Band-gaps in electrostatically controlled dielectric laminates subjected to incremental shear motions

1.7 MB·English

by  Gal Shmuel

BAND-GAPS IN ELECTROSTATICALLY CONTROLLED DIELECTRIC
LAMINATES SUBJECTED TO INCREMENTAL SHEAR MOTIONS
GALSHMUEL†ANDGALDEBOTTON†‡(cid:63)
†DepartmentofMechanicalEngineering
Ben-GurionUniversity
Beer-Sheva,84105
2
1
Israel
0
2

n ‡DepartmentofBiomedicalEngineering
a
J Ben-GurionUniversity

3
Beer-Sheva,84105

]
t Israel
f
o
s
t. ABSTRACT. The thickness vibrations of a finitely deformed infinite periodic laminate made
a
m outoftwolayersofdielectricelastomersisstudied. Thelaminateispre-stretchedbyinducing
- a bias electric field perpendicular the the layers. Incremental time-harmonic fields superim-
d
n posedontheinitialfinitedeformationareconsiderednext. UtilizingtheBloch-Floquettheorem
o
along with the transfer matrix method we determine the dispersion relation which relates the
c
[ incrementalfieldsfrequencyandthephasevelocity.

1 RangesoffrequenciesatwhichwavescannotpropagateareidentifiedwhenevertheBloch-
v
parameteriscomplex. Theseband-gapsdependonthephasesproperties,theirvolumefraction,
1
5 andmostimportantlyontheelectricbiasfield. Ouranalysisrevealshowtheseband-gapscan
7
0 be shifted and their width can be modified by changing the bias electric field. This implies
.
1 thatbycontrollingtheelectrostaticbiasfielddesiredfrequenciescanbefilteredout. Represen-
0
tative examples of laminates with different combinations of commercially available dielectric
2
1 elastomersareexamined.
:
v Keywords: dielectric elastomers; wave propagation; finite deformations; thickness vibra-
i
X
tions;non-linearelectroelasticity,band-gap,composite,Bloch-Floquetanalysis.
r
a
1. INTRODUCTION
Band-gaps corresponding to ranges of frequencies at which waves cannot propagate earned
the interest of the scientific community (e.g., Ziegler, 1977; Wang and Auld, 1985; Kushwaha
et al., 1994; Gei et al., 2004, 2009, to name a few). These band-gaps (BGs) or stop-bands
can be utilized for various applications such as filtering elastic waves, enhancing performance
of ultrasonic transducers, supplying a vibrationless ambient when needed for industrial and
e-mail: [email protected],tel: int. (972)8-6477105,fax: int. (972)8-6477106
∗
1
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 2
biomedical applications, and more. Several techniques are available for investigating wave
propagationincomposites,(e.g.,fortheplane-wavemethodinperiodiccompositesKushwaha
etal.,1993;TanakaandTamura,1998;SigalasandEconomou,1996)andtheeffectivemedium
methods (e.g., Sabina and Willis, 1988, for composites with random microstructre). As in this
work we are interested in the appearance of BGs in infinite periodic laminates, we find it
convenient to use the transfer matrix method (Wang and Auld, 1986), along with the Bloch-
Floquet theorem (Kohn et al., 1972). Specifically, we consider waves superposed on a pre-
deformedstateduetobiaselectricfieldactingonaninfiniteperiodiclaminatewitharepeating
sequence of of dielectric elastomers (DEs) layers. The motivation for the choice of DEs as the
composite constituents stems from their ability to undergo large deformations (Pelrine et al.,
2000),andtochangetheirmechanicalanddielectricpropertieswhensubjectedtoelectrostatic
fields (Kofod, 2008), thus effecting the way in which electroelastic waves propagate in the
matter (Shmuel et al., 2011; Gei et al., 2011). As will be shown in the sequel, by properly
tuningthebiaselectrostaticfielddifferentrangesoffrequenciescanbefilteredout.
The structure of this paper is as follows. Following the works of Dorfmann and Ogden
(2005);deBottonetal.(2007);DorfmannandOgden(2010);BertoldiandGei(2011);Rudykh
and deBotton (2011); Ponte Castan˜eda and Siboni (2011) and Tian et al. (2012), the back-
ground required for describing the static and dynamic responses of elastic dielectric laminates
is revisited in section 2. Particularly, the governing equations for small fields superimposed
on large deformations of electroactive elastomers are summarized. In this section we also
introduce an adequate extension of the transfer matrix method and the Bloch-Floquet theo-
remrequiredfortreatingthepropagationofincrementalelectroelasticwavessuperimposedon
finitely deformed DE laminates. Section 3 deals with the response an infinite periodic lam-
inate with alternating layers, whose behaviors are governed by the incompressible dielectric
neo-Hookean model (DH), to a fixed electric displacement field normal to the layers plane.
Subsequently, the response to small harmonic perturbations superimposed on the finitely de-
formed laminate is analyzed. The end result is given in terms of a dispersion relation relating
the incremental fields frequency and phase velocity to the Bloch-parameter. Several examples
areconsideredinsection4,wheretheinfluenceofthemorphologyofthelaminate,thecontrast
betweenthephasesproperties,andthebiaselectricfieldonthedispersionrelationisexamined.
Finally, realizable tunable isolators are studied by considering combinations of commercially
availableDEssuchasVHB4910,fluorosilicone730andELASTOSILRT-625.
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 3
2. THEORETICAL BACKGROUND
Consider a heterogeneous body occupying a volume region Ω R3 made out of N per-
0
⊂
fectly bonded different homogeneous phases. The external boundary of the body ∂Ω sepa-
0
rates it from the surrounding space R3 Ω , assumed to be vacuum. Each phase occupies a
0
\
volume region Ω(r)(r=1,2,...,N) and its boundary is ∂Ω(r). Let χ :Ω I R3 describe
0 0 0× →
a continuous and twice-differentiable mapping of a material point X from the reference con-
figuration of the body to its current configuration Ω with boundary ∂Ω by x = χ(X,t). The
correspondingvelocityandaccelerationaredenoted,respectively,byv=χ anda=χ ,and
,t ,tt
the deformation gradient is F=∇ χ. Vectors in the neighborhood of X are transformed into
X
vectors in the current configuration via dx=FdX. The volume change of a referential volume
elementdV isgivenbydv=JdV,wheredvisthecorrespondingvolumeelementinthecurrent
configuration, and J det(F)>0 due to material impenetrability. The conservation of mass
≡
implies that ρ =Jρ, where ρ and ρ are the material mass densities in the reference and the
L L
deformedconfigurations,respectively. AnareaelementdAwiththeunitnormalNintherefer-
ence configuration is transformed to a deformed area da with the unit normal n in the current
configurationaccordingtoNanson’sformulaNdA= 1FTnda. TherightandleftCauchy-Green
J
straintensorsareC=FTFandb=FFT.
Theelectricfieldinthecurrentconfiguration,denotedbye,isgivenintermsofagradientof
a scalar field, termedthe electrostaticpotential. Infree spacean inducedelectric displacement
field d is related to the electric field via the vacuum permittivity ε , such that d = ε e. In
0 0
dielectric bodies the connection between the fields is specified by an adequate constitutive
relation.
Intermsofa’total’stresstensorσ theequationsofmotionread
∇ σ =ρa, (1)
·
where in order to satisfy the balance of angular momentum symmetry of σ is required. The
terminology’total’stressisusedtoindicatethatσ accountsforbothmechanicalandelectrical
contributions. Thus,thetractiontonadeformedareaelementwithaunitnormalnisgivenby
σn=t. (2)
Whenidealdielectricsareconsiderednofreebodychargeispresent,andGauss’slawtakesthe
form
∇ d=0. (3)
·
Faraday’slawis
∇ e=0, (4)
×
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 4
whenaquasi-electrostaticapproximationisconsidered. Afulldescriptionofthesystemshould
consider interactions with fields outside the body, henceforth denoted by a star superscript.
Specifically,theseare
d(cid:63) = ε e(cid:63), (5)
0
(cid:20) (cid:21)
1
σ(cid:63) = ε e(cid:63) e(cid:63) (e(cid:63) e(cid:63))I , (6)
0
⊗ −2 ·
where I is the identity tensor. In vacuum the governing Eqs. (3)-(4) for d(cid:63) and e(cid:63) are equiva-
lent to Laplace’s equation for the electrostatic potential. As a consequence, σ(cid:63) known as the
Maxwellstress,isdivergence-free. Acrosstheouterboundary∂Ω,theelectricjumpconditions
are
(d d(cid:63)) n= w , (e e(cid:63)) n=0, (7)
e
− · − − ×
wherew isthesurfacechargedensity. Inordertoformulatethejumpinthestresswepostulate
e
a separation of the traction into a sum of a prescribed mechanical traction t , and an electric
m
traction t induced by the external stress such that t =σ(cid:63)n. Hence, the stress boundary con-
e e
ditionis
(σ σ(cid:63))n=t . (8)
m
−
The jump conditions across a charge-free internal boundary between two adjacent phases m
and f are
[[σ]]n=0, [[d]] n=0, [[e]] n=0, (9)
· ×
where[[ ]] ( )(b) ( )(a) denotesthejumpoffieldsbetweenthetwophases.
• ≡ • − •
A Lagrangian formulation of the governing equations and jump conditions is feasible by a
pull-back oftheEulerianfields
P=JσF T, D=JF 1d, E=FTe, (10)
− −
for the ’total’ first Piola-Kirchhoff stress, Lagrangian electric displacement and electric field,
respectively. Thus,thegoverningEqs.(1),(3)and(4)transform,respectively,to
∇ P=ρ a, ∇ D=0, ∇ E=0. (11)
X L X X
· · ×
Thecorrespondingreferentialjumpconditionsacrosstheexternalboundary∂Ω are
0
(P P(cid:63))N=t , (D D(cid:63)) N= w , (E E(cid:63)) N=0, (12)
M E
− − · − − ×
wheret dA=t daandw dA=w da. AcrossthereferentialinterfacesEqs.(9)become
M m E e
[[P]]N=0, [[D]] N=0, [[E]] N=0. (13)
· ×
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 5
FollowingDorfmannandOgden(2005),theconstitutiverelationisexpressedintermsofan
augmented energy density function (AEDF) Ψ with the independent variables F and D, such
thatthetotalfirstPiola-KirchhoffstressandtheLagrangianelectricfieldarederivedvia
∂Ψ ∂Ψ
P= , E= . (14)
∂F ∂D
The first of Eq. (14) should be modified when considering incompressible materials, as the
kinematic constraint yields an additional workless partof stress. Thelatter is accounted by in-
troducingaLagrangemultiplier p whichcandeterminedonlyfromtheequilibriumequations
0
togetherwiththeboundaryconditions. Thus,thetotalfirstPiola-Kirchhoffis
∂Ψ
P= p F T. (15)
0 −
∂F −
Based on the framework developed in Dorfmann and Ogden (2010), we superimpose an
infinitesimal time-dependent elastic and electric displacements x˙ = χ˙ (X,t) and D˙ (X,t), on
the pre-deformed configuration. Herein, and in the sequel, we use a superposed dot to denote
incremental quantities. Let the Eulerian quantities Σ,dˇ, and eˇ denote the push-forwards of
incrementsinthefirstPiola-Kirchhoffstress,theLagrangianelectricdisplacementandelectric
fields,respectively,namely
1 1
Σ= P˙FT, dˇ = FD˙, eˇ =F TE˙. (16)
−
J J
Intermsofthesevariables,theincrementalgoverningequationsread
∇ Σ=ρx˙ , ∇ dˇ =0, ∇ eˇ =0. (17)
,tt
· · ×
Foranincompressiblematerialtheincrementalconstraintis
∇ x˙ tr(h)=0, (18)
· ≡
where h=∇x˙ is the displacement gradient. Linearization of the incompressible material con-
stitutiveequationsintheincrementsyields
Σ = Ch+p hT p˙ I+Bdˇ, (19)
0 0
−
eˇ = BTh+Adˇ, (20)
where(cid:0)BTh(cid:1) =B h . ThequantitiesA,B andC arethepush-forwardsofthereferential
k ijk ij
reciprocaldielectrictensor,electroelasticcouplingtensor,andelasticitytensor,respectively. In
componentsform
1
A =JF 1A F 1, B =F B F 1, C = F C F , (21)
ij α−i 0αβ β−j ijk jα 0iαβ β−k ijkl J jα 0iαkβ lβ
where
∂2Ψ ∂2Ψ ∂2Ψ
A = , B = , C = . (22)
0αβ ∂D ∂D 0iαβ ∂F ∂D 0iαkβ ∂F ∂F
α β iα β iα kβ
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 6
Theincrementalouterfieldsaregivenby
d˙(cid:63) = ε e˙(cid:63), (23)
0
σ˙(cid:63) = ε (cid:2)e˙(cid:63) e(cid:63)+e(cid:63) e˙(cid:63) (cid:0)e(cid:63) e˙(cid:63)(cid:1)I(cid:3). (24)
0
⊗ ⊗ − ·
Hereagaind˙(cid:63) ande˙(cid:63) aretosatisfyEqs.(3)-(4),wheresubsequently∇ σ˙(cid:63) =0.
·
The push-forward of the increments of Eqs. (12) gives the following incremental jump con-
ditionsacrosstheexternalboundaryinthecurrentconfiguration
(cid:2)Σ σ˙(cid:63)+σ(cid:63)hT (∇ x˙)σ(cid:63)(cid:3)n = ˇt , (25)
m
− − ·
(cid:2)dˇ d˙(cid:63) (∇ x˙)d(cid:63)+hd(cid:63)(cid:3) n = wˇ , (26)
e
− − · · −
(cid:2)eˇ e˙(cid:63) hTe(cid:63)(cid:3) n = 0, (27)
− − ×
where ˇt da = ˙t dA and wˇ da = w˙ dA. Similarly, the push forward of the increments of
m M e E
Eqs.(13)resultsinthefollowingjumpsacrosstheinternalboundaries
[[Σ]]n=0, (cid:2)(cid:2)dˇ(cid:3)(cid:3) n=0 [[eˇ]] n=0. (28)
· ×
Several techniques are available in the literature for tackling the problem of wave propaga-
tion in periodic composites. In layered structures it is convenient to utilize the transfer-matrix
methodinconjunctionwiththeBloch-Floquettheorem(Adler,1990). Hereinweformulatean
adequate adjustment for treating the propagation of incremental electroelastic waves superim-
posedonafinitelydeformedmultiphaseDElaminates.
Consider a finitely deformed infinite DE-laminate made out of m-phase periodic unit-cells
subjected to incremental perturbations harmonic in time and normal to the layers plane. A
(p)
schematic illustration of the laminate is shown in Fig. 1. Let s , p = 1,..,m, denote an
n
incrementalstatevectorinthe p-phaseofthenthcellconsistingquantitieswhicharecontinuous
acrosstheinterfacebetweenthetwoadjacentlayers. Inviewofthelinearityoftheincremental
(p)
problemitcanbeshownthatthestatevectoratthetopofthe p-phases andthestatevector
t(n)
(p)
at the bottom of the p-phase s (see Fig. 1) are related via a unique non-singular transfer
b(n)
matrixT(p) suchthat
s(p) =T(p)s(p) . (29)
t(n) b(n)
Intermsofthestatevectors,thecontinuityconditionsacrosstheinterfaceare
(p) (p+1)
s =s . (30)
t(n) b(n)
(m) (1)
UtilizingEqs.(29)-(30)recursivelytherelationbetweens ands is
t b
st((bn)) =T(b)s(bb()n) =T(b)st((mn−)1) =T(b)T(m−1)sb(m(n−)1) =...=Ts(b1), (31)
whereT ∏m T(p) isthetotaltransfermatrix.
≡ p=1
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 7
s(m) =s(1)
t(n) b(n+1)
phase (m)
s(m−1) =s(m)
t(n) b(n)
phase (m-1)
cell (n+3)
cell (n+2)
phase (p+1)
cell (n+1) s(p) =s(p+1)
enlarged cell (n) t(n) b(n)
cell (n) phase (p)
s(p)
cell (n 1) b(n)
−
cell (n 2)
−
phase (2)
s(1) =s(2)
t(n) b(n)
phase (1)
s(1) =s(1)
b(n) t(n 1)
−
FIGURE 1. An illustration of an infinite laminate composed of periodic m-
layersunit-cells.
At one hand the continuity conditions across the interface between two successive cells n
andn+1are
(1) (m)
s =s . (32)
b(n+1) tn
At the other, the Bloch-Floquet’s theorem states that the state-vectors of the same phase in
adjacentcellsareidenticaluptoaphaseshiftintheform
s(p) =e ikBhs(p) , (33)
b(n+1) − b(n)
where 0 k <π/h is the Bloch-Floquet wavenumber in first irreducible Brillouin zone, rep-
B
≤
resenting the smallest region where wave propagation is unique (Kittel, 2005). Substitution of
Eq.(31)in(32),followedbyemployingthelatterwithEq.(33),yieldstheeigenvalueproblem
(cid:12) (cid:12)
det(cid:12)T e ikBhI(cid:12)=0, (34)
(cid:12) − (cid:12)
−
where I is an identity matrix with the dimension of the total transfer matrix. The solution of
Eq. (34) provides the dispersion relation describing the manner by which waves propagate in
thelaminate.
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 8
3. THICKNESS VIBRATIONS OF AN INFINITE PERIODIC TWO-PHASE DIELECTRIC
LAMINATE
Henceforth we restrict our attention to isotropic phases. Consequently, the phases AEDFs
canbewrittenintermsofthesixinvariants(DorfmannandOgden,2005)
I =tr(C)=C:I, I = 1(cid:0)I2 C:C(cid:1), I =det(C)=J2, (35)
1 2 2 1 − 3
I =D D, I =D CD, I =D C2D. (36)
4e 5e 6e
· · ·
Specifically, we consider phases which are characterized by incompressible dielectric neo-
Hookean(DH)model
µ(p) 1
(p)
Ψ (I ,I )= (I 3)+ I , (37)
DH 1 5e 2 1− 2ε(p) 5e
where µ(p) and ε(p) =ε ε(p) denote the phases shear moduli and dielectric constants, respec-
0 r
(p)
tively, and ε being the phases relative dielectric constant. The corresponding total stress
r
is
1
σ(p) =µ(p)b+ d d p(p)I, (38)
ε(p) ⊗ − 0
andthecurrentdisplacementandelectricfieldarerelatedviatheisotropiclinearrelation
d=ε(p)e. (39)
Calculation of the associated constitutive tensors A(p),B(p), andC(p) yields, in components
form,
(cid:18) (cid:19)
1 1 d 1
A(p) = δ , B(p) = δ d + i δ , C(p) =µ(p)δ b + δ d d , (40)
ij ε(p) ij ijk ε(p) ik j ε(p) jk ijkl ik jl ε(p) ik j l
whereδ arethecomponentsoftheKroneckerdelta.
ij
Consider a Cartesian coordinate system with unit vectors i ,i and i along x ,x and x ,
1 2 3 1 2 3
respectively. Let a DE laminate be composed of periodic two-phase unit-cells with thickness
H along the the x direction, infinite along the x and x directions as illustrated in Fig. 2.
2 1 3
For convenience we denote the phases a and b. The sum of the phases thicknesses equals the
unit-cellthicknessH =H(a)+H(b),suchthatv(a)=H(a)/H andv(b)=H(b)/H arethevolume
fractions of phase a and phase b, respectively. We assume that the composite is allowed to
expandfreelyinthex andx directions. Ofinterestisthecompositeresponsewhensubjected
1 3
toanoverallelectricdisplacementfieldd¯ =d i ,whered v(a)d(a)+v(b)d(b).
2 2 2 ≡ 2 2
Ineachphaseweassumedisplacementfieldscompatiblewiththehomogeneousuni-modular
(onaccountofincompressibility)diagonaldeformationgradients
(cid:20) (cid:21)
(cid:16) (cid:17) 1
[F](p) =diag λ(p),λ(p), λ(p)λ(p) − , (41)
1 2 1 2
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 9
x
2
x
1
x
3
(b) H(b)
H
(a) H(a)
FIGURE 2. An illustration of an infinite periodic laminate, composed of alter-
nating a and b phases, with initial thicknesses H(a) and H(b), respectively. The
electricbiasfieldisalongthex direction.
2
where henceforth p = a,b. In virtue of the perfect bonding between the phases, the stretch
ratiosinthex andx directionsinthephasesarethesame,i.e.,
1 3
(cid:16) (cid:17) 1 (cid:16) (cid:17) 1
λ(b) =λ(a), λ(b)λ(b) − = λ(a)λ(a) − . (42)
1 1 1 2 1 2
(b) (a) (a)
Eqs. (42) imply that λ =λ , and henceforth we define λ λ . From the symmetry of
2 2 ≡ 1
theprobleminthex x itfollowsthatλ(p)=λ 2. Thecorrespondingstressesineachphase
1− 3 2 −
are
σ(p) =σ(p) =µ(p)λ2 p(p), σ(p) = µ(p) + 1 (d(p))2 p(p). (43)
11 33 − 0 22 λ4 ε(p) 2 − 0
We assume mechanical traction-free boundary conditions at infinity. Utilizing the jump
conditions in the first of (7) and (8), along with the traction-free boundary conditions we have
that
(b) (a) (b) (a)
d =d =d , σ =σ =0. (44)
2 2 2 22 22
From the latter together with the second of Eq. (43) the expressions for the pressure in the
phasesare
µ(p) d2
p(p) = + 2 . (45)
0 λ4 ε(p)
Substitution of Eq. (45) into the first of Eq. (43) leads to a relation between the applied La-
grangianelectricdisplacementfieldD=λ2dandtheresultantstretchratio,namely
(cid:18) ε˘ (cid:19)1/6
λ = 1+D2 , (46)
2µ¯
where ε˘ = v(a)/ε(a)+v(b)/ε(b) and µ¯ = µ(a)v(a)+µ(b)v(b). The latter appropriately reduces
to the corresponding relation for a homogeneous specimen. In terms of the dimensionless
(cid:112)
quantities α = µ(a)/µ(b), β =ε(a)/ε(b), and Dˆ =D / µ(b)ε(b), denoting the shear contrast,
2
BAND-GAPSINDIELECTRICLAMINATESSUBJECTEDTOINCREMENTALSHEAR 10
permittivity contrast, and normalized Lagrangian electric displacement field, respectively, the
stretchratiois
(cid:32) (cid:33)1/6
v(b)+v(a)β 1
λ = 1+Dˆ2 − . (47)
v(b)+v(a)α
Fortheabovechoiceofconstitutiverelation,wealsohavethatEq.(40)reducesto
1
A(p) = A(p) =A(p) = , (48)
11 22 33 ε(p)
1 1
B(p) = B(p) =B(p) =B(p) = B(p) = d(p), (49)
121 211 323 233 2 222 ε(p) 2
C(p) = C(p) =C(p) =C(p) =C(p) =µ(p)λ2, (50)
1111 2121 2323 1313 3333
1
C(p) = C(p) =C(p) =µ(p)/λ4+ d(p)2. (51)
1212 2222 3232 ε(p) 2
The response of the laminate to a harmonic excitation superimposed on the aforementioned
finite deformation is addressed next. Let x˙(p) and ϕ(p) denote, respectively, the components
i
of the incremental displacement and the incremental electric potential in the phases, such that
eˇ(p)= ∇ϕ(p). Thefieldssoughtaretosatisfytheincrementalequationsofmotionalongwith
−
Gauss equation (17) and the incompressibility constraint (18). Following Tiersten (1963), we
consider fields that are functions of x and time alone, independent of x and x , compatible
2 1 3
with the simple-thickness mode (Mindlin, 1955). Consequently, in each phase the governing
equations(17)and(18)simplifyto
µ˜(p)h(p) =ρ(p)x˙(p), (52a)
12,2 1,tt
(cid:18) (cid:19)
3
µ˜(p) d2+p(p) h(p) p˙(p) 2d ϕ(p) =ρ(p)x˙(p), (52b)
−ε(p) 2 0 22,2− 0,2− 2 ,22 2,tt
µ˜(p)h(p) =ρ(p)x˙(p), (52c)
32,2 3,tt
and
dˇ(p) =0, (53a)
2,2
(p)
h =0, (53b)
22
(cid:16) (cid:17)
respectively,whereµ˜(p)=µ(p)/λ4,dˇ(p)= ε(p) d ϕ(p)+ 2 d h(p) . SubstitutionofEq.(53b)
2 − 2 ,2 ε(p) 2 22
(p)
intoEq.(53a)revealsthatϕ =0,hencetheincrementalelectricpotentialislinearinx ,say
,22 2
ϕ(p) = L(p)+x(p)L(p), where L(p) and L(p) are integration constants. More significantly, it
1 2 2 1 2
implies that the incremental electrical and mechanical fields are not coupled, as Eq. (52b) re-
mains a function of h(p) alone, independent of ϕ(p), together with Eq. (53a) which remains a
functionofϕ(p) alone,independentofh(p). Eq.(53b)statesthatalongthethicknesstheincre-
mental displacement is spatially constant. Together with Eqs. (52a) and (52c) the solution for