Analogy of Stress Singularities Analysis between Piezoelectric Materials and Anisotropic Materials

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ScienceDirect Procedia Materials Science 3 (2014) 1767 – 1772

20th European Conference on Fracture (ECF20)

Analogy of stress singularities analysis between piezoelectric materials and anisotropic materials Toru SASAKIa*, Toshimi KONDOa, Takeshi TANEb a

b

Departmeno of Mechanical Engineering, Nagaoka National College of Technology, Nagaoka, Niigata, 940-8532 JAPAN Departmeno of Mechanical Engineering, Kitakyusyu National College of Technology, Kitakyushu, Fukuoka, 802-0985 JAPAN

Abstract

In recent years, intelligent or smart structures and systems have become an emerging new research area. The joint structures of piezoelectric and piezoelectric materials in intelligent structures are often used. Piezoelectric materials have been extensively used as transducers and sensors due to their intrinsic direct and converse piezoelectric effects that take place between electric fields and mechanical deformation. They are playing a key role as active components in many branches of engineering and technology. Then it is known that stress singularity frequently occurs at interface due to a discontinuity of materials. The stress singularity fields are one of the main factors responsible for debonding under mechanical or thermal loading. So many investigations of stress singularities of piezoelectric materials have been conducted until now, but its experimental studies are not so much. In this paper, with view to establish experimental evaluation method of piezoelectric stress singularities, analogy of basic formulation between piezoelectric materials and anisotropic materials are shown. And the analysis of stress singularities in piezoelectric materials containing crack or wedge is performed. Next the analysis of stress singularities in anisotropic materials is performed. Then the analogy of their analysis theory is shown. © 2014 Ltd. Open access under CC BY-NC-ND © 2014Elsevier The Authors. Published by Elsevier Ltd. license. Selection and under responsibility of the University of Science and Technology (NTNU), (NTNU), Department Selection andpeer-review peer-review under responsibility ofNorwegian the Norwegian University of Science and Technology Department of of Structural Engineering Structural Engineering. Keywords: Stress singularity, Piezoelectric materials, Anisotropic materials, Analogy;

* Corresponding author. Tel: +81-258-34-9218; Fax: +81-258-34-9700. E-mail address: [email protected]

2211-8128 © 2014 Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.285

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1. Introduction Piezoelectric materials have become an important branch of modern engineering materials with the recent development of the intelligent materials and structures. There has been considerable research on the stress singularities of piezoelectric materials by using theoretical and numerically analysis method (Sosa,1992; Chue, C.H., Chen, C.D., 2003; Xu, X.L., Rajapakse, R.K.N.D, 2000;etc). However its experimental studies are not so much. For validation of their analysis model and inverse analysis, the establishment of experimental evaluation method of stress singularities is required. In this paper, basic formulations and analysis method of singularities problem for piezoelectric materials are shown. And basic formulations and analysis method of singularities problem for anisotropic materials are shown. Then the analogy of their analysis theory is shown. And the possibilities of application to evaluation method of stress singularities are discussed. 2. Basic formulation 2.1. Basic equation for piezoelectric materials The constitutive equation of piezoelectric materials for plane problems is given as follows(Sosa,1992):

where

­H x ° ®H y ° J ¯ xy

½ ° ¾ ° ¿

§ a11 ¨ a ¨ 12 ¨ 0 ©

­° E x ® E ¯° y

½° ¾ ¿°

§ 0 ¨ © b21

H i , J ij

a12 a 22 0

0 · ­V x ¸° 0 ®V y ¸ ° a 6 6 ¸¹ W ¯ xy

½ § 0 ° ¨ ¾¨ 0 ° ¨ b ¿ © 13

b22

b21 · °Dx ¸­ ¸®D ° y ¯ 0 ¸¹

½ ° ¾. ° ¿

(1)

­V x b1 3 · ° ¸ ®V y 0 ¹° W ¯ xy

½ ° § c1 1 ¾¨ ° © 0 ¿

0 · ­° D x ¸® c 2 2 ¹ ¯° D y

½° ¾. ¿°

(2)

0 b22

are normal and shear strains,

displacements, and

a ij , b ij , c ij

Ei

are electric fields,

V i , W ij

are normal and shear stresses,

Di

are electric

are reduced material constants for piezoelectric material. Elastic equilibrium and

Gauss’s law are given by wV

x

wx

wW



xy

wV

0,

wy

y

wW



wx

xy

wy

wD x

0,

wx

wD y



wy

0.

(3)

Furthermore the strains and electric field components satisfy the compatibility relations 2

wy

2

2

w H

2

w Hx



wx

y 2



w J

xy

wE x

0,

wxwy

wy



wE y

(4)

0.

wx

The equilibrium equations may be satisfied by introducing to the stress functions U ( x , y ) : 2

V

2

w U x

wy

V

,

2

w U y

wx

2

2

,

W

xy



w U wxwy

In addition, we introduce an induction function Dx

w< wy

,

Dy



w< wx

(5)

.

< ( x, y )

such that (6)

.

which satisfies Gauss’s law. Having to satisfy the compatibility relation, we obtain the following system the differential equations, 2

( 㻸 4 㻸 2  㻸 3 )U

in which

㻸2, 㻸3

0 ,

2

( 㻸 4 㻸 2  㻸 3 )<

0

.

and 㻸 4 are the differential operators of the second, third and fourth orders which have the form:

(7)

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Toru Sasaki et al. / Procedia Materials Science 3 (2014) 1767 – 1772

㻸2

c 22

㻸3

b22

㻸4

a 22

2

w

w

2

2

 c1 1

wx

3

  b 2 1  b1 3

w

4

wx

,

2

wy

3

w

wx

4



w

3

  2 a12  a 66

w



,

2

wxwy

4

2

wx wy

w

 a11

2

4

wy

4

½ ° ° ° ° ¾ ° ° .° ° ¿

(8)

Eq.(7) is reduced to the single sixth order differential equation as follows: 2

( 㻸 4 㻸 2  㻸 3 )U

(9)

0

Eq. (9) can be solved by means of complex variables. Set the characteristic equation of Eq.(9) is a 1 1 c1 1 P



6

2

U



2

 a 1 1 c 2 2  2 a 1 2 c1 1  a 6 6 c 1 1  b 2 1  b1 3  2 b 2 1 b1 3 P

 2 b 2 1 b 2 2  2 b1 3 b 2 2 P



2

 a 22 c 22  b22

2



0.

U ( zk ) 4

U (x  Pk y)

and P is a complex number,

  a 2 2 c1 1  2 a 1 2 c 2 2  a 6 6 c 2 2 ½ ° ¾ ° ¿

(10)

2.2. Basic equation for anisotropic materials The constitutive equation of anisotropic materials is given as follows(Lekhnitskii ,1981): ­Hx ° H ° y ° ® J yz °J ° zx J ° ¯ xy

E ij

where

ª E 11 « E « 12 « E 14 « « E 15 «E ¬ 16

½ ° ° ° ¾ ° ° ° ¿

si3 s3 j

s ij 

s33

E 12

E 14

E 15

E 16 º ­ V x ½

E 22

E 24

E 25

E 26

E 24

E 44

E 45

E 46

E 25

E 45

E 55

E 56

E 26

E 46

E 56

E 66

» » » » » » ¼

° ° ° ¾. ° ° ° ¿

(11)

(12)

.

are elastic compliance constants,

s ij

° V ° y ° ®W y z °W ° zx W ° ¯ xy

E ij

are reduced elastic compliance constants. Elastic equilibrium are

given by wV

x

wx

wW



xy

wW

0,

wy

xy

wx



wV

y

0,

wy

wW

zx

wx



wW

yz

wy

(13)

0.

Furthermore the strains satisfy the compatibility relations 2

2

wy wJ

zx

wy





2

w H

2

w Hx

wx wJ

y 2

w J



yz

xy

(14)

0.

wxwy

(15)

0.

wx

The equilibrium equations may be satisfied by introducing to the two stress functions F ( x , y ),\ 2

V

2

w F x

W zx

wy

w\ wy

, V

2

, W

w F y

yz

wx



2

w\ wx

( x, y )

:

2

, W

xy



w F wxwy

.

(16) (17)

.

Having to satisfy the compatibility relation, we obtain the following system the differential equations, L 4 F  L 3\

in which

0 , L 3 F  L 2\

L 2 , L 3 and L 4

0.

are the differential operators of the second, third and fourth orders which have the form:

(19)

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Toru Sasaki et al. / Procedia Materials Science 3 (2014) 1767 – 1772

L2

s 44

w

2

wx

L3

 s 24

L4

s 22

 2 s 45

2

w

wxwy

3

 2 s 26

4

w

w

 s55 w

 ( s 25  s 46 )

4

wx

2

3

wx w

w

2

wy

2

,

3

 ( s1 4  s 5 6 )

2

wx wy

4

 ( 2 s1 2  s 6 6 )

3

wx wy

w

w

3

wxwy

 s1 5

2

4

2

2

wx wy

 2 s1 6

w

w

3

wy

,

3

4

wxwy

3

 s1 1

w

4

wy

4

½ ° ° ° ° ¾ ° ° . ° ° ¿

(20)

Eq.(19) is reduced to the single sixth order differential equation as follows: 2

( L4 L2  L3 ) F

(21)

0.

Eq. (21) can be solved by means of complex variables. Set the characteristic equation of Eq.(21) is 2

l 4 ( Pˆ ) ˜ l 2 ( Pˆ )  l 3 ( Pˆ ) l 2 ( Pˆ ) l 3 ( Pˆ ) l 4 ( Pˆ )

F

F ( zk )

F ( x  Pˆ k y )

(22)

0.

s 5 5 Pˆ

2

 2 s 4 5 Pˆ  s 4 4 ,

s 1 5 Pˆ

3

2

s 1 1 Pˆ

4

 ( s 1 4  s 5 6 ) Pˆ 3

 2 s 1 6 Pˆ

and Pˆ is a complex number,

½ ° ¾ ° . ¿

 ( s 2 5  s 4 6 ) Pˆ  s 2 4 ,

 ( 2 s 1 2  s 6 6 ) Pˆ

2

 2 s 2 6 Pˆ  s 2 2

(23)

2.3. Analgy of basic formulation The solution of Eqs.(9) can be written by as 3

U

2 Re

k

where

( z k ), <

k

2 Re

1

¦

1, 2 , 3)

^

2

 ( b 2 1  b1 3 ) P k  b 2 2

(24)

O k U zc ( z k ).

1

k

x  P k y (k

zk

Ok

3

¦U

and complex constants

` c

P k  c 22 , 2

11

Ok

(k

are as follows; 1, 2 , 3).

(25)

And the solution of Eqs.(21) can be written by as 3

F ( x, y )

2 Re

3

¦

Fk ( z k ) ,

2 Re

1

k

where complex constants Oˆ k

\ ( x, y )

¦ k

Oˆ k

 l 3 ( Pˆ k ) l 2 ( Pˆ k )

Oˆ k F kc ( z k ) .

(26)

1

are as follows;  l 4 ( Pˆ k ) l 3 ( Pˆ k )

Then by introducing new function

Ik

(k

1, 2 , 3 ) .

(27)

which are defined as

(28) I k ( z z ) U kc ( z k ) F kc ( z k ), ( k 1, 2 , 3) . the components of stress, electric displacement, displacement, electric potential etc. are derived as Table.1. Table.1 show that the components of in-plane stress etc. are identical formulation, components of electric displacement and out-of-plane stress are analogical formulation. These analogies indicate that experiment of piezoelectric materials can be replaced by experiment of anisotropic elastic materials. It is very useful because experiment of piezoelectric materials is very sensitive for its environment and specimen. 3. Analogy of stress singularities analysis 3.1. Crack problem In the crack problem,we seek the expressions for the function I k in the following form(Lekhnitskii ,1981): Ik ( zk )

where

A ij

1 det A

f

¦ >A

1k

m

a m  A 2 k b m  A3 k c m

@

1

are cofactors of the following matrix.

]

m k

.

(22)

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Toru Sasaki et al. / Procedia Materials Science 3 (2014) 1767 – 1772

ª1 « P « 1 «O ¬ 1

A

1 º » P3 » O 3 »¼

1

P2 O2

And the coefficient

(23)

a m , bm , c m

are determined from the boundary condition.

Table 1. Basic formulations of piezoelectric material and anisotropic elastic material.

Piezoelectric Material

Anisotropic Elastic Material

3

V

3

¦

2 Re

x

k

2

P k I kc ( z k ) ,

V

¦

2 Re

x

k

1

1

3

V

3

¦ I c(z

2 Re

y

k

k

k

V

),

¦ I c(z

2 Re

y

k

k

1

k

P k I kc ( z k ).

W xy

k

3

¦ k

O k P k I kc ( z k ),

W zx

k

1

3

3

¦ k

O k I kc ( z k ).

W

k

2

½ ° a12 P k  a 22  b 22 O k P k ,° ¾ °  b1 3  c 1 1 O k P k , ° ( k 1, 2 , 3 ) . ¿



qk rk

 a12  b 21 O k ,



2



qˆ k rˆk

½ ° ° ° 1 2 { s 1 2 Pˆ k  s 2 2  s 2 6 Pˆ k  O k ( s 2 5 Pˆ k  s 2 4 )} , ° Pˆ k ¾ ° 1 2 { s 1 4 Pˆ k  s 2 4  s 4 6 Pˆ k  O k ( s 4 5 Pˆ k  s 4 4 )} , ° Pˆ k ° ° ( k 1, 2 , 3 ) . ¿ 1

2

Pˆ k

{ s 1 1 Pˆ k  s 1 2  s 1 6 Pˆ k  O k ( s 1 5 Pˆ k  s 1 4 )} ,



3

r Px

3

¦

2 Re

k

P k I k ( z k ),

r Px

2 Re

k

k

k

( z k ).

r Py

¦I

2 Re

k

1



3

¦

2 Re

k

O k I k ( z k ).

r Pz

k

2 Re

p k I k ( z k )  Z xy y  u 0 ,

u

2 R e ¦ pˆ k I k ( z k )  Z x y y  u 0 ,

2 Re

¦ k

1

k

1 3

v

3

q k I k ( z k )  Z xy x  v 0 .

v

2 Re

1

¦ k

2 Re

3

¦ k

rˆk I k ( z k )  Z x y x  v 0 ,

1



3

I

Oˆ k I k ( z k ).

1

3

¦ k

( z k ),



3

u

¦

2 Re

1



k

1



3

r Dn

Pˆ k I k ( z k ),

1

3

¦I

2 Re

¦

1

3

r Py

Oˆ k I kc ( z k ).

1

pˆ k

a11 P k

¦

2 Re

yz

1

pk

Oˆ k Pˆ k I kc ( z k ),

¦

2 Re

1

2 Re

Dy

Pˆ k I kc ( z k ) ,

1



3

2 Re

¦

2 Re

1

Dx

),

3

¦

2 Re

k

1

3

W xy

2

Pˆ k I kc ( z k ) ,

1

rk I k ( z k ).

w

2 Re

¦ k

1

rˆk I k ( z k ).

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Toru Sasaki et al. / Procedia Materials Science 3 (2014) 1767 – 1772

For piezoelectric materials, boundary condition are given as T r a c tio n fr e e :

Px

0 , Py

E le c tr ic a lly o p e n :

Dn

0, ½ ° ¾ 0 . ° ¿

on

(24)

crack .

And for anisotropic materials, boundary condition are given as T r a c tio n fr e e : Px

0 , Py

0 , Pz

0,

on

(25)

crack .

3.2. Wedge problem In the wedge problem, we seek the expressions for the function in the following form(Chue, C.H., Chen, C.D., 2003): Ik ( zk )

C k zk

O

 Dk zk

O

where is eigenvalue. The coefficient C k , D k , O are determined from the boundary condition. For piezoelectric materials, boundary condition are given as T r a c tio n fr e e : Px E le c tr ic a lly o p e n :

Py Dn

½° ¾ 0 . °¿

0,

on

(26)

edge 1

2

C o n tin u ity o f s tr e s s a n d e le c tr ic d is p la c e m e n t : Px

1

2

Px , P y

C o n tin u ity o f d is p la c e m e n t a n d e le c tr ic p o te n tia l : u

1

2

u ,v

1

2

Py , D n 1

2

v ,I

Dn , 1

2

I .

½° ¾ °¿

o n in te r fa c e

(27)

And for anisotropic materials, boundary condition are given as T r a c tio n fr e e : Px

Py 1

C o n tin u ity o f s tr e s s : P x

Pz

0,

2

1

Px , P y

C o n tin u ity o f d is p la c e m e n t : u

1

on 2

2

Py , D n 2

u ,v

1

(28)

e d g e.

1

Dn , 2

v ,w

1

½° ¾ w . °¿ 2

o n in te r fa c e

(29)

4. Conclusions Analogies of basic formulation and governing equation between piezoelectric materials and anisotropic materials were shown. The components of stress, electric displacement, displacement, electric potential etc. are derived by using these analogies. Analytical methods for crack and wedge problem were derived by unifying formulation. In the future work, we will establish experimental evaluation method of piezoelectric stress singularities by using these analogies. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 25820004. References Horacio Sosa, 1992. On the fracture mechanics of piezoelectric solids. International Journal of Solids and Structures 29, 2613–2622. Chue, C.H., Chen, C.D., 2003. Antiplane stress singularities in a bonded biomaterial piezoelectric wedge. Archive of Applied Mechanics 72, 673–685. Xu, X.L., Rajapakse, R.K.N.D, 2000. On singularities in composite piezoelectric wedges and junctions. International Journal of Solids and Structures 37, 3253–3275. Tong-Yi Zhang, Pin Tong, 1996. Fracture mechanics for a mode III crack in a piezoelectric material. International Journal of Solids and Structures 33, 343-359. Lekhnitskii, S.G., 1981. Theory of Elasticity of an Anisotropic Body, Mir Publishers, Moscow.