Fracture mechanics for piezoelectric ceramics

,

FRACTURE

0022 1992

SOY6 Y? $5.00 + 0.00 Pergamon Prt3?, L.td

MECHANICS FOR PIEZOELECTRIC CERAMICS

z. suo Department

of Mechanical

Engineering.

University

of California,

C.-M. Division

of Applied

Mechanics,

Stanford

Santa Barbara.

CA 93106-5070.

U.S.A.

Kuo University.

Stanford.

CA 94305-4040,

U.S.A

D. M. BAKNETT Department

of Materials

Science and Engineertng, Stanford CA 94305-2205. U.S.A.

University.

Stanford.

and J. R. WILLIS School of Mathematical

Sciences. University

of Bath, Bath BA2 7AY. U.K

WI:STIJDY cracks cithcr in pieroclectrics. or on intcrfxes between ptezoclectrics and other materials such as metal electrodes or polymer matrices. The projected applications include ferroelcctric actuators operating statically or cyclically. over the mllJor portion of the samples. in the linear rcgimc of the constitutive curve. but the clcvatcd ficld around defects causes the materials to undergo hysteresis locally. The fracture mcchamcs viewpoint is adopted& that IS. except for a region localized at the crack tip. the materinls xc taken to be linearly piezoclectric. The problem thus breaks into two subproblems: (i) determining the macroscopic held regarding the crack tip as a physically structureless point, and (ii) considering the hysteresis and other irrevcrsiblc processes near the crack tip at a relevant microscopic Icvrl. The lirst subproblcm. which prompts a phenomenological fracture theory, receives a thorough investigation in this paper. Griffith’s energy accounting is extended to include energy change due to both deformatmn nnd polarization. Four modes of square root smgularittcs arc identiiicd at the tip of II crack in a homogeneous picroelcctric. A new type of singularity is discovcrcd around interfxe crack tips. Spccilically, the singularittcs in general form two pairs: r ’ 2“’ and I’ ’ Iih. where i: and K xc real numbers depending on the constitutive constants. Also solved is a class of boundary value problems involving many cracks on the interfxc between half-spaces. Fracture mechanics ;tre established for fcrroclectrtc ceramics under smallscale hysteresis conditions, which fxihtates the experimental study of fracture rcsistancc and fatigue crack growth under combined mechanical and electrical Ionding. Both poled and unpolcd fcrroelcctric ceramics :tre discussed.

I. I .I . Pic,_ocli~c,tr-icit~, arld,fi~~~oclc~tri~

INTRODUCTION cwxtnic~s

PIEZOELEC‘TIIICS deform when subjected to an electric field and polarize when stressed. The stress 0, strain 7, electric induction D, and electric field E obey the constitutive relations 739

710

where C’ is the elasticity, 2:the permittivity and c the pieLoelectricity ; these tensors arc material-specific. For example. quartz is pieroelcctric. with the physical constants on the orders

Fcrroelectrics belong to ;I subgroup which, when in the form of ;I single domain, possess a spontaneous polarization that can be revcrscd by applying a high electric lield. the coercive field. Most ferroelcctrics are also linearly pieyoelcctric when operating around the spontaneous polariration. Figure 1 illustrates the idealized rcrroelectric behaviors. switching between two pic7oelectric branches. Since the 1940s. these propcrtics have been exploited to product polycrystalline pieloelectrics : macroscopic polarization is induced in ceramics by applying ;I high static field. which partially aligns the polar axes in the randomly oriented domains. a process called poling. The physical constants I’or poled bariLlm-titanate (RaTiO,) and lead-/irconatctitanate ;~re on the orders

Both i: and o l-or the ceramics arc substantially higher than Ibr quart/. Unpoled l.crroclectric ceramics do not have macroscopic pieLoclectr_icity but still have large pcrmittivity. so they make good capacitors. A dimensionless group is formed by the three types of moduli. varying for most materials in the range

Piezoelectricfracture mechanics I&!

= 0.1-I.

741

(1.4)

This number indicates the degree of the electromechanical coupling. Books that contain the background information include FATUZZO and MERZ (1967) on domain walls, JAFFE et ul. (1971) on ceramics processing, POHANKA and SMITH (1987) on fracture testing and polymer matrix composites, and MIKHAILOV and PARTON (I990)on electroelasticity. I .2. Cracking und,firtigue Ferroelectric ceramics are brittle (toughness z 1 MPa m’ ‘) and susceptible to cracking at all scales from domains to devices. When a ceramic is poled by a static field on the order of MV/m, cracks nucleate to relax the incompatible strains (CHUNG rt al., 1989). Ferroelectric thin films can sustain high fields without electric breakdown, but may delaminate from the substrate, or crack laterally. Actuators are usually layered with substrates, or embedded in polymer matrices. Interface debonding degrades the electrical performance. Many static and cyclic failure behaviors in multilayer actuators are described by WINZER et ul. (1989). There has recently been a resurgence in the effort to develop non-volatile memory devices based on ferroelectric hysteresis. The major material challenge has been fatigue : the hysteresis loops diminish with the cycling. Many fatigue mechanisms have been postulated. Microcracking is one mechanism responsible for fatigue, which grows with cycling, forming micro-capacitors. Another mechanism involves a-domains that grow from the surface and hinder the motion of 180 domain walls (HAYASHI et ~11.. 1964). The stress state plays an important role in these mechanisms, as shown by HE c~ful. (1991). PAN er ul. (1989) found that the number of cycles to failure can be greatly increased by carefully polishing the ceramic surfaces prior to electroding. Ferroelectric hysteresis degrades materials even if samples operate nominally in the linear piezoelectric regime. Under an alternating low amplitude field, the field concentrated around defects can cause materials to undergo hysteresis locally. Fatigue crack growth is likely, although no systematic study has been made. We will develop a mechanics framework to evaluate fatigue crack growth under small-scale hysteresis (see Fig. I). The terminology will be borrowed from fatigue crack growth in metals under small-scale yielding conditions, although the underlying physics may be entirely different for the two phenomena. Fracture testing has been conducted for ferroelectric ceramics (POHANKAand SMITH, 1987; FREIMAN and POHANKA, 1989). Ordinary fracture mechanics ignoring piezoelectricity has been used to interpret data. One theme of this paper is to introduce the electric intensity factor K,v.analogously to the stress intersity factor in the usual fracture mechanics, as a correlating parameter, so that failure testing can be interpreted in the full range of mechanical and electrical loadings. A second theme is to analyze interface cracks for applications in multilayer actuators.

Cracks in piezoelectrics have been investigated as an electroelastic problem by several authors. Contributions of the Soviet scientists have been reviewed by MIKHAILOV and

742

2. Sr 0 (‘i rri.

PARTON

(1990).

PARTON

(1976)

slits

that is. the potential across the slit :

These

boundary

piex~electric

conditions

ceramics,

took a crack to be ;I traction-free.

and the normal

cannot

componcnl

is 10; times

air or silicone ;I

oil). A crack may be thought drop.

potential DFEG

crack

(1980)

and PAK

(1990)

higher

another

arc continuous

grounds.

Specifically,

than the cnvironmcnt

of as ;I lo~v-capacitance

proposed

but permeuble.

induction

on physical

be defended

the permittivity

ofthe

set of boundary

rncdium

for (c.g.

carrying

conditions

on the

:

n,: = n,, = 0. Two

assumptions

face.

and

(ii)

ilrc involved.

the electric

namely

induction PAI(

(1.6)

: (i) no external

charge

in the environment (1990)

resides

is negligible.

on either On

crack

the basis

of

a

crack with its front coincidcnl with the poling axis, and Sosa :md PAK ( 1990) inccstigated the more general crack tip field using an eigenfunction analysis. SHINDO (31(I/. ( 1990) analyzed cracks in piezoelectric layers using integral equation methods. K~JO and BARNETT (1991) carried out an

these

boundary

asymptotic fxc

crack tip analysis,

boundary

allowable intcrf:lce

conditions.

conditions.

singularities cracks.

pointed

nnd found

various

for an interfxe

differ

which

In his calculation

studied

in one respect

we shall

illustrate

singularities

ctxck from

those

in the present

of the e1ectrostrictiL.c

stress

near ;I

out that

the crack 21s an

depending

on the crack

between bonded pie/;oelectrics. associated

with

The

the elListic

work.

crack tip. Mc.MI:I:KIN(; (IOXC))

(I .6) may not be appropriate for small crnck opening. elliptic flaw with lower pcrmitti\,ity. The paramelci

He modeled

(t:, X,,,)(Wll)

(1.7)

where C, and c,,, ;~re the pcrmittivitics of the fnw and the matrix respectively. are the semi-axes of the cllipsc (0 > /I). Assumption ( 1,6) cortaponda lo the case that parameter (1.7) is small. Howmer. it may not be sm~~ll e\‘cn if the pcrmittivity ratio is small. beca~~s~ c1.11 c;~n bc huge for crack-like flaws. Similar behavior emerges for pieroelectrics, which will not be disc~~ssed here. M(‘MtzlilNC; (I 987. 1990) also modeled cracks in dielectrics containing ;I conducting Iluid. The appropriate boundary condition is then

cn~rges,

and N and /I

of

conducting cracks in piemelcctrica h;~\ been carried 0~11 h! Srro ( 1093). We shall adopt ( I .h) in this article. since most results arc dircctcd towards cs;Lablishing ;I framew:ork for macroscopic fracture resting of fcrrocloctric ceramics. a here ~hc crack acparation is large except li,r ;I small /one near the crack tip. From the

An analysis

viewpoint lumped

of Ihc fracture

mechanics.

into the mncroscopically

this ~onc c;In hc treated as ;I sm:lll-scale

measured

toughness.

fcaturc.

743

Piezoelectric fracture mechanics

1.4. Plan

qf this paper

In Section 2, the elements of elasticity, electrostatics and thermodynamics are combined for ease of reference. In Section 3, Griffith’s energy accounting is extended to include the electrical energy. Examples are given for which exact energy release rates are readily obtained. It is found that an electrical loading usually provides a negative energy release rate. Whether this negative driving force will retard crack growth will depend on the fracture mechanism at the crack tip. To investigate the crack tip stress field, in Section 4 a general solution to the electroelastic problem is obtained, involving four analytic functions of different variables. The electroelastic field around a crack tip in a homogeneous piezoelectric is given in Section 5. It contains four modes of square root singularities, but only one mode (mode I) gives rise to the tension directly ahead of the crack. A class of boundary value problems is solved, including that of a crack in an infinite piezoelectric. The analysis is then extended to cracks on interfaces between different materials in Section 6. A new type of singularity, first reported by Kuo and BARNETT (1991), emerges from this analysis. Specifically, the four singularities form two pairs : r ’ ‘f’t and r I ?+h, The latter pair is peculiar, seemingly violating the singularity orders suggested by the J-integral. We demonstrate that these singularities are indeed consistent with path-independence of the J-integral. The solution for a finite crack on an interface is also given. We study the mechanics implications of our results in Section 7. Both unpoled and poled ferroelectric ceramics are treated. A pragmatic approach is described to evaluate fracture resistance, and fatigue crack growth of ferroelectrics.

? -.

BASK EQUATIONS OF THE CONTINUUM THEORY

Subject a material to a field of displacement u and electric potential 7 and the electric field E are derived from gradients : .I

is/ =

i(~,.,+u,,,), E, =

Attention is limited to small deformation The stress (Tand the electric induction they jump by 4

4. The strain

(2.1)

-4.z.

in this work. D are defined such that, across an interface,

b,T -or, ] = 1,. n,[DT -D,

] =

-w,

(2.2)

where n is the unit normal to the interface pointing from the + side ; t is the force and (1)the charge, per unit area. externally supplied on the interface. CTand D are divergence-free, i.e. 0

11.1

=

0,

D,,,

=

0.

Here the body force and extrinsic bulk charge are taken to be negligible. The principle of virtual work is

(2.3)

SUO(‘I d.

2.

744

(2.4)

where v is the volume and s the surface of the body. Any two among (2. I), (2.3) and (2.4) imply the third. Analogies may be drawn between the mechanical and electrical variables according to their roles in the above equations, such as 7 - E, II - c/j. (T - D and f - (II. However. it is ‘J and D together that define the physical distortion of the material. They arc assumed to be reversible, so that an energy function, $(y, D). exists, for which d$ = “,,d;>,,+E, We will focus on materials in the This class of nonliner materials operating around the remanent We shall formulate the theory u‘(Y. E), the electric enthalpy. is

dD,.

(2.5)

stable regime, where IJ is convex in the (7. D, $)-space. is general enough to include the poled ferroelectrics state, and electrostrictive ceramics. based on u and 4. A more convenient energy function defined by II’ = $ - E,D,.

(3.6)

Evidently. Uris not convex in the (;I. E. It,)-space : it has a saddle point at (E. ;3) = (0.0). A Legendre transformation shows that d\z, = c,, d;!,, - D, dE,. The stress and the electric induction fl,,

are therefore

= [:,,.;~c’“’ ii’.

(2.7)

given by

D, = -c:y/?E,.

(2.X)

Ferroelectrics are approximately linear when the loading amplitude is small compared to the depoling field. Linear piczoelectrics are described by a quadratic ,,‘ = .! c’ iits >l %I ,,‘,,S - &,,E,E, -- c,,,E,,~,,, where C is the stiffness, (: the permittivity reciprocal symmetries hold : C',,,, =

Other symmetries

implied

C',,,,,

by the symmetry

and I’ the piezoelectricity.

i:,, =

(2.10)

i:,,.

(Il.1 I)

(‘,,\ = (‘,\,.

are of less significance, and will not be invoked in the theoretical stable materials. both C‘ and I: arc positive definite, i.c.

This is the tensor form of ( I. I ). An alternative quantities is

For

for linear piezoelectrics

D, = GE\ +L’,,,;‘,,. expression

development.

(3.13)

t:,,E,E, > 0,

for any real non-Lero real tensor ;’ and vector E. Derived from (2.9) and (2.8). the constitutivc relations fl!, = C;,,,;!,, pcJ,,,E\.

The following

of the strain tensor, such as

c,,, < = c,,, 5 = c,,, .

C’,,,.,;‘,,;‘, > 0.

(2.0)

arc

(2.13)

using (Tand E as controlling

Piezoelectric

fracture

y,, = S r,,, ~,,+&E,.

745

mechanics

D, = CE,

(2.14)

+d,,\or,,

where S is the compliance, d is the piezoelectric strain tensor and a0 is the stress-free permittivity [which is different from i: in (2.13)]. The moduli in (2.13) and (2.14) are obviously related. Given a ceramic sample, the mechanical boundary conditions are taken to be n,o,, = t;, u, = u:‘, where the superscript ditions are

0 indicates

on s,,

(2.15a)

on s,,,

(2.15b)

the prescribed

n,D, = -co”, C#J= c$“,

values.

on s,,,,

[;c

!,,’

11,.r

u. !.I

(2.16b) are assumed

(2.15b) and (2.16b), the solution

(2.18)

A finite element procedure can be formulated using this variational four degrees of freedom, u ,, u?, u3 and 4, at each node. Since r,r is not the solution field need not minimize R. Although the relevant energy function, IV.is indefinite, uniqueness a fairly general class of nonlinear materials under small deformation. E, 4) and (5, y, G, d, & 6) be different solutions satisfying the conditions (2.15) and (2.16), so that the difference field does not do on the surface : jn,((~,,-d,,)(~,-z?,)-n,(D)-6,)(&$))ds

Next consider

the internal

virtual

(2.17)

field renders

6Q = 0.

s s,

to be negligible.

$,,4.,4., +e,,,~.ju,.,] dv- i,il’~,ds+S,,u”rldi.

Of all the fields (u, 4) satisfying

con-

(2.16a)

on sQ.

In the above, the stress and induction in the environment The normal n directs towards the environment. Define a functional

Q(u, 4) =

The electric boundary

= 0.

principle, with positive definite. can be stated for Let (0, 7. U, D, same boundary any virtual work

(2.19)

work

{(ol,-a,,)(y,,-~,i)+(D,-a,)(E,-~:,)}dv.

(2.20)

It must also vanish as implied by the principle of virtual work (2.4). However, the integrand is positive for stable materials. The contradiction proves uniqueness. As an example, for the linear material defined by (2.13), the integrand in (2.20) can be written as

746

z. SW <‘I (I/. C!,~\(;~,,-T,,)(:,\

which is positive because

-;:,

(2.3 I )

)+i:,,(n,-d,)(D,-n”,),

C and 1:are positive definite.

3.

ENERGY CONSIDERATIONS

Gritlith’s energy accounting is applicable to the non-linear reversible materials defined by (2.5). Let two such materials be bonded. with a crack arca A left on the interface. the crack facts being free of cxtcrnal charge and traction. Load the system by a generalized displacement A and voltage V, with Fand Q the associated for-cc and charge. The electric enthalpy W in the body. which is the volume integral of II‘, accumulates according to dl&‘= FdA-Qdl’--%d/I.

(3.1)

This equation defines ~7 as the driving force for the crack area A, as well as dcmonstrating that Fand Q drive A and V. When the crack area is fixed, dA = 0. so that (3. I ) recovers (2.4). with the identifications n

(3.2)

Equivalent definitions of 9 can be derived using Legendre example, using the potential energy n = W- FA gives

E‘ol

(3.3)

dFl = -AdF-QdC’Gd.4. The functions the body and depending on Consistent

transformations.

Wand Fi can be viewed as the total energy of the system. comprising its loading device, with the relevant function in any given problem the type of boundary conditions applied. with the above, 9 can be rewritten as

Y = -(c~M/!(:A),,.~ = -(in/iA)

,,,.

(3.4)

For example, if A is maintained by grips and c’ by electrodes, 9 is the c/~c~rc~~r in energy W associated with the creation of n unit crack arca. For a bimaterial with the interface along the .\--axis. the integral J =

(11’11, -/?,(T,,,ll,,,, -n,D,ci,.,)

ds

(3.5)

1‘ vanishes over contours not enclosing of external charge and traction

any singularity. 3 = J.

When the crack faces are free (3.6)

where the J-integral is taken over any path that begins at one point on the lower crack face and ends at another point on the upper crack face. These are extensions of the elastic version due to RIC‘E (1968). In general. for a given specimen 9 can be computed by analyzing the piezoelectric

Piexwlectric

fracture

mechanics

h

FIG. 2. Piezoelectric

slab loaded by separation

and voltage

problem, but rudimentary mechanics will suffice for the following examples. Figure 2 illustrates a cracked piezoelectric slab, of thickness h, electroded and gripped on the top and bottom surfaces. The wake far behind the crack tip is free from the strain and electric field. Far ahead of the crack tip, the non-zero components of strain and electric field are 1’1, = A/h,

E,. = V/h.

(3.7)

Since A and V are held fixed, (9 is the decrease in W for a unit advance of the crack. which in turn equals the energy in the volume h x 1 x 1 of the slab far ahead the tip, and !g = 1c.h= 2; (C,.,,.,.A’ -a?, if’-2r,.,.,

VA).

(3.8)

Figure 3 illustrates a double-cantilever-beam subject to applied moments and voltage. The top and bottom surfaces are electroded but traction-free. Since both load and voltage are fixed, 3 equals the decrease in energy fl for a unit crack advance : 93 = 96S,,,,M’/h3

-

;E;,

V’/h.

(3.9)

Here plane stress conditions are assumed, but plane strain can be treated by a suitable change of moduli. This experiment can be used to determine the effects of the electric field on the toughness of ferroelectric ceramics. The moment may be applied by inserting a wedge and the voltage can be either static or cyclic. Both examples can be treated using the J-integral as done for the elastic versions by RICE (1968). Observe that for normal operations the contribution to Y from the voltage is small and negative. This negative driving force is entirely physical ; for example, it manifests itself in a classical physics experiment, where a dielectric slab partially inserted in a

I

I-K. 3. Double-cantilever-beam

loaded by moments

I

and voltage

73x

2. Sr 0

C’,

(Il.

parallel-plate capacitor is driven in by this force [see FEYNMANclt u/. (I 964)]. However. whcthcr this force will retard crack growth should depend on the fracture mechanisms at the crack tip. We will discuss this point further in Section 7.

4. COMPILX VARIABLE SOLUTION Following a procedure of ESFWILRYc'/(I/. (1953) for anisotropic elasticity. we derive the general solution to the electroelastic problem. which involves four complex variables. For linear piezoelectrics. (3. I ) and (2.13) give n, = -l:,$(i,.> +1,,,,1r,.,.

ff,, = (‘,,,,lr,.,+c,,,,(l,,. Substituting

into (3.3). one obtains

four Navier-type

(C,,,,11,+rJ,,,4).\, =

0.

equations

( -l:)$(i, +l’,,,rr,

(4.1)

for u and (I, :

) I/ = 0.

(4.2)

Attention will be focussed on two-dimensional problems in the (.I-.j,)-plane. i.e. nothing varies with .Y,. function of a The general solution can bc obtained by considering an arbitrary linear combination of .Y and j‘ : (4.3) It is convenient, here and in the sequel, to take (~0, 4; to be a cdumn with the entities indicated, so that a is likewise ;I four-component column. Without loss of generality, one can always take ,, Cl

The numberp gives

and the column

a

=

I.

” = 1’. 5’

are determined

by substituting

(4.4) (4.3) into (4.2). which

where x and /I take on the values I and 2. This is an eigenvalue problem consisting of four equations; B nontrivial a exists if 17 is a root of the determinant polynomial. In Appendix A we prove that (4.5) admits no real root, so the eight roots form four conjugate pairs. Let /I,. I)~, I>, and /jJ bc the roots with positive imaginary part. a, the associnted columns, and =I = .X+p,r. The most gcncral real solution is obtained from a linear combination of four arbitrary functions : (4.6) For a given boundary value problem, the four functions. ,f’,>12. /; and f;. are sought to satisfy the boundary conditions. The stress and induction are given by

749

Piezoelectric fracture mechanics

jr~~,,D~) = 2 Re i b,.f‘j(=,), 1- I

(o,,,D,i

= -2

(4.7)

b is

where, for a pair (p, a), the associated

(4.8)

_,J> h, = (-&,,,U4+e?rpu,)i,l.

h, = (c2,#,+~,i,?Q4

This is derived from (4.1) and (4.3). Substituting expression : h, = -p

Re 2 b,p&(=,). x- I

‘(C I,,/iCI,+P,~,Ia,)i,(,

(4.8) into (4.5) gives an alternative

hJ = -p

‘(--E,,gI4+P

I,,,q.)i,,.

(4.9)

Define 4 x 4 matrices

A = [a,,a2,a3,a41,

B = [b,,bl,bl,b41.

(4.10)

Each of the as is determined by the eigenvalue problem up to a complex-valued normalization constant. Assuming that the four eigenvalues are distinct and following a method in STROH (1958), one can show that both A and B are non-singular. Define Y = iAB_‘,

(4.1 I)

where i = J1. We list several properties key to the theoretical development. their proofs being given in Appendix A. First, Y is Hermitian, and independent of the normalization for A. Next divide Y into blocks (4.12) where Y, , is the 3 x 3 upper left-hand They have different dimensions : Y,, - [rlu.wicity]

‘,

block and Y,, is the lower right-hand

Yd4 - [pu%Hitriz+q

element.

Y,J = Fi, - [piezoelectriciry]

‘,

(4. 13) For stable materials,

LOTHE and BARNETT (1975) Y, , is positive

Finally

consider

an in-plane

RE

showed that

definite, but Y,J < 0.

w

coordinate

rotation

%cl* &u, =

-sin6 cos 0 fl

cos0 sin0 0

01I ,

(4. 14)

(4.15)

where (*) indicates the new coordinates, and 0 is the rotation angle from x to x*. The four blocks of Y transform like second-rank tensors, vectors and scalars, respectively, i.e. YT, = RY,,R’

,

Yy4 = RY,4,

Y:, = Y,,.

(4.16)

750

z. sue (‘I oi

These arc extensions to the corresponding shorter proof is given in Appendix A. A bimaterial matrix is defined as

elasticity

theorems

of TIM; (1982),

F-/ = Y, + 72.

and ;I

(4.17)

where the overbar indicates complex conjugation. Hercaftcr the subscripts I and 2 attached to matrices and vectors distinguish the two materials; this notation should not cause confusion if one notes the context in which it appears. H maintains nil the properties of Y listed above. The general solution involves four analytic functions. each depending on its own complex variable. Following the one-complex-variable approach introduced in SrJO (1990), WCconsider a column of a single variable defined as f(z) = (,/\ (r),,f~(~),./~(~).,/1(-)).

r = .\-+(J-.

Im (I) > 0.

(4. IX)

The solution to any given boundary value problem can be given in terms of such ;I column. regardless of the precise value of <. To compute the field quantities from (4.6) function. This and (4.7). one must substitute z,, r2_ r3 or zI for each component approach combines standard matrix operations with the techniques of analytic functions of O/K’ variable, and thus bypasses some complexities arising from the use of four complex variables. Of particular importance are the quantities on the .\--axis : v(.\-) = [II,, c/5 ; = Af(.v) +x(3). t(.\-) = jcTJ,.0,;

= Bf’(.Y) + BT’(.Y).

(4.19) (4.10)

The general solution has the same structure 21sthe corresponding anisotropic &ISticity problem. Indeed, results in the following two sections were initially anticipated from the corresponding elasticity solutions given in Wrr.r.rs (1971) and SUO (1990). We should remark that the eigenvalue problem (4.5) can be recast in an alternative form using an eight-dimensional framcwork due to BARNETT and LOTHE (1975). Indeed the study by KLJO and BARNETT (1991) of interfacial cracks in bonded piczoelcctrics uses theeight-dimensional formalism rather than the four-dimensional framcwork of this section.

Results in this section apply to cracks in homogeneous piezoelectrics, and interface cracks for piezoelectric bimaterials having sufscient symmetry leading to a real valued H. Such interfaces have been discussed by QLJ and BASSAN (1989) for elastic materials.

5.1 Crock tip field

We begin with an asymptotic problem. A semi-infinite crack lies on the interface J’ = 0 between two half spaces, material I above, and 2 below. The crack tip coincides with the origin .Y = 0; the bonded interface is on .Y > 0. Denote L 21s the cracked

751

Piexxlectric fracture mechanics segment.

continuous

Assume the bond is perfect, across the bonded segment

so that

the displacement

and potential

are

:

v,(x) = V?(X),.u$L. There is no external

charge or traction t,(x)

on the bonded = f?(S),

(5.1)

interface,

so that

x$L.

(5.2)

As discussed in Section I, ferroelectric ceramics are much stiffer and more permittant than the environment, so that stress and electric induction outside of the ceramic are assumed to be negligible. and. in addition, there is no charge or traction on the crack faces. Thus. t,(x)

= t?(x) = 0,

.YE L.

(5.3)

Furthermore, the field vanishes far away from the crack tip, i.e. f’(z) = o(z”) as z goes to infinity. This is an eigenvalue problem : no length or load is involved. Let the columns defined in (4.18) be f,(z) and f?(z) for the two blocks. The continuity of t(x) across the x-axis, both the bonded and cracked segments, requires that B,f’,(x)+B,f’,(.u) The above is rearranged

= B,f’2(s)+&T;(x).

--x

<.Y<

+x.

(5.4)

-s

< .Y < +x.

(5.5)

in the form -= B~f’z(.u)-B,f’,(.u),

B,f’,(.u)-&T\(x)

The left-hand side of (5.5) is the boundary value of a function analytic in the upper half-plane, and the right-hand side is the boundary value of another function analytic in the lower half-plane. Hence, both functions can be analytically continued into the entire plane. Since both vanish at infinity to conform to a zero far field, they must vanish at any II = .u+ [y(Irn i > 0). Thus. B,f’,(z) = &F;(z),

I’ > 0.

(5.6)

Define the jump across the interface d(x) = v,(x)-vz(_~). Equations

(5.7)

(5.6) and (4.19) give id’(x) = HB,f’,(.u)-fl&f’,(x).

The solution

is simple if

(5.8)

H is real, that is H=f?.

When the crack is in a homogeneous

(5.9)

H is indeed real :

material,

H= Y+y=2ReY. H can be real for bimaterials having certain assume H is real. The interfaces for which section.

symmetry.

(5.10) In the rest of this section,

we

H is complex will be treated in the next

752

2. Sr 0

<‘I

trl.

Continuity of the displacement across the bonded implies that a function defined as

as inferred

(5.1 I)

1’ < 0

i Bzf:(:).

is analytic in the entire plane except on the crack. The traction condition (5.3) leads to a homogeneous Hilbcrt problem h+ (.\-)+h

from (5.X),

,I’ > 0,

B,f’, (z). h(r) =

interface.

(.\-) = 0.

charge

boundary

(5.12)

.Y-EL,

where superscripts + and - distinguish the boundary values when the crack is approached from above and below, respectively. The singular solution to (5.11) that gives bounded displacement and potential is h(z) = (Xx)

(5.13)

’ ‘k,

where the branch cut is along the crack lint. Note that the full field is obtained h(z) is determined. A few explicit results are given below. At a distance r ahead of the crack tip

;o:,.n, ; = (2711.) Since the traction ture. we write

’ 'k.

once

(5.14)

and the charge are real. k must be real. Following

Irwin’s nomencla-

(5.15)

k = :K,,, K,, K,,,,K,\ ;.

Thus the stress and the electric induction arc a linear superposition root singularities. At a distance I’ hrhiml the crack tip

of the four modes

of square

d(r) = (2q7c)

(5.16)

‘Hk.

The displacement jump and the voltage across the crack are asymptotically parabolas. Since. in general, H contains off-diagonal elements, 21K,, field will give rise to crack opening, and a K, field to voltage between the crack faces. The energy release rate can be computed by the closure integral t’(A - r)d(r) dr. where A is an arbitrary length. This is consistent with the definition in Section 3. and was derived by IRWIN (1957) for cracks in elastic materials. Using (5.14) and (5.16) we obtain (5.1X)

‘B = :k’fIk.

Since His indefinite [see (4.14)). 9 can be negative In deriving (5. IX). the identity

for k

with a large component

K,,..

Piezoelectric fracture mechanics

ty(l -t)

‘i dt = qn/sin qrr,

(IRe ql < 1)

753

(5.19)

S’0

has been used with q = l/2.

5.2. Full-fidd

.solutions

Having obtained the asymptotic fields, we turn our attention to a class of boundary value problems. Consider a set of cracks on the interface, collectively denoted by L. The only applied load is the tractioncharge column T(.u) prescribed on the crack faces : t,(x)

= t2(x) = -T(s),

SE L.

This problem is basic: other loadings can be solved by superposition. problem (5.12) becomes non-homogeneous : h’(s)+h

(x) = -T(s),

SEL.

(5.20) The Hilbert

(5.21)

This equation has many solutions. The auxiliary conditions that render a unique solution are: h(z) = o(z”) as 121+ w, h(.\-) is square root singular at the crack tips, and the net Burgers vector for every finite crack vanishes. From (5.8), the last statement leads to “[h+(.+h

(.y)]d.x = 0

(5.22)

i ‘,, for a finite crack in (s,,, x,,). Remarkably, the governing equations (5.21) and (5.22) do not involve any material constants. Thus the solution must be identical to that of a mode III crack in a homogeneous, isotropic, elastic body. The solutions to particular crack arrangements are available in handbooks. As an illustration, we consider the generalized Griffith problem: a crack of length 2a in an infinite piezoelectric, subjected to a uniform tractioncharge column T. The solution is

s

h(z) dz = ;T[(r’-u’)’

‘-z],

(5.23)

where the branch

cut for the square root coincides with the crack. The two columns, from (5.23) and (5.1 I). The field quantities are given by (4.6) and (4.7). with z2 substituted for each component function and each material. For example. one can confirm that f, and f,, are obtained

t(r)

= [x(x d(r)

-u2)

= (a’ -x2)

’ ’ - l]T, ‘HT,

1.~1> a, 1.~1< LI.

(5.24) (5.25)

Suppose that the traction and charge are caused by the remote fields {at,, 0: 1. The intensity factors are identical with the classical ones for a crack in an isotropic elastic material :

7.54

z. 4,

=

SUO

K, = /ntd&.

&bag;,,

(‘I

r/l.

K,,, = .&(T;~.

(5.26)

K,~ = &d.V

This result is surprising because it holds for pieLoe]ectric materials with ;lrbitrary anisotropy. However, the simplicity is accidental rather than a rule : it only works l’c)r cracks between identical half-spaces. or for certain special bimaterials.

6.

IYTEKFA(‘E CRACKS

Bimaterials with complex Hare examined in this section, which is the more common case for interfaces, be they dielectrics. pieroelectrics or conductors. The results in the above section up to (5.8) are still valid. Continuity of displacement across the bonded interface now requires that a function defined as

(6.1)

h(‘)= be analytic full field.

Consider gives

in the whole plane except on the crack lines. Once h(z) is found,

the asymptotic

problem h+(.\-)+A

This is a homogeneous

Hilbert

first. The traction ‘Hh

problem.

(.Y) =O.

condition

(5.3) (6.3)

.vEL.

A solution

h(z) = w:

charge-free

so is the

can be sought of the form

’ 2 +/I.

(6.3)

where w is a four-element column and t: a number, both to be determined. The branch cut coincides with the crack line, and the phase angle of : is measured from the positive .u-axis. Substituting (6.3) into (6.2) gives pw = (,“;i [fw,

(6.4)

This is a 4 x 4 eigenvaluc problem. The algebraic details of the solution are important for understanding the crack tip field: they are recorded in Appendix B. The fout eigenpairs have the structure (c,w).

(-i:.ii).

(-ir;.w,).

where X, 1~.w3 and w., are real, but w is complex. The general solution to (6.2) is then a linear combination the form (6.3) : cTii’K-_lJ w + c X:iRx

11 ,$

h(r) =

+

2,~‘2rc; cash 7~1:

(6.5)

(ik.,w4),

K$w,

of the four solutions

+ K,:

2\/‘2nZ

where Kis complex and K, and Kd are real : they are intensity

COS

*wj 71ti

of

(6.6) ’

factors. The arrangement

755

Piezoelectric fracture mechanics

ensures that the traction and the charge are real, and renders a simple expression for the tractionxharge interface. The novel feature of (6.6) is the presence of singularities of order y I.‘?+Xand rp 1,‘2ph-,which were previously found by Kuo and BARNETT (1991). The latter singularity is particularly surprising because it provides a term of order ym I lti m the integrand of the J-integral (3.5), which appears to conflict with path-independence of the J-integral (in elasticity, the integrand has order r ‘). This apparent paradox will be resolved below. It is convenient to display physical quantities in an eigenvector representation. For example, write the column t(x), .Y > 0 as -t = tw+tw+tjWj+f4W4. (6.7) Because t is real, the components for w and G must be complex conjugate, and C~and f4 are real. Consequently, the t-column in the interface a distance Yahead of the crack tip has three components : one is in the plane spanned by Re [w] and Im [w], the other two are along the wj- and w,-directions. These components vary with r according to [=_Kr” Joy.

t3 = ;g.

tj = $n-;.

(6.8)

The above may be taken as defining equations for the intensity factors. As r--f 0, the component t rotates in the plane spanned by Re [w] and Im [w]. This is clearly the analog of the corresponding result for isotropic elastic bimaterial, where t s a,.,.+ io,,.. The jump d a distance r behind the tip is d=

(H+A)

J

L 2x

&,

(I+ 2ic) coshz

-,t$,

+ (I--2ic:)sh% K&w3 Kg ‘w4 + ~~. _L _L _ + ~ ~~_ ~_ (1+2k.)cos7W (1-2ti)COSW

Substituting

the above crack tip field into the closure integral

1

~ .

(6.9)

(5.17) gives (6.10)

In deriving this, we have used the orthogonality described in Appendix B, and the identity (5.19) with 4 = l/2 k K, 1/2f k. The paradox related to the J-integral is resolved by this result, i.e., J = 3, just as in the case of elasticity, so that J is indeed finite and the term of order r- ’ ” In the integrand must integrate to zero. This is reflected in the orthogonality w:Hw, = 0 which, had it not been satisfied, would have introduced into the closure integral (5.17) a term of order r ’ -zK, with the consequence that a finite 3 could not have been defined. No normalization has yet been assigned to the eigenvectors. A different normalization changes Kl and K4 by real factors, and K by a complex factor. 6.2. Full~field solutions Now consider

Hilbert

problem

the boundary value problems analogous to (5.21) is

in Section

5.2 for complex

H. The

756

2. Sk<, <‘I C/l.

hi (s)+f? Writing

the above equation

’ Hh

(.\-) = -T(.\-),

in its components

h = /~,w+/~,W+/~,w,+i~,w,. one obtains

the decoupled

.YEL.

using the eigenvector

(6. I 1) representation.

‘r = Tw+ TW+ Tlw2 + T,w,,

(6.11)

equations

I1; + IL’ ‘“‘/I,

= - T.

/I; +p’““‘/j;

= ~ T;.

11; + C’f I,T,/I1 = _ % 11; +(J ““>/,,

zz -r,.

(6.13)

Furthermore. since they contain no explicit material dependence &sides I: and K. theit solutions must be identical to those for isotropic elastic bimaterials. One can derive the solution using the method of analytic functions or from the known solutions for isotropic elastic bimaterials. As an illustration, consider the generalized Griffith problem with a crack of length ?u on the interface between two half-spaces of pie7oelcctrics. Consulting EN(;I.ANI) (1965), where the corresponding elasticity problem ws solved, WC obtain

(6.14) where iz2 and 11~arc omitted. factors are

for they can bc obtained

K = T(1 +2k)cosh K, = T3(I

by substitutions.

The intensity

(nc)(ntr)’ ‘(?rr) “1

+k)COS(7Zh-)(Tcl/)’

‘(2~)

‘.

(6.15)

Without re-solving any problems. one can write down solutions for many cracks on the interface between half spaces of piezoelectrics, provided the corresponding solutions for isotropic elastic bimaterials are known.

7.

FRACTURE: MWHANI(.S

Unpoled BaTiO, makes good capacitors due to its large permittivity. The polycrystalline. multi-domain structure cancels the macroscopic piezoelectricity. This spccial class of materials is ideal for theoretical study since it has a simple macroscopic field, yet still retains the essential features of ferroelectric ceramics. In Section 5, we have established that the piezoelectric field around the crack tip is square root singular and linear in four intensity factors. But what is the crack tip’? Or what does the singularity mean physically? The assumed material linearity cannot hold at the crack tip. Specifically, fcrroelectric ceramics undergo hysteresis under sufficiently high loading. Furthermore, down to a small scale many other features. such as the presence of

757

Piezoelcctric fracture mechanics

grains and domains, will invalidate the solution. Consequently, the solution given above is only correct at some distance away from the crack tip. Because of the square root singularity, for large specimens, an annulus exists, smaller than the specimen size but larger than the crack process zone, in which the field is described by the singular field of Section 5. This so-called K-annulus is parameterized by the four intensity factors : any other information will be lost in the communication between the macroscopic loading and microscopic process. Consequently, the fracture criterion can be written in terms of the four intensity factors. Most fracture tests have been conducted with specimens under symmetric loading, such as double-cantilever-beams in Fig. 2. This will in general produce both mode I and mode IV fields at the crack tip. The general results in Section 5 are specialized to this case below, with matrix Y given by (A2). The stress and the electric induction are square root singular as the crack tip is approached. Let the crack plane be normal to the r-axis. The intensity factors K, and KIV are defined such that. at a distance Yahead of the crack tip, I or_ = J;i;, ~ ~ The crack separation

D, =

and voltage at a distance

$

r behind

(7.1) the crack tip are given by

(7.2) where v is the Poisson’s ratio, p the shear modulus and E the permittivity ; they are macroscopic average quantities of the polycrystalline ceramics. The Irwin-type relation connecting 9 and the intensity factors is (7.3) Under

static loading,

the fracture

criterion

should be of the form

9 = I(&K,vIK,),

(7.4)

where I is the fracture energy depending on the mixity of modes I and IV. Whether Klv will influence I depends on the magnitude of the dimensionless mixity, and on the crack tip process. In particular, the hysteresis strain associated with the domain switching as the crack grows is likely to play an important role. Under cyclic loading, a ferroelectric usually is subjected to low load, but the field concentrated at the crack tip generates hysteresis loops locally so that fatigue is possible. as observed experimentally by WINZER ef al. (1989). No systematic experiments or modeling have been done in this area. The present work would allow fatigue crack growth due to the small-scale hysteresis to be evaluated using the approach for lowstress high-cycle fatigue crack growth in metals. It is anticipated that experiments analogous to those described in PARIS and ERDOGAN (1963) for metal fatigue would establish Paris-type laws for ferroelectrics. Specifically, the crack extension per cycle, da/dN, must be a function of the loading range, i.e.

75x

K, = 0

K,#O FIG. 4. For a double-cantilever-beam. a mixed licld of modes I and IV is induced by the voltapc ;~lonc. When E is parallel with P,. the untracked portion undcr~ocs a pwoclcctric expansion along E. but thr cracked portion remams unchanged. The strain m~\lit cause\ stress ahwd of the crack up. and thus K, # 0

du/dN = f’(AK,, AKIV). This relation characterizes a material of the samples being tested.

property,

which is independent

(7.5) of the geometry

The additional complexity due to piezoelectricity can be illustrated by a special case of much practical significance : cracks in a plane normal to the poling axis of the poled ceramics. The relevant algebraic details are discussed in Appendix C. Most fracture tests have been conducted with specimens under symmetric mechanical loading. such as double-cantilever-beams in Fig. 2. This will in general produce both mode I and mode IV fields at the crack tip. The intensity factors are still defined by (7.1). The crack separation and voltage at a distance r behind the crack tip are given by

where CT, i: and c are effective moduli determined by the procedure The Irwin-type relation connecting Y and the intensity factors is

in Appendix

C.

Although the macroscopic field is much more involved in this case, the fracture mechanisms are expected to be qualitatively similar to those of unpoled materials. As illustrated in Fig. 4, the double-cantilever-beam has a mode I component under electric loading alone. This can be appreciated by noting that the material far behind the crack tip is not loaded, and the strain induced by the voltage ahead of the tip causes the stress intensity. However, finite element analysis by HE c/ cl/. ( 1991) indicates that K, due to Y is typically small.

Piezoelectricfracture mechanics 1.3.

Intufircc

759

dehondiny

Debonding at electrodes and other interfaces has been identified to be a major failure mode in piezoelectric and electrostrictive actuators (WINZER et al., 1989), although no systematic testing and modeling have been carried out. We believe that concepts used in elastic interface fracture mechanics are applicable for such a study [see RIVE (1988) and Sue (1991) for brief reviews]. A more extensive discussion on the fracture of layered materials has been presented by HIJTCHINSON and Suo ( 1992). In Section 6 we found that the crack tip field is linear in three intensity factors, one complex and two real, of different dimensions. Interfacial fracture in general should be of the mixed-mode type. The mode mixity must be chosen to reflect the state at the tip. A length scale, i, representative of the process zone size, may be identified, and the mode mixity can be defined by the dimensionless ratios Im (i”K)/IKI,

ihK,/(KI,

i

“KJ(KI,

(7.8)

the fracture toughness depending on these mixities. The experimental determination of the mixed-mode toughness presents a formidable task. Simplifications are possible for special crack orientations and loading conditions. For example, for unpoled capacitor materials, the mechanical and the electrical fields are decoupled at the macroscopic level, so that only Klv defined as in (7.1) is present under electrical loading. This fact simplifies the fracture mechanics for the ceramic-metal interfaces.

8.

CONCLUIXNG REMARKS

Microcracking, fatigue and fracture are among the properties that limit the use of ferroelectric ceramics as large strain actuators. A fracture mechanics theory is developed here to evaluate static toughness and fatigue crack growth for ferroelectric ceramics under small-scale hysteresis. It is likely that various toughening concepts for structural ceramics may find their electrical counterparts, so that ferroelectric materials may be engineered to exhibit superior performance. Future necessary work will include calibrating electromechanical fracture specimens using finite elements, failure testing under static and cyclic loads, modeling of fracture processes at the crack tip to understand the role of ferroelectric hysteresis and accounting for the presence of a conducting fluid, and analyzing typical structures such as piezoelectric films and multilayer actuators.

ACKNOWLEDGEMENTS ZS is funded by ONR/URI

contract

N-0-0014-86-K-0753

and NSF grant MSS-9011571.

REFERENCES BARNETT,D. M. and LOTHE,J. CHUNG,H.-T., SHIN, B.-C. and KIM, H.-G.

1975 1989

Phpica Status Solidi (h) 67, 105. J. Am. Ceram. Sot. 72, 321.

z.

760 DI:I~c;. W. I-. J. ENGI.ANII, A. H. ESHI:LRY. J. D., REAI). W. T. and SHOc’lCLI;Y. W. FAlUz%O. E. and MLK~, W. J. FFY~MAN, R. P., LI:IGHTO~. R. B. and SANDS, M. bklilMAN, S. W. and POHANU, R. C. HAYASHI, M.. IMAIXCIMI. S. and Arv:. R. HI:. M.-Y.. SUO. Z.. M~,Metww. R. M. :tnd Evauc. A. (i. HL I(‘l1lNSOS. .I. W. anti St0 Z. IK~IN. G. R. .I:\rt~,. B.. (‘ooa. W. R.. JI<. :tnd JAt-t I:. H. Kuo. C.-M. and BARNI~~‘I, D. M.

LOTHE. J. and BAKNETT. D. M. MCMIXINC;, R. M. M(,M~i~.t~rsc;. R. M. Mc.MI:I.~;ING. R. M. M1tc~~11.0~. G. K. :tnd PART-0~. V. 2. PAli. Y. E. PAN. W. Y.. DAM, C. Q.. ZtlA!Xti. Q. M. and CIWSS, L. E. PARIS, P. C. and ERIXGAN. F. PARTON, V. 2. POHANKA, R. C. and %ITH, P. L Qu. J. and BASSANI. J. L. RICI:. J. R. R1c.1:.J. R. SHINIX). Y.. OZAWA. E. and NOWACXI. J. P. SOSA. H. A. and PAK~C.Y. E. STKOH. A. N. Sr:o. Z. sue. z. SL,O. z. TIXG. T. (‘. T. TING, T. C. T. WILLIS. J. R. WINLLR, S. R.. SHANICAK. N. and RIT~IX. A. P.

Sue

(‘I

(II

Ph.D. Thesis, Stanford J. trppl. Mwlt. 32, 400. ACtrI t~rcttrll. I, 2s I. lY67 I’964

University.

IYXY

~~,,roc,/cc,/~~~,/~~,. North Holllund, New York. /.c,c~/ur~,.\ o/t b/t~~.tir~s,Vol. 2. p. I O-Y. Addison Wcslcy, Reading, MA. ./. :lut. C’c,rtr/!~.SOC. 72, 225X.

lY64

.loj~. .I. qpl.

1991

Manttscr-ipt in prup;rr;ttic~n

I YY’ IYS7 lY7l

.,l~/t.. q)/~l. &lcr,/~. 29 (in press). .I. ri/yl. !ZiCC~ll.24, 36 I Pic,:w/wrric~ C‘cwwic~s. Academic Pro\.

I YYO IYXY

.I. ri/y,l. .zlC~C~ll. 57, 647. .J. uppI. P/r~x 66, 6014.

P~I.L,.~.3, 63 7.

London.

1990 IYSX I YYO IYYI I YY3 I YX3 I986 lY7l IYXY

A few algebraic theorems of the complex vari;tblc rcprescnlalion itrc documcnbzd here. Fit-sl. following the method used by Estet_nu ~‘t nl. (lYS3) for the elasticity problem. WC prove thal

761

Piezoelectric fracture mechanics the eigenvalues j. and multiply

p of (4.5) are complex. Multiply the first equation in (4.5) by a, and sum over the second by u4 ; the difference of the two results is

(C&Jr~J,+ E+J:)i& = 0.

(Al)

If an eigenvalue p is real, the associated eigenvector a can be chosen as real, and, therefore, with C and E positive definite, the left-hand side of (Al) must be positive. Hence, by contradiction p cannot be real. Next consider the matrix Y defined by (4. I 1). For isotropic, non-piezoelectric dielectrics, the in-plane displacements, the anti-plane displacement, and the electric potential decouple. Unpoled ferroelcctric materials belong to this class. The general representation (4.6) fails because of the degeneracy associated with such symmetry, but Y can be found by a limiting process which gives

(W

0

0

0

I

-

E1

where v is the Poisson’s ratio, ,LL the shear modulus and t: the permittivity. Note that conductors can be treated by setting c = m. Several properties shown in this special case are general to all piezoelectrics. They have been mentioned in the text and their proofs are outlined below. We show that Y is Hermitian. The proof parallels that of STROH (I 958) for the elasticity problem. From (4.8) we have jj’2’$~‘.,,“’

= {Cz,,,,aj2’al’)+e,l,,L7:“a”’ J -+@n’,”

i-(2)?1) +e+, 32, u,(11, ,4z 4p 1

(0

and from (4.9) P’ ‘,a’Z’ . b’” = _jC’,,,,~a:“a~‘~+e,,,,~:“u:“--s,,,a~~’u:”+~~,,,,~~~‘cr~‘))r’,”~),“. Subtracting

(A4)

these two equations,

(ji’“_p”‘)$?‘.,,‘”

= {CX,,;;2’a,“’ + (e,,,,

+e,,,,)(a)“lJ’,”

+@lJ;‘))

-F,,,a(b’U:“}~1”;8”.

(AN

Now the right-hand side has the Hermitian hand side must also ; that is, ($~‘_,,‘I’)$~ or, since ($” -p”‘)

for indices (1) and (2), and so the left-

.,,“I = (p”‘_~‘?‘)a”’

has negative imaginary ,(2,.,,“‘=

From the definition

symmetry

.ij12,,

(A6)

part and so is not zero. _,(I).@?)

(A7)

(4. I I) of Y, the above implies that y’- = r,

(A8)

that is. Y is Hermitian. Only the reciprocal symmetry in (2.10) is invoked in the proof. Now consider the coordinate rotation (4.15). Since C’. e and i: are tensors, from (4.5) (c ,, ;, 1 transforms like a vector, {a,,~~, ul) a vector and a4 a scalar. Specifically,

(AC)) With the identification

that p = c2;‘;, and p* =
Equation

(4.8) implies that (h,,h,.

= 11sin Il+cos

/I,, ’ transforms

like a vector and h, a scalar. Dclinc

cos (I R=

R

0

0

I

L The transformation

-sin = ]I

can be summarized

II

sin0

0

0

cm lI

0

0

0

0

I

0

0

0

0

I I

(Al I)

as

A* = Lt. Note that fi

(Al())

II

l!I* = RR.

(Al?)

’ = R’. and (4.1 I) implies 2’* zz Ryd’.

(Al3)

This complctcs the proof of (4.16).

The structure of the cigenvalue problem

(6.4) is discussed hcrc. The parameter

1:satisfies

11 A- PZZ,H 11= 0. Sincc H is Hermitian.

the transpose of (BI

(HI)

) is

II//-(,‘“’

H

= 0.

(B’)

= 0.

CM)

Also. by complex conjugation. ,, H-e’“‘fi11

It follow~s that. ifr: satisfies (Bl). then so do -L
parts

:

N = n+icz..

(83)

where D is real and symmetric and I+~’is real and antisymmetric; in this context the symbola nothing to do with the clcctric induction or cncrgy used in the main text. Equation (6.4)

have

bccomcs n

‘M/w

=

-i/h.

(135)

whcrc

From the observation for 8:made above. if/i is an cipcnvnluc. so arc -/i. that (RI). written in terms of/L

!In must have the form

’ u’+i[i/

1)= 0.

/iand

-/;

It

li~llo\cs

(B7)

763

Piezoelectric fracture mechanics /?+2lJ/Y+c

= 0.

(B8)

where b = :tr[(D-‘W)‘],

c=

l\D-‘Wll.

(B9)

The above were originally presented by Kuo and Barnett at a workshop, and they also gave possible sets of roots to (BS), which lead to several different crack tip singularity structures. Since their initial presentation, the present writers have realized that only one set of the roots to (BS) is physically admissible because c is restricted as follows. Observe that // WI1 2 0,a property of any antisymmeric matrix of even order, and 11 D ((< 0,a consequence of (4.14), SO that (’ < 0. The roots of (BS) can therefore

be expressed

0310)

as [j1.4 = ~i[(b2-c)'~2+b]'g2.

/j,,> = -t[(h’-c)“2-h]‘~‘,

Thus, one pair is real and the other is purely imaginary. & = ; tanh’

[(p_,.)‘:2_,]‘;2,

In the notation

of the body of the text

n=~tan-‘[(b’_c)‘:‘ihJ’,‘.

(BI?)

These results have been included in Kuo and BARNETT(1991). Their numerical that E and ti are very small for several piezoelectric bicrystals. The associated eigenvectors satisfy fiw = ezni. Hw, It can be confirmed relations

@e = r

2x7HG,

that w is complex

W"

results indicate

z eZxrhHw,.

(Bl3)

the orthogonality

0

0

0

0

WTNii

0

0

0

0

0

w;Hw,

0

0

w':Hwt

0

(Bt4)

w:

Under a rotation of the interface fixed, H transforms according to

&q,

i+'Hw H[w,~,W~.W,l=

Ii W:

fiw 3 = (7-Zn,* Hw,,

but wi and wq are real. satisfying

@I-

@II)

:

but with the relative orientation

!.

of the two half-spaces

H* = I?HR', where a is defined by (Al I). Consequently, the eigenvalues a rotation, and the eigenvectors transform according to w* = Bw. These can be verified directly by substituting similar results for elastic materials.

(Bl5) E and ri arc invariant

under such

(Bl6)

(B15) into (Bl) and (6.4). TIW (1986) observed

APPENDIX

C

In the text, the crack problem has been solved in the sense that only algebraic operations remain to obtain explicit results. In general, Stroh’s eigenvalue problem must be solved numerically. Here explicit algebraic results are given for a class of materials of practical significance: the poled ceramics. The constitutive equation has transverse symmetry around the poling axis (axis-j) :

Hi n, n, = D,

0

LJ;,

0

0

(’ j ,

0

0

(‘3,

0

0

OI

0

(‘1 i

0

’ ,i

0

0

0

0

0

:

;‘Z:

0 (‘1,

00

1

C’,0 c

“ , <

0

0

0

0

0.

clq

-,‘2i ^, is 7. 1

-I-

?_, -, / 1

(i) Crack front coincides with the poling :txis. This case hw been studied by PAK (1990). The (.I-. ,L,)-piano coincides with the isotropic ( I. 2)plmc. The in-plane dcforrnution (u,, u,.) is decoupled from the anti-plane deformlltion and lhc potential (I(,, C/J).The formcr is identical with the ctasticity problem. We focus 011 lhc tatlcr. The cornpies ~~~?resci~t~lti~~n given by Pak is L/: = 2 Rc[./‘,(z)]. The two characteristic paper. wc wrilc

The tmtrix

roots itrc identical,

(/I = 2 Rc[J(z)].

; - .v+,I..

/J, = />- = i. Consistcnl

with the nowion

(0) of this

Y is then

This is lhc lower right-hand block of the 4 x 4 matrix. The upper left-hund block ii; identical lo thal in (442). The tornplctc solution is given cxpiisitly in the text. In particular, li>r ;L crack tip in ;I homogeneous malcrial,

Piezoelectric

Note that the determinant be written

fracture

is a third-order

polynomial

in p2. The characteristic

(P2+a’)(p4+2p52p2+54) The dimensionless numbers code is needed to determine

765

mechanics

= 0.

equation

can (W

X, p and 4 depend on the constitutive constants ; a short computer them. For stable materials, they arc real and restricted so that rr > 0,

The roots with positive imaginary Pj =

/I>

5 > 0.

-1.

(C9)

parts are

ix, PI.2 =

{

ii’(nfm).

(’

>

I

i’(i,z+n*).

/,

<

,.

(CIO)

where

(CI1) The expressions for the matrices are cumbersome in appcarancc, and it is probably best to proceed numerically at this point. Due to the symmetry of the problem and its role in (5.16), the real part of Y has the structure

ReY=

0

0 where the elements

should be computed

1

I

c,.

P

’

I

1

c

I:

numerically.

(C12)