Acta Mechanica 138, 97 111 (1999) ACTA MECHANICA �9 Springer-Verlag 1999

A new thermoelectroelast ic solution for piezoelectric materials with various openings

Q.-H. Qin and Y.-W. Mai, Sydney, Australia

(Received February 19, 1998; revised July 29, 1998)

Summary. A new solution is obtained for thermoelectroelastic analysis of an insulated hole of various shapes embedded in an infinite piezoelectric plate. Based on the exact electric boundary conditions on the hole boudnary, Lekhnitskii's formulation and conformal mapping, the solution for elastic and electric fields has been obtained in closed form in terms of complex potential. The solution has a simple unified form for various holes such as ellipse, circle, triangle and square openings. As an application of the solu- tion, the hoop stress and electric displacement (SED) and the solution for crack problems are discussed. Using the above results, the SED intensity factor and strain energy release rate can be obtained analyti- cally. One numerical example is considered to illustrate the application of the proposed formulation and compared with those obtained from impermeable model.

1 Introduction

The widespread use of piezoelectric materials in structural applications has generated renewed

interest in thermoelectroelastic behaviour. In particular, information on thermal stress con- centrations around material or geometrical defects in piezoelectric solids will have applica-

tions in composite structures. For isotropic materials, Evan-Iwanoski [1] used the complex variable approach to derive the stress solutions for an infinite isotropic plate with a triangular

inlay. Florence and Goodier [2] studied the thermal stress for an isotropic medium containing an insulated oval hole. Based on the complex variable method, Chen [3] studied the orthotro-

pic medium with a circular or elliptic hole, and obtained a complex form solution for the hoop stress around the hole. Z immermann [4] studied the compressibility of holes by way of

conformal mapping of a hole onto a unit circle. Kachanov et al. [5] developed a unified description concerning both cavities and cracks. For orthotropic plates with rectangular openings, work has been done by Jong [6], and Rajaiah and Naik [7]. Their results were based on the solutions given by Lekhnitskii [8], which are only approximate solutions due to the mathematical difficulties involved. Based on the Stroh formalism and complex conformal mapping, Hwu [9] obtained the stress fields for an anisotropic elastic plate with an hole of various shapes subjected to remote uniform mechanical loading. For plane piezoelectric mate-

rial without considering thermal effect, Sosa and Khutoryansky [10] obtained an analytical solution of piezoelectricity with an elliptic hole. Chung and Ting [11] also presented a general solution for piezoelectric plate with an elliptic inclusion. To the authors ' knowledge, however, very little work has been done for thermal stress and electric displacement disturbed by holes in piezoelectric materials. In this paper, an analytical solution for various openings embedded in an infinite plane piezoelectricity subjected to thermal, mechanical and electric loads is pre-

98 Q.-H. Qin and Y.-W. Mai

sented based on Lekhnitskii's complex potential approach and the exact boundary conditions on the hole boundary. The solution has a simple unified form for various holes such as ellipse, circle, triangle and square openings. As an application of the solution, the hoop SED and its concentration factor are analysed. Further, the solution for crack problems can be obtained

by setting parameter e and ~7 apprach to zero. As a consequence, the stress and induction

intensity factors and strain energy release rate can be derived analytically. Numerical results

for SED concentration factor and the electric displacement intensity factor are presented to illustrate the application of the proposed formulation.

2 Basic formulations

Let us consider a two-dimensional thermopiezoelectric plate, where the material is transver-

sely isotropic and coupling between in-plane stresses and in-plane electric fields take place. For a Cartesian coordinate system Oxyz , choose the z-axis as the poling direction, and denote

the coordinates x and z by xl and x2 in order to get a compacted notation. The plane strain

constitutive equations are expressed by [12]:

: k jHj, (1)

0-22 c12 c22 0 0 e22 c22 2

0-12 = 0 0 c33 el3 0 2e12 - 0 D1 0 0 e13 - - Z l l 0 - E l

D2 e21 e22 0 0 -z~2 -E2 ( g2

or inversely

Hi = gi jhj ,

r 2 2 0 0 P22 0"22 a22 2E12 = f33 P13 0 O'12 -- 0 0

- E l 0 0 P13 -/311 0 D1 0 -E~ P2I p22 0 0 -/322 D2 A2

and simply, in matrix form:

h = k H , H = Qh

/7 = CZ - 70, Z = F H + c~0,

(2)

(3)

(4)

heat conductivity and heat resistivity, 0 temperature change, and F = C -1, c~ = C-17. Equations (5) and (6) are completed by adding the following field equations:

h1,1 + h2,2 = 0 , (7.1)

O'11,1 @ 0"12,2 = 0 , 0"12,1 -I- 0-22,2 = 0 , (7.2)

D<I + D2,2 = 0 (7.3)

where 0-ij, eij, Dj and Ej are stress, strain, electric displacement and electric field, respectively,

cij is elastic stiffness, fly elastic compliance, eij and pij are piezoelectric constants, zij and/3ij dielectric permittivity, 92 and A2 pyroelectric constants, 7ij and aij stress-temperature con- stants and thermal expansion constants, Hj and hj are heat intensity and heat flux, kij and Qij

(~)

(6)

A new thermoelectroelastic solution for piezoelectric materials with various openings 99

together with the compatibility relations

e11,22 + e22,11 - 2c12,12 = 0, El,2 - E2,1 = 0 (8)

and the geometrical equations

u~,j + uj,~ Ej = - ~ , j , (9) H j = - O , j , e~j - 2 '

where commas indicate partial differentiation, ui is elastic displacement, and ~ the electric potential.

As was done in Sosa [13], introduce the stress function U and function of electric displace- ment, ~, in the form:

O-11 = U22 , 022 = U~ll: 0-12 = - g 1 2 , /91 : ~,2, D2 = --@,1 (10)

Obviously, these functions satisfy Eq. (7.2, 7.3). Inserting Eq. (10) into Eq. (4), and later into Eq. (8) leads to

L 4 U - L 3 ~ = -a110,22 -- 0~220,11 , L a U -k L 2 ~ = -A20,1 , (11)

where

0 4 0 4 0 4 L4 = f22 0x~l 4 -- f l l ~ § (2f12 + f33) 0x12 0x22 ,

0 3 0 3 0 2 0 2 (12)

L3 = P 2 2 ~ A v (P21 @P13) 035.10x22 , L2 =/~22 ~ - ~ - ~ i 1 0x22 �9

Since this is a linear problem, solutions to Eq. (11) is assumed to consist of the sum of par- ticular solutions, Up and %, and homogeneous part, Uh and r as

u = up + uh , ~ = Cp + ~h, (13)

where Ut~ and Ch will satisfy

L4Uh -- L 3 r = O, LaUh + L2r = 0. (14)

The particular solutions depend on the form of the known function 0 and are usually easy to find. A general solution to Eq. (14) has been discussed elsewhere [13]. The results are as follows:

3 3

k=t k=l

where "Re" stands for the real part of a complex function, and

Zk Xl _~_ ~kX2 ~f)hk(Zk ) , (P21 4- P1:3) #to 2 + P22 (16) ~- , : glzk(Zk), X/~ = 311#/~ 2 @ 322 '

while #k are three complex roots (with positive imaginary parts) satisfying the following char- acteristic equations:

f11/~11# 6 -r (f11/~22 + f33/~i1 + 2f12/~11 q- P~I q- P~3 d- 2p21P13) #4

+ (f22/311 + 2fr~'f122 + f33/322 -}- 2p21P22 + 2p13P22) #2 + f22~22 +P22 = O. (17)

100 Q.-H. Qin and Y.-W. Mai

Therefore the elastic displacements, electric potential, stress and electric displacement can be given as

2Re q~ ~5hk(Zk), k=l tk

o-22 2Re 1 ~5~k(zk ) , 0"12 k=l - - # k

( i s )

{} • } D1 2Re X~Pk , = D2 h k=l -X~

(19)

where

P~ = f ~ l ~ 2 + f12 - P21X~, q~ - f12#~ + f~2 P2zXk

#k #k tk = -( 13 + 11)c ) (20)

3 Boundary condit ions

Consider a plane thermopiezoelectric problem of a single insulated hole in a piezoelectric

plate subjected to uniform remote heat flow h0(hl0, h~0) and SED H o ( H l o , H2o), where /710 = 0 0 0 T {o-n%2D1 } ,//20 = {O-0Kr02D2~ contour of the hole used in this paper is repre-

sented by (see Fig. 1)

Xl = a(cos g? + r] cosjg?), x2 = a(e sing? - r] sinjg?), (21)

where 0 < e __< 1 and j is an integer. By an appropriate selection of the parameters e, j and ~7, we can obtain various special kind of holes, such as ellipse, circle, triangular, square and pen-

tagon. The hole surface is assumed to be free of tractions and also kept with zero heat flow. Besi-

des, on the hole boudnary the normal component of electric displacement and the electric

potential are continuous. We, therefore, have below boundary conditions:

At infinity:

h --~ h0, /7 ~ / / 0 . (22)

n ~ _ .

I o xl

f~ Fig. 1. Geometry of a particular hole (a = 1, e = 1, k = 4, 77 = 0.1)

A new thermoelectroelastic solution for piezoelectric materials with various openings l 0 1

On the hole boundary F (see Fig. 1):

hn = - h i sinco + h2 cosw = 0, a,~ = 0, 0~ ~

c r ~ = 0 , D ~ = - e o ~ - n ' ~ = q~ (23)

where n and m are, respectively, the normal and tangent to the hole boundary (Fig. 1). cr~,~

and o-~ are the normal and shear stresses along the boundary, c0 is the dielectric constant of

vakuum, superscript "c" indicates the quantity associated with the hole media.

It is convenient to represent the solution as the sum of a uniform heat flux in an unflawed

(which involves no thermal SED) and a corrective solution in which the boundary conditions

are

h -+ 0, at infinity h~ = hi0 sin ~ - h20 cos co, (24. ,-2)

and the others are the same as those of Eqs. (22) and (23). It can be shown that the electro-

elastic boundary conditions (23) can be expessed in terms of complex potentials in the form:

3 3

k = l k = l

3 / O@c 3 2Re E {X~gi/~} = - eo ~ ds, 2Re E { t ~ } = V) ~ ,

k = l k = l 0

(25.s-4)

where s is arc length of f , ~5 k = ~hk + ~Spk. Noting that [9]

On Om

Equation (25.3) can be further simplified as

3

2Re E {Xk~} = 2Re {igoF(z)}, k = l

(27)

where

= F(z) + F(z). (28)

4 General solutions

4.1 Conformal mapping

Since the conformal mapping is a fundamental tool used to find the comple• potentials, the transformation [9]

zk = + a2Ck 1 + aaCJ + a46 -5) (29)

in which

1 - i#~e 1 + i#~e 1 + i#~e 1 - i#~e a l - - ~ , a2 - - 2 a 3 = r ] 2 ' a 4 = V 2 (30)

will be used to map the region, Y2, occupied by piezoelectric material on to the outside of a unit circle in the i-plane. It is necessary that all the roots of equation dzk/dCk = 0 be located

102 Q.-H. Qin and Y.-W. Mai

inside the unit circle I Ckl = 1 in order to make the transformation single-valued and con- formal. To find the single-valued mapping of the region J2c, occupied by vacuum (or air), let

z = a(al~C + a 2 ~ 1 + a3~C j + a4~C-J),

where

l + e 1 - e alc -- : a2c -- ~ a3c = r] - -

2 2

(31)

1 - e l + e 2 ' a4c=r/ 2 (32)

The roots of d z / d ( = 0 represent the critical points for transformation (31), and can be,

then, obtained by solving

ale -- a2c~ 2 + ja3c~J-1 _ j a4c~- j - z = 0. (33)

The analytical solution of Eq. (33) is, in general, very difficulty to obtain for j > 1. For

convenience, we only consider the following cases:

(i) j = 1. The roots of Eq. (33) leads to

1 - e (34) r = + T ~ '

Thus, the mapping of the region X?c can be done by excluding a straight line F0 along xl and of length 2a~/1 - e 2 from the ellipse. In this case the function

z = a (a l c ( -[- a2c~ -1) (35)

will transform F and F0 into the ring of outer and inner circles with radii rout = 1 and

rim = V l - ~ e ' respectively.

(ii) j > 1 and e = 1. In this case the mapping function (31) becomes

z = a(alc~ + a 4 ~ J) . (36)

The roots of Eq. (33) can be, then, easily obtained as

@ _ J+.~/~. (37)

In practical, ~/is a small number. If we assume j r / < 1, the function (36) will transform the

trai t'i"e and (1 + nt~ anannuiar j

circle with inner and outer radii rin = J~/~ and rout = 1, respectively.

4.2 General solut ions

We first study the solution of temperature 0. The general solution for temperature and heat

flux can be expressed as [12]

0 = 2Re {9'(zt)}, hi = - 2 R e {kil + Tki2) 9 " ( z t ) } . (38)

Noting that along the hole boundary Xl and x2 are expressed by (21), and C = eiO, the boundary condition (24.2) suggests that the arbitrary function g(zt ) ought to be chosen in the

form

g(zt ) = bl f ~ - l ( z t ) dzt + bj f ~ J(zt) d z t , (39)

A new thermoelectroelastic solution for piezoelectric materials with various openings 103

where bl and bj are two complex numbers to be determined. In using the boundary condition (24.2), one need to evaluate g"(&) along the hole boundary. Knowing that Ct = e ~, and the

(see Fig. 1) is related to ~ by

a(sin~ +jr] sin j@ = &)cos w, a(ecos~ - j r ] cos j@ = -~)sin w, (40)

we have

d& = ie ir = i (ble ir + jb je-{ jr (41) dz~ 0(cos~ + ~- sincJ) ' g"(zt) o(eosco + ~- sinw)

On the use of (24.2), (38.2) and (41.2), one obtains:

ehl ~ + ih2 ~ hi ~ - ih2 ~ bl = - a 2k ' bk = ar] 2k '

where/~ = V/ k v k22 - k~2. To obtain the explicit expression of r integrating (38) yields

( 9(zt) = abl al~ 111Ct +

(42)

(43)

f o r j = 1,

aa2rbl& 2 jabla3~&j-1 g(zt) = a(blal~- + jbja3~) In & + 4

2 j - 1

-t a(ja4~-bl + a2~bj) ~ -(j+l) alTbjCtl-3 + abja4r&-2~ § j + l 1 - j 2

(44)

for other values of j, with

1 - i're 1 + i~-e 1 + i~re al-r - - , a2T -- , a 3 r : 7] - - ,

2 2 2

and ~- is the heat eigenvalue with positive imaginary part of

k22 T2 -H 2k12~- + kll = 0.

1 - iTe a4~=r] 2 (45)

(4~)

The particular solution for elastic displacement, stress, electric potential and electric dis- placement are thus given by [12]

,~ = 2 a e Ao~g(~) , ~,2 ~ = 2 R r Bo~g'(z~) , %o p Ao39(zt) ~ ) p --BO2T91(zt)

{ pl } = 21~e { -B03Tgt(Z12) } D2 p Bo39' ( zt ) '

(47)

(48)

where

{Aol } [ Cll @ T2C33 Ao2 = Ao3 L S y m m e t r i c

.Bo~} [~3~ ~3~ B02 = / C12 TC22

Bo3 L e21 Te22

T(C12 @ C33) T(r -~- e13) C33 H- T2C22 ~13 -H T 2 e 2 2

__(Zll 2- ,7-2X22

1{ 01 } {0} 7-r A02 -- ~22

--7)422) A03 92

-1{ ~11 } T'722 : T92

(49)

(50)

104 Q.-H. Qin and Y.-W. Mai

Substituting Eq. (47) and (48) into Eqs. (25) and (27), the boundary conditions can be rewritten as

Re ~hk(zk) + Bo2g(zt = 0, Re #~q)hk(zk) + TBo29(Zt = O, k = l k = l (51)

,} } Re Xk~hk(z~) -- Bo39(zt = Re {ieoF(z)}, 2Re t~h~(Z~) + Ao39(zt) = ~c. k = l k=l

The conditions (22) and (51) suggest that the homogeneous solutions #h~ be choose as

where

i=1

(5~)

for the region Y2. Due to the multivalued characteristics of logarithmic equation contained in 9(zt) and

Eq. (52), the requirement of single valuedness of elastic displacement and electric potential provides

2Re ck0 qk + 2 a R e Ao2 (blal~ +jbjaa~) = 0 (54) k=l t k A03

For different choices of the conformal mapping, the solution for F(z), may be assumed as

oo

r(~) = ~o + Z ~< ~ (5.5) k--i

for a crack, and

f(z) = ~ sk< ~ (ss)

for others, whose coefficients have the following relation [10]:

S_ k = kOo2kSk

where L)0 = v/(1 - e)/(1 + e) for elliptic hole, and Q0 = J + ~ for others. Using the above transformation, the boundary conditions (51) become

3 7

k--i m = l

3 7

k--I m = l

3 7

k--1 m = l

3 7

k--I m = l

(57)

(58)

(59)

(so)

A new thermoelectroelastic solution for piezoelectric materials with various openings 105

where

3

/~=1

3

k=l

3

k=l

3

k=l

(62)

3 3

k=l k=l

3 3

Z32 "= - - ~ ] E [Re (ck)~k) -- i e R e (ck#~X~)], /42 = -aT? E [Re (ckt~) - / e R e (ckp~r k=l k--1

(63)

k=l

3

~33 = J~03 ~* -- E {C'ko)~/~} , k=l

[ • ,1 123 = - /}026"r + {eko#k , k=1.

/43 = O, b* = a(bla> + jbja3~),

(64)

a a I14 -- j -- I [ j / ) o 2 b l a a r -- Bo2bjal~], /24 -- j _ $ [jBo2big3~ - Bo2bjas-] ,

a a 134 = j _ 1 [ jB0a/) laar -- Bo3bjalr], 144 - - j - 1 [jAo3bla3r - Aoabjal~],

aBo2 aBo2"r 115- j+ l [Jb la4"~+bja2~] ' 125- 7 7 1 [jbla4~+bja2~],

a B o 3 13~ -- j + 1 [jbia4~ + bja>-] ,

aAoa 145 - - j + 1 [jbla4r + bja2r],

(65)

(66)

116 = -aBo2bja4~/2 , 126 = -aBo2rbaa4~/2, 136 = aBoabja4~/2, 146 = -aAoabja4~/2,

(6r)

g17 = -aBo2b, a2r 127 = -aBo2rb la2r lar = aBo3bla2~/2, 14r = -aAoabla2r

(6s)

z , (~ ) : s /~ , z~(~) = 1/~J, Ea(~) : l n ( 1 / ~ ) , < ( ~ ) : S / ~ j-~ (69)

E'~(~) : 1 / ~ j+~ , E6(~) = 1 / s ET(~) = U s ~ : ~ i ~ .

Substituting Eqs. (52), (55) or (56) into Eqs. (58) - (61) , provides following equations:

2 Im(li3) = 0, (i = 1, 2, 3) ,

3

E dki = (5il/11 q- gij112 + ~i( j -1) /14 q- c5i(j+1)I15 Jr- (5i(2j)116 -1- ~5i2/17 , k=l

3

~ 2 #kdk{ = (~i1~21 -~- ~ij~22 ~- ~i(j 1)124 -}- (~i(j+1)Z25 -[- (~i(2j)[26 -Jr- ~i2t27, k=l

3

E (xkidki + ~kidki) = 5il/51 + (~ijl52 4- ~i(j_1)154 @ 5i(j+1)155 -}- 5i(2j)156 @ ~i2/57, k=i

(7o)

(71)

106 Q.-H. Qin and Y.-W. Mai

where 5ij is the Kronecker delta,

Xki = Xk - i eo tk , ~ki = O, lsi = 18i - ieol4i , (72)

3

Si = ~ ti dki -- (~il/41 -- (~ij142 -- (~i(j-1)144 - (~i(j+1)145 - (~i(2j)146 -- (~i2/47 (73) k=l

for a crack, and

i~o _ (74) Xki ~-~ Xk 1 -~04i (1 -~- LO04i) t k , ~ki 2ieoQo2itk 1 -- Q04i '

ic0 15i = 13i ~ 1 --~04i [2~02i[4i -- (1 + ~04i) 14i], (74)

s~ - 1 - oo 4i ( ~ 1 7 6 & - tk & ) + & ( / 4 1 - e0~141) + ~j(142 - oo2~l~2)

for the others. Thus the unknowns c~0 and dij can be solved from Eqs. (54), (70) and (71). Once the constants dij are obtained, the homogeneous solution, ~hk(Zk) , can be further writ-

ten in the form:

~hk(Zk) = (CRk + iczk) Zk + Cko in G + d k l & -1 + dk2~k -2 + dk(j-1)~k - ( j - l )

+ d k S k j + d]~(j+~)~k -(j+l) + dk(2j)~k -2j . (76)

The general solution for stress and electric displacement are, then, given by

~ = 2R~ 1 ~;~(z~) + ~0~ / ( z~ ) (77) ~ ~=~ - ~ - B o ~ r g ' ( z t )

= 2Re ~ ~ ( z k ) +

From Eqs. (77), (78) and the infinite boundary conditions (22.2), we can generating a

system of five equations for the six unknowns involved in the real and imaginary part of the constants c~. Because of the arbitrariness of the rotation at infinity [13], we can set one of these constants equal to zero. We will let cH = 0 in our analysis. Hence, using Eqs. (44),

(76) (78) and considering z~ --~ oc, yields

{ O.011 }0.020.02 : '~P~e E/c=13{ ~/~2 }--#kJ Ck' {DI~176 : ~ R e ~ ' ~ f ~ k ~ k } C k k : l _ --~/~ (79)

which compared with Eqs. (62) and (63), we known that

a a a /11 = - ~ (002 - ieo-~2), /21 ~--- - ~ (0.02 - ieol~ /31 = - ~ (D20 - ieD10)

(80) a~

a~ a~ (o~ + ie~0 ), I~ = T ( D ? + ieD~~ ~ = - T (0.0 + i e ~ ) , ~ = - T

As was pointed out by Sosa and Khutoryansky [10], an electric field will be induced by the applied load at infinity in the piezoelectric material, whose value is obtained by inserting

A new thermoelectroelastic solution for piezoelectric materials with various openings

Eq. (79) into Eq. (4) giving

E2 0 = 2Re ~ _ _ #k k = l

which compar ing with Eqs. (62) and (63) yields

at/ 141 = - ~a (ElO + ieE2O) ' 142 = - ~ (Ex ~ - ieE2 ~ .

Thus all unknown constants in Eq. (76) have completely determined.

107

(81)

(89_)

4.3 S E D along the hole boundary

Let n(w) and m(c~) be, respectively, the unit vector tangent and normal to the elliptical boundary (Fig. 1), one sees

n(w) = { - sinco cosw 0} T, m(aJ) = {cosw sinw 0} "2. (83)

With this notation, the surface traction and charge vector t~ at a point along the hole bound-

ary of which the normal is n, we have

t n = H 2 c o s co - H1 sin co, (84)

where f/1 = {O-11 O-12 D1} T, H 2 = {0-12 0-22 D2} T. The normal stress 0- ..... the shear stress 0 -~ and the normal electric displacement D.,~ along the boundary are given by

0-~,~ = nT(co) tn , 0-~ = roT(@ t~, D~ = (t~)a (85)

Similarly, for the surface traction and charge vector t~ on the surface of which the normal is m , one gets

t~n = / 7 2 sin cJ + /71 cos co. (86)

Thus the normal stress (often known as hoop stres) 0- ..... the shear stress 0- .... and the normal electric displacement D ~ are

0-~,~ = m T ( c J ) t ~ , 0 -~ = nT(a~)t,~, D~ = ( t~)3. (87)

4.4 Cracks

A crack of length 2a can be formed by letting e and r/in Eq. (21) approach to zero. The solu- tions for a crack in an infinite thermopiezoelectric plate can then be obtained f rom the formu- lation in Section 4 by setting e = r / = 0. In this case, Eqs. (44) and (76) can be simplified to

9(z ) - h2~ tn + - + J 4k

a

+ -2dk2 ( z 2 _ (89)

108 Q.-H. Qin and Y.-W. Mai

Thus, the asymptot ic form of SED, H2, ahead of the crack tip along the xl-axis can be given by:

1 (90) II2 = {a12 a22 D2} T = / 7 , x ~ v / ~ _ ~ ,

where

= k=l - - X l c (CkO - - d k l - 2d/~2) - Bo aB02 z~ J" (91)

With the usual definition, the SED intensity factors are given by

= lim X//27r(xl - a) H2 • H , ~ ? , (92) K x 1 ~+a v a

where K = {KII, K• 14o} ~, in w h i c h / ( i , K H are the usual stress intensity factors, and KD is the so-called "electric displacement intensity factor".

The energy release rate can be obtained by considering the work done in closing the crack

tip over an infinitesimal distance Ax, which can be calculated by [14]

A x

G = z~01im ~xl f H2T(x) Au(x ~- Ax) dx, (93)

0

where G denotes the energy release rate. Note that the integration variable x represents the distance ahead of the crack tip. By using the solution obtained previously and setting x2 = 0--, Ix~[ < a where • denotes the upper and lower surface of the crack, the jumps of elas- tic displacements and electric potential (EDPE), A u = {Aul Au2 Acp} ~ across the crack faces

can be given by

I ~ " ( 4ix~ dk2 Au = u(xl , 0 +) - u(x~, 0 ) = 2Re P~0

(0 < < a) ,

= = A T where Pk0 {Pk qk t~} r , A0 {A01 A02 0a} . The substitution of (90) and (94) into (93), we have

G = r r Re Pk0 - k=l a 2k

2idk~a ) A0h20x121~ v a2 - x12

(94)

(95)

5 Numerical example

For convenience we only consider a piezoelectric ceramic (BaTiO3) plate with an elliptic hole.

The material constants of the plate are as follows [15]:

C l l = 150Gpa, c12 = 66Gpa, c22 = 146Gpa, c33 = 44Gpa,

aZl = 8.53 • 1 0 6 / K , a22 = 1.99 x lO-6/K, A2 = 0.133 x 105N/CK,

e 2 1 = -4.35C/m 2 , e22 = 17.5C/m 2 , e~3 = l l .4C/m 2 , z n = 1115eo,

~22 = 1260Co, ao = 8.85 x 10 12C2/Nm2.

6

4

3

2

1

0

-1

-2

o present

20 40 60 80 1 O0

0)

Fig. 2. Concentration parameter ~ vs angle a;

100

3.5

-- - -e-- present _ ~ , , ~ 3

2.5

2

1.5

1

0.5

Oi I _ . t I I q

20 40 60 80

0) Fig. 3. Concentration parameter ~D VS angle ~2

o present

1.5 ~D

1.4

1.3

1.2

1.1

1

0.9

A new thermoelectroelastic solution for piezoelectric materials with various openings 109

I I _ I I _ , t

0.1 0.2 0.3 0.4 0.5 0.6

Fig. 4. The intensity factor parameter/3D vs ;q

110 Q.-H. Qin and Y.-W. Mai

Since the values of the coefficient of heat conduction both for BaTiO3 could not be found in the literature, the value k22/kll = 1.5, kll = 1 W/mK and ktx = 0 are assumed.

Figures 2 and 3 show the numerical results of {~ = ~,~,~/c~~ (when or~ 2 r 0 and others = 0)

and ~D = D,~/D2 ~ (when Dz ~ r 0 and others = 0) vs the angle co when e = 0.5 and j = 1. As it is evident from Figs. 2 and 3, both ~ and ~D increase along with the increase of angle co, and reach their maximum values at w = 90 ~ It can be found from the two figures that the

numerical results obtained from the present model (using the exact boundary condition) and the impermeable model (the model can be obtained simply by setting e0 = 0, which leads to the so-called approximate boundary condition, i.e., the hole surface is charge free and the

electric displacement vanishing inside the hole) are seen to agree within plotting accuracy. It

tells us that the impermeable assumption may be used in this case clue to the fact that the die- lectric constants of a piezoelectric material are much larger than that of vacuum. Figure 4

shows the numerical results for the coefficient of electric displacement intensity factor/3D vs

/3, where/3z) and/3 are defined by

K D ah2o92 (96) /3 D - D 2 0 ~ , /3= /~D20 .

It can be seen from Fig. 4 that there exists some discrepancy between the two models. Therefore one should be careful when the impermeable model is used in the crack problem.

6 Conclusions

The two-dimensional problem of a thermopiezoelectric sheet containing a hole of various

shapes is studied. A unified analytical solution for the hole problem is derived through use of the extended Lekhnitskii formalism, conformal mapping and exact electric boundary condi- tions. As an application of the solution, the hoop stress and the soIution for crack are discus-

sed. Based on the formulation, the stress and electric displacement intensity factors and strain energy release rate are derived analytically. Numerical results are obtained for stress concen-

tration along the hole boundary and electric displacement intensity factor. The results show that the impermeable model can achieve relatively good results for hole problems. However,

the results of electric displacement intensity factor indicate that there exists some discrepancy between the present model and the impermeable model. Therefore one should be careful when

the impermeable model is used in the crack problem.

Acknowledgements

The work was performed with the support of Australian Research Council (ARC) Foundation. This sup- port is gratefully acknowledged. Q. H. Q. is also greatly appreciated for the financial support under an ARC Queen Elizabeth II fellowship program.

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A new thermoelectroelastic solution for piezoelectric materials with various openings 111

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Authors' address: Q.-H. Qin and Y.-W. Mai, Department of Mechanical Engineering, University of Sydney, Sydney, NSW 2006, Australia