strength, fatigue and fracture -...

Factors Influencing Fatigue Mean Stress

Professor Stephen D. Downing Department of Mechanical Science and Engineering

University of Illinois at Urbana-Champaign

© 2011 Darrell Socie, All Rights Reserved

Fatigue and Fracture (Basic Course )

Factors Influencing Fatigue © 2011 Darrell Socie, All Rights Reserved 1 of 108

Factors Influencing Fatigue

Mean Stress Variable Amplitude Stress Concentrations Surface Finish


Mean Stresses

mean stress

stress range

stre

ss

2SSS minmax

mean+

=

Smax

Smin

max

min

SSR =


General Observations

Tensile mean stresses reduce the fatigue life or decrease the allowable stress range

Compressive mean stresses increase the fatigue life or increase the allowable stress range


Mechanism

Fatigue damage is a shear process

∆S

Tensile mean stresses open microcracks and make sliding easier

2Smean


Goodman 1890

Mechanics Applied to Engineering John Goodman, 1890

“.. whether the assumptions of the theory are justifiable or not …. We adopt it simply because it is the easiest to use, and for all practical purposes, represents Wöhlers data.

Sultimate = Smin + 2 ∆S


Goodman Diagram

Mean stress

Alte

rnat

ing

stre

ss

Su 0

Se

107 cycles

105 cycles

−

∆

=∆

−= ultimate

mean

1R SS1

2S

2S

R = -1 R = 1


Test Data ( 1941 )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.2

0.4

0.6

0.8

1.0

1.2

0

Mean Stress Ultimate Strength

Alte

rnat

ing

Stre

ss

Fatig

ue L

imit

J.O. Smith, The Effect of Range of Stress on the Fatigue Strength of Metals, Engineering Experiment Station Bulletin 334, University of Illinois, 1941


Compression

Se

Su 0 R = -1 R = 1

-Su R = - ∞

no influence

detrimental

beneficial


Modified Goodman ( no yielding )

Se

Su 0 R = -1 R = 1

-Su R = - ∞

Sys

Sys

-Sys

ysmean SS2S

<+∆


Mean Stress Influence on Life

0.1

1

10

100

1000

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 Mean Stress

Ultimate Strength

Rel

ativ

e Fa

tigue

Life


Stress Concentrations

Plastic Zone

∆ε

∆ε The elastic material surrounding the plastic zone around a stress concentration forces the material to deform in strain control


Mean Stresses at Notches

σ

ε

σ

ε

σ

ε

elastic plastic

Nominal Notch

Notch Nominal

Nominal mean stress is less than notch mean stress

Nominal mean stress is greater than notch mean stress

Nominal Notch


Morrow Mean Stress Correction

cf

'f

bf

mean'f )N2()N2(

E2ε+

σ−σ=

ε∆

10-5

10-4

0.001

0.01

0.1

1

Reversals, 2Nf

Stra

in A

mpl

itude

100 101 102 103 104 105 106 107

Emeanσ


Smith Watson Topper

σ

ε

∆ε

σmax cb

f'f

'f

b2f

2'f

max )N2()N2(E2

+εσ+σ

=ε∆

σ


Mean Stress Relaxation

Stadnick and Morrow, “Techniques for Smooth Specimen Simulation of Fatigue Behavior of Notched Members” ASTM STP 515, 1972, 229-252


Loading Histories


Test Results


Crack Growth Physics

σ

σ Maximum load

Minimum load

σ

ε

Mean stresses in plastic zone are small


Mean Stress Effects

From: Dowling and Thangjitham, An Overview and Discussion of Basic Methodology for Fatigue, ASTM STP 1389,2000, 3-38

( )γ−∆

=R1KC

dNda m

0 < γ < 0.5


Compression

Crack open

Crack closed


Compressive Stresses

Compressive stresses are not very damaging in crack growth

Stre

ss Crack opening level


Sources of Mean/Residual Stress

Loading History Fabrication Shot Peening Heat Treating


Loading History

Tension overloads produce favorable compressive residual stress

Compressive overloads produce unfavorable tensile residual stress

σ

ε

σ

ε


Fabrication


Cold Expansion

1965 Basic Cx process conceptualized (Boeing) The split sleeve is

slipped onto the mandrel, which is attached to the hydraulic puller unit.

The mandrel and sleeve are inserted into the hole with the nosecap held firmly against the workpiece.

When the puller is activated, the mandrel is drawn through the sleeve radially expanding the hole.

Courtesy of Fatigue Technology Inc.


Theory of Cold Expansion



Fatigue Life Improvement


100

200

300

0 108 107 106 105 104 103

Nom

inal

Stre

ss

Fatigue Life

Cold Expanded


Shot Peening

-600

-400

-200

0

200

0 0.2 0.4 0.6 0.8 Depth (mm)

Res

idua

l Stre

ss (M

Pa)

Residual stress in a shot peened leaf spring


Shot Peening Results

www.metalimprovement.com

500

1000

1500

500 0

0 1000 1500 2000 2500 Fatig

ue s

treng

th a

t 2x1

06 c

ycle

s, M

Pa

Ultimate Tensile Strength, MPa

2SS u

FL =

Smooth

Notched

Shot Peened Smooth & Notched


Heat Treating

200

600

400

0 0 5 10 15

Brin

ell H

ardn

ess

Depth, mm 0 5 10 15

-1500

-1000

-500

0

500

1000

Depth, mm R

esid

ual S

tress

, MP

a

Radial

Circumferential

Axial

50 mm diameter induction hardened 1045 steel shaft


Things Worth Remembering

Local mean stress rather than the nominal mean stress governs the fatigue life

Mean stress has the greatest effect on crack nucleation


Variable Amplitude Loading

How to you identify cycles ?

How do you assess fatigue damage for a cycle ?


Rainflow Cycle Counting

What could be more basic than learning to count correctly?

Matsuishi and Endo (1968) Fatigue of Metals Subjected to Varying Stress – Fatigue Lives Under Random Loading, Proceedings of the Kyushu District Meeting, JSME, 37-40


Rainflow strain

A

F

D

B

E

I, A H G

C

AD

BC

CB

DA

Counts 1/2 cycles


Rainflow and Hysteresis A

C B

D E F G

H I

ε

σ


Cumulative Damage

…. ….

…. ….

High - Low

Low - High nH nL

nL nH

∆SL

∆SH


Linear Damage

∆SH

∆SL

Nf H Nf L

Lf

L

Hf

H

F Nn

Nn

NnDamage +==∑

Miner’s Rule:


Nonlinear Damage

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0 0

Cycle ratio,

Dam

age

fract

ion

n Nf

0040.=ε∆

0200.=ε∆

ΣD = 0.7

ΣD = 1.3

ΣD ~ 1


Periodic Overload Results

10 100 103 104 105 106 107 108 109

10-3

10-4

0.01

0.1

Stra

in A

mpl

itude

Fatigue Life

….

….

The fatigue limit is reduced by a factor of 3 when a few large cycles are applied

Bonnen and Topper, “The Effects of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel” ASTM STP 1387, 2000, 213-231


Fatigue Damage Calculations

100

1000

10000

Stre

ss A

mpl

itude

, MP

a

100

Cycles 101 102 103 104 105 106 107

( )bf'f NS

2S

=∆

1

10

b1

'f

f S2SN

∆=

10SDamage ∆∝


Crack Growth Data

10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ck G

row

th R

ate,

m/c

ycle

mMPa,K∆

mKCdNda

∆=

∆KTH

Kc

1

3


Crack Growth Data

10-12 10-11 10-10 10-9 10-8 10-7 10-6

Crack Growth Rate, m/cycle

1

10

100

m

MP

a , K

∆

3SDamage ∆∝

( )2/m1SC

aaN2m

m

2/m1i

2/m1f

f

−π∆

−=

−−


Multiple Choice

Which cycles do the most fatigue damage ? (a) a few large cycles (b) a moderate number of intermediate cycles (c) a large number of small cycles


Fatigue Data

1

10

100

Am

plitu

de

100

Cycles 101 102 103 104 105 106 107

1000

n = 10

n = 5

n = 3 nSDamage ∆∝


Loading History

Time (Secs)

50 100 150 200 250 300

750

500

250

0

-250

-500

-750

Stra

in G

age

(ust

rain

)

Bracket.sif-Strain_b43

0

1500

-750

750

0

750 C

ount

s

rf_000.sif-Strain_b43


Slope = 3

0

1500

-750

750

3.15

% d

amag

e

Damage


Slope = 5

0

1500

-750

750

5.14

% d

amag

e

Damage


Slope = 10

0

1500

-750

750

20.78

% d

amag

e

Damage


Mechanisms and Slopes

100

103

104

Stre

ss A

mpl

itude

, MPa

100

Cycles 101 102 103 104 105 106 107

1 10

10-12 10-11 10-10 10-9 10-8 10-7 10-6

Crack Growth Rate, m/cycle

1

10

100

m

MPa

,

K ∆

1

3

Crack Nucleation

Crack Growth 10

100

103 104 105 106 107 108

Total Fatigue Life, Cycles

1

5 1

Equ

ival

ent L

oad,

kN

Structures

A combination of nucleation and growth


Equivalent Load

n

N

1i

ni

N

SS

∑=

∆=∆

Equivalent constant amplitude loading

Typically n ranges from 4 to 6 for structures

N cycles at an amplitude of does as much damage as the entire loading history

S∆


SAE Keyhole Specimen

Suspension

Transmission

Bracket


SAE Keyhole Test Data ManTen RQC 100

10

100

103 104 105 106 107 108


1

Suspension Transmission

Bracket

Constant Amplitude

5 1

Equi

vale

nt L

oad,

kN



Rainflow counting is employed to identify cycles

The slope of the fatigue curve ( damage mechanism) has a large influence on how much damage is caused by smaller cycles


Stress Concentration Factor

Applied stress Local stress

ρ+σ=σ

a21appliedlocal

Inglis Solution 1910

2a ρ


Kσ and Kε

Stre

ss (M

Pa)

Strain

KtS

Kte

σ ε

SK σ

=σ

eK ε

=ε


Neuber’s Rule St

ress

(MPa

)

Strain

KtS

Kte

σ

ε εσ=eKSK tt

Stress calculated with elastic assumptions

Actual stress


Neuber’s Rule for Fatigue

2222eKSK tt ε∆σ∆

=∆∆

Stress and strain amplitudes

Elastic nominal stress

E2S

2e ∆

=∆

Substitute and rearrange

22E

2SKt

ε∆σ∆=

∆

The product of stress times strain controls fatigue life


A Dilemma

Stress analysis and stress concentration factors are independent of size and are related only to the ratio of the geometric dimensions to the loads

Fatigue is a size dependent phenomenon

How do you put the two together ?


Similitude


Fatigue of Notches

From Dowling, Mechanical Behavior of Materials, 1999

104 105 106 107 108

100

400

300

200

Fatigue Life

Nom

inal

Stre

ss, M

Pa

d d

Kt = 3.1

Kt = 3.1

Kf = 2.2

104 105 106 107 108

100

400

300

200

Fatigue Life

dd dd

Kt = 3.1

Kt = 3.1

Kf = 2.2


Notch Size

Kt Kf Kt

Kf

Large Notch Small Notch


Microstructure Size

Kt Kf

Kt Kf

Low Strength High Strength


Stress Gradient

Kt Kf

Kt

Kf

Low Kt High Kt


Kt vs Kf

0

2

4

6

8

10

0 2 4 6 8 10 Kt

Kf

Kf = Kt

Experiments

Effe

ctiv

e st

ress

con

cent

ratio

n

Calculated stress concentration


Peterson’s Equation

ρα

+

−+=

1

1K1K tf

mmMPa2070025.08.1

u

σ

=α

0

1

2

3

4

10-4 10-3 10-2 0.1 1 10 102 103

Kf

ρα

No effect when ρ << α

Full effect when ρ >> α


Peterson’s Constant

1000 500 2000 1500

0.1

0.03

0.7

0.3

Ultimate Strength, MPa

α, m

m


Static Strength

hole Kt = 2.5

slot Kt = 5

diamond Kt = 20

edge Kt = 20


1018 Steel Test Data

0

20

40

60

80

100

0 2 4 6 8 10

load

, kN

displacement, mm

edge diamond slot hole


Notched SN Curve

100

1000

10000

100

Cycles

Stre

ss A

mpl

itude

, MPa

101 102 103 104 105 106 107

Smooth specimen data

Notched specimen Kf

Stress concentrations are not very important at short lives


Fatigue of Notches

From Dowling, Mechanical Behavior of Materials, 1999

104 105 106 107 108

100

400

300

200

Fatigue Life

Nom

inal

Stre

ss, M

Pa

d d

Kt = 3.1

Kt = 3.1

Kf = 2.2

104 105 106 107 108

100

400

300

200

Fatigue Life

dd dd

Kt = 3.1

Kt = 3.1

Kf = 2.2


Crack Growth Data

10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ck G

row

th R

ate,

m/c

ycle

mMPa,K∆

mKCdNda

∆=

∆KTH

Kc

a212.1KTH ππ

σ∆>∆

Nonpropagating cracks


Frost Data nucleation fracture

Rotating bending Notched bar

Notched plate

1 3 5 7 9 11 13 15

1.0

0.8

0.6

0.4

0.2

0

Kt

S nom

inal

S fat

igue

lim

it

tK1 nonpropagating cracks

Frost, “A Relation Between the Critical Alternating Propagation Stress and Crack Length for Mild Steel” Proceedings of the Institute for Mechanical Engineers, Vol. 173, No. 35, 1959, 811-836


Significance

DKS th

flπ

∆=

For Kt > 4, the notch acts like a crack with a depth D

Kt does not play a role for sharp notches !

A stress concentration behaves like a crack once a stress concentration becomes large (Kt > 4)


Cracks at Notches

D

a a D + a

KtS S S

a << D a >> D


Stress Intensity Factors

0 0.1 0.2 0.3 0.4 0.5 0.6

1.0

0

3.0

2.0

Da

DSK

aSKK t π=

( )aDSK +π=


Cracks at Holes

20 1 1 22

Once a crack reaches 10% of the hole radius, it behaves as if the hole was part of the crack


Specimens with Similar Geometry

Kt = 2.4 Kt = 10.7

Ultimate Strength 780 MPa Yield Strength 660 MPa

25 25

2.5 2.5


Test Results

1

100

1000

Nom

inal

Stre

ss A

mpl

itude

1 10 100 103 104 105 106 107


Strength Limited

Crack Growth Dominated Fatigue Strength Dominated

Threshold Stress Intensity Dominated

Kt = 2.4 Kt = 10.7



Fatigue may be thought of as a failure of the average stress concept, consequently, fatigue usually begins at stress concentrators which are most frequently located on the surface

The severity of a stress concentrator in fatigue is size dependent

Small stress concentrators are more effective in high strength materials


Modern View of the Fatigue Limit The fatigue limit is the stress where a crack may nucleate but will not grow through the first microstructural barrier such as the grain size, pearlite colony size, prior austenite grain size, eutectic cell size or precipitate spacing.

Slip Bands Crack


Intrinsic Flaws

Little effect of surface pit because it is smaller than the grain size

Large effect of defect because it is larger than the grain size


Surface Finish Influence

Method Stress-Life Strain-Life

Crack Growth

Physics Crack Nucleation

Microcrack Growth Macrocrack Growth

Size 0.01 mm

0.1 - 1 mm > 1mm

Influence of Surface Finish

Strong Moderate

None


Sources of Surface Effects Machining

Cutting Grinding

Corrosion General Pitting

Processing Cutting/Shearing Casting Forging Plating

Foreign Object Damage Nicks Scratches


Machining

σ1

Cracks start in machining marks not in the direction of the maximum principal stress

100 µm


Casting

100 µm

Surface flaw in gray cast iron


Nodular Iron Surface

Flake graphite formed on the surface of a nodular iron casting

Starkey and Irving, “A Comparison of the Fatigue Strength of Machined and As-cast Surfaces of SG Iron” International Journal of Fatigue, July, 1982, 129-136


Test Data 10-1

10-2

10-3

10-4

102 103 104 105 106 107 108 10 1

Stra

in A

mpl

itude

Fatigue Life, Reversals

Cast Surface

Machined Surface


Surface Reduction Factors

400 600 800 1000 1200 1400 1600


0.2

0.4

0.6

0.8

1.0

0

Surfa

ce F

acto

r Polished

Ground

Forged

Hot Rolled

Machined

5.0

2.5

1.7

1.2

1.0

Fatig

ue N

otch

Fac

tor


Noll and Lipson 1945

0

200

400

600

800

1000

400 600 800 1000 1200 1400 1600 1800 200 0 2000


Fatig

ue L

imit,

MPa

Ground

Machined

Hot Rolled Forged


Hiam and Pietrowski 1978

Driven for 1 or 2 years in Southern Ontario before making specimens to evaluate corrosion effects

Hiam and Pietrowski, “The Influence of Forming and Corrosion on the Fatigue Behavior of Automotive Steels”, SAE Paper 780040, 1978

Strain controlled fatigue testing


Kf for pitting

Hot RolledSurface

CorrodedSurface

950X 1.12 1.49

0.06% C HSLA 1.18 1.65

0.18% C HSLA 1.90

Surface finish factor predicts Kf = 1.6 for a Hot Rolled Surface

from Hiam and Pietrowski


Pit Depth Effects on Life 106

105

104

0.1 0.3 0.2 0 Pit Depth, mm

Fatig

ue L

ife, C

ycle

s

from Hiam and Pietrowski

950X Steel


Fatigue Notch Factor for Pits

1.0

0.1 0.3 0.2 0 Pit Depth, mm

1.1

1.2

1.3

1.4

1.5

Kf


Suspension


Spring Failures


Microscopic Examination

Corrosion Pits

Corrosion Pits


Chrome Plating

105 107 106 108 104

Stre

ss A

mpl

itiud

e, M

Pa

500

200

400

300

Life, Cycles

Almen, “Fatigue Loss and Gain by Electroplating” , Product Engineering, Vol. 22, No. 5, 1951, 109-116

Chrome plated

Shot peened and chrome plated

Base steel


Hard Chrome Plating

coating

Metals Handbook, Volume 9, Fractography and Atlas of Fractographs

In addition to cracks, coatings frequently have high tensile residual stresses


Galvanized Steel

3

0.5 0

1

2

1

225 MPa

305 MPa

240 MPa

Life Fraction

Cra

ck D

ensi

ty m

m-1

Vogt, Boussac, Foct, “Prediction of Fatigue Resistance of a Hot-dip Galvanized Steel” Fatigue and Fracture of Engineering Materials and Structures, Vol. 23, No. 1, 2001,33-40


Fatigue Limit for Galvanized Steel

100 10 100 1000

200

300

400

Fatig

ue L

imit

Coating Thickness, t µm

Uncoated Fatigue Limit

tKS TH

FLπ

∆=

Coatings can be modeled with a crack equal to the coating thickness


Anodized Aluminum

Rateick et. al. “Relationshipp of Microstructure to Fatigue Strength Loss in Anodized Aluminum-Copper Alloys” Aeromet 2004, June 2004


Pitting at Cu Rich Constituent


Upper Control Arm


Serial Number



Fatigue crack nucleation is a surface phenomena and everything about the surface affects the fatigue life

Most of the design rules are conservative having been developed for materials of the 1950’s

Fatigue and Fracture ( Basic Course )

strength, fatigue and fracture -...

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