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Procedia Materials Science 3 (2014) 1767 1772

Available online at www.sciencedirect.com

2211-8128 2014 Elsevier Ltd. Open access underCC 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

ScienceDirect

20th European Conference on Fracture (ECF20)

Analogy of stress singularities analysis between piezoelectric

materials and anisotropic materials

Toru SASAKIa*, Toshimi KONDO

a, Takeshi TANE

b

aDepartmeno of Mechanical Engineering, Nagaoka National College of Technology, Nagaoka, Niigata, 940-8532 JAPANbDepartmeno 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 conversepiezoelectric 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 analysisof stress singularities in piezoelectric materials containing crack or wedge is performed. Next the analysis of stresssingularities in anisotropic materials is performed. Then the analogy of their analysis theory is shown.

2014 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of

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:trsasaki@nagaoka-ct.ac.jp

2014 Elsevier Ltd. Open access underCC BY-NC-ND license.

Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department

of Structural Engineering

http://creativecommons.org/licenses/by-nc-nd/3.0/http://creativecommons.org/licenses/by-nc-nd/3.0/http://creativecommons.org/licenses/by-nc-nd/3.0/http://creativecommons.org/licenses/by-nc-nd/3.0/http://crossmark.crossref.org/dialog/?doi=10.1016/j.mspro.2014.06.285&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.mspro.2014.06.285&domain=pdf

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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 ofstress 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):

1 1 1 2 21

1 2 2 2 2 2

6 6 1 3

0 0

0 0 .

0 0 0

x x

x

y y

y

xy xy

a a bD

a a bD

a b

(1)

13 1 1

2 1 2 2 22

0 0 0.

0 0

x

x x

y

y y

xy

E Db c

E Db b c

(2)

where ,i ij

are normal and shear strains,i

E are electric fields, ,i ij

are normal and shear stresses,i

D are electric

displacements, and , ,ij i j i j

a b c are reduced material constants for piezoelectric material. Elastic equilibrium and

Gausss law are given by

0 , 0 , 0 .xy y xy yx x DD

x y x y x y

(3)

Furthermore the strains and electric field components satisfy the compatibility relations2 22

2 20 , 0 .

y xy yx xEE

x y y xy x

(4)

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

2 2 2

2 2, , .

x y xy

U U U

x yy x

(5)

In addition, we introduce an induction function ( , )x y such that

, .x yD Dy x

(6)

which satisfies Gausss law. Having to satisfy the compatibility relation, we obtain the following system thedifferential equations,

2 2

4 2 3 4 2 3( ) 0 , ( ) 0 .U (7)

in which2 3

, and4

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

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2 2

2 2 2 1 12 2

3 3

3 2 2 21 133 2

4 4 4

4 2 2 1 2 6 6 1 14 2 2 4

,

,

2 .

c cx y

b b bx x y

a a a ax x y y

(8)

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

4 2 3( ) 0U (9)

Eq. (9) can be solved by means of complex variables. Set ( ) ( )k k

U U z U x y and is a complex number,

the characteristic equation of Eq.(9) is

6 2 2 4

11 11 11 22 12 11 66 11 21 13 21 13 22 11 12 22 66 22

2 2

21 22 13 22 22 22 22

2 2 2

2 2 0 .

a c a c a c a c b b b b a c a c a c

b b b b a c b

(10)

2.2.Basic equation for anisotropic materials

The constitutive equation of anisotropic materials is given as follows(Lekhnitskii ,1981):

1 1 1 2 1 4 1 5 1 6

1 2 2 2 2 4 2 5 2 6

1 4 2 4 4 4 4 5 4 6

1 5 2 5 4 5 5 5 5 6

1 6 2 6 4 6 5 6 6 6

.

x x

y y

yz yz

zx zx

xy xy

(11)

3 3

3 3

.i j

i j i j

s ss

s (12)

where ijs are elastic compliance constants, ij are reduced elastic compliance constants. Elastic equilibrium aregiven by

0 , 0 , 0 .xy xy y yzx zx

x y x y x y

(13)

Furthermore the strains satisfy the compatibility relations2 22

2 20 .

y xyx

x yy x

(14)

0 .yzzx

y x

(15)

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

2 2 2

2 2, , .

x y xy

F F F

x yy x

(16)

, .zx yz

y x

(17)

Having to satisfy the compatibility relation, we obtain the following system the differential equations,

4 3 3 20 , 0 .L F L L F L (19)

in which2 3

,L L and4

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

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2 2 2

2 4 4 4 5 5 52 2

3 3 3 3

3 2 4 2 5 4 6 1 4 5 6 1 53 2 2 3

4 4 4 4 4

4 2 2 2 6 1 2 6 6 1 6 1 14 3 2 2 3 4

2 ,

( ) ( ) ,

2 ( 2 ) 2 .

L s s sx yx y

L s s s s s sx x y x y y

L s s s s s sx x y x y x y y

(20)

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

4 2 3( ) 0 .L L L F (21)

Eq. (21) can be solved by means of complex variables. Set ( ) ( )k k

F F z F x y and is a complex number,

the characteristic equation of Eq.(21) is2

4 2 3 ( ) ( ) ( ) 0 .l l l (22)

2

2 5 5 4 5 4 4

3 2

3 15 1 4 5 6 2 5 4 6 2 4

4 3 2

4 1 1 1 6 1 2 6 6 2 6 2 2

( ) 2 ,

( ) ( ) ( ) ,

( ) 2 (2 ) 2 .

l s s s

l s s s s s s

l s s s s s s

(23)

2.3.Analgy of basic formulation

The solution of Eqs.(9) can be written by as3 3

1 1

2 R e ( ) , 2 R e ( ).k k k z k

k k

U U z U z

(24)

where ( 1, 2 , 3)k k

z x y k and complex constantsk

are as follows;

2 2

21 13 22 11 22( ) , ( 1, 2, 3).

k k kb b b c c k (25)

And the solution of Eqs.(21) can be written by as3

1

( , ) 2 R e ( ) ,k k

k

F x y F z

3

1

( , ) 2 R e ( ) .k k k

k

x y F z

(26)

where complex constants k

are as follows;

3 2 4 3

( ) ( ) ( ) ( ) ( 1, 2 , 3) .k k k k k

l l l l k (27)

Then by introducing new functionk

which are defined as

( ) ( ) ( ), ( 1, 2 , 3) .k z k k k k

z U z F z k (28)

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 functionk

in the following form(Lekhnitskii ,1981):

1 2 31

1( ) .

d e t

m

k k k m k m k m k

m

z A a A b A cA

(22)

whereij

A are cofactors of the following matrix.

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1 2 3

1 2 3

1 1 1

A

(23)

And the coefficient , ,m m m

a b c are determined from the boundary condition.

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

Piezoelectric Material Anisotropic Elastic Material

3

2

1

3

1

3

1

2 R e ( ) ,

2 R e ( ) ,

2 R e ( ).

x k k k

k

y k k

k

xy k k k

k

z

z

z

3

2

1

3

1

3

1

2 R e ( ) ,

2 R e ( ) ,

2 R e ( ) ,

x k k k

k

y k k

k

xy k k k

k

z

z

z

3

1

3

1

2 R e ( ) ,

2 R e ( ).

x k k k k

k

y k k k

k

D z

D z

3

1

3

1

2 R e ( ) ,

2 R e ( ).

zx k k k k

k

yz k k k

k

z

z

2

1 1 1 2 2 1

2

1 2 2 2 22

1 3 1 1

,

,

,

( 1, 2 , 3 ) .

k k k

k k k k

k k k

p a a b

q a a b

r b c

k

2

1 1 1 2 1 6 1 5 1 4

2

1 2 2 2 2 6 2 5 2 4

2

1 4 24 4 6 4 5 4 4

1 { ( )} ,

1 { ( )} ,

1 { ( )} ,

( 1, 2 , 3 ) .

k k k k k

k

k k k k k

k

k k k k k

k

p s s s s s

q s s s s s

r s s s s s

k

3

1

3

1

2 R e ( ) ,

2 R e ( ).

x k k k

k

y k k

k

P z

P z

3

1

3

1

2 R e ( ) ,

2 R e ( ) ,

x k k k

k

y k k

k

P z

P z

3

1

2 R e ( ).n k k k

k

D z

3

1

2 R e ( ).z k k k

k

P z

3

0

1

2 R e ( ) ,k k k x y

k

u p z y u

3

0

1

2 R e ( ) .k k k x y

k

v q z x v

3

0

1

3

0

1

2 R e ( ) ,

2 R e ( ) ,

k k k xy

k

k k k x y

k

u p z y u

v r z x v

3

1

2 R e ( ).k k k

k

r z

3

1

2 R e ( ).k k k

k

w r z

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For piezoelectric materials, boundary condition are given as

: 0 , 0 ,.

: 0 .

x y

n

T r a c t io n f r e e P P o n c ra ck

E le c tr ic a ll y o p e n D

(24)

And for anisotropic materials, boundary condition are given as

: 0 , 0 , 0 , .x y z

T r a c ti o n fr e e P P P o n c r a c k (25)

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):

( )k k k k k k

z C z D z

where is eigenvalue. The coefficient , ,k k

C D are determined from the boundary condition.

For piezoelectric materials, boundary condition are given as

: 0 ,

: 0 .

x y

n

T ra c tio n fr e e P P o n e dg e

E le c tr ic a ll y o p e n D

(26)

1 2 1 2 1 2

1 2 1 2 1 2

: , , ,

: , , .

x x y y n nC o n ti n u it y o f s tr es s a n d e le ct r ic d is pla ce m e n t P P P P D D

o n i n t e r fa c e

C o n ti n u it y o f d i sp l a ce m e n t a n d e le c tr ic p o t en t ia l u u v v

(27)

And for anisotropic materials, boundary condition are given as

: 0 , .x y z

T r a c t io n fr e e P P P o n e d g e (28)

1 2 1 2 1 2

1 2 1 2 1 2

: , , ,

: , , .

x x y y n nC o n t in u i ty o f s t r e s s P P P P D D

o n i n t e r f a c e

C o n tin ui ty o f d is p la ce m e n t u u v v w w

(29)

4. Conclusions

Analogies of basic formulation and governing equation between piezoelectric materials and anisotropic materialswere 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, 26132622.

Chue, C.H., Chen, C.D., 2003. Antiplane stress singularities in a bonded biomaterial piezoelectric wedge. Archive of Applied Mechanics 72,

673685.

Xu, X.L., Rajapakse, R.K.N.D, 2000. On singularities in composite piezoelectric wedges and junctions. International Journal of Solids and

Structures 37, 32533275.

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.