Multiaxial Fatigue Contact Information - Illinoisfcp.mechse.illinois.edu/files/2015/10/Multiaxial-Fatigue.pdf · Multiaxial Fatigue Multiaxial Fatigue © 2003-2008 Darrell Socie,

1

Introduction

Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign

© 2003-2008 Darrell Socie, All Rights Reserved

Multiaxial Fatigue

Multiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 202

Contact Information

Darrell SocieMechanical Science and Engineering1206 West GreenUrbana, Illinois 61801

Office: 3015 Mechanical Engineering Laboratory

[email protected]: 217 333 7630Fax: 217 333 5634


When is Multiaxial Fatigue Important ?

Complex state of stress

Complex out of phase loading


Uniaxial Stress

one principal stressone direction

X

Z

Y


Proportional Biaxial

principal stresses vary proportionally but do not rotate

X

Z

Y1 = 2 = 3


Nonproportional Multiaxial

Principal stresses may vary nonproportionally and/or change direction

X

Z

Y

2


Crankshaft

Time

-2500

2500

90

45

0M

icro

stra

in

0


Shear and Normal Strains

-0.0025 0.0025

-0.005

0.005

-0.0025 0.0025

-0.005

0.005

0º 90º


Shear and Normal Strains

-0.0025 0.0025

-0.005

0.005

-0.0025 0.0025

-0.005

0.005

45º 135º


3D stresses

0.004 0.008 0.0120

0.001

0.002

0.003

0.004

0.005

50 mm

30 mm

15 mm

7 mm

Longitudinal Tensile StrainT

rans

vers

e C

ompr

essi

on S

trai

n

Thickness

100

x

z

y


Notch Stresses

t x z x z

7 0.01 -0.005 63.5 0

15 0.01 -0.003 70.6 14.1

30 0.01 -0.002 73.0 21.8

50 0.01 -0.001 75.1 29.3

x

z

y


Outline

State of Stress

Stress-Strain Relationships

Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models

Fracture Mechanics Models

Nonproportional Loading

Stress Concentrations

3


Book


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





State of Stress

Stress components

Common states of stress

Shear stresses


Stress Components

X

Z

Y

z

y

x

zyzx

xz

xyyx

yz

Six stresses and six strains


Stresses Acting on a PlaneZ

Y

X

Y’

X’Z’

x’

x’z’

x’y’


Principal Stresses

3 - 2( X + Y + Z ) + (XY + YZXZ -2XY - 2

YZ -2XZ )

- (XYZ + 2XYYZXZ - X2YZ - Y2

ZX - Z2XY ) = 0

1

3

2

13

1223

4


Stress and Strain Distributions

80

90

100

-20 -10 0 10 20

% o

f app

lied

stre

ss

Stresses are nearly the same over a 10° range of angles


Tension

x

1

1 2

3

2 = 3 = 0

1

231


Torsion

2

xy1

3

2

1

3

X

Y2

1 = xy

3

1


2 y

1 x

3

1 2

3

1 2

1

23

Biaxial Tension


1

2

3 3

2

1

Maximum shear stress Octahedral shear stress

Shear Stresses

231

13

232

221

231oct 3

1

oct2

3Mises: 1313oct 94.0

22

3


Maximum and Octahedral Shear

0.2

0.4

0.6

0.8

1.0

-1 -0.5 0 0.5 1

18%

6%

torsion tension biaxial tension

Octahedral shear

Maximum shear

3/1

5


Shear Stress Influence

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

1

1


State of Stress Summary

Stresses acting on a plane

Principal stress

Maximum shear stress

Octahedral shear stress


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





“About six months ago I wrote a paper, knowing that I should be very busy in the autumn and made a model to illustrate a point in it. But as I played with the model to learn how to use it, it grew too strong for me and took command and for the last six months I have been its obedient slave --- for the model explained the whole of my subject Fatigue.”

Jenkin 1922

“Fatigue in Metals,” The Engineer, Dec. 8, 1922


Elastic Stress Strain Relationships

x

y

z

xy

yz

xz

x

y

z

xy

yz

xz

E

1 1 2

1 0 0 0

1 0 0 0

1 0 0 0

0 0 01 2

20 0

0 0 0 01 2

20

0 0 0 0 01 2

2


von Misesyield surface

Tresca yieldsurface

Yield Surfaces

3xy

X

6


Equations

F

F

or

F

ys

ys

ys

1 1 3

2 1 2

3 2 3

0

0

0

F x y x y xy ys 2 2 2 23 0

Tresca

Mises


- planeyield cylinder axis

Mises Yield Surfaces

32 ys


Initial yield surface

Yield surface after loading

Isotropic Hardening

3


Cyclic Loading

stab

ilize

d st

ress

ran

ge

Strain Control Stress Control


Kinematic Hardening

3

A

B

C

B

A y

B y


Ratcheting

Cyclic torsion with a mean tension stress

7


Cyclic Mean Stress Relaxation

Cyclic strain with mean stress


Cyclic Creep

Cyclic stress with a mean stress


Plasticity Models

a

b

c

f

d

e

a

c

b

d

e

f

333

3 3 3


Why is modeling needed?

A B C D

A

B

CD

time

*

*


Flow and Hardening Rules

3

dij

n

ij

ij

dij

d dF

ijp

ij

F ijp f g( ) ( ) 0

Flow rule

Hardening rule


Multiaxial Example

-.006

0.003

x

-0.003

0.006xy

-500 500

x

-250

250xy

8


Multiaxial Example

-0.006 0.006xy

-250

250xy

-0.003 0.003x

-500

500x


Summary

Isotropic Hardening

Kinematic Hardening

Cyclic creep or ratcheting

Mean stress relaxation

Nonproportional hardening


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





10-10 10-8 10-6 10-4 10-2 100 102

Specimens StructuresAtoms Dislocations Crystals

Size Scale for Studying Fatigue

Understand the physics on this scale

Model the physics on this scale

Use the models on this scale


The Fatigue Process

Crack nucleation

Small crack growth in an elastic-plastic stress field

Macroscopic crack growth in a nominally elastic stress field

Final fracture


1903 - Ewing and Humfrey

Cyclic deformation leads to the development of slip bands and fatigue cracks

N = 1,000 N = 2,000

N = 10,000 N = 40,000 Nf = 170,000Ewing, J.A. and Humfrey, J.C. “The fracture of metals under repeated alterations of stress”, Philosophical Transactions of the Royal Society, Vol. A200, 1903, 241-250

9


Crack Nucleation


Slip Band in Copper

Polak, J. Cyclic Plasticity and Low Cycle Fatigue Life of Metals, Elsevier, 1991


Slip Band Formation

Loading Unloading

Extrusion

Undeformedmaterial

Intrusion


Slip Bands

Ma, B-T and Laird C. “Overview of fatigue behavior in copper sinle crystals –II Population, size, distribution and growthKinetics of stage I cracks for tests at constant strain amplitude”, Acta Metallurgica, Vol 37, 1989, 337-348


2124-T4 Cracking in Slip Bands

N = 60

N = 2000N = 1200

N = 300N = 240


2219-T851 Cracked Particle

10mJames & Morris, ASTM STP 811 Fatigue Mechanisms: Advances in Quantitative Measurement of Physical

Damage, pp. 46-70, 1983.

10


Initiation in Ti

Slip bands Hard alpha inclusion


Crack Initiation at Inclusions

Langford and Kusenberger, “Initiation of Fatigue Cracks in 4340 Steel”, Metallurgical Transactions, Vol 4, 1977, 553-559


Subsurface Crack Initiation

Y. Murakami, Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions, 2002


Fatigue Limit and Strength Correlation


Crack Nucleation Summary

Highly localized plastic deformation

Surface phenomena

Stochastic process


100 µm

bulksurface

10 µm

surface

20-25 austenitic steel in symmetrical push-pull fatigue (20°C, p/2= 0.4%) : short cracks on the surface and in the bulk

Surface Damage

From Jacques Stolarz, Ecole Nationale Superieure des MinesPresented at LCF 5 in Berlin, 2003

11


Stage I Stage II

loading direction

freesurface

Stage I and Stage II


Stage I Crack Growth

Single primary slip system

individual grain

near - tip plastic zone

S

S

Stage I crack is strongly affected by slip characteristics, microstructure dimensions, stress level, extent of near tip plasticity


Small Cracks at Notches

D acrack tip plastic zone

notch plastic zone

notch stress field

Crack growth controlled by the notch plastic strains


Small Crack Growth

1.0 mm

N = 900

Inconel 718 = 0.02Nf = 936


0

0.5

1

1.5

2

2.5

0 2000 4000 6000 8000 10000 12000 14000

J-603

F-495 H-491

I-471 C-399

G-304

Cycles

Cra

ck L

engt

h, m

m

Crack Length Observations


Crack - Microstructure Interactions

Akiniwa, Y., Tanaka, K., and Matsui, E.,”Statistical Characteristics of Propagation of Small Fatigue Cracks in Smooth Specimens of Aluminum Alloy 2024-T3, Materials Science and Engineering, Vol. A104, 1988, 105-115

10-6

10-7

0 0.005 0.01 0.015 0.02 0.0250.03 0.025 0.02 0.015 0.01 0.005

A B CD

F

E

Crack Length, mm

da/dN, mm/cycle

12


Strain-Life Data

Reversals, 2Nf

Str

ain

Am

plitu

de 2

10-5

10-4

0.01

0.1

1

100 101 102 103 104 105 106 107

10-3

10m100 m

1mmfracture Crack size

Most of the life is spent in microcrack growth in the plastic strain dominated region


Stage II Crack Growth

Locally, the crack grows in shear Macroscopically it grows in tension


Long Crack Growth

Plastic zone size is much larger than the material microstructure so that the microstructure does not play such an important role.


Material strength does not play a major role in fatigue crack growth

Crack Growth Rates of Metals


Maximum Load

monotonic plastic zone

Stresses Around a Crack


Stresses Around a Crack (continued)

Minimum Load

cyclic plastic zone

13


Crack Closure

S = 250

b

S = 175

c

S = 0

a


Crack Opening LoadDamaging portion of loading history

Nondamaging portion of loading history

Opening load


Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear

Mode I, Mode II, and Mode III


5 mcrac

k g

row

th d

ire

ctio

n

Mode I Growth


crack growth direction

10 m

slip bandsshear stress

Mode II Growth


1045 Steel - Tension

NucleationShear

Tension

1.0

0.2

0

0.4

0.8

0.6

1 10 102 103 104 105 106 107

Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf

100 m crack

14


Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf

f

Nucleation

Shear

Tension

1 10 102 103 104 105 106 107

1.0

0.2

0

0.4

0.8

0.6

1045 Steel - Torsion


1.0

0.2

0

0.4

0.8

0.6

1 10 102 103 104 105 106 107

Nucleation

TensionShear

304 Stainless Steel - Torsion

Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf


304 Stainless Steel - Tension

1 10 102 103 104 105 106 107

1.0

0.2

0

0.4

0.8

0.6 Nucleation

Tension

Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf


Nucleation

Tension

Shear

1.0

0.2

0

0.4

0.8

0.6

1 10 102 103 104 105 106 107

Inconel 718 - Torsion

Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf


Inconel 718 - Tension

Shear

Tension

Nucleation

1 10 102 103 104 105 106 107

1.0

0.2

0

0.4

0.8

0.6

Fatigue Life, 2Nf

Dam

age

Fra

ctio

n N

/Nf

Fatigue Made Easy

Similitude

Professor Darrell F. SocieMechanical Science and Engineering

University of Illinois

© 2002-2011 Darrell Socie, All Rights Reserved

15


Fatigue Analysis

MaterialData

ComponentGeometry

ServiceLoading

AnalysisFatigue

Life Estimate

?


The Similitude Concept

Why Fatigue Modeling Works !


What is the Similitude Concept

The “Similitude Concept” allows engineers torelate the behavior of small-scale cyclicmaterial test specimens, defined undercarefully controlled conditions, to the likelyperformance of real structures subjected tovariable amplitude fatigue loads under eithersimulated or actual service conditions.


Fatigue Analysis Techniques

Stress - Life

BS 7608, Eurocode 3

Strain - Life

Crack Growth


Life Estimation

MethodStress-LifeBS 7608

Strain-LifeCrack Growth

PhysicsCrack Nucleation

Crack GrowthMicrocrack GrowthMacrocrack Growth

Size0.01 mm

1 - 10 mm0.1 - 1 mm

> 1mm


Stress-Life Fatigue Modeling

P

FixedEnd

The Similitude Concept states that if theinstantaneous loads applied to the ‘test’structure (wing spar, say) and the testspecimen are the same, then the responsein each case will also be the same and canbe described by the material’s S-N curve.Due account can also be made for stressconcentrations, variable amplitude loadingetc.

100

1000

10000

Str

ess

Am

plitu

de, M

Pa

100

Cycles101 102 103 104 105 106 107

16


Fatigue Analysis: Stress-Life

MaterialData

ComponentGeometry

ServiceLoading

AnalysisFatigue

Life Estimate

SN curveKa, Ks, …

Kf

S , Sm


Stress-Life

Major Assumptions:Most of the life is consumed nucleating cracks

Elastic deformation

Nominal stresses and material strength control fatigue life

Accurate determination of Kf for each geometry and material


Stress-Life

Advantages:Changes in material and geometry can easily be

evaluated

Large empirical database for steel with standard notch shapes


Stress-Life

Limitations:Does not account for notch root plasticity

Mean stress effects are often in error

Requires empirical Kf for good results


BS 7608 Fatigue Modeling

The Similitude Concept states that if theinstantaneous loads applied to the ‘test’structure (welded beam on a bulldozer, say)and the test specimen (standard fillet weld)are the same, then the response in eachcase will also be the same and can bedescribed by one of the standard BS 7608Weld Classification S-N curves.

10

100

1000

105

Cycles

Str

ess

Ran

ge, M

Pa

108107106


Weld Classifications

D E

F2 G

17


Fatigue Analysis: BS 7608

MaterialData

ComponentGeometry

ServiceLoading

AnalysisFatigue

Life Estimate

Weld SN curve

Class

S


BS 7608

Major Assumptions:Crack growth dominates fatigue life

Complex weld geometries can be described by a standard classification

Results independent of material and mean stress for structural steels


BS 7608

Advantages:Manufacturing effects are directly included

Large empirical database exists


BS 7608

Limitations:Difficult to determine nominal stress and weld

class for complex shapes

No benefit for improving manufacturing process


Strain-Life Fatigue Modeling

The Similitude Concept states that if theinstantaneous strains applied to the ‘test’structure (vehicle suspension, say) and thetest specimen are the same, then theresponse in each case will also be the sameand can be described by the material’s e-Ncurve. Due account can also be made forstress concentrations, variable amplitudeloading etc.

10-5

10-4

0.001

0.01

0.1

1

Reversals, 2Nf

Str

ain

Am

plit

ude

100 101 102 103 104 105 106 107


Fatigue Analysis: Strain-Life

MaterialData

ComponentGeometry

ServiceLoading

AnalysisFatigue

Life Estimate

N curve curve

Kf

S , Sm

18


Strain-Life

Major Assumptions: Local stresses and strains control fatigue

behavior

Plasticity around stress concentrations

Accurate determination of Kf


Strain-Life

Advantages:Plasticity effects

Mean stress effects


Strain-Life

Limitations:Requires empirical Kf

Long life situations where surface finish and processing variables are important


Crack Growth Fatigue Modeling

The Similitude Conceptstates that if the stressintensity (K) at the tip of acrack in the ‘test’ structure(welded connection on an oilplatform leg, say) and thetest specimen are the same,then the crack growthresponse in each case willalso be the same and can bedescribed by the Parisrelationship. Account canalso be made for localchemical environment, ifnecessary.

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


Fatigue Analysis: Crack Growth

MaterialData

ComponentGeometry

ServiceLoading

AnalysisFatigue

Life Estimate

da/dN curve

K

S , Sm


Crack Growth

Major Assumptions:Nominal stress and crack size control fatigue life

Accurate determination of initial crack size

19


Crack Growth

Advantage:Only method to directly deal with cracks


Crack Growth

Limitations:Complex sequence effects

Accurate determination of initial crack size


Choose the Right Model

Similitude Failure mechanism

Size scale


Design Philosophy

Safe Life

Damage Tolerant


Safe Life

0

100

200

300

400

500

104 105 106 107 108 109

Str

ess

Am

plit

ude

, MP

a

Fatigue Life

99 90 11050

Percent Survival

0

100

200

300

400

500

104 105 106 107 108 109

Str

ess

Am

plit

ude

, MP

a

Fatigue Life

99 90 1105099 90 11050

Percent Survival

Choose an appropriate risk and replace critical partsafter some specified interval


Damage Tolerant

Inspect for cracks larger than a1 and repair

Cycles

Cra

ck s

ize

a1

a2

Safe Operating Life

Inspection

20


Inspection

A Boeing 777 costs $250,000,000

A new car costs $25,000

For every $1 spent inspecting and maintaining a B 777 you can spend only 0.01¢ on a car


Things Worth Remembering

Questions to askWill a crack nucleate ?

Will a crack grow ?

How fast will it grow ?

Similitude Failure mechanism

Size Scale


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





Fatigue Mechanisms Summary

Fatigue cracks nucleate in shear

Fatigue cracks grow in either shear or tension depending on material and state of stress


Stress Based Models

Sines

Findley

Dang Van


1.0

00 0.5 1.0 1.5 2.0

Shear stressOctahedral stress

Principal stress

0.5

She

ar s

tres

s in

ben

ding

1/2

Ben

ding

fat

igue

lim

it

Shear stress in torsion

1/2 Bending fatigue limit

Bending Torsion Correlation

21


Test Results

Cyclic tension with static tension

Cyclic torsion with static torsion

Cyclic tension with static torsion

Cyclic torsion with static tension


1.5

1.0

0.5

1.5-1.5 -1.0 1.00.5-0.5 0

Axi

al s

tres

s

Fat

igue

str

eng

th

Mean stressYield strength

Cyclic Tension with Static Tension


1.5

1.0

0.5

1.51.00.5

0

She

ar S

tres

s A

mpl

itude

She

ar F

atig

ue S

tren

gth

Maximum Shear StressShear Yield Strength

Cyclic Torsion with Static Torsion


1.5

1.0

0.5

3.00 2.01.0

Ben

ding

Str

ess

Ben

ding

Fat

igue

Str

eng

th

Static Torsion StressTorsion Yield Strength

Cyclic Tension with Static Torsion


1.5

1.0

0.5

1.5-1.5 -1.0 1.00.5-0.5 0

Tors

ion

shea

r st

ress

She

ar f

atig

ue s

tren

gth

Axial mean stress

Yield strength

Cyclic Torsion with Static Tension


Conclusions

Tension mean stress affects both tension and torsion

Torsion mean stress does not affect tension or torsion

22


Sines

)3(2 h

oct

)(

)(6)()()(6

1

meanz

meany

meanx

2yz

2xz

2xy

2zy

2zx

2yx


Findley

2

k n

maxf

tension torsion


Bending Torsion Correlation

1.0

00 0.5 1.0 1.5 2.0

Shear stressOctahedral stress

Principal stress

0.5

She

ar s

tres

s in

ben

ding

1/2

Ben

ding

fat

igue

lim

it

Shear stress in torsion

1/2 Bending fatigue limit


Dang Van

( ) ( )t a t bh

m

V(M)

ij(M,t) Eij(M,t)

ij(m,t)ij(m,t)


Isotropic Hardening

stab

ilize

d st

ress

ran

ge

Failure occurs when the stress range is not elastic


Oo

3

Co

Ro

CL

Oo

Loading path

Yield domain expands and

translates

a)

c) d)

b)3 3

3

11

11

2 2

2 2

Oo Oo

RL

OL

Multiaxial Kinematic and Isotropic

* stabilized residual stress

23


h

h

Loading path

Failurepredicted

taht= b

Dang Van ( continued )


Stress Based Models Summary

Sines:

Findley:

Dang Van: ( ) ( )t a t bh

)3(2 h

oct

2

k n

maxf


Model Comparison R = -1

Stress ratio,

Rel

ativ

e F

atig

ue L

ife

0.001

0.01

0.1

1

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Torsion Tension Biaxial Tension

Goodman

Findley

Sines


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





Strain Based Models

Plastic Work

Brown and Miller

Fatemi and Socie

Smith Watson and Topper

Liu


10 10 2 103 104 105

0.001

0.01

0.1

Cycles to failure

Pla

stic

oct

ahed

ral

shea

r st

rain

ran

ge Torsion

Tension

Octahedral Shear Strain

24


1

10

100

102 103 104

A

Fatigue Life, Nf

Pla

stic

Wor

k pe

r C

ycle

, M

J/m

3

T TorsionAxial0

90

180

135

45

30

T

A T

TT

TT

T

A

A

A A

A

Plastic Work


0.005 0.010.0

102

2 x102

5 x102

103

2 x103

Fa

tigu

e L

ife,

Cyc

les

Normal Strain Amplitude, n

= 0.03

Brown and Miller


Case A and B

Growth along the surface Growth into the surface


Uniaxial

Equibiaxial

Brown and Miller ( continued )


Brown and Miller ( continued )

max S n

1

max

', '( ) ( )

2

22 2

S A

EN B Nn

f n meanf

bf f

c


Fatemi and Socie

25


C F G

H I J

/ 3

Loading Histories


0

0.5

1

1.5

2

2.5

0 2000 4000 6000 8000 10000 12000 14000

J-603

F-495 H-491

I-471 C-399

G-304

Cycles

Cra

ck L

engt

h, m

m

Crack Length Observations


Fatemi and Socie

cof

'f

bof

'f

y

max,n )N2()N2(G

k12


Smith Watson Topper


SWT

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2


Liu

WI = (n nmax + (

b2f

2'fcb

f'f

'fI )N2(

E

4)N2(4W

WII = n n ) + (max

bo2f

2'fcobo

f'f

'fII )N2(

G

4)N2(4W

Virtual strain energy for both mode I and mode II cracking

26


Cyclic Torsion

Cyclic Shear Strain Cyclic Tensile Strain

Shear Damage Tensile Damage

Cyclic Torsion


Cyclic TorsionStatic Tension


Shear Damage Tensile Damage

Cyclic Torsion with Static Tension



Tensile DamageShear DamageCyclic TorsionStatic Compression

Cyclic Torsion with Compression


Cyclic TorsionStatic Compression

Hoop Tension


Tensile DamageShear Damage

Cyclic Torsion with Tension and Compression


Test Results

Load Case /2 hoop MPa axial MPa Nf

Torsion 0.0054 0 0 45,200 with tension 0.0054 0 450 10,300

with compression 0.0054 0 -500 50,000 with tension and

compression 0.0054 450 -500 11,200


Conclusions

All critical plane models correctly predict these results

Hydrostatic stress models can not predict these results

27


0.006Axial strain

-0.003

0.003

Sh

ear

stra

in

Loading History


Model Comparison

Summary of calculated fatigue lives

Model Equation LifeEpsilon 6.5 14,060Garud 6.7 5,210Ellyin 6.17 4,450

Brown-Miller 6.22 3,980SWT 6.24 9,930Liu I 6.41 4,280Liu II 6.42 5,420Chu 6.37 3,040

Gamma 26,775Fatemi-Socie 6.23 10,350

Glinka 6.39 33,220


Strain Based Models Summary

Two separate models are needed, one for tensile growth and one for shear growth

Cyclic plasticity governs stress and strain ranges

Mean stress effects are a result of crack closure on the critical plane


Separate Tensile and Shear Models

2

1 =

xy

3

1

Inconel 1045 steel stainless steel


Cyclic Plasticity

p

p

pp


Mean Stresses

eqf mean

fb

f fc

EN N

''( ) ( )2 2

R1

2)()(W maxnnI

cf

'f

bf

n'f

nmax )N2()S5.05.1()N2(

E

2)S7.03.1(S

2

cof

'f

bof

'f

y

max,n )N2()N2(G

k12

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2

28


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models






Mode I growth

Torsion

Mode II growth

Mode III growth


Mode I, Mode II, and Mode III

Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear


Mode I

Mode II

Mode I and Mode II Surface Cracks


= -1 = 0 = 1

= -1 = 0 = 1

= 193 MPa = 386 MPa10-3

10-4

10-5

10-6

da/d

N m

m/c

ycle

10 100 20020 50 10 100 20020 50K, MPa m

10-3

10-4

10-5

10-6

da/d

N m

m/c

ycle

Biaxial Mode I Growth

K, MPa m


Mode II

Mode III

Surface Cracks in Torsion

29


Transverse Longitudinal Spiral

Failure Modes in Torsion


1500

1000

Yie

ld S

tren

gth

, M

Pa

40

30

50

60

Har

dnes

s R

c

200 300 400 500

Shear Stress Amplitude, MPa

No Cracks

SpiralCracks

Transverse Cracks

Longi-tudinal

Fracture Mechanism Map


10-6

10-5

10-4

10-3

da/d

N,

mm

/cyc

le

5 10 1005020KI , KIII MPa m

KI

KIII

Mode I and Mode III Growth


1 10 100

10-7

10-6

10-5

10-4

10-3

10-2

da/d

N,

mm

/cyc

le

m

Mode I R = 0Mode II R = -1

7075 T6 Aluminum

Mode I R = -1Mode II R = -1

SNCM Steel

Mode I and Mode II Growth

KI , KII MPa



meqKCdN

da

25.04III

4II

4Ieq )1(K8K8KK

5.02III

2II

2Ieq K)1(KKK

5.02IIIII

2Ieq KKKKK

a)EF())1(2

EF()(K

5.0

2I

2IIeq

ak1GFKys

max,neq


Fracture Surfaces

Bending Torsion

30


10-9

10-8

10-7

10-6

10-5

10-4

0.2 0.4 0.6 0.8 1.0Crack length, mm

Cra

ck g

row

th r

ate,

mm

/cyc

le K = 11.0

K = 14.9

K = 10.0

K = 12.0

K = 8.2

Mode III Growth


0 5 10 15 20 25

5

10

15

20

25

tensile growth

shear growthno growth

KI

K

II

K

K

Growth of Inclined Cracks

From: Otsuka et.al. Engineering Fracture Mechanics, Vol 7, 1975

1010 Steel

Cracks grow in either tension or shear


Otsuka

sinK

2

3

2cosK

2cosK II

2I

1cos3K

2sinK

2cos

2

1K III

Tensile growth:

Shear growth:


Strain Energy Density

2III33

2II22III12

2I11 KaKaKKa2KaS

S

at o 0

2

2 0S

at o

S a K K a K K K K

a K K a K K

o Imean

I o IImean

I Imean

II

o IImean

II o IIImean

III

2 11 12

22 33

[

]

( ) ( ) ( )

( ) ( )

Strain energy density at the crack tip:

Necessary and sufficient conditions for crack growth:

Cyclic strain energy density:

Sih, G.C and Barthelemy, B.M. “Mixed Mode Fatigue Crack Growth Predictions” Engineering Fracture Mechanics, Vol. 13, 1980


Fracture Mechanics Models Summary

Multiaxial loading has little effect in Mode I

Crack closure makes Mode II and Mode IIIcalculations difficult


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models




31



In and Out-of-phase loading

Nonproportional cyclic hardening

Variable amplitude


x

xy

y

t

t

t

tx osint

xy osint

In-phase

Out-of-phase

x

x

xy

xy

x ocost

xy osint

In and Out-of-Phase Loading


xy

22xy

x

x

x

x

xy

2xy

In-Phase and Out-of-Phase


x x

x x

xy/2 xy/2

xy/2xy/2 cross

diamondout-of-phase

square

Loading Histories


in-phase

out-of-phase

diamond

square

cross

Loading Histories


Findley Model Results

MPa n,maxMPa

n,maxN/Nip

in-phase 353 250 428 1.0

90 out-of-phase 250 500 400 2.0

diamond 250 500 400 2.0

square 353 603 534 0.11

cross - tension cycle 250 250 325 16

cross - torsion cycle 250 0 250 216

x

xy/2cross

x

xy/2diamondout-of-phase

x

xy/2square

in-phase

x

xy/2

32


Nonproportional Hardening

t

t

t

tx osint

xy osint

In-phase

Out-of-phase

x

x

xy

xy

x ocost

xy osint


600

-600

300

-300

-0.003 -0.0060.003 0.006

Axial Shear

In-Phase


Axial

600

-600

-0.003 0.003

Shear

-300

300

0.006-0.006

90° Out-of-Phase


600

-600

-0.004 0.004

Proportional

-600

600

-0.004 0.004

Out-of-phase

Critical Plane

Nf = 38,500Nf = 310,000

Nf = 3,500Nf = 40,000


0 1 2 3 4

5 6 7 8 9

1011 12 13

Loading Histories


-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600

-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600

Case 3Case 2Case 1

Case 4 Case 5 Case 6

Sh

ear

Str

ess

( M

Pa

)

Stress-Strain Response

33


Stress-Strain Response (continued)

-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600

-300

-150

0

150

300

-600 -300 0 300 600

-300

-150

0

150

300

-600 -300 0 300 600-300

-150

0

150

300

-600 -300 0 300 600

Case 7 Case 9 Case 10

Case 13Case 12Case 11

Sh

ear

Str

ess

( M

Pa

)

Axial Stress ( MPa )


1000

200102 103 104

Equ

ival

ent

Str

ess,

MP

a

2000

Fatigue Life, Nf

Maximum Stress

Nonproportional hardening results in lower fatigue lives

All tests have the same strain ranges


Case A Case B Case C Case D

x x x x

y y y y

Nonproportional Example


Shear Stresses on Planes

= /2

= /2

= 0


Shear Stresses on Planes

= /2

= /2

= 0


Combined

= /2

= /2

34


Case A Case B Case C Case D

xy xy xy xy

Shear Stresses


Variable Amplitude

Time

-2500

2500

90

45

0

Mic

rost

rain

0


Cycle Counting and Damage?

-0.0025 0.0025

-0.005

0.005


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, AH

G

C

AD

BC

CB

DA

Counts 1/2 cycles


Rainflow and Hysteresis

A

CB

D EF G

HI

35


Principal Stress Directions


Signed Equivalent Stress

2 2 2

eq 1 2 1 3 2 3 max

1sgn

2


Torsion Stresses

eq


Torsion Stresses

45


Real Data

-60

-40

-20

0

20

40

60


Signed Equivalent Stress

-100

-80

-60

-40

-20

0

20

40

60

80

100

36


-0.003 0.003

-0.006

0.006

-300 300

x

-150

150

x

Simple Variable Amplitude History


-0.003 0.003x

-300

300

-0.005 0.005xy

-150

150

xy

Stress-Strain on 0° Plane


-0.003 0.00330

-300

300

-0.003 0.003

-300

30030º plane 60º plane

60

Stress-Strain on 30° and 60° Planes


Stress-Strain on 120° and 150° Planes

-0.003 0.003

-300

300

-0.003 0.003

-300

300150º plane120º plane

120 150


time

-0.005

0.005

She

ar s

trai

n,

Shear Strain History on Critical Plane


Fatigue Calculations

Load or strain history

Cyclic plasticity model

Stress and strain tensor

Search for critical plane

37


Analysis model Single event16 input channels2240 elements

An Example

From Khosrovaneh, Pattu and Schnaidt “Discussion of Fatigue Analysis Techniques for Automotive Applications”Presented at SAE 2004.


Biaxial and Uniaxial Solution10-2

1 10 100 1000 10000

Element rank based on critical plane damage

Da

ma

ge

Uniaxial solutionSigned principal stress

Critical plane solution

10-3

10-4

10-5

10-6

10-7

10-8


Nonproportional Loading Summary

Nonproportional cyclic hardening increases stress levels

Critical plane models are used to assess fatigue damage


Outline

State of Stress


Fatigue Mechanisms

Multiaxial Testing

Stress Based Models

Strain Based Models





Notches

Stress and strain concentrations

Nonproportional loading and stressing

Fatigue notch factors

Cracks at notches


z

z

MTMX

MY

P

Notched Shaft Loading

38


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125Notch Root Radius, /d

Str

ess

Con

cent

ratio

n F

acto

r

Tors

ion

Ben

ding

2.201.201.04

D/d

Dd

Stress Concentration Factors


a

r

r

r

r

r

Hole in a Plate


-4

-3

-2

-1

0

1

2

3

4

30 60 90 120 150 180

Angle

= 1

= 0

= -1

Stresses at the Hole

Stress concentration factor depends on type of loadingMultiaxial Fatigue © 2003-2008 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 225 of 202

0

0.5

1.0

1.5

1 2 3 4 5ra

r

Shear Stresses during Torsion


Torsion Experiments


Multiaxial Loading

Uniaxial loading that produces multiaxial stresses at notches

Multiaxial loading that produces uniaxial stresses at notches

Multiaxial loading that produces multiaxial stresses at notches

39


0.004 0.008 0.0120

0.001

0.002

0.003

0.004

0.005

50 mm

30 mm

15 mm

7 mm

Longitudinal Tensile Strain

Tra

nsve

rse

Com

pres

sion

Str

ain

Thickness

100

x

z

y

Thickness Effects


Multiaxial Loading





MX

MY

1 2 3 4

MX

MY

Applied Bending Moments


A

B

C

D

A

C

A’

C’

BD

B’ D’

Location

MX

MY

Bending Moments on the Shaft


Bending Moments

M A B C D2.82 1 12.00 3 21.41 2 11.00 20.71 2

M M 55

A B C D

M 2.49 2.85 2.31 2.84


Combined Loading

=

40


Maximum Tensile Stress Location

32

Kt = 31 =

Kt = 3.411 = 1.72

Kt = 41 =

45


Kt = 3 Kt = 4

In and Out of Phase Loading

In-phase Out-of-phase

Damage location changes with load phasing


Multiaxial Loading





t1 t2 t3 t4

MX

MT

t1

z

1T

T

t2

z

1

T

T

t3

z

t4

T

T

Torsion Loading

Out-of-phase shear loading is needed to produce nonproportional stressing

MX

MT


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125

Notch root radius,

Str

ess

conc

entr

atio

n fa

ctor

Kt Bending

Kt Torsion

Kf Torsion

Kf Bending

Dd

d

= 2.2Dd

Fatigue Notch Factors


1.0

1.5

2.0

2.5

1.0 1.5 2.0 2.5Experimental Kf

Cal

cula

ted

Kf conservative

non-conservative

Fatigue Notch Factors ( continued )

bendingtorsion

ra

1

1K1K T

f

Peterson’s Equation

41


Fracture Surfaces in Torsion

Circumferencial Notch

Shoulder Fillet


Neuber’s Rule

Str

ess

(MP

a)

Strain

KtS

Kte

eKSK tt

Stress calculated with elastic assumptions

Actual stress

ES2e

For cyclic loading

eS ee


Multiaxial Neuber’s Rule

ES2e

Define Neuber’s rule in equivalent variables

Stress strain curve

'n

1

'KE

Constitutive equation

xy

y

x

xy

y

x

f(E,K’,n’)

Five equations and six unknowns


Ignore Plasticity Theory

11

e2

e

2 e

e

11

e3

e

3 e

e

11

e2

e

2 S

S

11

e3

e

3 S

S


Hoffman and Seeger

1e

2e

1

2

S

S

1e

2e

1

2

e

e


Glinka

Strain energy density

Str

ess

(MP

a)

Strain

ee

eS

ije

ije

ije

ije

ijij

ijij

eS

eS

42


Koettgen-Barkey-Socie

xy

y

x

xye

ye

xe

T

S

S

fo(Eo,Ko,no)

E

K’ n’

Material Yield SurfaceStructural Yield Surface

Ko no

Eo

xy

y

x

xy

y

x

f(E,K’,n’)

Se


1.00 1.50 2.00 2.50 3.000.00

0.25

0.50

0.75

1.00

1.25

1.50

F

R

a

aR

Stress Intensity Factors

meqKCdN

da aFKI


0

20

40

60

80

100

0 2 x 10 6

Cra

ck L

eng

th,

mm

4 x 10 6 6 x 10 6 8 x 10 6

= -1 = 0 = 1

Cycles

Crack Growth From a Hole


Notches Summary

Uniaxial loading can produce multiaxial stresses at notches

Multiaxial loading can produce uniaxial stresses at notches

Multiaxial stresses are not very important in thin plate and shell structures

Multiaxial stresses are not very important in crack growth

Multiaxial Fatigue

Documents

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