3 multiaxial fatigue seminar - university of...

$: 3 Multiaxial Fatigue Seminar - University Of Illinoispublish.illinois.edu/fracturegroup/files/2014/07/3_Multiaxial_Fatigue.pdf · 2 Multiaxial Fatigue ©2001-2005 Darrell Socie, University$
1

Professor Darrell F. SocieDepartment of Mechanical and Industrial Engineering

University of Illinois at Urbana-Champaign

© 2001-2005 Darrell Socie, All Rights Reserved

Multiaxial Fatigue

2

Multiaxial Fatigue © 2001-2005 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

3


Outline

n State of Stress ( Chapter 1 )

n Fatigue Mechanisms ( Chapter 3 )

n Stress Based Models ( Chapter 5 )

n Strain Based Models ( Chapter 6 )

n Fracture Mechanics Models ( Chapter 7 )

nNonproportional Loading ( Chapter 8 )

nNotches ( Chapter 9 )

4


State of Stress

n Stress componentsnCommon states of stressn Shear stresses

5


Stress Components

X

Z

Y

σz

σy

σx

τzyτzx

τxz

τxyτyx

τyz

Six stresses and six strains

6


Stresses Acting on a PlaneZ

Y

X

Y’

X’Z’

σx’

τx’z’

τx’y’

θ

φ

7


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

8


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

9


Tension

τ

σσx

γ/2

εε1

σ1 σ2

σ3

σ2 = σ3 = 0

σ1

ε2 = ε3 = −νε1

10


Torsion

σ2 τ

στxyε1

ε3

ε2

γ/2

ε

σ1

σ3

X

Yσ2

σ1 = τxy

σ3

σ1

11


τ

σ

γ /2

ε

σ2 = σy

σ1 = σx

σ3

σ1 = σ2

σ3

ε1 = ε2

εν−ν−=ε

12

3

Biaxial Tension

12


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

3τ=σMises: 1313oct 94.0

223

τ=τ=τ

13


State of Stress Summary

n Stresses acting on a planen Principal stressnMaximum shear stressnOctahedral shear stress

14


Fatigue Mechanisms

nCrack nucleationn Fracture modesnCrack growthn State of stress effects

15


Crack Nucleation

16


Slip Bands

Loading Unloading

Extrusion

Undeformedmaterial

Intrusion

17


Slip Bands

18


Mode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear

Mode I, Mode II, and Mode III

19


Stage I Stage II

loading direction

freesurface

Stage I and Stage II

20


Case A and Case B

Growth along the surface Growth into the surface

21


5 µmcrac

k gr

owth

dire

ctio

n

Mode I Growth

22


crack growth direction

10 µm

slip bandsshear stress

Mode II Growth

23


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

24


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

25


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

26


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

27


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

28


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

29


Fatigue Mechanisms Summary

n Fatigue cracks nucleate in shearn Fatigue cracks grow in either shear or tension

depending on material and state of stress

30


Stress Based Models

n Sinesn FindleynDang Van

31


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

32


Test Results

nCyclic tension with static tensionnCyclic torsion with static torsionnCyclic tension with static torsionnCyclic torsion with static tension

33


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

34


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

35


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

36


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

37


Conclusions

n Tension mean stress affects both tension and torsionn Torsion mean stress does not affect tension

or torsion

38


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

39


Findley

∆τ2

+

=k nσ

maxf

tension torsion

40


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

41


Dang Van

τ σ( ) ( )t a t bh+ =

m

V(M)

Σij(M,t) E ij(M,t)

σij(m,t)εij(m,t)

42


τ

σh

τ

σh

Loading path

Failurepredicted

τ(t) + aσh(t) = b

Dang Van ( continued )

43


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

44


Strain Based Models

n Plastic Workn Brown and Millern Fatemi and Socien Smith Watson and Toppern Liu

45


10 10 2 10 3 10 4 10 50.001

0.01

0.1

Cycles to failure

Pla

stic

oct

ahed

ral

shea

r st

rain

ran

ge Torsion

Tension

Octahedral Shear Strain

46


1

10

100

102 103 104

A

Fatigue Life, Nf

Pla

stic

Wor

k pe

r Cyc

le, M

J/m

3

T TorsionAxial0o

90o

180 o

135 o

45o

30o

T

A T

TT

TT

T

A

A

A A

A

Plastic Work

47


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

48


Case A and B

Growth along the surface Growth into the surface

49


Uniaxial

Equibiaxial

Brown and Miller ( continued )

50


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

51


Fatemi and Socie

52


C F G

H I J

γ / 3

ε

Loading Histories

53


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

Leng

th, m

m

Crack Length Observations

54


Fatemi and Socie

cof

'f

bof

'f

y

max,n )N2()N2(G

k12

γ+τ

=

σσ

+γ∆

55


Smith Watson Topper

56


SWT

cbf

'f

'f

b2f

2'f1

n )N2()N2(E2

+εσ+σ

=ε∆

σ

57


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

58


Cyclic Torsion

Cyclic Shear Strain Cyclic Tensile Strain

Shear Damage Tensile Damage

Cyclic Torsion

59


Cyclic TorsionStatic Tension


Shear Damage Tensile Damage

Cyclic Torsion with Static Tension

60



Tensile DamageShear DamageCyclic TorsionStatic Compression

Cyclic Torsion with Compression

61


Cyclic TorsionStatic Compression

Hoop Tension


Tensile DamageShear Damage

Cyclic Torsion with Tension and Compression

62


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

63


Conclusions

n All critical plane models correctly predict these resultsnHydrostatic stress models can not predict

these results

64


0.006Axial strain

-0.003

0.003

She

ar s

trai

n

Loading History

65


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

66


Strain Based Models Summary

n Two separate models are needed, one for tensile growth and one for shear growthnCyclic plasticity governs stress and strain

rangesnMean stress effects are a result of crack

closure on the critical plane

67


Separate Tensile and Shear Models

τ

σ

σ2

σ1 = τxy

σ3

σ1

Inconel 1045 steel stainless steel

68


Cyclic Plasticity

∆ε∆γ∆εp

∆γp

∆ε∆σ∆γ∆τ∆εp∆σ∆γp∆τ

69


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

70


Fracture Mechanics Models

nMode I growthn TorsionnMode II growthnMode III growth

71


Mode I, Mode II, and Mode IIIMode Iopening

Mode IIin-plane shear

Mode IIIout-of-plane shear

72


σ

σ

σ σ

Mode I

Mode II

Mode I and Mode II Surface Cracks

73


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

74


Mode II

Mode III

Surface Cracks in Torsion

75


Transverse Longitudinal Spiral

Failure Modes in Torsion

76


1500

1000

Yie

ld S

tren

gth,

MP

a

40

30

50

60

Har

dnes

s R

c200 300 400 500

Shear Stress Amplitude, MPa

No Cracks

SpiralCracks

Transverse Cracks

Longi-tudinal

Fracture Mechanism Map

77


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

78


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

79


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

5.02

I2

IIeq π

ε∆+γ∆

ν+=ε∆

( ) ak1GFKys

max,neq π

σσ

+γ∆=ε∆

80


Fracture Surfaces

Bending Torsion

81


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

82


Fracture Mechanics Models Summary

nMultiaxial loading has little effect in Mode InCrack closure makes Mode II and Mode III

calculations difficult

83


Nonproportional Loading

n In and Out-of-phase loadingnNonproportional cyclic hardeningn Variable amplitude

84


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

85


θε1

γ xy

22τxy

σx

εx

∆σx

∆εx

∆γxy

2∆2τxy

In-Phase and Out-of-Phase

86


εx εx

εx εx

γxy/2 γxy/2

γxy/2γxy/2 cross

diamondout-of-phase

square

Loading Histories

87


in-phase

out-of-phase

diamond

square

cross

Loading Histories

88


Findley Model Results

∆τ/2 MPa σn,maxMPa ∆τ/2 + 0.3 σ

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

εx

γxy/2cross

εx

γxy/2diamondout-of-phase

εx

γxy/2square

in-phase

εx

γxy/2

89


Nonproportional Hardening

t

t

t

tεx = εosin(ωt)


In-phase

Out-of-phase

εx

εx

γxy

γxy

εx = εocos(ωt)


90


600

-600

300

-300

-0.003 -0.0060.003 0.006

Axial Shear

In-Phase

91


Axial

600

-600

-0.003 0.003

Shear

-300

300

0.006-0.006

90° Out-of-Phase

92


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

93


0 1 2 3 4

5 6 7 8 9

1011 12 13

Loading Histories

94


-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

95


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

tress

( M

Pa

)

Axial Stress ( MPa )

96


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

97


Case A Case B Case C Case D

σx σx σx σx

σy σy σy σy

Nonproportional Example

98


Case A Case B Case C Case D

τxy τxy τxy τxy

Shear Stresses

99


-0.003 0.003

-0.006

0.006γ xy

-300 300

σx

-150

150

εx

τ xy

Simple Variable Amplitude History

100


-0.003 0.003εx

-300

300σ x

-0.005 0.005γxy

-150

150τ xy

Stress-Strain on 0° Plane

101


-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

102


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

103


time

-0.005

0.005

She

ar s

trai

n, γ

Shear Strain History on Critical Plane

104


Fatigue Calculations

Load or strain history

Cyclic plasticity model

Stress and strain tensor

Search for critical plane

105


Nonproportional Loading Summary

nNonproportional cyclic hardening increases stress levelsnCritical plane models are used to assess

fatigue damage

106


Notches

n Stress and strain concentrationsnNonproportional loading and stressingn Fatigue notch factorsnCracks at notches

107


τθzσθ

σz

MTMX

MY

P

Notched Shaft Loading

108


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

109


λσ

λσ

σ σθ

a

r

σr

τrθ

σr

σθ

σθ

τrθ

Hole in a Plate

110


-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

111


0

0.5

1.0

1.5

1 2 3 4 5ra

τrθ

σ

Shear Stresses during Torsion

112


Torsion Experiments

113


Multiaxial Loading

nUniaxial loading that produces multiaxial stresses at notchesnMultiaxial loading that produces uniaxial

stresses at notchesnMultiaxial loading that produces multiaxial

stresses at notches

114


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

115


MX

MY

1 2 3 4

MX

MY

Applied Bending Moments

116


A

B

C

D

A

C

A’

C’

BD

B’ D’

Location

MX

MY

Bending Moments on the Shaft

117


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

118


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

119


Kt = 3 Kt = 4

Plate and Shell Structures

120


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 BendingKt Torsion

Kf Torsion

Kf Bending

Dd

ρ

ρ

d

= 2.2Dd

Fatigue Notch Factors

121


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

122


Fracture Surfaces in Torsion

Circumferencial Notch

Shoulder Fillet

123


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

∆= aFK I πσ∆=

124


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

125


Notches Summary

nUniaxial loading can produce multiaxial stresses at notchesnMultiaxial loading can produce uniaxial

stresses at notchesnMultiaxial stresses are not very important in

thin plate and shell structuresnMultiaxial stresses are not very important in

crack growth

126


Final Summary

n Fatigue is a planar process involving the growth of cracks on many size scalesnCritical plane models provide reasonable

estimates of fatigue damage

127

University of Illinois at Urbana-Champaign

Multiaxial Fatigue

3 multiaxial fatigue seminar - university of...

Documents