Mechanics of Materials 72 (2014) 33–45

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Mechanics of Materials

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Thermoelectromechanical stability of dielectric elastomersundergoing temperature variation

http://dx.doi.org/10.1016/j.mechmat.2013.05.0130167-6636/� 2013 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors. Tel./fax: +86 451 86414825 (Y. Liu), tel./fax:+86 451 86402328 (J. Leng).

E-mail addresses: yj_liu@hit.edu.cn (Y. Liu), lengjs@hit.edu.cn (J. Leng).

Liwu Liu a, Yanju Liu a,⇑, Kai Yu b, Jinsong Leng b,⇑a Department of Astronautical Science and Mechanics, Harbin Institute of Technology (HIT), P.O. Box 301 , No. 92 West Dazhi Street, Harbin 150001, PR Chinab Centre for Composite Materials, Science Park of Harbin Institute of Technology (HIT), P.O. Box 3011, No. 2 YiKuang Street, Harbin 150080, PR China

a r t i c l e i n f o

Article history:Received 26 February 2012Received in revised form 25 November 2012Available online 14 June 2013

Keywords:Dielectric elastomersThermoelectromechanical stabilityDielectric constant

a b s t r a c t

In this paper, the influence of both temperature and deformation on dielectric constant isconsidered during the establishment of free energy function of dielectric elastomers. A con-stitutive model of the thermodynamic systems undergoing adiabatic process is derived tostudy its thermoelectromechanical stability. The relations between different work conju-gated parameters of dielectric elastomer are theoretically described, including the relationsbetween nominal electric field and nominal electric displacement, entropy and tempera-ture. Under different temperatures and electric fields, the allowable energy range of dielec-tric elastomer is calculated. Furthermore, the electric-induced variation of dielectricelastomer’s temperature and entropy is also studied under various principal planar stretchratios. These simulation results should offer assistances in guiding the design and fabrica-tion of excellent actuators featuring dielectric elastomers.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Large deformation will be generated when an electricfield is applied on a thin dielectric elastomer film (Pelrineet al., 2000; Brochu and Pei, 2010; O’Halloran et al., 2008;Zhang et al., 2010). Due to its excellent properties, such ashigh elastic energy density, fast response, low cost, lightweight, high efficiency, and easy to be processed, dielectricelastomers show great application potential in the area ofsmart bionics, medical devices, aeronautics and astronau-tics. Various devices based on dielectric elastomers suchas actuators, sensors, tactile display and energy harvestersetc. have also been designed (Gallone et al., 2004, 2010;Carpi et al., 2008, 2010a; Keplinger et al., 2010; Kolloscheand Kofod, 2010; Plante and Dubowsky, 2006; Adrian Kohet al., 2011; Liu et al., 2010b). The principal failure modesof the dielectric elastomers include electric breakdown,

material rupture, loss of tension, electromechanical insta-bility (Adrian Koh et al., 2011), would definitely hindertheir wide applications. Hence, a well established theoreti-cal model on the failure mechanism of dielectric elastomersis important. Recently, the nonlinear mechanical perfor-mance (Suo et al., 2008; Liu et al., 2009b,c), electromechan-ical stability (Liu et al., 2009d,e; Zhao et al., 2007, 2008b;Zhou et al., 2008; Liu et al., 2008, 2010a,b,c; Kong et al.,2011; Li et al., 2012), dynamical performance (Zhao et al.,2011; Zhu et al., 2010b) and failure of application devices(Adrian Koh et al., 2011; Moscardo et al., 2008; Liu et al.,2014) of dielectric elastomers are hotspots in all theoreticalresearches on electroactive materials.

Zhao and Suo have proposed an analytical method basedon thermodynamics and continuum mechanics to predictthe electromechanical stability of dielectric elastomers(Zhao and Suo, 2007). Based on the method, the electrome-chanical instability of Mooney–Rivlin type dielectric elasto-mers and the allowable area of the energy harvester werestudied (Liu et al., 2009a, 2010a). The stable area of dielec-tric elastomers was investigated by research groups of

34 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

Díaz-Calleja and Suo respectively (Díaz-Calleja et al., 2008;Adrian Koh et al., 2009). The bifurcation and chaos perfor-mance of dielectric elastomer’s thermodynamic systemwas also studied by Zhu et al. (2010a). The electromechan-ical stability of dielectric elastomer undergoing homoge-neous and inhomogeneous large deformation wasinvestigated by Zhao and Suo (2008a) and He et al.(2009). Suo et al. studied electromechanical stability ofdielectric elastomers of interpenetrating networks (Suoand Zhu, 2009). Zhao and Suo proposed the theory of dielec-tric elastomers capable of giant deformation for actuation(Zhao and Suo, 2010). Besides, the constitutive relationand electromechanical instability of different dielectricelastomer actuators, such as planar, rolled, tubular andhemispheric actuators, were also studied by Zhao and Suo(2007), Moscardo et al. (2008), Zhu et al. (2010c). However,the influence of temperature to dielectric elastomers hasnot been considered in the above mentioned theoreticalanalyses. Recent experiments indicate that the temperatureplays a considerable role in the dielectric property of dielec-tric elastomers (Jean-Mistral et al., 2010). In Suo’s researchgroup, a model was proposed to show that the snap-through instability is markedly affected by both the exten-sion limit of polymer chains and the polarization saturationof dipoles. The model is essentially useful for the future re-search of high-performance dielectric elastomer transduc-ers (Li et al., 2011a; Liu et al., 2012a,b). In addition, somepotential applications of stretchable dielectric elastomersare predicted by Carpi et al. (2010b). In Zhuo’s researchgroup, Li proposed a model to study the polarization mode,and characterize the behavior when the dipolar alignmentis constrained by deformation. The conditional polarizationwould modify the final state as well as the path to instabil-ity during actuation due to the mechanism of induced elec-trostriction, indicating a new route to optimize theperformance of elastic dielectrics (Li et al., 2011b).

Suo reviews the theory of dielectric elastomers devel-oped within continuum mechanics and thermodynamics.The theory couples large deformation and electric poten-tial, and describes nonlinear and nonequilibrium behavior.

F2

T

Q

L1

Fig. 1. Thermoelectromechanical coupli

It also enables the finite element method to simulate trans-ducers of realistic configurations, and suggests alternativeroutes to achieve giant voltage-induced deformation(Suo, 2010).

In this paper, a constitutive model of the thermody-namic systems under adiabatic process is deduced in orderto study the thermoelectromechanical stability of dielec-tric elastomers. The allowable energy range of dielectricelastomer’s thermodynamic system is predicted and theelectric-induced changes of dielectric elastomer’s tempera-ture and entropy are also evaluated.

2. Fundamental theory

After being uniformly coated with compliant electrodeson the both surfaces and applied a voltage across, a thindielectric elastomer film will endure shrinkage in thicknessand expansion in area. As a result of increased tempera-ture, the material’s modulus will decrease and then gener-ate faster deformation during its thickness reduction andarea expanding. The decreased thickness of dielectric elas-tomer film results in a higher electric filed in the material.Under the coupling effect of thermal, electric and mechan-ical fields, such positive feedback continues. When the in-duced electric filed surpass the critical filed, the dielectricelastomer film will breakdown, resulting in the thermo-electromechanical instability.

As we know, a dielectric elastomer is actually a networkof polymer chains. Each polymer chain consists of manymonomers. The polymer chains are crosslinked by covalentbonds. The covalent bonds give solid-like behavior to therubber. If these crosslinks are removed, the rubber be-comes a polymer melt of liquid form. In fact, a dielectricelastomer is very similar to liquid at the level of mono-mers, where the scales are not large enough yet to includeany crosslink. Like liquids, the polymers are denselypacked and it is difficult to change the rubber’s volume.Also like liquids, the polymers can move relative to one an-other, which contributes to dielectric elastomers’ strongabilities in shape alternation (Liu et al., 2011a). For this

F1

U

L2

L3

ng system of dielectric elastomer.

Table 1The typical parameters of dielectric elastomer.

p a (F/m K) b (F/m) er Nk (J/kg K) c0 (J/kg K)0.9 �0.05921 �0.0048 �0.0533 5.54 3.66 1.7 � 103

1.1 �0.0484

Fig. 2. Nominal electric field vs. nominal electric displacemen

Fig. 3. Nominal electric field of dielectric elastomer

L. Liu et al. / Mechanics of Materials 72 (2014) 33–45 35

reason, most dielectric elastomers can be taken as incom-pressible materials.

When a dielectric elastomer undergoes large deforma-tion suffering different physical fields, the change in theshape of the elastomer is much larger than that of the vol-ume. Consequently, the elastomer is often taken to be

t of dielectric elastomer under different temperatures.

under different temperatures and stretches.

Fig. 4. Entropy of dielectric elastomer under different temperatures and stretches.

Fig. 5. Nominal electric field and nominal electric displacement of dielectric elastomer under different stretches.

36 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

incompressible, that is, the volume of the material remainsunchanged during the deformation.

As is shown in Fig. 1, for the isotropic, homogeneousand incompressible thermoelastic solid, the free energyfunction can be expressed as a function with four variables,k1, k2, D� and T, namely Wðk1; k2;D

�; TÞ, where k1 and k2 arestretches in the principal planar directions, D� donates

nominal electric displacement, T represents current tem-perature. Under the coupling effect of thermal, electricand mechanical fields, the small variation of the four inde-pendent variables are dk1, dk2, dD� and dT respectively. Thevariation of free energy of dielectric elastomer’s thermo-electromechanical coupling system can be expressed asfollowing

Fig. 6. Nominal electric field of dielectric field under different temperatures and stretches.

Fig. 7. The entropy of dielectric elastomer under different temperatures and stretches.

Table 2Critical nominal electric field and critical stretch of dielectric elastomerunder different temperatures and stretches.

p Critical nominal electric field Critical stretch

273 K 373 K 273 K 373 K

0.9 0.747ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p0.913

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p1.22

1 0.695ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p0.851

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p1.26

1.1 0.652ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p0.799

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p1.31

L. Liu et al. / Mechanics of Materials 72 (2014) 33–45 37

dW ¼ @Wðk1; k2;D�; TÞ

@k1dk1 þ

@Wðk1; k2;D�; TÞ

@k2dk2

þ @Wðk1; k2;D�; TÞ

@D�dD� � @Wðk1; k2;D

�; TÞ@T

dT ð1Þ

Therefore, the nominal stress, nominal electric field andentropy are

s1 ¼@Wðk1; k2;D

�; TÞ@k1

ð2Þ

38 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

s2 ¼@Wðk1; k2;D

�; TÞ@k2

ð3Þ

E� ¼ @Wðk1; k2;D�; TÞ

@D�ð4Þ

S ¼ � @Wðk1; k2;D�; TÞ

@Tð5Þ

Wðk1; k2; T;D�Þ ¼ Uðk1; k2; TÞ þ Vðk1; k2; T;D

�Þ ð6Þ

where Uðk1; k2; TÞ is thermoelastic energy per unit volume.Vðk1; k2; T;D

�Þ is electric filed energy density.In Jean-Mistral’s experiment, it is indicated that the

dielectric constant of dielectric elastomer is related withthe temperature and stretch (Jean-Mistral et al., 2010).Therefore, in the following study, a model of electric fieldenergy density function with variable dielectric constantis introduced to analyze the mechanical performance andelectromechanical stability of dielectric elastomers. Con-sidering the influence of temperature and stretch, we pro-pose the expression of dielectric constant eðT; k1; k2Þ asfollowing:

eðT; k1; k2Þ ¼ er þ aðT � T0Þ þ bk1k2 ð7Þ

where er is the relative dielectric constant of dielectric elas-tomer under the reference temperature without deforma-tion, a is a phenomenology parameter, T0 is thetemperature of referenced state, b is the coefficient ofelectrostriction.

Expression (7) is the result of the Taylor expansion ofeðT; k1; k2Þ. The dielectric constant model we suggestedonly considers the simplest circumstances, and we ignorethe influence of high order term of the temperature. Thatis to say, the dielectric constant in this paper is only takento be a linear function of temperature. This assumption isreasonable as in many researchers’ experimental studies,the dielectric constant of insulating oil liquid is linear withthe temperature variation, and also the ideal silicon rubberdielectric elastomer could be taken as ideal liquid dielec-tric (Hmmer et al., 2002). Therefore, in the research below,all the parameters (a, b and so on) are studied base on theJean-Mistral’s experiment (Jean-Mistral et al., 2010).

Of course, we can also use model with more clear phys-ical significance to describe the temperature’s influence todielectric constant. According to the research of Jean-Mis-tral and the Debye equation function (Jean-Mistral et al.,2010), we propose the expression of dielectric constanteðT; k1; k2Þ as follow:

eðT; k1; k2Þ ¼ er þa0

Tþ b0k1k2 ð8Þ

The dielectric constant is the inverse proportion func-tion of the temperature. Due to the thermal force field,the dipoles reorientation, which is the molecular origin ofthe macroscopic dielectric permittivity, is strongly re-stricted. In Eq. (8), a0 ¼ Nl2

3je0, N is the dipole density of a

dielectric elastomer, l is the dipole moment of a dielectricelastomer, j is the Boltzmann constant, e0 is the dielectricpermittivity of a vacuum, b0 is the coefficient of electro-striction a dielectric elastomer.

The influence of temperature on the permittivity of ac-rylic dielectric elastomers was studied in detail by Shenget al. First, it was experimentally observed that the permit-tivity changed in demonstrating a peak value as the tem-perature alternation. Theoretical studies then showedthat the Cole–Cole equation was a better choice to describethe temperature induced permittivity increase, while thepermittivity decrease after the peak value could be wellcaptured by Debye equation. Furthermore, the mathemat-ical model for the permittivity of VHB 4910 was estab-lished, which would provide assistances for the furtherstudies of the electromechanical stability of dielectric elas-tomers (Sheng et al., 2013).

The elastomer is taken to be network of long polymersobeying the Gaussian statistics, so that the elastic behaviorof the elastomer is neo-Hookean. For an ideal dielectricelastomer, the dielectric energy per unit volume is D2

2e, andthe dielectric constant e is a constant independent of defor-mation (Zhu et al., 2010b). Considering the electrostrictionof dielectric elastomers, the dielectric constant is a func-tion of the stretches, the dielectric energy per unit volumeis D�2

2eðk1 ;k2Þk�2

1 k�22 (Suo, 2010). Hence, the electric field energy

density function of incompressible dielectric elastomerscan be written as follows (Zhao and Suo, 2007)

Vðk1; k2; T;D�Þ ¼ D�2

2 er þ aðT � T0Þ þ bk1k2½ � k�21 k�2

2 ð9Þ

Considering the influence of temperature, a proposedthermoelastic strain energy function is Horgan and Sacco-mandi (2006)

Uðk1; k2; TÞ ¼12

NkT k21 þ k2

2 þ k�21 k�2

2 � 3� �

þ c0 ðT � T0Þ � T lnTT0

� �ð10Þ

Equation (9) describes the coupling influence of tem-perature and stretch of the thermodynamical system. Thefirst item in Eq. (10) is the neo-Hookean thermoelastic en-ergy. N is the number of polymer chains per unit volume ofdielectric elastomer, kT is the temperature in units of en-ergy and NkT = l(T). Here l(T) is the temperature relatedshear modulus under infinitesimal deformation. The sec-ond item donotes the thermal contribution of temperatureto the free energy of the entire thermodynamic system, c0

is the specific heat of dielectric elastomers.The second term of Eq. (10) c0 T � T0ð Þ � T ln T

T0

h iis

called the thermal contribution. We can describe the deri-vation of the thermal contribution simply as followings(Liu et al., 2011b), the thermal contribution can be ex-pressed as n(T) = Q(T) � TS, where Q(T) is the internal en-ergy (Horgan and Saccomandi, 2006). As the temperaturerises, internal energy follows Q(T) = c0(T � T0). Accordingto the relation of specific heat for constant volume and en-tropy, c0 ¼ T @S

@T, we obtain S ¼ c0 ln TT0

, where c0 is the spe-

cific heat of polar dielectric, T0 and T are thetemperatures in reference and current states respectively.Therefore, the thermal contribution of the thermodynamicsystem related to temperature is nðTÞ ¼ c0½ðT � T0Þ�T ln T

T0�, which reflects the effect of temperature on the free

energy of polar dielectric thermo-electric coupling system.

Fig. 8. The allowable energy range of dielectric elastomer in plane of nominal electric field and nominal electric displacement.

Fig. 9. The allowable energy range of dielectric elastomer in plane of temperature and entropy.

L. Liu et al. / Mechanics of Materials 72 (2014) 33–45 39

And the expression of thermal contribution is widely appli-cable to the thermodynamic systems with great tempera-ture changes.

3. Constitutive model of dielectric elastomers

Considering equations (2)–(6), the nominal stress ofdielectric elastomer’s thermodynamic system in the twoplanar principal directions, the nominal electric field inthe thickness direction and the entropy are obtainedrespectively, as is shown in the followings

s1 ¼ NkTðk1 � k�31 k�2

2 Þ �D�2k�3

1 k�22

½er þ aðT � T0Þ þ bk1k2�

� D�2bk�21 k�1

2

2½er þ aðT � T0Þ þ bk1k2�2ð11Þ

s2 ¼ NkTðk2 � k�21 k�3

2 Þ �D�2k�2

1 k�32

½er þ aðT � T0Þ þ bk1k2�

� D�2bk�11 k�2

2

2½er þ aðT � T0Þ þ bk1k2�2ð12Þ

Fig. 10. The allowable energy range of dielectric elastomer in plane of nominal electric field and nominal electric displacement.

Fig. 11. The allowable energy range of dielectric elastomer in plane of temperature and entropy.

Table 3Entropy variation and allowable energy range under differenttemperatures.

p Entropy variation (J/kg K) Allowable energy range (J/kg)273–373 K 273–373 K

0.9 1.67 1671 1.22 1221.1 1.29 129

40 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

E� ¼ 1½er þ aðT � T0Þ þ bk1k2�

D�k�21 k�2

2 ð13Þ

S ¼ c0 lnTT0� 1

2Nkðk2

1 þ k22 þ k�2

1 k�22 � 3Þ

þ aD�2k�21 k�2

2

2½er þ aðT � T0Þ þ bk1k2�2ð14Þ

L. Liu et al. / Mechanics of Materials 72 (2014) 33–45 41

Fig. 1 represents the schematic graph of the thermo-electromechanical system of dielectric elastomer. InFig. 1, mechanical load F1 and F2 are applied. Before thedeformation, dielectric elastomer’s dimensions in thesetwo directions are L1 and L2 respectively. Dimensions afterdeformation will be k1L1 and k2L2 respectively. The nomi-nal stress of dielectric elastomer in the planar directionare s1 = F1/L2L3, s2 = F2/L1L3. The true stress of dielectricelastomer are r1 ¼ F1=k2k3L2L3, r2 ¼ F2=k1k3L1L3. There-fore, the relation between true stress and nominal stressare r1 ¼ s1=k2k3, r2 ¼ s2=k1k3. Similarly, the nominal elec-tric field of dielectric elastomer is defined as E� ¼ U

L3. The

nominal electric displacement is defined as D� ¼ QL1L2

. Thus,the corresponding true electric field of dielectric elastomercan be expressed as E ¼ U

k3L3¼ E�

k3, while the true electric dis-

placement is D ¼ Qk1L1k2L2

¼ D�

k1k2.

Based on the above mentioned definitions, the truestress and true electric field can be given respectively as

r1 ¼ NkTðk21 � k�2

1 k�22 Þ �

D�2k�21 k�2

2

½er þ aðT � T0Þ þ bk1k2�

� D�2bk�11 k�1

2

2½er þ aðT � T0Þ þ bk1k2�2ð15Þ

r2 ¼ NkTðk22 � k�2

1 k�22 Þ �

D�2k�21 k�2

2

½er þ aðT � T0Þ þ bk1k2�

� D�2bk�11 k�1

2

2½er þ aðT � T0Þ þ bk1k2�2ð16Þ

E ¼ 1½er þ aðT � T0Þ þ bk1k2�

D�k�11 k�1

2 ð17Þ

4. Thermoelectromechanical stability of dielectricelastomers

During the fabrication process of actuators, the dielectricelastomers are subjected to pre-stretch, most commonly,unequal biaxial stretch. Therefore, in the following study,to simplify the calculation, we set k2 ¼ pk1 ¼ pk. Here p isthe ratio between principal planar stretches of dielectricelastomer. Sometimes, this pre-stretch could be an equalbiaxial one, such as while making the spherical devices fea-turing isotropic dielectric elastomers (Fox and Goulbourne,2008). This can be treated as a special case of p = 1.

According to equations (11)–(14), the dimensionlessnominal electric displacement, the nominal electric fieldand the entropy of dielectric elastomer can be obtainedas followings

D�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0er

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ aðT�T0Þþpbk2

er

h ip�2 þ p�1bk2

2½erþaðT�T0Þþpbk2 �

h i TT0ðk6 � p�2Þ � s1k

5

NkT0

" #vuuutð18Þ

E�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

p�2þ p�1bk2

2½erþaðT�T0 Þþpbk2 �

h i1þaðT�T0 Þþpbk2

er

h i TT0ðp�4k�2�p�6k�8Þ�s1p�4k�3

NkT0

" #vuuut ð19Þ

SNk¼ c0

Nkln

TT0� 1

2½ð1þ p2Þk2 þ p�2k�4 � 3�

þ aD�2p�2k�4

2Nk½er þ aðT � T0Þ þ pbk2�2ð20Þ

Then by submitting Eq. (18) into Eq. (20), the entropycan be expressed as a function of stretch and temperature:

SNk¼ c0

Nkln

TT0� 1

2½ð1þ p2Þk2 þ p�2k�4 � 3�

þ ap�2

½2er þ 2aðT � T0Þ þ 3p�1bk2�Tðk2 � p�2k�4Þ � sk

Nk

� �ð21Þ

Further, to investigate the relation of entropy and nom-inal electric field, we submit Eq. (13) into Eq. (20),

SNk¼ c0

Nkln

TT0� 1

2ð1þ p2Þk2 þ p�2k�4 � 3

þ aE�2p2k4

2Nkð22Þ

By referring to Jean-Mistral’s experimental research(Jean-Mistral et al., 2010) of polyacrylate very-high-bond(VHB) dielectric elastomer, some material parameters usedfor the simulation are shown in Table 1. The representativevalue of a and b shown in Table 1 are calculated accordingto experimental results. Further, an initial temperature ofT0 = 273 K and a maximum temperature of Tmax = 373 Kare also selected to precede the analysis. These are alsothe temperatures under which the dielectric elastomerwas experimentally investigated by Jean-Mistral et al.

Based on Eqs. (18) and (19), under the condition ofequal biaxial condition (p = 1), the relation of nominal elec-tric field and nominal electric displacement under differenttemperature is presented in Fig. 2, and the nominal electricfield under different temperatures and stretches is de-scribed in Fig. 3. Clearly, on each curve, the voltage first in-creases, reaches a peak, and then decreases. With theincrease of temperature, the critical nominal electric fieldincreases, namely the thermoelectromechanical stabilityincreases. The entropy of dielectric elastomer under differ-ent temperatures and stretches are presented in Fig. 4.With the increase of temperature and decrease of stretch,the entropy increases.

Similarly, under unequal biaxial state (p – 1), the rela-tion among different performance parameters of dielectricelastomer’s thermodynamic system is given in Figs. 5–7. Asshown in Fig. 5, with increase of stretch, the critical nom-inal electric field of dielectric elastomer decreases. FromFig. 6, it can be seen that with the increase of stretch, crit-ical stretch decreases.

Table 2 summarizes certain values of critical nominalelectric field and critical stretch of dielectric elastomer un-der different temperatures and stretches. As an example, inthe state of equal biaxial stretch, the critical nominal elec-tric field of dielectric elastomer is 0.695

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

pand

0.851ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

pwhen the temperatures are 273 and

373 K respectively. The critical stretch is 1.26, which isthe same as that obtained by Zhao and Suo (2007).

Fig. 12. The electric-induced temperature and entropy variation of dielectric elastomer.

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6-0.11-0.10-0.09-0.08-0.07-0.06-0.05-0.04-0.03

β (F

/m)

p0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

1.1

1.2

1.3

1.4

1.5

1.6

λC

p

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.61.01.52.02.53.03.54.04.55.0

ΔS (J

/kgK

)

p0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ΔT (K

)

p

Fig. 13. Temperature and entropy variation of dielectric elastomer under different stretches.

42 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

5. Allowable energy range of dielectric elastomers underthe coupling effect of electric field and temperature

The allowable energy range of dielectric elastomer isdetermined by the critical conditions which the elastomers

could encounter during their working process. The mainfailure mode of soft materials such as dielectric elastomeris electromechanical instability (Zhao and Suo, 2010).Therefore, the E�EMI value is the most important issue todetermine the allowable energy range.

L. Liu et al. / Mechanics of Materials 72 (2014) 33–45 43

Considering biaxial stretch that the principle planarstretch ratio is set as p, i.e. k2 ¼ pk1 ¼ pk, according to Eq.(19), at a constant s, when

s1NkT0¼ T

T0

f12bþ4½erþaðT�T0Þ�gk�p�2f30bþ16½erþaðT�T0Þ�gk�5

15bþ6½erþaðT�T0Þ�, the function

E�ðk; sÞ reaches its maximum value. This maximum nominalelectric field corresponds to the critical voltage for the onsetof the electromechanical instability. By submitting it intoEqs. (18) and (19), the formulations of the dimensionlessnominal electric displacement nominal electric field, thenominal displacement and the entropy can be inducedrespectively.

D�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0er

p

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þaðT�T0Þþpbk2

er

h ip�2þ p�1bk2

2 erþaðT�T0Þþpbk2½ �

� � TT0

3bþ2½erþaðT�T0Þ�15bþ6½erþaðT�T0Þ�

ðk6þ5p�2Þ� �vuuuut ð23Þ

E�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNkT0=er

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

p�2 þ p�1bk2

2 erþaðT�T0Þþpbk2½ �

� �1þ aðT�T0Þþpbk2

er

h i TT0

3bþ 2½er þ aðT � T0Þ�15bþ 6½er þ aðT � T0Þ�

ðp�4k�2 þ 5p�6k�8Þ� �vuuut ð24Þ

SNk¼ c0

Nkln

TT0� 1

2½ð1þ p2Þk2 þ p�2k�4 � 3�

þ Taðp�2k2 þ 5p�4k�4Þ½2er þ 2aðT � T0Þ þ 3pbk2�

� 3bþ 2½er þ aðT � T0Þ�15bþ 6½er þ aðT � T0Þ�

ð25Þ

The allowable energy range on the plane of dielectricelastomer’s nominal electric field and nominal electric dis-placement is the encircled range by four curves, which re-flect the four critical conditions as initial temperature T0,maximum temperature T, zero electric field E� = 0, as wellas electromechanical instability E�EMIðT0Þ. Due to the factthat the dielectric elastomer gets more stable with theincreasing temperature, the EMI curve under the lowertemperature, i.e. initial temperature is selected as the con-trolling curve. Further, based on Eqs. (21) and (25), theallowable energy range on the plane of dielectric elasto-mer’s temperature and entropy can also be determined.

Figs. 8 and 9 show the allowable energy range (thehatching range) in terms of nominal electric field vs. dis-placement and temperature vs. entropy respectively. Here,equal biaxial stretch is considered, i.e. p = 1, and the criticalstretch value is 1.26 obtained from the EMI analysis. Sim-ilarly, the allowable energy ranges of dielectric elastomerundergoing unequal biaxial stretches in which p = 0.9 and1.1 are plotted in Figs. 10 and 11. These results would beof great help in analyzing and understanding the physicalprocess of dielectric elastomer undergoing different tem-peratures and electric fields.

Table 3 illustrates the entropy variation and allowableenergy range under different conditions. For the conditionof equal biaxial stretch, namely k1 ¼ k2 ¼ k, the entropy

variation is 1.22 J/kgK and the range is 122 J/kg. Evidentlywhen dielectric elastomer undergoes equal biaxial stretch,allowable energy range is minimum.

6. Electric-induced temperature and entropy variationof dielectric elastomers

Applying an electric field to polar dielectric materials,represented by ferroelectric polymer materials such as P(VDF-TrFE) or P (VDF-TrFE-CFE), large variation in tempera-ture and entropy will be induced due to the electrocaloric ef-fect, which shows potential in designing and fabrication ofcooling devices (Neese et al., 2008; Lu and Zhang, 2009). Inthis session, the temperature and entropy change of dielec-tric elastomer under high electric field will be evaluated.

When the electric filed applied on dielectric elastomerincreases from a low voltage, the temperature and theentropy variation will be generated. The plane of workconjugated parameters entropy-temperature is applied to

demonstrate this process. In this plane, the allowablerange of dielectric elastomer’s thermoelectromechanicalsystem is encircled by four curves T, T + DT, E� and E�EMI .In the allowable range, each point in a plane represents aspecified temperature and voltage state of dielectric elas-tomer, each curve represents a temperature and voltagevariation process, and each cycle represents a possiblethermodynamics energy cycle. The maximum energy cycleprocess can be described in Fig. 12 since the temperaturevariation has reached a maximum value.

As shown in Fig. 12, for example, under the condition ofequal biaxial, when the referenced temperatures are 273and 283 K respectively, the temperature changes of dielec-tric elastomer are 0.2 and 0.2 K, the entropy changes are1.22 and 1.23 J/kgK respectively. Under the condition ofunequal biaxial (p = 0.9), the temperature changes ofdielectric elastomer are 0.28 and 0.28 K, the entropychanges are 1.67 and 1.70 J/kgK respectively.

Under different stretch ratios in the principal plane,Fig. 13 represents the coefficient of electrostriction, thecritical stretch, and the temperature and entropy variationof dielectric elastomer with electric field applied. Clearly,when 0:5 6 p < 1, with the increase of stretch ratio inthe principal plane, the temperature and entropy variationdecreases. To obtain a maximum temperature and entropychange, dielectric elastomer application cooling deviceswith small p should be selected.

7. Conclusion

In this paper, we firstly proposed the expression ofdielectric constant relying on temperature and stretch,established free energy function of dielectric elastomer’s

44 L. Liu et al. / Mechanics of Materials 72 (2014) 33–45

thermodynamic system under an adiabatic process, anddeduced constitutive relation. Under equal and unequalbiaxial conditions, the thermoelectromechanical stabilityof dielectric elastomer was studied. As a result, with the in-crease of temperature and decrease of the ratio betweenprincipal planar stretches, the stability of dielectric elasto-mer materials or structures increases. Then, the thermody-namic performance of dielectric elastomer undergoingvarious temperatures and electric fields is studied. Theallowable energy range of dielectric elastomer under thecondition of equal and unequal biaxial stretches is de-scribed. Finally, the electric-induced temperature and en-tropy variation of dielectric elastomer is evaluated. When0:5 6 p < 1, with the decrease of the ratio between princi-pal planar stretches, the temperature and the entropy var-iation increases. Numerical results will offer great help inguiding the design and fabrication of dielectric elastomerapplication devices, such as dielectric elastomer actuatorsoperating in variable ambient temperatures.

Acknowledgments

The one-year visit of Liwu Liu to Harvard Universitywas supported by the China Scholarship Council Founda-tion. The author would like to appreciate the esteemedguidance of Prof. Zhigang Suo at Harvard University inthe theories of dielectric elastomers. This work is sup-ported by the National Natural Science Foundation of Chi-na (Grant Nos. 11102052, 11272106, 11225211,51205076), China Postdoctoral Science Foundation (GarntNo. 2012M520032) and the Fundamental Research Fundsfor the Central Universities (Grant No. HIT.NSRIF.2013030).

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