FATIGUE

Fatigue • Fatigue = failure under cyclic stress.

• Stress varies with time. -- key parameters are sa, sm,

and frequency

s max

s min

s

time

s m

sa

• Key points: Fatigue... --can cause part failure, even though smax < sc.

--causes ~ 90% of mechanical engineering failures.

Adapted from Fig. 8.18,

Callister 7e. (Fig. 8.18 is

from Materials Science in

Engineering, 4/E by Carl.

A. Keyser, Pearson

Education, Inc., Upper

Saddle River, NJ.) tension on bottom

compression on top

counter motor

flex coupling

specimen

bearing bearing

Fatigue failures are widely studied

because it accounts for 90% of all service

failures due to mechanical causes.

Fatigue failures occur when metal is

subjected to a repetitive or fluctuating

stress and will fail at a stress much lower

than its tensile strength.

Fatigue failures occur without any plastic

deformation (no warning).

Fatigue surface appears as a smooth

region, showing beach mark or origin of

fatigue crack.

Factors causing fatigue failure

1) A maximum tensile stress of sufficiently high value.

2) A large amount of variation or fluctuation in the applied stress.

3) A sufficiently large number of cycles of the applied stress.

Basic factors

Additional factors

• Stress concentration

• Corrosion

• Temperature

• Overload

• Metallurgical structure

• Residual stress

• Combined stress

Stress cycles

Ds

Ds

Nomenclature

The S-N curve • Engineering fatigue data is

normally represented by means of

S-N curve, a plot of stress Stress

against the Number of cycle, N.

• Stress can be σa, σmax, σmin

•σm and R should be mentioned.

• S-N curve is concerned chiefly

with fatigue failure at high numbers

of cycles (N > 104 cycles)

high cycle fatigue (HCF).

• N < 104 cycles low cycle fatigue (LCF).

• Fatigue limit or endurance limit is normally defined at 107

or 108 cycles. Below this limit, the material presumably can

endure an infinite number of cycle before failure.

• Nonferrous metal, i.e., aluminium, do not have fatigue limit

fatigue strength is defined at ~ 108 cycles.

•AISI 4130 STEEL, completely reversed axial fatigue,

• Fatigue limit, se:

--no fatigue if s < se

Adapted from Fig.

8.19(a), Callister 7e.

Fatigue Design Parameters

se

case for steel (typ.)

N = Cycles to failure 10

3 10

5 10

7 10

9

unsafe

safe

s= stress amplitude

• Sometimes, the

fatigue limit is zero!

Adapted from Fig.

8.19(b), Callister 7e.

case for Al (typ.)

N = Cycles to failure 10

3 10

5 10

7 10

9

unsafe

safe

S = stress amplitude

Fatigue Ratio • Fatigue strength is frequently correlated with tensile properties.

Note: for smooth and polished specimen.

Ex: fatigue ratio ~ 0.5 for cast and wrought steels, (SU (MPa) = 3.45 x HB )

fatigue ratio ~ 0.35 for non-ferrous (SU versus hardness: no simple relation)

Relation between rotating, bending unnotched fatigue strength and tensile strength. (a) Alloy (.)

and carbon (x) steels; (b) wrought copper alloys.

Fatigue ratio= fatigue strength / tensile strength

Estimating Fatigue Life

where

In some cases, we need to know the number of failure cycles for a given stress.

b

f

C

a N10s

e

UTSSb

s

8.0log

3

1

e

UTSSC

s

28.0

log

ba

bC

N1

10 s

For ferritic materials fatigue life can be estimated analitically.

at 103 cyles, fatigue strength is 0.8 Su, and

at 106 cycles fatigue strength is 0.5 Su.

After arranging and simplifying we get

Endurance Limit Modifying Factors

• Material: composition, basis of failure, variability

• Manufacturing: method, heat treatment, fretting corrosion, surface

condition, stress concentration

• Environment: corrosion, temperature, stress state, relaxation times

• Design: size, shape, life, stress state, stress concentration, speed,

fretting, galling

Factors effecting Endurance Limit

se = ka kb kc kd ke kf s′e

ka = surface condition modification factor

kb = size modification factor

kc = reliability factor

kd = temperature modification factor

ke = effect of stress concentration

kf = miscellaneous-effects modification factor

s′e = rotary-beam test specimen endurance limit

se = endurance limit at the critical location of a machine part in

the geometry and condition of use

The surface of a rotating-beam specimen is highly polished, with a final

polishing in the axial direction to smooth out any circumferential

scratches. The surface modification factor depends on the quality of the

finish of the actual part surface and on the tensile strength of the part

material.

Surface Factor ka

ka decreases

Size Factor kb

For bending and torsion

Larger specimens increases surface area subjected to cyclic load

higher possibility to find defects on surface.

Statistical nature of fatigue • Because the S-N fatigue data is

normally scattered, it should be

therefore represented on a

probability basis.

• Considerable number of

specimens are used to obtain

statistical parameters.

• At σ1, 1% of specimens would be

expected to fail at N1 cycles.

• 50% of specimens would be

expected to fail at N2 cycles.

mean stress, stress range

Haig-Solderberg diagram • In Haig-Solderberg diagram is

a plot of alternating stress σa and

mean stress σm.

• The Goodman relationship may

be expressed by

where x = 1 for the Goodman line,

x = 2 for the Gerber parabola,

σe = the fatigue limit for completely

reversed loading.

• If the design is based on the yield strength σy, (based on

Solderberg line), then the σu is replaced by σy in this equation.

Example: A 4340 steel bar is subjected to a fluctuating axial load that

varies from a maximum of 330 kN tension to a minimum of 110 kN

compression. The mechanical properties of the steel are:

σu = 1090 MPa, σy = 1010 MPa, σe = 510 MPa

Determine the bar diameter to give infinite fatigue life based on

a safety factor of 2.5.

Solution: Cylindrical cross section of the bar = A, the variation of stress will be

D=38.7 mm

Cumulative damage rule The percentage of fatigue life consumed by operation at one operating

stress level depends on the magnitude of subsequent stress levels

the cumulative rule called Miner’s rule.

Where n1, n2,..nk = the number of cycles of operation at

specific overstress levels.

N1, N2,..Nk = the life (in cycles) at this same

overstress level.

• Crack grows incrementally

typ. 1 to 6

asD~

increase in crack length per loading cycle

• Failed rotating shaft --crack grew even though

Kmax < Kc --crack grows faster as • Ds increases

• crack gets longer

• loading freq. increases.

Fatigue Mechanism

mKC

dN

daD

Structural features of fatigue

The fatigue process can be divided into the following processes;

1) Crack initiation: the early development of fatigue damage

(can be removed by a suitable thermal anneal).

2) Slip band crack growth: the deepening of the initial crack on

plane of high shear stress (stage I crack growth)

3) Crack growth on planes of high tensile stress: growth of

well-defined crack in direction normal to maximum tensile stress (stage II)

4) Ultimate ductile failure: occurs when the crack reaches sufficient length

so that the remaining cross section cannot support the applied load.

Initiation of fatigue crack and slip band crack growth (stage I)

• Fatigue cracks are normally initiated at a free surface. Slip lines

are formed during the first few thousand cycles of stress.

• Back and forth fine slip movements of fatigue could build up notches

or ridges at the surface. act as stress raiser initiate crack.

• In stage I, the fatigue crack tends to propagate initially along slip planes

(extrusion and intrusion of persistent slip bands) and later take the direction

normal to the maximum tensile stress (stage II).

• The crack propagation rate in stage I is generally very low on the order of

nm/cycles giving featureless surface.

Static deformation

Fatigue deformations leading to

Surface notch (intrusion) Slip band extrusion

PSB: Persistant slip bands

Stable crack growth (stage II)

• Compression load reverses the slip direction in the end zones

crushing the crack surface to form a resharpened crack tip.

• Crack tip blunting occurs during tensile load at 45o and crack grows

longer by plastic shearing.

Crack closed

Crack opening

Crack at

Maximum load

Crack closing

Crack closed

And, crack

opening again!

Structural features

• The fracture surface of stage II crack propagation frequently shows

a pattern of ripples or fatigue striations.

• Each striation is produced by a single stress cycle and represents

the successive position of an advancing crack front normal to the

greatest tensile stress.

Fretting fatigue striations on the fracture surface of an

aluminum head gasket from a reciprocating gas compressor. Fracture surface of a rotating steel shaft that

experienced fatigue failure.

Fatigue failure surface. A crack formed

at the top edge. The smooth region also

near the top corresponds to the area

over which the crack propagated slowly.

Rapid failure occurred over the area

having a dull and fibrous texture

Fatigue crack propagation

For design against fatigue failure, fracture mechanics is utilised to

monitor the fatigue crack growth rate in the stage II Paris regime.

da/dN= C (DK)m

da/dN : fatigue crack grwth rate

DK : stress intensity factor range (Kmax-Kmin)

C, m : material variables, depends on

environment, frequency, temperature stress ratio

Fatigue Life Prediction

maYCdN

dasD

Df

m

fm

a

a

N

mm

a

dadNCY

0

2

2

0

s

D

2

22

22

11

2

2

0

mmm

f

mmf

aaCYmN

sm ≠2

Example: The fatigue crack markings shown in Figure were found in a

fractured part. Determine the time to rupture of this part if the loading

frequency is 10 Hz, the maximum stress applied to the part is 300 MPa,

and the minimum stress is zero. The initiation stage of the flaw is 50%

of the life of the part.

Mech. B. Of Mat. Chawla

a1= 2 mm a2= 10 mm

Fracture occurs when af = 14 mm. Assuming that we have plane strain,

We now find the parameters for the Paris equation:

Integrating between the limits, af = 14 mm and ai = 0.02 mm, we get:

Each cycle corresponds to 0.1 s, so t = 0.1N

The total time is equal to the initiation time plus the propagation time:

t total= 2t = 98.6 × 2 s

= 197.2 s ~ 3 minutes

maYCdN

dasD

N=986 cycles

Improving Fatigue Life I. Impose a compressive

surface stress (to suppress surface

cracks from growing)

N = Cycles to failure

moderate tensile s m Larger tensile s m

S = stress amplitude

near zero or compressive s m Increasing

sm

--Method 1: shot peening

put surface

into compression

shot --Method 2: carburizing

C-rich gas

Adapted from

Fig. 8.24, Callister 7e.

• 3. Surface rolling

- Compressive stress is introduced in

between the rollers during sheet rolling.

• 4. Thermal stress

- Quenching or surface treatments

introduce volume change giving

compressive stress.

III. Polishing

- Reducing surface scratches

II. Remove stress

concentrators. Adapted from

Fig. 8.25, Callister 7e.

bad

bad

better

better

Design for fatigue

1) Infinite-life design: Keeping the stress at some fraction of the fatigue limit

of the material.

4) Damage tolerant design: Use fracture mechanics to determine

whether the existing crack will grow large enough to cause failure.

3) Fail-safe design: The fatigue cracks will be detected and repaired

before it actually causes failure. For aircraft industry.

2) Safe-life design: Based on the assumption that the material has

flaws and has finite life. Safety factor is used to compensate for

environmental effects, varieties in material production/manufacturing.

• Flaws produce stress concentrations that cause

premature failure.

• Sharp corners produce large stress concentrations

and premature failure.

• Failure type depends on T and stress:

- for noncyclic s and T < 0.4Tm, failure stress decreases with:

- increased maximum flaw size,

- decreased T,

- increased rate of loading.

- for cyclic s:

- cycles to fail decreases as Ds increases.

SUMMARY