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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 1970–1981

0022-50

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

Band-gaps in electrostatically controlled dielectric laminatessubjected to incremental shear motions

Gal Shmuel a, Gal deBotton a,b,n

a Department of Mechanical Engineering, Ben-Gurion University, Beer-Sheva 84105, Israelb Department of Biomedical Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel

a r t i c l e i n f o

Article history:

Received 5 December 2011

Received in revised form

12 May 2012

Accepted 31 May 2012Available online 15 June 2012

Keywords:

Dielectric elastomers

Wave propagation

Finite deformations

Thickness vibrations

Non-linear electroelasticity

Band-gap

Composite

Bloch-Floquet analysis

96/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jmps.2012.05.006

esponding author. Tel.: þ972 8 647 7105; fa

ail address: debotton@bgumail.bgu.ac.il (G. d

a b s t r a c t

The thickness vibrations of a finitely deformed infinite periodic laminate made out of

two layers of dielectric elastomers is studied. The laminate is pre-stretched by inducing

a bias electric field perpendicular to the layers. Incremental time-harmonic fields

superimposed on the initial finite deformation are considered next. Utilizing the Bloch-

Floquet theorem along with the transfer matrix method we determine the dispersion

relation which relates the incremental fields frequency and the phase velocity.

Ranges of frequencies at which waves cannot propagate are identified whenever the

Bloch-parameter is complex. These band-gaps depend on the phases properties, their

volume fraction, and most importantly on the electric bias field. Our analysis reveals

how these band-gaps can be shifted and their width can be modified by changing the

bias electric field. This implies that by controlling the electrostatic bias field desired

frequencies can be filtered out. Representative examples of laminates with different

combinations of commercially available dielectric elastomers are examined.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Band-gaps corresponding to ranges of frequencies at which waves cannot propagate earned the interest of the scientificcommunity (e.g., Ziegler, 1977; Wang and Auld, 1985; Kushwaha et al., 1994; Gei et al., 2004, 2009, to name a few). Theseband-gaps (BGs) or stop-bands can be utilized for various applications such as filtering elastic waves, enhancingperformance of ultrasonic transducers, supplying a vibrationless ambient when needed for industrial and biomedicalapplications, and more. Several techniques are available for investigating wave propagation in composites: for the plane-wave method in periodic composites see, for example, Kushwaha et al. (1993), Sigalas and Economou (1996) and Tanakaand Tamura (1998), and for composites with random microstructures the effective medium method was employed inSabina and Willis (1988). As in this work we are interested in the appearance of BGs in infinite periodic laminates, we findit 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-deformed state due to bias electric field acting on an infiniteperiodic laminate with a repeating sequence of dielectric elastomer (DE) layers. The motivation for the choice of DEs as thecomposite constituents stems from their ability to undergo large deformations (Pelrine et al., 2000), and to change theirmechanical and dielectric properties when subjected to electrostatic fields (Kofod, 2008), thus effecting the way in whichelectroelastic waves propagate in the matter (Gei et al., 2011; Shmuel et al., 2012). As will be shown in the sequel, by

properly tuning the bias electrostatic field different ranges of frequencies can be filtered out.

ll rights reserved.

x: þ972 8 647 7106.

eBotton).

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1971

The structure of this paper is as follows. Following the works of Dorfmann and Ogden (2005, 2010), deBotton et al.(2007), Bertoldi and Gei (2011), Rudykh and deBotton (2011), Ponte Castaneda and Siboni (2012) and Tian et al. (2012), thebackground 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 aresummarized. In this section we also introduce an adequate extension of the transfer matrix method and the Bloch-Floquettheorem required for treating the propagation of incremental electroelastic waves superimposed on finitely deformed DElaminates. Section 3 deals with the response of an infinite periodic laminate with alternating layers, whose behaviors aregoverned by the incompressible dielectric neo-Hookean (DH) model, to a fixed electric displacement field normal to thelayers plane. Subsequently, the response to small harmonic perturbations superimposed on the finitely deformed laminateis analyzed. The end result is given in terms of a dispersion relation relating the incremental fields frequency and phasevelocity to the Bloch-parameter. Several examples are considered in Section 4, where the influence of the morphology ofthe laminate, the contrast between the phases properties, and the bias electric field on the dispersion relation areexamined. Finally, realizable tunable isolators are studied by considering combinations of commercially available DEs suchas VHB 4910, fluorosilicone 730 and ELASTOSIL RT-625.

2. Theoretical background

Consider a heterogeneous body occupying a volume region O0 � R3 made out of N perfectly bonded differenthomogeneous phases. The external boundary of the body @O0 separates it from the surrounding space R3

\O0, assumed tobe vacuum. Each phase occupies a volume region OðrÞ0 ðr¼ 1;2, . . . ,NÞ and its boundary is @OðrÞ0 . Let v : O0 � I-R3 describea continuous and twice-differentiable mapping of a material point X from the reference configuration of the body to itscurrent configuration O with boundary @O by x¼ vðX,tÞ. The corresponding velocity and acceleration are denoted,respectively, by v¼ v,t and a¼ v,tt , and the deformation gradient is F¼rXv. Vectors in the neighborhood of X aretransformed into vectors in the current configuration via dx¼ F dX. The volume change of a referential volume element dV

is given by dv¼ J dV , where dv is the corresponding volume element in the current configuration, and J� detðFÞ40 due tomaterial impenetrability. The conservation of mass implies that rL ¼ Jr, where rL and r are the material mass densities inthe reference and the deformed configurations, respectively. An area element dA with the unit normal N in the referenceconfiguration is transformed to a deformed area da with the unit normal n in the current configuration according toNanson’s formula N dA¼ ð1=JÞFTn da. The right and left Cauchy–Green strain tensors are C¼ FTF and b¼ FFT.

The electric field in the current configuration, denoted by e, is given in terms of a gradient of a scalar field, termed theelectrostatic potential. In free space an induced electric displacement field d is related to the electric field via the vacuumpermittivity E0, such that d¼ E0e. In dielectric bodies the connection between the fields is specified by an adequateconstitutive relation.

In terms of a ‘total’ stress tensor r the equations of motion read

r � r¼ ra, ð1Þ

where in order to satisfy the balance of angular momentum symmetry of r is required. The terminology ‘total’ stress isused to indicate that r accounts for both mechanical and electrical contributions. Thus, the traction t on a deformed areaelement with a unit normal n is given by

rn¼ t: ð2Þ

When ideal dielectrics are considered no free body charge is present, and Gauss’s law takes the form

r � d¼ 0: ð3Þ

Faraday’s law is

r � e¼ 0, ð4Þ

when a quasi-electrostatic approximation is considered. A full description of the system should consider interactions withfields outside the body, henceforth denoted by a star superscript. Specifically, these are

d%

¼ E0e%, ð5Þ

r% ¼ E0½e% � e%�1

2ðe% � e%ÞI�, ð6Þ

where I is the identity tensor. In vacuum the governing equations (3) and (4) for d% and e% are equivalent to Laplace’sequation for the electrostatic potential. As a consequence, r% known as the Maxwell stress, is divergence-free. Across theouter boundary @O, the electric jump conditions are

ðd�d%

Þ � n¼�we, ðe�e%Þ � n¼ 0, ð7Þ

where we is the surface charge density. In order to formulate the jump in the stress we postulate a separation of thetraction into a sum of a prescribed mechanical traction tm, and an electric traction te induced by the external stress suchthat te ¼ r%n. Hence, the stress boundary condition is

ðr�r%Þn¼ tm: ð8Þ

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–19811972

The jump conditions across a charge-free internal boundary between two adjacent phases a and b are

1rUn¼ 0, 1dU � n¼ 0, 1eU� n¼ 0, ð9Þ

where 1U� ðÞðbÞ�ðÞðaÞ denotes the jump of fields between the two phases.A Lagrangian formulation of the governing equations and jump conditions is feasible by a pull-back of the Eulerian fields

P¼ JrF�T, D¼ JF�1d, E¼ FTe, ð10Þ

for the ‘total’ first Piola–Kirchhoff stress, Lagrangian electric displacement and electric field, respectively. Thus, thegoverning equations (1), (3) and (4) transform, respectively, to

rX � P¼ rLa, rX � D¼ 0, rX � E¼ 0: ð11Þ

The corresponding referential jump conditions across the external boundary @O0 are

ðP�P%

ÞN¼ tM , ðD�D%

Þ � N¼�wE, ðE�E%

Þ � N¼ 0, ð12Þ

where tM dA¼ tm da and wE dA¼we da. Here P%, D% and E% are, respectively, formal pull-backs of r%, d% and e% as obtainedby the application of Eq. (10) with an arbitrary deformation gradient F% in the surrounding space such that at the boundaryF%

¼ FðIþM� NÞ where M is any vector tangent to the boundary (e.g., the projections defined in deBotton and Hariton,2002). It can be easily shown that with this definition for F% Eq. (12) is essentially equivalent to Eqs. (7) and (8). Across thereferential interfaces Eqs. (9) become

1PUN¼ 0, 1DU � N¼ 0, 1EU� N¼ 0: ð13Þ

Following Dorfmann and Ogden (2005), the constitutive relation is expressed in terms of an augmented energy density

function (AEDF) C with the independent variables F and D, such that the total first Piola–Kirchhoff stress and theLagrangian electric field are derived via

P¼@C@F

, E¼@C@D

: ð14Þ

The first of Eq. (14) should be modified when considering incompressible materials, as the kinematic constraint yields anadditional workless part of stress. The latter is accounted by introducing a Lagrange multiplier p0 which can be determinedonly from the equilibrium equations together with the boundary conditions. Thus, the total first Piola–Kirchhoff is

P¼@C@F�p0F�T: ð15Þ

Based on the framework developed in Dorfmann and Ogden (2010), we superimpose an infinitesimal time-dependentelastic and electric displacements _x ¼ _vðX,tÞ and _DðX,tÞ, on the pre-deformed configuration. Herein, and in the sequel, weuse a superposed dot to denote incremental quantities. Let the Eulerian quantities R, d

ˇ, and eˇ denote the push-forwards of

increments in the first Piola–Kirchhoff stress, the Lagrangian electric displacement and electric fields, respectively, namely

R¼1

J_PFT , d

ˇ¼

1

JF _D, eˇ ¼ F�T _E: ð16Þ

In terms of these variables, the incremental governing equations read

r � R¼ r _xtt , r � dˇ¼ 0, r � eˇ ¼ 0: ð17Þ

For an incompressible material the incremental constraint is

r � _x � trðhÞ ¼ 0, ð18Þ

where h¼r _x is the displacement gradient. Linearization of the incompressible material constitutive equations in theincrements yields

R¼ Chþp0hT� _p0IþBd

ˇ, ð19Þ

eˇ ¼ BThþAdˇ, ð20Þ

where ðBThÞk ¼ Bijkhij. The quantities A,B and C are the push-forwards of the referential reciprocal dielectric tensor,electroelastic coupling tensor, and elasticity tensor, respectively. In components form

Aij ¼ JF�1ai A0abF�1

bj , Bijk ¼ FjaB0iabF�1bk , Cijkl ¼

1

JFjaC0iakbFlb, ð21Þ

where

A0ab ¼@2C

@Da@Db, B0iab ¼

@2C@Fia@Db

, C0iakb ¼@2C

@Fia@Fkb: ð22Þ

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1973

The incremental outer fields are given by

_d%

¼ E0 _e% , ð23Þ

_r% ¼ E0½ _e% � e%þe% � _e%�ðe% � _e% ÞI�: ð24Þ

Here again _d% and _e% are to satisfy Eqs. (3) and (4), where subsequently r � _r% ¼ 0.The push-forward of the increments of Eqs. (12) gives the following incremental jump conditions across the external

boundary in the current configuration

½R� _r%þr%hT�ðr � _xÞr%�n¼ t

ˇm, ð25Þ

½dˇ� _d%

�ðr � _xÞd%

þhd%

� � n¼�wˇ e, ð26Þ

½eˇ� _e%�hTe%� � n¼ 0, ð27Þ

where tˇm da¼ _tM dA and wˇ e da¼ _wE dA. Similarly, the push forward of the increments of Eqs. (13) results in the following

jumps across the internal boundaries

1RUn¼ 0, 1dˇU � n¼ 0, 1eˇU� n¼ 0: ð28Þ

Several techniques are available in the literature for tackling the problem of wave propagation in periodic composites.In layered structures it is convenient to utilize the transfer-matrix method in conjunction with the Bloch-Floquet theorem(Adler, 1990). Herein we formulate an adequate adjustment for treating the propagation of incremental electroelasticwaves superimposed on finitely deformed multiphase DE laminates.

Consider a finitely deformed infinite DE-laminate made out of m-phase periodic unit-cells subjected to incrementalperturbations harmonic both in time and along the normal to the layers plane. A schematic illustration of the laminate isshown in Fig. 1. Let sðpÞn , p¼ 1,: :,m, denote an incremental state vector in the p-phase of the nth cell consisting of quantitieswhich are continuous across the interface between the two adjacent layers. In view of the linearity of the incrementalproblem it can be shown that the state vector at the top of the p-phase sðpÞtðnÞ and the state vector at the bottom of thep-phase sðpÞbðnÞ (see Fig. 1) are related via a unique non-singular transfer matrix TðpÞ such that

sðpÞtðnÞ ¼ TðpÞsðpÞbðnÞ: ð29Þ

In terms of the state vectors, the continuity conditions across the interface are

sðpÞtðnÞ ¼ sðpþ1ÞbðnÞ : ð30Þ

Utilizing Eqs. (29)–(30) recursively the relation between sðmÞt and sð1Þb is

sðmÞtðnÞ ¼ TðmÞsðmÞbðnÞ ¼ TðmÞsðm�1ÞtðnÞ ¼ TðmÞTðm�1Þsðm�1Þ

bðnÞ ¼ � � � ¼ Tsð1ÞbðnÞ, ð31Þ

where T�Qm

p ¼ 1 TðpÞ is the total transfer matrix.

Fig. 1. An illustration of an infinite laminate composed of periodic m-layers unit-cells.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–19811974

At one hand the continuity conditions across the interface between two successive cells n and nþ1 are

sð1Þbðnþ1Þ ¼ sðmÞtðnÞ: ð32Þ

At the other, the Bloch-Floquet’s theorem states that the state-vectors of the same phase in adjacent cells are identical upto a phase shift in the form

sðpÞbðnþ1Þ ¼ e�ikBhsðpÞbðnÞ, ð33Þ

where 0rkBop=h is the Bloch-Floquet wavenumber in the first irreducible Brillouin zone, representing the smallestregion where wave propagation is unique (Kittel, 2005). Substitution of Eq. (31) in (32), followed by employing the latterwith Eq. (33), yields the eigenvalue problem

det9T�e�ikBhI9¼ 0, ð34Þ

where I is an identity matrix with the dimension of the total transfer matrix. The solution of Eq. (34) provides thedispersion relation describing the manner by which waves propagate in the laminate.

3. Thickness vibrations of an infinite periodic two-phase dielectric laminate

Henceforth we restrict our attention to isotropic phases. Consequently, the phases AEDFs can be written in terms of thesix invariants (Dorfmann and Ogden, 2005)

I1 ¼ trðCÞ ¼ C : I, I2 ¼12ðI

21�C : CÞ, I3 ¼ detðCÞ ¼ J2, ð35Þ

I4e ¼D � D, I5e ¼D � CD, I6e ¼D � C2D: ð36Þ

Specifically, we consider phases which are characterized by the incompressible dielectric neo-Hookean (DH) model

CðpÞDHðI1,I5eÞ ¼mðpÞ

2ðI1�3Þþ

1

2EðpÞI5e, ð37Þ

where mðpÞ and EðpÞ ¼ E0EðpÞr denote the phases shear moduli and dielectric constants, respectively, and EðpÞr being the phasesrelative dielectric constant. The corresponding total stress is

rðpÞ ¼ mðpÞbþ 1

EðpÞ d� d�pðpÞ0 I, ð38Þ

and the current electric displacement and electric fields are related via the isotropic linear relation

d¼ EðpÞe: ð39Þ

Consider a Cartesian coordinate system with unit vectors i1,i2 and i3 along x1,x2 and x3, respectively. Let a DE laminatebe composed of periodic two-phase unit-cells with thickness H along the x2 direction, infinite along the x1 and x3 directionsas illustrated in Fig. 2. For convenience we denote the phases a and b. The sum of the phases thicknesses equals the unit-cell thickness H¼HðaÞ þHðbÞ, such that vðaÞ ¼HðaÞ=H and vðbÞ ¼HðbÞ=H are the volume fractions of phase a and phase b,respectively. We assume that the composite is allowed to expand freely in the x1 and x3 directions. The electrostaticexcitation is accomplished by means of a potential drop between two soft electrodes that are attached to the laminatefaces parallel to the interfaces between the layers. Specifically, in Fig. 1 the normal to the electrodes plane is along the x2

direction. We find it advantageous to determine the composite response to the induced overall electric displacement fieldd ¼ d2i2, where d2 � vðaÞdðaÞ2 þvðbÞdðbÞ2 .

Fig. 2. An illustration of an infinite periodic laminate, composed of alternating a and b phases, with referential thicknesses HðaÞ and HðbÞ , respectively. The

electric bias field is along the x2 direction.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1975

In each phase we assume displacement fields compatible with the homogeneous uni-modular (on account ofincompressibility) diagonal deformation gradients

½F�ðpÞ ¼ diag½lðpÞ1 ,lðpÞ2 ,ðlðpÞ1 lðpÞ2 Þ�1�, ð40Þ

where henceforth p¼ a,b. In virtue of the perfect bonding between the phases, the stretch ratios in the x1 and x3 directionsin the phases are the same, i.e.,

lðbÞ1 ¼ lðaÞ1 , ðlðbÞ1 lðbÞ2 Þ�1¼ ðlðaÞ1 lðaÞ2 Þ

�1: ð41Þ

Eqs. (41) imply that lðbÞ2 ¼ lðaÞ2 , and henceforth we define l� lðaÞ1 . From the symmetry of the problem in the x12x3 it followsthat lðpÞ2 ¼ l�2. The corresponding stresses in each phase are

sðpÞ11 ¼ sðpÞ33 ¼ m

ðpÞl2�pðpÞ0 , sðpÞ22 ¼

mðpÞ

l4þ

1

EðpÞðdðpÞ2 Þ

2�pðpÞ0 : ð42Þ

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

dðbÞ2 ¼ dðaÞ2 ¼ d2, sðbÞ22 ¼ sðaÞ22 ¼ 0: ð43Þ

From the latter together with the second of Eq. (42) the expressions for the pressure in the phases are

pðpÞ0 ¼mðpÞ

l4þ

d22

EðpÞ : ð44Þ

Substitution of Eq. (44) into the first of Eq. (42) leads to a relation between the applied Lagrangian electric displacementfield D¼ l2d and the resultant stretch ratio, namely

l¼ 1þD22

�Em

� �1=6

, ð45Þ

where �E ¼ vðaÞ=EðaÞ þvðbÞ=EðbÞ and m ¼ mðaÞvðaÞ þmðbÞvðbÞ. The latter appropriately reduces to the corresponding relation for ahomogeneous specimen. In terms of the dimensionless quantities a¼ mðaÞ=mðbÞ, b¼ EðaÞ=EðbÞ, and D ¼D2=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimðbÞEðbÞ

p, denoting

the shear contrast, permittivity contrast, and normalized Lagrangian electric displacement field, respectively, the stretchratio is

l¼ 1þD2 vðbÞ þvðaÞb�1

vðbÞ þvðaÞa

!1=6

: ð46Þ

Before we proceed and tackle the wave problem, we note that in practice DEs are commonly actuated by applying avoltage drop between two thin and flexible electrodes. Thus, from a practical viewpoint it is useful to relate the resultingstretch with the applied voltage drop instead of the electric displacement appearing in (46). To this end we note that interms of the referential electric field the potential drop over a single unit-cell of the periodic laminate is DV ¼�E2H. Fromthe pull-back operators (10), the constitutive relation (39), and the first of (43) it follows that

E2 ¼D2 �E=l2: ð47Þ

Thus, solving Eq. (45) and substitution of the values of D2 and l into Eq. (47) yields the potential drop DV . Thecorresponding overall potential drop between the two electrodes is E2 times the laminate thickness.

Calculation of the associated incremental constitutive tensors AðpÞ,BðpÞ, and CðpÞ yields, in components form,

AðpÞij ¼1

EðpÞ dij, BðpÞijk ¼1

EðpÞ ðdikdjþdidjkÞ, CðpÞijkl ¼ mðpÞdikbjlþ

1

EðpÞ dikdjdl, ð48Þ

where dij are the components of the Kronecker delta. For the above choice of loading path, Eq. (48) further reduces to

AðpÞ11 ¼AðpÞ22 ¼A

ðpÞ33 ¼

1

EðpÞ , ð49Þ

BðpÞ121 ¼ BðpÞ211 ¼ B

ðpÞ323 ¼ B

ðpÞ233 ¼

1

2BðpÞ222 ¼

1

EðpÞd2, ð50Þ

CðpÞ1111 ¼ CðpÞ2121 ¼ C

ðpÞ2323 ¼ C

ðpÞ1313 ¼ C

ðpÞ3333 ¼ m

ðpÞl2, ð51Þ

CðpÞ1212 ¼ CðpÞ2222 ¼ C

ðpÞ3232 ¼ m

ðpÞ=l4þ

1

EðpÞd2

2: ð52Þ

The response of the laminate to a harmonic excitation superimposed on the aforementioned finite deformation isaddressed next. Let _xðpÞi and jðpÞ denote, respectively, the components of the incremental displacement and theincremental electric potential in the phases, such that eˇ

ðpÞ¼ �rjðpÞ. The fields sought are to satisfy the incremental

equations of motion along with Gauss equation (17) and the incompressibility constraint (18). Following Tiersten (1963),

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–19811976

we consider fields that are functions of x2 and time alone, independent of x1 and x3, compatible with the simple-thicknessmode (Mindlin, 1955). Consequently, in each phase the governing equations (17) and (18) simplify to

~mðpÞhðpÞ12;2 ¼ rðpÞ _xðpÞ1,tt , ð53aÞ

~mðpÞ� 3

EðpÞ d22þpðpÞ0

� �hðpÞ22;2�

_pðpÞ0;2�2d2jðpÞ,22 ¼ rðpÞ _xðpÞ2,tt , ð53bÞ

~mðpÞhðpÞ32;2 ¼ rðpÞ _xðpÞ3,tt , ð53cÞ

and

dˇ ðpÞ

2;2 ¼ 0, ð54aÞ

hðpÞ22 ¼ 0, ð54bÞ

respectively, where ~mðpÞ ¼ mðpÞ=l4 and dˇðpÞ

2 ¼�EðpÞðd2jðpÞ,2 þð2=EðpÞÞd2hðpÞ22Þ. Substitution of Eq. (54b) into Eq. (54a) reveals that

jðpÞ,22 ¼ 0, hence the incremental electric potential is linear in x2, say jðpÞ ¼ LðpÞ1 þxðpÞ2 LðpÞ2 , where LðpÞ1 and LðpÞ2 are integration

constants. More significantly, it implies that the incremental electrical and mechanical fields are not coupled, as Eq. (53b)

remains a function of hðpÞ alone, independent of jðpÞ, together with Eq. (54a) which remains a function of jðpÞ alone,

independent of hðpÞ. Eq. (54b) states that along the thickness the incremental displacement is spatially constant. Togetherwith Eqs. (53a) and (53c) the solution for the incremental problem is

_xðpÞ1 ¼ ðUðpÞ1 sin kðpÞx2þUðpÞ2 cos kðpÞx2Þe

�iot ,

_xðpÞ2 ¼UðpÞ3 e�iot ,

_xðpÞ3 ¼ ðUðpÞ4 sin kðpÞx2þUðpÞ5 cos kðpÞx2Þe

�iot , ð55Þ

where kðpÞ ¼o=c ðpÞ, c ðpÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~mðpÞ=rðpÞ

q, o is the angular excitation frequency, and UðpÞr ðr¼ 1;2, . . . ,5Þ are integration

constants. Without loss of generality the constants UðpÞ3 can be set to be zero. The motion described by the solution

equation (55) corresponds to displacement gradients which retain only shear components. For completeness, the solution

for the incremental pressure emerges from Eq. (53b), which reduces to � _pðpÞ0;2 ¼ rðpÞ _xðpÞ2,tt , and implies that _pðpÞ0 ¼ PðpÞe�iot ,

where PðpÞ are integration constants. The incremental state vector of the nth cell takes the form

sðpÞðnÞ ¼ f _x

ðpÞn1 , _xðpÞn3 ,SðpÞn12,SðpÞn22,SðpÞn32,jðpÞn ,d

ˇðpÞ

n2gT: ð56Þ

The components of the transfer matrix TðpÞ relating the top sðpÞt nð Þ and bottom sðpÞbðnÞ state vectors are given in the Appendix.

Note that the continuity condition for _xðpÞn2 is identically satisfied when we set UðpÞ3 to zero. The corresponding solution of the

eigenvalue problem equation (34) resulting from the Bloch-Floquet theorem can be expressed in terms of thetrigonometric expression

kB ¼1

harccos Z, ð57Þ

where

Z¼ 1

2hcos kðbÞhðbÞ coskðaÞhðaÞ�

1

2kðbÞ=kðaÞ þkðaÞ=kðbÞ� �

sin kðbÞhðbÞ sin kðaÞhðaÞ� �

, ð58Þ

with hðpÞ ¼HðpÞ=l2 and h¼ hðaÞ þhðbÞ. Eq. (57) is an extension of the solution given in Wang et al. (1986) to the class offinitely deformed infinite periodic DE laminates. The dependency on the bias field becomes evident when Z is rephrased interms of the electric bias field, the referential geometry and the frequency, namely

Z¼ 1

2H1þD2

2

�Em

� �1=3

cos oHðbÞ

ffiffiffiffiffiffiffiffirðbÞmðbÞ

scos oHðaÞ

ffiffiffiffiffiffiffiffirðaÞmðaÞ

s"

�1

2

arðbÞ

rðaÞ þr að Þ

arðbÞ

� �sin oHðbÞ

ffiffiffiffiffiffiffiffirðbÞmðbÞ

ssin oHðaÞ

ffiffiffiffiffiffiffiffirðaÞmðaÞ

s #: ð59Þ

In virtue of the uncoupled nature of the incremental problem, the incremental electric quantities indeed do not appearin Eq. (59). We conclude this section noting that waves of certain frequencies cannot propagate when the pertainedsolution of kB is complex. These cases will be examined in the following section.

0

0.5

1

1.5

2

2.5

3

0

� � �

�=�=5

kB

0

0.5

1

1.5

2

2.5

3�=�=10

kB

0

0.5

1

1.5

2

2.5

3�=�=20

kB

2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10

Fig. 3. Variations of the Bloch wavenumber kB as functions of the normalized frequency o ¼oH=c , for (a) a¼ b¼ 5, (b) a¼ b¼ 10 and (c) a¼ b¼ 20. The

red, blue and black curves correspond to vðbÞ ¼ 0:2,0:5 and 0.8, respectively. (For interpretation of the references to color in this figure caption, the reader

is referred to the web version of this article.)

0

2

4

6

8

10

0

v(b)

=5

0

2

4

6

8

10=10

0

2

4

6

8

10=20

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

v(b) v(b)

� ��

Fig. 4. Prohibited frequencies as functions of vðbÞ at D ¼ 3, for (a) a¼ b¼ 5, (b) a¼ b¼ 10 and (c) a¼ b¼ 20.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1977

4. Numerical examples

With the help of a few examples we explore the dispersion relation characterizing the dynamic response of thecomposite. We examine its dependency on the phases volume fractions and the contrast between the phases shear anddielectric moduli. In particular we are interested in the influence of the electric bias field on the dynamic response of thecomposite. To this end we make the following choice for the properties of phase b

rðbÞ ¼ 1000 kg=m3, mðbÞ ¼ 200 kPa, EðbÞr ¼ 2, ð60Þ

and determine dispersion diagrams for various choices of phase a properties and volume fractions at different values of thedimensionless bias field D. In all the forthcoming examples rðaÞ ¼ rðbÞ is assumed.

Fig. 3 shows variations of the Bloch wavenumber kB as functions of the normalized frequency o ¼oH=c wherec ¼

ffiffiffiffiffiffiffiffiffiffim=r

p, r ¼ rðaÞvðaÞ þrðbÞvðbÞ, for a representative value of the bias field D ¼ 3. Fig. 3a, b and c correspond to a¼ b¼ 5;10

and 20, respectively. The red, blue and black curves correspond to vðbÞ ¼ 0:2,0:5 and 0.8, respectively. The period of thedispersion curves becomes smaller with an increase of the softer phase volume fraction. An inverse effect is revealed whenthe contrast is enhanced, that is longer periods for higher values of a and b.

Fig. 4 displays regions of prohibited normalized frequencies o as functions of the volume fraction of phase b at a fixedvalue of the bias field D ¼ 3. Fig. 4a, b and c correspond to a¼ b¼ 5, 10 and 20, respectively. Herein and henceforth theband-gaps are the blue regions in the plots. We observe how an increase of the softer phase volume fraction results inappearance of additional thinner bands. Conversely, when the contrasts are increased, that is higher values of a and b,there are fewer thicker bands. The latter is evident in Fig. 5 which displays the prohibited frequencies as functions of a forvðbÞ ¼ 0:5 and D ¼ 3, with different values of b. Specifically, Fig. 5a corresponds to b¼ 0:1 and Fig. 5b corresponds to b¼ 10.Similar trends are observed when the dielectric contrast is investigated. Fig. 6 displays the prohibited frequencies o asfunctions of b for vðbÞ ¼ 0:5, with different values of a. Fig. 6a corresponds to a¼ 0:1 and Fig. 6b to a¼ 10. Figs. 5 and 6reveal that whenever the phases have non-monotonous ratios of shear moduli and dielectric contrasts, i.e., a41 and bo1or ao1 and b41, the band gaps become thinner and denser.

Fig. 7 illustrates the influence of the bias field by examining the variations of the BGs as functions of the normalizedparameter D. Values of vðbÞ ¼ 0:8 and a¼ b¼ 5 were chosen in Fig. 7a, vðbÞ ¼ 0:5 and a¼ b¼ 10 in Fig. 7b, and vðbÞ ¼ 0:2 and

0

2

4

6

8

10= 0.1 = 10

0

2

4

6

8

10

2 4 6 8 102 4 6 8 10� �

��

Fig. 5. Prohibited frequencies as functions of a at D ¼ 3 and vðbÞ ¼ 0:5, for (a) b¼ 0:1 and (b) b¼ 10.

0

2

4

6

8

10

2

�

=0.1

0

2

4

6

8

10

�

=10

4 6 8 10

�

2 4 6 8 10

�

Fig. 6. Prohibited frequencies as functions of b at D ¼ 3 and vðbÞ ¼ 0:5, for (a) a¼ 0:1 and (b) a¼ 10.

0

2

4

6

8

10

0

v(b)=0.8, 5

0

2

4

6

8

10

0

v(b)=0.5, 10

0

2

4

6

8

10

0

v(b)=0.2, 20

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3

�

D

� �

D D

Fig. 7. Prohibited frequencies as functions of D for (a) vðbÞ ¼ 0:8,a¼ b¼ 5, (b) vðbÞ ¼ 0:5,a¼ b¼ 10 and (c) vðbÞ ¼ 0:2,a¼ b¼ 20.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–19811978

a¼ b¼ 20 in Fig. 7c. The influence of the electric field is evident as the BGs are shifted toward lower frequencies when thefield is increased. This phenomena suggest that control of BGs and filtering of desired frequencies are feasible by activating DE

laminates with adequate bias electric fields.

Finally, we consider realizable laminates made out of commercially available DEs. We examine three materials: VHB-4910 by 3M, ELASTOSIL RT-625 by Wacker, and fluorosilicone 730 by Dow Corning. Approximate physical properties of

Table 1Physical properties of commercially available DEs.

Material Density (kg/m3) Shear modulus (kPa) Relative dielectric constant

VHB-4910 960 406 4.7

ELASTOSIL RT-625 1020 342 2.7

Fluorosilicone 730 1400 167 6.9

0

0.5

1

1.5

2

2.5

3

0

� �

kB

VHB-4910 and ELASTOSIL RT-625

0

0.5

1

1.5

2

2.5

3

kB

VHB-4910 and fluorosilicone 730

2 4 6 8 10 0 2 4 6 8 10

Fig. 8. Variations of the Bloch wavenumber kB as functions of the normalized frequency o ¼oH=c for combinations of VHB-4910 with (a) ELASTOSIL RT-

625, and with (b) fluorosilicone 730. In both cases there is an equal volume fraction of the two constituents. The blue, black and red dotted curves

correspond to D ¼ 0:5,1 and 1.5, respectively. (For interpretation of the references to color in this figure caption, the reader is referred to the web version

of this article.)

0

2

4

6

8

10

0

VHB-4910 and ELASTOSIL RT-625

E2[kV/mm]

0

2

4

6

8

10

0

VHB-4910 and Fluorosilicone 730

E2[kV/mm]

20 40 60 80 100 120 20 40 60 80 100 120

� �

Fig. 9. Prohibited frequencies as functions of E2 for combinations of VHB-4910 with (a) ELASTOSIL RT-625, and (b) fluorosilicone 730. In both cases there

is an equal volume fraction of the two constituents.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1979

these elastomers, as reported in the literature (e.g., Kofod and Sommer-Larsen, 2005; Kornbluh and Pelrine, 2008), aresummarized in Table 1. We investigate different combinations of the two-phase laminate and assume that vðbÞ ¼ vðaÞ.

Fig. 8 displays the dispersion relation in terms of Bloch wavenumber kB as a function of the normalized frequency o.Fig. 8a corresponds to a combination of VHB-4910 and ELASTOSIL RT-625, while Fig. 8b to VHB-4910 and fluorosilicone730. In terms of the dimensionless parameters in Fig. 8a a¼ 1:19 and b¼ 1:74, and in Fig. 8b a¼ 2:44 and b¼ 0:68. Theblue, black and red curves correspond to D ¼ 0:5,1 and 1.5, respectively.

To highlight the dependence of the BGs on the imposed voltage, we show in Fig. 9 the prohibited frequencies asfunctions of the referential electric field E2. Again, Fig. 9a corresponds to the combination of VHB-4910 and ELASTOSIL RT-625, and Fig. 9b to VHB-4910 and fluorosilicone 730. Wider bands are depicted in Fig. 9b in comparison with Fig. 9a, sincethe latter corresponds to a laminate with higher shear contrast between the phases.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–19811980

We note that for most real-life applications the first BG is the important one. To gain some perspective about thefrequency range associated with this BG in laminated DEs, in a way of an example we consider the case of a 10 mm thickunit-cell and factor out the velocity normalization. From Fig. 4a for a¼ b¼ 5 we determine that the first gap opensapproximately at 9.6 MHz at a dilute volume fraction of phase b and closes at 2.14 MHz at a dilute volume fraction ofphase a. Its maximal width is attained at vðbÞ 0:2. Consider further the specific volume fraction vðbÞ ¼ 0:8 for the samecontrasts as depicted in Fig. 7a. In this case when E2 ¼ 0 the BG ranges between 4.5 and 5.2 MHz, while at E2 ¼ 155 kV=mmthe gap is between 2.7 and 3 MHz. This implies that by changing the electric bias field the mean value of the first BGfrequency is reduced by a factor of 2 with no overlap between the initial and the final frequency ranges. Following Fig. 7b,for volume fraction vðbÞ ¼ 0:5 and contrast a¼ b¼ 10, the first BG ranges between 5 and 8 MHz when no electric field isapplied, and between 4 and 6.5 MHz when E2 ¼ 140 kV=mm. In contrast with the previous composite, this time the meanvalue of the first BG decreases by 20% in response to the applied electrostatic bias field, with a common range between 5and 6.5 MHz throughout.

These variations in the effects of the bias field are related to the way in which the BG mechanism is influenced by theshear contrast. Essentially, when the shear contrast is low, the relative role of the electric field is larger. This trend may alsobe seen in Fig. 9 for the commercially available constituents. Taking the dimensional frequencies associated withH¼ 10 mm, for the VHB-4910 and ELASTOSIL RT-625 composite the first BG average thickness is approximately 0.19 MHz,whereas for the VHB-4910 and fluorosilicone 730 layered composite the average thickness of the first band is 1 MHz. Thus,for the VHB-4910 with RT-625 composite, in which the contrast between the shear moduli is a¼ 1:19, the mean frequencyof the first BG is reduced by 6.5 times its thickness due to the application of the electrostatic bias field. Contrary, for theVHB-4910 with fluorosilicone 730 composite, in which the contrast between the shear moduli is a¼ 2:44, the variation inits mean frequency due to the application of the electrostatic bias field is smaller and equals approximately 2.9 times theband thickness. We conclude that to render the first BG electrostatically tunable, a lower shear contrast is to be chosen,while in order to widen the gap a high shear contrast is preferred. In passing, we recall that by choosing thinner unit-cellsthe actual frequencies increase, while thicker ones lead to lower BG frequencies.

5. Concluding remarks

Motivated by the ability of DEs to undergo large deformations and change their mechanical and electrical propertieswhen electrostatically excited, the feasibility of inducing and controlling BGs by the electrical bias field is explored. To thisend we first considered the static response of an infinite periodic laminate to a bias electric displacement field.Subsequently, the propagation of small waves superposed on the pre-strained composite is addressed. Applications ofthe transfer matrix method along with the Bloch-Floquet theorem resulted in a dispersion relation between the wavefrequencies, the phase velocities, and the Bloch-parameter. Numerical analysis of these relations was conducted forspecific two-phase laminates in order to examine how the stop-bands, associated with complex Bloch-parameter, vary asfunctions of the geometrical, mechanical and electric properties of the laminate. Most importantly, this analysis reveals theinfluence of the electrostatic bias fields on the BGs. Our findings demonstrate how when the concentration of the softerphase is increased, the bands become thinner and additional band gaps appear. An inverse effect is observed when thecontrast between the phases properties is increased (larger values of a and b). The primary conclusion of our analysis isdepicted in Fig. 7, demonstrating how variations in the bias electric displacement field lead to modifications in the widthand shifts in the range of the prohibited frequencies. We note that for lower values of a and b the effect of the bias electricfield becomes more evident in the sense that the shift in the BGs range is larger. From a practical viewpoint, while stoppinglarger bands of frequencies is feasible by choosing a higher contrast, lower contrasts will allow to actively adjust thedesired band at a higher precision. The analysis was concluded with two examples in which we chose phases properties tobe identical to those of commercially available DEs. The results show how tunable stop-bands are achievable by properlyadjusting the electrostatic bias field.

Appendix A. The transfer matrix

The components of the transfer matrices TðpÞ appearing in Section 3:

T ðpÞ11 ¼ TðpÞ22 ¼ T ðpÞ33 ¼ TðpÞ55 ¼ cos kðpÞhðpÞ, ð61Þ

T ðpÞ13 ¼ TðpÞ52 ¼ sin kðpÞhðpÞ=mkðpÞ, ð62Þ

T ðpÞ31 ¼ TðpÞ13 ¼ T ðpÞ52 ¼�mkðpÞ sin kðpÞhðpÞ, ð63Þ

T ðpÞ66 ¼�hðpÞ=dEðpÞ, ð64Þ

T ðpÞ44 ¼ TðpÞ66 ¼ 1, ð65Þ

with all the remaining components being zero.

G. Shmuel, G. deBotton / J. Mech. Phys. Solids 60 (2012) 1970–1981 1981

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