Analysis of multilayer electro-active spherical balloons

Accepted Manuscript

Analysis of multilayer electro-active spherical balloons Eliana Bortot PII: DOI: Reference:

S0022-5096(16)30870-5 10.1016/j.jmps.2017.02.001 MPS 3057

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Journal of the Mechanics and Physics of Solids

Received date: Revised date: Accepted date:

30 November 2016 20 January 2017 3 February 2017

Please cite this article as: Eliana Bortot, Analysis of multilayer electro-active spherical balloons, Journal of the Mechanics and Physics of Solids (2017), doi: 10.1016/j.jmps.2017.02.001

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Analysis of multilayer electro-active spherical balloons Eliana Bortot

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Faculty of Mechanical Engineering, Technion - Israel Institute of Technology, 32000 Haifa, Israel

Abstract

An electro-active spherical balloon is susceptible to electromechanical instability which, for certain material models, can trigger substantial size change. Hence, the electro-active balloon can conveniently be employed for application as actuator or generator. Practical applications, however, require proper

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electrode protection from aggressive agents and electric safety. For this purpose, the active membrane can be sandwiched between two soft protective passive layers. In this paper, the theory of nonlinear electro-elasticity for heterogeneous soft dielectrics is applied to the investigation of the electromechanical response of multilayer electro-active spherical balloons, formed either by the active membrane only (single-layer balloon) or by the coated active membrane (multilayer balloon). Numerical results showing the influence of the soft passive layers on the electromechanical response of the active membrane are

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presented.

Keywords: Finite electro-elasticity, Electro-active polymers, Dielectric elastomers, Electromechanical

1. Introduction

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instability, Composite material, Multilayer electro-active balloon, Passive layers.

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Spherical elastomeric balloons exhibit a well known nonlinear behavior, characterized by a nonmonotonic relation between the inflation pressure and the balloon volume. Since the 70’s, when Alexan-

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der (1971) presented the first experimental results following the theory by Green and Adkins (1960), many studies have been conducted on this topic, see, e.g., Haughton and Ogden (1978); Johnson and Beatty (1995); deBotton et al. (2013).

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Among Electro-Active Polymers (EAPs), a broad class of materials able to undergo large strains

when subjected to an electric stimulus, Dielectric Elastomers (DEs) are a new promising group (Pelrine et al., 1998, 2000). Their peculiar properties, such as lightness, reliability, fast response and low cost,

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make these materials particularly attractive for the realization of electromechanical transducers (Schlaak et al., 2016; Madsen et al., 2015; Carpi and Smela, 2009; Carpi et al., 2008). These devices can be simply Email address: [email protected] (Eliana Bortot)

Preprint submitted to Elsevier

February 3, 2017

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obtained by coating the surfaces of a dielectric elastomer membrane with compliant electrodes. When a voltage is applied between the electrodes, the elastomeric membrane reduces its thickness and expands its area. Dielectric elastomer transducers are susceptible to electromechanical instability, due to the fact 15

that, as the voltage increases, the elastomeric layer thins down inducing in turn a higher electric field. This effect is positive, but it may conversely result in a catastrophic thinning of the elastomer (Zurlo

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et al., 2016; Keplinger et al., 2012) and in the failure of the device. The electromechanical instability strongly depends on the material model and on the boundary conditions (Plante and Dubowsky, 2006; Bertoldi and Gei, 2011; De Tommasi et al., 2013a; Gei et al., 2014; Khan et al., 2013; Ask et al., 2015; 20

Wang et al., 2016).

The particular configuration of a spherical balloon, consisting of a layer of soft dielectric material, is especially interesting for the realization of both actuators and generators. The apex of the balloon is

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under equi-biaxial stress; this condition enhances the electric-induced deformation and maximizes the energy density (Huang et al., 2013; Bortot and Gei, 2015). As the elastomeric balloon is characterized 25

by mechanical instability, so the soft dielectric balloon is characterized by electromechanical instability. For certain material behaviors, this instability can be exploited to trigger substantial change in the volume of the balloon, the so-called snap-through phenomenon. For these reasons, the behavior of a single-layer electro-active balloon has been investigated in many works (Mockensturm and Goulbourne,

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2006; Wissler and Mazza, 2007; Zhu et al., 2010; Rudykh et al., 2012; Li et al., 2013; Vertechy et al., 2013; De Tommasi et al., 2013b; Dorfmann and Ogden, 2014), proposing its application as a pump or

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as an energy harvesters.

These possible applications suggest, from a practical viewpoint, the need to ensure proper electrode protection from aggressive agents and electrical safety. For this purpose, the active membrane can be

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sandwiched between two passive layers providing opportune insulation. This issue, being related to the safety and the service life of the electromechanical transducers, is not at all trivial and it has began to

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gain attention in the last years. Calabrese et al. (2016) proposed a wearable active bandage capable of dynamically modulating the pressure exerted on the limbs. The bandage consists of two active layers embedded between two soft passive layers; experimental and numerical investigations showed that the

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passive layers not only ensure electrical safety, but also play a key role in the transmission of actuation

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from the active layers to the load. A recent paper by Atashipour and Sburlati (2016) deals with the analysis of a coated spherical piezoceramic sensor. In order to extend the working life of the sensor, the authors proposed the application of a protective coating layer and investigated the electro-elastic response of the coated sensor in the framework of small-strain electro-elasticity. Whereas for single-layer electro-active balloons some experimental investigations have been conducted

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(Fox and Goulbourne, 2008; Ahmadi et al., 2013; Li et al., 2013), at the best of our knowledge, multilayer

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balloons have not been yet designed or tested. The aim of this work is to investigate the electromechanical behavior of an elastomeric multilayer electro-active balloon and, more specifically, to understand how the passive layers with their mechanical characteristics influence the nonlinear electro-elastic response of the spherical active membrane. 50

The application of inner and outer coatings to the spherical active membrane implies the presence

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of interfaces. Since the single-layer electro-active balloon itself is a membrane coated with compliant electrodes, the interface failure will take place at the electrodes on the passive layer side, or on the active layer one. However, throughout this work, we make the simplifying assumption of perfectly bonded interfaces. 55

The paper is organized as follows. In Section 2, we briefly recall the nonlinear theory of electroelasticity for heterogeneous soft dielectrics. Section 3 is dedicated to the modeling of the electromechan-

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ical response of the spherical electro-active balloon. In Subsection 3.1, the response to an electromechanical excitation of an electro-active balloon, made up of a single active membrane, is examined. Then, in Subsection 3.2, the electromechanical behavior of a multilayer balloon, realized by embedding the 60

active membrane between two soft passive layers, is analyzed. In order to understand the influence of the passive layers on the response of the active membrane, we have conducted a numerical investigation of the electromechanical response of single-layer and multilayer balloons. The results of the numerical

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investigation are presented in Section 4. The results, presented at first in non-dimensional form, are specialized in Subsection 4.1 to an active membrane made up of a commercial dielectric elastomer, the VHB-4910. Finally, concluding remarks are pointed out in Section 5.

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2. Finite electro-elasticity

In this section, we provide a summary of the equations governing the nonlinear deformation of heterogeneous soft dielectrics, following the approach by Maugin (1988); Dorfmann and Ogden (2005);

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McMeeking and Landis (2005); Suo et al. (2008). Throughout this work, the standard notation of continuum electromechanics is adopted and quasi-static electromechanic conditions are considered. Examine a heterogeneous electro-elastic body, occupying the volume region B 0 ⊂ R3 , composed of

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n perfectly bonded homogeneous phases, B 0 = ∪nj=1 Bj0 . A generic interface between phases j and k

0 0 0 (j, k = 1, ..., n) is denoted by ∂Bin(j,k) , so that ∂Bin = ∪j,k ∂Bin(j,k) denotes the set of all internal

interfaces between the heterogeneities in B 0 . The body is separated from the surrounding vacuum,

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R3 − B 0 , by its external boundary ∂B 0 . The motion of such multi-phase electro-elastic body is described by a sufficiently smooth function χ(X, t), mapping a reference point X in the undeformed configuration

B 0 to its deformed position x = χ(X, t) in the current configuration B at time t. Hence, the associated deformation gradient tensor with respect to the reference configuration B 0 is defined by F = Gradχ. The 4

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ratio of an infinitesimal volume element in the current configuration dv on its counterpart in the reference 80

configuration dV is equal to the determinant of the deformation gradient tensor, J = dv/dV = det F. For an incompressible material the deformation gradient must satisfy the constraint J = det F = 1. The right and left Cauchy-Green strain tensors, providing a measure of the deformation, are defined as C = FT F and B = FFT , respectively.

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The quantities of interest to define the current electrostatic state of the dielectric are the electric field E and the electric displacements D. Electromagnetic interactions are governed by Maxwell equations. Under the hypotheses of electrostatics, the local form of Maxwell equations with respect to the current configuration B reduces to curlE = 0,

divD = 0.

(1)

The first of equations (1) implies that the electric field is conservative and, therefore, it can be derived

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as the gradient of an electrostatic potential ϕ, specifically, E(x) = −gradϕ(x).

In the quasi-static case, the balance of linear momentum with respect to the current configuration B reads

divσ + ρf = 0 ,

(2)

where ρ is the current mass density, f is the mechanical body force vector, and σ is the total stress

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tensor, incorporating mechanical and electrical stresses. In absence of mechanical body force, Eq. (2) reduces to the equilibrium equation in the current configuration B

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divσ = 0 .

(3)

Across the outer boundary of the body ∂B the following jump conditions take place

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(σ − σ ? )n = tm ,

(D − D? ) · n = −ωe ,

(E − E? ) × n = 0,

(4)

where n is the outward current unit normal vector, tm is a prescribed mechanical traction, ωe is the

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surface charge density, D? is the outer electric displacement field, and σ ? is the Maxwell stress outside

(5)

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the body defined in terms of the outer electric field E? as   1 σ ? = 0 E? ⊗ E? − (E? · E? )I , 2

here I is the identity tensor and 0 is the vacuum permittivity (8.854 pF/m).

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Across an internal charge-free boundary ∂Bin(a,b) between two phases (a) and (b) the jump conditions

read

[[σ]]n = 0,

[[D]] · n = 0,

[[E]] × n = 0,

(6)

where [[•]] = (•)(a) −(•)(b) denotes the jump operator, and where the outward current unit normal vector n points from (a) towards (b). 5

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In general, it is advantageous to recast the electro-elastic problem in Lagrangian formulation by 105

appropriate pull-back operations. To this end, we define the total first Piola-Kirchhoff stress, the nominal electric displacement and electric fields, respectively, as D0 = JF−1 D.

E0 = FT E,

Thus, the governing equations (1) and (3) turn to CurlE0 = 0,

DivD0 = 0,

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P = J σF−T ,

DivP = 0.

(7)

(8)

Accordingly, the jump conditions across the outer boundary of the body (4) become (P − P? )n0 = tM ,

(D − D0? ) · n0 = −ωE ,

(E0 − E0? ) × n0 = 0,

(9)

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conditions across an internal surface (6) read [[P]]n0 = 0,

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where tM dA = tm da, ωE dA = ωe da and n0 is the outward referential unit normal vector. The jump

[[D0 ]] · n0 = 0,

[[E0 ]] × n0 = 0.

(10)

Following the formulation proposed by Dorfmann and Ogden (2005), the total first Piola-Kirchhoff stress and the nominal electric displacement field for an incompressible material are derived from an

∂W − p0 F−T , ∂F

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P=

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augmented energy density function W (F, E0 ) via

D0 = −

∂W , ∂E0

(11)

where p0 is an unknown Lagrange multiplier associated with the incompressibility constraint, that can be determined only by imposing the boundary conditions.

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3. Finite electro-elastic deformation of electro-active spherical balloons

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This section is devoted to the modeling of the electromechanical response of a spherical electro-active balloon. Following Rudykh et al. (2012); Dorfmann and Ogden (2014), we model first the behavior of an

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electro-active balloon made up of a single active membrane, that is a single-layer electro-active balloon. 120

Then, we specialize the model to a multilayer balloon, formed by embedding the active membrane between two soft passive layers. 3.1. Single-layer electro-active balloon Consider a thick-walled spherical balloon made up of an isotropic, hyperelastic and incompressible elastomeric membrane, whose inner and outer surfaces are coated with compliant electrodes (Fig. 1).

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The balloon is characterized in its undeformed configuration B 0 by inner and outer radii Ri and Ro , 6

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Figure 1: Three-dimensional sketch of a single-layer electro-active spherical balloon in the reference configuration.

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respectively; the initial thickness is thereby H = Ro − Ri (Fig. 2a). Here and hereafter, we indicate with the notations (•)i and (•)o quantities related to the inner and outer surfaces, respectively. The balloon

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undergoes a finite deformation due to a combination of i ) a pressure Pi applied to the inner surface, and ii ) a radial electric field induced by applying an electric potential ∆φ between the stretchable electrodes 130

on its surfaces (see Fig. 2b).

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The resultant deformation is described by the mapping

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r = R3 − Ri3 + ri3


1/3

,

θ = Θ,

ψ = Ψ,

(12)

where (R, Θ, Ψ ) and (r, θ, ψ) are the referential and the current spherical polar coordinate systems,

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respectively. With respect to the aforementioned coordinate systems, the principal stretches are λr =

dr = dR



R r

2

,

λθ =

r , R

λψ =

r . R

(13)

The deformation gradient matrix, hence, admits the diagonal representation

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  F = diag λ−2 , λ, λ ,

with λ=

r = R



1+

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ri3 − Ri3 R3

1/3

(14)

.

(15)

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Figure 2: Section of the single-layer electro-active spherical balloon in the (a) reference and (b) current configurations.

For later use, we define the stretches at the inner and outer surfaces of the deformed balloon as

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λi = ri /Ri and λo = ro /Ro , respectively. These stretches are mutually related via (16)

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(λ3o − 1) = t3 (λ3i − 1),

where t = Ri /Ro is the radius ratio of the balloon, such that 0 < t < 1; the limit cases t → 0 and t → 1 correspond to a spherical cavity in an infinite medium and to a thin-walled spherical shell, respectively. For the purpose of describe the balloon electromechanical behavior, we choose to derive λo as a function

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λo = (1 + t3 (λ3i − 1))1/3 .

(17)

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of λi

Under the hypothesis of ideal dielectric behavior, we assume that the constitutive response of the

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dielectric elastomer is governed by a Gent strain-energy function " # tr(FT F) − 3  µJm 0 W (F, E ) = − 1− − F−T E0 · F−T E0 , 2 Jm 2

(18)

where µ is the shear modulus of the material,  = 0 r is the strain independent permittivity with r the

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relative dielectric constant, and Jm is the dimensionless locking parameter related to the strain-stiffening

exhibited by polymers stemming from the limited extensibility of the polymer chains. In the limit of Jm → ∞ the Gent model reduces to the neo-Hookean one.

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The total stress resulting from Eq. (18) is µ

σ=

T

1−

tr(F F)−3 Jm

B + E ⊗ E − p0 I,

(19)

in the neo-Hookean limit it turns to σ = µB + E ⊗ E − p0 I. The balloon is electrically loaded by connecting to a battery the electrodes coating its inner and

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outer surfaces. In this way, the voltage ∆φ = ϕ(ro ) − ϕ(ri ) is applied through the balloon thickness and a radial electric field is induced, Er = −∂ϕ/∂r. Thereby, the radial electric field, satisfying Maxwell equations and the electric boundary condition, is given by Er =

∆φ ri ro . r2 ri − ro

(20)

For next use, it can be advantageous to rewrite Eq. (20) as a function of the stretch

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(1 − t)tλi λo Ro2 , tλi − λo R2 λ2

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Er = Er0

(21)

where Er0 = ∆φ/H is the nominal radial electric field.

Due to the spherical symmetry of the deformation, the only component of the equilibrium equation (3) not identically satisfied is

dσrr σrr − σθθ +2 = 0. dr r

(22)

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Integration of Eq. (22), making use of the mechanical boundary conditions at the inner and outer surfaces σrr (ri ) = −Pi and σrr (ro ) = 0, leads to the following relation between the pressure and the electro-mechanical deformation

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σrr (ro ) − σrr (ri ) = Pi =

Z

ro

ri

2

σθθ (r) − σrr (r) dr. r

(23)

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In order to solve the integral in Eq. (23), it is advantageous to operate a change of variable according

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to

dr 1 dλ =− 3 . r λ −1 λ

(25)

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Eq. (24) allows to rewrite Eq. (23) in the alternative form Z λo σrr (λ) − σθθ (λ) σrr (ro ) − σrr (ri ) = Pi = 2 dλ. λ(λ3 − 1) λi

(24)

Hence, integrating Eq. (25) along with Eq. (17), we obtain the internal pressure Pi as a function of the

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stretch at the inner surface λi and of the nominal radial electric field Er0 , namely λo 6 µJm X 2 ln(λ − ηj )ηj3 − ln(λ − ηj ) − 3 ln(λ − ηj )ηj − Jm ln(λ − ηj )ηj Pi = − 2 j=1 3(ηj3 − ηj ) − Jm ηj λi   λo 1 (t − 1)2 (t4 λ4i − λ4o ) 0 2 + 2µJm ln + 2 Er , λi 2t λ2i λ2o (tλi − λo ) 9

(26)

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where ηj are the six roots of the polynomial equation 1 − (3 + Jm )λ4 + 2λ6 = 0. In the neo-Hookean limit, Eq. (26) simplifies to Pi =

µ 2





1 + 4λ3i 1 + λ3o − 4 λo λ4i

+

1 (t − 1)2 (t4 λ4i − λ4o ) 0 2 Er . 2t2 λ2i λ2o (tλi − λo )

(27)

The behavior of an electro-active balloon is characterized by electromechanical instability.

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The internal pressure is a function of the applied electric field and of the stretch at the inner surface, 2

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Pi = Pi (Er0 , λi ). While this function is monotonic with respect to the electric field, it is non-monotonic with respect to the stretch. At a given electric field, hence, the onset of electromechanical instability corresponds to the maximum of the function Pi (λi ). Thus, deriving Eq. (26) with respect to λi at 2

constant Er0 and imposing the result equal to zero, we can obtain the electric field as a function of the 2

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critical stretch at the inner surface, Er0 (λci ), namely 2 dPi (Er0 , λi ) 2 = 0 −→ Er0 (λci ). 02 dλi

(28)

Er

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Substituting back the resulting expression into Eq. (26), we obtain the electromechanical instability equation, connecting the critical pressure Pic and the critical stretch at the inner surface λci .

(29)

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In the neo-Hookean limit1 , employing Eq. (27), we obtain the following relation "


5 + 4t3 λ3ci − 1 1 + 4λ3ci − 1 + tλco (λco λci + tλ2ci ) + t3 2λ3ci − 1 + Pic =µ − 4 4 2λci 2λco #

λco 1 + λ3ci + (2t3 + t6 (λ3ci − 1)) (λco − λci )λ6ci − λco , λ4co λ4ci (−1 + t2 λco λ2ci + t3 (2 − t3 + tλ2co λ4ci + t2 λco λ2ci (λ3ci − 1))) where λco is a function of λci , according to Eq. (17).

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Vice versa, inverting Eq. (26), the electric field can be written as a function of the internal pressure 2

2

and of the stretch at the inner surface, Er0 = Er0 (Pi , λi ). While this function is monotonic with respect to the pressure, it is non-monotonic with respect to the stretch. At a given internal pressure, the onset

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2

of electromechanical instability corresponds, thereby, to the maximum of the function Er0 (λi ). Thus, inverting Eq. (26), deriving the resultant expression with respect to λi at constant Pi and imposing the

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result equal to zero, we can obtain the internal pressure as a function of the critical stretch at the inner

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surface, Pi (λci ), namely

2 dEr0 (Pi , λi ) dλi

Pi

= 0 −→ Pi (λci ).

(30)

Substituting back, we obtain the electromechanical instability equation, connecting the critical electric 2

field Er0 c and the critical stretch at the inner surface λci . 1 The

relations for the Gent model are not given here for conciseness.

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In the neo-Hookean limit, inverting Eq. (27), we obtain the following relation

λco 1 + λ3ci + (2t3 + t6 (λ3ci − 1)) (λco − λci )λ6ci − λco 2µt2 (λco − tλci ) 2 , Er0 c = (t − 1)2 λ2co λ2ci (−1 + t2 λco λ2ci + t3 (2 − t3 + tλ2co λ4ci + t2 λco λ2ci (λ3ci − 1)))

(31)

where λco is a function of λci , according to Eq. (17). 190

This method is essentially an extension of the one proposed by Haughton and Ogden (1978) for

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the mechanical instability. Alternatively, electromechanical stability can be analyzed with other two methods. The criterion proposed by Zhao and Suo (2007), according to the principle of minimum energy, is based on the loss of positive definiteness of the Hessian matrix of the system free-energy. Note that this method is suitable only for isotropic electro-elastic response. The criterion proposed by 195

Bertoldi and Gei (2011), using the theory of incremental electro-elasticity, detects bifurcation along the principal deformation path and is based on the loss of positive definiteness of the tangent electro-elastic

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constitutive tensor. This latter method is more general and suitable also for anisotropic electro-elasticity. 3.2. Multilayer electro-active balloon

Consider next a thick-walled multilayer spherical balloon, formed by coating the active spherical 200

membrane with two soft protective passive layers (see Fig. 3). In this way the active membrane is insulated from the surrounding space.

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The three layers are perfectly bonded, and each one is constituted by an homogeneous, hyperelastic, incompressible material, whose response is described by the strain energy function (18). In the reference (a)

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configuration, the active layer is characterized by inner and outer radii Ri

(a)

and Ro . The active

membrane surfaces are coated with compliant electrodes; then the layer is embedded between two passive (pi)

layers, so that the inner passive layer is characterized by outer radii Ro (po)

(a)

= Ro .

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layer by inner radii Ri

(a)

= Ri , and the outer passive

Here and henceforth, we indicates with superscripts (a), (pi) and (po) quantities related to the active,

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inner and outer passive layers, respectively. (l)

(l)

Each layer l is characterized by a radius ratio t(l) = Ri /Ro . Since at the interface the layers share

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the same radius, these ratios reads (a)

t(a) =

Ri

, (a)

Ro

(pi)

t(pi) =

Ri

(a)

Ri

(a)

,

t(po) =

Ro

(po)

Ro

.

(32) (pi)

Besides the layer radius ratios t(l) , we define the multilayer balloon radius ratio as T = Ri

(po)

/Ro

.

Taking into account the layer radius ratios (32), the balloon radius ratio reads (pi)

T =

Ri

(po)

Ro

= t(a) t(pi) t(po) ,

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(33)

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Figure 3: Section of the multilayer electro-active spherical balloon, obtained by embedding the active membrane between two soft passive layers, in the (a) reference and (b) current configurations.

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and it is subjected to the condition T > 0, so as to exclude the limit case of spherical cavity in an infinite medium2 . On the basis of the radius ratios (32), we can define for each layer (l) the reference thickness (a)

H

(a)

= (1 − t

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H (l) as a function of the outer radius of the active layer Ro , namely (a)

)Ro(a) ,

H

(pi)

(pi)

= (1 − t

)t

(a)

Ro(a) ,

H

(po)

=



1 t(po)

 − 1 Ro(a) .

(34)

Since the cavity of the balloon must not vanish, the thickness of the inner passive layer must be smaller (a)

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than the inner radius of the active membrane, that is H (pi) < Ri

(pi)

and, consequently, Ri

> 0.

The multilayer balloon is deformed by applying a potential difference ∆φ across the active layer thickness, and a pressure Pi to the surface of the balloon cavity, i.e., to the inner surface of the inner

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passive layer. At the bounds of the multilayer balloon, the mechanical boundary conditions read (pi)

σrr (ri

σrr (ro(po) ) = 0.

) = −Pi ,

(35)

Each layer undergoes a deformation according to the mapping (12). Perfect bonding between the layers requires that, at each active-passive layer interface, the two layers

share the same stretch (a)

(po)

λ(pi) = λi , o 2 In

λi

= λ(a) o .

(36)

the numerical analysis to follow, the multilayer balloon radius ratio has been kept larger than 10%, namely T > 0.1.

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Furthermore, at the active-passive layer interfaces, continuity of tractions requires that (a)

(po)

σrr (ro(pi) ) = σrr (ri ),

σrr (ro(a) ) = σrr (ri

).

(37)

The equilibrium of each layer is governed by Eq. (22). Making use of the boundary (35) and of the interface conditions (37), integrating Eq. (22) for each layer and summing each contribution, we obtain

Pi =

Z

ro(a)

(a) ri

2 (a) (a) (σ − σrr )dr + r θθ

Z

ro(pi)

2 (pi) (pi) (σ − σrr )dr + r θθ

(pi) ri

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the following relation between the pressure and the deformation of the multilayer balloon Z

ro(po)

(a) ri

2 (po) (po) (σ − σrr )dr. r θθ

(38)

As for the single-layer electro-active balloon, it is advantageous to rephrase Eq. (38) in terms of the stretch Pi =

Z

λ(a) o (a)

(a)

2

λi

(a)

σrr − σθθ dλ + λ(λ3 − 1)

Z

λ(pi) o (pi)

λi

(pi)

2

(pi)

σrr − σθθ dλ + λ(λ3 − 1)

Z

(po) λo

(po)

2

(a)

λi

(po)

σrr − σθθ dλ. λ(λ3 − 1)

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(39)

Through Eq. (16) and of Eq. (36), we obtain the following additional relations for the passive layers (pi)

λi

= (1 + t(pi)

−3

(a) 3

(λi

3

− 1))1/3 ,

3

λ(po) = (1 + t(po) (λ(a) − 1))1/3 . o o

(40)

Eq. (39) can thus be integrated to yield the internal pressure Pi as a function of the electromechanical (a)

response of the active layer, i.e., as a function of the stretch at inner surface of the active layer λi

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and of the nominal radial electric field Er0 . Otherwise, the internal pressure Pi can be described as a (pi)

function of the stretch at the inner surface of the inner passive layer λi

, giving us a measure of the

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deformation of the multilayer balloon cavity.

The electromechanical instability of the multilayer balloon can be examined by following the same procedure introduced for the single-layer balloon.

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Indeed, as for the single-layer balloon, the internal pressure is a function of the applied electric field 2

(a)

and of the stretch at the inner surface of the active layer, Pi = Pi (Er0 , λi ). While this function

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is monotonic with respect to the electric field, it is non-monotonic with respect to the stretch. At a given electric field, hence, the onset of electromechanical instability corresponds to the maximum of the (a)

function Pi (λi ). Thus, after integration of Eq. (39), deriving the resultant expression with respect to (a)

2

at constant Er0 and imposing the result equal to zero, we can obtain the electric field as a function

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λi

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2

(a)

of the critical stretch at the inner surface of the active layer, Er0 (λci ), namely 2 (a) dPi (Er0 , λi ) 2 = 0 −→ Er0 (λci(a) ). (a) 02 dλ i

(41)

Er

Substituting back the result into Eq. (39), we obtain the electromechanical instability equation, con(a)

necting the critical pressure Pic and the critical stretch at the inner surface of the active layer λci .

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In the neo-Hookean limit3 , we obtain the following relation for the critical internal pressure

(a) 3

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Pic

( (a) 3 (po) 3 µ(po) /µ(a) (1 + 4λco ) 1 (a) 1 + 4λco = µ + + (a) 4 (po) 4 2 λco λco   (a) 3 (µ(pi) /µ(a) − 1) 1 + 4λci (a) (a) (a) 2 (a) 2 + 2(λco + tλci )(λco + t2 λci ) (a) 4 λci " (a) 3 (a) 3 (pi) 3 (µ(po) /µ(a) + µ(pi) /µ(a) )λco λci (1 + λci )  − 2 (a) 4 (a) 2 3 (a) 3 (a) 3 4 (a) 2 (a) 4 6 (a) (a) 6 (a) 6 (pi) 7 λci t(pi) λco − t(a) λco λci − 2t(a) λco λci − t(a) λco λci + t(a) λco λci (a) 3

(µ(pi) /µ(a) − 1)λco (1 + λci )

(42)

− 2 (a) 4 (a) 2 3 (a) 3 (a) 3 4 (a) 2 (a) 4 6 (a) (a) 6 − t(a) λco λci − 2t(a) λco λci − t(a) λco λci + t(a) λco λci   #) 3 3 (po) 3 (po) 7 (a) 3 (a) 7 (a) 3 λci t(a) λco (1 + λco ) + µ(po) /µ(a) λco t(po) (1 + λco ) ,   2 (a) 4 (a) 2 3 (a) 3 (a) 3 4 (a) 2 (a) 4 6 (a) (a) 6 (a) 6 (a) 4 (po) 7 λco − t(a) λco λci − 2t(a) λco λci − t(a) λco λci + t(a) λco λci λco λco (a)



(a) 6

λco

(pi)

(po)

where λco , λci , and λco 250

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(a) 4

λci

(a)

are functions of λci , according to Eqs. (17) and (39).

Vice versa, the electric field can be written as a function of the internal pressure and of the stretch 2

2

(a)

at the inner surface of the active layer, Er0 = Er0 (Pi , λi ). While this function is monotonic with respect to the pressure, it is non-monotonic with respect to the stretch. At a given pressure, the onset 2

(a)

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of electromechanical instability corresponds, thereby, to the maximum of the function Er0 (λi ). Hence, (a)

inverting Eq. (26), deriving the resultant expression with respect to λi

the result equal to zero, we can obtain the internal pressure as a function of the critical stretch at the (a)

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255

at constant Pi and imposing

inner surface, Pi (λci ), namely

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2 (a) dEr0 (Pi , λi ) (a) dλ i

(a)

Pi

= 0 −→ Pi (λci ).

(43)

Substituting back, we obtain the electromechanical instability equation, connecting the critical electric 2

(a)

relations for the Gent model are not given here for conciseness.

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3 The

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field Er0 c and the critical stretch at the inner surface of the active layer λci .

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In the neo-Hookean limit, we obtain the following relation for the critical electric field 2

(a)

(a)

2t(a) (t(a) λci − λco ) µ(a) = (t(a) − 1)2 (a) " (a) 5 (a) 5 (pi) 3 (µ(po) /µ(a) + µ(pi) /µ(a) )λco λci (1 + λci )  − 3 2 (a) 4 (a) 2 3 (a) 3 (a) 3 4 (a) 2 (a) 4 6 (a) (a) 6 (pi) 7 (a) 6 λci t(pi) λco − t(a) λco λci − 2t(a) λco λci − t(a) λco λci + t(a) λco λci (a) 5

λci



(a)

(pi)

(a) 2

(a) 6 λco

(a) 3

(µ(pi) /µ(a) − 1)λco (1 + λci )

−

2 (a) 4 (a) 2 t(a) λco λci

 3

−

3 (a) 3 (a) 3 2t(a) λco λci

−

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2 Er0 c

4 (a) 2 (a) 4 t(a) λco λci

+

(44)

− 6 (a) (a) 6 t(a) λco λci

 # 3 (po) 7 (a) 5 (a) 3 (a) 7 (po) 3 λci t(a) λco (1 + λco ) + µ(po) /µ(a) λco t(po) (1 + λco )   , 2 (a) 4 (a) 2 3 (a) 3 (a) 3 4 (a) 2 (a) 4 6 (a) (a) 6 (a) 2 (po) 7 (a) 6 λco λco λco − t(a) λco λci − 2t(a) λco λci − t(a) λco λci + t(a) λco λci (po)

where λco , λci , and λco

(a)

are functions of λci , according to Eqs. (17) and (39).

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4. Numerical investigation of the electromechanical response of electro-active spherical balloons

The main goal of the numerical investigation is to show how the presence of the protective passive layers influence the electromechanical response of the active membrane and the deformation of the balloon.

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For this purpose, we first compare the electromechanical response of a single-layer balloon with that

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of a multilayer one, consisting of three identical layers. Then, for pure electric and pure mechanical loading cases, we focus our attention on how the behavior of the multilayer balloon is affected by the shear modulus and the thickness of the passive layers.

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To this end, we introduce the following dimensionless parameters s (l) (pi) (a) Pi e (l) = H , α(pi) = µ , E 0 = Er0 , P = , H i (a) µ(a) µ(a) µ(a) Ro

α(po) =

µ(po) . µ(a)

(45)

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For the locking parameter, we assume the same value Jm = 50 for both the active and the passive elastomeric membranes. In all the following plots, we denote by continuous and dashed lines the results

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obtained for the Gent and the neo-Hookean models, respectively. Results for a single-layer balloon are presented in Fig. 4. Fig. 4a shows the electric field as a function

275

of the stretch at the inner surface for fixed values of the internal pressure, in respect of a balloon e = 0.2. Fig. 4b shows the internal pressure as a function of the stretch at characterized by thickness H

the inner surface for fixed values of the electric field, in respect of a balloon characterized by dimensionless e = 0.2. Left- and right-hand parts of Fig. 4c show the influence of the balloon thickness on the thickness H

electromechanical response for pure electric and pure mechanical loading cases, respectively. Increasing 15

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Figure 4: Electromechanical response of a single-layer balloon, formed by an active membrane with dimensionless thickness e = 0.2. For (a) electric-control loading, the electric field is plotted as a function of the stretch at the inner surface for fixed H

values of the internal pressure. For (b) pressure-control loading, the internal pressure is plotted as a function of the stretch

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at the inner surface for fixed values of the electric field. The black curves are referred to the electromechanical instability. e on the electromechanical response; on the left-hand side, the electric field (c) Influence of the dimensionless thickness H

is plotted as a function of the stretch at the inner surface for pure electric loading (Pi = 0), and, on the right-hand side, 2

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the pressure is plotted as a function of the stretch at the inner surface for pure mechanical loading (E 0 = 0). Continuous

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and dashed lines are referred to Gent and neo-Hookean models, respectively.

the electric field at a fixed value of the internal pressure, the balloon gradually expands until a critical electric field. At this state, a further increase in the electric field leads to transition of the balloon to a

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new stable state characterized by larger volume. This instability phenomenon is known as snap-through. Note that this effect cannot be recovered by the neo-Hookean model, since it does not account for the lock-up mechanism. Similarly, the mechanical excitation, induced by increasing the internal pressure at a

285

fixed electric field, causes the balloon expansion up to a critical pressure. At this state, a further increase in the internal pressure results in the snap-through of the balloon. Experimental evidence presented by Li et al. (2013) shows that the pressure versus volume curve is N-shaped—the pressure increases up to a maximum, decreases to a minimum and then increases monotonically—supporting the Gent model.

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When the balloon is mechanically pre-expanded, the critical value of the electric field which triggers 290

the snap-through decreases if the internal pressure is increased. At the same way, when the balloon is electrically pre-expanded, the critical value of the pressure which triggers the snap-through decreases if the applied electric field is increased. For pure electric loading (Pi = 0), due to an increase in the thickness of the balloon, the value of the critical electric field is slightly reduced, less of 1% when the 2

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thickness is doubled. For pure mechanical loading (E 0 = 0), the increase in the thickness of the balloon implies a strong increase in the critical pressure, approximately 132% when the thickness is doubled. A thicker balloon, in fact, requires a larger pressure to reach the same level of inflation of a thinner balloon.

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In both cases, the increase in the balloon thickness corresponds to an increase in the critical stretch.

e (a) = Figure 5: Electromechanical response of a multilayer electro-active balloon constituted by three identical layers, H

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e (pi) = H e (po) = 0.2 and α(p) = α(pi) = α(po) = 1. For (a) electric-control loading, the electric field is plotted as a function H

of the stretch at the inner surface of the active and of the inner passive layers for fixed values of the internal pressure. For

(b) pressure-control loading, the internal pressure is plotted as a function of the stretch at the inner surface of the active and of the inner passive layers for fixed values of the electric field. The black curves are referred to the electromechanical instability. Continuous and dashed lines are referred to Gent and neo-Hookean models, respectively. Note the change in scale with respect to Figs. 4a and 4b.

Results for a multilayer balloon, made up of three identical layers, characterized by the same thickness 17

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300

e (a) = H e (pi) = H e (po) = 0.2 and the same shear modulus α(pi) = α(po) = 1, are presented in Fig. 5. For H fixed values of the internal pressure, left-hand part of Fig. 5a shows the electric field as a function of the

stretch at the inner surface of the active layer, while right-hand part of Fig. 5a shows the electric field as a function of the stretch at the surface of the balloon cavity. For fixed values of the electric field, left-hand part of Fig. 5b shows the internal pressure as a function of the stretch at the inner surface of

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the active layer, while right-hand side part of Fig. 5b shows the internal pressure as a function of the stretch at the surface of the balloon cavity. As for the single-layer balloon, increasing the electric field at a fixed value of the internal pressure, the balloon gradually expands until the critical electric field is achieved. At this state, a further increase in the electric field leads to the snap-through. Similarly, increasing the internal pressure at a fixed electric field, the balloon expands up to a critical pressure. At this state, a further increase in the internal pressure results in the snap-through of the balloon. When the multilayer balloon is mechanically pre-expanded, the critical value of the electric field which triggers

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the snap-through decreases if the internal pressure is increased. At the same way, when the multilayer balloon is electrically pre-expanded, the critical value of the pressure which triggers the snap-through decreases if the applied electric field is increased. However, with respect to the single-layer balloon, it is necessary to apply larger electric field/internal pressure in order to obtain the same deformation of 2

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the balloon cavity. For pure electric (Pi = 0) and pure mechanical (E 0 = 0) loading cases, there is a

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five-fold increment in the critical electric field and in the critical pressure, respectively. The multilayer balloon, being three times thicker than the single-layer one, requires a larger pressure and a larger electric

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field to reach the same level of inflation of the single-layer balloon. Due to the presence of the passive layers, constraining its deformation, the active layer locks up at a lower stretch. Hence, the passive 320

layers provoke a stiffening of the active membrane, making the electro-active balloon less deformable.

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Results for different values of the passive layer shear modulus α(p) , concerning a multilayer balloon e (a) = 0.2 and by two identical passive layers with the same formed by an active membrane with H

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e (p) = H e (pi) = H e (po) = 0.2, are presented in Fig. 6. Figs. 6a and 6c show, thickness of the active layer H

for pure electric loading (Pi = 0), the electric field as a function of the stretch at the inner surface of the

325

active and of the inner passive layers, respectively. Figs. 6b and 6d show, for pure mechanical loading 2

AC

(E 0 = 0), the internal pressure as a function of the stretch at the inner surface of the active and of the inner passive layers, respectively. For pure electric and pure mechanical loading cases, respectively, the critical electric field and the critical pressure increase proportionally to the increase in the shear modulus of the passive layers α(p) . An increase in the shear modulus of the passive layers with respect

330

to that of the active one results, thus, in an proportional increase of the overall stiffness of the multilayer balloon. Hence, when the passive layers are softer than the active membrane, the multilayer balloon is more deformable and its electromechanical response is closer to that of the balloon formed by the active

18

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e (a) = 0.2 and by two identical passive Figure 6: Multilayer electro-active balloon formed by an active membrane with H

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e (p) = H e (pi) = H e (po) = 0.2. Electromechanical behavior for different layers with the same thickness of the active layer H values of the passive layer shear modulus α(p) : (a) and (c) electric field, for pure electric loading (Pi = 0), versus the

stretch at the inner surface of the active and of the inner passive layers, respectively; (b) and (d) internal pressure, for pure mechanical loading (E 0 = 0), versus the stretch at the inner surface of the active and of the inner passive layers, respectively. Black curves in (c) and (d) are referred to the deformation of the balloon formed by the active layer only. Continuous and dashed lines are referred to Gent and neo-Hookean models, respectively.

19

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e (a) = 0.2 and by two identical passive layers Figure 7: Multilayer electro-active balloon formed by an active layer with H

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with the same shear modulus of the active layer α(p) = α(pi) = α(po) = 1. Electromechanical behavior for different values e (p) : (a) and (c) electric field, for pure electric loading (Pi = 0), versus the stretch at the of the passive layer thickness H inner surface of the active and of the inner passive layers, respectively; (b) and (d) internal pressure, for pure mechanical

loading (E 0 = 0), versus the stretch at the inner surface of the active and of the inner passive layers, respectively. Black

curves in (c) and (d) are referred to the deformation of the balloon formed by the active layer only. Continuous and dashed lines are referred to Gent and neo-Hookean models, respectively.

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e (a) = 0.2 and by two unequal passive Figure 8: Multilayer electro-active balloon formed by an active membrane with H

layers. The passive layers, characterized by the same shear modulus of the active layer α(p) = α(pi) = α(po) = 1, have e (pi) and different thicknesses. Electromechanical behavior for two different combination of the passive layer thicknesses H

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e (po) : (a) electric field, for pure electric loading (Pi = 0), and (b) internal pressure, for pure mechanical loading (E 0 = 0), H

versus the stretch at the inner surface of the active layer. Black curves are referred to the balloons formed by two identical e (p) =0.4 and 0.2. Continuous and dashed lines are referred to Gent and passive layers with dimensionless thickness H

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neo-Hookean models, respectively.

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e (a) = 0.4 and by two identical passive Figure 9: Multilayer electro-active balloon formed by an active membrane with H

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layers with the same shear modulus of the active layer α(p) = α(pi) = α(po) = 1. Electromechanical behavior for different e (p) : (a) and (c) electric field, for pure electric loading (Pi = 0), versus the stretch at values of the passive layer thickness H the inner surface of the active and of the inner passive layers, respectively; (b) and (d) internal pressure, for pure mechanical

loading (E 0 = 0), versus the stretch at the inner surface of the active and of the inner passive layers, respectively. Black

curves in (c) and (d) are referred to the deformation of the balloon formed by the active layer only. Continuous and dashed lines are referred to Gent and neo-Hookean models, respectively. Note the change in scale with respect to Fig. 7.

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membrane only.

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e (p) , concerning a multilayer balloon formed Results for different values of the passive layer thickness H

e (a) = 0.2 and by two identical passive layers with the same shear modulus by an active membrane with H of the active layer α(p) = α(pi) = α(po) = 1, are presented in Fig. 7. Figs. 7a and 7c show, for pure

electric loading (Pi = 0), the electric field as a function of the stretch at the inner surface of the

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active and of the inner passive layers, respectively. Figs. 7b and 7d show, for pure mechanical loading 2

(E 0 = 0), the internal pressure as a function of the stretch at the inner surface of the active and of 340

the inner passive layers, respectively. For pure electric and pure mechanical loading cases, respectively, e (p) the critical electric field and the critical pressure increase as the thickness of the passive layers H

increases. As a result of the increase in the passive layer thickness, the lock-up stretch at the inner surface of the active layer decreases, whereas the critical stretch at the surface of the balloon cavity

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increases markedly. These findings can be understood in light of the fact that the larger passive layer thickness results not only in a stronger constraint to the active membrane deformation, but also in a global stiffening of the multilayer balloon. As the passive layer thickness increases, the cavity of the multilayer balloon becomes smaller causing the stiffening of the multilayer balloon. The thickness of the passive layers, hence, strongly affects the electromechanical response of the multilayer balloon. Given the high impact of this parameter, besides the case of identical passive layers, it is also important to investigate for unequal passive layers the role played by the thickness of each of the two layers on the

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response of the balloon.

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e (a) = H e (pi) = H e (po) = With respect to the multilayer balloon formed by three identical layers, i.e. H

0.2 and α(p) = α(pi) = α(po) = 1, if the thickness of only one of the passive layers is increased with respect to that of the active one, the behavior of the balloon is indeed considerably different depending on which one of the two layers is modified, as depicted in Fig. 8. When the thickness of the inner passive

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355

layer is doubled, the behavior is close to that of balloon whose both passive layers have double thickness

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with respect to the active layer. Whereas, if the thickness of the outer passive layer is doubled, the behavior does not differ largely from that of a balloon whose both passive layers have the same thickness of the active layer. The addition of material at the inner surface of the active membrane results in the reduction of the balloon cavity and in a strong increase of the multilayer balloon stiffness. The

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360

addition of material on the outer surface of the active membrane, instead, does not affect so markedly the electromechanical response of the multilayer balloon. To conclude the analysis of the electromechanical behavior of the multilayer balloon, it is interesting

to take into account the response of a multilayer balloon formed by a thicker active membrane. For this 365

e (p) , concerning a multilayer balloon purpose, results for different values of the passive layer thickness H e (a) = 0.4 and by two identical passive constituted by an active membrane with dimensionless thickness H 23

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layers with the same shear modulus of the active layer α(p) = α(pi) = α(po) = 1, are presented in Fig. 9. Figs. 9a and 9c show, for pure electric loading (Pi = 0), the electric field as a function of the stretch at the inner surface of the active and of the inner passive layers, respectively. Figs. 9b and 9d show, for 2

370

pure mechanical loading (E 0 = 0), the internal pressure as a function of the stretch at the inner surface of the active and of the inner passive layers, respectively. Comparison of Fig. 9 with Fig. 7 reveals that

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the increase in the active layer thickness, for the pure electric loading , involves a reduction in the critical electric field of approximately 34%, whereas, for the pure mechanical loading, it implies an increase in the critical pressure of approximately 60%. Note that, when the thickness of the passive layers is equal 375

to that of the active layer, the snap-through is not attained for the pure mechanical loading; due to the large overall thickness and the small size of the cavity, the balloon is stiffer and undergoes small strains. As for the single-layer balloon, the increase in the active layer thickness results in an increment of the

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critical stretch.

4.1. Electromechanical response of electro-active balloons based on a commercially available soft dielectric 380

elastomer

To conclude the numerical investigation, we specialize the previous analysis to an active layer made up of a commercially available dielectric elastomer, the acrylic VHB-4910. The VHB-4910 is a polyacrylate

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elastomer produced by 3M, available as a precast 1mm thick adhesive foam. In the following, we assume typical material properties for the VHB-4910 (see, e.g., Bortot and Gei (2015)), namely

385

r = 4.5.

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µ = 35kPa,

(46)

The limiting uniaxial stretch for the VHB is assumed equal to 7 (Plante and Dubowsky, 2006; Koh et al.,

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2011), thereby the Gent parameter is set to Jm = 50. The active membrane of the balloon, being realized with one layer of VHB-4910, is 1mm thick and we set its radius ratio at 0.8. Thereby, the VHB active membrane is characterized by outer and inner undeformed radii Ro = 5mm and Ri = 4mm, respectively.

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e = 0.2 (cf. curves for H e = 0.2 Fig. 4c). We These values correspond to a dimensionless thickness H consider a single-layer balloon, consisting of the VHB active membrane, and a multilayer one, formed by

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sandwiching the VHB active membrane between two identical passive layers. For the multilayer balloon we take into account four possible setups, examined in the previous dimensionless analysis, characterized by different features of the passive layers. Specifically, the passive layers have i ) same thickness and

395

e (p) = 0.2 in Fig. 7), shear modulus, H (p) = H (a) = 1mm and µ(p) = µ(a) = 35kPa (cf. curves for H

ii ) double thickness and same shear modulus, H (p) = 2H (a) = 2mm and µ(p) = µ(a) = 35kPa (cf. e (p) = 0.4 in Fig. 7), iii ) same thickness and lower shear modulus, H (p) = H (a) = 1mm and curves for H µ(p) = 0.25µ(a) = 8.75kPa (cf. curves for α(p) = 0.25 in Fig. 6), and iv ) same thickness and higher shear

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modulus, H (p) = H (a) = 1mm, and µ(p) = 1.5µ(a) = 52.5kPa (cf. curves for α(p) = 1.5 in Fig. 6), with respect to the active layer. 400

With respect to pure electric and pure mechanical loading cases, numerical data concerning the critical state for the single-layer and the multilayer balloons, with 1mm thick active membrane, are reported in Table 1.

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Comparing the critical state of the single-layer balloon with that of the multilayer one i ), numerical data reveal that the presence of two passive layers, with the same thickness and the same shear modulus 405

of the active layer, involves a two-fold increase in the critical electric field, for pure electric loading, and a five-fold increase in the critical pressure, for pure mechanical loading. Looking at data concerning the multilayer balloons ii ) and iv ), we can deduce that the presence of passive layers with larger thickness or with higher stiffness has, in practice, similar effects on the critical state of the balloon. For both

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multilayer balloons, the critical electric field and the critical pressure increase approximately three-fold and eight-fold, respectively. When the passive layers forming the multilayer balloon are softer than the active layer, setup iii ), the critical state of the multilayer balloon is not too dissimilar with respect to that of the single-layer balloon, particularly in the case of pure electric loading. Instead, for pure mechanical loading, the critical pressure increases seven times with respect to that of the single-layer balloon.

Focusing our attention on data for pure mechanical load (right-hand side of Table 1), we note that the

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critical pressure for all the multilayer balloons is markedly higher with respect to that of the single-layer

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balloon. This is due to the fact that the total thickness of the multilayer balloon is larger and thereby, in absence of electrical load, a higher pressure is needed to deform the balloon. The multilayer balloon, having a smaller cavity, is actually stiffer. The critical pressure can be reduce through an electrically driven pre-expansion of the balloon.

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The active membrane of the balloon can be realized by sticking together two VHB layers, so that its

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thickness is 2mm. In this case, the VHB active membrane, having a radius ratio of 0.6, is characterized by outer and inner undeformed radii Ro = 5mm and Ri = 3mm, respectively. These values correspond

multilayer balloon formed by two identical passive layer with the same thickness and shear modulus of

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425

e = 0.4 (cf. curves for H e = 0.4 in Fig. 4c). We take into account a to a dimensionless thickness H

e (p) = 0.4 the active layer, i.e., setup i ) H (p) = H (a) = 2mm and µ(p) = µ(a) = 35kPa (cf. curves for H in Fig. 9).

With respect to pure electric and pure mechanical loading cases, numerical data concerning the crit-

ical state for the single-layer and the multilayer balloons, consisting of the 2mm thick active membrane, 430

are reported in Table 2. For both pure electric and pure mechanical loadings, comparing Tables 1 and 2, data show that the

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Er0 = 0MV/m

Pi = 0kPa

µ

(p)

=H =µ

H µ

(p)

=µ

(a)

H µ

(p)

=H

(a)

Multilayer iv ) H µ

(p)

=H

Er0 c [MV/m]

44.84

[MV/m]

(a) λci

Er0 c

[MV/m]

(a)

9.6

9.8

1.35

λci

1.49

1.53

47.49

Pic [kPa]

49.9

51.5

1.30

(a) λci

1.42

1.47

1.54

1.56

(pi) λci

1.76

1.83

62.16

63.05

Pic [kPa]

82.5

86.8

1.22

(a) λci

1.32

1.40

1.92

1.98

(pi) λci

2.26

2.27

29.34

29.57

Pic [kPa]

19.6

20.2

1.32

(a) λci

1.45

1.49

1.60

(pi) λci

1.80

1.87

Pic [kPa]

70.1

72.4

1.29

(a) λci

1.45

1.49

1.56

(pi) λci

1.75

1.82

1.31

(pi) λci

Er0 c

Pic [kPa]

1.50

1.58 [MV/m]

(a) λci

(a)

= 1.5µ

Er0 c

Gent

1.50

(a) λci

(a)

= 0.25µ

(p)

1.34

(pi) λci

(a)

Multilayer iii) (p)

λci

(pi) λci

(a)

= 2H

20.42

(a) λci

(a)

Multilayer ii) (p)

20.29

[MV/m]

neo-Hookean

55.88 1.50

(pi) λci

1.53

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Multilayer i) H

Gent

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Single-layer

(p)

neo-Hookean Er0 c

56.37

Table 1: Comparison, at the critical state, between single-layer and multilayer balloons, consisting of a 1mm thick VHB-

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4910 active membrane, in the case of pure electric (left-hand side) and pure mechanical (right-hand side) loadings. The VHB active membrane is characterized by outer and inner radii Ro = 5mm and Ri = 4mm, respectively. For the multilayer balloon we take into account four possible setups characterized by different features of the passive layers,

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specifically, i) H (p) = H (a) = 1mm and µ(p) = µ(a) = 35kPa, ii) H (p) = 2H (a) = 2mm and µ(p) = µ(a) = 35kPa, iii)

Multilayer i) =H

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H

(p)

µ

(p)

=µ

(a)

(a)

Er0 = 0MV/m

Pi = 0kPa neo-Hookean

Gent

Er0 c [MV/m]

19.97

20.14

λci

1.51 48.76

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Single-layer

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H (p) = H (a) = 1mm and µ(p) = 0.25µ(a) = 8.75kPa and iv ) H (p) = H (a) = 1mm, and µ(p) = 1.5µ(a) = 52.5kPa.

Er0 c (a) λci (pi) λci

[MV/m]

neo-Hookean

Gent

Pic [kPa]

21.5

22.1

1.53

λci

1.68

1.74

50.76

Pic [kPa]

1.26 3.03

116.0

—–

1.35

(a) λci

1.50

—–

3.46

(pi) λci

3.45

—–

Table 2: Comparison, at the critical state, between single-layer and multilayer balloons, consisting of a 2mm thick VHB4910 active membrane, in the case of pure electric (left-hand side) and pure mechanical (right-hand side) loadings. The VHB active membrane is characterized by outer and inner radii Ro = 5mm and Ri = 3mm, respectively. For the multilayer balloon we take into account the setup characterized by passive layers with the same thickness and shear modulus of the active layer, specifically, i) H (p) = H (a) = 2mm and µ(p) = µ(a) = 35kPa.

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critical stretches at the cavity surface are markedly larger when the active membrane is 2mm thick. For pure electric loading there is no relevant difference in the critical electric field nor for the singlelayer balloon, neither for the multilayer one i ). The critical stretches at the cavity surface otherwise 435

increases markedly when the active membrane is 2mm thick. For pure mechanical loading, instead, the critical pressure is higher for both the single-layer and the multilayer balloons. According to the Gent

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model, there is no instability for the multilayer balloon under pure mechanical load. These facts are related to the larger thickness of the balloons with 2mm thick active membrane. An electrically driven pre-expansion of the balloon can definitely reduce the critical pressure.

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5. Conclusions

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Electro-active spherical balloons are promising electromechanical transducers, suitable for application as actuators or generators. A critical issue, in practical applications, is to provide the insulation of the active membrane, so as to guarantee electrical safety and electrode protection from aggressive agents. This goal can be achieved simply embedding the active membrane between two soft passive layers. In 445

order to understand the influence of the protective layers on the response of the active membrane, we have investigated the electromechanical response of electro-active balloons formed either by the single

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active membrane or by the multilayer system, where the active membrane is coated by the protective passive layers.

An electro-active balloon, when subjected to increasing electric field (pressure), undergoes an expansion until a critical value of the electric field (pressure) is attained. At this stage, a further increase in

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the electric field (pressure) results in the snap-through of the balloon. Comparing the electromechanical

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response of a single-layer balloon with that of a multilayer one, we have shown that, with respect to the single-layer balloon, the activation of a multilayer balloon requires higher electric field (pressure) to achieve the same deformation of the balloon cavity. Indeed, the multilayer balloon is stiffer, being its overall thickness larger and its cavity smaller. Furthermore, the deformation of the active layer is

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constrained due to the presence of the passive layers.

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For pure electric and pure mechanical loading cases, we have shown that the electromechanical behavior of the multilayer balloon is strongly influenced by the shear modulus and—most importantly— by the thickness of the passive layers. In particular, the critical electric field and the critical pressure

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increase proportionally to the increase in the passive layer shear modulus. An increase in the passive layer thickness involves an increase in the critical electric field and in the critical pressure, an increase in the critical stretch at the surface of the balloon cavity, and a reduction in the lock-up stretch at the inner surface of the active layer. An important fact to take into account is that the change in the thickness of only one of the passive layers produces different results depending on which one of the layers is modified. 27

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465

Accordingly, a change in the thickness of the inner passive layer results in a stronger modification of the electromechanical behavior of the balloon with respect to the same change operated on the outer passive layer. The thickening of the inner passive layer results in the reduction of the balloon cavity and in a strong increase of the balloon stiffness, while the thickening of the outer passive layer does not affect markedly the electromechanical response of the multilayer balloon. These first results suggest that an improvement in the response of the multilayer balloon can be achieved through the prestretch of the

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three layers.

Our findings, showing that the electromechanical response of a multilayer balloon can be properly adjusted by modifying the properties of its constituent layers, provide new design tools for the realization of multilayer balloons whose passive layers ensure the insulation of the active membrane from the surrounding environment.

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Acknowledgment

This work has been supported in part at the Technion by a fellowship from the Lady Davis Founda-

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tion. The author gratefully acknowledges Prof. Gal deBotton for his comments on this work.

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URL http://arxiv.org/abs/1610.03257

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