ZAMM � Z. Angew. Math. Mech. 82 (2002) 1, 43––49

Shindo, Y.; Moribayashi, H.; Narita, F.

Scattering of Antiplane Shear Wavesby a Circular Piezoelectric Inclusion Embeddedin a Piezoelectric Medium Subjected to a Steady-State Electrical Load

The dynamic theory of antiplane piezoelectricity is applied to solve the problem of a circular piezoelectric inclusionembedded in an infinite piezoelectric matrix subjected to horizontally polarized shear waves and a steady-state inplaneelectrical load. The problem is formulated by means of the wave function expansion method. Numerical calculations arecarried out for the dynamic stress and electric field concentrations, and the results are presented in graphical form.

Key words: elasticity, piezoelectricity, wave propagation, circular inclusion, antiplane shear, inplane electric field, stressconcentration, electric field concentration, smart material systems

MSC (2000): 74J20, 78A45

1. Introduction

Significant progress has recently been made in the development of piezoelectric ceramics as a result of the extensiveresearch efforts [1, 2]. In view of their intrinsic electro-mechanical coupling characteristics and the potential for use inapplications involving smart and adaptive material systems, piezoelectric ceramics are receiving increased attentionfrom the scientific community. The mechanical reliability of piezoelectric ceramics becomes increasingly important asthey are used in more and more sophisticated areas. The piezoelectric ceramics have a tendency to develop criticalcrack growth because of stress concentrations induced by mechanical and/or electrical loads. Also, the original defectsembedded in the material, e.g. delamination, inclusions, and voids, have a dominant influence on the failure of compo-nents. Therefore, it is of vital importance to study the electroelastic fields as a result of the presence of defects orinhomogeneities in these quasi-brittle solids. Based on the use of exact electric boundary conditions at the rim of thehole, Sosa [3] has analyzed the problem of a piezoelectric body with an elliptic cavity within the framework of two-dimensional electro-elasticity. In the work by Wang [4], a solution was developed for an infinite, piezoelectric mediumcontaining a piezoelectric, ellipsoidal inclusion. Pak [5] has analyzed the piezoelectric antiplane problem in which acircular piezoelectric inclusion embedded in a piezoelectric matrix is subjected to a far-field mechanical and an electri-cal load.

Interest in the wave propagation problems involving piezoelectric composites has also grown appreciably becauseof various applications. The problem of the propagation of elastic waves has been extensively studied by many authors[6––10]. In this study, the dynamic theory of linear piezoelectricity is applied to investigate the scattering of horizon-tally polarized shear waves by a circular piezoelectric inclusion in an infinite piezoelectric matrix under a harmonicinplane electric field. The wave function expansion method is used to formulate the problem. This analysis also enablesus to examine the dynamic stress and electric field concentrations due to the presence of an inclusion. This study isuseful in designing piezoelectric composites and in reducing the problem of dielectric breakdowns that frequently occurduring a poling process. Numerical results for the dynamic stress and electric field concentrations are computed, andthey are plotted in terms of the piezoelectric material constants, frequency, and applied electric field.

2. Problem statement and method of solution

We consider a single circular piezoelectric inclusion of radius a in an infinite piezoelectric matrix subjected to an anti-plane elastic wave and a harmonic inplane electric field. In order to study the dynamic stress and electric field concen-trations around the circular piezoelectric inclusion, we first consider the scattered field due to the inclusion. The geome-try is depicted in Fig. 1 where ðx; y; zÞ is the Cartesian coordinate system with origin at the center of the piezoelectriccylinder and (r; q; z) is the corresponding cylindrical coordinate system. The matrix and the inclusion are assumed tohave different material properties, but they are assumed to have the same material orientation in that they have bothbeen poled along the z-direction. It is also assumed that the circular piezoielectric inclusion is perfectly bonded withthe piezoelectric matrix. The piezoelectric boundary value problem simplifies considerably if we only consider the out-of-plane displacement and the in-plane electric fields such that

udr ¼ ud

q ¼ 0 ; udz ¼ ud

z ðr; q; tÞ ðd ¼M; IÞ ; ð1Þ

Edr ¼ Ed

r ðr; q; tÞ ; Edq ¼ Ed

qðr; q; tÞ Edz ¼ 0 ðd ¼M; IÞ ð2Þ

Shindo, Y. et al.: Scattering of Shear Waves in a Piezoelectric Medium 43

# WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0044-2267/02/0101-0043 $ 17.50þ.50/0

where ðudr ; u

dq; u

dz Þ and ðEd

r ; Edq ; E

dz Þ are the components of displacement and electric field vectors, and the superscripts

M and I refer to the quantities in the matrix and the inclusion, respectively.Constitutive relations for crystal class 6 mm can be written as [11]

sdrz ¼ 2cd

44edrz � ed

15Edr ;

sdqz ¼ 2cd

44edqz � ed

15Edq

ðd ¼M; IÞ ; ð3Þ

Ddr ¼ 2ed

15edrz þ Ed

11Edr ;

Ddq ¼ 2ed

15edqz þ Ed

11Edq

ðd ¼M; IÞ ð4Þ

where sdrz; sd

qz and Ddr ; D

dq are the components of the stress tensor and electric displacement vector, cd

44 is the elasticstiffness constant measured in a constant electric field, Ed

11 is the dielectric constant measured at constant strain, anded15 is the piezoelectric constant. The shear strains are defined as

2edrz ¼ ud

z; r ;

2edqz ¼

1

rudz; q

ðd ¼M; IÞ ð5Þ

where a comma denotes partial differentiation with respect to the coordinate or the time t. The electric field compo-nents may be written in terms of an electric potential jdðr; q; tÞ as

Edr ¼ �jd

; r ;

Edq ¼ � 1

rjd; q

ðd ¼M; IÞ : ð6Þ

The governing equations are obtained as

cd44r2ud

z þ ed15r2jd ¼ rdu

d

z; tt ;

ed15r2ud

z � Ed11r2jd ¼ 0

ðd ¼M; IÞ ð7Þ

where r2 ¼ @2=@r2 þ ð1=rÞ ð@=@rÞ þ ð1=r2Þ ð@2=@q2Þ is the two-dimensional Laplacian operator in the variables r and q,and rd is the mass density. The continuity conditions across the matrix-inclusion interface are

uMz ¼ uIz ;sMrz ¼ sIrz ;

jM ¼ jI ;

DMr ¼ DIr

ðr ¼ aÞ : ð8Þ

Let us consider a plane shear (longitudinal shear, SH) wave polarized in the z-direction and propagating in thepositive x-direction. Thus,

uiz ¼ uz0 exp½iðkMSHx� wtÞ ;

ji ¼ eM15EM11

� �uz0 exp½iðkMSHx� wtÞ

ð9Þ

44 ZAMM � Z. Angew. Math. Mech. 82 (2002) 1

Fig. 1. Piezoelectric inclusion subjected to an incident wave and a harmonicinplane electric field

where the superscript i stands for the incident component, w is the circular frequancy of the wave, uz0 is the amplitudeof the incident SH wave, kMSH ¼ w=cMSH is the wave number in the piezoelectric matrix, cMSH ¼ ðCccM44=rMÞ1=2 is the shearwave speed, and CccM44 ¼ cM44 þ ðeM15Þ

2=EM11 is the piezoelectrically stiffened elastic constant. The shear stresses sixz, siyz,electric fields Eix, E

iy, and electric displacements Dix, D

iy associated with fields of eq. (9) take the form

sixz ¼ t0m exp½iðkMSHx� wtÞ ; siyz ¼ 0 ;

Eix ¼ E0m exp½iðkMSHx� wtÞ ; Eiy ¼ 0 ; Dix ¼ 0 ; Diy ¼ 0ð10Þ

where

t0m ¼ CccM44iuz0kMSH ; E0m ¼ � eM15

EM11

� �iuz0k

MSH : ð11Þ

Consider now the case where the piezoelectric matrix is subjected to harmonic electrical load Ex ¼ E0e exp½�iwt.In this case, it can be shown that

uez ¼ 0 ; je ¼ �E0ex exp½�iwt ð12Þ

where the superscript e stands for the electrically induced component. The shear stresses sexz, seyz, electric fields Eex, Eey,

and electric displacements Dex, Dey corresponding to je can be obtained as

sexz ¼ �eM15E0e exp½�iwt ; seyz ¼ 0 ;

Eex ¼ E0e exp½�iwt ; Eey ¼ 0 ;

Dex ¼ EM11E0e exp½�iwt ; Dey ¼ 0 :

ð13Þ

The expressions uiz þ uez and ji þ je represent the displacement and electric potential fields in the piezoelectricceramic in the absence of a piezoelectric inclusion, respectively. The latter displacement and electric potential solutionsare the scattered waves which are denoted by usz and js. The complete solutions are given by

uMz ¼ uiz þ uez þ usz ; ð14Þ

jM ¼ ji þ je þ js ð15Þ

where the superscript s denotes the scattered component within the piezoelectric matrix. The conditions (8) on r ¼ acan be reduced to the following boundary conditions:

usz þ uiz þ uez � utz ¼ 0 ; ssrz þ sirz þ serz � strz ¼ 0 ;

js þ ji þ je � jt ¼ 0 ; Dsr þDir þDez �Dtr ¼ 0ð16Þ

where the superscript t denotes the transmitted component within the circular piezoelectric inclusion.The displacement and electric potential fields in the piezoelectric matrix and the circular piezoelectric inclusion

may be expressed [12] in the forms

usz ¼P1l¼�1

AlHlðkMSHrÞ exp½ilq exp½�iwt ; ð17Þ

js ¼P1l¼0

Blr�l exp½ilq exp½�iwt þ e

M15

EM11

P1l¼�1

AlHlðkMSHrÞ exp½ilq exp½�iwt ; ð18Þ

utz ¼P1l¼�1

ClJlðkISHrÞ exp½ilq exp½�iwt ; ð19Þ

jt ¼P1l¼0

Dlrl exp½ilq exp½�iwt þ e

M15

EM11

P1l¼�1

ClJlðkISHrÞ exp½ilq exp½�iwt ð20Þ

where Al, Bl, Cl, and Dl are the unknowns to be solved, Hlð Þ is the lth order Hankel function of the first kind andJlð Þ is the lth order Bessel function of the first kind, kISH is the wave number in the piezoelectric inclusion. Eqs. (17),(18) and (19), (20) are solutions of the governing eq. (7) for d ¼M and d ¼ I, respectively. Note that kISH ¼ w=cISH,cISH ¼ ðCccI44=rIÞ

1=2, CccI44 ¼ cI44 þ ðeI15Þ2=EI11 where cISH is the shear wave speed and CccI44 is the piezoelectrically stiffened elas-

tic constant. In what follows, the time factor exp½�iwt will be omitted from all the field quantities.

Shindo, Y. et al.: Scattering of Shear Waves in a Piezoelectric Medium 45

From boundary conditions (16), the unknowns Al, Bl, Cl, and Dl are found to be

Al ¼ ð�lÞI A�l ¼iluz0PlQl

� ðlþ 1Þ ðeI15Þ2 FJlðkISHaÞ

CccM44QlE0edl1 ;

Bl ¼ iluz0alEeI15EM11

alPlHlðkMSHaÞ þQlJlðkMSHaÞQl

� �� al ðlþ 1Þ ðeI15Þ

4 F 2HlðkMSHaÞ JlðkISHaÞCccM44E

M11Ql

þ EM11=EI11 � 1

EM11=EI11 þ 1

a

( )E0edl1 ;

Cl ¼ ð�1Þl C�l ¼ iluz0PlHlðkMSHaÞ þQlJlðkMSHaÞ

QlJlðkISHaÞ

� �� ðlþ 1Þ ðeI15Þ

2 FHlðkMSHaÞCccM44Ql

E0edl1 ;

Dl ¼ �iluz0a�lEeI15EM11

alPlHlðkMSHaÞ þQlJlðkMSHaÞQl

� �þ a�l ðlþ 1Þ ðeI15Þ

4 F 2HlðkMSHaÞ JlðkISHaÞCccM44E

M11Ql

� 2EM11=EI11

2EM11=EI11 þ 1

a

( )E0edl1

ð21Þ

where dl1 is the Kronecker delta, and

Pl ¼ �JlðkISHaÞ@

@aJlðkMSHaÞ þ

CccI44CccM44

JlðkMSHaÞ@

@aJlðkISHaÞ �

ðeI15Þ2

CccM44EM11

Fl

aJlðkMSHaÞ JlðkISHaÞ ;

(22)

Ql ¼ JlðkISHaÞ@

@aHlðkMSHaÞ �

CccI44CccM44

HlðkMSHaÞ@

@aJlðkISHaÞ þ

ðeI15Þ2

CccM44EM11

Fl

aHlðkMSHaÞ JlðkISHaÞ ;

al ¼1 ðl ¼ 0Þ ;2 ðl 6¼ 0Þ ;

�ð23Þ

E ¼ EM11=EI11 � eM15=eI15

EM11=EI11 þ 1

;

(24)

F ¼ ðeM15=eI15 � EM11=EI11Þ

2

EM11=EI11 þ 1

:

Special cases of the piezoelectric inclusion can be deduced from eqs. (21), (22), and (24) by an appropriate limit-ing process. When cM44=c

I44 ! 0, eM15=e

I15 ! 1, EM11=E

I11 ! 1, and rM=rI ¼ constant, the inclusion reduces to a rigid

inclusion with a finite mass density. In addition, when cM44=cI44 ! 1, eM15=e

I15 ! 1, EM11=E

I11 ! 1, and rM=rI ! 1, the

inclusion reduces to a hole.

3. Numerical results and discussion

To examine the effect of the material constants, the frequency, and the applied electric field on the dynamic stress andelectric field concentrations, the scattered field due to a circular piezoelectric inclusion has been computed numerically.As an example, lead zirconate titante (PZT4) piezoceramic is considered; its material properties are given in Table 1[13]. The computations were validated by increasing the truncation limit and checking that the difference was negligible.

The maximum dynamic stress and electric field in the matrix depend on various material constants and fre-quency, and should be determined case by case. We first consider the case where the applied electric field is zero. Fig. 2displays the dynamic stress concentration jsxz=t0mj ¼ jsMxz=t0mj in the matrix at q ¼ p=2 against the normalized fre-quency W ¼ aw=ðcM44=rMÞ1=2 for various ratios of the elastic constants cM44=c

I44, while letting eM15 ¼ eI15 ¼ 12:7 Cm�2,

EM11 ¼ EI11¼ 64:6� 10�10 CV�1 m�1 and cI44 ¼ 2:56� 1010 Nm�2. It is evident that the ratio of elastic constants cM44=c

I44

plays a major role in the behavior of jsxz=t0mj. The peak values of jsxz=t0mj are always larger than the static one andthey occur at different frequencies for different cM44=c

I44. In addition, as W ! 0, the dynamic stress concentration tends

to the static stress concentration [5]. The maximum stress concentration for W ¼ 0 and cM44=cI44 > 1 occurs at q ¼ p=2.

However, the maximum stress concentration for W ¼ 0 and cM44=cI44 < 1 occurs at q ¼ 0 where the shear stress in the

matrix is equal to the constant shear stress of the inclusion. The value of jsyz=t0mj ¼ jsMyz=t0mj is zero at q ¼ 0 or atq ¼ p=2. The corresponding dynamic electric field concentration jeM15Ex=t0mj ¼ jeM15EMx =t0mj is shown in Fig. 3. Similarto the dynamic stress concentration, it is observed that the sensitivity of the dynamic electric field concentration to the

46 ZAMM � Z. Angew. Math. Mech. 82 (2002) 1

Tab l e 1: Material properties of PZT4

c44 (�1010 N/m2) r (kg/m3) e15(C/m2) E11 (�10�10 C/V m)

PZT4 2.56 7500 12.7 64.6

frequency increases with the increase of the ratio of elastic constants cM44=cI44. The value of jeM15Ey=t0mj ¼ jeM15EMy =t0mj is

zero at q ¼ 0 or at q ¼ p=2. Fig. 4 shows the dynamic stress concentration in the matrix at q ¼ p=2 as a function ofthe normalized frequency W for various ratios of the piezoelectric constants eM15=e

I15, while holding

cM44 ¼ cI44 ¼ 2:56� 1010 Nm�2, EM11 ¼ EI11 ¼ 64:6� 10�10 CV�1 m�1, and eI15 ¼ 12:7 Cm�2. The negative piezoelectric con-stant implies the reversal of the poling direction. The dynamic stress concentration tends to increase with W and peaksat different frequencies depending on the ratio eM15=e

I15. The dynamic electric field concentration exhibits characteristics

similar to those of the dynamic stress concentration. The effect of the piezoelectric constant on the static stress andelectric field concentrations may be found in [5]. Fig. 5 exhibits the variation of the dynamic stress concentration in thematrix at q ¼ p=2 against the normalized frequency W for various ratios of the dielectric constants EM11=E

I11, when we let

cM44 ¼ cI44 ¼ 2:56� 1010 Nm�2, eM15 ¼ eI15 ¼ 12:7 Cm�2, and EI11 ¼ 64:6� 10�10 CV�1 m�1. The dynamic stress concentra-tion approaches unity as the normalized frequency W ! 0 and the ratio of dielectric constants has no effect on thestatic stress concentration. The corresponding dynamic electric field concentration is shown in Fig. 6. The dynamicelectric field concentration is seen to decrease monotonically with EM11=E

I11.

Next, to provide a parametric study of the effect of the normalized applied electric field E*0m ¼ eM15E0e=t0m onthe dynamic stress and electric field concentrations, the value of E*0m was varied from �0.5 to 1.0 in the consideredexamples. Fig. 7 is a plot of the dynamic stress concentration in the matrix at q ¼ p=2 versus the normalized frequency

Shindo, Y. et al.: Scattering of Shear Waves in a Piezoelectric Medium 47

Fig. 2. Dynamic stress concentration at q ¼ p=2 versusfrequency W for various ratios of the elastic constantscM44=c

I44

Fig. 3. Dynamic electric field concentration at q ¼ p=2versus frequency W for various ratios of the elastic con-stants cM44=c

I44

Fig. 4. Dynamic stress concentration at q ¼ p=2 versusfrequency W for various ratios of the piezoelectric con-stants eM15=e

I15

Fig. 5. Dynamic stress concentration at q ¼ p=2 versusfrequency W for various ratios of the dielectric constantsEM11=E

I11

Fig. 6. Dynamic electric field concentration at q ¼ p=2 versus fre-quency W for various ratios of the dielectric constants EM11=E

I11

W for a hole. The dynamic stress concentration drops rapidly beyond the first maximum and exhibits oscillations ofapproximately constant period as W increases. At W ¼ 0:53, the dynamic stress concentration is about 2.1 for E*

0m ¼ 0,which is 5 percent larger than the static value. jsxz=t0mj depends on E*0m as well as on W. The maximum values ofjsxz=t0mj, depending on the values of E*0m, range from 0.69 at W ¼ 1:67 for E*0m ¼ 1:0 to 3.09 at W ¼ 0:51 forE*0m ¼ �0:5. The dynamic electric field concentration for a hole is also displayed graphically in Fig. 8. The frequencyat which the overshoot occurs is coincident with the peak of jsxz=t0mj as shown in Fig. 7 for E*0m ¼ 0. jeM15Ex=t0mjdepends on E*0m as well as on W. jeM15Ex=t0mj reaches a maximum value of 2.03 at W ¼ 0:50 for E*0m ¼ �0:5. Fig. 9shows the variation of the dynamic stress concentrations in the matrix at q ¼ 0 and q ¼ p with the normalized fre-quency for a rigid inclusion. The dynamic stress concentrations jsxz=t0mj are independent of the applied electric field.It is noted that jsxz=t0mj approaches the value of 2.1 on the incident side of the inclusion when W � 1. Also, as Wbecomes large, jsxz=t0mj at q ¼ 0 approaches zero, where q ¼ 0 is the center of the shadow side of the inclusion. Themagnitude of jsxz=t0mj is always larger on the incident side of the inclusion. The variation of jeM15Ex=t0mjwith fre-quency is similar to that presented in Fig. 9 for the dynamic stress concentration. Fig. 10 shows the variation of thedynamic stress concentration in the matrix at q ¼ p=2 with the normalized frequency W for the case where cM44=c

I44 ¼ 1,

EM11=EI11 ¼ 1 and eM15=e

I15 ¼ �5. It shows the pronounced effect of E*0m on the magnitude of jsxz=t0mj. The lower the

normalized applied electric field, the higher the overshoot becomes. In addition, the dynamic stress concentration tendsto increase with the frequency reaching a peak and then to decrease in magnitude. The dynamic electric field concen-trations exhibit the same general trend as the dynamic stress concentrations do.

4. Conclusion

The dynamic electroelastic problem for a piezoelectric ceramic having a circular piezoelectric inclusion has been ana-lyzed theoretically. The traditional concept of elastic wave scattering problems for the dynamic stress concentrationstudies is extended to include the piezoelectric effects and the results are expressed in terms of the dynamic stress andelectric field concentrations. The dynamic stress and electric field concentrations are found to depend on the piezoelec-tric material constants, the frequency, and the applied electric field. Tuning of the material constants are required toproduce maximum sensitivity for the piezoelectric composite sensor. The study reveals the phenomena of very highstress and electric field concentrations that can be induced by tailoring piezoelectric material properties, and the impor-tance of the electromechanical coupling terms upon the resulting dynamic stress and electric field concentrations.

48 ZAMM � Z. Angew. Math. Mech. 82 (2002) 1

Fig. 7. Dynamic stress concentration at q ¼ p=2 versusfrequency W for a hole

Fig. 8. Dynamic electric field concentration at q ¼ p=2versus frequency W for a hole

Fig. 9. Dynamic stress concentration at q ¼ 0 and q ¼ pversus frequency W for a rigid inclusion

Fig. 10. Dynamic stress concentration at q ¼ p=2 versusfrequency W for different values of the applied electricfield E*

0m

References

1 Zhang, T. Y.; Fu, C.; Tong, P.: Linear electro-elastic analysis of a cayity or crack in a piezoelectric material. Int. J. SolidsStructures 35 (1998), 2121––2149.

2 Meguid, S. A.; Zhong, Z.: Electroelastic analysis of a piezoelectric elliptical inhomogeneity. Int. J. Solids Structures 34 (1997),3401––3414.

3 Sosa, H.; Khutoryansky, N.: New developments concerning piezoelectric materials with defects. Int. J. Solids Structures 33(1996), 3399––3414.

4 Wang, B.: Three-dimensional analysis on an ellipsoidal inclusion in a piezoelectric material. Int. J. Solids Structures 29 (1992),293––308.

5 Pak, Y. E.: Circular inclusion problem in antiplane piezoelectricity. Int. J. Solids Structures 29 (1992), 2403––2419.6 Wills, J. R.: A polarization approach to the scattering of elastic waves. II. Multiple scattering from inclusions. J. Mech. Phys.

Solids 28 (1980), 307––327.7 Miklowitz, J.; Achenbach, J. D. (eds.): Modern problems in elastic wave propagation. Proc. Intern. Union of Theoretical and

Applied Mechanics Sympos. Northwestern Univ., Evanston, Ill., Sept. 12––15, 1977. Wiley & Sons, New York––Chichester––Bris-bane 1978.

8 Chen, Y. M.: Scattering of plane elastic waves by a rigid circular cylindrical inclusion imbedded in a random medium. J. Math.Anal. Appl. 31 (1970), 182––195.

9 Gomes, J. C.; Villamizar, V.: Response of a viscous incompressible fluid cylinder to an incident plane compressional elasticwave. SIAM J. Appl. Math. 56 (1996), 372––387.

10 Bose, S. K.; Mal, A. K.: Longitudinal shear waves in a fiber-reinforced composite. Int. J. Solids Structures 9 (1973),1075––1085.

11 Berlincourt, D. A.; Curran, D. R.; Jaffe, H.: Piezoelectric and piezomagnetic materials and their function in transducers.In: Mason, W. P. (ed.): Physical Acoustics. Vol. 1. Academic Press, New York 1964.

12 Morse, P. M.; Feshbach, H.: Methods of theoretical physics. II. McGraw-Hill Book Company, New York 1953.13 Narita, F.; Shindo, Y.: Scattering of antiplane shear waves by a finite crack in piezoelectric laminates. Acta Mech. 134 (1999),

27––43.

Received October 26, 1999, revised January 19, 2001, accepted January 22, 2001

Address: Dr. Yasuhide Shindo, Dr. Hirokazu Moribayashi, Fumio Narita, Department of Materials Processing, GraduateSchool of Engineering, Tohoku University, Aoba-yama 02, Sendai 980-8579, Japan

Shindo, Y. et al.: Scattering of Shear Waves in a Piezoelectric Medium 49