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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
Kyoungsoo Park ([email protected]) 8/8/2011 3
Motivation
Fatigue fracture along the interface between
crystals in a metal
http://www.larrylawson.net/fatigue.htm
Kyoungsoo Park ([email protected]) 8/8/2011 4
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
Kyoungsoo Park ([email protected]) 8/8/2011 5
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
Kyoungsoo Park ([email protected]) 8/8/2011 6
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
Kyoungsoo Park ([email protected]) 8/8/2011 7
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
Kyoungsoo Park ([email protected]) 8/8/2011 8
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
Kyoungsoo Park ([email protected]) 8/8/2011 9
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
Kyoungsoo Park ([email protected]) 8/8/2011 10
Proposed Fatigue Crack Growth Model
Traction-Separation Relationship
( , )
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
Kyoungsoo Park ([email protected]) 8/8/2011 11
Fatigue Crack Growth Model
Traction-Separation Relationship
( , )
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
nb
Co = 0 δno = 0
( , )
1 1n n tnn c c T
n n n
DT C C D
D
Kyoungsoo Park ([email protected]) 8/8/2011 12
Fatigue Crack Growth Model
Traction-Separation Relationship
( , )
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
,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
Kyoungsoo Park ([email protected]) 8/8/2011 13
Fatigue Crack Growth Model
Traction-Separation Relationship
( , )
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
,maxnTnb n nD
no o n nC D
nb
( , )
1n n tnn T
n n n
DT D
D
Cc = 0, Co = 0
Kyoungsoo Park ([email protected]) 8/8/2011 14
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
Kyoungsoo Park ([email protected]) 8/8/2011 15
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
Kyoungsoo Park ([email protected]) 8/8/2011 16
Effect of Crack Opening
Parameters: kn = 10, kt = 10, CC = 0.3, CO = 0.1
Cyclic loading
Monotonic loading
Displacement (m)
Str
ess (
Pa)
Kyoungsoo Park ([email protected]) 8/8/2011 17
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)
Kyoungsoo Park ([email protected]) 8/8/2011 18
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
Kyoungsoo Park ([email protected]) 8/8/2011 19
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.
Kyoungsoo Park ([email protected]) 8/8/2011 20
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)
Kyoungsoo Park ([email protected]) 8/8/2011 21
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
Kyoungsoo Park ([email protected]) 8/8/2011 22
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
Kyoungsoo Park ([email protected]) 8/8/2011 23
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