On Fatigue Crack Growth Using Cohesive Zone...

8/8/2011

On Fatigue Crack Growth Using

Cohesive Zone Model

Kyoungsoo Park1; Glaucio H. Paulino2;

Robert H. Dodds Jr.2

1 School of Civil & Environmental Engineering

Yonsei University, Seoul, Korea

2 Department of Civil & Environmental Engineering

University of Illinois at Urbana-Champaign

Kyoungsoo Park ([email protected]) 8/8/2011 2

Contents

Motivation

Previous fatigue crack growth models

Based-on stress intensity factors

Nonlinear cohesive zone model

Constitutive model of fatigue crack growth

Numerical examples

Simple mode-I problem

Double cantilever beam

Summary and Future research


Motivation

Fatigue fracture along the interface between

crystals in a metal

http://www.larrylawson.net/fatigue.htm


Previous Fatigue Crack Growth Models

Linear Elastic Fracture Mechanics

Based on Paris type relations

NASGRO, Southwest Research Institute (www.swri.org)

AFGROW (www.afgrow.net), LexTech, Inc.

Nonlinear cohesive zone model

Nguyen, Repetto, Ortiz, Radovitzky , 2001, A cohesive model of fatigue crack growth, IJF 110, 315-369

Maiti, Geubelle, 2005, A cohesive model for fatigue failure of polymers , EFM 72, 691-708

Roe, Siegmund, 2003, An irreversible cohesive zone model for interface fatigue crack growth simulation, EFM 70, 209-23

Ural, Krishnan, Papoulia, 2009, A cohesive zone model for fatigue crack growth allowing for crack retardation, IJSS 46, 2453-2462


Model by Ngyen et al (2001)

Unloading/Reloading Relationship

Unloading:

Reloading:

When the softening starts?

Stiffness Decaying Factor, K+

Under uniform separation,

max

max

TK

0 /

0fN

NK e K

0

O Nguyen, EA Repetto, M Ortiz and RA Radovitzky , 2001, A cohesive model of fatigue

crack growth, IJF 110, 315-369

( 0)

( 0)

KT

K

/ ( 0)

( ) / ( 0)

f

f

KK

K K


Reduction of Cohesive Stiffness

What if we change the load frequency or magnitude?

Crack Closure due to Wedge Effect

,nc f n

n

dTk F N T

d

1 11j j j j j j

c c f c n nk k N k

n

nnT

Maiti and Geubelle, 2006, Cohesive modeling of

fatigue crack retardation in polymers: Crack closure

effect, EFM 73, 22-41

Maiti and Geubelle, 2005, A

cohesive model for fatigue failure

of polymers , EFM 72, 691-708

Model by Maiti and Geubelle (2005, 2006)

1

c f c nk N k 0n


Model by Roe and Siegmund (2003)

Unloading/Reloading Relationship

What if permanent deformation is negligible?

What if the local cohesive traction does not reach the cohesive strength?

Evolution of Damage Parameter

,max ,max( )n n n n nT T k

max

0

(1 )n

ek D

ma

0

x max,0(1 )f

f

TD

DH

0D

Initial slope in the

exponential potential

D = 0

Roe and Siegmund, 2003, An irreversible cohesive zone model for interface fatigue

crack growth simulation, EFM 70, 209-232

0/n 0n


Model by Ural et al (2009)

Traction-Separation Relationship

Damage Evolution

No Damage Before Cohesive Strength

Relatively lower cohesive strength is used in computational

simulation

( )T F k (1 )

( )(1 )

c

n c

kF k

k k

( ) (1 )cC k k

c n

A Ural, VR Krishnan, KD Papoulia, 2009, A cohesive zone model for fatigue crack

growth allowing for crack retardation, IJSS 46, 2453-2462.

( ) 0, 0

( ) 0, 0

, ,0 ( ) 0

, 0

DBk T C T C

COk T C T Ck

OA BC ODT C

ABT C


Remarks

Previous Models

Boundary between reloading and softening is not clear

Model parameters should be free from the number of cycles

Damage may occur before the cohesive traction reaches cohesive strength

Little explanation on how the model parameters relate to the Paris “Law”

Goals

Tackle the limitations in the previous models

Provide a general fatigue crack growth model, which also captures Paris-type relations


Proposed Fatigue Crack Growth Model


( , )

1n tn T

n

T D

1 ( , )

11

c n n tnn c o T

o n n n

C DT C C D

C D

Softening condition

Unloading/Reloading

DT: Damage associated with the rate of the cohesive traction

Dn: Damage associated with the rate of the cohesive separation

CC: Crack closure effect (CC=0.3)

CO: Crack opening effect nT

n

,maxnT

,maxc nC T

nb n nD

no o n nC D

nbno

Softening

Contact

Unloading

Softening

:

: , 0

: , 0

:

:

nb n nb

no n nb n

no n nb n

n no

n n

Softening

Reloading

Unloading

Contact

Complete failure

Reloading


Fatigue Crack Growth Model


( , )

1n tn T

n

T D

1 ( , )

11

c n n tnn c o T

o n n n

C DT C C D

C D

Softening condition

Unloading/Reloading





n

,maxnT

,maxc nC T

nb n nD

no o n nC D

nb

Co = 0 δno = 0

( , )

1 1n n tnn c c T

n n n

DT C C D

D




( , )

1n tn T

n

T D

1 ( , )

11

c n n tnn c o T

o n n n

C DT C C D

C D

Softening condition

Unloading/Reloading





n

,maxnTnb n nD

no o n nC D

nbno

( , )1

11

n n tnn o T

o n n n

DT C D

C D

Cc = 0




( , )

1n tn T

n

T D

1 ( , )

11

c n n tnn c o T

o n n n

C DT C C D

C D

Softening condition

Unloading/Reloading





n

,maxnTnb n nD

no o n nC D

nb

( , )

1n n tnn T

n n n

DT D

D

Cc = 0, Co = 0


Evolution of Damage Parameters

Damage Associated with Rate of Separation, Dn

Damage Associated with Rate of Traction, DT

Material Parameters

Higher values of kn and kt (inversely proportional to

damage), higher resistance in fatigue crack growth

max( )/

( / )0

tT

ReloadingTD

Unloading Softening

/ ( )

, 0 ( )/

, 0 ( )0

( )0

n n n nb

no n nb nn n nn

no n nb n

n no

Softening

ReloadingD

Unloading

Contact


Simple Mode – I Problem

Problem Description

Displacement Control

E = 10.34 MPa, v = 0.3, t=0.04

GI = 98.1N/m, σmax = 0.1MPa, α=0.3

kn = 10, kt = 10, CC = 0.3, CO = 0.0

Displacement

Str

ess (

Pa)

Cyclic loading

Monotonic loading

-0.1

Am

plit

ud

e (

mm

)

1 1.1 1.2

1.3 1.3 1.3 1.5

0.1

0.1


Effect of Crack Opening

Parameters: kn = 10, kt = 10, CC = 0.3, CO = 0.1

Cyclic loading

Monotonic loading

Displacement (m)

Str

ess (

Pa)


kn = 10

CC = 0.0

CO = 0.0

Constant Loading Amplitude

Effect of Damage Measure for the Rate of Traction

Monotonic loading

kt = 500

kt = 200

kt = 100

Displacement (m)

Str

ess (

Pa)


kt = 200

CC = 0.0

CO = 0.0

Constant Displacement Amplitude

Displacement (m)

Str

ess (

Pa)

Monotonic loading

kn = 30

kn = 50

kn = 100

Effect of Damage Measure for the Rate of Separation


Double Cantilever Beam

Problem Description

Displacement Control

E = 70 GPa, v = 0.33 , σmax = 6.66MPa

GI = 675 N/m, α = 0.3

kn = 5/10/20, kt = 40/80/∞, CC = 0.3 , CO = 0.0

Constant loading amplitude (5, 7.5, 10 kN)

P

P

h = 18.72 mm

L = 216 mm

a0 = 127 mm

A Ural, VR Krishnan, KD Papoulia, 2009, A cohesive zone model for fatigue crack growth

allowing for crack retardation, IJSS 46, 2453-2462.


Computational Results

Number of cycles C

rack t

ip location

Number of cycles

Cra

ck t

ip location

Load Magnitude Rate of Separation

5 kN

7.5 kN

10 kN

kn = 5

kn = 10

kn = 20

kn = 10, kt = ∞ (fixed) Pmax = 10 kN, kt = ∞ (fixed)


Effect of Fracture Energy

log(ΔK)

log (

da/d

N)

Thre

shold

regio

n

Rapid

-unsta

ble

gro

wth

Paris Region

Stable growth

mda

C KdN

m

C

GI = 675 N/m

GI = 800 N/m

log(ΔK)

log (

da/d

N)

10.619

12 20.673 0.815 0.429

a h aK P

h h a h

Foote and Buchwald, 1985, IJF 29, 125-134


Comparison with Paris Equation

log(ΔK)

log (

da/d

N)

kn = 5

kn = 10

kn = 20

Paris’ Parameters

• m = 3 / C = 1.6x10-7, 0.75x10-7, 0.37x10-7

GI = 675 N/m, kt = ∞ (fixed) GI = 800 N/m, kn = 10 (fixed)

kt = 40

kt = 80

kt = ∞

Paris’ Parameters

• m = 3, 2.5, 2

• C = 1.04x10-7, 10.7x10-7, 112x10-7

log(ΔK)

log (

da/d

N)

kn is related to C


Summary and Future Research

Previous fatigue crack models show several limitations

Proposed model is based on two damage measures

Define softening and unloading/reloading conditions

Fatigue damage occurs before the cohesive traction reaches the cohesive strength

Extension to mixed-mode problems

Further research needed on crack closure, bulk plasticity and temperature effects


Thank you for your attention !

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