Dynamic response of a crack in a functionally gradedinterface of two dissimilar piezoelectric half-planes

J. Chen, Z. Liu, Z. Zou

Summary In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneouspiezoelectric materials bonded through a functionally graded interfacial region is considered.Integral transforms are employed to reduce the problem to Cauchy singular integral equations.Numerical results illustrate the effect of the loading combination parameter k, materialproperty distribution and crack configuration on the dynamic stress and electric displacementintensity factors. It is found that the presence of the dynamic electric field could impede ofenhance the crack propagation depending on the time elapsed and the direction of appliedelectric impact.

Keywords Functionally graded material, Piezoelectricity, Stress intensity factor, Crack

1IntroductionDue to their intrinsic coupling between mechanical deformation and electric fields, piezo-electric materials are widely used as sensors and actuators to monitor and control the dynamicbehavior of intelligent structural systems. Conventional piezoelectric devices are homogeneousor multi-layered, and they are susceptible to cracking and debonding due to uneven stressdistributions or metal-to-ceramic bonds at which stress concentration occurs. In order toimprove the integrity and reliability of piezoelectric devices, considerable research has beendirected toward the development of functionally graded piezoelectric materials (FGPMs), [1–3].In FGPMs, the material properties are continuously varied in one or more directions in order toeliminate mismatch of material properties at the interface and, thus, enhance the bondingstrength and significantly improve the fracture toughness.

Piezoelectric components are often subjected to impact mechanical and electrical loads inoperation. Furthermore, in comparison with static loads, dynamic loads are likely to producehigher stress and electric fields in piezoelectric devices. It is of great importance to investigatethe transient response of cracked piezoelectric materials and structures, therefore many studiesinvestigated the fracture behavior of conventional piezoelectric materials and structures. Thedynamic crack problem in a piezoelectric strip subjected to electro-mechanical impact wasconsidered in [4], [5] and [6], while [7] analyzed a cracked piezoelectric laminate subjected toelectromechanical impact loads. Paper [8] analyzed the interface crack between two piezo-electric layers under mechanical and electrical impacts, whereas [9] analyzed the interfacecrack of bonded piezoelectric and elastic half-space under transient electromechanical loads.Transient response of a smart laminate was investigated in [10] in which two piezoelectriclayers were bonded to an elastic layer. In [11], the mode III Yoffe-type moving crack in a FGPMstrip was considered, while in [12] a transient mode I crack in a FGPM plate, was investigated.

Archive of Applied Mechanics 72 (2003) 686 – 696 � Springer-Verlag 2003

DOI 10.1007/s00419-002-0238-5

686

Received 4 December 2001; accepted for publication 9 July 2002

J. Chen (&), Z. LiuDepartment of Engineering Mechanics, Shanghai Jiaotong University,Shanghai 200030, P.R. Chinae-mail: chenjian130@yahoo.com.cn

Z. ZouDepartment of Astronautics Engineering and Mechanics,Harbin Institute of Technology, Harbin 150001, P.R. China

This work is supported by the National Natural Science Foundation ofChina through Grant No. 10132010.

In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous pi-ezoelectric materials bonded through a thin layer of functionally graded interfacial region(FGIR) is considered. Integral transform technique is used to reduce the problem to thesolution of singular integral equations. Numerical results are shown graphically to illustrate theeffects of the loading combination parameter, material property distribution and crack con-figuration on the dynamic stress and electric displacement intensity factors (DSIF and EDIF).

2Formulation of the problemLet a crack lie in the y ¼ 0 plane of right-handed cartesian coordinate system (x; y; z) and bebounded between lines x ¼ �a as in Fig. 1. Two dissimilar homogeneous piezoelectric materials1 and 4 are bonded through a FGIR, which consists of medium 2 with thickness h1 and medium 3with thickness h2. All of these materials exhibit transversely isotropic behavior and are poledin the z-direction. An anti-plane shear impact and an electric displacement impact are imposedon the crack surfaces at t ¼ 0, represented in Fig. 1 by the Heaviside unit step function HðtÞ.

Crack problems in nonhomogeneous piezoelectric materials do not appear to be analyticallytractable for arbitrary variations of material properties. Usually, one tries to generate the formsof nonhomogeneities for which the problem becomes tractable. Similar to the treatment of thecrack problem for isotropic nonhomogeneous interface in [13, 14], we assume the materialproperties are described by:

c441 ¼ c440ebh1 ; e151 ¼ e150ebh1 ; e111 ¼ e110ebh1 ; q1 ¼ q0ebh1 ; ð1Þ

c442 ¼ c440eby; e152 ¼ e150eby; e112 ¼ e110eby; q2 ¼ q0eby ; ð2Þ

c443 ¼ c440eby; e153 ¼ e150eby; e113 ¼ e110eby; q3 ¼ q0eby ; ð3Þ

c444 ¼ c440e�bh2 ; e154 ¼ e150e�bh2 ; e114 ¼ e110e�bh2 ; q4 ¼ q0e�bh2 ; ð4Þ

where c44k; e15k; e11k; qk are the shear modulus, the piezoelectric coefficient, the dielectric para-meter and the mass density, respectively, while subscripts k ¼ 1; 2; 3; 4 refer to the half-plane 1,medium 2, medium 3 and half-plane 4, respectively.

The constitutive relation for the anti-plane state of piezoelectric medium can be written as

rzxk ¼ c44kowk

oxþ e15k

o/k

ox; rzyk ¼ c44k

owk

oyþ e15k

o/k

oy; ð5Þ

Dxk ¼ e15kowk

ox� e11k

o/k

ox; Dyk ¼ e15k

owk

oy� e11k

o/k

oy; ð6Þ

where rzik;Dik; ði ¼ x; yÞ are the anti-plane shear stress and in-plane electric displacement; wk

and /k are the mechanical displacement and electric potential, respectively.Substituting Eqs. (1)–(6) into the equations of dynamic elastic equilibrium and Gauss’ Law

of electrostatics

orzxk

oxþ

orzyk

oy¼ q

o2wk

ot2; ð7Þ

Fig. 1. Geometry of the crackproblem

687

oDxk

oxþ

oDyk

oy¼ 0 ; ð8Þ

we can obtain the governing equations

c440r2wk þ e150r2/k ¼ q0

o2wk

ot2; ð9Þ

e150r2wk � e110r2/k ¼ 0 ; ð10Þ

for k ¼ 1; 4 and

c440 r2wk þ bo

oywk

� �þ e150 r2/k þ b

o

oy/k

� �¼ q0

o2wk

ot2; ð11Þ

e150 r2wk þ bo

oywk

� �� e110 r2/k þ b

o

oy/k

� �¼ 0 ; ð12Þ

for k ¼ 2; 3. Here, r2 ¼ o2=ox2 þ o2=oy2 is the two-dimensional Laplace operator.From Eqs. (9)–(12), we can get

r2wk ¼ c�22

o2wk

ot2; r2wk ¼ 0; ðk ¼ 1; 4Þ ð13Þ

r2wk þ bo

oywk ¼ c�2

2

o2wk

ot2; r2wk þ b

o

oywk ¼ 0; ðk ¼ 2; 3Þ ð14Þ

/kðx; y; tÞ ¼e150

e110wkðx; y; tÞ þ wkðx; y; tÞ; ðk ¼ 1–4Þ ð15Þ

where c2 ¼ffiffiffiffiffiffiffiffiffiffiffiffil0=q0

pis the shear wave speed with l0 ¼ c440 þ e2

150=e110.The boundary and continuity conditions of the problem represented in Fig. 1 can be written

as

w1ðx; h1; tÞ ¼ w2ðx; h1; tÞ; /1ðx; h1; tÞ ¼ /2ðx; h1; tÞ ; ð16Þ

rzy1ðx; h1; tÞ ¼ rzy2ðx; h1; tÞ; Dy1ðx; h1; tÞ ¼ Dy2ðx; h1; tÞ ; ð17Þ

rzy2ðx; 0þ; tÞ ¼ rzy3ðx; 0�; tÞ; Dy2ðx; 0þ; tÞ ¼ Dy3ðx; 0�; tÞ ; ð18Þ

w3ðx;�h2; tÞ ¼ w4ðx;�h2; tÞ; /3ðx;�h2; tÞ ¼ /4ðx;�h2; tÞ ; ð19Þ

rzy3ðx;�h2; tÞ ¼ rzy4ðx;�h2; tÞ; Dy3ðx;�h2; tÞ ¼ Dy4ðx;�h2; tÞ ; ð20Þ

rzy2ðx; 0; tÞ ¼ �sðxÞHðtÞ; Dy2ðx; 0; tÞ ¼ �DðxÞHðtÞ; jxj < a ð21Þ

w2ðx; 0þ; tÞ ¼ w3ðx; 0�; tÞ; /2ðx; 0þ; tÞ ¼ /3ðx; 0�; tÞ; jxj > a ð22Þ

It is worth noting that the electric impermeability condition is adopted on the crack surface.The validity of this assumption has been discussed in [15], [16] and [17], among others.

3Derivation of the singular integral equationsLet the Laplace transform pair be written as:

f �ðpÞ ¼Z1

0

f ðtÞe�pt dt; f ðtÞ ¼ 1

2pi

ZBr

f �ðpÞept dp ð23Þ

688

in which Br stands for the Bromwich path of integration. The time-dependence in Eqs. (13)–(15) is eliminated by the application of transformation (23). Applying the Fourier transform,we obtain

w�kðx; y; pÞ ¼1

2p

Z1

�1

½Akða; pÞem10y þ Bkða; pÞem20y�e�iax da; ðk ¼ 1; 4Þ ð24Þ

w�kðx; y; pÞ ¼1

2p

Z1

�1

½Ckða; pÞen10y þ Dkða; pÞen20y�e�iax da; ðk ¼ 1; 4Þ ð25Þ

w�kðx; y; pÞ ¼1

2p

Z1

�1

½Akða; pÞem11y þ Bkða; pÞem21y�e�iax da; ðk ¼ 2; 3Þ ð26Þ

w�kðx; y; pÞ ¼1

2p

Z1

�1

½Ckða; pÞen11y þ Dkða; pÞen21y�e�iax da; ðk ¼ 2; 3Þ ð27Þ

/�kðx; y; pÞ ¼e150

e110w�kðx; y; pÞ þ w�kðx; y; pÞ; ðk ¼ 1–4Þ ð28Þ

where Akða; pÞ;Bkða; pÞ;Ckða; pÞ and Dkða; pÞðk ¼ 1–4Þ are the unknowns to be determined and

m10 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ p2

c22

s; n10 ¼ jaj ; ð29Þ

m20 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ p2

c22

s; n20 ¼ �jaj ; ð30Þ

m11 ¼ �b2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

� �2

þa2 þ p2

c22

s; n11 ¼ �

b2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

� �2

þa2

s; ð31Þ

m21 ¼ �b2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

� �2

þa2 þ p2

c22

s; n21 ¼ �

b2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

� �2

þa2

s: ð32Þ

The condition that w1;/1;w4 and /4 vanish at infinity requires that A1ða; pÞ;C1ða; pÞ;B4ða; pÞand D4ða; pÞ must be zero.

We introduce the following displacement and potential discontinuity functions

g1ðx; pÞ ¼o½w�2ðx;0;pÞ�w�3ðx;0;pÞ�

ox jxj < a,0 jxj > a ,

�ð33Þ

g2ðx; pÞ ¼o½/�2ðx;0;pÞ�/�3ðx;0;pÞ�

ox jxj < a,0 jxj > a .

�ð34Þ

From (33), (34) and the boundary and continuity conditions (16)–(20), we obtain

A2ða; pÞ

¼ i

aðm21 �m20Þeh1m21 ½m11ðm21 �m10Þeh2m11 �m21ðm11 �m10Þeh2m21 �

ðm11 �m21Þ½ðm11 �m10Þðm21 �m20Þeðh1þh2Þm21 � ðm11 �m20Þðm21 �m10Þeðh1þh2Þm11 �

�Za

�a

g1ðu; pÞeiau du ; ð35Þ

689

B2ða;pÞ

¼ � i

aðm11�m20Þeh1m11 ½m11ðm21�m10Þeh2m11 �m21ðm11�m10Þeh2m21 �

ðm11�m21Þ½ðm11�m10Þðm21�m20Þeðh1þh2Þm21 �ðm11�m20Þðm21�m10Þeðh1þh2Þm11 �

�Za

�a

g1ðu;pÞeiau du ; ð36Þ

C2ða; pÞ ¼i

aðn21 � n20Þeh1n21 ½n11ðn21 � n10Þeh2n11 � n21ðn11 � n10Þeh2n21 �

ðn11 � n21Þ½ðn11 � n10Þðn21 � n20Þeðh1þh2Þn21 � ðn11 � n20Þðn21 � n10Þeðh1þh2Þn11 �

�Za

�a

�e150

e110g1ðu; pÞ þ g2ðu; pÞ

� �eiau du ; ð37Þ

D2ða; pÞ

¼ � i

aðn11 � n20Þeh1n11 ½n11ðn21 � n10Þeh2n11 � n21ðn11 � n10Þeh2n21 �

ðn11 � n21Þ½ðn11 � n10Þðn21 � n20Þeðh1þh2Þn21 � ðn11 � n20Þðn21 � n10Þeðh1þh2Þn11 �

�Za

�a

�e150

e110g1ðu; pÞ þ g2ðu; pÞ

� �eiau du ; ð38Þ

Substituting Eqs. (24)–(28) into Eqs. (5) and (6) in the Laplace transform domain, we obtain

r�zy2ðx; 0þ; pÞ ¼ limy!0þ

1

2p

Z1

�1

�l0½A2ða; pÞm11em11y þ B2ða; pÞm21em21y�

þ e150½C2ða; pÞn11en11y þ D2ða; pÞn21en21y��

e�iax da ; ð39Þ

D�y2ðx; 0þ; pÞ ¼ limy!0þ

1

2p

Z1

�1

�e110½C2ða; pÞn11en11y þ D2ða; pÞn21en21y�e�iax da : ð40Þ

By means of Eqs. (35)–(38) and (21) in the Laplace transform domain, we can obtain thefollowing integral equations:

1

2p

Za

�a

g1ðu; pÞ l0h1ðu; x; pÞ �e2

150

e110h2ðu; xÞ

� �þ e150g2ðu; pÞh2ðu; xÞ

� �du ¼ � sðxÞ

p; jxj < a

ð41Þ

1

2p

Za

�a

½e150g1ðu; pÞ � e110g2ðu; pÞ�h2ðu; xÞdu ¼ �DðxÞp

; jxj < a ð42Þ

where

h1ðu;x;pÞ

¼ limy!0

Z1

�1

½m11ðm21�m10Þeh2m11�m21ðm11�m10Þeh2m21 �iaðm11�m21Þ½ðm11�m20Þðm21�m10Þeðh1þh2Þm11�ðm11�m10Þðm21�m20Þeðh1þh2Þm21 �

�½m11ðm21�m20Þeh1m21þym11�m21ðm11�m20Þeh1m11þym21 �eiaðu�xÞþby da ; ð43Þ

690

h2ðu;xÞ

¼ limy!0

Z1

�1

½n11ðn21� n10Þeh2n11 � n21ðn11� n10Þeh2n21 �iaðn11� n21Þ½ðn11� n20Þðn21� n10Þeðh1þh2Þn11 � ðn11� n10Þðn21� n20Þeðh1þh2Þn21 �

� ½n11ðn21� n20Þeh1n21þyn11 � n21ðn11� n20Þeh1n11þyn21 �eiaðu�xÞþby da ; ð44Þ

Following the method developed by Erdogan, [18], Eqs. (41) and (42) can be modified asfollows:

1

p

Za

�a

g1ðu; pÞc440

u� xl0k1ðu; x; pÞ �

e2150

e110k2ðu; xÞ

� �þ e150g2ðu; pÞ

1

u� xþ k2ðu; xÞ

� �� �du

¼ � 2sðxÞp

; jxj < a ; ð45Þ

1

p

Za

�a

e150g1ðu;pÞ1

u�xþk2ðu;xÞ

� �� e110g2ðu;pÞ

1

u�xþk2ðu;xÞ

� �� �du¼�2DðxÞ

p; jxj< a ;

ð46Þ

where

k1ðu; x; pÞ ¼Z1

0

½M1ða; u; x; pÞ þM1ð�a; u; x; pÞ � sinðu� xÞa�da ; ð47Þ

M1ða;u;x;pÞ

¼ ½m11ðm21�m10Þeh2m11 �m21ðm11�m10Þeh2m21 �½m11ðm21�m20Þeh1m21 �m21ðm11�m20Þeh1m11 �iaðm11�m21Þ½ðm11�m20Þðm21�m10Þeðh1þh2Þm11 �ðm11�m10Þðm21�m20Þeðh1þh2Þm21 �

� eiaðu�xÞ ; ð48Þ

k2ðu; xÞ ¼Z1

0

½M2ða; u; xÞ þM2ð�a; u; xÞ � sinðu� xÞa�da ; ð49Þ

M2ða; u; xÞ

¼ ½n11ðn21 � n10Þeh2n11 � n21ðn11 � n10Þeh2n21 �½n11ðn21 � n20Þeh1n21 � n21ðn11 � n20Þeh1n11

iaðn11 � n21Þ½ðn11 � n20Þðn21 � n10Þeðh1þh2Þn11 � ðn11 � n10Þðn21 � n20Þeðh1þh2Þn21 �

� eiaðu�xÞ ; ð50Þ

4Solution and intensity factorsTo solve the system of integral equations (45) and (46) we define the following normalizedquantities:

s ¼ u

a; r ¼ x

a; G1ðs; pÞ ¼ g1ðu; pÞ; G2ðs; pÞ ¼ g2ðu; pÞ ; ð51Þ

K1ðs; r; pÞ ¼ k1ðu; x; pÞ; K2ðs; rÞ ¼ k2ðu; xÞ : ð52Þ

691

Then Eqs. (45) and (46) become

1

p

Z1

�1

G1ðs; pÞc440

s� rþ l0K1ðs; r; pÞ �

e2150

e110K2ðs; rÞ

� �þ e150G2ðs; pÞ

1

s� rþ K2ðs; rÞ

� �� �ds

¼ � 2sðarÞp

; ð53Þ

1

p

Z1

�1

e150G1ðs; pÞ1

s� rþ K2ðs; rÞ

� �� e110G2ðs; pÞ

1

s� rþ K2ðs; rÞ

� �� �ds ¼ � 2DðarÞ

p;

ð54Þ

and the continuity condition (22) may be expressed as

Z1

�1

G1ðs; pÞds ¼ 0;

Z1

�1

G2ðs; pÞds ¼ 0 : ð55Þ

Based on the numerical method, [18], G1ðs; pÞ and G2ðs; pÞ are expressed as

G1ðs; pÞ ¼Rðs; pÞffiffiffiffiffiffiffiffiffiffiffiffi

1� s2p ; G2ðs; pÞ ¼

Tðs; pÞffiffiffiffiffiffiffiffiffiffiffiffi1� s2p : ð56Þ

Expanding Rðs; pÞ and Tðs; pÞ in forms of Chebyshev polynomials

Rðs; pÞ ¼X1i¼0

CiTiðsÞ; Tðs; pÞ ¼X1i¼0

DiTiðsÞ ; ð57Þ

we find that a system of linear algebraic equations can be obtained by using the Gauss-Chebyshev formula, [18],

XN

l¼1

C440

sl � rmþ l0K1ðsl; rm; pÞ �

e2150

e110K2ðsl; rmÞ

� �Rðsl; pÞ

N

þXN

l¼1

e1501

sl � rmþ K2ðsl; rmÞ

� �Tðsl; pÞ

N¼ � 2sðarmÞ

p; ð58Þ

XN

l¼1

e1501

sl � rmþ K2ðsl; rmÞ

� �Rðsl; pÞ

N�XN

l¼1

e1101

sl � rmþ K2ðsl; rmÞ

� �Tðsl; pÞ

N

¼ � 2DðarmÞp

; ð59Þ

with

XN

l¼1

pN

Rðsl; pÞ ¼ 0 ; ð60Þ

XN

l¼1

pN

Tðsl; pÞ ¼ 0 ; ð61Þ

692

where

TNðslÞ ¼ 0; sl ¼ cos2l� 1

2Np

� �; l ¼ 1; . . . ;N ; ð62Þ

UN�1ðrmÞ ¼ 0; rm cosm

Np

� ; m ¼ 1; . . . ;N � 1 : ð63Þ

We obtain thus 2N equations with 2N unknown quantities to be solved.The dynamic stress and electric displacement intensity factors are determined as

K�IIIðaÞ ¼ limx!a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx� aÞ

pr�zyðx; 0; pÞ; K�IIIð�aÞ ¼ lim

x!�a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�x� aÞ

pr�zyðx; 0; pÞ ; ð64Þ

K�DðaÞ ¼ limx!a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðx� aÞ

pD�yðx; 0; pÞ; K�Dð�aÞ ¼ lim

x!�a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�x� aÞ

pD�yðx; 0; pÞ : ð65Þ

By virtue of the property of Chebyshev polynomials (see [18])

1

p

Z1

�1

TnðuÞdu

ðu� xÞffiffiffiffiffiffiffiffiffiffiffiffiffi1� u2p ¼ � jxj

xffiffiffiffiffiffiffiffiffiffiffiffiffix2 � 1p x� jxj

ffiffiffiffiffiffiffiffiffiffiffiffiffix2 � 1p

x

" #n

n ¼ 0; 1; . . . ; jxj > 1 ; ð66Þ

we obtain

K�IIIðaÞ ¼ �ffiffiffiap

2c440Rð1; pÞ þ e150Tð1; pÞ½ �; K�IIIð�aÞ ¼

ffiffiffiap

2c440Rð�1; pÞ þ e150Tð�1; pÞ½ � ;

ð67Þ

K�DðaÞ ¼ �ffiffiffiap

2e150Rð1; pÞ � e110Tð1; pÞ½ �; K�Dð�aÞ ¼

ffiffiffiap

2e150Rð�1; pÞ � e110Tð�1; pÞ½ � :

ð68Þ

The Laplace inversion transformation of Eqs. (67) and (68) is carried out by the numericalmethod developed in [19]. When the Laplace transform f �ðpÞ can be evaluated at discrete pointgiven by

p ¼ ðxþ 1þ kÞ; k ¼ 0; 1; 2; . . . ð69Þ

we determine coefficients Cm from the following set of equations:

d� f �½ðkþ xþ 1Þd� ¼Xk

m¼0

Cmk!

ðkþ xþ 1Þðkþ xþ 2Þ � � � ðkþ xþ 1þmÞðk�mÞ! ;

ð70Þ

where d � 0 and x � �1. If the coefficients are calculated up to CM�1, an approximate value off ðtÞ can be found as

f ðtÞ ¼XM�1

m¼0

CmPð0;xÞm ½2 expðdtÞ � 1� ; ð71Þ

where Pð0;xÞm ðzÞ is a Jacobi polynomial and M is the number of terms employed. The parameters

x; d and M must be selected such that f ðtÞ can be best described within a particular range oftime t. In this paper, we present the values of KIIIða; tÞ;KIIIð�a; tÞ;KDða; tÞ and KDð�a; tÞwithin the range of c2t=a of ð0; 8Þ with the values of x ¼ 0; d ¼ 0:2;M ¼ 10:

5Results and discussionsIn this section, we investigate the dynamic response of the crack under the uniform shearimpact sðxÞ ¼ s0 and uniform electrical impact DðxÞ ¼ D0. Due to the symmetry, only the

693

numerical results as the right crack tip will be presented. The material at y ¼ 0 plane isassumed to be the piezoelectric material BaTiO3 of which the material constants arec440 ¼ 4:4� 1010 N=m2, e150 ¼ 11:4 C=m2, e110 ¼ 128:3� 10�10 C/Vm, and q0 ¼ 5700 kg=m3,[20]. The loading combination parameter k is determined as

k ¼ D0e150

s0e110

which is used to reflect the combination between the shear impact �s0HðtÞ and the electricalimpact D0HðtÞ.

In obtaining the numerical results, the number of terms N in (58)–(61) was increased untilthe values of intensity factors are computed upto three or four digits accuracy beyond thedecimal point. The largest number used was N ¼ 20. In addition, for the evaluation of thekernels in (47)–(51) which are bounded in the respective closed domains of definition of theirarguments, the Gauss–Legendre quadrature formula was employed to evaluate the relatedinfinite integrals.

For the case where only the shear impact is acting, it is found that the EDIFs are zero at anytime t. Figure 2 shows the variation of the DSIF with time for h1 ¼ h2 ¼ 0:5a. The results aresimilar to that for an inhomogeneous material without piezoelectric effect. They demonstratethat for inhomogeneous piezoelectric materials under mechanical loading alone, the piezo-electric effect in the DSIF is negligible. However, the normalized DSIF is stongly dependent onthe material property distribution, and it increases with the increase of bðh1 þ h2Þ.

For the combined load, it is found that the loading combination parameter k has no effect onthe normalized dynamic EDIF, and this means that the mechanical load has insignificantcontribution to EDIF. However, as shown in Fig. 3, the normalized dynamic EDIF is dependenton the material property distribution bðh1 þ h2Þ.

Figure 4 shows the effects of the loading combination parameter k on the DSIF forh1 ¼ h2 ¼ 0:5a and bðh1 þ h2Þ ¼ lnð5:0Þ. The general trend of these curves is that they all riseat first, reach a maximum and then oscillate about their static values. The direction of theapplied electric impact loading plays a great role in the behavior of the DSIF. It may lead to anincrease or decrease in the resulting DSIF at different loading stages, depending on the di-rection of applied electric field. Moreover, in the early stages of the electromechanical loading,the suddenly applied electric impact retards the crack growth if k > 0 or promotes it if k < 0.

Fig. 2. Normalized DSIF under mechanical impact alone forh1 ¼ h2 ¼ 0:5a

Fig. 3. Normalized dynamic EDIF under combined impactðk 6¼ 0Þ for h1 ¼ h2 ¼ 0:5a

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Figure 5 shows the normalized DSIF for different geometrical parameters h1=a in the case ofh1 þ h2 ¼ a; bðh1 þ h2Þ ¼ lnð5:0Þ and k ¼ �0:5. As the crack’s distance h1 from the stiffer half-plane 1 decreases, the DSIF increases.

6ConclusionIn this paper, the dynamic response of a functionally graded piezoelectric layer between twohalf-planes containing a finite crack subjected to electromechanical impact has been consid-ered. The analysis is based on the use of Laplace and Fourier transforms. From the results of thepresent study, following conclusions have been drawn:

(i) The DSIF is a function of the material parameters distribution, the loading combinationparameter and the distance from bonded surfaces.

(ii) The loading combination parameter has more influence on the normalized DSIF than onthe normalized EDIF.

(iii) The direction of the electric impact loading affects significantly the resulting DSIF atdifferent loading stages.

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