5 Multiaxial Fatigue.ppt - University Of Illinoisfcp.mechse.illinois.edu/files/2014/07/5-Multiaxial-Fatigue.pdf · 5 Multiaxial Fatigue.ppt - University Of Illinois ... axial

Professor Darrell F. SocieDepartment of Mechanical Science and Engineering

University of Illinois at Urbana-Champaign

© 2001-2012 Darrell Socie, All Rights Reserved

Multiaxial Fatigue

Multiaxial Fatigue © 2001-2012 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 125

Contact Information

Darrell SocieMechanical Science and Engineering1206 West GreenUrbana, Illinois 61801USA

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


Outline

State of Stress ( Chapter 1 )

Fatigue Mechanisms ( Chapter 3 )

Stress Based Models ( Chapter 5 )

Strain Based Models ( Chapter 6 )

Fracture Mechanics Models ( Chapter 7 )

Nonproportional Loading ( Chapter 8 )

Notches ( Chapter 9 )


State of Stress

Stress componentsCommon 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


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

12

3

Biaxial Tension


1

2

3 3

2

1

Maximum shear stress Octahedral shear stress

Shear Stresses

231

13

232

221

231oct 3

1

oct23

Mises: 1313oct 94.022

3


State of Stress Summary

Stresses acting on a plane Principal stressMaximum shear stressOctahedral shear stress


Fatigue Mechanisms

Crack nucleation Fracture modesCrack growth State of stress effects


Crack Nucleation


Slip Bands

Loading Unloading

Extrusion

Undeformedmaterial

Intrusion


Slip Bands


Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear

Mode I, Mode II, and Mode III


Stage I Stage II

loading direction

freesurface

Stage I and Stage II


Case A and Case B

Growth along the surface Growth into the surface


5 mcrac

k gr

owth

dire

ctio

n

Mode I Growth


crack growth direction

10 m

slip bandsshear stress

Mode II Growth


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

Frac

tion

N/N

f


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

Frac

tion

N/N

f


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

Frac

tion

N/N

f


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

Frac

tion

N/N

f


Fatigue Life, 2Nf

Dam

age

Frac

tion

N/N

ff

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


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

Frac

tion

N/N

f


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 FindleyDang Van


1.0

00 0.5 1.0 1.5 2.0

Shear stressOctahedral stress

Principal stress

0.5

She

ar s

tress

in b

endi

ng

1/2

Ben

ding

fatig

ue li

mit

Shear stress in torsion

1/2 Bending fatigue limit

Bending Torsion Correlation


Test Results

Cyclic tension with static tensionCyclic torsion with static torsionCyclic tension with static torsionCyclic 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

tress

Fatig

ue s

treng

th

Mean stressYield strength

Cyclic Tension with Static Tension


1.5

1.0

0.5

1.51.00.50

She

ar S

tress

Am

plitu

deS

hear

Fat

igue

Stre

ngth

Maximum Shear StressShear Yield Strength

Cyclic Torsion with Static Torsion


1.5

1.0

0.5

3.00 2.01.0

Ben

ding

Stre

ssB

endi

ng F

atig

ue S

treng

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 stre

ss

She

ar fa

tigue

stre

ngth

Axial mean stressYield strength

Cyclic Torsion with Static Tension


Conclusions

Tension mean stress affects both tension and torsion

Torsion mean stress does not affect tension or torsion


Sines

)3(2 h

oct

)(

)(6)()()(61

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

tress

in b

endi

ng

1/2

Ben

ding

fatig

ue li

mit

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)


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


Strain Based Models

Plastic Work Brown and Miller Fatemi and Socie Smith Watson and Topper Liu


10 102 103 104 1050.001

0.01

0.1

Cycles to failure

Pla

stic

oct

ahed

ral

shea

r stra

in ra

nge Torsion

Tension

Octahedral Shear Strain


1

10

100

102 103 104

A

Fatigue Life, Nf

Pla

stic

Wor

k pe

r Cyc

le, M

J/m

3T Torsion

Axial0

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 x103Fa

tigue

Life

, 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

', '( ) ( )

22

2 2

S AE

N B Nnf n mean

fb

f fc


Fatemi and Socie


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

mCrack 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(

E4)N2(4W

WII = n n ) + (max

bo2f

2'fcobo

f'f

'fII )N2(

G4)N2(4W

Virtual strain energy for both mode I and mode II cracking


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


0.006Axial strain

-0.003

0.003S

hear

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(

E2

)S7.03.1(S2

cof

'f

bof

'f

y

max,n )N2()N2(G

k12

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2


Fracture Mechanics Models

Mode I growth TorsionMode II growthMode III growth


Mode I, Mode II, and Mode IIIMode 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


Transverse Longitudinal Spiral

Failure Modes in Torsion


1500

1000

Yiel

d S

treng

th, M

Pa

40

30

50

60

Har

dnes

s R

c

200 300 400 500Shear 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, m

m/c

ycle

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


Fracture Mechanics Models

meqKCdNda

25.04III

4II

4Ieq )1(K8K8KK

5.02III

2II

2Ieq K)1(KKK

5.02IIIII

2Ieq KKKKK

a)EF())1(2

EF()(K5.0

2I

2IIeq

ak1GFKys

max,neq


Fracture Surfaces

Bending Torsion


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 ra

te, m

m/c

ycle K = 11.0

K = 14.9

K = 10.0

K = 12.0

K = 8.2

Mode III Growth


Fracture Mechanics Models Summary

Multiaxial loading has little effect in Mode ICrack closure makes Mode II and Mode III

calculations difficult


Nonproportional Loading

In and Out-of-phase loadingNonproportional 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 1.090 out-of-phase 250 500 400 2.0diamond 250 500 400 2.0square 353 603 534 0.11cross - tension cycle 250 250 325 16cross - torsion cycle 250 0 250 216

x

xy/2cross

x

xy/2diamondout-of-phase

x

xy/2square

in-phase

x

xy/2


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


Axial600

-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

Shea

r Stre

ss (

MPa

)Stress-Strain Response


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

Shea

r Stre

ss (

MPa

)

Axial Stress ( MPa )


1000

200102 103 104

Equ

ival

ent S

tress

, 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 Dx x x x

y y y y

Nonproportional Example


Case A Case B Case C Case D

xy xy xy xy

Shear Stresses


-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

train

,

Shear Strain History on Critical Plane


Fatigue Calculations

Load or strain history

Cyclic plasticity model

Stress and strain tensor

Search for critical plane


Nonproportional Loading Summary

Nonproportional cyclic hardening increases stress levels

Critical plane models are used to assess fatigue damage


Notches

Stress and strain concentrationsNonproportional loading and stressing Fatigue notch factorsCracks at notches


z

z

MT MX

MY

P

Notched Shaft Loading


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125Notch Root Radius, /d

Stre

ss C

once

ntra

tion

Fact

or

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 loading


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


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

Tran

sver

se C

ompr

essi

on S

train

Thickness

100

x

z

y

Thickness Effects


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 DM 2.49 2.85 2.31 2.84


t1 t2 t3 t4

MX

MT

t1z

1T

T

t2z

1

T

T

t3 z

t4

T

T

Torsion Loading

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

MX

MT


Kt = 3 Kt = 4

Plate and Shell Structures


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125Notch root radius,

Stre

ss c

once

ntra

tion

fact

or

Kt BendingKt 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

ra1

1K1K Tf

Peterson’s Equation


Fracture Surfaces in Torsion

Circumferencial Notch

Shoulder Fillet


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

meqKCdNda

aFKI


0

20

40

60

80

100

0 2 x 10 6

Cra

ck L

engt

h, m

m

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


Final Summary

Fatigue is a planar process involving the growth of cracks on many size scales

Critical plane models provide reasonable estimates of fatigue damage

University of Illinois at Urbana-Champaign

Multiaxial Fatigue

5 Multiaxial Fatigue.ppt - University Of Illinoisfcp.mechse.illinois.edu/files/2014/07/5-Multiaxial-Fatigue.pdf · 5 Multiaxial Fatigue.ppt - University Of Illinois ... axial

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