EM 388F Term Paper:

Discussion of Fracture Criterions under Impermeable and Permeable Crack Surface of Piezoelectric Materials

RONG JIAO

April 27, 2008

Outline of this topic

Fundamental equations for piezoelectric material Impermeable crack surface J-integral or Total Potential Energy Release Rate (TPERR )

Mechanical Strain Energy Release Rate (MSERR) Permeable crack surface Total Potential Energy Release Rate (TPERR )

Mechanical Strain Energy Release Rate (MSERR)

Fundamental equations for piezoelectric materials Field equations: 0, ijij f

Boundary equations:

bii qD ,

ijij Tn sii qnD

Constitutive equations: kkijklijklij EesC kikklikli EseD

Crack surface

Impermeable or permeable?

1x

a2

12

222D

2x

r

ij if

iD

bqiT

sq

ijklC ij

kije

~stress tensor ~body force

~electric displacement

~body charge

~traction vector

~surface change

~elastic module ~dielectric constants

~piezoelectric constants iE ~electric field

Impermeable crack surface:

Boundary condition: Traction free condition:

0)()( 1212 xx 0)()( 2222 xx ),( aax

Charge free condition:

0)()( 22 xDxD ),( aax

Stress intensity factors (SIF) : aK I 22 aK II 12

Electric displacement intensity factor (EDIF): aDK e 2

SIF is not a good fracture criterion for the impermeable crack since they do

not consider the influence of the electric field on fracture.

J-integral (TPERR: Total Potential Energy Release Rate)

C

iiiip

ipiijij dsx

DndxEDdsx

undxsJ

12

12

(Park and Sun,1995, International Journal of Fracture 70, 203~216)

222

72222

2222

11 )(1056.81034.5)(1048.12

DDa

GI

Why TPERR is not a good fracture criterion here?

1. Negative last term always block the crack growth

2. Do not agree with the experiment

Mechanical Coupling Electric, Negative?

(Mode I)

MSERR (Mechanical Strain Energy Release Rate) Fracture is mechanical process and it’s more reasonable to

only use the mechanical energy release rate as the fracture criterion.

222222

2211 1067.2)(1048.1

2D

aG M

I (Mode I)

(Park and Sun,1995,International Journal of Fracture 70, 203~216)

The negative electric term vanishes. Positive electric field decreases the critical stress and hence accelerate crack growth while negative electric field increases the fracture stress hence blocks crack growth. These conclusions match experimental results very well.

Permeable crack surface

Boundary condition:

Traction free condition:

0)()( 1212 xx 0)()( 2222 xx ),( aax

),( aax )()( 1212 xDxD )()( 1212 xx

Charge continuous condition:

Stress intensity factors (SIF) : aK I 22 aK II 12

Energy calculations for permeable crack surface:( Xu and Rajapakse, 2001, International Journal of Solids and Structures, 38: 7643)

3

102222322123

2212122

1222

221

))((

)()()(Im

2 k kkkk

kkkkkkkkM

DDAqAp

ApAqApAqaG

3

1

20223

022122221 )())((Im

2 kkkkkkk

E DDAsDDAsAsa

G

23

13

3

1122221

02

Im

)(ImD

A

AAD

kkk

kkkk

)( 10

2 xD ~distributed normal eclectic displacement k kiA ~complex coefficient in the complex potentials

Electric displacement intensity factor (EDIF): aDDK e )( 022

Energy analysis for impermeable and permeable crack(Liu and Chen, 2002, International Journal of Fracture, 116: 15-20 )

Mechanical part of TPERR (MSERR)

1. It’s constant for permeable crack under different electric loading.

2. It’s always bigger than that for impermeable crack, which implies the permeable boundary enhance the crack growth.

3. For the impermeable crack, MSERR is influenced by the applied electric loading: Positive electric load increases MSERR while negative decreases MSERR.

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0a (d e g re e )

- 6

- 4

- 2

0

2

4

6

8

1 0

1 2

1 4

GM

R ed ~ Im p erm eab le

E 2 = 2 .0 e -3 V /m

E 2 = -2 .0 e -3 V /m

E 2 = 0

B lu e~ P erm eab le

S tress2 2 = 0 .6 M p a

Energy analysis for impermeable and permeable crack(Liu and Chen,2002)(Continue)

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0a (d e g re e )

-1 0

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

GE

R ed ~ Im p erm eab le

E 2 = 2 .0 e -3 V /m

E 2 = -2 .0 e -3 V /m

E 2 = 0

B lu e~ P erm eab le

S tre ss2 2 = 0 .6 M p a

Electric part of TPERR 1. It’s almost zero for permeable crack

under different electric loading and this is the reason why TPERR could be used as a fracture criterion for the permeable crack.

2. For the impermeable crack, it is affected by the applied electric loading and this is the reason why TPERR could not be used as a fracture criterion for the impermeable crack.

Comments?

End and thank You!