Dielectric elastomer composites

Journal of the Mechanics and Physics of Solids 60 (2012) 181–198

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Dielectric elastomer composites L. Tian a, L. Tevet-Deree b, G. deBotton b, K. Bhattacharya a,n a b

Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, United States The Pearlstone Center for Aeronautical Studies, Department of Mechanical Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel

a r t i c l e i n f o

abstract

Article history: Received 30 December 2010 Received in revised form 1 August 2011 Accepted 10 August 2011 Available online 18 August 2011

The coupled electromechanical response of electroactive dielectric composites is examined in the setting of small deformation and moderate electric field. In this setting, the mechanical stress depends quadratically on the electric field through a combination of material electrostriction and Maxwell stress. It is rigorously shown that the macroscopic mechanical stress of the composite also depends quadratically on the macroscopic electric field. It is further demonstrated that the effective electromechanical coupling can be computed from the examination of the uncoupled electrostatic and elastic problems. The resulting expressions suggest that the effective electromechanical coupling may be very large for microstructures that lead to significant fluctuations of the electric field. This idea is explored through examples involving sequential laminates. It is demonstrated that the electromechanical coupling – the macroscopic strain induced in the composite through the application of a unit electric field – can be amplified by many orders of magnitude by either a combination of constituent materials with high contrast or by making a highly complex and polydisperse microstructure. These findings suggest a path forward for overcoming the main limitation hindering the development of electroactive polymers. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Electroactive dielectrics EAP Electromechanical coupling Maxwell stress Homogenization

1. Introduction 1.1. Background Electroactive polymers (EAP) are soft materials that can change their shape in response to electrical stimulation. They have attracted the attention of various researchers for their potential as light-weight and flexible actuators for applications in robotic manipulators, active damping, conformal control surfaces and energy recovery. In comparison with other types of active materials such as piezoelectrics, magnetostrictive materials and shape-memory alloys, electroactive polymers can undergo large deformation (Kornbluh et al., 2000; Wax and Sands, 1999; Pelrine et al., 1992; Zhang et al., 1998), have short response time (Kornbluh et al., 2000), have lower density and higher resilience (Bar-Cohen, 2001). However, they generally have low actuation force and low mechanical energy density. Broadly speaking, there are three classes of electroactive polymers: dielectric elastomers, ionic polymers and ferroelcetric/ liquid crystal elastomers. The first class is the most developed and closest to application (Pelrine et al., 2000; Lacour et al., 2004). Roughly speaking, they actuate by squeezing a piece of elastomer between electrodes. The second class, the ionic polymers including gels and conductive polymers, actuate by the differential deformation induced by the electric-field-induced diffusion of ions. They tend to operate under small fields, but are slow and require a controlled environment. Therefore they are

n

Corresponding author. Tel.: þ1 626 395 8306; fax: þ1 626 583 4963. E-mail address: [email protected] (K. Bhattacharya).

0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2011.08.005

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limited in their application. Ferroelectric as well as liquid crystal elastomers are spontaneously polarized and deform by the reorientation of the polarization by an applied electric field (Bauer et al., 2004; Warner and Terentjev, 2003). They are extremely promising, but require careful material development. The current work concerns the first class, but also have implications for the third. While they are simple to manufacture and have a number of promising applications, dielectric elastomers are limited by the large electric fields ðH100 MV=mÞ they require for meaningful actuation. The reason for this is poor electromechanical coupling. This is governed by the ratio of dielectric to elastic modulus, and this ratio is generally limited across a broad class of materials (flexible polymers have low dielectric modulus while high dielectric modulus polymers are stiff). A number of researchers have sought to overcome this limitation by making composites (Zhang et al., 2002; Huang and Zhang, 2004; Huang et al., 2004). The experimental results are extremely encouraging, and remarkably, the observed effective electromechanical coupling is significantly larger than that can be expected from the ratio of effective dielectric to effective elastic moduli. The reason for this high enhancement was pointed out by Li (2003) and Li et al. (2004). They pointed out that the electro-mechanical coupling is nonlinear, and hence the effective behavior of the composite depends on the mean square of the electric field rather than the square mean. It follows that the contrast between component properties promotes the enhancement of the effective electromechanical coupling (deBotton and Tevet-Deree, 2006). The goal of this work is to make this insight quantitative and rigorous. Specifically, we seek to understand the effective behavior of electrostrictive composites and the effective electro-mechanical coupling in these materials. We limit ourselves to small strains. Electromechanical coupling may be broadly classified as either piezoelectric or electrostrictive based on symmetry. The piezoelectric effect is a linear (and more generally odd) coupling between the mechanical stress/strain fields and the electric fields/displacement current. Piezoelectric systems are reasonably well understood, including heterogeneous systems of polycrystals and ceramics (Benveniste, 1993). Electrostriction, in contrast, is nonlinear and the strains depend quadratically (and more generally in an even manner) on the electric field. Electrostriction can result either from intrinsic material properties or as a result of change in electrostatic interaction due to deformation. Electrostrictive materials, especially heterogeneous electrostrictive materials, are relatively less well understood and are the subject of the current work. 1.2. Model The governing equations characterizing the response of elastic dielectrics were developed in the pioneering work of Toupin (1956). We obtain the small strain model by expanding the equations under the assumption of small strain and pffiffiffi moderate electric field, i.e., assuming that the strains are OðeÞ and electric fields are Oð eÞ. This was pointed out formally by Toupin, and proved rigorously in Tian (2007) and Tian and Bhattacharya. In this small strain formulation, the displacement u and electrostatic potential j are governed by a pair of coupled partial differential equations,

r  Mrj ¼ 0,

ð1Þ

r  ðC ru þArjrjÞ ¼ 0,

ð2Þ

subject to appropriate boundary conditions. Above, M is the second order tensor of dielectric modulus, C is the fourth order tensor of elastic modulus, and A is a fourth order tensor describing the electromechanical coupling. It may be written as Aijkl ¼ Amat ijkl þMjl dik 

e0 2

dij dkl :

ð3Þ

where Amat describes the inherent material electrostriction and the rest the contribution of the change in the electrostatic field due to the deformation.1 To see the latter, note that ðAAmat Þrjrj ¼ rj  M rj

e0 2

2

9rj9 I

is the Maxwell stress. Eq. (1) above is the usual equation governing the electrostatic field in a dielectric medium. Eq. (2) describes the deformation driven by the electric field in addition to any boundary conditions. Note that the coupling is not symmetric and this comes from the fact that the two equations arise at different orders in the approximation (Tian, 2007; Tian and Bhattacharya). The equations are nonlinear; however, the first is linear in the electrostatic potential and the second in the displacement. Finally, in a medium with no material electrostriction (i.e. when Amat ¼ 0) as in dielectric elastomers, the coupling comes solely from the Maxwell stress that depends on the dielectric constant. Therefore, the displacement induced by an applied electric field depends on the ratio of the dielectric constant to the elastic modulus. We refer the reader to Tian (2007) and Tian and Bhattacharya for further details. We are interested in a heterogeneous medium where the material constants M, C and A vary spatially on a scale that is much smaller than that of the body. Here one expects the electrostatic potential and displacement to oscillate rapidly around smoothly varying macroscopic fields. We are interested in developing equations for these macroscopic fields, and understanding the nature of electrostatic coupling in these media. 1

e0 is the dielectric constant of vacuum.

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183

We show rigorously in Section 2 that in the periodic setting, these macroscopic electrostatic potential and displacement are governed by exactly the same Eqs. (1) and (2) above, but with the material constants now replaced with new effective material constants M H , C H and AH . Further, we show that the effective constants are given by MijH ¼ /Mik gkj S,

ð4Þ

H ¼ /Cijmn Gmnkl S, Cijkl

ð5Þ

AH ijkl ¼ /Amnpq gpk gql Gmnij S,

ð6Þ

where gij and Gijkl are the electrostatic and elastic concentration tensors and /  S denotes average over a representative volume element (unit cell). The concentration tensors are defined as follows. Consider a purely dielectric medium with the given modulus M, and independently consider a purely elastic medium (i.e., no electrostriction) with given modulus C. In these linear heterogeneous media, the point-wise electric field and strain depend linearly on the macroscopic electric field and strain, and the concentration tensors relate the local to the macroscopic fields: Ei ðxÞ ¼ gij ðxÞE j ,

eij ðxÞ ¼ Gijkl ðxÞe kl :

ð7Þ

The problem at hand has some similarity with that of thermoelasticity, and the pioneering work of Levin (1967). As in that formulation of thermoelasticity, our governing equations (1) and (2) are one-way coupled, i.e., we can solve for the electric field independent of the elastic moduli, and we can regard the elastrostriction as a given eigenstress or residual stress in the elasticity problem. Therefore, it is not surprising that we can obtain the effective dielectric and elastic moduli by studying the uncoupled problems and that these are given by (4) and (5). However, unlike in thermoelasticity, the problem at hand is nonlinear since the electrostriction has a quadratic dependance on the electric field. So, we cannot readily use Levin’s formula and it is not obvious that the effective equations will have the same form, and that there is a notion of effective electrostrictive coefficient. In this paper, we establish these facts. An important consequence of our result is that the effective constants may be determined by looking independently at the electrostatic and the elastic problem with no coupling. Indeed, the effective elastic and dielectric constants are those of the uncoupled media. This is not surprising for reasons mentioned above. The somewhat unexpected result is that the effective electrostrictive coefficient AH can also be obtained from the uncoupled concentration tensors. A second consequence is that the effective electrostrictive coefficient is unrelated to the effective dielectric coefficient even in a medium with no material electrostriction. Notice from (3) that in the absence of material electrostriction, the electrostrictive coefficient is directly related to the dielectric modulus. However, there is no analogous relation between the effective electrostrictive coefficient and effective dielectric coefficient. A third consequence is that the effective electrostrictive coefficient depends nonlinearly on the concentrations, and consequently can be extremely large. We explore this further in Section 3. The proof of the results above is given in Section 2, and uses the methodology of two-scale convergence. While it is rigorous, it is quite technical. Therefore, we provide a more accessible argument following Tevet-Deree (2008) (and inspired by Levin, 1967) here. We consider the periodic setting for specificity, though we can proceed similarly in the random case or with affine boundary conditions. Since Eq. (2) is linear in the displacement, we solve it by superposition of two problems, one governed by the macroscopic strain and one governed by the electric field: u ¼ ue þum where

r  C rum ¼ 0, /rum S ¼ e ,

ð8Þ

r  ðC rue þ ArjrjÞ ¼ 0, /rue S ¼ 0

ð9Þ e

for j given by the solution of Eq. (1). The second of these equations, Eq. (9), is an equation for u , and may be rewritten as

r  C rue ¼ r  ðArjrjÞ ¼ ðr  ðAggÞÞEE

ð10Þ

where we have recalled the definition of g from (7). It follows that ue is proportional to EE, and consequently we can write ~ rue ðxÞ ¼ GðxÞEE ~ Consequently, for some G. C rue þArjrj ¼ ðC G~ þAggÞEE: The left hand side of the equation above is the stress in the body solely due to the application of the electric field, and this is proportional to the square of the macroscopic electric field. It follows that the average is also proportional to EE, and this allows to define the effective coupling modulus: AH EE :¼ /C rue þ ArjrjS:

ð11Þ

We now seek to characterize this effective coupling modulus. Multiplying the first of Eq. (9) with um , integrating (averaging) over the representative volume element we obtain 0 ¼ /um  ðr  ðC rue þ ArjrjÞÞS,

ð12Þ

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0 ¼ /r  ðum  ðC rue þArjrjÞÞS/rum  ðC rue þ ArjrjÞS,

ð13Þ

0 ¼ /rum S  /C rue þ ArjrjS/C rue  rum Sþ /Arjj  rum S:

ð14Þ

Above we have used the divergence theorem and the periodic setting to obtain the last identity. The major symmetry of C, Eq. (8), the divergence theorem, the periodic setting, and the second equation of (9) implies that /C rue  rum S ¼ /rue  C rum S ¼ /rue S  /C rum S ¼ 0: Substituting this back in (14) and recalling the second of (8), we obtain

e  /C rue þ ArjrjS ¼ /Arjj  rum S: Now recall the definition of AH in (11) as well as those of g, G in (7). We conclude that AH EE  e ¼ /AggGSEE  e : Since this holds for any macroscopic electric field E and any macroscopic strain e , we conclude that relation (11) holds.

1.3. Laminates We have noted above that the effective coupling can potentially be extremely large in composite media. We explore this further in Section 3 by calculating the effective behavior of sequentially laminated composites made of two dielectric materials. A laminated composite, or a rank-one laminated composite, consists of alternating layers of two homogeneous materials. Sequentially laminated composites are obtained by iterating this procedure with one or both the layers themselves being sequentially laminated composites. If the constituent layers are rank-N laminates, then the newly constructed composite is a rank-(N þ1) laminate. We note that in this multi-hierarchical structure, it is assumed at each lamination stage that the characteristic size (thickness) of the constituent phases is an order of magnitude smaller than that of the enclosing laminate. Thanks to this assumption of scale separation, the electrical and mechanical fields are constant in each phase, and may be obtained through the solution of an algebraic problem. Consequently, it is possible to obtain the effective electromechanical coupling relatively simply for this class of composites. We note that the sequentially laminated composites of rank greater than one are not periodic, and thus the rigorous homogenization theorem of Section 2 do not strictly hold for these microstructures. However, we believe based on the arguments presented above that the theorem and the resulting formulae can be extended to a broader class of microstructures. In Section 3, we begin by examining the nature of the strains and electric fields in rank-one and rank-two laminates made of two materials. We use the insights gained from these examinations to construct examples that show extremely large electromechanical coupling in two situations. We specifically focus on the macroscopic longitudinal strain induced in one direction due to the application of a macroscopically transverse electric field consistent with experiments. Further, we limit ourselves to two constituent materials that are isotropic that have equal ratio of dielectric to shear moduli. It turns out that this ratio determines the response of a monolithic material, and thus choosing identical ratios enables us to narrow on the effective behavior of the composites. Further, many materials of interest in dielectric elastomers have similar ratio, and thus this assumption is consistent with the experimental situation. In this setting, we first demonstrate through the construction of rank two laminates that the electromechanical coupling can grow unboundedly (in a power law) with the contrast in the properties (dielectric and elastic moduli) of the constituent materials – see Fig. 5. Second, we demonstrate through the construction of a particular sequence of laminates that the electromechanical coupling can again grow unboundedly (in a power law) with the rank of the laminate – see Fig. 6. These results show that a composite material made of a compliant weak dielectric and a stiff high dielectric material with an extremely polydisperse microstructure provide vast possibilities of improving the electromechanical coupling. This is broadly consistent with experimental observations (Zhang et al., 2002; Huang and Zhang, 2004; Huang et al., 2004) and well as the observations of Li (2003) and Li et al. (2004). Indeed, an implicit understanding of these facts guided the experimental efforts. The results here reveal the extent of improvement that is possible. Further, they show that the key issue is the polydispersity of the microstructure and the resulting extreme fluctuation of the electric field instead of the near percolative nature of composites that is responsible for the large improvement in the coupling. The results presented in this paper demonstrate the exciting possibilities of electric elastomer composites. However, it is important to keep in mind the limitations of the analysis presented here. Our model is limited to small strains, and this assumption can potentially breakdown as the electrostrictive enhancement increases. So it is worth revisiting our results in the context of finite deformation. This is especially so since composite media undergoing finite deformations can suffer various instabilities. The manner in which these instabilities affect the overall behavior is an important and interesting question that remains to be examined in the future. Further, there are physical limitations such as dielectric breakdown, mechanical failure and constitutive nonlinearities. Finally, there are practical difficulties associated with the synthesis of fine-structured composites. All of these can limit the overall electromechanical coupling. Understanding these and evolving strategies for overcoming them remains a topic of current research.

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185

2. Homogenization of the small-strain dielectric model 2.1. Introduction of the main results Consider a dielectric composite with a periodic microstructure with period e occupying a domain O in RN . The dielectric, elasticity and electromechanical coupling tensors are M e ðxÞ ¼ Mðx=eÞ, C e ðxÞ ¼ Cðx=eÞ and Ae ðxÞ ¼ Aðx=eÞ, respectively. We assume that two flexible electrodes are attached to the body on two parts G0 and G1 of the boundary, and that these electrodes are maintained at the potential g0 and g1 respectively. We assume that the body is in vacuum, and therefore extend M e to all of space by setting it equal to e0 I outside O. The electrostatic field is given by 8 r  ðMe rje Þ ¼ 0 in RN \G, > > > e > < j ¼ g0 on G0 , ð15Þ e ¼g > j on G1 , 1 > > > : je 2 L2 ðRN Þ, rje 2 L2 ðRN Þ: loc We assume that the tractions are prescribed on a part G2 of the boundary while displacements are prescribed on its complement G3 . The displacement ue ðxÞ is given by 8 e e e e e > < r  ðC ru þA rj rj Þ ¼ 0 in O, ðC e rue þ Ae rje rje Þ  n ¼ f > : ue ¼ 0

on G2 ,

ð16Þ

on G3 :

We prove the following results. Theorem 2.1. In the limit e-0, je , the solution of (15), two-scale converges to j0 ðxÞ, the solution of the equation 8 r  ðMH rx j0 Þ ¼ 0 in RN \G, > > > 0x > < j ¼ g0 on G0 , j0 ¼ g 1 on G1 , > > > > : j0 2 L2 ðRN Þ, rj0 2 L2 ðRN Þ: loc

ð17Þ

In Eq. (17), MijH

8 R > < 1 Mik ðyÞgkj ðyÞ dy Y ¼ 9Y9 > :e I 0

in O, ð18Þ else,

where gkj ðyÞ ¼ dkj 

@w^ j ðyÞ @yk

ð19Þ

is the ‘‘electrostatic concentration’’ tensor and w^ j , the unit cell solution, is Y-periodic with zero average in Y satisfying   @w^ j @Mij @ Mik ðyÞ in Y: ¼  @yi @yk @yi

ð20Þ

Assuming further that rje is uniformly bounded in L4 ðOÞ, we derive the following homogenized equation for the deformation field. Theorem 2.2. In the limit e-0, ue , the solution of Eq. (16), two-scale converges to u0 ðxÞ, the solution of the equation 8 H 0 H 0 0 > < r  ðC ru þ A rj rj Þ ¼ 0 in O, H 0 H 0 0 on G2 , ðC ru þ A rj rj Þ  n ¼ f > : 0 on G3 : u ¼0

ð21Þ

In Eq. (21), H Cijkh ¼

1 9Y9

Z Y

Cijlm ðyÞGlmkh ðyÞ dy,

ð22Þ

@wkh l ðyÞ @ym

ð23Þ

where Glmkh ðyÞ ¼ dkl dhm 

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L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

is the ‘‘mechanical concentration’’ tensor and wkh , the unit cell solution, is Y-periodic with zero average satisfying ! @wkh @Cijkh @  Cijlm l in Y ¼ @yj @ym @yj

ð24Þ

and AH ijkh ¼

1 9Y9

Z Y

Almpq ðyÞGlmij ðyÞgpk ðyÞgqh ðyÞ dy:

ð25Þ

For later use, we recall the following results from Allaire (1992) and Cioranescu and Donato (1999) concerning the theory of two-scale convergence. Let O be an open set in RN and Y ¼ ½0; 1N be the closed unit cube. We denote by C1 ] ðYÞ the space of infinitely differentiable functions in RN that are periodic in Y. Then L2] ðYÞ and H1] ðYÞ are the completions of C1 ] ðYÞ with respect to L2 ðYÞ and H1 ðYÞ norms, respectively. Definition 2.1. A sequence of functions ve in L2 ðOÞ is said to two-scale converge to a limit v0 ðx,yÞ belonging to L2 ðO  YÞ if, for any function cðx,yÞ 2 DðO; C1 ] ðYÞÞ, we have Z Z Z  x lim ve ðxÞc x, dx ¼ v0 ðx,yÞcðx,yÞ dy dx:

e

e-0 O

O Y

Allaire (1992) demonstrated that the set of test functions in Definition 2.1 can be extended to an ‘‘admissible’’ test function set Aad . Thus, a function cðx,yÞ 2 L1 ðO  YÞ, periodic in y, is an admissible test function if cðx,yÞ is measurable and Z Z Z    x   lim c x,  dx ¼ 9cðx,yÞ9 dy dx: e-0 O

e

O Y

Allaire (1992) also proved that if cðx,yÞ 2 L1 ðO; C] ðYÞÞ or L1] ðY; CðOÞÞ, then cðx,yÞ 2 Aad . If cðx,yÞ ¼ c1 ðxÞc2 ðyÞ, in which c1 ðxÞ 2 Ls ðOÞ, c2 ðyÞ 2 Lr] ðYÞ, r and s satisfying 1=r þ 1=s ¼ 12, then again cðx,yÞ 2 Aad . For a two-scale convergence sequence ve the following property holds. Proposition 2.3. Let ve be a sequence of functions in L2 ðOÞ, which two-scale converges to a limit v0 ðx,yÞ 2 L2 ðO  YÞ. Then ve R converges also to vðxÞ ¼ Y v0 ðx,yÞ dy in L2 ðOÞ weakly. Furthermore, limJve JL2 ðOÞ Z Jv0 JL2 ðOYÞ ZJvJL2 ðOÞ : e-0

ð26Þ

An important result is the following (Cioranescu and Donato, 1999). Proposition 2.4. If ve ðxÞ is a bounded sequence in H1 ðOÞ that converges weakly to a limit vðxÞ in H1 ðOÞ, then ve two-scale converges to vðxÞ, and there exists a function v1 ðx,yÞ 2 L2 ðO; H1] ðYÞ=RÞ such that, up to a subsequence, rve two-scale converges to rx vðxÞ þ ry v1 ðx,yÞ. 2.2. Two-scale convergence of the electric field potential In this section, we will prove Theorem 2.1. Since the domain for Maxwell equation (15) is unbounded and the extended matrix Mðx=eÞ is not exactly periodic, we write down the entire proof although the method we use is quite standard. Proof. For each e the solution of Eq. (15) exists and satisfies Jrx je JL2 ðRN Þ rc, where c depends on MðyÞ, e0 , the boundary condition, but not on e. Now, for any integer n, je is uniformly bounded in H1 ðBn \GÞ. According to Proposition 2.4, there exists a function j0n 2 H1 ðBn \GÞ, such that je , j0n in H1 ðBn \GÞ, and another function j1n ðx,yÞ 2 L2 ðBn \G; H1] ðYÞ=RÞ such that rje two-scale converges to rx j0n þ ry j1n . Increase n, we get another j0n and j1n , but they are the same on their common region. In this way, we can find a subsequence je and two functions j0 ðxÞ 2 D1 ðRN \GÞ and j1 ðx,yÞ 2 L2loc ðRN \G; H1] ðYÞ=RÞ such that rje , rj0 in L2 ðRN Þ, and rje two-scale converges to rx j0 þ ry j0 for any test function cðx,yÞ 2 Aad that has a compact support with respect to variable x. Actually, for any compact set K  RN ,

je -j0 in L2 ðKÞ, rje -rj0 in L2 ðKÞ, rje two-scale converges to rx j0 þ ry j1 : From Proposition 2.3 it follows that for any compact set K, Jrx j0 þ ry j1 JL2 ðKYÞ r Jrje JL2 ðKÞ rJrje JL2 ðRN Þ :

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187

Therefore, j1 2 L2 ðRN \G; H1] ðYÞ=RÞ. We recall that je is the solution of Eq. (15) if for any c 2 DðRN \GÞ, Z M e wðOÞrx je  rx c dx ¼ 0:

ð27Þ

RN

0

1

0

1

Hence, for any f 2 DðRN \GÞ and f ðx,yÞ 2 DðRN \G; C1 ] ðYÞÞ, by using f ðxÞ þ ef ðx,x=eÞ as a test function in Eq. (27) 0¼

Z

Z  x  x   x  x 0 1 1 dx þ ðM e ðxÞe0 IÞrx je  rx f ðxÞ þ ry f x, þ erx f x, dx: e0 rx je  rx f0 ðxÞ þ ry f1 x, þ erx f1 x, 

RN

e

e

e

O

e

ð28Þ

1

Since erx f ðx,x=eÞ-0 in L2 and rje two-scale converges to rx j0 ðxÞ þ ry j1 ðx,yÞ, the first term of Eq. (28) converges to Z Z 1 e0 ðrx j0 ðxÞ þ ry j1 ðx,yÞÞ  ðrx f0 ðxÞ þ ry f1 ðx,yÞÞ dy dx: 9Y9 RN Y As to the second term, first rje two-scale converges to rx j0 ðxÞ þ ry j1 ðx,yÞ on the region O. Second MðyÞ 2 L1 ] ðYÞ, 1 0 1 ry f1 ðx,yÞ 2 DðRN \G; C1 ] ðYÞÞ, so MðyÞry f ðx,yÞ and HðyÞr x f ðxÞ are both in L] ðY; CðOÞÞ, and can be looked as the test functions for the two-scale convergence of rje on O, the second term becomes Z Z 1 0 1 ðHðyÞe0 IÞðrx j0 ðxÞ þ ry j1 ðx,yÞÞ  ðrx f ðxÞ þ ry f ðx,yÞÞ dy dx: 9Y9 O Y Consequently, from Eq. (28) we have that Z Z 0 1 MðyÞwðOÞðrx j0 ðxÞ þ ry j1 ðx,yÞÞ  ðrx f ðxÞ þ ry f ðx,yÞÞ dy dx ¼ 0:

ð29Þ

RN Y

Following the procedure outlined in Chapter 9 of Cioranescu and Donato (1999), the existence and the uniqueness of the 0 solution for Eq. (29) can be deduced. Choosing f  0 in Eq. (29), ry  ðMðyÞwðOÞðrx j0 ðxÞ þ ry j1 ðx,yÞÞ ¼ 0: 2 y

1

ð30Þ

1

1

2O, j ðx,yÞ  0. However, if If x= 2O, Eq. (30) becomes r j ðx,yÞ ¼ 0 with j ðx,yÞ periodic in y with zero average. Thus if x= x 2 O, Eq. (30) becomes ry  ðMðyÞry j1 ðx,yÞÞ ¼ ry  ðMðyÞrx j0 ðxÞÞ:

ð31Þ

If we define w^ j to be the unit cell solution of Eq. (20), then j1 can be expressed as

j1 ðx,yÞ ¼ wðOÞw^ j ðyÞ

@ j0 : @xj

ð32Þ

1

Next, by choosing f ðx,yÞ  0 in Eq. (29), we end up with Z rx  MðyÞwðOÞðrx j0 ðxÞ þ ry j1 ðx,yÞÞ dy ¼ 0:

ð33Þ

Y

Finally, Eq. (17) for j0 ðxÞ together with Eq. (18) for Mij0 are obtained by plugging Eq. (32) in Eq. (33) with the aid of definition (19) for the electrostatic concentration tensor. & 2.3. The local strong convergence of the electric field Assume that rje is locally uniformly bounded on O in L4 norm, then rje converges strongly on the smaller region O. This will turn out to be crucial for the derivation of the homogenized equation for the deformation field. Since this strong convergence is only valid locally on O, we cannot use analogous results in the theory of two-scale convergence. Here, we prove it by combining Tartar’s idea with two-scale convergence. Proposition 2.5. Assume there exists a compact set K*O and a constant c, such that Jrx je JL4 ðKÞ oc, then  x rje rj0 ry j1 x, -0 in L2 ðOÞ:

ð34Þ

e

N Proof. Construct function xðxÞ 2 C1 0 ðR Þ satisfying suppðxÞ  K and xðxÞ  1 in O. First we prove that there exists a constant c such that for any Y 2 ðDðK\GÞÞN ,

e

lim supJxðxÞrje xðxÞrj0 þ rw^ i Yi JL2 ðKÞ r cJry j1 þ ry w^ i Yi JL2 ðOYÞ ,

ð35Þ

e-0 e

where w^ i ðxÞ ¼ wðOÞw^ i ðx=eÞ. In fact,

e0 JxðxÞrje xðxÞrj0 þ rw^ ei Yi J2L2 ðKÞ r

Z

M

ZK

¼ Z þ O

M

x

e

rw^ i

x

e

Yi  rw^ j

x

e

K

x

M



wðOÞ xrje xrj0 þ wðOÞrw^ i

e

x

e

Z

Yj dx þ2

e

 

Yi  xrje xrj0 þ wðOÞrw^ i

x

e



Yi dx

wðOÞðxrje xrj0 Þ  ðxrje xrj0 Þ dx

M O

x

x  x ðxrje xrj0 Þ  rw^ i Yi dx:

e

e

ð36Þ

188

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

Taking HðyÞw^ i ðyÞYi Yj as the test function for the two-scale convergence of w^ j ðx=eÞ and je , in the limit e-0 the second term on the right hand side of Eq. (36) becomes Z Z MðyÞrw^ i ðyÞYi  rw^ j ðyÞYj dy dx, ð37Þ O Y

and the last term becomes Z Z 2 MðyÞxry j1  rw^ i ðyÞYi dy dx:

ð38Þ

O Y

The first term on the right hand side of Eq. (36) may be rewritten as Z Z x  x M wðOÞrje  x2 ðrje rj0 Þ dx M wðOÞrj0  x2 ðrje rj0 Þ dx e e K K Z Z x x wðOÞrje  rxðje j0 Þ2x dx M wðOÞrj0  x2 ðrje rj0 Þ dx ¼ M

e

K

e

K

Z Z 2 - MðyÞrj0  x ry j1 ðx,yÞ dy dx

ð39Þ

ð40Þ

O Y

Z Z ¼

2

MðyÞry j1  x ry j1 dy dx:

ð41Þ

O Y

Above, Eq. (39) follows from Eq. (27), Eq. (40) is because je j0 -0 in L2 ðKÞ, and Eq. (41) results from Eq. (31). We substitute Eqs. (37), (38) and (41) back in Eq. (36) to obtain e

JxðxÞrje xðxÞrj0 þ rw^ i Yi J2L2 ðKÞ r cJry j1 þ ry w^ i Yi J2L2 ðOYÞ :

ð42Þ

Now consider  2 x  ri j0  2 xðxÞrje xðxÞrj0 þ wðOÞrw^ i

e

L ðKÞ

 x 2   r xðxÞrje xðxÞrj0 þ wðOÞrw^ i Yi  2

e

1

2 y ^ i Yi JL2 ðOYÞ þ

rcJry j þ r w

L ðKÞ

 2 x    ðYi þ ri j0 Þ 2 þ rw^ i

e

L ðOÞ

 x2 X   JYi þ ri j0 J2L4 ðOÞ rw^ i  4

e

i

rcJry w^ i ðyÞri j0 þ ry w^ i ðyÞYi J2L2 ðOYÞ þ c

L ðOÞ

X JYi þ ri j0 J2L4 ðOÞ

ð43Þ

ð44Þ

i

rc

X JYi þ ri j0 J2L4 ðOÞ : i

Above, Eq. (43) results from Eq. (42), and Eq. (44) comes from Eq. (32) and the L4 boundedness of w^ i ðyÞ. If we choose a sequence Yi -ri j0 in L4 ðOÞ, then  2 x  ri j0  2 -0 as e-0: xðxÞrje xðxÞrj0 þ rw^ i

e

L ðKÞ

Finally, since xðxÞ ¼ 1 in O   x2   rje ðxÞrj0 ðxÞry j1 x,  2

e

L ðOÞ

-0

as e-0:

&

This result enables us to prove the two-scale convergence of the Maxwell stress. Consider the electromechanical N 1 coupling tensor Ae ðxÞ ¼ Aðx=eÞ, AðyÞ is Y periodic in y and is L1 bounded. For any test function vðx,yÞ 2 ðC1 0 ðO; H] ðYÞ=RÞÞ , Z Z  x   x  x dx ¼ lim Ae rje rje rx j0 ry j1 x, v x, dx lim Ae rje rje v x, e-0 O e e-0 O e e Z Z   x  x   x  x v x, dx ¼ lim Ae rje rx j0 þ ry j1 x, v x, dx þ lim Ae rje rx j0 þ ry j1 x, e-0 O e-0 O e e e e Z Z 1 0 1 0 1 AðyÞðrx j þ ry j ðx,yÞÞðrx j þ ry j ðx,yÞÞvðx,yÞ dy dx, ð45Þ 9Y9 O Y in which Eq. (45) results from Proposition 2.5. The last step is because rj0 rj0 two-scale converges to itself, and we can use AðyÞrw^ j rw^ i vðx,yÞ as the test function for this convergence since AðyÞrw^ j rw^ i vðx,yÞ 2 L1] ðY; CðO ÞÞ.

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

189

2.4. The two-scale convergence of the deformation field Before proving Theorem 2.2, we recall some basic notation, assumption and existence result in homogenization theory of elasticity (Cioranescu and Donato, 1999). In component form Eq. (2) is  8  @ @u @j @j > > ¼0 cijkh k þAijkh > > @xj @xh @xk @xh > <  @u @j @j nj ¼ fi cijkh k þ Aijkh > > > @x @x > h k @xh > : u¼0

in O, on G2 ,

ð46Þ

on G3 :

Definition 2.2. Let a1 , a2 2 R such that 0 o a1 o a2 , and let O be an open set of RN . We denote by Me ða1 , a2 , OÞ the set of the fourth order tensor C ¼ ðcijkh Þ satisfying 8 cðxÞijkh 2 L1 ðOÞ 8i,j,k,h ¼ 1, . . . ,N, > > > > < cijkh ¼ cijhk ¼ ckhij 8i,j,k,h ¼ 1, . . . ,N, 8 x 2 O, > a1 9x92 rC xx > > > : 9C x9 r a 9x9 2

8symmetric matrix x, 8matrix x:

Define the space V ¼ fv9v 2 H1 ðOÞ, v ¼ 0 on G3 g. Set V ¼ ðVÞN , and equip V with norm !1=2 N X Jrvi J2L2 ðOÞ , JvJV ¼ i¼1

then V is a Hilbert space with inner product ðu,vÞV ¼

N X

ðrui , rvi ÞL2 ðOÞ

8u,v in V:

i¼1

Thanks to the symmetries of C 2 Me ða1 , a2 , OÞ, for any f ¼ ðf1 , . . . ,fN Þ 2 ðH1=2 ðG2 ÞÞN the weak form of Eq. (16) is Z Z CðxÞeðuÞeðvÞ dx þ AðxÞrjrjrv dx ¼ /f ,vSðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN , O

ð47Þ

O

for any v 2 V, where   1 @ui @uj þ eij ðuÞ ¼ 2 @xj @xi is the linearized strain tensor. Define the bilinear form Z Lu ðu,vÞ ¼ CðxÞeðuÞeðvÞ dx, O

and note that it is a bounded bilinear map since CðxÞ 2 L1 ðOÞ. Moreover, since CðxÞ 2 Me ða1 , a2 , OÞ, Z a1 9eðvÞ92 dx rLu ðv,vÞ, 8v 2 V: O

From the second Korn’s inequality, Z 2 Lu ðv,vÞ Z a1 9eðvÞ9 dx ZcJvJ2H1 ðOÞ : O

In addition, since f 2 ðH1=2 ðG2 ÞÞN  V 0 and AðxÞrjrj 2 L2  V 0 , Lax–Milgram theorem applies and a unique solution of Eq. (47) exists and satisfies JuJV rcðJf JðH1=2 ðG2 ÞÞN þ JAðxÞrjðxÞrjðxÞJL2 ðOÞ Þ:

ð48Þ

Now, let us prove Theorem 2.2 in this framework. Proof. Since rx je is locally uniformly bounded in L4 norm, from Eq. (48), ue is uniformly bounded in ðH1 ðOÞÞN . Thus we can find a subsequence and a function u0 such that ue -u0 in ðH1 ðOÞÞN . Moreover, there exists u1 ðx,yÞ 2 L2 ðO; H1] ðYÞ=RÞN such that up to a subsequence, rue two-scale converges to rx u0 þ ry u1 ðx,yÞ. Now consider N v0 ðxÞ 2 ðC1 0 ðOÞÞ

and

N 1 v1 ðx,yÞ 2 C1 0 ðO; H] ðYÞ=RÞÞ ,

190

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

then vðxÞ ¼ v0 ðxÞ þ ev1 ðx,x=eÞ 2 ðH10 ðOÞÞN . Using this as a test function for Eq. (47), we get Z Z C e ðxÞeðue Þeðve Þ dx þ Ae ðxÞrje rje rve dx ¼ /f ,ve SðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN : O

First, Z

ð49Þ

O

C e ðxÞeðue Þeðve Þ dx ¼ O

Z

C e ðxÞeðve Þeðue Þ dx: O

Second, eðue Þij ¼ 12 ð@uei =@xj þ@uej =@xi Þ two-scale converges to ! 1 @u0j @u1 @uj 1 @u0i þ þ i þ ¼ ðex ðu0 ÞÞij þðey ðu1 ÞÞij , 2 @xj @xi @yj @yi where ðex ðu0 ÞÞij ¼

! @u0j 1 @u0i þ , 2 @xj @xi

ðey ðu1 ÞÞij ¼

! @u1j 1 @u1i þ : 2 @yj @yi

2

Since CðyÞ 2 ðL1 ðYÞÞN , usage of C e eðve Þ as the test function for the two-scale convergence of eðue Þ results in Z Z Z 1 CðyÞðex ðu0 Þ þ ey ðu1 ÞÞðex ðv0 Þ þ ey ðv1 ÞÞ dy dx: lim C e ðxÞeðue Þeðve Þ dx ¼ e-0 O 9Y9 O Y From the two-scale convergence of the Maxwell stress (Section 2.3) we have that Z Z Z  x 1 dx ¼ AðyÞðrx j0 þ ry j1 ðx,yÞÞðrx j0 þ ry j1 ðx,yÞÞðrx v0 ðxÞ þ ry v1 ðx,yÞÞ dy dx: lim Ae rje rje rv x, e-0 O e 9Y9 O Y In addition, lim/f ,ve SðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN ¼ /f ,v0 SðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN : e-0

Hence, by passing to the limit e-0 in Eq. (49) we finally get Z Z 1 CðyÞðex ðu0 Þ þ ey ðu1 ÞÞðex ðv0 Þ þ ey ðv1 ÞÞ dy dx 9Y9 O Y Z Z 1 AðyÞðrx j0 þ ry j1 ðx,yÞÞðrx j0 þ ry j1 ðx,yÞÞðrx v0 ðxÞ þ ry v1 ðx,yÞÞ dy dx þ 9Y9 O Y ¼ /f ,v0 SðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN :

ð50Þ

Let us show that (50) is a variational equation in the space Hu :¼ ½H1 ðOÞN  ½L2 ðO; H1] ðYÞ=RÞN , and that the hypotheses of the Lax–Milgram theorem are fulfilled. Indeed, endowing the space Hu with the norm JVJ2Hu ¼ Jv0 J2ðH1 ðOÞÞN þJv1 J2ðL2 ðO;H1 ðYÞ=RÞÞN , ]

8V ¼ ðv0 ,v1 Þ 2 Hu ,

the bilinear form defined by Z Z 1 CðyÞðex ðu0 Þ þ ey ðu1 ÞÞðex ðv0 Þ þ ey ðv1 ÞÞ dy dx Lu ðU,VÞ ¼ 9Y9 O Y is continuous on Hu . Since CðyÞ 2 Me ða1 , a2 ,YÞ, Z Z X N a ðex ðu0 Þij þey ðu1 Þij Þ2 dy dx: Lu ðU,UÞ Z 9Y9 O Y i,j ¼ 1 Observe that Z Z X N

ð51Þ

Z Z ðex ðu0 Þij þ ey ðu1 Þij Þ2 dy dx ¼ Jex ðu0 ÞJ2L2 ðOÞ þ Jey ðu1 ÞJ2L2 ðOYÞ þ 2 ex ðu0 Þij ey ðu1 Þij dy dx:

O Y i,j ¼ 1

O Y

The last term in the above equation vanishes since Z Z Z Z Z Z Z Z @u1 @ ex ðu0 Þij ey ðu1 Þij dy dx ¼ ex ðu0 Þij i dy dx ¼ ðex ðu0 Þij u1i Þ dy dx ¼ ex ðu0 Þij u1i nj dSy dx ¼ 0: @yj O Y O Y O Y @yj O @Y

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

191

Above, the first equality is due to the symmetry of ex ðu0 Þij , the third equality follows from the divergence theorem, and the last one results from the periodicity of u1 ðx,yÞ with respect to variable y. Therefore, Eq. (51) becomes Lu ðU,UÞ Z

a 9Y9

ðJex ðu0 ÞJ2L2 ðOÞ þ Jey ðu1 ÞJ2L2 ðOYÞ Þ Z cðJu0 J2ðH1 ðOÞÞN þJu1 J2ðL2 ðO;H1 ðYÞ=RÞN Þ ¼ cJUJ2Hu : ]

This demonstrates the coerciveness of Lu ðU,UÞ. Furthermore, the following mappings are linear and continuous on Hu , 1 : ðv0 ,v1 Þ-/f ,v0 SðH1=2 ðG2 ÞÞN ,ðH1=2 ðG2 ÞÞN ; Z Z AðyÞðrx j0 þ ry j1 ðx,yÞÞðrx j0 þ ry j1 ðx,yÞÞðrx v0 ðxÞ þ ry v1 ðx,yÞÞ dy dx: 2 : ðv0 ,v1 ÞO Y

Now, we can apply Lax–Milgram theorem to obtain the existence and the uniqueness of ðu0 ,u1 Þ 2 Hu as a solution of Eq. (50). Choosing first v0  0 and afterwards v1  0, we conclude that Eq. (50) is equivalent to the following problem: 8 ry  ðCðyÞry u1 Þ ¼ ry  ðCðyÞrx u0 þAðyÞðrx j0 þ ry j1 Þðrx j0 þ ry j1 ÞÞ, > > > R R > < rx  CðyÞðrx u0 þ ry u1 Þ dy ¼ rx  AðyÞðrx j0 þ ry j1 Þðrx j0 þ ry j1 Þ dy, Y Y R

0 1 0 1 0 1 > on G2 , > Y CðyÞðr x u þ r y u Þ þ AðyÞðr x j þ r y j Þðrx j þ ry j Þ dy n ¼ f > > : u ¼ 0 on G : 3

ð52Þ

Recalling that

ry j1 ¼ 

N X

ry w^ j

j¼1

@j0 , @xj

then  0  0  @j @ j1 @j @j1 þ þ @xa @ya @xb @yb  0

AðyÞðrx j0 þ ry j1 Þðrx j0 þ ry j1 Þ ¼ AðyÞijab ¼ Aijab

 0  0 @j @w^ @j0 @j @w^ @j  k  h @xa @ya @xk @xb @yb @xh

¼ Aijab gak gbh

@j0 @j0 : @xk @xh

ð53Þ

kh ~ kh ¼ ðw~ kh ~ kh Let wkh ¼ ðwkh 1 , . . . , w N Þ be the Y-periodic with zero 1 , . . . , wN Þ be the zero average periodic solution of Eq. (24), and w average solution of ! @Bijkh @ @w~ kh l  Cijlm in Y, ð54Þ ¼ @yj @ym @yj

where, for convenience, we define Bijkh ðyÞ ¼ Aijab ðyÞgak ðyÞgbh ðyÞ:

ð55Þ

kh

In terms of wkh and w~ , u1 ðx,yÞ in Eq. (52) can be expressed as u1 ðx,yÞ ¼ wkh ðyÞ

@u0k @ j0 @ j0 ðxÞ þ w~ kh ðyÞ ðxÞ ðxÞ: @xh @xk @xh

ð56Þ

We note that to resolve the precise distribution of the fields within the unit cell Y the periodic solutions w^ k , wkh and w~ kh must be determined. However, in order to obtain the homogenized solution it is sufficient to determine w^ k and wkh . Thus, substituting Eq. (56) into the second of Eq. (52), we get " # ! Z @u0l @wkh @u0k @w~ kh @ 1 @j0 @j0 @j0 @j0  Cijlm ðyÞ þ l þ l dy ¼ 0: ð57Þ þ BðyÞijkh @xj 9Y9 Y @xm @ym @xh @ym @xk @xh @xk @xh Consider the third term in Eq. (57). The periodicity requirement together with the symmetry of C result in Z Z @Clmij ðyÞ kh @w~ kh Cijlm ðyÞ l dy ¼  w~ l dy: @ym @y m Y Y From Eq. (24) for wijp it follows that ! Z Z Z @wijp @w~ kh @ @ l Cijlm ðyÞ dy ¼ Clmpq ðyÞ w~ kh l dy ¼ @ym @yq Y Y @ym Y @yq

! @w~ kh l Clmpq ðyÞ wijp dy, @ym

192

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

where in the second equality once again the periodicity of the functions was exploited. Finally, from Eq. (54) for w~ kh l , Z Z Z ij kh @Bpqkh ðyÞ ij @w @w~ Cijlm ðyÞ l dy ¼  wp dy ¼ Bpqkh ðyÞ p dy: @y @y @yq m q Y Y Y Upon substitution of the last equality back in Eq. (57) we end up with " # ! ! Z @wkh @u0k @wijp @j0 @j0 @ 1  Cijlm dkl dhm þ l þBpqkh dip djq þ dy ¼ 0: @xj 9Y9 Y @ym @xh @yq @xk @xh

ð58Þ

H We complete the proof with the homogenized Eq. (21) for u0 ðxÞ by defining Cijkh and AH ijkh as in Eqs. (22) and (25), respectively. &

3. Sequentially laminated composites 3.1. Formulation A simple laminated composite, denoted a rank-1 laminate, is constructed by layering two materials in an alternating manner as shown in Fig. 1(a). A rank-2 laminate is constructed by layering a rank-1 laminate with another rank-1 laminate, or with one of the original phases, as illustrated in Fig. 1(b). A sequentially laminated composite (SLC) of rank-N is constructed by iterating this procedure N times. It is assumed that the characteristic size of the layers at each successive lamination step is an order of magnitude larger than the size of the layers in the previous rank. This assumption of ‘‘scale separation’’ leads to a solution that involves a constant field within each layer of the laminate. Thus, the effective coupling tensor defined by Eq. (25) may be obtained by the following exact expression: n X

AH ijkl ¼

ðrÞ lðrÞ Amnpq GðrÞ g ðrÞ g ðrÞ , mnij pk ql

ð59Þ

r¼1 ðrÞ

where l are the volume fractions of the phases composing the laminate. We note that expression (59) was obtained in Tevet-Deree (2008) by following a method which is reminiscent of the one introduced in Levin (1967) for the coupled thermoelastic problem in heterogeneous materials. ðiÞ ðmÞ ðiÞ Consider a rank-1 laminate made out of two anisotropic phases with volume fractions l and l ¼ 1l , respectively. ð1Þ ð1Þ ^ ^ The normal to the layers is n , and the unit vector along the interface is m (Fig. 1(a)). The mean electric field in the laminate is ðiÞ

E ¼ ð1l ÞE

ðmÞ

ðiÞ ðiÞ

þl E :

ð60Þ

The continuity of the electric potential requires that ðE E

ðmÞ

where a

E

ð1Þ

E

ðmÞ

E

ðiÞ

ðiÞ

¼ að1Þ ðE  n^

ð1Þ

Þn^

ð1Þ

ðmÞ

ðiÞ

^ E Þ  m

ð1Þ

¼ 0. Alternatively, this may be expressed in the form

,

ð61Þ

is a scalar. From Eqs. (60) and (61) it follows that ðiÞ ð1Þ ¼ E þ að1Þ l n^ , ðiÞ

¼ Eað1Þ ð1l Þn^

ð1Þ

:

ð62Þ

When the interface is charge-free, the continuity condition on the electric displacement field is ðMðmÞ E

ðmÞ

ðiÞ

M ðiÞ E Þ  n^

ð1Þ

¼ 0:

ð63Þ

matrix

nˆ

1

ˆ 1 m

inclusion 1

x2

core

x1

2

Fig. 1. A rank-1 (a) and a rank-2 (b) laminated composites.

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

193

Eqs. (62) and (63) can be solved to find

að1Þ ¼

ðMðmÞ M ðiÞ Þn^ ðiÞ

ð1Þ

E

ðiÞ

ðl M ðmÞ þð1l ÞM ðiÞ Þn^

ð1Þ

 n^

ð1Þ

ð64Þ :

Substituting Eq. (64) in Eq. (62), and recalling the definition of g ðrÞ , it follows that ð1Þ n^  ðM ðmÞ M ðiÞ Þn^

ðiÞ

g ðmÞ ¼ I þ l

ð1Þ

ð1Þ ð1Þ ðiÞ ðiÞ ðl MðmÞ þ ð1l ÞM ðiÞ Þn^  n^ ,

ð1Þ n^  ðM ðmÞ MðiÞ Þn^

ðiÞ

g ðiÞ ¼ Ið1l Þ

ðiÞ

ðiÞ

ðl M ðmÞ þ ð1l ÞM ðiÞ Þn^

ð1Þ

ð1Þ

 n^

ð1Þ

ð65Þ :

The effective dielectric tensor of the rank-1 laminate is then ðmÞ

M ð1Þ ¼ l

ðiÞ

M ðmÞ g ðmÞ þ l M ðiÞ g ðiÞ :

ð66Þ

An analogous procedure is followed for the mechanical problem. Since the strain fields are uniform in each layer, the macroscopic strain tensor is

e ¼ ð1lðiÞ ÞeðmÞ þ lðiÞ eðiÞ :

ð67Þ

The displacement continuity condition may be written as ^ ð1Þ  ðeðmÞ eðiÞ Þm ^ ð1Þ ¼ 0: m

ð68Þ

Hence, in each phase ðiÞ ðiÞ ð1Þ ^  n^ ð1Þ , ^ ð1Þ  n^ ð1Þ þ n^ ð1Þ  m ^ ð1Þ Þ þ oð1Þ eðmÞ ¼ e þ oð1Þ 1 l ðm 2 l n ðiÞ ðiÞ ^ ð1Þ  n^ ð1Þ þ n^ ð1Þ  m ^ ð1Þ Þoð1Þ ^ ð1Þ  n^ ð1Þ , eðiÞ ¼ e oð1Þ 1 ð1l Þðm 2 ð1l Þn

ð69Þ

ð1Þ where oð1Þ 1 and o2 are scalars. The traction continuity condition implies that

ðC ðmÞ eðmÞ C ðiÞ eðiÞ Þn^

ð1Þ

¼ 0:

ð70Þ ð1Þ

ð1Þ

^ The scalar products of this expression with n^ and m give two equations from which explicit expressions for oð1Þ 1 and ð1Þ oð1Þ can be obtained. Note that these are linear in the average strain e . Substituting these expressions for oð1Þ 2 1 and o2 back in Eq. (69) yields a linear relation between the strain in the phases and the average strain, and this gives the elastic concentration tensors GðrÞ . The effective elasticity tensor of the rank-1 laminate is ðmÞ

C ð1Þ ¼ l

ðiÞ

C ðmÞ GðmÞ þ l C ðiÞ GðiÞ :

ð71Þ ð1Þ

The expression for the macroscopic electromechanical coupling tensor A is obtained by combining the results for the electrostatic and the mechanical problems in Eq. (59). ðmÞ In the limit of incompressible phases oð1Þ and pðiÞ need to be added to the 2 ¼ 0, arbitrary hydrostatic pressures p expression for the stress in each phase (deBotton and Hariton, 2002). In this limit,

oð1Þ ¼

^ ðC ðmÞ C ðiÞ Þe m ðiÞ

ðiÞ

^ ½ðl C ðmÞ þ ð1l ÞC ðiÞ Þm

ð1Þ

ð1Þ

 n^

 n^

ð1Þ

ð1Þ

^ m

ð1Þ

 n^

ð1Þ

ð72Þ :

Consider next a rank-2 laminate (Fig. 1(b)) consisting of layers of the former rank-1 laminate as the core phase together with layers of another phase that we denote the shell. We follow the prescription described above for the rank-1 laminate and determine the constants að2Þ and oð2Þ , the effective dielectric and elastic tensors, and the corresponding macroscopic electromechanical coupling tensor Að2Þ . We iterate this procedure for higher rank laminates. Importantly, if a rank-ðN1Þ laminate with effective electromechanical coupling tensor AðN1Þ is used as the core phase in a rank-N laminate, then the effective electromechanical coupling tensor of the newly formed composite is ðNÞ ðN1Þ ðN1Þ ðN1Þ ðN1Þ ðsÞ ðsÞ ðsÞ ðsÞ Amnpq Gmnij gpk gql þ l AðsÞ mnpq Gmnij gpk gql ,

¼l AðNÞ ijkl ðN1Þ

ðsÞ

ðN1Þ

ð73Þ

and l ¼ 1l are the volume fractions of the core and the shell phases, and g ðN1Þ , g ðsÞ , GðN1Þ and GðsÞ are where l the appropriate electrostatic and elastic concentration tensors of the core and the shell phases that are determined independently from the uncoupled electrostatic and elastic problems. Iterative application of Eq. (66), (71) and (73) allows the determination of the macroscopic dielectric moduli, elastic moduli, and the electromechanical coupling tensor of sequentially laminated composites.

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Fig. 2. Transverse actuation strain of a two-phase rank-1 laminate normalized by the actuation strain of the phases as a function of the volume fraction lð1Þ of the stiff phase and lamination angle yð1Þ .

3.2. Electrically induced strain in rank-1 and rank-2 laminates A typical experiment in a dielectric elastomer consists of application of an electric field with no external tractions, and measurement of the induced strain. This electrically induced strain may be obtained from Eq. (2) to be e ¼ C 1 A : E  E. For a homogenous isotropic incompressible material with no material electrostriction, this strain depends simply on the ratio between the dielectric and the shear moduli. Thus, two such materials exhibit the same ‘‘electrically induced strain response’’ if the ratios between these two moduli are the same for both materials. Here we exploit this observation and consider only two-phase sequentially laminated composites in which the ratios between the dielectric and the shear moduli of the two phases are identical (i.e., M ðiÞ =mðiÞ ¼ MðmÞ =mðmÞ ). This is broadly consistent with many experimental observations. More importantly, under this assumption, any difference between the macroscopic strain responses of the composites and those that would develop in the phases are solely due to the heterogeneity and the spatial arrangement of the phases. Finally, following common experimental practice, we examine the longitudinal induced or ‘‘actuation’’ strains ðe 11 Þ due to a transverse field ðE 2 Þ (e.g., Bhattacharya et al., 2001; McMeeking and Landis, 2005). We begin with a rank-1 laminate made out of a stiff phase with a high dielectric constant and a compliant phase with a low dielectric constant. Specifically, we choose dielectric moduli Mð1Þ ¼ 103 e0 and M ð2Þ ¼ 10e0 , and shear moduli mð1Þ ¼ 103 MPa and mð2Þ ¼ 10 MPa. Fig. 2 shows the longitudinal strain e11 as a function of the lamination angle y and ð1Þ volume fraction of the stiff more dielectric material l . In this figure, as well as below, the strain is normalized by the corresponding strain eðhomÞ that would be seen in a homogeneous material subjected to the same electric field.2 Fig. 2 11 shows that the longitudinal strain strongly depends on the lamination angle y, but only a little on the volume fraction of the phases. More accurately, the actuation strain depends on the relative angle between the direction of the applied electric field and the interface. This is because the rank-1 laminate is highly anisotropic, and the actuation strain in some directions is larger than the corresponding strain in homogeneous materials while it is smaller in others. The maximal ð1Þ strain is achieved at l ¼ 0:5 and y ¼ 0:66p, and is 12% larger than eðhomÞ .3 11 Consider next a rank-2 laminate (Fig. 1(b)). As shown in Fig. 3, there are two ways of constructing this laminate. In the first shown in Fig. 3(a) (and hereafter called ‘‘tree (a) laminate’’), we laminate a rank-1 laminate with material 1 to obtain a particulate microstructure of the stiff and high dielectric inclusions surrounded by compliant and low dielectric matrix. In the second shown in Fig. 3(b) (tree (b) laminate), we laminate a rank-1 laminate with material 2 to obtain a particulate microstructure with compliant and low dielectric inclusions in a stiff with high dielectric matrix. There are two independent structural parameters for each level in the trees shown in Fig. 3, the volume fraction ðlÞ and the lamination ð1Þ angle ðyÞ. Thus, the four parameters that define the microstructure of the rank-2 composite are: l the volume fraction of ð2Þ ð1Þ the soft phase in the core, l the volume fraction of the core in the composite, y the lamination angle of the layers in the ð2Þ core laminate, and y the lamination angle of the layers in the final rank-2 composite. These four parameters are shown in Fig. 1 and schematically represented on the two trees shown in Fig. 3. Along the tree (a) laminate, the volume fraction of ð1Þ ð2Þ the stiff inclusion phase is ð1l Þl , while along the tree (b) laminate the volume fraction of the compliant inclusion ð1Þ ð2Þ phase is l l . We begin with material parameters M ð1Þ ¼ 103 e0 , M ð2Þ ¼ 10e0 , mð1Þ ¼ 103 MPa and mð2Þ ¼ 10 MPa so that the contrast ratio is 102 . We numerically compute the macroscopic-induced strain in the x1 direction due to an applied macroscopic electric

2 For the particular choice of dielectric and shear moduli, the strain in the homogeneous material eðhomÞ ¼ 2:21% in response to an applied electric 11 field E2 ¼ 100 MV=m. 3 The strain is 2.48% for an applied field of E2 ¼ 100 MV=m.

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

θ(2) (2)

(2)

θ(1) (1)

(2)

λ

λ

θ(2)

θ(1)

λ

(1)

λ

(2)

(1)

(1)

195

(1)

(2)

Fig. 3. Two trees representing two ways of constructing a rank-2 laminate with two phases (phase one being stiff and phase two being compliant) together with schematic representation of the four structural parameters.

Table 1 The structural parameters at which the best actuation strains attained for rank-2 laminates. M ð1Þ =M ð2Þ ¼ mð1Þ =mð2Þ

lð1Þ

Tree a

10 102 103 104 105

0.569 0.531 0.584 0.690 0.759

Tree b

10 102 103 104 105

0.769 0.984 0.992 0.992 0.992

yð1Þ (deg)

lð2Þ

yð2Þ (deg)

e 11 =eðhomÞ 11

61.9 60.3 63.1 62.8 60.1

0.819 0.964 0.992 0.997 0.992

21.4 14.6 27.5 42.3 52.9

1.15 4.66 37.43 225.56 559.59

 57.4  79.3  67.6  63.2  61.0

0.506 0.023 0.010 0.009 0.008

48.4 31.5 32.0 30.1 30.0

1.23 4.42 25.97 102.80 286.88

ðsÞ

ðsÞ

field in the x2 direction for each value of the microstructure parameters l and y . We then vary the parameters systematically till we find the maximum of the induced strain.4 We find that the optimal value of (normalized) strain is 4.66 and 4.42 for the tree (a) and tree (b) laminates respectively. The corresponding optimal values of the structural parameters are listed in Table 1. We observe that we have a four-fold increase in the possible actuation strain for a rank-2 laminate compared to a homogenous material or an optimal rank-1 laminate. To understand this further, we increase the contrast ratio to 103 (M ð1Þ ¼ 104 e0 , Mð2Þ ¼ 10e0 , mð1Þ ¼ 104 MPa and mð2Þ ¼ 10 MPa). The optimal (normalized) induced strains increase to 37.43 and 25.97 respectively for the tree (a) and tree (b) laminates with the structural parameters shown in Table 1. A detailed examination of the electric and the strain field provide a good insight into the significant enhancement we observe with rank-2 lamination and with contrast. We focus on tree (a) and assume a large contrast ratio. Fig. 4 schematically represents the electric field in the tree (a) laminate and the deformation schematically represents the ensuing strains. The core (rank-1 laminate) consists of a stiff, high dielectric (red) and compliant, low dielectric material (white) material. The shell (green) consists of a compliant low dielectric material. Since the contrast is large, the effective dielectric constant of the core rank-1 laminate is significantly higher than that of the shell layer. Therefore, most of the electric field is concentrated in the low dielectric shell (green), and thus its magnitude is inversely proportional to its volume fraction. Quantitatively, let us adapt formulas (Eqs. (62) and (64)) for the rank-2 laminate on hand. We need to change the superscript (1) to (2) (representing the rank-2 laminate). Further, the superscript ðiÞ refers to the core rank-1 laminate and superscript (m) refers to the shell layer. In our situation, the MðiÞ bM ðmÞ . Thus, we see from Eq. (64) that

að2Þ

MðiÞ n^

1 ðiÞ

1l

ð2Þ

ð2Þ M ðiÞ n^

E

ð2Þ  n^ :

ð2Þ Substituting this in Eq. (62) and looking at a macroscopic field in the n^ direction, we see that

EðmÞ

E 1l

ðiÞ

¼

E ð2Þ

1l

,

as we argued earlier. ð2Þ It follows that we can make this field large by making l large. In this situation, we can view the rank-2 laminate as a stack of (green) micro-actuators. The large electric field in this micro-actuators squeezes them and tries to make them expand laterally. This would be resisted by the core rank-1 laminate that contains the stiff phase if it were internally 4 We note that the variation of this strain with the structural parameters is highly non-concave and so there is always the possibility that a numerical search misses the global optimum. We have attempted to avoid this to the best of our abilities. Further, even if our search miss the optimum, it provides lower bounds and the insights we seek.

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layered parallel to the rank-2 layers. However, an angle between the layering directions allows the rank-2 laminate to effectively shear through the rotation of the stiff phase as shown in Fig. 4. In summary, there are two important elements to the enhancement of the electromechanical response of a tree (a) rank-2 laminate. First, the ratio of the less dielectric layer has to be small and second, the core laminate containing the stiff layer has to have a soft deformation mode consistent with the induced deformation of the less dielectric layer. These insights will prove useful as we study even higher-rank laminates later. The tree (b) rank-2 laminate obtains enhancement in a similar manner – field concentration and soft deformation modes, though the resulting geometric attributes are different. 3.3. Strain enhancement due to contrast The study of rank-2 laminates shows that contrast between the two constituents exploited properly can lead to a significant enhancement of effective electromechanical coupling. Fig. 5 shows the (normalized) macroscopic-induced electric field in the x1 direction due to an applied macroscopic electric field along the x2 direction for various contrast ratio and for both tree (a) and tree (b). For each case, the structural parameters were optimized to obtain the maximum macroscopic strain; the optimal values are listed in Table 1. The squares in Fig. 5 correspond to rank-2 laminates with stiff inclusions (tree (a)) while the circles to rank-2 laminates with compliant inclusions (tree (b)). It is clear from the figure that the induced strain increases according to some power (close to 1/2) of the contrast for both types of rank-2 laminates though the tree (b) laminates seem to be growing slower. We do not know if this power-law will persist for even larger contrasts, and the reason for the specific power. These issues defy a simple analysis since the optimal structural parameters change with contrast, and remain an interesting question for the future. In any case, our results clearly establish the promise of high contrast composites. 3.4. Strain enhancement due to microstructural complexity The marked ability of rank-2 laminates to intensify the electromechanical coupling motivates the study of higher order laminates. We consider two-phase composites with a contrast ratio of 103; i.e., M ð1Þ =Mð2Þ ¼ mð1Þ =mð2Þ ¼ 103 . There are

Fig. 4. Schematic representation of the fields in a rank-2 laminate, involving lateral expansion of the soft (green) layers together with shear of the core rank-1 laminate (red-white layers). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

1000 Stiff inclusions (tree a) Soft inclusions (tree b)

_

11

/

(hom) 11

100

10

1 1

10

100 (1)

1000 (2)

(1)

104

105

(2)

h /h = μ /μ

Fig. 5. The transverse actuation strain of rank-2 laminates normalized by the actuation strain of the phases as functions of the contrast between the phases moduli.

L. Tian et al. / J. Mech. Phys. Solids 60 (2012) 181–198

197

numerous ways to construct high rank laminates, and two possible trees are shown in Fig. 6 along with the corresponding microstructural parameters. Unfortunately, these are too many parameters to systematically optimize. Therefore, we rely on the insights we gained from the study of rank-2 laminates and make specific choices. Consider the tree Fig. 6(a) which begins with the ‘‘circled’’ rank-2 laminate shown in Fig. 1(b). Based on the insights of ð1Þ ð2Þ Section 3.2, we set the opening angle between the two lamination directions to be 451 with y ¼ 451 and y ¼ 901. We ð2Þ recall that in Section 3.2 l was chosen to be large to ensure that the intensity of the electric field in the layers of the soft and less dielectric phase (2) will be large. In the current hierarchical structure such a choice will lead to an opposite result ð2Þ since large l means that the overall dielectric response of the circled rank-2 composite is high. Consequently, the contrast between the dielectric responses of the circled rank-2 composite and the layers of the stiffer high-dielectric phase ð3Þ ð3Þ (1) to be laminated at the next stage (at a lamination angle y and volume fraction ð1l Þ) will be small. It turns out that ð2Þ to ensure the amplification of the electric field in the circled rank-2 composite we must choose small l . We now layer ð3Þ ð3Þ this rank-2 laminate with the stiff, high dielectric phase (1) with y ¼ 451 and small volume fraction l to end up with a rank-3 laminate in which there is a large contrast between the dielectric responses of the stiff layers and the compliant rank-2 laminate. We take this rank-3 laminate and then laminate it with the compliant, low dielectric phase (2) with yð4Þ ¼ 901 and high volume fraction ð1lð4Þ Þ (to ensure high contrast of dielectric responses at the next step). We continue ðJÞ ðJÞ this process, laminating with the stiff, high dielectric phase (1) with y ¼ 451 and small volume fraction l for odd J, and ðJÞ with the compliant, low dielectric phase (2) with y ¼ 901 and small volume fraction for even J. We find out that the ðJÞ simple choice l ¼ l0 for both odd and even J with small l0 leads to significant amplification of the electromechanical coupling. The resulting effective electromechanical coupling, specifically the (normalized) longitudinal strain in the x1 direction due to an applied electric field in the transverse direction is shown in Fig. 7 as the square symbols for various l0 . The lines drawn in the figure are exponential fits of the form ðhomÞ e 11 =e11 ffi k1 ek2 N ,

ð74Þ

θ(Ν)

(odd-N)

θ(Ν)

λ

λ(even-N)

(2)

(1)

(1) (3)

(2)

λ

θ(3)

(1)

(2)

λ

(2)

θ(1)

(1)

λ

(1)

(2)

θ(2)

(3)

θ(3)

(2)

θ(2)

(1)

λ

(1)

(2)

λ

θ(1)

(2)

(1)

λ

(2)

Fig. 6. Two trees representing two possible ways to construct two-phase rank-N laminates together with schematic representation of the corresponding structural parameters. Trees a and b correspond to odd and even rank laminates, respectively.

10

20

10

16

10

12

_

11

/

(hom) 11

0=

0.005

8

10

0=

4

0.00015

10

0=

0

10

1

5

9

13

0.15

17 Rank

21

25

29

Fig. 7. Transverse actuation strain of two-phase rank-N laminates normalized by the actuation strain of the phases as a function of the lamination rank.

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where k1 and k2 are the curve fit parameters. It is clear that the electromechanical coupling increases exponentially with the rank of lamination. The reason for this exponential increase is simple to understand. As the rank of lamination increases, the internal electric field is amplified exponentially giving rise to a finer and finer set of microactuators subjected to higher and higher fields. At the same time, there is enough flexibility at each layer for the compliant phase to deform by the rotation of the stiff phase. We note from Fig. 7 that the rate k2 of exponential growth depends sensitively on l0 . Our numerical exploration suggests that the best growth rate is k2 ffi 1:79 for the value l0 ¼ 0:005. Decreasing the volume fraction to l0 ¼ 0:00015 makes the composite considerably stiffer and drops the growth to k2 ffi 0:61. Conversely, increasing the volume fraction to l0 ¼ 0:15 reduces the amplification of the electric field and drops the growth to k2 ffi 0:71. The results for tree (b) are similar, and shown as the filled circles in Fig. 7. Tree (b) is constructed in a manner similar to ðNÞ tree (a), but with one additional lamination step with layers of the soft phase at volume fraction l ¼ 0:5. We again have exponential growth, but the results trail those of tree (a). Finally, we note that the high-rank composites considered in this section have not been systematically optimized even with the class of sequentially laminated composites. Nonetheless, they clearly make the case that it is possible to dramatically increase the electromechanical coupling by orders of magnitude by increasing the microstructural complexity.

Acknowledgments This work draws from the doctoral thesis of Tian at the California Institute of Technology and Tevet-Deree at the Ben-Gurion University. This work was supported by the United States-Israel Binational Science Foundation (Grant no. 2004146) as well as the US National Science Foundation (ITR Grant ACI-0204932). References Allaire, G., 1992. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518. Bar-Cohen, Y. (Ed.), 2001. Electroactive Polymer (EAP) Actuators as Artificial MusclesSPIE Press, Bellingham, WA. Bauer, F., Fousson, F., Zhang, Q.M., Lee, L.M., 2004. Ferroelectric copolymers and terpolymers for electrostrictors: synthesis and properties. IEEE Trans. Dielectr. Electr. Insul. 11 (2), 293–298. Benveniste, Y., 1993. Universal relations in piezoelectric composites with eigenstress and polarization-fields 1. Binary media—local-fields and effective behavior. J. Appl. Mech. Trans. ASME 60, 265–269. Bhattacharya, K., Li, J.Y., Xiao, Y., 2001. Electromechanical models for optimal design and effective behavior of electroactive polymers. In: Bar-Cohen, Y. (Ed.), Electroactive Polymer (EAP) Actuators as Artificial MusclesSPIE Press, pp. 309–330 (Chapter 12). Cioranescu, D., Donato, P., 1999. An Introduction to Homogenization. Oxford University Press, New York. deBotton, G., Hariton, I., 2002. High-rank nonlinear sequentially laminated composites and their possible tendency towards isotropic behavior. J. Mech. Phys. Solids 50, 2577–2595. deBotton, G., Tevet-Deree, L., 2006. Electroactive polymer composites - analysis and simulation. In: Armstrong, W.D. (Ed.), Smart Structures and Materials 2006: Active Materials: Behavior and Mechanics, Proceedings of SPIE, vol. 6170, San Diego, CA, pp. 2401–2410. Huang, C., Zhang, Q.M., 2004. Enhanced dielectric and electromechanical responses in high dielectric constant all polymer percolative composites. Adv. Func. Mater. 14 (5), 501–506. Huang, C., Zhang, Q.M., deBotton, G., Bhattacharya, K., 2004. All organic dielectric percolative three component composite materials with high electromechanical response. Appl. Phys. Lett. 84 (22), 4391–4393. Kornbluh, R., Pelrine, R., Pei, Q., Oh, S., Joseph, J., 2000. Ultrahigh strain response of field-actuated elastomeric polymers. In: Bar-Cohen, Y. (Ed.), Smart Structures and Materials 2000: Electroactive Polymer Actuators and Devices (EAPAD), Proceedings of SPIE, vol. 3987; 2000, pp. 51–64. Lacour, S.P., Prahlad, H., Pelrine, R., Wagner, S., 2004. Mechatronic system of dielectric elastomer actuators addressed by thin film photoconductors on plastic. Sensors Actuators A: Phys. 111, 288–292. Levin, V.M., 1967. Thermal expansion coefficients of heterogeneous materials. Mekh. Tverd. Tela 2 (1), 88–94 (English translation—Mechanics of Solids 2(1):58–61). Li, J.Y., 2003. Exchange coupling in p(VDF-TRFE) copolymer based all-organic composites with giant electrostriction. Phys. Rev. Lett. 90, 217601–217604. Li, J.Y., Huang, C., Zhang, Q.M., 2004. Enhanced electromechanical properties in all polymer percolative composites. Appl. Phys. Lett. 84, 3124–3126. McMeeking, R.M., Landis, C.M., 2005. Electrostatic forces and stored energy for deformable dielectric materials. J. Appl. Mech. Trans. ASME 72, 581–590. Pelrine, R., Eckerle, J., Chiba, S., 1992. Review of artificial muscle approaches. In: Proceedings of the Third International Symposium on Micro Machine and Human Science, Japan. Pelrine, R., Kornbluh, R., Pei, Q.B., Joseph, J., 2000. High speed electrically actuated elastomers with strain greater than 100%. Science 287 (5454), 836–839. Tevet-Deree, L., 2008. Electroactive Polymer Composites—Analysis and Simulation. Ph.D. Thesis, Ben-Gurion University. Tian, L., 2007. Effective Behavior of Dielectric Elastomer Composites. Ph.D. Thesis, California Institute of Technology. Tian, L., Bhattacharya, K. Small-strain models for electroactive polymers. Preprint. Toupin, R.A., 1956. The elastic dielectric. J. Ration. Mech. Anal. 5, 849–915. Warner, M., Terentjev, E.M., 2003. Liquid Crystal Elastomers. Oxford University Press, Oxford, UK. Wax, S., Sands, R., 1999. Electroactive polymer actuators and devices. In: Bar-Cohen, Y. (Ed.), Smart Structures and Materials 1999: Electroactive Polymer Actuators and Devices (EAPAD), Proceedings of SPIE, vol. 3669; 1999, pp. 2–10. Zhang, Q., Bharti, V., Zhao, X., 1998. Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly(vinylidene fluoridetrifluoroethylene) copolymer. Science 280, 2101–2104. Zhang, Q.M., Li, H., Poh, M., Xia, F., Cheng, Z.Y., Xu, H., Huang, C., 2002. An all organic composite actuator material with a high dielectric constant. Nature 419, 284–287.