Fatigue Analysis on the Web - Mechanical Science …Fatigue Analysis on the Web - Mechanical Science...

Multiaxial Fatigue

Professor Darrell F. Socie Mechanical Science and Engineering

University of Illinois

© 2004-2013 Darrell Socie, All Rights Reserved

Fatigue and Fracture

Multiaxial Fatigue © 2003-2013 Darrell Socie, University of Illinois at Urbana-Champaign, All Rights Reserved 1 of 134

When is Multiaxial Fatigue Important ?

Complex state of stress Complex out of phase loading


Uniaxial Stress

one principal stress one direction

X

Z

Y


Proportional Biaxial

principal stresses vary proportionally but do not rotate

X

Z

Y σ1 = ασ2 = βσ3


Nonproportional Multiaxial

Principal stresses may vary nonproportionally and/or change direction

X

Z

Y


Crankshaft

Time

-2500

2500

90 45 0

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 90º γ γ

ε ε


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 0.012 0

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


Book


Outline

State of Stress Stress-Strain Relationships Fatigue Mechanisms Multiaxial Testing Stress Based Models Strain Based Models Fracture Mechanics Models Nonproportional Loading Stress Concentrations


State of Stress

Stress components Common states of stress Shear stresses


Stress Components

X

Z

Y

σz

σy

σx

τzyτzx

τxz

τxyτyx

τyz

Six stresses and six strains


Stresses Acting on a Plane Z

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

τ12 τ23


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

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


Maximum and Octahedral Shear

0.2

0.4

0.6

0.8

1.0

-1 -0.5 0 0.5 1

18%

6%

torsion tension biaxial tension

Octahedral shear

Maximum shear

σ3/σ1


State of Stress Summary

Stresses acting on a plane Principal stress Maximum shear stress Octahedral shear stress


Outline



The Fatigue Process

Crack nucleation Small crack growth in an elastic-plastic

stress field Macroscopic crack growth in a nominally

elastic stress field Final fracture


Slip Bands

Ma, B-T and Laird C. “Overview of fatigue behavior in copper sinle crystals –II Population, size, distribution and growth Kinetics of stage I cracks for tests at constant strain amplitude”, Acta Metallurgica, Vol 37, 1989, 337-348


5 µmcrac

k gr

owth

dire

ctio

n

Mode I Growth


crack growth direction

10 µm

slip bands shear stress

Mode II Growth


1045 Steel - Tension

Nucleation Shear

Tension

1.0

0.2

0

0.4

0.8

0.6

1 10 10 2 10 3 10 4 10 5 10 6 10 7

Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f

100 µm crack


Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f f

Nucleation

Shear

Tension

1 10 10 2 10 3 10 4 10 5 10 6 10 7

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 10 2 10 3 10 4 10 5 10 6 10 7

Nucleation

Tension Shear

304 Stainless Steel - Torsion

Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f


304 Stainless Steel - Tension

1 10 10 2 10 3 10 4 10 5 10 6 10 7

1.0

0.2

0

0.4

0.8

0.6 Nucleation

Tension

Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f


Nucleation

Tension

Shear

1.0

0.2

0

0.4

0.8

0.6

1 10 10 2 10 3 10 4 10 5 10 6 10 7

Inconel 718 - Torsion

Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f


Inconel 718 - Tension

Shear

Tension

Nucleation

1 10 10 2 10 3 10 4 10 5 10 6 10 7

1.0

0.2

0

0.4

0.8

0.6

Fatigue Life, 2N f

Dam

age

Frac

tion

N/N

f


Outline



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 Findley Dang Van


1.0

0 0 0.5 1.0 1.5 2.0

Shear stress Octahedral 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 tension Cyclic torsion with static torsion Cyclic tension with static torsion Cyclic torsion with static tension


1.5

1.0

0.5

1.5 -1.5 -1.0 1.0 0.5 -0.5 0

Axi

al s

tress

Fa

tigue

stre

ngth

Mean stress Yield strength

Cyclic Tension with Static Tension


1.5

1.0

0.5

1.5 1.0 0.5 0

She

ar S

tress

Am

plitu

de

She

ar F

atig

ue S

treng

th

Maximum Shear Stress Shear Yield Strength

Cyclic Torsion with Static Torsion


1.5

1.0

0.5

3.0 0 2.0 1.0

Ben

ding

Stre

ss

Ben

ding

Fat

igue

Stre

ngth

Static Torsion Stress Torsion Yield Strength

Cyclic Tension with Static Torsion


1.5

1.0

0.5

1.5 -1.5 -1.0 1.0 0.5 -0.5 0

Tors

ion

shea

r stre

ss

She

ar fa

tigue

stre

ngth

Axial mean stress Yield 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

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 torsion


Bending Torsion Correlation

1.0

0 0 0.5 1.0 1.5 2.0

Shear stress Octahedral 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

τ σ( ) ( )t a t bh+ =

m

V(M)

Σ ij (M,t) E ij (M,t)

σ ij (m,t) ε ij (m,t)


Isotropic Hardening

σ

ε

stab

ilized

stre

ss ra

nge

Failure occurs when the stress range is not elastic


Oo

σ3

Co

Ro

ρ∗

CL

Oo

Loading path

Yield domain expands and

translates

a)

c) d)

b) σ3 σ3

σ3

σ1 σ1

σ1 σ1

σ2 σ2

σ2 σ2

Oo Oo

RL

OL

Multiaxial Kinematic and Isotropic

ρ* stabilized residual stress


τ

σh

τ

σh

Loading path

Failure predicted

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


Model Comparison R = -1

Stress ratio, λ

Rel

ativ

e Fa

tigue

Life

0.001

0.01

0.1

1

10

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Torsion Tension Biaxial Tension

Goodman

Findley

Sines


Outline



Strain Based Models

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


10 10 2 10 3 10 4 10 5 0.001

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

Plas

tic W

ork

per C

ycle

, MJ/

m3

T Torsion Axial 0

90

180

135

45

30

T

A T

T T

T T

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

', '( ) ( )

22

2 2+ =−

+S AE

N B Nnf n mean

fb

f fc

σ σε


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


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 Torsion Static Tension


Shear Damage Tensile Damage

Cyclic Torsion with Static Tension



Tensile Damage Shear Damage Cyclic Torsion Static Compression

Cyclic Torsion with Compression


Cyclic Torsion Static Compression

Hoop Tension


Tensile Damage Shear 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 results

Hydrostatic stress models can not predict these results


0.006 Axial strain

-0.003

0.003 S

hear

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 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

τ

σ

σ 2

σ 1 = τ xy

σ 3

σ 1

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(

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


Outline



Nonproportional Loading

In and Out-of-phase loading Nonproportional cyclic hardening Variable amplitude


εx γxy

εy

t

t

t

t εx = 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 ν 1 +

∆γ

γ ν 1 +

In and Out-of-Phase Loading


θ ε 1

γ xy

2 2 τ xy

σ x

ε x

∆σ x

∆ε x

∆γ xy

2 ∆2τ xy

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 σ n,max MPa ∆τ/2 + 0.3 σ

n,max N/N ip

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

ε x

γ xy /2 cross

ε x

γ xy /2 diamond out-of-phase

ε x

γ xy /2 square

in-phase

ε x

γ xy /2


Nonproportional Hardening

t

t

t

t εx = εosin(ωt)


In-phase

Out-of-phase

εx

εx

γxy

γxy

εx = εocos(ωt)



600

-600

300

-300

-0.003 -0.006 0.003 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 = 38,500 Nf = 310,000

Nf = 3,500 Nf = 40,000


0 1 2 3 4

5 6 7 8 9

10 11 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

200 10 2 10 3 10 4

Equi

vale

nt S

tress

, MPa

2000

Fatigue Life, N f

Maximum Stress

Nonproportional hardening results in lower fatigue lives

All tests have the same strain ranges


Case A Case B Case C Case D σ x σ x σ x σ x

σ y σ y σ y σ y

Nonproportional Example


Case A Case B Case C Case D

τxy τxy τxy τxy

Shear Stresses


-0.003 0.003

-0.006

0.006

-300 300

σx

-150

150

εx

Simple Variable Amplitude History


-0.003 0.003 εx

-300

300

-0.005 0.005 γxy

-150

150

τ xy

Stress-Strain on 0° Plane


-0.003 0.003 ε30

-300

300

-0.003 0.003

-300

300 30º plane 60º plane

ε60

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

300 150º plane 120º plane

ε120 ε150


time

-0.005

0.005

She

ar s

train

, 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 event 16 input channels 2240 elements

An Example

From Khosrovaneh, Pattu and Schnaidt “Discussion of Fatigue Analysis Techniques for Automotive Applications” Presented at SAE 2004.


Biaxial and Uniaxial Solution 10-2

1 10 100 1000 10000

Element rank based on critical plane damage

Dam

age

Uniaxial solution Signed principal stress

Critical plane solution

10-3

10-4

10-5

10-6

10-7

10-8


Nonproportional Loading Summary

Nonproportional cyclic hardening increases stress levels

Critical plane models are used to assess fatigue damage


Outline



Notches

Stress and strain concentrations Nonproportional loading and stressing Fatigue notch factors Cracks at notches


τ θ z σ θ

σ z

M T M X

M Y

P

Notched Shaft Loading


0

1

2

3

4

5

6

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

Stre

ss C

once

ntra

tion

Fact

or

Tors

ion

Ben

ding

2.20 1.20 1.04

D/d

D d

ρ

Stress Concentration Factors


λσ

λσ

σ σ θ

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 5 r a

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 0.012 0

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


Multiaxial Loading





M X

M Y

1 2 3 4

M X

M Y

Applied Bending Moments


A

B

C

D

A

C

A’

C’

B D

B’ D’

Location

MX

MY

Bending Moments on the Shaft


Bending Moments

∆ M A B C D 2.82 1 1 2.00 3 2 1.41 2 1 1.00 2 0.71 2

∆ ∆M M= ∑ 55

A B C D ∆ M 2.49 2.85 2.31 2.84


Combined Loading

σ1 = 1.72σ

σ2 = −0.72σ

λ = −0.41 σ

τ = σ

σ

=

τ = σ


Maximum Tensile Stress Location

σ

σ

σ

τ = σ

σ

32°

Kt = 3 σ1 = σ

Kt = 3.41 σ1 = 1.72σ

Kt = 4 σ1 = τ

τ

45°

τ τ = σ


Kt = 3 Kt = 4

In and Out of Phase Loading

In-phase Out-of-phase

Damage location changes with load phasing


Multiaxial Loading





t 1 t 2 t 3 t 4

M X

M T

t 1

σ z

σ 1 σ T

σ T

t 2

σ z

σ 1

σ T

σ T

t 3

σ 1 = σ z

t 4

σ T

σ 1 = σ T

Torsion Loading

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

MX

MT


0

1

2

3

4

5

6

0.025 0.050 0.075 0.100 0.125 Notch root radius,

Stre

ss c

once

ntra

tion

fact

or

Kt Bending Kt Torsion

Kf Torsion

Kf Bending

D d

ρ

ρ d

= 2.2 D d

Fatigue Notch Factors


1.0

1.5

2.0

2.5

1.0 1.5 2.0 2.5 Experimental Kf

Cal

cula

ted

Kf conservative

non-conservative

Fatigue Notch Factors ( continued )

bending torsion

ra1

1K1K Tf

+

−+=

Peterson’s Equation


Fracture Surfaces in Torsion

Circumferencial Notch

Shoulder Fillet


Neuber’s Rule St

ress

(MPa

)

Strain

KtS

Kte

σ

ε

εσ=eKSK tt

Stress calculated with elastic assumptions

Actual stress

ε∆σ∆=∆ ES2e

For cyclic loading

εσ=eS ee


Multiaxial Neuber’s Rule

ε∆σ∆=∆ ES2e

Define Neuber’s rule in equivalent variables

Stress strain curve

'n1

'KE

σ∆

+σ∆

=ε∆

Constitutive equation

γεε

=

τσσ

xy

y

x

xy

y

x

f(E,K’,n’)

Five equations and six unknowns


Ignore Plasticity Theory

11

e2

e

2 ee

ε=ε

11

e3

e

3 ee

ε=ε

11

e2

e

2 SS

σ=σ

11

e3

e

3 SS

σ=σ


Hoffman and Seeger

1e

2e

1

2

SS

=σσ

1e

2e

1

2

ee

=εε


Glinka

Strain energy density

Stre

ss (M

Pa)

Strain

∆ee

∆eS ∆σ

∆ε

∑∑ ∆∆

∆∆=

ε∆σ∆

ε∆σ∆

ije

ije

ije

ije

ijij

ijij

eSeS


Koettgen-Barkey-Socie

γεε

=

xy

y

x

xye

ye

xe

TSS

fo(Eo,Ko,no)

E

K’ n’

Material Yield Surface Structural Yield Surface

Ko no

Eo

γεε

=

τσσ

xy

y

x

xy

y

x

f(E,K’,n’)

σ

εε

Se


1.00 1.50 2.00 2.50 3.00 0.00

0.25

0.50

0.75

1.00

1.25

1.50

F

λ = −1

λ = 0

λ = 1 σ

λσ

σ

λσ

R

a

a R

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 structures

Multiaxial stresses are not very important in crack growth

Multiaxial Fatigue

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Fatigue Analysis on the Web - Mechanical Science …Fatigue Analysis on the Web - Mechanical Science...

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