Structured Cohesive Zone Crack Model
Michael P Wnuk
College of Engineering and Applied Science
University of Wisconsin - Milwaukee
Preliminary Propagation of Crack in Visco-elastic or
Ductile Solid
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
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
E1
E2 τ2
1 = E1/E2 2 – relaxation time
Creep Compliance for Standard Linear Solid
1 21
1( ) 1 1 exp( / )J t t
E
1 2( ) 1 1 exp( / )t t
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
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
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
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
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
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
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
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
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)
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
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
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
Necessary Conditions Determining Nature of
Crack Propagation
dR/da > 0, stable crack growth
dR/da < 0, catastrophic crack growth
dR/da = 0, Griffith case
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
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
=
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 )
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
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
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
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
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
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
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
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
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
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
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
*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
*New Law of Physics of Fracture Discovered:
Ten Commandments from God and one equation
from Wnuk
1log
2
dY m
dX Y