multiaxial fatigue

1

Introduction

Professor Darrell F. SocieUniversity of Illinois at Urbana-Champaign

© 2003 Darrell Socie, All Rights Reserved

Multiaxial Fatigue

Multiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 14

Contact Information

Darrell SocieMechanical 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 stressComplex 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

Yσ1 = ασ2 = βσ3


Nonproportional Multiaxial

Principal stresses may vary nonproportionallyand/or change direction

X

Z

Y

2


Crankshaft

Time

-2500

2500

90450

Mic

rost

rain

0


Shear and Normal Strains

-0.0025 0.0025

-0.005

0.005

-0.0025 0.0025

-0.005

0.005

0º 90ºγ γ

εε


Shear and Normal Strains

-0.0025 0.0025

-0.005

0.005

-0.0025 0.0025

-0.005

0.005

45º 135º

ε

γ

ε

γ


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 StrainTr

ansv

erse

Com

pres

sion

Stra

in

Thickness

100

x

z

y


Notch Stresses

29.375.1-0.0010.015021.873.0-0.0020.013014.170.6-0.0030.0115

063.5-0.0050.017σzσxεzεxt

x

z

y

State of Stress



Multiaxial Fatigue

3


Outline

State of StressStress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations


State of Stress

Stress componentsCommon 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

4


Tension

τ

σσx

γ/2

εε1

σ1 σ2

σ3

σ2 = σ3 = 0

σ1

ε2 = ε3 = −νε1


Torsion

σ2 τ

στxyε1

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

−=ε12

3

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

oct23

τ=σMises: 1313oct 94.022

3τ=τ=τ


Shear Stress Influence

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

σσ1

στ

σ

1σ


State of Stress Summary

Stresses acting on a planePrincipal stressMaximum shear stressOctahedral shear stress

5

Stress Strain Relationships



Multiaxial Fatigue


Outline

State of Stress Stress-Strain RelationshipsFatigue MechanismsMultiaxial TestingStress Based Models Strain Based ModelsFracture Mechanics ModelsNonproportional LoadingStress Concentrations


“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

( ) ( )

σσστττ

ν ν

ν ν νν ν νν ν ν

ν

ν

ν

εεεγγγ

x

y

z

xy

yz

xz

x

y

z

xy

yz

xz

E

=+ −

−−

−−

−

−

1 1 2

1 0 0 01 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


σ2

von Misesyield surface

Tresca yieldsurface

Yield Surfaces

σ1

3τxy

σX


Equations

F

F

orF

ys

ys

ys

1 1 3

2 1 2

3 2 3

0

0

0

= − − =

= − − =

= − − =

σ σ σ

σ σ σ

σ σ σ

F x y x y xy ys= + − + − =σ σ σ σ τ σ2 2 2 23 0

Tresca

Mises

6


π - planeyield cylinder axis

Mises Yield Surfaces

32 σys

σ1

σ3

σ2

σ1


σ

ε

Initial yield surface

Yield surface after loading

Isotropic Hardening

σ

3τ


Cyclic Loading

σ

ε ε

σ

stab

ilize

d st

ress

rang

e

Strain Control Stress Control


Kinematic Hardening

σ

ε

σ

τ3

A

B

C

σB

σ = σA y

ασ − 2σB y


Ratcheting

γ

ε

Cyclic torsion with a mean tension stress


Cyclic Mean Stress Relaxation

ε

σ

Cyclic strain with mean stress

7


Cyclic Creep

ε

σ

Cyclic stress with a mean stress


Plasticity Models

σ

ε

a

b

c

f

d

e

σ

a

σ

c

σ

b

σ σσde

f

3τ3τ3τ

3τ 3τ 3τ


Why is modeling needed?

σ

A B C D

A

B

CD

ε

ε

time

ε*

ε*


Flow and Hardening Rules

σ

τ3φ

dσij

n

σij

αij

dαij

d d Fijp

ij

ε λ∂

∂σ=

F ijp= + =f g( ) ( )σ ε 0

Flow rule

Hardening rule


Multiaxial Example

-.006

0.003

εx

-0.003

0.006γxy

-500 500

σx

-250

250τxy


Multiaxial Example

-0.006 0.006γxy

-250

250τxy

-0.003 0.003εx

-500

500σx

8


Summary

Isotropic HardeningKinematic HardeningCyclic creep or ratchetingMean stress relaxationNonproportional hardening

Fatigue Mechanisms



Multiaxial Fatigue


Outline



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 nucleationSmall crack growth in an elastic-plastic stress fieldMacroscopic crack growth in a nominally elastic stress fieldFinal 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


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

10


Fatigue Limit and Strength Correlation


Crack Nucleation Summary

Highly localized plastic deformationSurface phenomenaStochastic process


100 µm

bulksurface

10 µm

surface

20-25 austenitic steel in symmetrical push-pull fatigue (20°C, ∆εp/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


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

SStage 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

11


Small Crack Growth

1.0 mm

N = 900

Inconel 718∆ε = 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


Strain-Life Data

Reversals, 2Nf

Stra

in A

mpl

itude

∆ε 2

10-5

10-4

0.01

0.1

1

100 101 102 103 104 105 106 107

10-3

10µm100 µm

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.

12


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

Crack Growth Rates of Metals


Maximum Load

monotonic plastic zone

σ

Stresses Around a Crack

σ

ε


Stresses Around a Crack (continued)

Minimum Load σ

ε

cyclic plastic zone

σ


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

13


5 µmcrac

k gr

owth

dire

ctio

n

Mode I Growth


crack growth direction

10 µm

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

Frac

tion

N/N

f

100 µm crack


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


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

14


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

Stress Based Models



Multiaxial Fatigue


Outline



Fatigue Mechanisms Summary

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


Stress Based Models

SinesFindleyDang Van

15


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

Bend

ing

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

Shea

r Stre

ss A

mpl

itude

Shea

r 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

Bend

ing

Stre

ssBe

ndin

g Fa

tigue

Stre

ngth

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

Shea

r fat

igue

stre

ngth

Axial mean stressYield strength

Cyclic Torsion with Static Tension

16


Conclusions

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


Sines

β=σα+τ∆ )3(2 h

oct

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 torsionMultiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 93 of 14

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

Bend

ing

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)


Isotropic Hardening

σ

ε

stab

ilize

d st

ress

rang

e

Failure occurs when the stress range is not elastic

17


Oo

σ3

Co

Ro

ρ∗

CL

Oo

Loading path

Yield domain expands and

translates

a)

c) d)

b)σ3 σ3

σ3

σ1σ1

σ1σ1

σ2 σ2

σ2 σ2

Oo Oo

RL

OL

Multiaxial Kinematic and Isotropic

ρ* stabilized residual stressMultiaxial Fatigue - Lecture 0 © 2003 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 97 of 14

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 h

oct

∆τ2

+

=k nσ

maxf

Strain Based Models



Multiaxial Fatigue


Outline



Strain Based Models

Plastic WorkBrown and MillerFatemi and SocieSmith Watson and TopperLiu

18


10 10 2 103 104 1050.001

0.01

0.1

Cycles to failure

Plas

tic o

ctah

edra

lsh

ear s

train

rang

e Torsion

Tension

Octahedral Shear Strain


1

10

100

102 103 104

A

Fatigue Life, Nf

Plas

tic W

ork

per 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 x103

Fatig

ue L

ife, C

ycle

s

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

', '( ) ( )

22

2 2+ =−

+S AE

N B Nnf n mean

fb

f fc

σ σε

19


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

m

Crack 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

20


Liu

∆WI = (∆σn ∆εn)max + (∆τ ∆γ)

b2f

2'fcb

f'f

'fI )N2(

E4)N2(4W σ

+εσ=∆ +

∆WII = (∆σn ∆εn ) + (∆τ ∆γ)max

bo2f

2'fcobo

f'f

'fII )N2(

G4)N2(4W τ

+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

21


Conclusions

All critical plane models correctly predict these resultsHydrostatic stress models can not predict these results


0.006Axial strain

-0.003

0.003

Shea

r stra

in

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 rangesMean 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

22


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(

E2

)S7.03.1(S2

ε++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



Multiaxial Fatigue


Outline




Mode I growthTorsionMode II growthMode III growth


Mode I, Mode II, and Mode IIIMode 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

23


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

Yiel

d St

reng

th,M

Pa

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

24



( )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()(K5.0

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 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 IIIcalculations difficult

Nonproportional Loading



Multiaxial Fatigue


Outline


25


Nonproportional Loading

In and Out-of-phase loadingNonproportional 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

γ

ν1+

∆γ

γ

ν1+

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,maxMPa ∆τ/2 + 0.3 σn,max N/Nipin-phase 353 250 428 1.090° out-of-phase 250 500 400 2.0

diamond 250 500 400 2.0square 353 603 534 0.11cross - tension cycle 250 250 325 16cross - torsion cycle 250 0 250 216

εx

γxy/2cross

εx

γxy/2diamondout-of-phase

εx

γxy/2square

in-phase

εx

γxy/2

26


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


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

27


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

,MPa

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 Dσx 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

28


-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

300

σ 120

-0.003 0.003

-300

300150º plane120º plane

σ 150

ε120 ε150


time

-0.005

0.005

Shea

r stra

in, γ

Shear Strain History on Critical Plane


Fatigue Calculations

Load or strain history

Cyclic plasticity model

Stress and strain tensor

Search for critical plane


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

Dam

age

Uniaxial solutionSigned principal stress

Critical plane solution

10-3

10-4

10-5

10-6

10-7

10-8

29


Nonproportional Loading Summary

Nonproportional cyclic hardening increases stress levelsCritical plane models are used to assess fatigue damage

Stress Concentrations



Multiaxial Fatigue


Outline



Notches

Stress and strain concentrationsNonproportional loading and stressingFatigue notch factorsCracks at notches


τθzσθ

σz

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, ρ/d

Stre

ss C

once

ntra

tion

Fact

or

Tors

ion

Bend

ing

2.201.201.04

D/d

Dd

ρ

Stress Concentration Factors

30


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

τrθσ

Shear Stresses during Torsion


Torsion Experiments


Multiaxial Loading

Uniaxial loading that produces multiaxial stresses at notchesMultiaxial loading that produces uniaxial stresses at notchesMultiaxial 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

31


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

t1σz

σ1σT

σT

t2σz

σ1

σT

σT

t3σ1 = 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 0.125Notch root radius,

Stre

ss c

once

ntra

tion

fact

or

Kt BendingKt Torsion

Kf Torsion

Kf Bending

Dd

ρ

ρ

d

= 2.2Dd

Fatigue Notch Factors

32


1.0

1.5

2.0

2.5

1.0 1.5 2.0 2.5Experimental Kf

Cal

cula

ted

K f

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

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 notchesMultiaxial loading can produce uniaxial stresses at notchesMultiaxial stresses are not very important in thin plate and shell structuresMultiaxial stresses are not very important in crack growth

Multiaxial Fatigue

multiaxial fatigue

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