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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:
 ij, j  f i  0
Di , i  q b

Boundary equations:
 ij n j  Ti
Di ni  q s

Constitutive equations:  ij  Cijkl s kl  ekij E k

Crack surface
Impermeable or permeable?
 ij ~stress tensor
force
Di ~electric displacement Ti ~traction vector
q b ~body charge
q s ~surface change
 ij ~dielectric constants
C ijkl~elastic module
e kij ~piezoelectric constants Ei ~electric field
Di  eikl s kl   ik Ek
x2

 22
 12
D2
r
f i ~body

2a
x1
Impermeable crack surface:
Boundary condition:

Traction free condition:
 12  ( x)   12  ( x)  0

 22  ( x)   22  ( x)  0
x  (  a, a )
Charge free condition:


D2 ( x )  D 2 ( x )  0
x  (  a, a )
Stress intensity factors (SIF) :

K I   22
a
K II   12 a
Electric displacement intensity factor (EDIF): K e  D2 a

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)

u p  
 



J     ij sij dx 2  ni ip
ds    Di Ei dx 2  ni Di
ds 
x1  
x1 
C

GI 
a

1.48  10
2
11
 2


 2
( 22
)  5.34  10  2  22
D22
 8.56  10 7 ( D22
)
Mechanical
Coupling
Electric, Negative?
(Park and Sun,1995, International Journal of Fracture 70, 203~216)
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

(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.
G IM 
a
1.48  10
2
11
 2


( 22
)  2.67  10  2  22
D22

(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:
 22  ( x)   22  ( x)  0

x  (  a, a )
Charge continuous condition:


D2 ( x1 )  D2 ( x1 )

 12  ( x)   12  ( x)  0
 2  ( x1 )   2  ( x1 )
Stress intensity factors (SIF) :

K I   22
a
x  (  a, a )
K II   12 a
Energy calculations for permeable crack surface:
( Xu and Rajapakse, 2001, International Journal of Solids and Structures, 38: 7643)
3
D2 
0

Im   k ( Ak1 22
 Ak 2 12 )
k 1
3
Im   k Ak 3
 D2

k 1
0
~distributed normal eclectic displacement
 k Aki ~complex coefficient in the complex potentials
D2 ( x1 )

Electric displacement intensity factor (EDIF):
G
M
 2
 2
 

 q k Ak1 ( 22 )  p k Ak 2 ( 12 )  (q k Ak 2  p k Ak1 ) 12 22

Im  
 


0
2
k 1 
( p k Ak 3 12 22  q k Ak 3 22 )( D2  D2 )
G 
E
K e  ( D2  D20 ) a
a
a
2
3
3


Im  ( s k Ak1 22
 s k Ak 2 12 )( D2  D20 )  s k Ak 3 ( D2  D20 ) 2
k 1






Energy analysis for impermeable and permeable crack
(Liu and Chen, 2002, International Journal of Fracture, 116: 15-20 )
14
Red~Impermeable
Blue~Permeable
12
E2=2.0e-3V/m
10
E2=0
E2=-2.0e-3V/m
8
Stress22=0.6Mpa
GM
6
4
2
0
-2
-4
-6
0
20
40
60
80 100 120 140 160 180
a(degree)
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.
Energy analysis for impermeable and permeable crack
(Liu and Chen,2002)(Continue)
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.
0
-1
-2
-3
GE
-4
-5
Red~Impermeable
Blue~Permeable
-6
E2=2.0e-3V/m
-7
E2=0
E2=-2.0e-3V/m
-8
Stress22=0.6Mpa
-9
-10
0
20 40
60
80 100 120 140 160 180
a(degree)
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.
End and thank You!
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