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![Page 1: Structured Cohesive Zone Crack Model Michael P Wnuk College of Engineering and Applied Science University of Wisconsin - Milwaukee.](https://reader036.fdocument.pub/reader036/viewer/2022062722/56649f315503460f94c4ce3f/html5/thumbnails/1.jpg)
Structured Cohesive Zone Crack Model
Michael P Wnuk
College of Engineering and Applied Science
University of Wisconsin - Milwaukee
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Preliminary Propagation of Crack in Visco-elastic or
Ductile Solid
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Constitutive Equations of Linear Visco-elastic Solid
1
0
2
0
( , )( , ) ( )
( , )( , ) ( )
tij
ij
t
e xs t x G t d
e xs t x G t d
0
( ) ( )t
relE t J d t
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Wnuk-Knauss equation for the Incubation Phase
0
2
11
0
( )( )
(0)G
a a const
KJ tt
J K
Mueller-Knauss-Schapery equation for the Propagation
Phase 2
0
( / )
(0)
o
Go
KJ a
J Ka
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E1
E2 τ2
1 = E1/E2 2 – relaxation time
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Creep Compliance for Standard Linear Solid
1 21
1( ) 1 1 exp( / )J t t
E
1 2( ) 1 1 exp( / )t t
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Solution of Wnuk-Knauss Equation for Standard
Linear Solid
0
2
11
0
( )( )
(0)G
a a const
KJ tt
J K
0
2G
Ea
2
0
Gn
1 1 21 1 exp( / )t n
11 2
1
ln1
tn
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Range of Validity of Crack Motion Phenomenon
0
0
11
threshold G
GG
1 = E1/E2
(0)
( )glassy
threshold G Grubbery
JJ
J J
0
2G
Ea
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Solution of Mueller-Knauss-Schapery equation for a
Moving Crack in SLS
1 21 1 exp( / )n
tx
1
2 1
ln1
o na
x
1
1
1
ln1
dxndx
x = a/a0 = t/2
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Crack Motion in Visco-elastic Solid
2
1 2
/
1
/ 11
1ln
1
t x
t
d dznz
2 11
11
ln1
x
t t dznz
2 1 1 11
1 1 1 1
(1 ) 1ln ln ln
(1 ) 1 1
x x n nnt t x
x n n
x = a/a0 = /a0 = t/2
t = /a a = da/dt
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n=4t1=0.375τ2 1
n=4t2=0.277τ2/δ
n=8.16t2=1.232τ2/δn=6.25
t2=0.720τ2/δ
n=6.25t1=0.744τ2
NONDIMENSIONAL TIME IN UNITS OF (τ2)
1.5 1.0 0.5 0 0.5 1.0 1.5
2
3
4
5
6
NONDIMENSIONAL TIME IN UNITS OF (τ2/ )δ
n=8.16t1=1.26τ2
2
0
Gn
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Critical Time / Life Time
1 2 1 11 2 2
1 1 1 1
1ln ln ln
1 1 1cr
n nnT t t
n n
t1 = incubation timet2 = propagation time= /a0
n = (G/0)2
1 = E1/E2
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0. 01 0.1 1 100.3
0.4
0.5
0.6
0.7
ß1 =10ß1 = 100
LOGARITHM (TIME/τ2)
NO
ND
IME
NS
ION
AL
LO
AD
, s=
σ o/σ
G
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0.01 0.1 1 100.3
0.4
0.5
0.6
0.7
LOGARITHM (CRITICAL TIME/(τ2/δ))
β1 =10β1 =100NO
ND
IME
NS
ION
AL
LO
AD
, s=
σ o/σ
G
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Material Parameters:•Process Zone Size •Length of Cohesive Zone at Onsetof Crack Growth Rini
Material Ductility
iniR
111 1
1 1
4( , ) ( ) ln
2Y
y
R R xxu x R R R x
E R R x
Profile of the Cohesive Zone (R << a)
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Wnuk’s Criterion for Subcritical Crack Growth in
Ductile Solids
2 1( ) ( ) / 2u P u P
01
1
( ) ( )4( ) ( )( ( ) ) ln
2 ( ) ( )
R Ru P R R
E R R
1 1
0 02 0
1 1
4 4( )
x x
dRu P R R
E E da
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Governing Differential Equation
1( ) ln2 2 4 Y
EdR R RR R R
da R R
ini
ini
RY
R
aX
R
1
2 4 Y
EM
11( 1) ln
2 1
Y YdYM Y Y Y
dX Y Y
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Wnuk-Rice-Sorensen Equation for Slow Crack Growth in
Ductile Solids
ini
ini
RY
R
aX
R
1 1( ) ln(4 )
2 2
dYM Y
dX
iniR
1 1( ) 1.1 ln(4 )
2 2M
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Necessary Conditions Determining Nature of
Crack Propagation
dR/da > 0, stable crack growth
dR/da < 0, catastrophic crack growth
dR/da = 0, Griffith case
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Auxiliary Relations
1
2
1
8
8
( ) 2 2 ( ) 2 2 ( )( )
Y tip
Ytip
Y
Y
J
RE
J RE
a R a Y Xa
a X
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Terminal Instability Point
1( , ) ( )
2T
ij ij i i
V S
a dV Tu dS SE a
( , ) ( )
( , ) ( )APPL MAT
APPL MAT
R a R a
R a dR a
a da
2
2
( , ) ( , )APPLR a a
a a
transition transition
dY Y
dX X
=
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Rough Crack Described by Fractal Geometry
Solution of Khezrzadeh, Wnuk and Yavari (2011) 11
1 11 1
4( , ) ( ) ( ) ln
2
f ff fY
y f f
R R xxu x R R R x
E R R x
12
1
1
12 3 22
1
1/2
( , , )
( )
( , , ) ( ) ( )
( )( ) 4 0.829 1.847 1.805 1.544
12
2 2 ( )( )
f
ftip tip
R N X Y R
N X Y N X
N
Y XX
X
1 ( 1)sin( )( )
2 (1 )
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Governing Differential Equation for Stable Growth of
Fractal Crack
1 1 1( ) ln 4 ( , , ) /
( , , ) 2 2 ini
dRM N X Y R R
da N X Y
= (2-D)/2D – fractal dimension
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10 11 12 13 141
1.2
1.4
1.6
1.8
ρ =20
ρ =40
ρ =80
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
MA
TE
RIA
L R
ES
IST
AN
CE
TO
CR
AC
K, Y
=R
/Rin
i
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S
TA
BIL
ITY
IN
DE
X, S
11 12 13 14
0.04
0.02
0
0.02
0.04
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
ρ =20
ρ =40
ρ =80
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0.29 0.3 0.3110
11
12
13
14
0.32
ρ =20
ρ =40
ρ =80
NONDIMENTIONAL TIME
NONDIMENSIONAL CRACK LENGTH, X=a/R
ini
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10 11 12 13 141
1.5
2
2.5
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
EF
FE
CT
IVE
MA
TE
RIA
L R
ES
IST
AN
CE
, Y=
R/R
ini
α =0.40
α =0.45
α =0.50
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10 11 12 13 14
0.3
0.32
0.34
0.36
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
AP
PL
IED
LO
AD
, β=
σ/σ Y
α =0.40
α =0.45
α =0.50
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α =0.40
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
11 12 13 140.01
5 103
0
5 103
0.01
ST
AB
ILIT
Y I
ND
EX
, S
α =0.45α =0.50
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α =0.40
NO
ND
IME
NS
ION
AL
CR
AC
K L
EN
GT
H, X
=a/
Rin
i
0.28 0.3 0.32 0.34 0.36 0.3810
11
12
13
14
α =0.45α =0.50
NONDIMENSIONAL TIME
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LO
AD
ING
PA
RA
ME
TE
R, Q
=πσ
/2σ Y
CRACK LENGTH, a
No growth range
STABLE GROWTH UNSTABLE GROWTH
0da
dQ
0da
dQ0
da
dQ
fQ
0Q
dQ
da0a fa
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UNSTABLE GROWTH
iQ
0a
NO GROWTH
INITIATION LOCUS(Local Instability)
RESERVE STRENGTH USED BY SMART MATERIALS WITH ENHANCED THOUGHNESS
STEADY STATE TOUGHNESSUPPER BOUND
STABLE GROWTH
I
II
III
(Global Instability)
fQ
faCRACK LENGTH, a
LO
AD
ING
PA
RA
ME
TE
R, Q
=πσ
/2σ Y
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10 11 12 13 14
0.11
0.12
0.13
NONDIMENSIONAL CRACK LENGTH, X=a/Rini
ρ =20
ρ =40
ρ =80
NO
ND
IME
NS
ION
AL
SL
OP
ES
, ∂R
APP
L/∂
a an
d dR
MA
T/d
a
0.1
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*New mathematical tools are needed to describe fracture process at the
nano-scale range*More research is needed in the nano range of fracture
and deformation
example: fatigue due to short cracks
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.02
0.04
0.06
0.08
0.1
Q
Q
Q
Q
Q
X X0 Q X0( ) f0 X0 Q X0( ) f1 X0 Q x1 X0( )( ) f2 X0 Q x2 X0( )( ) f3 X0 Q x3 X0( )( )
min
max
min
2 3
2
2 3
2
2
3 2
2
3 2
Q
Q
Q
Q
X X qdq
N Xq
X qX dq
Xq
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*New Law of Physics of Fracture Discovered:
Ten Commandments from God and one equation
from Wnuk
1log
2
dY m
dX Y