Electromechanical stability of compressible dielectric elastomer actuators

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Smart Mater. Struct. 20 (2011) 115015 (12pp) doi:10.1088/0964-1726/20/11/115015

Electromechanical stability ofcompressible dielectric elastomeractuatorsLiwu Liu1, Yanju Liu1,5, Jinsong Leng2,5 and Kin-tak Lau3,4

1 Department of Astronautical Science and Mechanics, Harbin Institute of Technology (HIT),PO Box 301, No. 92 West Dazhi Street, Harbin 150001, People’s Republic of China2 Centre for Composite Materials, Science Park of Harbin Institute of Technology (HIT),PO Box 3011, No. 2 YiKuang Street, Harbin 150080, People’s Republic of China3 Centre of Excellence in Engineered Fibre Composites, Faculty of Engineering andSurveying, University of Southern Queensland, Toowoomba, Queensland, Australia4 Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon,Hong Kong SAR, People’s Republic of China

E-mail: yj liu@hit.edu.cn and lengjs@hit.edu.cn

Received 9 May 2011, in final form 21 September 2011Published 14 October 2011Online at stacks.iop.org/SMS/20/115015

AbstractThe constitutive relation and the electromechanical stability of Varga–Blatz–Ko-typecompressible isotropic dielectric elastomers undergoing large deformation are investigated inthis paper. Free energy in any form, which consists of elastic strain energy and electric fieldenergy, can be applied to analyze the electromechanical stability of dielectric elastomers. Theconstitutive relation and the electromechanical stability are analyzed by applying a new kind offree energy model, which consists of elastic strain energy, composed of the Varga model as thevolume conservative energy and the Blatz–Ko model as the volume non-conservative energy,and electric field energy with constant permittivity. The ratio between the principal planarstretches, the ratio between the thickness and length direction stretches, and the power exponentof the stretch are defined to characterize the mechanical loading behavior and compressiblebehavior of the dielectric elastomer. Along with the increase of these parameters, whichdetermine the shape or volume of the elastomer, and the Poisson ratio, the critical nominalelectric field is higher, which indicates a more stable dielectric elastomer electromechanicalsystem. In contrast, with the decrease of the dimensionless material parameter α of the Vargaelastic strain energy, the critical nominal electric field increases. The coupling system becomesmore stable. We further demonstrate that the critical nominal electric field of the compressibledielectric elastomer electromechanical coupling system is significantly influenced by the ratiobetween the principal planar stretches.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Electroactive polymer materials have been extensivelydeveloped and intensively studied in recent years [1–8].Dielectric elastomer (DE) is regarded as the most promisingelectroactive polymer material for transducers; it is capableof very large recoverable deformation, high elastic energy

5 Authors to whom any correspondence should be addressed.

density, high efficiency, high responsive speed, good durabilityand reliability [1–5]. It has attracted much attention inrecent years [1–10]. Dielectric elastomers usually work ascapacitors with variable capacitances. When an electric fieldis applied across the two electrodes, the induced charge causesan electrostatic attraction between them. The electrical forcebetween the electrodes, also known as Maxwell stress, leadsto a reduction in the membrane thickness, which in turnresults in elongation in the plane of the membrane. Their

0964-1726/11/115015+12$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 20 (2011) 115015 L Liu et al

potential applications include smart actuators and sensors,energy harvesters, smart space robots, Braille display, adaptiveoptics, etc. All of these indicate a good prospect fordielectric elastomers, especially in the areas of mechatronics,biomimetics, aeronautics and astronautics [1–8].

Electromechanical instability significantly influences thedielectric elastomer actuation and can lead to failures of thewhole system. When a piece of dielectric elastomer membraneis sandwiched between two compliant electrodes with a highelectric field, due to the electrostatic force between the twoelectrodes, the membrane expands in plane and contractsout-of-plane with its thickness being reduced. The reducedthickness in turn results in a stronger electric field whichsqueezes the membrane to a greater extent. When the electricfield exceeds a critical value, the dielectric elastomer breaksdown and the actuator ceases to function. We call this processthe electromechanical instability [9, 32].

Recently, a lot of research has been conducted focusingon the nonlinear mechanics, electromechanical instabilityand failures of incompressible dielectric elastomers [8–38].Zhao and Suo proposed a theory to investigate theelectromechanical instability. Their study proved that pre-stretch can improve the critical nominal electric field andenhance the electromechanical stability. The numerical valuesof the critical electric field which were obtained basedon the proposed theory coincide well with experimentalobservations [9]. An elastic strain energy function withtwo material constants was employed in the authors’previous work to analyze the stability performance ofdielectric elastomers [10]. Norrisa used the Ogden elasticstrain energy model to analyze the stability of dielectricelastomers [11]. The relations among the critical actualelectric field, nominal strain and pre-stretching of dielectricelastomers were obtained accurately. Dıaz-Calleja’s groupinvestigated the electromechanical stability performance ofneo-Hookean silicone by applying the neo-Hookean elasticstrain energy, in which the stable and unstable domainshelp us to thoroughly understand the electromechanicalperformance of neo-Hookean silicone [12]. Furthermore, theyresearched the deformation and furcation of incompressiblesilicone under an electric field [13]. Based on the previousresearch, Liu et al investigated the electromechanicallystable domain of Mooney–Rivlin type silicone rubber andformulated the Hessian matrices under two specific loadingconditions [14]. Zhu and Suo studied the dynamic andchaotic performance of dielectric elastomer balloons byapplying the neo-Hookean elastic strain energy coupled withthe electric field energy with invariable permittivity [15].Zhao and Suo proposed that the permittivity could be fittedas a linear function depending on the stretch. Based onthis, they analyzed the mechanical performance of dielectricelastomers undergoing large deformation and studied theirelectromechanical stability by applying the neo-Hookeanelastic strain energy function [16]. Furthermore, Liuet al proposed a nonlinear expression for the permittivity.Accordingly they derived the analytic expression of theelectromechanical stability parameters on the basis of thefree energy function in any form [17–19]. Adrian Koh

et al made a theoretical analysis of the failure of dielectricelastomer energy harvesting, and computed the amount ofenergy generated in one cycle and the corresponding maximalefficiency [20, 21]. We studied a typical failure model of aMooney–Rivlin type silicone energy harvester, illustrated theallowable areas under the two conditions of equal-biaxial andunequal-biaxial loadings, respectively, calculated the energyproduction in one cycle of an energy harvester, designed a newharvester, and conducted its primary tests [22]. Moscardo et alinvestigated the failure of dielectric elastomer rolling actuatorsthrough the neo-Hookean elastic strain energy coupledwith the electric field energy with invariable permittivity,and demonstrated the optimal design theory [23]. Zhuet al analyzed the large deformation and electromechanicalinstability of a dielectric elastomer tube actuator [24]. Zhaoand Suo proposed a programmable design method for dielectricelastomer actuators [25]. Xu et al, using the total stressconcept, obtained explicit results for the equilibrium state andcritical electric field for dielectric elastomer actuators [26].Suo et al proposed a nonlinear field theory of deformabledielectrics [27]. Suo and Zhu studied the mechanicalbehavior, large deformation behavior, and stability behavior ofdielectric elastomers in interpenetrating networks [30]. Zhaoand Suo built a dielectric elastomer theory concerning giantelectrical actuation deformation, and predicted the possibilityof manufacturing a dielectric elastomer undergoing giantdeformation [31]. The electromechanical stability of dielectricelastomers undergoing inhomogeneous large deformation wasinvestigated by He et al [35]. Li et al studied dielectricelastomers in conditions of polarization saturation [35, 36].

All of the related papers above are based upon theassumption of incompressible dielectric elastomers. As weknow, a dielectric elastomer is actually a network of polymerchains. Each polymer chain consists of many monomers.The polymer chains are crosslinked by covalent bonds. Thecovalent bonds give the solid-like behavior of the rubber. Ifthese crosslinks are removed, the rubber becomes a polymermelt in liquid form. In fact, a dielectric elastomer is verysimilar to a liquid at the level of monomers, where the scalesare not large enough yet to include any crosslinks. Likeliquids, the polymers are densely packed and it is difficult tochange the rubber’s volume. Also, like liquids, the polymerscan move relative to one another, which contributes to thedielectric elastomers’ strong abilities in shape variation [40].For this reason, most dielectric elastomers can be taken asincompressible materials. Moreover, from an experimentalpoint of view, although the bulk moduli of the elastomers arenot infinitely great, they are usually much larger than the shearmoduli [41]. Therefore, considering dielectric elastomers asincompressible materials is a reasonable approximation.

In most of the application circumstances, there is littledifference between using compressible models and usingincompressible ones. As a simple example, in the uniaxialtensile test, the volume change is less than one per cent ofthe initial volume [42, 43]. However, in the static watercompression experiment in which elastomers are subjected toa very large pressure, the volume change is relatively large,and can be 10%–20% of the initial volume. Furthermore,

2

Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 1. Compressible dielectric elastomer undergoing electromechanical coupling deformation.

the relationship between pressure change and volume changetends to be nonlinear. In this case, the incompressiblemodel can no longer fit [44–46]. In many circumstanceswith special load and boundary conditions, such as imitatingthe three dimensional stress state or circumstances withrigid boundary conditions, we must use the compressiblemodel. Therefore, the strict constitutive model of elastomersshould be compressible [43]. It should be emphasizedthat dissipative effects are not considered when studyingcompressible dielectric elastomers. Moreover, for somespecific elastomer materials, such as dielectric elastomercomposite materials mixed up with particles, foam elastomers,and multiaperture rubber elastomers, the compressibility cannever be ignored.

This paper proposes an analytical method for the constitu-tive relation, large deformation and electromechanical stabilityof compressible dielectric elastomers. The electromechanicalstability is analyzed through a new kind of free energy model,which consists of the Varga–Blatz–Ko elastic strain energy andthe electric field energy density with constant permittivity. TheVarga–Blatz–Ko model consists of the Varga model and Blatz–Ko model as the isochoric and distortional parts of the elasticstrain energy respectively. The relation between the nominalelectric field and the nominal electric displacement is derivedand the critical stability parameters are obtained.

2. Free energy of compressible dielectric elastomer

In figure 1, the mechanical loads T1, T2 and T3 are applied tothe elastomer in the directions of length, width and thicknessrespectively. It is supposed that the original size of thereference state is x1, x2 and x3, and then the compressibledielectric elastomer deforms to a current state with the sizesX1, X2 and X3 in the three directions. λ1, λ2 and λ3 are theprincipal stretch rates. By coating compliant electrodes on bothsurfaces, we applied a voltage U to the dielectric elastomer,and therefore an amount of charge Q is accumulated on theelectrodes. When the sizes of the elastomer change by smallamounts �x1, �x2 and �x3, the mechanical loads perform thework T1�x1, T2�x2 and T3�x3. Similarly, the electric voltage

performs the work U�Q. Therefore, the Helmholtz freeenergy for the system is written as �A = T1�x1 + T2�x2 +T3�x3+U�Q, and the Helmholtz free energy density functionis expressed as �W = s1�λ1 + s2�λ2 + s3�λ3 + E∼�D∼.Therefore, the equations of state of the compressible dielectricelastomer thermodynamics system are

s1(λ1, λ2, λ3, D∼) = ∂W (λ1, λ2, λ3, D∼)

∂λ1(1a)

s2(λ1, λ2, λ3, D∼) = ∂W (λ1, λ2, λ3, D∼)

∂λ2(1b)

s3(λ1, λ2, λ3, D∼) = ∂W (λ1, λ2, λ3, D∼)

∂λ3(1c)

E∼(λ1, λ2, λ3, D∼) = ∂W (λ1, λ2, λ3, D∼)

∂ D∼ . (1d)

Once the free energy function (λ1, λ2, λ3, D∼) is knownfor a given elastic dielectric, equations (1) constitute theequations of state. The stretches are defined as λ1 = X1/x1,λ2 = X2/x2 and λ3 = X3/x3; the nominal stresses canbe derived from the un-deformed state by dividing the pre-stretch forces by the areas before deformation: s1 = T1/x2x3,s2 = T2/x1x3 and s3 = T3/x1x2; the true stresses areσ1 = T1/X2 X3, σ2 = T2/X1 X3 and σ3 = T3/X1 X2. Thus,we have the relations between each nominal and true value:σ1 = s1/λ2λ3, σ2 = s2/λ1λ3, σ3 = s3/λ1λ2.

Similarly, the nominal electric field is defined as E∼ = Ux3

and the nominal electric displacement is D∼ = Qx1 x2

, so thatthe true electric field is related to the nominal electric field byE = U

λ3 X3= E∼

λ3. Also, the true electric displacement is related

to the nominal electric displacement by D = Qλ1x1λ2x2

= D∼λ1λ2

.The free energy function hfor the electromechanical cou-

pling thermodynamics system can be expressed as [28, 29, 32]

W (λ1, λ2, λ3, D∼) = U(λ1, λ2, λ3)+V (λ1, λ2, λ3, D∼) (2)

where U(λ1, λ2, λ3) and V (λ1, λ2, λ3, D∼) are the elasticstrain energy and the electrical field energy density functionsrespectively.

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

The compressible model of elastomers is normallymodified on the basis of the incompressible model, whichis usually described by two parts: the partial elastic strainenergy and the hydrostatic pressure strain energy; thismeans the volume conservative part and the volume non-conservative. The two parts, volume conservative and volumenon-conservative, are expressed by the first, second and thirdinvariants of the unimodular deformation tensors. And thehydrostatic stress components in total stress only depend onthe volume conservative strain energy part.

In the case where the volume of the compressibledielectric elastomer does not remain constant, in thispaper, we assume that the elastic strain energy is thesuperposition of two parts: U(λ1, λ2, λ3) and U (λ1, λ2, λ3),representing the volume conservative and non-conservativeenergies respectively.

For the volume conservative energy, the Varga model withtwo material parameters is employed, the specific form is givenby [39]

U(λ1, λ2, λ3) = μ

2[α(λ1 + λ2 + λ3 − 3)

+ β(λ−11 + λ−1

2 + λ−13 − 3)] (3)

where μ is the shear modulus under small deformation, α andβ are the dimensionless material parameters, which are relatedto the material of the dielectric elastomer and the structure ofits application structure. Moreover, α + β = 1; these aredetermined by experimental data.

For the elastic strain energy of the homogeneous isotropiccompressible elastomer, Blatz and Ko provided the functionas [39]

U(λ1, λ2, λ3) = μ

2

{1 − 2ν

να[(λ1λ2λ3)

− 2ν1−2ν − 1]

+ 1 − 2ν

νβ[(λ1λ2λ3)

2ν1−2ν − 1]

}(4)

where μ and β are identical with those in equation (2). ν is thePoisson ratio for infinitesimal deformation, and λ1λ2λ3 �= 1for homogeneous isotropic elastic compressible materials.

The electric field energy density is expressed as [4, 33–37]

V (λ1, λ2, λ3, D∼) = D∼2

2ελ−1

1 λ−12 λ3 (5)

where ε is the permittivity, for VHB acrylic acid dielectricelastomer, ε ≈ 4.68ε0 [47], ε0 = 8.85 × 10−12 F m−1 [32]is the permittivity of vacuum.

Thus, we give the specific free energy of compressibledielectric elastomer as follows:

W (λ1, λ2, λ3, D∼) = μ

2

{α(λ1 + λ2 + λ3 − 3)

+ 1 − 2ν

να[(λ1λ2λ3)

− 2ν1−2ν − 1]

+ β(λ−11 + λ−1

2 + λ−13 − 3)

+ 1 − 2ν

νβ[(λ1λ2λ3)

2ν1−2ν − 1]

}+ D∼2

2ελ−1

1 λ−12 λ3. (6)

3. Constitutive relations of compressible dielectricelastomer

Substituting equation (6) into (1), we obtain the nominal stressand nominal electric field of compressible dielectric elastomeras

si (λ1, λ2, λ3, D∼) = μ

2{α[1 − 2(λ jλz)

− 2ν1−2ν λ

− 11−2ν

i ]

+ β[2(λ jλz)2ν

1−2ν λ4ν−11−2ν

i − λ−2i ]}

− D∼2

2ελ−1

1 λ−12 λ3λ

−1i (7a)

E∼(λ1, λ2, λ3, D∼) = D∼

ελ−1

1 λ−12 λ3. (7b)

The corresponding true stress and true electric field areexpressed as

σi (λ1, λ2, λ3, D∼) = μ

2λ−1

j λ−1z {α[1 − 2(λ jλz)

− 2ν1−2ν λ

− 11−2ν

i ]

+ β[2(λ jλz)2ν

1−2ν λ4ν−11−2ν

i − λ−2i ]} − D∼2

2ελ−2

1 λ−22 (8a)

E(D) = D

ε(8b)

where i , j , z = 1–3, and i �= j �= z.

4. Electromechanical stability of compressibledielectric elastomer

4.1. Electric field and electric displacement

For a compressible dielectric elastomer, the Hessian matrix canbe written as

H =⎡⎢⎣

∂s1(λ1,λ2,λ3,D∼)∂λ1

∂s1(λ1,λ2,λ3,D∼)∂λ2

∂s1(λ1,λ2,λ3,D∼)∂λ3

∂s1(λ1,λ2,λ3,D∼)∂ D∼

∂s2(λ1,λ2,λ3,D∼)∂λ1

∂s2(λ1,λ2,λ3,D∼)∂λ2

∂s2(λ1,λ2,λ3,D∼)∂λ3

∂s2(λ1,λ2,λ3,D∼)∂ D∼

∂s3(λ1,λ2,λ3,D∼)∂λ1

∂s3(λ1,λ2,λ3,D∼)∂λ2

∂s3(λ1,λ2,λ3,D∼)∂λ3

∂s3(λ1,λ2,λ3,D∼)∂ D∼

∂ E∼(λ1,λ2,λ3,D∼)∂λ1

∂ E∼(λ1,λ2,λ3,D∼)∂λ2

∂ E∼(λ1,λ2,λ3,D∼)∂λ3

∂ E∼(λ1,λ2,λ3,D∼)∂ D∼

⎤⎥⎦(9)

where∂s1(λ1, λ2, λ3, D∼)

∂λ1= μ

2

{α

2

1 − 2ν(λ2λ3)

− 2ν1−2ν λ

2ν−21−2ν

1

+ β

[8ν − 2

1 − 2ν(λ2λ3)

2ν1−2ν λ

6ν−21−2ν

1 − 2λ−31

]}

− D∼2

ελ−3

1 λ−12 λ3

∂s2(λ1, λ2, λ3, D∼)

∂λ2= μ

2

{α

2

1 − 2ν(λ1λ3)

− 2ν1−2ν λ

2ν−21−2ν

2

+ β

[8ν − 2

1 − 2ν(λ1λ3)

2ν1−2ν λ

6ν−21−2ν

2 − 2λ−32

]}

− D∼2

ελ−1

1 λ−32 λ3

∂s3(λ1, λ2, λ3, D∼)

∂λ3= μ

2

{α

2

1 − 2ν(λ1λ2)

− 2ν1−2ν λ

2ν−21−2ν

3

+ β

[8ν − 2

1 − 2ν(λ1λ2)

2ν1−2ν λ

6ν−21−2ν

3 − 2λ−33

]}

∂ E∼(λ1, λ2, λ3, D∼)

∂ D∼ = λ−11 λ−1

2 λ3

ε

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

∂s1(λ1, λ2, λ3, D∼)

∂λ2= ∂s2(λ1, λ2, λ3, D∼)

∂λ1

= μ

2

[4ν

1 − 2ν

(αλ

− 11−2ν

1 λ− 1

1−2ν

2 λ− 2ν

1−2ν

3

+ βλ4ν−11−2ν

1 λ4ν−11−2ν

2 λ2ν

1−2ν

3

)] + D∼2

2ελ−2

1 λ−22 λ3

∂s1(λ1, λ2, λ3, D∼)

∂λ3= ∂s3(λ1, λ2, λ3, D∼)

∂λ1

= μ

2

[4ν

1 − 2ν(αλ

− 11−2ν

1 λ− 2ν

1−2ν

2 λ− 1

1−2ν

3

+ βλ4ν−11−2ν

1 λ2ν

1−2ν

2 λ4ν−11−2ν

3 )

]+ D∼2

2ελ−2

1 λ−12

∂s1(λ1, λ2, λ3, D∼)

∂ D∼ = ∂ E∼(λ1, λ2, λ3, D∼)

∂λ1

= −λ−21 λ−1

2 λ3 D∼

ε∂s2(λ1, λ2, λ3, D∼)

∂λ3= ∂s3(λ1, λ2, λ3, D∼)

∂λ2

= μ

2

[4ν

1 − 2ν

(αλ

− 2ν1−2ν

1 λ− 1

1−2ν

2 λ− 1

1−2ν

3

+ βλ2ν

1−2ν

1 λ4ν−11−2ν

2 λ4ν−11−2ν

3

)] − D∼2

2ελ−1

1 λ−22

∂s2(λ1, λ2, λ3, D∼)

∂ D∼ = ∂ E∼(λ1, λ2, λ3, D∼)

∂λ2

= −λ−11 λ−2

2 λ3 D∼

ε∂s3(λ1, λ2, λ3, D∼)

∂ D∼ = ∂ E∼(λ1, λ2, λ3, D∼)

∂λ3

= λ−11 λ−1

2 D∼

ε∗ . (10)

To obtain a stable state for the compressible dielectricelastomer, the determinant of the Hessian matrix shouldbe positive. When the compressible dielectric elastomer’selectromechanical coupling system reaches the critical state,det(HC) = 0. Solving equation (10), we can get thecritical electromechanical stability parameters of compressibledielectric elastomers, such as the critical nominal electric fieldE∼

max(t0), the critical true electric field E∼max(t0), the critical

nominal stress sC(t0), the critical true stress σC(t0) and thecritical stretch λC(t0).

For compressible dielectric elastomers, the stretch ineach principal direction is relatively independent. In orderto analyze the electromechanical stability of compressibledielectric elastomers, we prescribe λ1 = λ, λ2 = m(t)λand λ3 = n(t)λk(t) , where m(t), n(t), k(t) are time-dependent functions. m(t) is the ratio between principalplanar stretches. n(t) is the ratio between the thickness andlength direction stretches, which determines the shape of theelastomer. n(t) is the power exponent of the stretch, whichdetermines the volume of the elastomer. These parameters,which determine the shape or volume of the elastomer, cancharacterize the mechanical loading behavior and compressiblebehavior of dielectric elastomer. In the study of quasi-staticdeformation, at a certain time t0, substituting the simplification

into equation (7), we obtain the nominal electric field and thenominal electric displacement as follows:

D∼√

με=

{m(t0)

n(t0)

{α[λ3−k(t0) − 2(m(t0)n(t0))

− 2ν1−2ν λ

2−k(t0 )−8ν

1−2ν

]

+ β[2(m(t0)n(t0))2ν

1−2ν λ4k(t0 )ν+2−k(t0 )

1−2ν − λ1−k(t0 )]}

− 2s1m(t0)

μn(t0)λ3−k(t0)

}1/2

E∼√

μ/ε

={

n(t0)

m(t0)

{α[λk(t0)−1 − 2(m(t0)n(t0))

− 2ν1−2ν λ

k(t0 )−2−4k(t0 )ν

1−2ν

]

+ β[2(m(t0)n(t0))

2ν1−2ν λ

k(t0 )−2+8ν

1−2ν − λk(t0)−3]}

− 2s1n(t0)

μm(t0)λk(t0)−1

}1/2

(11a)D∼

√με

={

m2(t0)

n(t0)

{α[λ3−k(t0) − 2m(t0)

− 11−2ν n(t0)

− 2ν1−2ν λ

2−k(t0 )−8ν

1−2ν

]

+ β[2m(t0)

4ν−11−2ν n(t0)

2ν1−2ν λ

4k(t0 )ν+2−k(t0 )

1−2ν − λ1−k(t0)]}

− 2s2m2(t0)

μn(t0)λ3−k(t0)

}1/2

E∼√

μ/ε

={

n(t0){α[λk(t0)−1 − 2m(t0)

− 11−2ν n(t0)

− 2ν1−2ν λ

k(t0 )−2−4k(t0 )ν

1−2ν

]

+ β[2m(t0)

4ν−11−2ν n(t0)

2ν1−2ν λ

k(t0 )−2+8ν

1−2ν − λk(t0)−3]}

− 2s2n(t0)

μλk(t0)−1

}1/2

(11b)D∼

√με

={

2s3m(t0)

μλ2 − m(t0)

× {α[λ2 − 2m(t0)

− 2ν1−2ν n(t0)

− 11−2ν λ

2−k(t0 )−8ν

1−2ν

]+ β

[2m(t0)

2ν1−2ν n(t0)

4ν−11−2ν λ

4k(t0 )ν−k(t0 )+21−2ν

− n(t0)−2λ2−2k(t0)

]}}1/2

E∼√

μ/ε=

{2

s3n2(t0)

m(t0)μλ2k−2 − n2(t0)

m(t0)

× {α[λ2k−2 − 2m(t0)

− 2ν1−2ν n(t0)

− 11−2ν λ

k(t0 )−2−4k(t0 )ν

1−2ν

]

+ β[2m(t0)

2ν1−2ν n(t0)

4ν−11−2ν λ

8ν+k(t0 )−21−2ν − n(t0)

−2λ−2]}}1/2

.

(11c)

In the following, we assume that the tensile forces appliedon the dielectric elastomer are equal, and the original sizes inthe three principal directions are equal too; we have T1 = T2 =T3, x1 = x2 = x3, therefore s1 = s2 = s3.

5

Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 2. Nominal electric displacement and nominal electric field. Variables: s1μ

, ν. (λ1 = λ2 = λ, α = β = 12 , m(t0) = 1, n(t0) = 1,

k(t0) = −1.)

Figure 3. Stretch and nominal electric field. Variables: s1μ

, ν. (λ1 = λ2 = λ, α = β = 12 , m(t0) = 1, n(t0) = 1, k(t0) = −1.)

4.2. Influence of the Poisson ratio on the electromechanicalstability of compressible dielectric elastomers

Figures 2 and 3 illustrate the electromechanical stabilityperformance of different compressible dielectric elastomermaterials or structures under loading conditions of λ1 = λ2 =λ, λ3 = λ−1 and α = β = 1

2 , m(t0) = n(t0) = 1,

k(t0) = −1. (To simplify the labels in the figures, we usem, n, k instead of m(t0), n(t0), k(t0).) Figures 2(a)–(d) showthe relation between the nominal electric displacement and thenominal electric field of the compressible dielectric elastomerwith different values of ν ( 1

6 ,15 , 1

4 , 13 ) and different values of

s1μ

(0, 0.25) respectively. The critical points for instabilityare marked as ‘|’ in figures 2–7. Evidently, along with the

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 4. Nominal electric displacement and nominal electric field. Variables: s1μ

, n(t0). (λ1 = λ2 = λ, α = β = 12 , m(t0) = 1, k(t0) = −1,

ν = 1/4.)

Figure 5. Nominal electric displacement and nominal electric field. Variables: s2μ

, k(t0). (λ1 = λ2 = λ, α = β = 12 , m(t0) = 1, n(t0) = 1,

ν = 1/4.)

increase of ν, the peaks of the nominal electric field increaseand the electromechanical stability improves. However, thecomparative stability performance of this kind of dielectricelastomer is even lower when s1

μincreases. For s1

μ= 0, with

ν of 16 and 1

3 , the corresponding critical nominal electric fieldsare 0.4982

√μ/ε and 1.006

√μ/ε. For representative values of

μ = 1 × 106 Pa, ε = 4 × 10−11 F m−1, the critical nominalelectric fields are E∼

max ≈ 0.79×108, 1.28×108 V m−1, which

are approximately equal to the magnitudes of the reportedbreakdown fields [9].

Figure 3 plots the relation between the nominal electricfield and the stretch when the Poisson ratio bears differentvalues. Evidently, the nominal electric field of compressibledielectric elastomer increases along with the increasing stretch;when it reaches the critical value, the electric field decreasesand tends to a stable value. In a special example, when ν = 1

4

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 6. Nominal electric displacement and nominal electric field. Variables: s1μ

, s2μ

, m(t0). (λ1 = λ2 = λ, α = β = 12 , n(t0) = 1,

k(t0) = −1, ν = 1/4.)

Figure 7. The critical nominal electric field, the critical stretch, the critical area strain and the critical thickness strain. Variables: s1μ

, s2μ

, m(t0).

(λ1 = λ2 = λ, α = β = 12 , n(t0) = 1, k(t0) = −1, ν = 1/4.)

and the nominal stress s1μ

= 0, 0.25, the corresponding critical

stretch is λC = 1.41, 1.71 respectively. And the correspondingdirection equals 50%, 66%. For incompressible dielectricelastomer, the experimental value of thickness strain is lessthan 40% [9, 32], but for compressible dielectric elastomer, itcan be larger. Clearly, the critical stretch increases along withthe increase of nominal stress. In another example, When thenominal stress s1

μ= 0, no matter what values the Poisson ratio

takes, say 16 , 1

5 , 14 , or 1

3 , the critical stretch is always λC = 1.41.

This demonstrates that the change of Poisson ratio will notinfluence the value of the critical stretch; however, the criticalstretch increases along with the nominal stress increasing.

4.3. Influence of the ratio between thickness direction stretchand length direction stretch on the electromechanical stabilityof compressible dielectric elastomers

Figure 4 illustrates the electromechanical stability performanceof different compressible dielectric elastomer materials or

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 8. The critical nominal electric field, the critical stretch, the critical area strain and the critical thickness strain. Variable: n(t0).(λ1 = λ2 = λ, α = β = 1

2 , m(t0) = 1, k(t0) = −1, ν = 1/4.)

structures under loading conditions of λ1 = λ2 = λ, materialparameter conditions of α = β = 1, ν = 1

4 , shape parametercondition of m(t0) = 1, and volume parameter condition ofk(t0) = −1. Along with the increase of the ratio between thethickness direction stretch and the length direction stretch n(t0)

or decrease of s1μ

, the critical nominal electric field increasesand the electromechanical stability improves.

4.4. Influence of the power exponent of stretch on theelectromechanical stability of compressible dielectricelastomers

Figure 5 shows the relation between the nominal electric fieldand the nominal electric displacement when the stretch powerexponent k(t0) bears different values (−2.125, −2.25, −2.325,−2.5) and s2

μtakes different values (0, 0.05). It is evident that

the peak of the nominal electric field increases along with theincreasing of the stretch power exponent k(t0), which meansthat its electromechanical stability performance improves;otherwise the critical nominal electric field declines along withs2μ

increasing, which means that the electromechanical stabilityperformance goes down.

4.5. Influence of the ratio between principal planar stretcheson the electromechanical stability of compressible dielectricelastomers

Figure 6 plots the relation between the nominal electricfield and the nominal electric displacement of differentcompressible dielectric elastomer materials or structures forvarious values of the ratio between principal planar stretchesm(t0), with the loading condition λ1 = λ2 = λ, material

parameter conditions α = β = 1, ν = 14 , shape parameter

condition n(t0) = 1, and volume parameter condition k(t0) =−1. Along with the increase of m(t0) or decrease of s1

μor s2

μ,

the critical nominal electric field increases and the stabilityof the compressible dielectric elastomer electromechanicalcoupling system improves.

5. The critical nominal electric field, the criticalstretch, the critical area strain and the criticalthickness strain of compressible dielectric elastomer

For compressible dielectric elastomer, at time t0, we calculatedthe parameters of the compressible dielectric elastomer’selectromechanical stability, such as the critical nominal electricfield E∼

max(t0), the critical stretch λC(t0), the critical area strainSC(t0), and the critical thickness strain hC(t0). E∼

max(t0) andλC(t0) are determined by equation (11), SC(t0) and hC(t0)depend on the following relations:

SC(t0) = m(t0)λC(t0)2 (12)

hC(t0) = n(t0)λ−k(t0 )C . (13)

Figures 7–11 show the changing law of the parameterscharacterizing the compressible dielectric elastomer’s stabilityperformance under different material parameters. As shown inthese figures, with the increase of m(t0), n(t0), k(t0) and ν, thecritical electric field of the compressible dielectric elastomerincreases, and the compressible dielectric elastomer becomesmore stable. However, along with the decrease of α, thecritical electric field increases, and the compressible dielectricelastomer’s electromechanical coupling system becomes morestable.

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 9. The critical nominal electric field, the critical stretch, the critical area strain and the critical thickness strain. Variable: k(t0).(λ1 = λ2 = λ, α = β = 1

2 , m(t0) = 1, n(t0) = 1, ν = 1/4.)

In figure 7 the nominal stresses in directions 1 and 2have different influences on the electromechanical instabilityby selecting different values of m(t0). For 0 < m(t0) � 1,we have E∼

max(s1)� E∼

max(s2)where the critical nominal electric

field of the system is determined by E∼max(s2)

(E∼max = E∼

max(s2));

on the other hand, when 1 < m(t0) � 2, we have E∼max(s1)

�E∼

max(s2), thus the E∼

max(s1)dominate the system’s critical

nominal field (E∼max = E∼

max(s1)), as marked in figure 7 with

blue lines. Along with the increase of m(t0), the critical stretchand the critical area strain experience a drop before increase;however, the critical thickness strain increases. Evidently,as figure 8 illustrates, the critical stretch and critical areastrain increase along with the increasing stretch ratio n(t0);however, the critical thickness strain increases. As shownin figure 9, along with increase of k(t0), the critical stretchand the critical area strain experience a drop before increase;however, the critical thickness strain increases. The curvein figure 10 suggests that these critical values are insensitiveto the change of Poisson ratio, and the change of Poissonratio will not influence the values of the critical stretch, thecritical area strain and the critical thickness strain. Figure 11illustrates the changing law of the parameters characterizingthe compressible dielectric elastomer’s stability performanceunder different values of the material parameter α. Along withthe increase of α, the critical nominal electric field and criticalthickness strain decrease, indicating lower system stability,while the critical stretch and critical area strain are improved.

6. Conclusions

This paper investigates the constitutive relation and elec-tromechanical stability of Varga type compressible dielec-

tric elastomer through the free energy function, whichis composed of the electric field energy with invariablepermittivity and the elastic strain energy, which consistsof the Varga model as the volume conservative energyand the Blatz–Ko model as the volume non-conservativeenergy. Furthermore, the relations between the nominalelectric field and nominal electric displacement of com-pressed dielectric elastomers under different loading con-ditions are obtained. Finally, the critical stability param-eters, such as the critical nominal electric field, criticalstretch, critical area strain and critical thickness strain, areyielded. The results show that the critical nominal electricfield increases along with either increase of the ratios ofprincipal stretches, m(t0), n(t0), the power exponent ofstretch, k(t0), and the Poisson ratio, ν, or decline of thematerial constant α, which means that a thermodynamicsystem based on compressible dielectric elastomers is morestable. This indicates that, to make dielectric elastomertransducers with excellent properties, it is essential to raisethe ratios of the planar pre-stretches properly. When0 < m(t0) � 1, the critical nominal electric field,E∼

max, equals E∼max(s2)

. Otherwise, if 1 < m(t0) � 2,E∼

max has the same value as E∼max(s1)

. We think that theseresults can be used to facilitate the design and manufac-ture of transducers based on compressible dielectric elas-tomer, which advances the understanding of these promisingmaterials.

Acknowledgments

The one-year visit of Liwu Liu to Harvard University wassupported by the China Scholarship Council Foundation. The

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Smart Mater. Struct. 20 (2011) 115015 L Liu et al

Figure 10. The critical nominal electric field, the critical stretch, the critical area strain and the critical thickness strain. Variable: ν.(λ1 = λ2 = λ, α = β = 1

2 , m(t0) = 1, n(t0) = 1, k(t0) = −1.)

Figure 11. The critical nominal electric field, the critical stretch, the critical area strain and the critical thickness strain. Variable: α.(λ1 = λ2 = λ, m(t0) = 1, n(t0) = 1, k(t0) = −1, ν = 1

4 .)

author hereby acknowledges Professor Zhigang Suo’s foresightand sagacious guidance on the theory of dielectric elastomers.

References

[1] Pelrine R, Kornbluh R, Pei Q B and Joseph J 2000 High-speedelectrically actuated elastomers with strain greater than100% Science 287 836–9

[2] Brochu P and Pei Q B 2010 Advances in dielectric elastomersfor actuators and artificial muscles Macromol. RapidCommun. 31 10–36

[3] O’Halloran A, O’Malley F and McHugh P 2008A review on dielectric elastomer actuators, technology,applications, and challenges J. Appl. Phys.104 071101

[4] Gallone G, Carpi F, Rossi D, Levita G and Marchetti A 2007Dielectric constant enhancement in a silicone elastomer

11

Smart Mater. Struct. 20 (2011) 115015 L Liu et al

filled with lead magnesium niobate—lead titanate Mater.Sci. Eng. C 27 110–6

[5] Carpi F, Gallone G, Galantini F and Rossi D 2008Silicone-poly (hexylthiophene) blends as elastomers withenhanced electromechanical transduction properties Adv.Funct. Mater. 18 235–41

[6] Keplinger C, Kaltenbrunner M, Arnold N and Bauer S 2010Rontgen’s electrode-free elastomer actuators withoutelectromechanical pull-in instability Proc. Natl Acad. Sci.107 4505–10

[7] Zhang H, During L, Kovacs G, Yuan W, Niu X and Pei Q 2010Interpenetrating polymer networks based on acrylicelastomers and plasticizers with improved actuationtemperature range Polym. Int. 59 384–90

[8] Suo Z G 2010 Theory of dielectric elastomers Acta Mech.Solid. Sin. 23 549–78

[9] Zhao X H and Suo Z G 2007 Method to analyzeelectromechanical stability of dielectric elastomers Appl.Phys. Lett. 91 061921

[10] Liu Y J, Liu L W, Zhang Z, Shi L and Leng J S 2008 Commenton ‘Method to analyze electromechanical stability ofdielectric elastomers’ Appl. Phys. Lett. 93 106101

[11] Norrisa A N 2007 Comments on ‘Method to analyzeelectromechanical stability of dielectric elastomers’ Appl.Phys. Lett. 92 026101

[12] Dıaz-Calleja R, Riande E and Sanchis M J 2008 Onelectromechanical stability of dielectric elastomers Appl.Phys. Lett. 93 101902

[13] Dıaz-Calleja R, Riande E and Sanchis M J 2009 Effect of anelectric field on the bifurcation of a biaxially stretchedincompressible slab rubber Eur. Phys. J. E 30 417–26

[14] Liu Y J, Liu L W, Sun S H, Shi L and Leng J S 2009 Commenton ‘Electromechanical stability of dielectric elastomers’Appl. Phys. Lett. 94 096101

[15] Zhu J, Cai S Q and Suo Z G 2010 Nonlinear oscillation of adielectric elastomer balloon Polym. Int. 59 378–83

[16] Zhao X H and Suo Z G 2008 Electrostriction in elasticdielectrics undergoing large deformation J. Appl. Phys.104 123530

[17] Liu Y J, Liu L W, Yu K, Sun S H and Leng J S 2009 Aninvestigation on electromechanical stability of dielectricelastomer undergoing large deformation Smart Mater. Struct.18 095040

[18] Liu Y J, Liu L W, Sun S H and Leng J S 2010Electromechanical stability of Mooney–Rivlin-typedielectric elastomer with nonlinear variable dielectricconstant Polym. Int. 59 371–7

[19] Leng J S, Liu L W, Liu Y J, Yu K and Sun S H 2009Electromechanical stability of dielectric elastomer Appl.Phys. Lett. 94 211901

[20] Adrian Koh S, Zhao X H and Suo Z G 2009 Maximal energythat can be converted by a dielectric elastomer generatorAppl. Phys. Lett. 94 262902

[21] Adrian Koh S, Keplinger C, Li T F, Bauer S and Suo Z G 2011Dielectric elastomer generators: how much energy can beconverted IEEE/ASME Trans. Mechatronics 16 33–41

[22] Liu Y J, Liu L W, Zhang Z, Jiao Y, Sun S H and Leng J S 2010Analysis and manufacture of an energy harvester based on aMooney–Rivlin-type dielectric elastomer Europhys. Lett.90 36004

[23] Moscardo M, Zhao X H, Suo Z G and Lapusta Y 2008 Ondesigning dielectric elastomer actuators J. Appl. Phys.104 093503

[24] Zhu J, Stoyanov H, Kofod G and Suo Z G 2010 Largedeformation and electromechanical instability of a dielectricelastomer tube actuator J. Appl. Phys. 108 074113

[25] Zhao X H and Suo Z G 2008 Method to analyze programmabledeformation of dielectric elastomer layers Appl. Phys. Lett.93 251902

[26] Xu B X, Mueller R, Classen M and Gross D 2010 Onelectromechanical stability analysis of dielectric elastomeractuators Appl. Phys. Lett. 97 162908

[27] Suo Z G, Zhao X H and Greene W H 2008 A nonlinear fieldtheory of deformable dielectrics J. Mech. Phys. Solids56 476–86

[28] Zhou J X, Hong W, Zhao X H, Zhang Z and Suo Z G 2008Propagation of instability in dielectric elastomers Int. J.Solids Struct. 45 3739–50

[29] Liu Y J, Liu L W, Zhang Z and Leng J S 2009 Dielectricelastomer film actuators: characterization, experiment andanalysis Smart Mater. Struct. 18 095024

[30] Suo Z G and Zhu J 2009 Dielectric elastomers ofinterpenetrating networks Appl. Phys. Lett. 95 232909

[31] Zhao X H and Suo Z G 2010 Theory of dielectric elastomerscapable of giant deformation of actuation Phys. Rev. Lett.104 178302

[32] Liu Y J, Liu L W, Sun S H and Leng J S 2009 Stability analysisof dielectric elastomer film actuator Sci. Chin. E 52 2715–23

[33] Liu L W, Liu Y J, Zhang Z, Li B and Leng J S 2010Electromechanical stability of electro-active silicone filledwith high permittivity particles undergoing largedeformation Smart Mater. Struct. 19 115025

[34] He T H, Cui L L, Chen C and Suo Z G 2010 Nonlineardeformation analysis of a dielectric elastomermembrane-spring system Smart Mater. Struct. 19 085017

[35] Li B, Chen H L, Qiang J H, Hu S L, Zhu Z C and Wang Y Q2011 Effect of mechanical pre-stretch on the stabilization ofdielectric elastomer actuation J. Phys. D: Appl. Phys.44 155301

[36] Li B, Liu L W and Suo Z G 2011 Extension limit, polarizationsaturation, and snap-through instability of dielectricelastomers Int. J. Smart Nano Mater. 2 59–67

[37] Liu L W, Liu Y J, Li B, Yang K, Li T F and Leng J S 2011Thermo-electro-mechanical stability of dielectric elastomersSmart Mater. Struct. 20 075004

[38] Kong X H, Li Y B, Liu L W and He X D 2011Electromechanical stability of semi-crystalline polymer ThinSolid Films 519 5017–21

[39] Blatz P J and Ko W L 1986 On the use of the Blatz–Koconstitutive model in nonlinear finite element analysis Trans.Soc. Rheol. 6 223–51

[40] http://imchanica.org/node/10589[41] Peng S T J and Landel R F 1975 Stored energy function and

compressibility of compressible rubber like materials underlarge strain J. Appl. Phys. 46 2599

[42] Gent A N and Lindley P B 1959 Internal rupture of bondedrubber cylinders in tension Proc. R. Soc. 249 195

[43] Bischoff J E et al 2001 A new constitutive model for thecompressibility of elastomers at finite deformations RubberChem. Technol. 74 541–59

[44] Bridgman P W 1945 The compression of sixty-one solidsubstances to 25 000 kg cm−1, determined by a new rapidmethod Proc. Am. Acad. Arts Sci. 76 9

[45] Adams L H and Gibson R E 1930 The compressibility ofrubber J. Wash. Acad. Sci. 20 213

[46] Li X F and Yang X X 2005 A review of elastic constitutivemodel for rubber materials China Elastomer. 15 50–8

[47] Wissler M and Mazza E 2007 Electromechanical coupling indielectric elastomer actuators Sensors Actuators AA138 384–93

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