pdf_m11ekm fatigue and fracture_autumn 2010

7/31/2019 Pdf_m11ekm Fatigue and Fracture_autumn 2010

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Fracture:

Metals& Alloys

M11EKM

Dr Phil SwansonMF220

[email protected]

Fatigue crack & subsequent fast fracture surfaceOrigin is an inclusion. Crack grew undercyclic loading until it reached critical size, prior to

fast fracture

Fatigue crack initiation & growth in steel castingCrack initiation at internal shrinkage porosityin steel casting. Visible beach marksindicate periodic

crack arrest points

Ryan Spirit of St Louis replica crash

Coventry Airshow May 2003Fatigue failure at a weld adjoining a wing tie

bar and undercarriage leg junction

Cracking from multiple weld defects(shrinkage & gas porosity + oxide inclusions )

DH 106 Comet 1: Several crashes due tofatigue cracks in pressurised stressed skin.

Cracks initiated from rivet holes adjacent tosquare windows (stress concentration)

Fracture modes: Ductile

Extensive plastic deformationprior to fracture

Extensive energy absorption:Tough

Brittle Little/no plastic deformation

prior to fracture

Low toughness Creep

High temperature & pro-longed stressing

Gradual accumulation ofplastic strain under load withtime at elevated temp

Fatigue Progressive crack growth

under cyclic stressing Crack initiates at surface or

internal defect

Ductile vs. Brittle: Stress-strain characteristics

Fatigue fracturesurface:Marine diesel enginecrankshaft

Smooth area is the crack

growth region

Rough smaller surface

corresponds to final

ductile overloadOrigin of crack


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Ductile failure a) Necking localised thinning after yield

b) Cavity Formation Inclusions pull away from metal

matrix producing voids/cavities. Material in-between cavities is

subject to excessive work-hardening

c) Coalescence of adjacent cavitiesto form a crack

d) Crack propagation in plain strain

(3D stress system)

e) Fracture. fracture surface usuallyshows: Shear lips at section periphery Dull voided central section

Torn-voided ductile fracture

in low alloy steel x2500

Characteristic cup & cone

ductile failure test piece

Creep Gradual accumulation of

plastic strain with time atconstant load or stress

Elevated temperature phenomena(above RCT)

Mechanisms Grain boundary sliding

Dislocation climb Void formation @ grain boundaries

Diffusion contributes to all the abovemechanisms

Creep is a function of : Time

Temperature Stress Microstructure

Primary Creep

Strain rate decreases: due to work-hardening

Secondary Creep

Steady state: abalance betweenwork-hardening and thermally induceddiffusional recovery

Tertiary Creep

Void formation and grain boundary

sliding accelerate rupture

Typical creep strain vs. time profile


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Creep mechanisms

Dislocations can climb over obstacles via diffusionObstacles: Precipitates, Dispersoids, Dislocation pile ups/entanglements

Grain boundary slidingGrains rotation to align close packed preferredslip planes with maximum shear stress

Stress directed atomic diffusionpromotes grain elongation.

Diffusion contributes to both climb, boundary sliding & void formation

Dislocations can

climb up and down

Creep mechanismsVoids on transverse grain boundaries accelerate

creep rate

Creation of voids at an inclusion trapped

at the grain boundary

Creation of a void at a triple point where

three grains are in contact

Ferritic stainless steel ruptured at 600 C:Grain boundaries exhibiting dimples on g.b. and g.b. slip

Creep cavities formed at grain boundaries in

an austentic stainless steel


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Creep:Ductile rupture at elevated temperature

- grain boundary cracking & voids

Grain boundary voids form atboundary triple points and on boundaries

at 90 deg to applied stress

app

app

Grain boundary voidsFormed at 600C inNi based alloy(Nimonic 105)

Brittle Fracture

Occurs with no appreciable plasticdeformation, typically crack propagates froma defect or stress concentration

Crack propagation is very fast (~2000 m/s)

Crack propagates nearly perpendicular to thedirection of the applied stress

Crack often propagates by cleavage breaking of atomic bonds along specificcrystallographic planes (cleavage planes).

Typically brittle fractures occurs at stresses

below the yield stress, the crack beinginitiated at a defect with the material

Brittle fracture in a mild steel

Flat cleavage fracture surface in high alloy steel


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Brittle Fracture

21

02

=

tTip

a

Griffthshowed that the stress at the crack tip canbe found from the expression

0 = Applied stress

t = Crack tip radius

For a through thickness cracklength 2a in an infinite thin

plate (plane stress conditions)

a

E

2=

E = Elastic modulus = Surface energy

= Fracture stress

As stress increases stored elastic energy is sufficient to initiateinstantaneous crack propagation

Griffith used an elastic energy balanceto calculate.

Brittle (Fast) Fracture Griffith approach only works with a

perfectly elastic material (i.e. nocrack tip plasticity) In metals there is always localised

crack tip plasticity as tip > y

The Griffith criteria can be modifiedto account for localised plasticyielding by substitutingG1c for 2 ( = crack surface energy)

G1c = Energy absorbed in generatingunit area of crack (J.m-2)

G1c = Fracture Toughness or criticalstrain energy release rate. It is ameasure of the amount of plasticwork that must be done before thecrack extends

Crack tip plastic zone

visualisation through a thick section body

Surface: 2D stressing, plane stress (greater plasticity)

Interior: 3D stressing, plane strain (minimal plasticity)

)1.(.

.

2

1

=

a

GE cModified GriffithEqn for plane strain

Plastic zone is large in the vicinity of

surface (plane stress)

Plastic zone isSmall in bulk

material

(plane strain)


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Stress Intensity Approach to Fast FractureK1c represents the critical value of the stress intensity factor (K) for fast fracture,

that is. K1c= Plain Strain Fracture Toughness

(Stress intensity factor K is mathematically related to strain energy release rate, G)

The expression for K1c is written as:

Y = Geometry factor =

E

KcGc

2

=

cFic aYK ... =

W

af

Penny shaped crack

Edge crack

Y values for simple shapes are

tabulated, for complex shapes they needto be determined by finite element

techniques

Kc = Plane stress fracture toughness (thin sheet)K1c = Plane strain fracture toughness (thick plate)

Kc > K1c

K1c is the minimum value of the fracture toughness

corresponding to plane strain (thick plate- 3D stresses)

Y =

Y = 1

Internal crack

w

w

2

E

KG

c

c

)1( 212

1

=

E

KcGc

2

=

Crack initiation due to micro-structuralphenomena

Elastic incompatibility between

adjacent grains can cause crack initiation

at interface (grain boundary)Often if adjacent grains are:

Different phases (structure & composition) Different crystallographic orientationElastic modulus of adjacent grains which

Differ in structure and/or composition

will differ

Blocking of a slip plane

by a hard second phase particle

can generate sufficiently highstress as to nucleate a crack.

Crack orientation depends onmatrix crystal structure andcohesion strength at interface

with hard second phase particle

In amorphous polymers,

subject to sufficient stresses

the disorganised moleculartangle becomes aligned as

orientated filaments in directionof max stress...voids nucleatein-between fibrous filaments.

Further stressing causes void

coalescence and crack formation


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Crack initiation due to micro-structural

phenomena

Ductile solids containing, rigid

particles (e.g. inclusions), may sufferfracture at inclusion interface, generating

voidsshear and deformation ofmaterial between voids causes very high

localised plastic straining and voidgrowth leading to void coalescence

(diffusion at high temps can accelerate this)

At sufficiently high temps,

deformation of poly-crystallinematerials (& some semi-crystallinepolymers), can occur by boundary

sliding.

Cracks often initiate at boundarytriple points

Cyclic loading can cause

local deformation on preferredslip planes within individualGrains.

Grains at surface are especially

vulnerable.Adjacent slip steps sliding back& forth can generate surface

striations that can initiate a crack

Slip bands

Fatigue

Low cycle fatigue (high strain fatigue):

Failure occurs in less than 104 cycles

max>Yand the process is under strain control.

Thermal stresses frequently give rise to low cycle thermal fatigue failures.

High cycle fatigue:

Failure occurs in more than 104 cycles

max


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FATIGUE

Fatigue of un-cracked componentsNo pre-existing cracks initiation controlledGear teeth, crankshafts, axles

Fatigue of cracked componentsCracks pre-exist propagation controlled fractureLarge welded structures, bridges, pressure vessels

Fracture Mechanics

High cycle FatigueMax stress below yield, > 104 cycles to failure

Rotating or vibrating systems(wheels axles, engine components)

Basquin LawS/N approach

Low cycle FatigueMax stress above yield, < 104 cycles to failure

Airframes (fighters), nuclear components,components subject to occasional overloadCoffin Mason..S/N approach

Ductile fatigue crack propagation surface

(striations) in Nickel alloy (650C)

Cyclic loading: S/N Curves

Endurance limit (allowable stress

amplitude) is usually quoted for afatigue life of 107 cycles

Standard tests on polished test pieces

loaded in;Tension

Torsion

Reversed bending

One of the standard depictions of a materials fatigue performance

is the experimentally derived S/N curve

Plots allowable stress amplitude vs. number of cycles to failure

The line displayed on S/N plot corresponds to a failure probability of 50% (P=0.5)

S/N curves usually derived at a mean stress (m) = 0

Endurance limits for non zero mean stresses can be calculated using

equations produced by Goodman, Gerber or Soderburg

a0= Endurance Limit usually quoted for N= 107cycles


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S/N Curves: Distribution of data

S/N method

Uses blanket safety factors to account for features such as:-poor surface finish-stress concentrations-welds

The use of safety factors leads to inefficient over-design (heavy)

Cannot deal satisfactorily with cracked components

Two main types of S/N curves Fatigue Life curve

(Most materials)

Fatigue Limit curve

(Steels & certain Cu alloys Only)

For most materials, the allowablestress amplitude gradually decreaseswith increasing number of cycles

Endurance limit quoted at N= 107 cycles

Endurance limit

a0

m = 0

m = 0

Steels show a Fatigue LimitIf the stress amplitude is less thanfatigue limit, then fatigue life is theoreticallyinfinite (given P values)Can destroy fatigue limit by:

-Overstressing-Over heating-Corrosion

P= 0.5

P= 0.5Fatigue limit in

steels & Cu alloysis a

consequenceof their

excellent

work-hardeningcapability


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Fatigue Cycle

Reminder:Stress range: = max - minStress amplitude: a = (max - min)Mean stress: m = (max + min)

Load ratio: R = min/max

Usual to quote fatigue stresses as:m a

Loading frequency: Hz (cycles/sec)

Cyclic stress profile modelled on sine curveMethods available to deal with more complex

loading cycles e.g.Miners RulebasicRainflow analysiscomplex

For metals the waveform shape and

loading frequency have little effect

For polymers the waveform shape andloading frequency are important

Polymers:

Thermal insulators plus high internal friction

Higher loading frequencies cause reduction infatigue strength due to heat build up; can causelocal melting in extreme cases

S/N Curves

=

=

=

=

TS

maa

y

maa

m

m

1|

1|

0

0

Derived from many experimental testsusing polished, stress free test pieces

Usually S/N curves are generated at zeromean stressMethods to correct for non zero meanStress:

Endurance limit often quoted at N = 107 cycles

a0 = allowable stress amplitude at zeromean stress

a0Soberberg equation

Goodman equation

=

=

=

=

=

=

2

0

0

0

1|

1|

1|

y

maa

TS

maa

y

maa

m

m

m

Gerber equation


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Comparison of different correction methods

for a0

For a material with an endurance limit of a0 = 500 MPa, calculate (using thethree different equations given previously) the allowable stress amplitude,a when the mean stress is raised to 100 MPa and comment on whichcorrection method would be the safest

Given: Tensile strength = 1000 MPa, Yield Strength = 750 MPa

Design factors to address fatigue

Reduction of stress concentrations by:

Using well blended, generous radii at changes of sections

Improving surface finish (no coarse machining marks)

Using cleaner material without inclusions

Ensuring low porosity and other defect levels at the surface

Compressive residual stress from shot peening or heat treatment e.g. carburising.

Metallurgical defects can detract from fatigue strength

Shrinkage porosity that occurs during solidification

Trapped gas bubbles

The accumulation of non-metallic impurities in the melt

Insufficient bonding of the grains in sintered materials (powder metallurgy)


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Crack Initiation & Propagation Crack Init iation:

-Usually from a defect

-Most common at free surfaces

-Can be initiated in the absence of adefect by dislocation interactions

Propagation: Two stagesStage 1:

-Crack growth on crystal planessubject to high shear stresses

-Crack changes direction from grain tograin following planes of highest shearstress

Stage 2:-Crack growth independent ofcrystallographic orientation

-Propagation rate much faster than

stage 1

Stage I Stage II

Stage I slip

line cracks

Crack PropagationStages 1 and 2:

In reality stage 1 only occupies a small portion of thefracture surface

Crack Propagation:SEM image

Nickel alloy (Inconel 718), Fatigue tested at 600 C

C.P.

C.P.Fatigue crack propagation surface

revealing stage 1 of crack propagation:

Crack growth changes direction in

adjacent grains following crystal planes

on which the shear stress is highest

Striations: increments of crack growthper loading cycle.

Only visible at high magnification (SEM)


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Fatigue corrosionof a shot-peened automotive

suspension spring

Crack initiated at peened layer-core interface

High Cycle: Fatigue Fracture Surfaces

Fatigue fracture surfaces are typified

by the schematic diagram below

Fast fracture face

(rough surface)

ac

origin

fatigue

fracture

face

(smooth)

Beach marks

Radials

Crack growth from origin, typicallysurface defect

Beach marksare visible with naked eye,

caused by periodic crack arrest orminor changes in crack direction

Radialsmay be visible, caused by joiningtogether of lengths of crack fronton differentcrystal planes

Striations are present between the beachmarks but only visible at high magnification

Crack grows from origin, across the section, until it reaches critical size (a c),then final fracture occurs

Final fracturesurface


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Fatigue failure surfaces

Fatigue failure in rotating shaft

Originating at stress concentration

Crack growth across > 90% of section

Eventual ductile overload (rupture)

Shaft was subject to low stresses (small

final overload area)

Fatigue failure in rotating shaft originating at surface

Crack growth across 10% of section

Ductile overload failure surface covers90% of section

Shaft was subject to high stresses (largefinal overload area)

Crack Tip Behaviour-High Cycle

Small plastic zone at crack tip

Towards max crack openingreaches max displacement, locallyexpanding the plastic zone

This creates new crack surfacearea

As cycle reverses crack goes intocompression and the newlyformed crack surface folds forwardadvancing the crack tip

Maximum applied stress belowyield stress

At crack tip local stress exceedsyield


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Crack Tip Behaviour

-Low Cycle

Large plastic zone at crack tip

Inclusions in large crack tip plasticzone separate from surroundingmetal grain

Void nucleation & growth as stressreaches max

Voids coalesce to extend crack

Maximum applied stress aboveyield stress Whole sample is plastic in highest

stress part of the cycle

Miners Rule: Cumulative damage An empirical method to assess the fatigue life of a component that is stressed at

several stress amplitudes

Miners Rule expresses the number of cycles at a particular stress amplitude (n) as afraction of the Fatigue life (N) at that stress amplitude

1i fi

i

N

n

For a stress spectrum compose of 3 stress amplitudes 13

3

2

2

1

1

++

Nn

N

n

N

n

n1 = Numbers of cycles at stress amplitude level 1

N1 = Fatigue life at stress amplitude level 1etc

It can been viewed that is the fraction of the fatigue life used up at stress

amplitude level 1 1

1

N

n

Miners Rule should not be used for Critical components without experimental testing

in the actual service atmosphere

It does not take into account work-hardening in the plastic zone ahead of the crack;Stresses that in isolation would cause damage, may not do so if the follow a higher stressamplitude that has caused crack tip work hardening


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Calculation An alloy tie bar is subject to a regular stress cycle at 1 Hz, maximum stress = 300 MPa,

minimum stress = 50 MPa. Alloy tensile strength = 900 MPa

If the endurance limit a0 (for a failure probability of P = 0.001) at 107 cycles = 450 MPa

1. Calculate the allowable stress amplitude

2. How long will it take to reach its maximum safe service life

Fracture Mechanics ApproachFracture MechanicsModern heavily analytical method based on stress intensity factor (K ) and Plain StrainFracture Toughness (K1c)

Paris Law:

)...( aYK =

mKA

dN

da= .

Crack growth rate

A and m are experimentallydetermined material constants

m = 2 - 7

Change in stress intensity factor

over the stress cycle

Can deal with complex geometries via use of Y factor,

Although for complex shapes, Y needs to be determined by FE methods

Can calculate remaining fatigue life of cracked components

Complex but use is justified for critical components( )

=

f

i

a

a

mm

mm

da

aYA

N

22

1

= max - minaYK ... =

Steady state

Crack growthregime


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Crack Growth Rate:

Can determine from fatigue

fracture surface

Scanning electron microscope

(SEM) analysis can measure the

increment of crack growth per

cycle, the distance between

subsequent striations

dN

da:Crack growth per loading cycle

Crack path follows path ofleast resistance through material.Lengths of crack front

on adjacent planes can join togetherCrack growth surface from austenitic

stainless steel screw fastener

dN

da

dN

da

Crack growth rate vs. K

mKA

dN

da= .

For steel shownA = 1.62 x 10-12 m.(MN.m-3/2)3.2

m = 3.2

A and m are experimentallyderived material constants

Steady state

growth

Kth

Kth = Threshold Kbelow which nocrack growth occurs

A B C

Crack growth regimes: A, B,C

A: SlowB: Steady stateC: HighNote:

Origin

suppressed


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Crack Growth in Regime A

(around threshold)

Concept of the threshold stress intensity factor Kth:

When K is close to Kth, the rate of crack growth is so slow that thecrack is often assumed to be growing in an undetectable rate

An operational definition for Kth often used is that if the rate of crackgrowth is 10-8 mm/cycle (10-11m/cycle) or less the conditions are

assumed to be at or below Kth

An important point is that these extremely slow crack growth rates representan average crack advance of less than one atomic spacing per cycle !?

How is this possible? What actually occurs is that there are many cycleswith no crack advance?

What actually occurs is that there are many cycles with no crack advance

Crack Growth in Regime B(Paris Law: steady state)

aYK ... =

mKA

dN

da= .

mKA

dadN

=

.

In regime B, the crack growth can be described by:

Re-arranging & Integrating this expression enables, Nf (cycles to failure) to be calculated

22 ....

1mm

mmaY

a

A

Substitute

Nf=

Nf =

f

o

a

a

mm

mm a

a

YA 22

.

.....

1

Integrate between initial crack size (ao) and final (critical) crack size, af

daa

YA

f

o

a

a

m

m

mm

2

2

.

.....

1

==

21

21

.

.....

1 21

21

2

m

a

m

a

YA

m

o

m

f

m

mm

Nf =

21

.

.....

12

1

02

1

2

m

aa

YA

mm

f

m

mm


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Crack Growth in Regime B

(Paris Law: steady state)

If m>2 If m= 2

Nf =

21

.

.....

12

1

02

1

2m

aa

YA

mm

f

m

mm

022

ln.....

1

a

a

YA

f

Nf =

If min is compressive (i.e. negative), use min = 0(as crack will not open in compression)

Often a0 = crack detection confidence limit and a f = critical crack size for fast fracture (ac)

12

=

Y

Ka

ICcRemember

Example

A structural component made from a high strength Al alloy is subject to cyclic loading,with max = 210 MPa and min = 70 MPa: K1c = 25 MN.m

-3/2

The NDT detection limit = 2 mm and Y = 1.2

Calculate the number of cycles to failure assuming that there is a Edge crack presentat the limit of detection

Given:

Units of A = 6.87 x10-12 m. (MN.m-3/2)-3

312.1087.6 Kx

dN

da=


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Crack Detection and Fracture

How do we determine the initial crack length?

Cracks can detected using various techniques, ranging from simplevisual inspection to more sophisticated techniques based on ultrasonicor X-rays. If no cracks are detected by our inspection, we must assumethat a crack just at the resolution of our detected system exists.

How do we determine the final crack length?

We know that eventually the crack can grow to a length at which the

material fails immediately, i.e. Kmax K1c

A very important principle that comes from this analysis: even if acomponent has a detected crack, it need not be removed from service!Using this frame work, the remaining life time can be assessed!

This is called crack-tolerant or damage-tolerant design approach.

12

=

Y

Ka

ICc

Fatigue Design Determine typical service stress spectra

Estimate useful fatigue life based on laboratory tests or analysesplus add factor of safety

At the end of the expected life, the component is retired from service, evenif no failure has occurred and the component has considerable residual life

Emphasis on prevention of crack initiation (good quality material, surfacefinish, surface treatment, design to eliminate stress concentrations)

For critical applications design should include multiple load paths so even ifan individual member of a component fails, there should be sufficientstructural integrity to operate safely

Mandatory periodic inspection


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S/N vs. Fracture Mechanics

S/N curves have been used formany years and much data havebeen collected.

It is simple to use, provided all thenecessary safety factors areknown.

It is a widely accepted andunderstood technique.

However, it can lead to veryconservative use of material dueto the many safety factorsinvolved.

Its major drawback is that itcannot satisfactorily deal withcracked components and thuscannot be used to assess theremaining life of crackedcomponents or set defectdetection limits for inspection.

The LEFM approach is relatively

new and thus materials data mightnot be readily available.

It is not easy to use as itsprinciples must be wellunderstood if errors are not to bemade.

The Y factor for complex shapesmust be calculated by finite orboundary element techniques.

However it can give designs whichmake optimum use of material. Itcan deal with crackedcomponents both by assessingremaining fatigue life and by

setting defect detection limits.

In general the S/N method ischeaper and quicker and is usedfor inexpensive and non-criticalstructures while the moreexpensive and time consumingLEFM approach is used for safetycritical or expensive components.

AppendixRyan Crash

Coventry Air Show 2003

Replica of Spirit of StLouis crashes justafter take off

Right wing foldsbackward during agentle right turn at 300ft

Prior to take of aircraftwas subject to buffetingdue to propwash fromadjacent larger aircraft


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Tubular steel wishbonestrut (in red) mountedfrom lower edge offuselage

braced to upper fuselageby diagonal bracing strut(yellow)

locates & supports upperend of right landing gearshock strut (blue)

Also supports lower endof forward wing strut(green)

Wishbone strut is welded

at its apex.

strut extends outboardbeyond apex where it iswelded to diagonalbracing strut that is itselfattached to upper

fuselage

What Happened?


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Fracture Surfaces

Fatigue crack originated in weld in vicinity ofwishbone apexWeld was exposed & painted with primer

Weld metal adjacent to failure location showedevidence of slag inclusions, shrinkageporosity, gas porosity & oxide inclusions

Out-board strut section failure surface

In-board strut section failure surface

AppendixLow cycle Fatigue

fracture of bicycle gearderailleur pulley

Fatigue striations on fracturedthread on bike gear connector

Fatiguefailure @

threaded

hole


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Striations visible propagated on failed thread.Many inclusion dimples plus visible striations

Higher mag.

Final overload

Striations plus secondary cracking in Al bike gear connector threaded hole


25/25

Affect of stress amplitude on fatigue

striations (in Al alloy)

Al alloy (7475): Increase in striation spacing due to

increase in applied stress amplitude Al alloy: centre of image shows a band ofmore widely spaced striations due to 45 cyclesof higher stress amplitude loading

High cycle fatigue of un-cracked components

High cycle fatigue life (Nf) isrelated to cyclic stress range ()by the Basquin equation

Where b and C1 are experimentally

derived constants

Typical b values 0.07-0.13

a = (/2)

Dividing by the Elastic modulusand taking logs gives:

1. CNbf =

E

CNb

fel1logloglog +=

High cycle fatigue behaviour of un-cracked components is controlled by crack initiation

This relationships applies to:Un-cracked components subject to high cycle fatigue (max < y)Stressed at constant stress amplitude about a mean stress of zero

pdf_m11ekm fatigue and fracture_autumn 2010

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