Socie 3.0 Multiaxial Fatigue

8/11/2019 Socie 3.0 Multiaxial Fatigue

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Professor Darrell F. Socie

Department of Mechanical and Industrial Engineering

University of Illinois at Urbana-Champaign

2001 Darrell Socie, All Rights Reserved

Multiaxial Fatigue


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

Contact Information

Darrell Socie

Mechanical Engineering1206 West Green

Urbana, Illinois 61801

USA

[email protected]

Tel: 217 333 7630Fax: 217 333 5634


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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 )


4/127


5/127


Stress Components

X

Z

Y

z

y

x

zyzx

xz

xyyx

yz

Six stresses and six strains


6/127


Stresses Acting on a PlaneZ

Y

X

Y

XZ

x

xz

xy


7/127


Principal Stresses

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

2YZ -

2XZ )

- (XYZ + 2XYYZXZ -X2

YZ -Y2

ZX -Z2

XY ) = 0

1

3

2

13

1223


8/127


Stress and Strain Distributions

80

90

100

-20 -10 0 10 20

%o

fap

pliedstress

Stresses are nearly the same over a 10range of angles


9/127


Tension

x

/2

11 2

3

2=3= 0

1

2= 3= 1


10/127


Torsion

2

xy1

3

2

/2

1

3

X

Y2

1 = xy

3

1


11/127


/2

2= y

1= x

3

1= 2

3

1= 2

=

1

2

3

Biaxial Tension


12/127


1

2

3 3

21

Maximum shear stress Octahedral shear stress

Shear Stresses

2

31

13= ( ) ( ) ( )232221231oct 3

1 ++=

oct

2

3=Mises: 1313oct 94.0

22

3==


13/127


State of Stress Summary

Stresses acting on a plane

Principal stressMaximum shear stress

Octahedral shear stress


14/127


Fatigue Mechanisms

Crack nucleation

Fracture modes

Crack growth

State of stress effects


15/127


Crack Nucleation


16/127


Slip Bands

Loading Unloading

Extrusion

Undeformed

material

Intrusion


17/127


Slip Bands


18/127


Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear

Mode I, Mode II, and Mode III


19/127


StageI StageII

loading direction

freesurface

Stage I and Stage II


20/127


Case A and Case B

Growth along the surface Growth into the surface


21/127


5 mcrackgrowthdirection

Mode I Growth


22/127


crack growth direction

10 m

slip bandsshear stress

Mode II Growth


23/127


1.0

0.2

0

0.4

0.8

0.6

1 10 102

103

104

105

106

107

Nucleation

Tension

Shear

304 Stainless Steel - Torsion

Fatigue Life, 2Nf

Damage

FractionN/Nf


24/127


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

Damage

FractionN/Nf


25/127


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

Damage

FractionN/Nf


26/127


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

Damage

FractionN/Nf


27/127


Fatigue Life, 2Nf

Damage

FractionN/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


28/127


1045 Steel - Tension

Nucleation

Shear

Tension

1.0

0.2

0

0.4

0.8

0.6

1 10 102

103

104

105

106

107

Fatigue Life, 2Nf

Damage

FractionN/Nf


29/127


Fatigue Mechanisms Summary

Fatigue cracks nucleate in shear

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


30/127


Stress Based Models

Sines

FindleyDang Van


31/127


1.0

0

0 0.5 1.0 1.5 2.0

Shear stress

Octahedral stressPrincipal stress

0.5

Shearstressinbendin

g

1/2Bend

ingfatiguelim

it

Shear stress in torsion1/2 Bending fatigue limit

Bending Torsion Correlation


32/127


Test Results

Cyclic tension with static tension

Cyclic torsion with static torsionCyclic tension with static torsion

Cyclic torsion with static tension


33/127


1.5

1.0

0.5

1.5-1.5 -1.0 1.00.5-0.5 0

A

xialstress

Fatiguestrength

Mean stress

Yield strength

Cyclic Tension with Static Tension


34/127


1.5

1.0

0.5

1.51.00.5

0

ShearStre

ssAmplitude

ShearFatigueStrength

Maximum Shear Stress

Shear Yield Strength

Cyclic Torsion with Static Torsion


35/127


36/127


1.5

1.0

0.5

1.5-1.5 -1.0 1.00.5-0.5 0

Torsion

shearstress

Shearfatiguestrength

Axial mean stress

Yield strength

Cyclic Torsion with Static Tension


37/127


Conclusions

Tension mean stress affects both tension

and torsion

Torsion mean stress does not affect tensionor torsion


38/127


Sines

==

)3(

2h

oct

=++

++++++

)(

)(6)()()(6

1

meanz

meany

meanx

2yz

2xz

2xy

2zy

2zx

2yx


39/127


40/127


Bending Torsion Correlation

1.0

0

0 0.5 1.0 1.5 2.0

Shear stress

Octahedral stressPrincipal stress

0.5

Shearstressinbendin

g

1/2Bend

ingfatiguelim

it

Shear stress in torsion

1/2 Bending fatigue limit


41/127


Dang Van

( ) ( )t a t bh+ =

m

V(M)

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

ij(m,t)

ij(m,t)


42/127


h

h

Loading path

Failurepredicted

(t) + ah(t) = b

Dang Van ( continued )


43/127


44/127


Strain Based Models

Plastic Work

Brown and Miller Fatemi and Socie

Smith Watson and Topper

Liu


45/127


10 102 103 104 105

0.001

0.01

0.1

Cycles to failure

Plastico

ctahedral

shearstrainrange Torsion

Tension

Octahedral Shear Strain


46/127


1

10

100

102 103 104

A

Fatigue Life, Nf

Plas

ticWorkp

erCycle,MJ/m3

T Torsion

Axial0o

90o

180o

135o

45o

30o

T

A T

T

T

T

T

T

A

A

AA

A

Plastic Work


47/127


0.005 0.010.0

102

2 x102

5 x102

103

2 x103

Fatigue

Life,

Cycles

Normal Strain Amplitude, n

= 0.03

Brown and Miller


48/127


Case A and B

Growth along the surface Growth into the surface


49/127


Uniaxia

l

Equibiaxial

Brown and Miller ( continued )


50/127


Brown and Miller ( continued )

( ) $ max

= + S n

1

max

', '

( ) ( )2

2

2 2+ =

+S A E N B Nnf n mean

fb

f fc


51/127


Fatemi and Socie


52/127


C F G

H I J

/ 3

Loading Histories


53/127


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

CrackLength,mm

Crack Length Observations


54/127


Fatemi and Socie

cof

'f

bof

'f

y

max,n )N2()N2(G

k12

+

=

+


55/127


Smith Watson Topper


56/127


SWT

cbf'f'fb2f

2'

f1n )N2()N2(E2

++=


57/127


Liu

WI = (nn)max + ( )

b2f

2'fcb

f'f

'fI )N2(

E

4)N2(4W

+= +

WII =(nn ) + ( )max

bo2f

2'

fcobof'f'fII )N2(G

4)N2(4W

+= +

Virtual strain energy for both mode I and mode II cracking


58/127


Cyclic Torsion

Cyclic Shear Strain Cyclic Tensile Strain

Shear Damage Tensile Damage

Cyclic Torsion


59/127


Cyclic TorsionStatic Tension


Shear Damage Tensile Damage

Cyclic Torsion with Static Tension


60/127



Tensile DamageShear DamageCyclic TorsionStatic Compression

Cyclic Torsion with Compression

Cyclic Torsion with Tension


61/127


Cyclic TorsionStatic Compression

Hoop Tension


Tensile DamageShear Damage

y

and Compression


62/127


63/127


Conclusions All critical plane models correctly predict

these results

Hydrostatic stress models can not predict

these results


64/127


0.006Axial strain

-0.003

0.003

She

arstrain

Loading History


65/127


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 9,930Liu I 6.41 4,280

Liu II 6.42 5,420

Chu 6.37 3,040

Gamma 26,775Fatemi-Socie 6.23 10,350

Glinka 6.39 33,220


66/127


Strain Based Models Summary Two separate models are needed, one for

tensile growth and one for shear growth

Cyclic plasticity governs stress and strainranges

Mean stress effects are a result of crackclosure on the critical plane


67/127


68/127


Cyclic Plasticity

p

p

p

p


69/127


Mean Stresses eq

f meanf

bf f

c

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++

+=+

co

f

'

f

bo

f

'f

y

max,n

)N2()N2(Gk12 +

=

+

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2

++

=


70/127


Fracture Mechanics ModelsMode I growth

Torsion

Mode II growth

Mode III growth


71/127


Mode I, Mode II, and Mode IIIMode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear


72/127


Mode I

Mode II

Mode I and Mode II Surface Cracks


73/127


= -1 = 0 = 1

= -1 = 0 = 1

= 193 MPa = 386 MPa10-3

10-4

10-5

10-6

da/dNmm/cycle

10 100 20020 50 10 100 20020 50

K, MPa m

10-3

10-4

10-5

10-6

da/d

Nmm/cycle

Biaxial Mode I Growth

K, MPa m


74/127


Mode II

Mode III

Surface Cracks in Torsion


75/127


Transverse Longitudinal Spiral

Failure Modes in Torsion

Fracture Mechanism Map


76/127


1500

1000YieldStren

gth,

MPa

40

30

50

60

HardnessRc

200 300 400 500

Shear Stress Amplitude, MPa

No Cracks

Spiral

Cracks

TransverseCracks

Longi-

tudinal

Fracture Mechanism Map

Mode I and Mode III Growth


77/127


10-6

10-5

10-4

10-3

da/dN

,mm/cycle

5 10 1005020

KI , KIII MPa m

KI

KIII

Mode I and Mode III Growth

Mode I and Mode II Growth


78/127


1 10 100

10-7

10-6

10-5

10-4

10-3

10-2

da/dN

,mm/cycle

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


79/127


Fracture Mechanics Models

( )meqKCdN

da=

[ ]25.0

4III

4II

4Ieq )1(K8K8KK ++=

[ ] 5.02III2II2Ieq K)1(KKK +++=

[ ]5.02

IIIII

2

Ieq KKKKK ++=

a)EF())1(2

EF()(K

5.0

2I

2IIeq

+

+=

( ) ak1GFKys

max,n

eq

+=

Fracture Surfaces


80/127


Bending Torsion

Mode III Growth


81/127


10-9

10-8

10-7

10-6

10-5

10-4

0.2 0.4 0.6 0.8 1.0Crack length, mm

Crackgrowthrate,mm/

cycle K = 11.0

K = 14.9

K = 10.0

K = 12.0

K = 8.2

Fracture Mechanics Models Summary


82/127


y

Multiaxial loading has little effect in Mode I

Crack closure makes Mode II and Mode IIIcalculations difficult

Nonproportional Loading


83/127


p p g

In and Out-of-phase loading

Nonproportional cyclic hardening Variable amplitude

In and Out-of-Phase Loading


84/127


xxy

y

t

t

t

tx= osin(t)

xy= (1+)osin(t)

In-phase

Out-of-phase

x

x

xy

xy

x= ocos(t)

xy= (1+)osin(t)

1+

1+

g

In-Phase and Out-of-Phase


85/127


1

xy

22xy

x

x

x

x

xy

2

2xy

Loading Histories


86/127


x x

x x

xy/2 xy/2

xy/2xy/2 cross

diamondout-of-phase

square

Loading Histories


87/127


in-phase

out-of-phase

diamond

square

cross

Findley Model Results


88/127


/2 MPa n,max

MPa /2 + 0.3 n,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

x

xy/2cross

x

xy/2diamondout-of-phase

x

xy/2square

in-phase

x

xy/2

Nonproportional Hardening


89/127


t

t

t

tx= osin(t)

xy= (1+)osin(t)

In-phase

Out-of-phase

x

x

xy

xy

x= ocos(t)

xy= (1+)osin(t)

In-Phase


90/127


600

-600

300

-300

-0.003 -0.0060.003 0.006

Axial Shear

90 Out-of-Phase


91/127


Axial

600

-600

-0.003 0.003

Shear

-300

300

0.006-0.006

Critical Plane


92/127


600

-600

-0.004 0.004

Proportional

-600

600

-0.004 0.004

Out-of-phase

Nf = 38,500

Nf = 310,000

Nf = 3,500

Nf = 40,000

Loading Histories


93/127


0 1 2 3 4

5 6 7 8 9

1011 12 13

Stress-Strain Response


94/127


-300

-150

0

15 0

30 0

-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

15 0

30 0

-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

ShearStress(MPa)

Stress-Strain Response (continued)


95/127


-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

ShearStr

ess(MPa)

Axial Stress ( MPa )

Maximum Stress


96/127


1000

200102 103 104

EquivalentStress,M

Pa

2000

Fatigue Life, Nf

Nonproportional hardening results in lower fatigue lives

All tests have the same strain ranges


97/127

Shear Stresses


98/127


Case A Case B Case C Case D

xy xy xy xy

Simple Variable Amplitude History


99/127


-0.003 0.003

-0.006

0.006

xy

-300 300

x

-150

150

x

xy

Stress-Strain on 0 Plane


100/127


-0.003 0.003x

-300

300

x

-0.005 0.005xy

-150

150xy

Stress-Strain on 30 and 60 Planes


101/127


-0.003 0.00330

-300

300

-0.003 0.003

-300

30030 plane 60 plane

6

0

3

0

60

Stress-Strain on 120 and 150 Planes


102/127


-0.003 0.003

-300

300

1

20

-0.003 0.003

-300

300150 plane120 plane

150

120 150

Shear Strain History on Critical Plane


103/127


time

-0.005

0.005

Shearstrain,

Fatigue Calculations


104/127


Load or strain history

Cyclic plasticity model

Stress and strain tensor

Search for critical plane

Nonproportional Loading Summary


105/127


Nonproportional cyclic hardening increases

stress levels

Critical plane models are used to assessfatigue damage

Notches


106/127


Stress and strain concentrations

Nonproportional loading and stressing

Fatigue notch factors

Cracks at notches

Notched Shaft Loading


107/127


z

z

MTMX

MY

P

6

Stress Concentration Factors


108/127


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125Notch Root Radius, /d

StressC

oncentrationFactor

To

rsion

Ben

ding

2.201.201.04

D/d

Dd

Hole in a Plate


109/127


a

r

r

r

r

r

4

Stresses at the Hole


110/127


-4

-3

-2

-1

0

1

2

3

4

30 60 90 120 150 180

Angle

= 1

= 0

= -1

Stress concentration factor depends on type of loading

Shear Stresses during Torsion


111/127


0

0.5

1.0

1.5

1 2 3 4 5ra

r

Torsion Experiments


112/127


Multiaxial Loading


113/127


Uniaxial loading that produces multiaxialstresses at notches

Multiaxial loading that produces uniaxialstresses at notches

Multiaxial loading that produces multiaxialstresses at notches

Longitudinal Tensile Strain

Thickness Effects


114/127


0.004 0.008 0.0120

0.001

0.002

0.003

0.004

0.005

50 mm

30 mm

15 mm

7 mm

g

TransverseCom

pressionStra

in

Thickness

100

x

zy

Applied Bending Moments


115/127


MX

MY

1 2 3 4

MX

MY

A

Bending Moments on the Shaft


116/127


B

C

D

A

C

A

C

BD

B D

Location

MX

MY

Bending Moments


117/127


M A B C D

2.82 1 1

2.00 3 21.41 2 1

1.00 2

0.71 2

M M= 55

A B C DM 2.49 2.85 2.31 2.84

Torsion Loading


118/127


t1 t2 t3 t4

MX

MT

t1

z

1T

T

t2

z

1

T

T

t3

1= z

t4

T

1= T

Out-of-phase shear loading is needed to producenonproportional stressing

MX

MT

Plate and Shell Structures


119/127


Kt= 3 Kt= 4

6

Fatigue Notch Factors


120/127


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125

Notch root radius,

Stressconce

ntrationfactor

Kt Bending

Kt Torsion

Kf Torsion

Kf Bending

Dd

d

= 2.2

D

d

Fatigue Notch Factors ( continued )

bending


121/127


1.0

1.5

2.0

2.5

1.01.5 2.0 2.5Experimental Kf

Ca

lculatedKf

conservative

non-conservative

bending

torsion

r

a1

1K1K Tf+

+=

Petersons Equation

Fracture Surfaces in Torsion


122/127


Circumferencial Notch

Shoulder Fillet

1 50

Stress Intensity Factors


123/127


1.00 1.50 2.00 2.50 3.000.00

0.25

0.50

0.75

1.00

1.25

1.50

F

= 1

= 0

= 1

R

a

a

R

( )meqKCdN

da= aFKI =


124/127

Notches Summary

Uniaxial loading can produce multiaxial


125/127


Uniaxial loading can produce multiaxialstresses at notches

Multiaxial loading can produce uniaxialstresses at notches

Multiaxial stresses are not very important in

thin plate and shell structures

Multiaxial stresses are not very important incrack growth

Final Summary

Fatigue is a planar process involving the


126/127


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

Critical plane models provide reasonableestimates of fatigue damage

Multiaxial Fatigue


127/127

University of Illinois at Urbana-Champaign

Multiaxial Fatigue

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Socie 3.0 Multiaxial Fatigue

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