3.0 Multiaxial Fatigue - University Of Illinoisfcp.mechse.illinois.edu/files/2014/07/3.0-Multiaxial-fatigue.pdf · 3.0 Multiaxial Fatigue - University Of Illinois ... axial

Professor Darrell F. SocieDepartment of Mechanical and Industrial Engineering

University of Illinois at Urbana-Champaign

© 2001 Darrell Socie, All Rights Reserved

Multiaxial Fatigue

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

Contact Information

Darrell SocieMechanical 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 components

Common states of stressShear stresses


Stress Components

X

Z

Y

σz

σy

σx

τzyτzx

τxz

τxyτyx

τyz

Six stresses and six strains


Stresses Acting on a PlaneZ

Y

X

Y’

X’Z’

σx’

τx’z’

τx’y’

θ

φ


Principal Stresses

σ3 - σ2( σX + σY + σZ ) + σ(σXσY + σYσZσXσZ -τ2XY - τ2

YZ -τ2XZ )

- (σXσYσZ + 2τXYτYZτXZ - σXτ2YZ - σYτ2

ZX - σZτ2XY ) = 0

σ1

σ3

σ2

τ13

τ12τ23


Stress and Strain Distributions

80

90

100

-20 -10 0 10 20

θ

% o

f app

lied

stre

ss

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


Tension

τ

σσx

γ/2

εε1

σ1 σ2

σ3

σ2 = σ3 = 0

σ1

ε2 = ε3 = −νε1


Torsion

σ2 τ

σττττxy ε1

ε3

ε2

γ/2

ε

σ1

σ3

X

Yσ2

σ1 = τxy

σ3

σ1


τ

σ

γ/2

ε

σ2 = σy

σ1 = σx

σ3

σ1 = σ2

σ3

ε1 = ε2

εν−

ν−=ε1

23

Biaxial Tension


σ1

σ2

σ3 σ3

σ2

σ1

Maximum shear stress Octahedral shear stress

Shear Stresses

231

13

σ−σ=τ ( ) ( ) ( )2

322

212

31oct 31 σ−σ+σ−σ+σ−σ=τ

oct2

3 τ=σMises: 1313oct 94.022

3 τ=τ=τ


State of Stress Summary

Stresses acting on a planePrincipal stress

Maximum 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 µmcrac

k g

row

th d

ire

ctio

n

Mode I Growth


crack growth direction

10 µm

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

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 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


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


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

tres

s in

ben

ding

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 tension

Cyclic torsion with static torsionCyclic 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

sF

atig

ue s

tren

gth

Mean stressYield strength

Cyclic Tension with Static Tension


1.5

1.0

0.5

1.51.00.50

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

engt

h

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

Tor

sion

she

ar s

tres

s

She

ar fa

tigue

str

engt

h

Axial mean stress

Yield strength

Cyclic Torsion with Static Tension


Conclusions

Tension mean stress affects both tension and torsionTorsion mean stress does not affect tension or torsion


Sines

β=σα=τ∆

)3(2 hoct

β=σ+σ+σα

+τ∆+τ∆+τ∆+σ∆−σ∆+σ∆−σ∆+σ∆−σ∆

)(

)(6)()()(61

meanz

meany

meanx

2yz

2xz

2xy

2zy

2zx

2yx


Findley

∆τ2

+

=k nσ

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

fatig

ue li

mit

Shear stress in torsion

1/2 Bending fatigue limit


Dang Van

τ σ( ) ( )t a t bh+ =

m

V(M)

Σij(M,t) Eij(M,t)

σij(m,t)εij(m,t)


τ

σh

τ

σh

Loading path

Failurepredicted

τ(t) + aσh(t) = b

Dang Van ( continued )


Stress Based Models Summary

Sines:

Findley:

Dang Van: τ σ( ) ( )t a t bh+ =

β=σα=τ∆

)3(2 hoct

∆τ2

+

=k nσ

maxf


Strain Based Models

Plastic Work

Brown and MillerFatemi and Socie

Smith Watson and TopperLiu


10 102 103 104 105

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


1

10

100

102 103 104

A

Fatigue Life, Nf

Pla

stic

Wor

k pe

r C

ycle

, MJ/

m3

T TorsionAxial0o

90o

180o

135o

45o

30o

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, ∆εn

∆γ = 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 )

( )∆ ∆γ ∆ε$maxγ α α α= + S n

1

∆γ∆εmax

', '( ) ( )

2

22 2+ =

−+S A

EN B Nn

f n meanf

bf f

cσ σ

ε


Fatemi and Socie


C F G

H I J

γ / 3

ε

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

γ+τ=

σσ

+γ∆


Smith Watson Topper


SWT

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2

+εσ+σ=ε∆σ


Liu

∆WI = (∆σn ∆εn)max + (∆τ ∆γ)

b2f

2'fcb

f'f

'fI )N2(

E4

)N2(4Wσ+εσ=∆ +

∆WII = (∆σn ∆εn ) + (∆τ ∆γ)max

bo2f

2'fcobo

f'f

'fII )N2(

G4

)N2(4Wτ+γτ=∆ +

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 ∆γ/2 σhoop MPa σaxial 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 resultsHydrostatic stress models can not predict these results


0.006Axial strain

-0.003

0.003S

hear

str

ain

Loading History


Model Comparison

Summary of calculated fatigue lives

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

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

Gamma 26,775 Fatemi-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 growthCyclic plasticity governs stress and strain ranges

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


Separate Tensile and Shear Models

τ

σ

σ2

σ1 = τxy

σ3

σ1

Inconel 1045 steel stainless steel


Cyclic Plasticity

∆ε∆γ∆εp

∆γp

∆ε∆σ∆γ∆τ∆εp∆σ∆γp∆τ


Mean Stresses

∆ε eqf mean

fb

f fc

EN N=

−+

σ σε

''( ) ( )2 2

[ ]

−

γ∆τ∆+ε∆σ∆=∆R1

2)()(W maxnnI

cf

'f

bf

n'f

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

E

2)S7.03.1(S

2ε++

σ−σ+=ε∆+γ∆

cof

'f

bof

'f

y

max,n )N2()N2(G

k12

γ+τ=

σσ

+γ∆

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2

+εσ+σ=ε∆σ


Fracture Mechanics Models

Mode I growth

TorsionMode II growth

Mode III growth


Mode I, Mode II, and Mode III

Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear


σ

σ

σ σ

Mode I

Mode II

Mode I and Mode II Surface Cracks


λ = -1λ = 0λ = 1

λ = -1λ = 0λ = 1

∆σ = 193 MPa ∆σ = 386 MPa10-3

10-4

10-5

10-6

da/d

Nm

m/c

ycle

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

10-3

10-4

10-5

10-6

da/d

Nm

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

Yie

ld S

tren

gth,

MP

a

40

30

50

60

Har

dnes

sR

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 1005020∆KI , ∆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 ν−∆+∆+∆=∆

[ ] 5.02III

2II

2Ieq K)1(KKK ∆ν++∆+∆=∆

[ ] 5.02IIIII

2Ieq KKKKK ∆+∆∆+∆=∆

a)EF())1(2

EF()(K

5.0

2I

2IIeq π

ε∆+γ∆ν+

=ε∆

( ) ak1GFKys

max,neq π

σσ

+γ∆=ε∆


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 r

ate,

mm

/cyc

le ∆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 I

Crack closure makes Mode II and Mode IIIcalculations difficult


Nonproportional Loading

In and Out-of-phase loading

Nonproportional cyclic hardeningVariable amplitude


εx

γxy

εy

t

t

t

tεx = εosin(ωt)

γxy = (1+ν)εosin(ωt)

In-phase

Out-of-phase

εx

εx

γxy

γxy

εx = εocos(ωt)

γxy = (1+ν)εosin(ωt)ε

∆γ

ε

γν1+

∆γ

γν1+

In and Out-of-Phase Loading


θε1

γ xy

22τxy

σx

εx

∆σx

∆εx

∆γxy

2∆2τxy

In-Phase and Out-of-Phase


εx εx

εx εx

γxy/2 γxy/2

γxy/2γxy/2 cross

diamondout-of-phase

square

Loading Histories


in-phase

out-of-phase

diamond

square

cross

Loading Histories


Findley Model Results

∆τ/2 MPa σn,max

MPa ∆τ/2 + 0.3 σn,max

N/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

εx

γxy/2cross

εx

γxy/2diamondout-of-phase

εx

γxy/2square

in-phase

εx

γxy/2


Nonproportional Hardening

t

t

t

tεx = εosin(ωt)


In-phase

Out-of-phase

εx

εx

γxy

γxy

εx = εocos(ωt)



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

She

ar S

tres

s (

MP

a )

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

She

ar S

tres

s (

MP

a )

Axial Stress ( MPa )


1000

200102 103 104

Equ

ival

ent S

tres

s,M

Pa

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

σx σx σx σx

σy σy σy σy

Nonproportional Example


Case A Case B Case C Case D

τxy τxy τxy τxy

Shear Stresses


-0.003 0.003

-0.006

0.006γ xy

-300 300

σx

-150

150

εx

τ xy

Simple Variable Amplitude History


-0.003 0.003εx

-300

300σ x

-0.005 0.005γxy

-150

150τ xy

Stress-Strain on 0° Plane


-0.003 0.003ε30

-300

300

-0.003 0.003

-300

30030º plane 60º plane σ 60

σ 30

ε60

Stress-Strain on 30° and 60° Planes


Stress-Strain on 120° and 150° Planes

-0.003 0.003

-300

300σ 1

20

-0.003 0.003

-300

300150º plane120º plane

σ 150

ε120 ε150


time

-0.005

0.005

She

ar s

trai

n, γ

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 levelsCritical plane models are used to assess fatigue damage


Notches

Stress and strain concentrations

Nonproportional loading and stressingFatigue notch factors

Cracks at notches


τθzσθ

σz

MTMX

MY

P

Notched Shaft Loading


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125Notch Root Radius, ρ/d

Str

ess

Con

cent

ratio

n F

acto

r

Tor

sion

Ben

ding

2.201.201.04

D/d

Dd

ρ

Stress Concentration Factors


λσ

λσ

σ σθ

a

r

σr

τrθ

σr

σθ

σθ

τrθ

Hole in a Plate


-4

-3

-2

-1

0

1

2

3

4

30 60 90 120 150 180

Angle

σθ

σ

λ = 1

λ = 0

λ = -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

τrθ

σ

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

Tra

nsve

rse

Com

pres

sion

Str

ain

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 D

∆M 2.49 2.85 2.31 2.84


t1 t2 t3 t4

MX

MT

t1

σz

σ1σT

σT

t2

σz

σ1

σT

σT

t3

σ1 = σz

t4

σT

σ1 = σT

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 0.125

Notch root radius,

Str

ess

conc

entr

atio

n fa

ctor

Kt Bending

Kt Torsion

Kf Torsion

Kf Bending

Dd

ρ

ρd

= 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


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

λ = −1

λ = 0

λ = 1σ

λσ

σ

λσ

R

a

aR

Stress Intensity Factors

( )meqKCdNda ∆= aFKI πσ∆=


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

λ = -1 λ = 0 λ = 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 structuresMultiaxial 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

Documents

3.0 Multiaxial Fatigue - University Of Illinoisfcp.mechse.illinois.edu/files/2014/07/3.0-Multiaxial-fatigue.pdf · 3.0 Multiaxial Fatigue - University Of Illinois ... axial