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