Archive of Applied Mechanics 71 (2001) 63±73 Ó Springer-Verlag 2001

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium W. Q. Chen, T. Shioya, H. J. Ding

Summary An exact, three-dimensional analysis is developed for a penny-shaped crack in an in®nite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric ®eld are obtained. A numerical calculation is ®nally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Key words Penny-shaped crack, antisymmetric loading, fundamental solution, piezoelectric half-space, integral equation

1 Introduction The fracture mechanics of piezoelectric materials has gained considerable interest recently [1±11]. For a transversely isotropic piezoelectric medium, the eigenstrain formulation and Cauchy's residue theorem was utilized to derive an explicit expression for a penny-shaped crack subjected to electromechanical loadings, [7]. Exact closed-form solutions have been obtained for the stress and induction ®led of a spheroidal inclusion in an in®nite piezoelectric matrix subjected to spatially homogeneous mechanical and electrical remote loadings; as a limiting case, solutions for a penny-shaped crack were presented in terms of elementary functions, [8]. The potential theory method developed for elasticity, [12], was generalized to piezoelasticity based on the general solution proposed in [13], and the problem of a pennyshaped crack subjected to symmetric normal mechanical forces and surface electric charges was analysed, [9]. In that analysis, the symmetric crack problem was formulated as a mixed boundary value problem of a piezoelectric half-space, and the potential theory method was then employed. Elementary function expressions for the elastoelectric ®eld were derived for the point loading case. The same method has also been used to obtain the exact solution of a symmetric external circular crack, [10]. Instead of using the potential theory method to obtain the governing equations, [9], the analysis in this paper starts directly from the fundamental solution of a transversely isotropic Received 30 August 1999; accepted for publication 1 March 2000 W. Q. Chen (&), H. J. Ding Department of Civil Engineering, Zhejiang University, Hangzhou, 310027, P.R. China e-mail: [email protected] T. Shioya Department of Aeronautics and Astronautics, The University of Tokyo, Tokyo, 113-8656, Japan The work was supported by the Natural Science Foundation of China (No. 19872060) and the Scienti®c Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Financial support from the Japanese Ministry of Education, Science, Sports and Culture is also acknowledged.

63

64

piezoelectric half-space. The same technique has been described in [12, 14] for transversely isotropic elasticity, but no application in piezoelasticity was found yet. The present fundamental solution corresponds to a concentrated force and a point electric charge applied on the surface of a piezoelectric half-space. In fact, it can be extracted from the general Green's functions for a two-phase piezoelectric space, [13]. In this paper, however, it is rewritten in a complex manner suitable for the present study. Three integral equations for the general mixed boundary value problem are then derived through the direct integration of the fundamental solution. The problem of a penny-shaped crack subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane is investigated by utilizing the third integral equation. The exact solution of the resulting integral equation can be obtained by using available results from the elasticity theory, [14]. Closed-form expressions for the elastoelectric ®eld are then presented. Numerical examples are ®nally given to show the coupling effect in piezoelectric materials. It is mentioned here that by employing the results for axisymmetric loading as well as for antisymmetric loading, one can deal with the problem of a penny-shaped crack subjected to arbitrary surface electromechanical loading combinations.

2 Basic formulations and the fundamental solution for a piezoelectric half-space 2.1 Basic formulations for a transversely isotropic piezoelectric medium One of the most important piezoelectric materials in technology is poled ceramics exhibiting transverse isotropy with the unique axis aligned along the poling direction. In Cartesian coordinates (with the z-axis being normal to the plane of isotropy), the linear constitutive relations of a transversely isotropic piezoelectric body are, [9, 10, 13],   ou ov ow oU ou ow oU ‡ e31 ; sxz ˆ c44 ‡ ‡ e15 ; rx ˆ c11 ‡ …c11 2c66 † ‡ c13 ox oy oz oz oz ox ox   ou ov ow oU ov ow oU ry ˆ …c11 2c66 † ‡ c11 ‡ c13 ‡ e31 ; syz ˆ c44 ‡ ‡ e15 ; ox oy oz oz oz oy oy   ou ov ow oU ou ov rz ˆ c13 ‡ c13 ‡ c33 ‡ e33 ; sxy ˆ c66 ‡ ; ox oy oz oz oy ox     ou ow oU ov ow oU Dx ˆ e15 ‡ e11 ; Dy ˆ e15 ‡ e11 ; oz ox ox oz oy oy ou ov ow oU e33 ; Dz ˆ e31 ‡ e31 ‡ e33 ox oy oz oz

…1†

where rij , sij , U and Di are the normal and shear stress components, the electric potential and electric displacement components, respectively; u; v and w are the displacement components in the x-, y- and z-directions, respectively; cij , eij and eij are the elastic, dielectric, and piezoelectric constants, respectively. The governing equations of piezoelasticity for transverse isotropy can be found in [15]. By introducing the complex displacement U ˆ u ‡ iv, the general solution proposed in [13] is rewritten in the following form, [9, 10]:

UˆK

3 X

! Fi ‡ iF4 ;

wˆ

iˆ1

3 X

ai1

iˆ1

oFi ; ozi

Uˆ

3 X iˆ1

ai2

oFi ; ozi

…2†

where K ˆ o=ox ‡ io=oy, zi ˆ si z, s24 ˆ c66 =c44 , and s2i …i ˆ 1; 2; 3† are eigenvalues, see [9, 10], and

ai1 ˆ

c11 e11 m3 s2i ‡ c44 e33 s4i ; …m1 m2 s2i †si

ai2 ˆ

m1 ˆ e11 …c13 ‡ c44 † ‡ e15 …e15 ‡ e31 †; m3 ˆ c11 e33 ‡ c44 e11 ‡ …e15 ‡ e31 †2 ;

c11 e15 m4 s2i ‡ c44 e33 s4i ; …m1 m2 s2i †si m2 ˆ e33 …c13 ‡ c44 † ‡ e33 …e15 ‡ e31 †;

m4 ˆ c11 e33 ‡ c44 e15

…c13 ‡ c44 †…e15 ‡ e31 † ;

…3†

It is noted here that the general solution given in Eq. (2) is only valid for distinct s2i , while different forms should be adopted for other cases, [9, 13]. Moreover, Fi …z† satis®es the following quasi-harmonic equation:

  o2 D ‡ 2 Fi ˆ 0; ozi

i ˆ 1; 2; 3; 4 ;

…4†

where D ˆ o2 =ox2 ‡ o2 =oy2 . From Eqs. (1) and (2), the following expressions for stresses and electric displacements are derived: 3 X

rz ˆ

iˆ1 3 X

r1 ˆ

iˆ1

sz ˆ K

c1i

o2 Fi ; oz2i

c3i

3 X iˆ1

2

o Fi ; oz2i

Dz ˆ

3 X iˆ1

c2i

o2 Fi ; oz2i

65

r2 ˆ 2c66 K2 …F1 ‡ F2 ‡ F3 ‡ iF4 †;

! o o c1i si Fi ‡ is4 c44 F4 ; ozi oz4

DˆK

3 X iˆ1

…5† o o c2i si Fi ‡ is4 e15 F4 ozi oz4

! ;

where

r1 ˆ rx ‡ ry ; r2 ˆ rx ry ‡ 2isxy ; sz ˆ sxz ‡ isyz ; D ˆ Dx ‡ iDy ; c1i ˆ c13 ‡ c33 si ai1 ‡ e33 si ai2 ; c2i ˆ e31 ‡ e33 si ai1 e33 si ai2 ; c3i ˆ 2‰…c66 c11 † ‡ c13 si ai1 ‡ e31 si ai2 Š :

…6†

2.2 Fundamental solution for a half-space It is considered that a piezoelectric half-space z  0 is subjected to a concentrated force with components Tx , Ty and P in the x-, y- and z-directions, respectively, and a point charge Q is applied at the point (q0 ; /0 ; 0) of the surface z ˆ 0 (Hereafter, cylindrical coordinates are alternatively used for the sake of convenience). The solution can be obtained from the general Green's functions for a two-phase piezoelectric medium, [13]. Here, in contrast to the notations employed in [13], it is rewritten as follows: Di   …T K ‡ TK†v…zi † ‡ …Ai1 P ‡ Ai2 Q† ln…Ri ‡ zi †; 2 i  TK†v…z  …T K F4 ˆ 4† ; 4ps4 c44 Fi ˆ

i ˆ 1; 2; 3;

…7†

where T ˆ Tx ‡ iTy , the overbar indicates the complex conjugate value, and

v…zi † ˆ zi ln…Ri ‡ zi † 8 9 > < D1 > = D2 ˆ > : > ; D3

Ri ;

Ri ˆ

8 9 > <0> = 1 1 0 ; M > 2p : > ; 1

q q2 ‡ q20 2qq0 cos…/ /0 † ‡ z2i ;

8 9 > < A11 > = A21 ˆ > > : ; A31

8 9 > <1> = 1 1 0 ; M > 2p : > ; 0

8 9 > < A12 > = A22 ˆ > > : ; A32

8 9 > <0> = 1 1 1 M ; > 2p : > ; 0 …8†

with

2

c11 4 M ˆ c21 c11 s1

c12 c22 c12 s2

3 c13 c23 5 : c13 s3

…9†

The following expressions for the displacements and the electric potential can then be obtained from Eqs. (2) and (7): 3  X ai1 Di

66

 1 1 ‡ ai1 …Ai1 P ‡ Ai2 Q† ; wˆ 2 Ri …Ri ‡ zi † Ri iˆ1  3  X ai2 Di 1 1  …Ts ‡ Ts† ‡ ai2 …Ai1 P ‡ Ai2 Q† ; Uˆ 2 Ri …Ri ‡ zi † Ri iˆ1 " # 3 3 X s 1X T s2 T Uˆ …Ai1 P ‡ Ai2 Q† Di ‡ ‡ Ri …Ri ‡ zi † 2 iˆ1 Ri Ri …Ri ‡ zi †2 iˆ1 " # 1 T s2 T ; ‡ ‡ 4ps4 c44 R4 R4 …R4 ‡ z4 †2  …Ts ‡ Ts†

where s ˆ qei/

…10†

q0 ei/0 . When z ˆ 0, the above equations give

    P Q T P Q T ‡ g2 ‡ G3 Re ; U ˆ g3 ‡ g4 ‡ G4 Re ; R R s R R s 2 P Q G1 T G2 Ts U ˆ H1 ‡ H2 ‡ ‡ ; s s 2 R 2 R3 w ˆ g1

…11†

where

Rˆ

q q2 ‡ q20 2qq0 cos…/ /0 † ;

Re‰xŠ denotes the real part of x, and

H1 ˆ

3 X

Ai1 ;

H2 ˆ

3 X

iˆ1

3 X

G1 ˆ

iˆ1

3 X 1 ‡ Di ; G2 ˆ 2ps4 c44 iˆ1

g3 ˆ

Ai2 ;

Ai1 ai2 ;

g4 ˆ

iˆ1

g1 ˆ

3 X

Ai1 ai1 ;

g2 ˆ

iˆ1 3 X iˆ1

Ai2 ai2 ;

3 X

1 2ps4 c44

Di ;

iˆ1 3 X

…12†

Ai2 ai1 ;

iˆ1

G3 ˆ

3 X iˆ1

Di ai1 ;

G4 ˆ

3 X

Di ai2 :

iˆ1

Now, integrating Eq. (11) over the entire surface plane, the following three integral equations are obtained that are valid for the general boundary value problem of a piezoelectric half-space:

Z2pZ1

Z2pZ1 p…q0 ; /0 † q…q0 ; /0 † q0 dq0 d/0 ‡ g2 q0 dq0 d/0 w…q; /; 0† ˆ g1 R R 0 0 0 0 2 2p 1 3 ZZ s…q0 ; /0 † q0 dq0 d/0 5 ; ‡ G3 Re4 s

…13†

0 0

Z2pZ1

Z2pZ1 p…q0 ; /0 † q…q0 ; /0 † q0 dq0 d/0 ‡ g4 q0 dq0 d/0 U…q; /; 0† ˆ g3 R R 0 0 0 0 2 2p 1 3 ZZ s…q ; / † 0 0 q0 dq0 d/0 5 ; ‡ G4 Re4 s 0 0

…14†

Z2pZ1 U…q; /; 0† ˆ H1 0 0

G1 ‡ 2

p…q0 ; /0 † q0 dq0 d/0 ‡ H2 s

Z2pZ1 0 0

Z2pZ1

q…q0 ; /0 † q0 dq0 d/0 s

0 0

s…q0 ; /0 † G2 q0 dq0 d/0 ‡ 2 R

Z2pZ1 0 0

s…q0 ; /0 †s2 q0 dq0 d/0 ; R3

…15†

where p…q; /†; s…q; /†; q…q; /† are the distributed normal force, complex tangential force and the surface electric charge, respectively. They are known on some part of the surface, and unknown on the rest. Though deriving the integral equations (13)±(15) is straightforward, these equations can play key roles in solving crack problems for transversely isotropic piezoelectric materials. As in the elastic case, [12, 14], the symmetric problem of a penny-shaped crack in a piezoelectric solid can be solved directly from Eqs. (13) and (14), and results will be identical to those obtained by potential theory method, [9]. Details are omitted and the antisymmetric problem will be considered immediately, based on Eq. (15).

3 The antisymmetric crack problem and the complete solution It is now considered that a penny-shaped crack is located in the plane z ˆ 0 of an in®nite transversely isotropic piezoelectric medium and subjected to a couple of concentrated normal forces and a couple of point electric charges at the points (r; w; 0 ) that are antisymmetric with respect to the crack plane, Fig. 1. The radius of the crack is denoted as a. Obviously, the problem can also be turned into a mixed boundary value problem of a piezoelectric half-space z  0, subjected to the following conditions on the plane z ˆ 0: U ˆ rz ˆ Dz ˆ 0 for a < q < 1; rz ˆ Pd…q r; / w†=q Dz ˆ Qd…q sz ˆ 0 for 0  q < a :

r; /

w†=q for 0  q < a;

…16†

Under these conditions, Eq. (15) becomes

G1 2

Z2p Z1 0

a

s…q0 ; /0 † G2 q0 dq0 d/0 ‡ 2 R

Z2p Z1 0

a

s…q0 ; /0 †s2 q0 dq0 d/0 ˆ R3 qe

L i/

re

iw

;

…17†

where L ˆ …H1 P ‡ H2 Q†. It is shown that the above equation has the same structure as that for pure elasticity, [14], so that its solution can be obtained in a similar way as

Fig. 1. Penny-shaped crack and the concentrated mechanical and electric loadings

67

" # Lei/ 1 p b ‡ ; s…q; /; 0† ˆ pG1 q q2 a2 …1 f†3=2

…18†

for q > a, where b ˆ G2 =…G1 G2 † and f ˆ reiw =…qei/ †. Following the Fabrikant's method, [14], expressions for the functions Fi can be obtained through a tedious derivation

Fi ˆ Ni fRe‰ f …zi †Š ‡ bf0 …zi †g ‡ Li ln…Si ‡ zi †; 68

i ˆ 1; 2; 3;

F4 ˆ

L Im ‰ f …z4 †Š ; pG1 s4 c44 …19†

where Ni ˆ 2LDi =G1 and Li ˆ Ai1 P ‡ Ai2 Q, and Im‰xŠ represents the imaginary part of x, and

s)    2 a b b l22 a2 1 ; f …z† ˆ aE sin l2 l22 b2 a2 b2 l2 a s q q l22 q2 ‡ ln l22 b2 ‡ l22 q2 ; l22 b2     p q a2 l21 z 1 a ln l2 ‡ l22 q2 ; sin ‡ f0 …z† ˆ a a l2 q Si ˆ q2 ‡ r2 2qr cos…/ w† ‡ z2i ; (

z



…20†

with

b2 ˆ qrei…/

w†

;

E‰x; yŠ stands for the incomplete elliptic integral of the second kind, and

1 l1 ˆ 2

q …q ‡ a†2 ‡ z2

q q q 1 2 2 …q ‡ a†2 ‡ z2 ‡ …q a†2 ‡ z2 : …q a† ‡ z ; l2 ˆ 2 …21†

The expressions for the elastoelectric ®eld are then obtained by substitution of Eq. (19) into Eqs. (2) and (5) 3  X Uˆ Ni KRe‰ f …zi †Š ‡ Ni bKf0 …zi † ‡

 Li t iL KIm‰ f …z4 †Š; Si …Si ‡ zi † pG1 s4 c44 iˆ1     3 X o Ni b Li 1 a sin ‡ ; ai1 Ni Re‰ f …zi †Š wˆ Si ozi a l2i iˆ1     3 X o Ni b a Li ‡ ; sin 1 ai2 Ni Re‰ f …zi †Š Uˆ Si ozi a l2i iˆ1 s ( ) 3 X o2 Ni b a2 l21i Li zi rz ˆ ; c1i Ni 2 Re‰ f …zi †Š ‡ ozi l22i l21i S3i a iˆ1 s ( ) 3 X o2 Ni b a2 l21i Li zi ; c2i Ni 2 Re‰ f …zi †Š ‡ Dz ˆ ozi l22i l21i S3i a iˆ1 s ( ) 3 X o2 Ni b a2 l21i Li zi ; c3i Ni 2 Re‰ f …zi †Š ‡ r1 ˆ ozi l22i l21i S3i a iˆ1 ( ) 3 X Li t…2Si ‡ zi † 2iLs4 2 2 2 r2 ˆ 2c66 Ni K Re‰ f …zi †Š ‡ Ni bK f0 …zi † K Im‰ f …z4 †Š ; 2 3 pG1 Si …Si ‡ zi † iˆ1

…22†

 2LDi o o sz ˆ c1i Si K Re‰f …zi †Š ‡ bK f0 …zi † G1 ozi ozi iˆ1  3 X 2LDi o o c2i Si K Re‰f …zi †Š ‡ bK f0 …zi † Dˆ G1 ozi ozi iˆ1

 t Si  t Si

3 X

iLi o K Im‰f …z4 †Š; pG1 oz4 iLi e15 o K Im‰f …z4 †Š ; pG1 c44 oz4

…22†

where

t ˆ qei/

reiw ; 69

and

q q 1 …q a†2 ‡ z2i ; …q ‡ a†2 ‡ z2i l1i ˆ 2 q q 1 …q ‡ a†2 ‡ z2i ‡ …q a†2 ‡ z2i : l2i ˆ 2

…23†

The explicit expressions for various derivatives of f and f0 have been calculated in [14]. They are listed in the Appendix for the sake of integrity of the paper. It is seen from the boundary conditions (16), that the normal stress, rz as well as the normal electric displacement Dz in the plane z ˆ 0 vanish in the case of normal antisymmetric loading, as does the mode I stress intensity factor and the electric displacement intensity factor. On the other hand, the mode II and III stress intensity factors can be obtained in the following way:

n kII ‡ ikIII ˆ lim …q q!a

a†1=2 e

i/

sz jzˆ0

o

:

…24†

From Eq. (18) it is directly obtained that

kII ‡ ikIII

" L p b ‡ ˆ pG1 a 2a ‰1

#

1 …r=a†ei…/

w† Š3=2

:

…25†

4 Numerical examples For the sake of simplicity, the concentrated mechanical force P and/or the point electric charge Q are assumed to be applied at the point …0:5a; 0; 0† in the following numerical examples. As the ®rst example, the distribution curves of the nondimensional normal displacement x ˆ ac44 w…q; /; 0†=P and electric potential

W ˆ ae33 U…q; /; 0†=P in the plane z ˆ 0 versus the nondimensional radial coordinate q=a are shown in Figs. 2 and 3, respectively, for several different values of the circular angle /. Only the concentrated force P is acting at the point …0:5a; 0; 0†. The piezoelectric material used is PZT-4, the material constants of which can be found in [16], for example. From the two ®gures, the singularity of x or W at the point where the force is applied is clearly seen. The coupling effect in piezoelectric materials is shown in the distribution curves of x…/ ˆ p=4† in Fig. 4 for four different cases for the material PZT-4. A nondimensional load parameter

K ˆ …c44 =e33 †Q=P is introduced so that the results for different loading combinations are readily compared. The curve that is indicated as ``elastic'' in Fig. 4 corresponds to the elastic material ignoring the piezoelectric effect in PZT-4. It is shown that the coupling effect has a signi®cant effect on the

70

Fig. 2. Distribution along radial direction of the nondimensional displacement x ˆ ac44 w=P in the plane z ˆ 0 due to a concentrated force P applied at the point (0:5a; 0; 0)

Fig. 3. Distribution along radial direction of the nondimensional electric potential W ˆ ae33 U=P in the plane z ˆ 0 due to a concentrated force P applied at the point (0:5a; 0; 0)

Fig. 4. Distribution along radial direction (/ ˆ p=4) of the nondimensional displacement x ˆ ac44 w=P in the plane z ˆ 0 for different loading conditions

displacements. The results also illustrate that, for peizoelectric materials, it is possible to control the deformation by combining the mechanical and electric loadings according to a speci®ed ratio, K. The corresponding curves of W…/ ˆ p=4† are also given in Fig. 5. To get the knowledge of the piezoelectric effect on the material toughness, the nondimensional mode II and III stress intensity factors along the crack tip …q ˆ a† are shown in Figs. 6 and 7, respectively, for three different materials. In particular, PZT-4(E) denotes the corresponding elastic one of the piezoelastic material PZT-4, which is also transversely isotropic. The Poisson's ratio of the isotropic elastic material is taken to be 0.3. It is seen that the piezoelectric effect plays an important role in determining the value of the stress intensity factors. For PZT-4, the nondimensional stress intensity factors jII and jIII are further shown in

Fig. 5. Distribution along radial direction (/ ˆ p=4) of the nondimensional electric potential W ˆ ae33 U=P in the plane z ˆ 0 for different loading conditions

Fig. 6. The nondimensional mode II stress pintensity factor jII ˆ pa 2a kII =P along the crack tip due to a concentrated force P applied at the point (0:5a; 0; 0)

Fig. 7. The nondimensional mode III stress p intensity factor  jIII ˆ pa 2a kIII =P along the crack tip due to a concentrated force P applied at the point (0:5a; 0; 0)

Figs. 8 and 9, respectively, for different loading combinations. It is apparent that by adopting a proper value of the load parameter K, one is able to determine the mode II and III stress intensity factors within a safe range.

5 Conclusion This paper extends Fabrikant's results [12, 14] to piezoelasticity in a rather straightforward way. At ®rst, the fundamental solution of a concentrated mechanical force and a point electric charge acting on the surface of a transversely isotropic piezoelastic half-space is rewritten in a complex and concise form. Three integral equations are then derived for the general boundary

71

Fig. 8. The nondimensional mode II stress intensity factor jII at the crack tip for different loading conditions

72

Fig. 9. The nondimensional mode III stress intensity factor jIII at the crack tip for different loading conditions

value problem by integrating the fundamental solution directly. The problem of a pennyshaped crack subjected to concentrated normal forces and electric charges antisymmetric with respect to the crack plane is then analyzed. By using previous results in [14], the exact expressions for the whole piezoelastic ®eld are derived. Within the regime of linear piezoelasticity, numerical results show that the piezoelectric characteristic has an important effect on the fracture mechanism of piezoelectric materials. Utilizing the theorem of superposition for the linear theory of piezoelectricity, the solution of a penny-shaped crack subjected to a one-sided normal loading can be obtained, [14]. This can be accomplished by simply adding the current antisymmetric solution to the symmetric one, [9].

References

1. Pak, Y.E.: Crack extension force in a piezoelectric material. J Appl Mech 57 (1990) 647±653 2. Sosa, H.A.: On the fracture mechanics of piezoelectric solids. Int J Solids Struct 29 (1992) 2613±2622 3. Suo, Z.; Kuo, C.M.; Barnett, D.M.; Willis, J.R.: Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 40 (1992) 739±765 4. Zhang, T.Y.; Hack, J.E.: Mode III cracks in piezoelectric materials. J Appl Phys 71 (1992) 5865±5870 5. Zhong, Z.; Meguid, S.A.: Analysis of a circular arc-crack in piezoelectric materials. Int J Fracture 84 (1997) 143±158 6. Sosa, H.A.; Pak, Y.E.: Three dimensional eigenfunction analysis of a crack in a piezoelectric material. Int J Solids Struct 26 (1990) 1±15 7. Huang, J.H.: A fracture criterion of a penny-shaped crack in transversely isotropic piezoelectric media. Int J Solids Struct 34 (1997) 2631±2644 8. Kogan, L.; Hui, C.Y.; Molkov, V.: Stress and induction ®eld of a spheroidal inclusion or a penny-shaped crack in a transversely isotropic piezoelectric material. Int J Solids Struct 33 (1996) 2719±2737 9. Chen, W.Q.; Shioya, T.: Fundamental solution for a penny-shaped crack in a piezoelectric medium. J Mech Phys Solids 47 (1999) 1459±1475 10. Chen, W.Q.; Shioya, T.: Green's functions of an external circular crack in a transversely isotropic piezoelectric medium. JSME Int J A42 (1999) 73±79

11. Chen, W.Q.: Exact solution of a semi-in®nite crack in an in®nite piezoelectric body. Arch Appl Mech 69 (1999) 309±316 12. Fabrikant, V.I.: Applications of potential theory in mechanics: a selection of new results. The Netherlands: Kluwer Academic Publishers 1989 13. Ding, H.J.; Chen, B.; Liang, J.: On the Green's functions for two-phase transversely isotropic piezoelectric media. Int J Solids Struct 34 (1997) 3041±3057 14. Fabrikant, V.I.: Mixed boundary value problem of potential theory and their applications in engineering. The Netherlands: Kluwer Academic Publishers 1991 15. Tiersten, H.F.: Linear piezoelectric plate vibrations. New York: Plenum Press 1969 16. Dunn, M.L.; Taya, M.: Electroelastic ®eld concentrations in and around inhomogeneities in piezoelectric solids. J Appl Mech 61 (1994) 474±475

73

Appendix o f …z† ˆ oz Kf …z† ˆ

(

1 a2 1 1 t

o K f …z† ˆ 2 …l2 oz 2 K2 f …z† ˆ 2 1 t

b2

    a b ; aE sin 1 l2 a

s! l22 q2 ; l22 b2 l2 qei/ l21 †…l22

b2 l2

s) l22 a2 ; l22 b2

o2 f …z† ˆ 2 oz2 …l2

l22 l21 †…l22

s l22 q2 ; b2 † l22 b2

s l22 a2 ; b2 † l22 b2

s! s l22 q2 qei/ …a2 b2 l42 † l22 q2 ‡ ; t l22 …l22 l21 †…l22 b2 † l22 b2 l22 b2

   q  1 ei/ 1 a 1 a2 l21 =a ; sin ; Kf0 …z† ˆ q a l2 p p a2 l21 o qei/ l22 a2 o2 ; ; f …z† ˆ K f0 …z† ˆ 2 2 0 l2 …l2 l21 † a…l22 l21 † oz2 oz p  q  2i/ a2 l21 2e2i/ e 2 2 K f0 …z† ˆ 2 1 : a2 l1 =a 2 a…l2 l21 † q o f0 …z† ˆ oz