neuber’s rule:-

• Substituting in the actual stresses and strains:-

• Or use effective stress concentration factor kf

• Intersect with cyclic stress- strain curve

kkkt .

Ektav /2

Procedure• Calculate local stresses, strains• Use FE or Neuber’s rule• Determine time to crack initiation

from tests on smooth specimens• Fracture mechanics analysis for

crack propagation

Further reading

• Hertzberg, R. W., Deformation and fracture mechanics of engineering materials. Wiley, New York, 1989.

Fracture mechanics approach

• Probably the most elegant and effective way of analysing fatigue is by directly considering the mechanics of the growing crack

• We can do this using fracture mechanics • It seems reasonable to suppose that the growth

of the crack will depend on the stress distribution around the crack tip, and very little else

• It makes sense, therefore, to relate crack growth to K (in practice, to K and R)

Fracture mechanics approach• Use a precracked specimen and

measure crack growth and load• Plot log crack growth rate against log

K• All results should superimpose to form

a single line, which is straight over a wide range

• Full curve from one test

da/dN vs K • If we plot the rate of crack growth against K, we

find that all the data throughout the growth of the crack comes together to form a single curve

• At low K the growth rate decreases towards the fatigue crack growth threshold Kth

• At high K the growth rate increases towards final fracture at K1c

• The rest of the graph is typically more or less linear (on log axes)

• Different for different R ratios

Paris Law• The linear part of the graph, which comprises

most of the fatigue life, is described by the Paris Law:-

• C and m are material constants which also depend on R ratio; m is usually around 3 or 4.

• Watch out for units! The equation is empirical and not dimensionally consistent.

• Note K is Kmax-Kmin; Kmin taken as zero if actual value is negative- crack closed in compression

mKCdN

da

Life prediction• The Paris equation can be integrated to find

the number of cycles to reach a given crack length

• This can be done in a general way - much easier to substitute in known values first

• Start by substituting

• and then

aYK

2/2/ mmmm aCYdN

da

Life prediction• Rearranging,

• Integrating both sides,

• and hence

dNCdaaY mmmm 2/2/

ff

i

Nmma

a

mm dNCdaaY0

2/2/

f

i

a

a

mmmmf daaY

CN 2/

2/

1

Design calculations• The same approach of separating da and

dN and integrating each side can be used to calculate the fatigue life, the critical initial crack length etc

• Starting from the basic Paris equation and substituting the various constants as early as possible the maths is relatively simple

• Calculate the final crack length from K1c, and assume linear crack growth up to K1c

Example (1)

• A large steel plate is cyclically loaded with a maximum stress of 100MPa and a minimum stress of -50MPa. The largest surface crack in the material has a length 2a=2mm, K1c=25MPam and the Paris parameters for the material are 1x10-12 and 3, assuming in MPa and a in metres. Estimate how long it will take for the sheet to fail in fatigue, assuming plane strain and Y=1.

Example (2)

• The same steel sheet is to be inspected for cracks every million cycles. What is the minimum crack length that must be detected to avoid failure before the next inspection?

Complications• R- ratio• Short cracks • Frequency effects• Temperature effects• Creep• Environmental effects• Loading sequence effects

Frequency effects• The loading frequency and/or strain rate can affect

crack initiation and growth• This complicates testing and design• Polymers are often particularly sensitive, because

their mechanical properties depend on the movement of their molecules, which takes time

• Various effects are possible in different polymers• Hysteretic heating is also important• Metals can have frequency dependent behaviour

because of environmental effects

Temperature effects• The temperature is particularly important for

many polymers, in the same way as frequency• Again, for metals the temperature can affect

environmental interactions which can be important in fatigue crack growth

• Corrosion in wet environments and oxidation at high temperatures are examples

• Oxidation or corrosion products can cause crack closure and hence reduce the effective K.

Short cracks• Several types of short crack:-• Physically short crack, a 1mm • Microstructurally short crack, a grain size• Mechanically short crack, a plastic zone size• Geometrically short crack, a < notch size• Chemically short crack, with length dependance of

corrosion fatigue• Anomalous behaviour is likely• Crack growth usually enhanced- Paris law may be

non- conservative

Effects of porosity• Boundary

element model of the centre of a tensile specimen with a void

kt=2.06

K=(2/)((r+a))

K=1.13 x 2(a)

K=1.12(2/) x 2(a)

Crack length aVoid radius rApplied stress

0.1mm quarter- circular crack

2mm diameter void, half symmetry model

Boundary element model

0.02

0.03

0.04

0.05

0.06

0 0.0005 0.001 0.0015 0.002

Stress intensity K (MPam)

Crack length a (m)

Interpolation

Circular crack a>>r

Semicircular crack a<<r

Simulation results

Crack growth around a void

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5

Fatigue life (x105 cycles)

Void size (mm)

Cycles for crack to grow from 0.02mm to 2mm

Effect of overloads• Occasional overloads can break through

microstructural barriers eg grain boundaries• Plasticity induced closure may retard long cracks• Residual compressive stresses may contribute• Compressive underloads may flatten fracture

surface asperities and increase crack growth, especially at low K

• Substantial sequence effects possible- Miner’s rule may not be obeyed

-6

-5.5

-5

-4.5

-4

-3.5

-3

5.7 5.8 5.9 6 6.1

Log K (Pam)

Log da/dN (mm/cycle)

Retardation afterload- shedding

Intermittentcrack growth?

Overloads: PMMA

-10

-9

-8

-7

-6

-5

5.5 5.6 5.7 5.8 5.9

Baseline (large)

SOL (large)

Baseline (small)

SOL (small)

Log K (Pam)

Log da/dN (m/cycle)

Overloads: Palacos R

0

1

2

3

4

5

6

7

0 10000

Cycles after overload

da/dN (normalised)

Pronounced accelerationafter overload

Environment effects• Chemical reactions at the crack tip can greatly

enhance fatigue crack growth• Dissolution of the metal at the crack tip can

occur when the surface oxide film is ruptured• Effects depend on:-• electrochemical conditions • rate of diffusion to crack tip • loading frequency and waveform• Can cause deviation from Paris law

Design considerations• Consider possibility of fatigue at all times• Cyclic stresses may arise from accidental loading,

thermal expansion, etc etc• Use simple S-N calculations if in doubt• Use local stress- strain method to predict crack

initiation accurately• Use fracture mechanics to predict crack growth• Test critical components and assemblies• Avoid stress raisers and other risk factors• Remember tolerances

Fatigue checklist• Stress raisers• Rough surface finish• High cyclic stresses• Material property changes• Notch sensitive material • Tensile residual stresses • Environmental factors