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Plane problems in piezoelectric media with defects
PLANE
hl&anictd
PROBLEMS IN PIEZOELECTRIC MEDIA WITH DEFECTS Engimxriry and bltvhanics Depxtment. Phdudelphu. PA 1910-k. U.S.A.
Drew1 Universit).
Abstract-A two-&mw&nal eltxtrcwlt~sticnn;llysis is perfwmed on a trmwersel~ iwtropic piczoelectric matrriaf c~~n~~inin~ dekts. A general solution is provided in terms of complex potentials. withemphasis lxinp @ml on slresscc,n~~ntrations that arise in the vicinity ofcircular and elliptical h&s. It is shown that for this grnrc of problem both mechanical ;md electrical urishks are responvlhle for the pe;lC stresses.
For
pietotlectric
dlXndc!j,
ceramics have been the ideal niatcrials used in the fabrication 01
elcctromcchanical dcviccs [see Pohankn and Smith twin
dis;tdvantagc, however, is their brittlcncss:
critical cr;rck growth
(I%%) for ;ln updiltcd rcvicw]. Their
piczoccramics have ;I tcndcncy to dcvclop
bocitusc of stress concentrations
induced by both mechanical and
olcctricnl loitds. Yet, dcfccts arc not limited only to cr;tcks : voids. inclusions, ~~~I~iIllin~~ti~~ns and pot-&tics applic:tlions
conrlitions,
may bc prcscnt and contrihutc
ol’piczoclwtric
ttlakri;tls
to failure 35 \vt‘ll. IkCausc the IlCW t%ljOr
involve IilrgCr con~poncnki imtlcr more SCvCfCIWtliIlg
thcrc is ;Inatur;d incrcasc of the likclihootl
of frlilurc.
As itn CXiImplC O~C GIII
It is. thcrcforc, impcr;rlivc that ill1 iln;llySis bc cite the so-called “adaptive struclurcs”. dsvclopcd which is capahtc of&scribing p~l~n~~ttl~nil st~ctlitsmechanisms that lriggcr crnck
YS wll pmp~igation in piCLoclectric media, 1
as stress hchavior in the vicinity of holes or
inclusions. In ;I rcccnt articlc Sosa and l’ak ( 199O)t study Lhc inlluoncc that clcctric liclds huvc on the distribution
of strcsscs in the ncighbourhood
isotropic pic%ocfcc:cfric mitt&d.
of ;I crack cmbcddcd in it transvcrscly
The aualysis is carried out for the particular CilsC
with its Icading edge assumed to bc straight rind p:tritlkl
ofu crack
to the poling direction (or axis of
transverse isotropy). as is shown in Fig. I ;I. The study rcvcals thar near the crack the stresses contained in the s-_V phnc
arc idepcndcnt of the clcctric lield. This
for the slux~r stresses in thr :-direction.
is not true, however.
It is concluded in the study that clcctromcchunical
interaction is strongly in!lurnccd by the crack’s orientation. The prcscnt work has been motivated by the i~ft)r~~n~~~tion~ditrticlc and represents ;m intcrmcdiatc step towards dcvcloping ii description media. Towards
this end WCconsider the same mittAd
of crack propagarion in piczoclcctric ils that rcfcrcnccd by Sosi\ iind Pak.
Our point of depnrturc. howcvcr. will hc two-fold : (I) the defect is no longer iI crack, but a cylindrical cavily of elliptical shupr; (2) fhc gcncrator of the cylinder (which in the particular case of rhc crack becomes the crack front) is along an axis other than the axis of transverse isotropy,
as represented in Fig.
I b. This
new defect orientation
posts math-
cmsticill diflicultics not prcscnt in previous iln;llyscs which can be circumvented by resorting to ;1 two-dimensional
model. In this manner we ilrc Icad to ;I more complctc and intcrcsting
coupling phcnomcnon bctwecn the mcchrtnical and the clcctrical variables. A plant strain formulation of the piczoclcctric problem solved within
the formalism
of the complex variables technique is provided. While some work has been done in the area of fracture mechanics of piczoclectric materials. in particular from an cxpcrimental standpoint.
t The
art&
it appears that only the work of Dccg (1980) has thcorctically addrcsscd the
also provides a w&w
of the thcorcticnl and eapcrimwtai
rcscxch pcrfwncd
in this ZKFZI.
Fig. ital. Piezoeiectric mcttsnctl with
:I crack whose Ieadinp edge is paraiiri
to the poling directron.
ib)
prohlcm of defects other th;m cracks.t
Although
the theory d~clopcd
;rpplicahlc to the study of the crack problem, our ;lttcsntion will rightly the study of stress cc)nccntr;Ltions around clliptic;ll qu;rntiljling
in this urticle is bc focusscd on
itnd circular holes, which will include
the clfcct thnt the clcctrictil vnri:rbles have on thcsc strcsscs. The crack probicm
will bc studicd indcpcndcntty ;md prcscntcd clscwhcro.
2. C;OVEKI\;ING EQUA’I-IONS The theory of piczoclcctricity consists of the simultaneous clcctric fields existing in anisotropic, picxo&xtric
cfIixt
study of dcfotmations
nonconducting clustic media. The description
of
and the
is achieved by means of two n~~~h~Ini~~ltand two elcctricrtl variublcs:
the strain and stress tsnsors and the electric licld ild electric displacement vectors dcnotcd by E,,. a,,, E, and D,, rcspcctivcly. As ;I oonssqucnco. thcrc ;lrc four possible milnncrs of describing clcctromcchanical interaction. In thcorctical analysts it is customary to choose ;t rcprsscntation in which the strain ;md clcctric licld ilrc the indcpcndont variables. In oxpcriment4 iinitlyscs, howtzvcr, constitutivc rclrttions hearing the stress rend the electrical ficfd as independent variables xc prcfcrrcd. In the end the choice is dictittcd by the particulx
ofa form
problem that one has in mind. The present study mrtkcs USC arc the indcpendcnt quantities. Thus,
electric displnccmtxlt
in which stresses and following Bcrlincourt et NI.
( 1964). WC write t tn contrast. the prtA&xt of a amity crnbcdded in an cktstic isotropic extensively. See MoMwking (1989) for rcfcrencr”s.
diclcctric has hccn twatcd more
Plane problems1x3 pwzzoelectncmedia
193
6, = $h,+g,,,D1, Et =
where &, is the compliance
-,tfrkibkl
+
(1)
p:k Dk
tensor of the material measured at zero electric displacement.
g,,, is the piezociectric tcnsor. and Ek is the dielectric im~rme~bility zero stress. Although to be quite convenient In the MKS [E] = mm-‘. [s”] = m2 N-l.
tensor measured at
f 1) is not the most widely used form of constitutive ivhen formulating
two-dimensional
relation. it proves
boundary value problems.
system the aforementioned variables are measured in the following [E] = Vm-’
[a] = iUrn_‘, [g] = Vm N-’
where + represents the ektric
= NC-‘.
= m’C_‘.
[D] = Cm-’
[p”] = Nm’C-L
potential given by E = -grad
in the Introduction. we will focus on transversely isotropic and with reference to the co~~rdin~t~ system shown in Fig.
units:
= NV-‘m-’ [4] = V
= V’N-‘. (b. As previously
mentioned
piezoelectrics. In such a case. I. eqn (I) titkcf the foli~~~in~
(‘b)
From (2) it is clear that no coupling exists between the mechanical and electrical vnriablcs containccl in the .v -,Yplunc. A more complete state ofclcctromcchanical ohscrvcd by reducing (I) into ;1 two-dimensjon~l
interaction c;m be
model. Since. xcording
to (2). the S-J
plant is the isotropic plane. one can employ either the I--Z or the _P-: pklne for tho study of plant clcotromcchanical phcnomcna. Choosing
the former,
the plant strain conditions
require that? c,., = c,, = c,, = E, = 0 which allows us to write
(3)
Substituting
(3) and (1) into (7) yields the plane strain constitutive
notation we introduce the following
which are known as the &l/4
definitions
ttrtrtrritrl
cvttstmts.
equations. To minimize
:
An additional step towards compactness
in notation is achieved by renaming the coordinates such that s + x1 and -_-, .v>. Hence, the two-din~ensionill
constitutive
equations can now be written as
(hh)
I:or ;I ctuiipletc formulation Ihc equations ofcl;Mic
of the piczoclcctric prohlcm wc need lo supplement (6) with
ccluilihrium
ilIld
Gauss’
Law of Iilectrostatics.
which in two climcn-
sions and in the :~hscnrc of hotly forces and (ice electric volume ch;trgc are giscn by
(73 c)
Furthcrmorc.
the strain and electric licld components satisfy the comp;~tibility reliltions
The solution stress function
to the system of equations furnished
Ufr,,
by (6) (8) is sought by means of ;t
s?) which satistics the cktstic c~l~~ilibri~~r~~ equations when dcfincd as
In addition, we introduce an induction function $(s,.s~)
such lhal
(IO) which satisfies (7~). Next, introducing
(9) and (IO) into
where the commas denote difl’erentiation. into (6b) and (Sb) yields
Similarly.
(&I).
and later into @a) leads to
substituting
(9) and (IO) succcssivcly
195 (1’)
Equations ( I I ) and (I 2) can be expressed in compact form by writing : L,c:-
LJ$ = 0
L,U+
L-,!) = 0
(13)
where L, (i = 4.3.2) are differential operators of order four. three. and two. reflecting the elastic. piezoelectric, and dielectric properties of the material. respectively, and given by
(14) Thus.
the plunc piezoelectric problem is governed by a system of two partial ditl’erential
equations coupled in Ii and 4. If we eliminate $. (13) is reduced to a single sixth order partial ditTcrcntial equation for the stress function.
namely
(L.lL:+LJL,)u=o or
writtcil
Equation
W
explicitly
(IO) can be solved by means of complex variablcst : WCcxprcss the solution
by
nicans of ;I function U(z) dclincd as
U(z) = U(.K,+/Is?).
11 =
a+$.
i =
J-
I
(17)
where z is a gcncralizcd complex variable, kt is a complex parameter, and CYand /I are real numhcrs. By introducing (17) into (16). and using the chain rule of ditTercntiation, an cxprcssion of the form i*)C/‘.’ = 0 is obtained. A nontrivial solution follows by setting the characteristic equation (that is, the quantity cncloscd within
Owing to the particular
material symmetry
bmces) equal to zero, namely
of the piczoclcctric under investigation.
the
polynomial is cxprcsscd in tcrms of cvcn powers of 1’. This allows us to solve (I 8) analytically, rcndcring
whcrc [I,. r L and /I2 depend on the material constants. Once the roots {lk. k = I, 2.3 arc known, the solution is written as
t WC extend the i&as cicveloped hy Lckhnitskii
(1981) in the framework
of anisotropic
elasticity.
H. %lSA
196
U(.K,..K,) = 1.2
2 L;(:,) k=
I
where
and .s? denotes the rest part of a given complex expression. The next step is to find the function $ using one of theequations (13). If we consider LJC- = -L&,
assuming solutions
of the form C(.r, +jca.~2) and I&X, +pk.r2), we obtain
(22,
h(jtt)U” = -S(jft,3/;: where h(jt&.) = (&i +h,,)jj;+h:2. intog2tion
j(jjk)
= &,jji+&:.
(‘3)
of (22) yieldst
$J,(=,,= 2, c/,‘(=,) where
i., ( jfk ) = Thus.
the solution
& j'k)
l
for the electric: induction becomes
(36) k-t
k-t Alternatively,
we could have obtained $ by using LJU = L&,
leading to
Hut i.,( jjA) = ;I( jjk). since by (18). u( j&)d( jck)+ h’(jjk) = 0: hence the same function
tbI is
obtained. With
the aid of (30) and (26) we can write expressions
displacement components. derivatives,
it is convsnicnt
Towards
for the stress and electric
this end and in order to reduce the order of the
to introduce
new functions
cp, of the complex variable :
(hereafter called the compicx potentials) which arc defined as d t’k
whcrc k = 1,1.3, and no summation
(27)
is implied over rcpcated indices. The use of (9). (20)
and ($7) leads to the stress components
Plane problems in piaoektric
/lIcp;(zt).
0:;
=
497
media
‘R
(33)
k=. I
Likewise (IO). (26) and (27) yield
Finally.
using the constitutive
equations in conjunction
to find expressions for the elastic displacement. The results are summarized below. The components
with (28) and (19) allows us
the electric field and the electric potential.
of strain result in
(30) llsing the strain tlisplaccrncnt rclntionship
I:,,
the intcgralion
=
L(4,
(31)
+ q.,)
of the norrn;Il strains rcndcrs 1
1
”
I
=
c
2.9
/‘k
‘pk
(=k) + (lJ.Y: +
tf,,,
If> =
k-l
whcrc the constants
2.9
c (/k’pk(:k)-(I,.\-,
+l’,,
(32)
k-l
OJ, u,,
and I’” rcprcscnt rigid body displaccmcnts and
(33)
Similarly, electric lirld :
using (28) and (39) in conjunction
Finally. intcgnrtion
of K = -grad
with (6b) gives the components
of the
(b Icads to the clcctric potential :
(35)
whcrc $,, is ;L rcferencc potential. Kccapitulating. the plant strain piezoelcctric finding three complex potential
is a function
potentials.
problem
has bean rcduccd to one of
cp,. rp2 and (p,, in some region ST of the medium.
of a difrcrcnt generalized
complex
variable :k = .yl +/ig2.
Each Alter-
natively. the complex potentials can bc viewed as functions of the ordinary complex variable = x’,k’ + i~\~’ where
=k
Using this point of view the functions cp, , (17:and cp?must be determined in regions Q,. R1 and R,. respectively. obtained from R by the atline transformations (36). We should note. however. that the problem as formulated
is still undetermined.
The
complex potentials need to be determined subject to certain boundary and jump conditions on the boundary
(or surfaces of discontinuit!)
are of mechanical
r’R. The piezoelectric boundary conditions
nature (prescribed elastic displacement
1 or surface traction T) and of
electrical nature (prescribed electric field or electric displacement). ?QO and iQ,
Thus, calling X2,. (1S&.
the parts of the boundary A2 where t, u. D and Q, are prescribed. we can write
in the most general case (see Erineen and bfaqin.
an = f
on X2,
u=ii
on XE,
n - iD!j = I,;.
on c’R,,
l ff, i = 0
whore rt;. is ;a prtscrihcd surfrm
charge
1959)
tfcrlsity
(37a 4)
on K&
:rnct
n is ttw
o~it~~~lr~l
unit normal to 22. We
note that (37d) is ;I conscqucncc of
if we impose boundary and jump conditions in terms oft and iI only (as will bc done in lhc prcsenl study). we can write
?lJ
f’s,
=--
J
f> ds,
0
Ltu l’.\-,
= -
Jf/
ds,
II, = -
11
where /, and I? are the rectangular C;lrtesi:m components oft,
J
n,,ds
(3Y)
I,
I),, is the normal component
of D, and ds is an clement of arc length on iQ. Or in terms of the complex potentials we can rewrite (38) ati
(39)
3. INFISlTE
PIEZOELECTRIC
XlEDlUXi
WIT?1 AN ELLIPTICAL
CAVITY
Consider an infinite space filled with transvcrscly isotropic piezoelectric material and containing i1 hole of elliptical shape. The axes of the cavity of length 2n and 26 are assumed
Plane problems in pinoektric
w?
media
In Fig. 2. Elliptical
hole in an infinite
piezocktric
to be ~lilced along the ;IWS of elastic symmetry Furthermore.
m&urn.
of the material
11s shown
in Fig.
7.
cikll f the boundary of the hole with outward unit normal n. Mechitnic;tl or
electrical loads applied at remote distitnces from the hole induce deformations
and electric
ticlds in the region of space R tilled with mutter described by (6). The induced or applied fields also exist in the region inside the cavity $2”’ (filled with vacuum) dcscribcd by the c(~nstitittive relation I)“,’
= t:,, I;,““
where I:,, is the diclcctric constant (or ~rr~~ittivity~ In W’
(40) of vxuum
v?fp’*’
Liz
(I:,, = X.X.5 x IO ” N V
‘).
Lilpl:lCCcqu;ttion :
the governing equation is simply the two-dimensional
(41)
1)
antI the normal component of the electric displacement can be e:xpressctl as
Dw.“=
_-I
l?f/J“” ‘(1 .
I!n
The depicted situ~ltion constitutes a two-domain boundary value problem. Hence. once the electric displacement and electric potential
are found in both Q and R”‘,
the electric
boundary conditions become
(43
where the quantities on the Icft-hand side
of
(43) are evirluitted within
the piezoclcctric. A
signiticant simplification to the original two-domain problem is achicvcd by noglocting the surroundings (the v;tcuuIII in thiscxc)ofQ. This process is permissible bcxtusc thedielectric constants
in the picxocloctric mitteriid
are significantly
larger than E,. Consequently,
assuming that 11; = 0. the boundary conditions at the SurfLIce of a traction-free cavity can be cxprcsscd as (WC also Pak, 1990)
t=o
on
r.
(43)
D*n=O The problem is now merely reduced to one of finding the complex potentials in the region R. Towards
this end we assume a general solution of the form
H. Sts~
500
where .-lk. .-I:. B,, B: are real constants and
is a hoiomorphi~
function up to infinity
with real coetkients
enforced at the rim of the hole require that Ai+i.-l: Furthermore,
(I$‘. The boundary conditions
= 0 for cpkto be single valued.
the constant BI and BP arc determined from the far field loading conditions,
as is described at the end of this section. To find the hol~~n~orphi~ functions we make use equal to zero. That
of 139). with
their right-hand sides set
is (46)
or substituting
Equation
the general espression for cpr:
(47) can bc solved for (p: by mc;lns of ;i conformal
1ri~nslbrI~~~llitIn which
maps the cxtcrior of three cllipscs (CWCfor each root jtr ) containctl in the zk plant into the exterior ofthc unitcirclt is (SIX Lckhnitskii.
(~)fbl)l~il~~~~ry;‘) Iocatcd in the cl-plant. The relcv:tnt t~lnsf(~rn~~Iti~)r~
1% I )
Note that both :k and cl travel on r ;tnrl ;‘. rcspcctivcly, in 3 counterclockwise scnsc. furthermore, the three points on the contours of&. map into ;I sin& point on the contour of the unit circle. which is drscribcd by iA = d = e“‘. 0 < Ii 6 2~. Assigning a notation ot (I$‘( in) to the functions cp:‘(zl) after (48) is opplicd. the boundary conditions (47) bamnc
(49)
where
Plane problems in
plno&ctric
me&n
501
with r,. & and r, being their complex conjugates. To solve (49) for the functions multiply
both sides by do o-<
0:
we
and integrate over ;‘. where { is any point outside the unit
circle. By observing that
do = - ?;cD;(;;
).
we obtain
Salving for the function (P:‘. and rcg;lrding 5 as ck when X-takes the values I. 2 or 3. we can e\press the solution as
k = 1.2.3 bk
(53)
ivhcrc A, ,, A, :. CIC.. ;lrc the clcmcnts of the matrix
f~‘inally. to obtain (pF(zl). (53) is invcrtcd by substituting
each CAby
(56) yiclcling the three coniplcx potentials
To dctcrminc u and D, the derivative of (57) with respect to zk is evaluated. which results in
To loading can bc making
solve for the constants Sk. Bf. one must invoke the rcmolc clectromcchanical conditions. In the most gcncrnl cast. three mechanical and two clcctrical variables cnforccd. If thcsc loads arc the stress and electric displaccmcnt components. by use of (28). (79) and (5X) when ):I 4 ZC. a system of five equations in the six
unknowns Bk. BZ is obtained. Without loss of generality one can arbitrarily set one of these constants equal to zero. Thus. in the rcmaindcr of this article it is assumed that L?: = 0. Otherwise, a sixth equation can be formulated in terms of the remote rigid rotation. That is. one can impose the condition ZCU= II:,, -u,,~ = 0. as 1~1+ 33. Once explicit solutions
501
H. SOSA
for & and l34f are obtained in terms of the remote load and mltterial properties. ir is a simple matter to show rhat (50) renders
That
is. I,. I, and Ii depend on the applied load and the geometry of the cavity. In the
following
section. the stress distribution
around rlliptical
and circular holes will be found
in terms of the complex potentiats given by (57).
In this section it is nssurned that the piczoelectric medium is a PZT-4 material constants that can be found in Bcrlincourt
ceramic with
(‘I ul. (IYGJ). The corresponding reduced
material constants obtained from (5) arc
111, =
1’1.3 x IO ‘? (m’
b,, = 7.00 x IO’.
N
‘)
822 = 9.82 x IO’ (V’ N
‘).
To ;tpprociatc the ordsr ofm;1gnitud~1 of the variahlcs involvccl in this typr of problem wc note that ths poling process in csromics (th;~t is, the process through which the piczoctcctric elti~t
is induced) t;tkcs ptacc itt ctectris fictd lcvcls of IO” V m
‘. Furthtrmorc.
typical ~lppli~~lti~~ns invotvc clcctrical dispt~lc~rn~nts of the order of IO ’ to 10 ’ C m ‘. whils the strasscs can vary bctwecn IO”
and IO’ N m ‘. As :t ctosurc for the theory dcvctopcd in the previous sections two cxamplcs arc
presented which can bc rcgitrded ilbout Ihc theory ol’ctc~tro~tasticity
as
of'
with
ft~~i~l~i~~~nt~~t importance: in drawing contlusions J~cc~s.
Quite often it has been claimed that stress anutyscs in piczoclcctrics can bc implcmcntcd ncgtccting fhc ctl&t
of the ctectrical variubtcs. As cvidcncc. note that it is not unusual to
tind fracture mechanics anatyscs in piczoetcctricity based on convcntionat rtpproathcs drawn from the theory of elasticity. While this upproach ccrrainly simplifies the study, it may also product misleading results. Moreover. discrcpancics tend to become more pronounced for stress tcvcts nc;lr crack tips or cavities. The purpose of this cxamptc is to exhibit the difTcrcnccs that may arise when using it purely Astic mod4 versus iIn rlcctroclastic model. Consider the etliptical cavity shown in Fib.p 2 with boundary conditions given by (33). For simplification
WC consider fur field mechanical toilding in the .r,-direction
: CT’,;’= p.
WC took for the stresses along the _r?-axis. and. more spccificalty, the maximum vrttues at *X2= h. If in (6) we ncgtcct the terms containing the ctcctrical variables. the probtcm is reduced to one of purely anisotropic elasticity governed by L,U = 0. Usins ;L procedure similar to the one described in Section 2, we took for two complex potentials and subsequently compute the stress component (T, !. The results arc displayed in the second column of Table I for four diffcrcnt ratios of rr,‘h.
Pkme problems in ptrzorlsctrtc Table
I. Valwsofla,,),~,
~(at
uh
Elastic
Efrctroelrtsttc
3 I I 3
I.62 2.870 6.610 IV.7
I .7Ji 3.230 7.700 23.26
I I9
503
media v: = h) “b D&krence 7.5 12.5 16.5 18.0
If under the same lortdinp conditions the electricaf terms a-e retained. the problem falls in the domain of the theory described in Sections 1 and 3. Solving for the three potentials and the corresponding stresscs for this electroelastic case leads to the results shown in the third column of Table
I. The last column in the table provides the percentage differences
bet\\-eenthe purely elastic and ektroefastic
GISCS at the point of maximum stress. Note
that these differences ;tre by no means negligible. which clearly indicates that a stress analysis should take account of both electrical and mechanical effects.
We WC this simple configuration
to illustrate
stress and electric field variations
at the
rim of the hole when remote mcchanicnl or efcctrical loaf is applied in the .r:-direction. thiscaso it isconvcnient to introduce polarcoordinatcs.
Thus.
r 3 CL 0 g 0 < 2~. the applied far field load, the boundary potctttiitls
WI1
hc cspwwd
I:ig. 3. n,, vxriation
Fig. 4. II,> variation
letting zA = r(cos II+/{, conditions
In
sin 0).
and the complex
as
on the rim (>(‘:I circular hole subjcctcd to rcmotc mcchnnical lo:tding.
on the rim of a circulnr hale suhjrstcd to remote mechanicnl loading.
Fig. 5. &, and E,, variations on the rim of II circular hole
the maximum
value of the hoop stress is almost
subjected
10
remote mechumcal louding.
16% Icss than the maximum
stress whsn
;I load in the s,-direction is applied (see Table I); (2) tho maximum values of c,, occur at 0 = 0 , ISO’. while D,, rcachcs its maximums at (1 = 65 . I I4 ‘. The components of the induced clcctric field arc shown in Fig. 5. Obscrvc that according to the prcscnt normalization. applied strcsscs of order IO’ N m ’ induct clcctric displnccmcnts of order IO ’ C m -- ’ and clcctric fields of order IO’ V rn. ‘. When an electrical load in the form of D,, is applied it will produce stresses and :ln electric &Id.
Figures 6 and 7 show the normalized
Fig. 8 represents the distribution
of the components
values of rr,, and D,,. rcspectivcly. while of the clcctric field. It is clear that an
applied clcctric displaccmcnt of order IO _ ’ C m -’ produces strcsscs of order IO” N m _’ we note that while D,r and E,, are and electric fields of order IO5 V m- ‘. Furthcrmorc. maximum
at 0 = 0 ‘, I80 , u. achieves its maximum
at (I = 90 .
Ftg. 1. &, and E,, varlanvns on the rim
ofa circular
hole subjected to remote
electricalloading.
Obviously. other loading conditions can be analyscd. At this point, however, it is of more fundamental importance to pursue qualitative results. rather than present a collection of diverse loading and geometric configurations.
5. A
pli\nc
strain
CONCLUSIONS
piczoclcctric problem hiIs been formulated
and solved by means of
complex variahlcs theory. The analysis shows that stresses. displacomcnts, electric field components. etc.. can lx cxprcsscd in terms of three complex potentink A ~tcriv~~ti~~r1 of thcsc p~3tc~~ti~~ls has been illl~str;~tc~t by mans of 3 problem in which an elliptical cavity is cmbtddcd in an infinite possihlc loading and boundary conditions hccn shown that slrcss
illl~liySl3
clcctrical clkcts arc not
tilkW
pic/oclcctric medium. Within
in the vicinity of ;I lloli: into account.
this context,
have also been discussed. Furthcrmorc,
it has
can product incorrect results if
hl&anictd
PROBLEMS IN PIEZOELECTRIC MEDIA WITH DEFECTS Engimxriry and bltvhanics Depxtment. Phdudelphu. PA 1910-k. U.S.A.
Drew1 Universit).
Abstract-A two-&mw&nal eltxtrcwlt~sticnn;llysis is perfwmed on a trmwersel~ iwtropic piczoelectric matrriaf c~~n~~inin~ dekts. A general solution is provided in terms of complex potentials. withemphasis lxinp @ml on slresscc,n~~ntrations that arise in the vicinity ofcircular and elliptical h&s. It is shown that for this grnrc of problem both mechanical ;md electrical urishks are responvlhle for the pe;lC stresses.
For
pietotlectric
dlXndc!j,
ceramics have been the ideal niatcrials used in the fabrication 01
elcctromcchanical dcviccs [see Pohankn and Smith twin
dis;tdvantagc, however, is their brittlcncss:
critical cr;rck growth
(I%%) for ;ln updiltcd rcvicw]. Their
piczoccramics have ;I tcndcncy to dcvclop
bocitusc of stress concentrations
induced by both mechanical and
olcctricnl loitds. Yet, dcfccts arc not limited only to cr;tcks : voids. inclusions, ~~~I~iIllin~~ti~~ns and pot-&tics applic:tlions
conrlitions,
may bc prcscnt and contrihutc
ol’piczoclwtric
ttlakri;tls
to failure 35 \vt‘ll. IkCausc the IlCW t%ljOr
involve IilrgCr con~poncnki imtlcr more SCvCfCIWtliIlg
thcrc is ;Inatur;d incrcasc of the likclihootl
of frlilurc.
As itn CXiImplC O~C GIII
It is. thcrcforc, impcr;rlivc that ill1 iln;llySis bc cite the so-called “adaptive struclurcs”. dsvclopcd which is capahtc of&scribing p~l~n~~ttl~nil st~ctlitsmechanisms that lriggcr crnck
YS wll pmp~igation in piCLoclectric media, 1
as stress hchavior in the vicinity of holes or
inclusions. In ;I rcccnt articlc Sosa and l’ak ( 199O)t study Lhc inlluoncc that clcctric liclds huvc on the distribution
of strcsscs in the ncighbourhood
isotropic pic%ocfcc:cfric mitt&d.
of ;I crack cmbcddcd in it transvcrscly
The aualysis is carried out for the particular CilsC
with its Icading edge assumed to bc straight rind p:tritlkl
ofu crack
to the poling direction (or axis of
transverse isotropy). as is shown in Fig. I ;I. The study rcvcals thar near the crack the stresses contained in the s-_V phnc
arc idepcndcnt of the clcctric lield. This
for the slux~r stresses in thr :-direction.
is not true, however.
It is concluded in the study that clcctromcchunical
interaction is strongly in!lurnccd by the crack’s orientation. The prcscnt work has been motivated by the i~ft)r~~n~~~tion~ditrticlc and represents ;m intcrmcdiatc step towards dcvcloping ii description media. Towards
this end WCconsider the same mittAd
of crack propagarion in piczoclcctric ils that rcfcrcnccd by Sosi\ iind Pak.
Our point of depnrturc. howcvcr. will hc two-fold : (I) the defect is no longer iI crack, but a cylindrical cavily of elliptical shupr; (2) fhc gcncrator of the cylinder (which in the particular case of rhc crack becomes the crack front) is along an axis other than the axis of transverse isotropy,
as represented in Fig.
I b. This
new defect orientation
posts math-
cmsticill diflicultics not prcscnt in previous iln;llyscs which can be circumvented by resorting to ;1 two-dimensional
model. In this manner we ilrc Icad to ;I more complctc and intcrcsting
coupling phcnomcnon bctwecn the mcchrtnical and the clcctrical variables. A plant strain formulation of the piczoclcctric problem solved within
the formalism
of the complex variables technique is provided. While some work has been done in the area of fracture mechanics of piczoclectric materials. in particular from an cxpcrimental standpoint.
t The
art&
it appears that only the work of Dccg (1980) has thcorctically addrcsscd the
also provides a w&w
of the thcorcticnl and eapcrimwtai
rcscxch pcrfwncd
in this ZKFZI.
Fig. ital. Piezoeiectric mcttsnctl with
:I crack whose Ieadinp edge is paraiiri
to the poling directron.
ib)
prohlcm of defects other th;m cracks.t
Although
the theory d~clopcd
;rpplicahlc to the study of the crack problem, our ;lttcsntion will rightly the study of stress cc)nccntr;Ltions around clliptic;ll qu;rntiljling
in this urticle is bc focusscd on
itnd circular holes, which will include
the clfcct thnt the clcctrictil vnri:rbles have on thcsc strcsscs. The crack probicm
will bc studicd indcpcndcntty ;md prcscntcd clscwhcro.
2. C;OVEKI\;ING EQUA’I-IONS The theory of piczoclcctricity consists of the simultaneous clcctric fields existing in anisotropic, picxo&xtric
cfIixt
study of dcfotmations
nonconducting clustic media. The description
of
and the
is achieved by means of two n~~~h~Ini~~ltand two elcctricrtl variublcs:
the strain and stress tsnsors and the electric licld ild electric displacement vectors dcnotcd by E,,. a,,, E, and D,, rcspcctivcly. As ;I oonssqucnco. thcrc ;lrc four possible milnncrs of describing clcctromcchanical interaction. In thcorctical analysts it is customary to choose ;t rcprsscntation in which the strain ;md clcctric licld ilrc the indcpcndont variables. In oxpcriment4 iinitlyscs, howtzvcr, constitutivc rclrttions hearing the stress rend the electrical ficfd as independent variables xc prcfcrrcd. In the end the choice is dictittcd by the particulx
ofa form
problem that one has in mind. The present study mrtkcs USC arc the indcpendcnt quantities. Thus,
electric displnccmtxlt
in which stresses and following Bcrlincourt et NI.
( 1964). WC write t tn contrast. the prtA&xt of a amity crnbcdded in an cktstic isotropic extensively. See MoMwking (1989) for rcfcrencr”s.
diclcctric has hccn twatcd more
Plane problems1x3 pwzzoelectncmedia
193
6, = $h,+g,,,D1, Et =
where &, is the compliance
-,tfrkibkl
+
(1)
p:k Dk
tensor of the material measured at zero electric displacement.
g,,, is the piezociectric tcnsor. and Ek is the dielectric im~rme~bility zero stress. Although to be quite convenient In the MKS [E] = mm-‘. [s”] = m2 N-l.
tensor measured at
f 1) is not the most widely used form of constitutive ivhen formulating
two-dimensional
relation. it proves
boundary value problems.
system the aforementioned variables are measured in the following [E] = Vm-’
[a] = iUrn_‘, [g] = Vm N-’
where + represents the ektric
= NC-‘.
= m’C_‘.
[D] = Cm-’
[p”] = Nm’C-L
potential given by E = -grad
in the Introduction. we will focus on transversely isotropic and with reference to the co~~rdin~t~ system shown in Fig.
units:
= NV-‘m-’ [4] = V
= V’N-‘. (b. As previously
mentioned
piezoelectrics. In such a case. I. eqn (I) titkcf the foli~~~in~
(‘b)
From (2) it is clear that no coupling exists between the mechanical and electrical vnriablcs containccl in the .v -,Yplunc. A more complete state ofclcctromcchanical ohscrvcd by reducing (I) into ;1 two-dimensjon~l
interaction c;m be
model. Since. xcording
to (2). the S-J
plant is the isotropic plane. one can employ either the I--Z or the _P-: pklne for tho study of plant clcotromcchanical phcnomcna. Choosing
the former,
the plant strain conditions
require that? c,., = c,, = c,, = E, = 0 which allows us to write
(3)
Substituting
(3) and (1) into (7) yields the plane strain constitutive
notation we introduce the following
which are known as the &l/4
definitions
ttrtrtrritrl
cvttstmts.
equations. To minimize
:
An additional step towards compactness
in notation is achieved by renaming the coordinates such that s + x1 and -_-, .v>. Hence, the two-din~ensionill
constitutive
equations can now be written as
(hh)
I:or ;I ctuiipletc formulation Ihc equations ofcl;Mic
of the piczoclcctric prohlcm wc need lo supplement (6) with
ccluilihrium
ilIld
Gauss’
Law of Iilectrostatics.
which in two climcn-
sions and in the :~hscnrc of hotly forces and (ice electric volume ch;trgc are giscn by
(73 c)
Furthcrmorc.
the strain and electric licld components satisfy the comp;~tibility reliltions
The solution stress function
to the system of equations furnished
Ufr,,
by (6) (8) is sought by means of ;t
s?) which satistics the cktstic c~l~~ilibri~~r~~ equations when dcfincd as
In addition, we introduce an induction function $(s,.s~)
such lhal
(IO) which satisfies (7~). Next, introducing
(9) and (IO) into
where the commas denote difl’erentiation. into (6b) and (Sb) yields
Similarly.
(&I).
and later into @a) leads to
substituting
(9) and (IO) succcssivcly
195 (1’)
Equations ( I I ) and (I 2) can be expressed in compact form by writing : L,c:-
LJ$ = 0
L,U+
L-,!) = 0
(13)
where L, (i = 4.3.2) are differential operators of order four. three. and two. reflecting the elastic. piezoelectric, and dielectric properties of the material. respectively, and given by
(14) Thus.
the plunc piezoelectric problem is governed by a system of two partial ditl’erential
equations coupled in Ii and 4. If we eliminate $. (13) is reduced to a single sixth order partial ditTcrcntial equation for the stress function.
namely
(L.lL:+LJL,)u=o or
writtcil
Equation
W
explicitly
(IO) can be solved by means of complex variablcst : WCcxprcss the solution
by
nicans of ;I function U(z) dclincd as
U(z) = U(.K,+/Is?).
11 =
a+$.
i =
J-
I
(17)
where z is a gcncralizcd complex variable, kt is a complex parameter, and CYand /I are real numhcrs. By introducing (17) into (16). and using the chain rule of ditTercntiation, an cxprcssion of the form i*)C/‘.’ = 0 is obtained. A nontrivial solution follows by setting the characteristic equation (that is, the quantity cncloscd within
Owing to the particular
material symmetry
bmces) equal to zero, namely
of the piczoclcctric under investigation.
the
polynomial is cxprcsscd in tcrms of cvcn powers of 1’. This allows us to solve (I 8) analytically, rcndcring
whcrc [I,. r L and /I2 depend on the material constants. Once the roots {lk. k = I, 2.3 arc known, the solution is written as
t WC extend the i&as cicveloped hy Lckhnitskii
(1981) in the framework
of anisotropic
elasticity.
H. %lSA
196
U(.K,..K,) = 1.2
2 L;(:,) k=
I
where
and .s? denotes the rest part of a given complex expression. The next step is to find the function $ using one of theequations (13). If we consider LJC- = -L&,
assuming solutions
of the form C(.r, +jca.~2) and I&X, +pk.r2), we obtain
(22,
h(jtt)U” = -S(jft,3/;: where h(jt&.) = (&i +h,,)jj;+h:2. intog2tion
j(jjk)
= &,jji+&:.
(‘3)
of (22) yieldst
$J,(=,,= 2, c/,‘(=,) where
i., ( jfk ) = Thus.
the solution
& j'k)
l
for the electric: induction becomes
(36) k-t
k-t Alternatively,
we could have obtained $ by using LJU = L&,
leading to
Hut i.,( jjA) = ;I( jjk). since by (18). u( j&)d( jck)+ h’(jjk) = 0: hence the same function
tbI is
obtained. With
the aid of (30) and (26) we can write expressions
displacement components. derivatives,
it is convsnicnt
Towards
for the stress and electric
this end and in order to reduce the order of the
to introduce
new functions
cp, of the complex variable :
(hereafter called the compicx potentials) which arc defined as d t’k
whcrc k = 1,1.3, and no summation
(27)
is implied over rcpcated indices. The use of (9). (20)
and ($7) leads to the stress components
Plane problems in piaoektric
/lIcp;(zt).
0:;
=
497
media
‘R
(33)
k=. I
Likewise (IO). (26) and (27) yield
Finally.
using the constitutive
equations in conjunction
to find expressions for the elastic displacement. The results are summarized below. The components
with (28) and (19) allows us
the electric field and the electric potential.
of strain result in
(30) llsing the strain tlisplaccrncnt rclntionship
I:,,
the intcgralion
=
L(4,
(31)
+ q.,)
of the norrn;Il strains rcndcrs 1
1
”
I
=
c
2.9
/‘k
‘pk
(=k) + (lJ.Y: +
tf,,,
If> =
k-l
whcrc the constants
2.9
c (/k’pk(:k)-(I,.\-,
+l’,,
(32)
k-l
OJ, u,,
and I’” rcprcscnt rigid body displaccmcnts and
(33)
Similarly, electric lirld :
using (28) and (39) in conjunction
Finally. intcgnrtion
of K = -grad
with (6b) gives the components
of the
(b Icads to the clcctric potential :
(35)
whcrc $,, is ;L rcferencc potential. Kccapitulating. the plant strain piezoelcctric finding three complex potential
is a function
potentials.
problem
has bean rcduccd to one of
cp,. rp2 and (p,, in some region ST of the medium.
of a difrcrcnt generalized
complex
variable :k = .yl +/ig2.
Each Alter-
natively. the complex potentials can bc viewed as functions of the ordinary complex variable = x’,k’ + i~\~’ where
=k
Using this point of view the functions cp, , (17:and cp?must be determined in regions Q,. R1 and R,. respectively. obtained from R by the atline transformations (36). We should note. however. that the problem as formulated
is still undetermined.
The
complex potentials need to be determined subject to certain boundary and jump conditions on the boundary
(or surfaces of discontinuit!)
are of mechanical
r’R. The piezoelectric boundary conditions
nature (prescribed elastic displacement
1 or surface traction T) and of
electrical nature (prescribed electric field or electric displacement). ?QO and iQ,
Thus, calling X2,. (1S&.
the parts of the boundary A2 where t, u. D and Q, are prescribed. we can write
in the most general case (see Erineen and bfaqin.
an = f
on X2,
u=ii
on XE,
n - iD!j = I,;.
on c’R,,
l ff, i = 0
whore rt;. is ;a prtscrihcd surfrm
charge
1959)
tfcrlsity
(37a 4)
on K&
:rnct
n is ttw
o~it~~~lr~l
unit normal to 22. We
note that (37d) is ;I conscqucncc of
if we impose boundary and jump conditions in terms oft and iI only (as will bc done in lhc prcsenl study). we can write
?lJ
f’s,
=--
J
f> ds,
0
Ltu l’.\-,
= -
Jf/
ds,
II, = -
11
where /, and I? are the rectangular C;lrtesi:m components oft,
J
n,,ds
(3Y)
I,
I),, is the normal component
of D, and ds is an clement of arc length on iQ. Or in terms of the complex potentials we can rewrite (38) ati
(39)
3. INFISlTE
PIEZOELECTRIC
XlEDlUXi
WIT?1 AN ELLIPTICAL
CAVITY
Consider an infinite space filled with transvcrscly isotropic piezoelectric material and containing i1 hole of elliptical shape. The axes of the cavity of length 2n and 26 are assumed
Plane problems in pinoektric
w?
media
In Fig. 2. Elliptical
hole in an infinite
piezocktric
to be ~lilced along the ;IWS of elastic symmetry Furthermore.
m&urn.
of the material
11s shown
in Fig.
7.
cikll f the boundary of the hole with outward unit normal n. Mechitnic;tl or
electrical loads applied at remote distitnces from the hole induce deformations
and electric
ticlds in the region of space R tilled with mutter described by (6). The induced or applied fields also exist in the region inside the cavity $2”’ (filled with vacuum) dcscribcd by the c(~nstitittive relation I)“,’
= t:,, I;,““
where I:,, is the diclcctric constant (or ~rr~~ittivity~ In W’
(40) of vxuum
v?fp’*’
Liz
(I:,, = X.X.5 x IO ” N V
‘).
Lilpl:lCCcqu;ttion :
the governing equation is simply the two-dimensional
(41)
1)
antI the normal component of the electric displacement can be e:xpressctl as
Dw.“=
_-I
l?f/J“” ‘(1 .
I!n
The depicted situ~ltion constitutes a two-domain boundary value problem. Hence. once the electric displacement and electric potential
are found in both Q and R”‘,
the electric
boundary conditions become
(43
where the quantities on the Icft-hand side
of
(43) are evirluitted within
the piezoclcctric. A
signiticant simplification to the original two-domain problem is achicvcd by noglocting the surroundings (the v;tcuuIII in thiscxc)ofQ. This process is permissible bcxtusc thedielectric constants
in the picxocloctric mitteriid
are significantly
larger than E,. Consequently,
assuming that 11; = 0. the boundary conditions at the SurfLIce of a traction-free cavity can be cxprcsscd as (WC also Pak, 1990)
t=o
on
r.
(43)
D*n=O The problem is now merely reduced to one of finding the complex potentials in the region R. Towards
this end we assume a general solution of the form
H. Sts~
500
where .-lk. .-I:. B,, B: are real constants and
is a hoiomorphi~
function up to infinity
with real coetkients
enforced at the rim of the hole require that Ai+i.-l: Furthermore,
(I$‘. The boundary conditions
= 0 for cpkto be single valued.
the constant BI and BP arc determined from the far field loading conditions,
as is described at the end of this section. To find the hol~~n~orphi~ functions we make use equal to zero. That
of 139). with
their right-hand sides set
is (46)
or substituting
Equation
the general espression for cpr:
(47) can bc solved for (p: by mc;lns of ;i conformal
1ri~nslbrI~~~llitIn which
maps the cxtcrior of three cllipscs (CWCfor each root jtr ) containctl in the zk plant into the exterior ofthc unitcirclt is (SIX Lckhnitskii.
(~)fbl)l~il~~~~ry;‘) Iocatcd in the cl-plant. The relcv:tnt t~lnsf(~rn~~Iti~)r~
1% I )
Note that both :k and cl travel on r ;tnrl ;‘. rcspcctivcly, in 3 counterclockwise scnsc. furthermore, the three points on the contours of&. map into ;I sin& point on the contour of the unit circle. which is drscribcd by iA = d = e“‘. 0 < Ii 6 2~. Assigning a notation ot (I$‘( in) to the functions cp:‘(zl) after (48) is opplicd. the boundary conditions (47) bamnc
(49)
where
Plane problems in
plno&ctric
me&n
501
with r,. & and r, being their complex conjugates. To solve (49) for the functions multiply
both sides by do o-<
0:
we
and integrate over ;‘. where { is any point outside the unit
circle. By observing that
do = - ?;cD;(;;
).
we obtain
Salving for the function (P:‘. and rcg;lrding 5 as ck when X-takes the values I. 2 or 3. we can e\press the solution as
k = 1.2.3 bk
(53)
ivhcrc A, ,, A, :. CIC.. ;lrc the clcmcnts of the matrix
f~‘inally. to obtain (pF(zl). (53) is invcrtcd by substituting
each CAby
(56) yiclcling the three coniplcx potentials
To dctcrminc u and D, the derivative of (57) with respect to zk is evaluated. which results in
To loading can bc making
solve for the constants Sk. Bf. one must invoke the rcmolc clectromcchanical conditions. In the most gcncrnl cast. three mechanical and two clcctrical variables cnforccd. If thcsc loads arc the stress and electric displaccmcnt components. by use of (28). (79) and (5X) when ):I 4 ZC. a system of five equations in the six
unknowns Bk. BZ is obtained. Without loss of generality one can arbitrarily set one of these constants equal to zero. Thus. in the rcmaindcr of this article it is assumed that L?: = 0. Otherwise, a sixth equation can be formulated in terms of the remote rigid rotation. That is. one can impose the condition ZCU= II:,, -u,,~ = 0. as 1~1+ 33. Once explicit solutions
501
H. SOSA
for & and l34f are obtained in terms of the remote load and mltterial properties. ir is a simple matter to show rhat (50) renders
That
is. I,. I, and Ii depend on the applied load and the geometry of the cavity. In the
following
section. the stress distribution
around rlliptical
and circular holes will be found
in terms of the complex potentiats given by (57).
In this section it is nssurned that the piczoelectric medium is a PZT-4 material constants that can be found in Bcrlincourt
ceramic with
(‘I ul. (IYGJ). The corresponding reduced
material constants obtained from (5) arc
111, =
1’1.3 x IO ‘? (m’
b,, = 7.00 x IO’.
N
‘)
822 = 9.82 x IO’ (V’ N
‘).
To ;tpprociatc the ordsr ofm;1gnitud~1 of the variahlcs involvccl in this typr of problem wc note that ths poling process in csromics (th;~t is, the process through which the piczoctcctric elti~t
is induced) t;tkcs ptacc itt ctectris fictd lcvcls of IO” V m
‘. Furthtrmorc.
typical ~lppli~~lti~~ns invotvc clcctrical dispt~lc~rn~nts of the order of IO ’ to 10 ’ C m ‘. whils the strasscs can vary bctwecn IO”
and IO’ N m ‘. As :t ctosurc for the theory dcvctopcd in the previous sections two cxamplcs arc
presented which can bc rcgitrded ilbout Ihc theory ol’ctc~tro~tasticity
as
of'
with
ft~~i~l~i~~~nt~~t importance: in drawing contlusions J~cc~s.
Quite often it has been claimed that stress anutyscs in piczoclcctrics can bc implcmcntcd ncgtccting fhc ctl&t
of the ctectrical variubtcs. As cvidcncc. note that it is not unusual to
tind fracture mechanics anatyscs in piczoetcctricity based on convcntionat rtpproathcs drawn from the theory of elasticity. While this upproach ccrrainly simplifies the study, it may also product misleading results. Moreover. discrcpancics tend to become more pronounced for stress tcvcts nc;lr crack tips or cavities. The purpose of this cxamptc is to exhibit the difTcrcnccs that may arise when using it purely Astic mod4 versus iIn rlcctroclastic model. Consider the etliptical cavity shown in Fib.p 2 with boundary conditions given by (33). For simplification
WC consider fur field mechanical toilding in the .r,-direction
: CT’,;’= p.
WC took for the stresses along the _r?-axis. and. more spccificalty, the maximum vrttues at *X2= h. If in (6) we ncgtcct the terms containing the ctcctrical variables. the probtcm is reduced to one of purely anisotropic elasticity governed by L,U = 0. Usins ;L procedure similar to the one described in Section 2, we took for two complex potentials and subsequently compute the stress component (T, !. The results arc displayed in the second column of Table I for four diffcrcnt ratios of rr,‘h.
Pkme problems in ptrzorlsctrtc Table
I. Valwsofla,,),~,
~(at
uh
Elastic
Efrctroelrtsttc
3 I I 3
I.62 2.870 6.610 IV.7
I .7Ji 3.230 7.700 23.26
I I9
503
media v: = h) “b D&krence 7.5 12.5 16.5 18.0
If under the same lortdinp conditions the electricaf terms a-e retained. the problem falls in the domain of the theory described in Sections 1 and 3. Solving for the three potentials and the corresponding stresscs for this electroelastic case leads to the results shown in the third column of Table
I. The last column in the table provides the percentage differences
bet\\-eenthe purely elastic and ektroefastic
GISCS at the point of maximum stress. Note
that these differences ;tre by no means negligible. which clearly indicates that a stress analysis should take account of both electrical and mechanical effects.
We WC this simple configuration
to illustrate
stress and electric field variations
at the
rim of the hole when remote mcchanicnl or efcctrical loaf is applied in the .r:-direction. thiscaso it isconvcnient to introduce polarcoordinatcs.
Thus.
r 3 CL 0 g 0 < 2~. the applied far field load, the boundary potctttiitls
WI1
hc cspwwd
I:ig. 3. n,, vxriation
Fig. 4. II,> variation
letting zA = r(cos II+/{, conditions
In
sin 0).
and the complex
as
on the rim (>(‘:I circular hole subjcctcd to rcmotc mcchnnical lo:tding.
on the rim of a circulnr hale suhjrstcd to remote mechanicnl loading.
Fig. 5. &, and E,, variations on the rim of II circular hole
the maximum
value of the hoop stress is almost
subjected
10
remote mechumcal louding.
16% Icss than the maximum
stress whsn
;I load in the s,-direction is applied (see Table I); (2) tho maximum values of c,, occur at 0 = 0 , ISO’. while D,, rcachcs its maximums at (1 = 65 . I I4 ‘. The components of the induced clcctric field arc shown in Fig. 5. Obscrvc that according to the prcscnt normalization. applied strcsscs of order IO’ N m ’ induct clcctric displnccmcnts of order IO ’ C m -- ’ and clcctric fields of order IO’ V rn. ‘. When an electrical load in the form of D,, is applied it will produce stresses and :ln electric &Id.
Figures 6 and 7 show the normalized
Fig. 8 represents the distribution
of the components
values of rr,, and D,,. rcspectivcly. while of the clcctric field. It is clear that an
applied clcctric displaccmcnt of order IO _ ’ C m -’ produces strcsscs of order IO” N m _’ we note that while D,r and E,, are and electric fields of order IO5 V m- ‘. Furthcrmorc. maximum
at 0 = 0 ‘, I80 , u. achieves its maximum
at (I = 90 .
Ftg. 1. &, and E,, varlanvns on the rim
ofa circular
hole subjected to remote
electricalloading.
Obviously. other loading conditions can be analyscd. At this point, however, it is of more fundamental importance to pursue qualitative results. rather than present a collection of diverse loading and geometric configurations.
5. A
pli\nc
strain
CONCLUSIONS
piczoclcctric problem hiIs been formulated
and solved by means of
complex variahlcs theory. The analysis shows that stresses. displacomcnts, electric field components. etc.. can lx cxprcsscd in terms of three complex potentink A ~tcriv~~ti~~r1 of thcsc p~3tc~~ti~~ls has been illl~str;~tc~t by mans of 3 problem in which an elliptical cavity is cmbtddcd in an infinite possihlc loading and boundary conditions hccn shown that slrcss
illl~liySl3
clcctrical clkcts arc not
tilkW
pic/oclcctric medium. Within
in the vicinity of ;I lloli: into account.
this context,
have also been discussed. Furthcrmorc,
it has
can product incorrect results if