Dynamic electromechanical performance of viscoelastic dielectric elastomersJunjie Sheng, Hualing Chen, Lei Liu, Junshi Zhang, Yongquan Wang, and Shuhai Jia

Citation: Journal of Applied Physics 114, 134101 (2013); doi: 10.1063/1.4823861 View online: http://dx.doi.org/10.1063/1.4823861 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/114/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Viscoelastic effects on frequency tuning of a dielectric elastomer membrane resonator J. Appl. Phys. 115, 124106 (2014); 10.1063/1.4869666 Cyclic performance of viscoelastic dielectric elastomers with solid hydrogel electrodes Appl. Phys. Lett. 104, 062902 (2014); 10.1063/1.4865200 Viscoelastic deformation of a dielectric elastomer membrane subject to electromechanical loads J. Appl. Phys. 113, 213508 (2013); 10.1063/1.4807911 Energy harvesting of dielectric elastomer generators concerning inhomogeneous fields and viscoelasticdeformation J. Appl. Phys. 112, 034119 (2012); 10.1063/1.4745049 Model of dissipative dielectric elastomers J. Appl. Phys. 111, 034102 (2012); 10.1063/1.3680878

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Dynamic electromechanical performance of viscoelastic dielectricelastomers

Junjie Sheng (盛俊杰),1,2 Hualing Chen (陈花玲),1,2,a) Lei Liu (刘磊),1,2

Junshi Zhang (张军诗),2,3 Yongquan Wang (王永泉),1,2 and Shuhai Jia (贾书海)1,21School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China2State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University,Xi’an 710049, China3School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China

(Received 18 July 2013; accepted 16 September 2013; published online 1 October 2013)

Because of their viscoelasticity, dielectric elastomers (DEs) are able to produce a large time-

dependent electromechanical deformation. In the current study, we use the Euler-Lagrange

equation to characterize the influence of temperature, excitation frequency, and viscoelasticity on

the dynamic electromechanical deformation and stability of viscoelastic dielectrics. We investigate

the time-dependent dynamic performance, hysteresis, phase diagram, and Poincare map associated

with the viscoelastic dissipative process. The results show that the dynamic response has strong

temperature and frequency dependencies. It is observed that the natural frequency of the DE

decreases with increasing temperature and the maximal amplitude increases at higher temperatures.

At relative low frequencies, the amplitude is very small and the viscoelasticity has a significant

effect on the oscillation of the system. Furthermore, the results show that the viscoelasticity has a

relatively major influence on the dynamic performance for DEs that have very low relaxation

times.VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4823861]

I. INTRODUCTION

Because of their fast response time, softness, lightweight,

low-cost, and high energy density, dielectric elastomers

(DEs) have been developed for use in high performance

applications such as artificial muscles, Braille displays, life-

like robots, tunable lenses, and power generators.1–6 They

usually consist of a soft elastomeric membrane sandwiched

between two compliant electrodes. Most of the existing stud-

ies on DEs have focused on quasi-static deformation,2,6

neglecting the effect of inertia and viscoelasticity.

However, to perform as an electromechanical actuator, a

DE is often subject to transient, time-dependent forces, and

voltages7–9 and is mostly expected to deform at high fre-

quencies in applications,10 where inertia can play a signifi-

cant role in the dynamic application. Applications exploiting

the dynamic behavior of DEs have long been realized

through experiments investigating, for example, vibrotactile

displays for mobile applications,11 frequency tuning,12

pumps,13 and acoustic actuator.14 Recently, research has

been also carried out on modeling the nonlinear vibrations of

hyperelastic DE membranes.15–19 Zhu et al.15 studied the

resonant behavior of a pre-stretched DE membrane and sub-

sequently analyzed the nonlinear oscillations of a DE bal-

loon.16 Based on simple geometrical and spherical capacitor

assumptions, Yong et al.17 investigated the dynamics of a

thick-walled DE spherical shell. Li et al.18 analyzed the elec-

tromechanical and dynamic of tunable pure-shear DE-based

resonator and identified the safe operation range for failure

prevention while actuating the resonator. Xu et al.19 obtained

an analytical model for the DE by the Euler-Lagrange

equation to study the dynamic analysis of a DE with stretch-

ing deformation. Soares et al.20 presented a mathematical

model for the nonlinear vibration analysis of a radially

pre-stretched hyperelastic annular membrane under finite

deformations and compared results obtained using the finite

element method.

Although the above studies have attempted to model the

nonlinear dynamical behavior of the DE to probe its time-

dependent performance, all neglected the viscoelasticity of

DEs. The time-dependency can cause dissipation in the sys-

tem and significantly affect its dynamic performance and

coupling efficiency.21,22 In particular, experiments have

shown that viscoelasticity can significantly influence the elec-

tromechanical transduction and its application.4,8,21–23

Moreover, the dielectric constant and the shear modulus of

DEs present a deformation and temperature-dependence

behavior.24–27 Meanwhile, the viscoelastic relaxation of a DE

also depends on temperature and frequency.28 Thus, a study

on temperature effects and how that affects the dynamic per-

formance of a DE is expected to take temperature into

account in a new model.

Based on the theory of non-equilibrium thermodynamics

of viscoelastic DEs,29–31 and the nonlinear vibration

modeling,15–19 we aim to characterize the dynamic electro-

mechanical actuation of viscoelastic DE, to predict how tem-

perature, frequency, and viscoelastic processes affect its

dynamic performance and to present a physical interpretation

of the dynamic deformation.

In this study, the influence of the temperature, fre-

quency, and viscoelasticity on the dynamic characteristics of

a DE membrane is considered using the Euler-Lagrange

equation to take into account the temperature- and

a)Author to whom correspondence should be addressed. Email:

hlchen@mail.xjtu.edu.cn

0021-8979/2013/114(13)/134101/8/$30.00 VC 2013 AIP Publishing LLC114, 134101-1

JOURNAL OF APPLIED PHYSICS 114, 134101 (2013)

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deformation-dependent permittivity and the temperature de-

pendence of the modulus. Through solving the governing

equations, the dynamic responses of a viscoelastic DE film

subject to a cyclic electric field are investigated. Then, the

dynamic oscillation, phase diagrams, and Poincare maps of

the viscoelastic elastomer are studied. A detailed analysis

shows the influence of the temperature, frequency, and vis-

coelasticity on the dynamic stability and the hysteresis.

II. EXPERIMENTS

Very-high-bond (VHB) poly-acrylic film is the most

widely studied dielectric material because of its low cost,

large deformation, and commercial availability as transparent

adhesive tapes from the 3M Company.32 In our experiment,

we selected VHB 4910, for which the original thickness is

1mm, to study its viscoelasticity, where the material parame-

ters in the proposed model are expected to be obtained.

By stretching the specimen at a prescribed nominal

strain of 100%, we are able to record the stress relaxation

with time at different temperatures by the dynamic mechani-

cal analysis (DMA Q800 V7.5, TA Instruments, New Castle,

Delaware, USA) on rectangular cross-sectioned specimens

under the single-cantilever clamping mode. Temperatures

for the specimen were set at 20K intervals of 273, 293, 313,

333, 353, and 373K in a flowing nitrogen atmosphere.

Figure 1 plots the relaxation curves of the viscoelastic

dielectric at the nominal strain of 100% for six of the differ-

ent temperatures. The relaxation curves for the measured

nominal stress can be fitted with the following multi-

exponentials equation:

r ¼r1 þ r1 exp�t

s

� �þ r2 exp

�t

s

� �� �2

þ r3 exp�t

s

� �� �3

; (1)

where r is the nominal stress, r1 the long term nominal

stress,33 t the time, r1, r2, and r3 are the parameters describ-

ing the distribution of the relaxation stress, and s is the aver-age relaxation time, which depends on the relaxation

behavior. Using multi-exponentials in data fitting is required

because the stress relaxation of DE has a distribution of

relaxation times. The values of the various fitting parameters

are given in Table I.

The dependence of the relaxation time on temperature

can be appropriately expressed by the Arrhenius law34

sðTÞ ¼ s1 þ s0 expEa

RT

� �; (2)

where s1 is the relaxation time at infinitely high tempera-

ture, s0 is the pre-factor, T is the absolute temperature, R is

the gas constant, and Ea is the activation energy of the relax-

ation process.

A simple curve fitting procedure has been used for deter-

mining the parameters, s1, s0, and Ea, in Eq. (2). Figure 1(b)

shows good agreement between Eq. (2) and the calculated

relaxation times in Table I, and the coefficient of determination

R2 > 0:99. The experimental data obtained from Fig. 1(a) can

be fitted to the Eq. (2) using a common set of parameters:

Ea ¼ 3:201� 104 J/mol, s1 ¼ 63:64 s, s0 ¼ 1:766� 10�5 s,

and the coefficient of determination R2 > 0:998. We use these

relaxation parameters in the following analysis.

III. MODELING OF THE VISCOELASTIC DIELECTRICELASTOMER

A thin film of viscoelastic dielectric deforms when sub-

ject to both in-plane biaxial force and a voltage-induced

force across its surfaces. Figure 2 illustrates a membrane of a

DE, sandwiched between two compliant electrodes. As the

material is incompressible, the DE deforms from its original

configuration 2L� 2L� 2H to the new configuration

2L=ffiffiffik

p � 2L=ffiffiffik

p � 2kH in terms of an equal biaxial expan-

sion. Here, we define k as the stretch ratio in the thickness

direction. The stretch k is homogeneous through the DE and

is a function of time. The nominal electric field is ~E ¼U=ð2HÞ and the nominal electric displacement is defined by~D ¼ Q=ð2L� 2LÞ, the true electric field by E ¼ U=ð2kHÞ,and the true electric displacement by D ¼ kQ=ð2L� 2LÞ.The true electric field relates to the true electric displacement

as D ¼ eE with e the permittivity of the DE.

FIG. 1. (a) Curve fitting with Eq. (1)

of the nominal stress relaxation at dif-

ferent temperatures. (b) Curve fitting

with Eq. (2) of the relaxation process.

TABLE I. Fitting parameters for the application of the Eq. (1) to the nomi-

nal stress relaxation response of VHB 4910.

Temperature (K) r1=MPa r1=MPa r2=MPa r3=MPa s=s

273 0.07232 0.1667 �0.3869 0.3568 87.216

293 0.04802 0.08028 �0.1834 0.1696 72.377

313 0.04289 0.04068 �0.09253 0.08661 67.802

333 0.03987 0.02688 �0.05977 0.05603 65.573

353 0.03798 0.01801 �0.04134 0.03803 64.747

373 0.03632 0.01038 �0.02460 0.02240 63.875

134101-2 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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Experiments have verified that the electromechanical

responses of DEs are highly rate-dependent, which implies

that the deformation and actuation of DEs are highly reliant on

the rates of mechanical or electrical activation.6,7 This rate-

dependence is mainly induced by the viscoelasticity of the

elastomeric polymer matrix and consequently might influence

the electromechanical actuation.23 Viscoelastic relaxation can

be represented by a rheological model of springs and dash-

pots.29,30 As the DE exhibits elastic deformation and inelastic

deformation, the latter of which is time-dependent, here, we

first model this material by assuming it to be composed of two

molecular chain networks, A and B, as sketched in Fig. 3.

Network A is an ideal hyperelastic chain and deforms reversi-

bly, whereas network B relaxes in time and dissipates energy.

The viscous deformation is represented by a dashpot.

We sketched an equal bi-axially deformed dielectric mem-

brane. For the spring-dashpot network along the bottom, the

overall deformation (k) arises from both spring and dashpot.

We adopt the well-established multiplication rule that29–31,35

k ¼ ken; (3)

where ke represents stretch due to the bottom spring, and nrepresents stretch due to the dashpot.

We assume that the middle planes of the DE ð0; 0; 0Þshow no normal displacement. In the following, the equation

of motion for the DE is obtained by the Euler-Lagrange

equation19

@‘

@k� d

dt

@‘

@ _k

� �¼ 0; ‘ ¼ V �W; (4)

where ‘ is the Lagrangian, with V the kinetic energy and Wthe potential of the conservative forces in the system.

The kinetic energy can be obtained in the form19

V ¼ 2

3qHL4

_k2

k3þ 4

3qH3L2 _k

2; (5)

where q is the density of the elastomer.

By noting that such DE actuators can operate at different

temperatures, the temperature dependence of the elastic

modulus and the dielectric constant is included. We repre-

sent the elasticity of each network by the neo-Hookean

model, in which the free energy associated with the deforma-

tion of an incompressible thermo-viscoelastic DE can be

written as26

Wdef ¼ lAðTÞ2

2

kþ k2 � 3

� �þ lBðTÞ

22nkþ k2

n2� 3

!

þ q0c0 T � T0 � T lnT

T0

� �� �; (6)

where lAðTÞ and lBðTÞ are the temperature-dependent shear

moduli of each elastomer network, and lAðTÞ=lBðTÞ ¼ 3=7,which is suggested by fitting the experimental data.31 In the

following statement, we define the instantaneous modulus

lðTÞ ¼ lAðTÞ þ lBðTÞ ¼ YðTÞ=3, where YðTÞ signifies the

isothermal elastic modulus in small deformation at tempera-

ture T.26

Subject to the electric voltage, the electrostatic energy

takes the form36

We ¼ 1

2e0erðk; TÞ

~E2

k2; (7)

where e0erðk; TÞ ¼ e, e0 ¼ 8:85� 10�12 F/m the permittivity

of the vacuum, and erðk; TÞ the relative permittivity, which

is a function of temperature and deformation25

erðk; TÞ ¼ e1 þ A

T

� �ð1þ að2k�1=2 � 2Þ þ bð2k�1=2 � 2Þ2

þ cð2k�1=2 � 2Þ3Þ; (8)

with e1 ¼ 2:1, A ¼ 960, a ¼ �0:1658, b ¼ 0:04086, and

c ¼ �0:003027.Because of the homogeneity, the potential W is obtained

by multiplying the sum of the free energy density owing to

the deformation and the electrostatic energy with the volume

W ¼ 8HL2

lAðTÞ2

2

kþ k2 � 3

� �þ lBðTÞ

22nkþ k2

n2� 3

!

þq0c0 T� T0 � T lnT

T0

� �� �� 1

2e0erðk;TÞ

~E2

k2

266664

377775:

(9)

FIG. 2. Schematics of a viscoelastic DE subject biaxial force in the plane

and voltage through the thickness applied via stretchable electrodes. In the

reference state, a membrane of a DE is un-deformed and has dimensions

2L� 2L� 2H. In the deformed state, subject to voltage U, the membrane

attains dimensions 2l� 2l� 2h, where k1 ¼ k2 ¼ l=L and k ¼ k3 ¼ h=H.Because of symmetries in the x–y plane, the incompressibility relation fur-

ther leads to k1 ¼ k2 ¼ 1=ffiffiffik

p.

FIG. 3. Schematic illustration of a viscoelastic model for DE, consisting of

two parallel elements: one a spring A and the other a spring B connected to

a dashpot in series. lA and lB are the shear moduli of each elastomer net-

work, respectively. g is the viscosity of the dashpot.

134101-3 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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Substituting the expressions of V in Eq. (5) and W in Eq. (9) into the Euler-Lagrange equation (4), we obtain the govern-

ing equation

€k � 3

2

1

kþ c1k4_k2

þ c2

ð1þ c1k3Þ lAðTÞðk4 � kÞ þ lBðTÞð�nkþ k4n�2Þ þ e ~E

2 � k2e0@er@k

~E2

� �¼ 0; (10)

where c1¼2H2=L2, c2¼6=ðqL2Þ, and

@erðk; TÞ=@k ¼ e1 þ A

T

� �ð�ak�3=2 � 2bk�3=2ð2k�1=2 � 2Þ

� 3ck�3=2ð2k�1=2 � 2Þ2Þ:

The rate of deformation in the dashpot is described by

dn=dt.31 Specifically, we set �@Wdef =@n ¼ gdn=dt.30 We

relate this deformation to the stress on the dashpot and write

dndt

¼ lB

gðk2n�3 � k�1Þ: (11)

The viscoelastic relaxation time, defined as the ratio of

the viscosity of the dashpot and the modulus of spring B,

g=lB ¼ sðTÞ, is a function of temperature, which can be cal-

culated from Eq. (2).

Equations (10) and (11) constitute a complete set of equa-

tions of state for the specific Neo-Hookean model for thermo-

viscoelastic elastomers, which we shall use in the following

analysis on the dynamic performance of viscoelastic elastomer.

IV. DYNAMIC ANALYSIS OF VISCOELASTICDIELECTRIC ELASTOMER

We now use the governing equation for viscoelastic DEs

to simulate the vibration, oscillation, and viscoelasticity behav-

ior of the viscoelastic DE. A membrane of a DE is subjected to

a time-dependent voltage UðtÞ. Because of the nonlinearity of

the system, the dynamic response under time-dependent

electric loading can be very complicated. We applied a cyclic

load ~E ¼ UðtÞ=ð2HÞ ¼ E0 sinð2pftÞ in Eq. (10), with f the fre-quency of the applied electric loading. In the following study,

the amplitude of the nominal electric field is kept constant at

E0 ¼ 12� 103 kV/m,19 so that the material is not subject to in-

stantaneous instability within the selected frequency range.

In the following, the dynamic response of the DE is

investigated using numerical solutions of Eqs. (10) and (11).

The voltage is applied at time t ¼ 0. The dashpot does not

move instantaneously, thence the initial value of the internal

variable is nð0Þ ¼ 1. Assume that the system is activated

from the un-deformed configuration and the voltage is

instantaneously applied; the initial condition is then given by

kð0Þ ¼ 1 and _kð0Þ ¼ 0. In calculations, we set parameters

for the application at: H ¼ 1:0� 10�3 m, L ¼ 5:0� 10�3 m,

q¼1:2�103 kg/m3,19 c1 ¼ 2H2=L2 ¼ 0:08, c2 ¼ 6=ðqL2Þ¼ 200, lAðTÞ¼ 0:3YðTÞ=3, lBðTÞ¼ 0:7YðTÞ=3, and YðTÞ¼ 0:2001ð1000=TÞ2�1:078ð1000=TÞþ1:518 MPa.26,31

A. Effect of temperature

From experiments, temperature is seen to have great

influence on the dielectric and mechanical properties of the

DEs,25–27 therefore, can greatly affect the actuation perform-

ance of the DE.37 Recently, static thermo-electro-elastic

models have been presented to analyze the effect of tempera-

ture on transduction devices based on DEs. However, the

temperature effects are seldom analyzed in the dynamic

response context of the viscoelasticity DE.

The frequency dependence of the amplitude at three dif-

ferent temperatures, T ¼ 290, 300, and 310K is shown in

Fig. 4. A large peak appears around f ¼ 421Hz for

T ¼ 290K, which lies in the natural frequency range identi-

fied using numerical solutions. The DE resonates strongest

when the frequency of excitation is around the natural fre-

quency. As expected, with increasing temperature, the peak

amplitude increases and the natural frequency decreases.

The maximum values for the amplitude are about 1.238 at

421Hz for 290K, 1.330 at 374Hz for 300K, and 1.512 at

334Hz for 310K. The DE also resonates when the frequency

of excitation is several times the natural frequency or a frac-

tion of the natural frequency.15 In addition, results show the

existence of natural frequencies of other orders, for instance,

the peak at f ¼ 223 Hz and f ¼ 883Hz for 290K, the peak

at f ¼ 197Hz and f ¼ 781Hz for 300K, and the peak at f ¼177 Hz and f ¼ 703Hz for 310K.

Figure 5 illustrates the vibrations and hysteresis loop of

the viscoelastic DE at the three different temperatures,

T ¼ 290K, 300K, and 310K with frequency f ¼ 335Hz. It

is seen that the system experiences strong oscillations under

these three temperatures. As the temperature increases, the

amplitude of the total stretch, k, increases while the ampli-

tude of the viscous stretch, n, is markedly smaller than that

FIG. 4. Oscillating amplitude of the viscoelastic DE is plotted as a function

of the frequency of excitation at T ¼ 290, 300, and 310K.

134101-4 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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of k. This is because the voltage was varied on a time

scale much faster than the viscoelastic relaxation time, the

total stretch is relatively large (Figs. 5(a)–5(c)), whereas

the viscous effect is negligible due to the fast loading rate

(Figs. 5(d)–5(f)). Furthermore, when the temperature

increases from 290K to 310 K, the hysteresis loop can

be observed on both the ~E � k (Fig. 5(i)), and ~E � ~D=e(Fig. 5(l)) plots at 310 K, indicating more dissipation of

energy at higher temperatures.

The effect of temperature on the phase paths and the

Poincare maps are plotted in Fig. 6. The k� _k phase paths

are presented in a closed region in Figs. 6(a)–6(c). The

Poincare map is used to better detect the stability transition

of the system from low temperature (290K) to high tempera-

ture (310K). The Poincare maps for each of the three tem-

peratures at the frequency of 335Hz all form closed loops.

The results show that the viscoelastic DE system experiences

a nonlinear quasi-periodic oscillation and the dynamic oscil-

lations of the viscoelastic DE under these temperatures are

stable.

B. Effect of frequency

To study the effect of frequency of the applied voltage

on the dynamic response, the temperature is held at a con-

stant value of T ¼ 300K. A set of representative results is

presented in Figs. 7 and 8.

The dynamic response of the viscoelastic DE at low fre-

quency (1Hz) for a fixed temperature of 300K is plotted in

Fig. 7. When the DE is actuated by a voltage alternating at a

relatively low frequency, the magnitude of the total stretch kis very small and the mean value of the total stretch drifts

away from the initial equilibrium state as shown in Fig. 7(a),

whereas the viscous effect becomes significant and the vis-

cous stretch n in Fig. 7(b) is now comparable with the stretch

k. No clear limit cycle is seen in the ~E � k plot in Fig. 7(c)

and in the ~E � ~D=e plot in Fig. 7(d), indicating insignificant

hysteresis. Fig. 7(e) is the k� _k phase diagram for the visco-

elastic DE. The Poincare map for the 1Hz is disordered

(Fig. 7(f)), and the dynamic response of the system at 1Hz

will undergo an aperiodic motion.

FIG. 5. Total stretch k (a), (b), and (c),

and the viscous stretch n (d), (e), and (f)

in response to the applied cyclic

nominal electric field ~E ¼ E0 sinð2p ftÞ.The corresponding nominal electric

field-stretch responses are shown in

(g), (h), (i) and the nominal electric

field-nominal electric displacement

responses in (j), (k), (l) for the fre-

quency f ¼ 335Hz. Three different

temperatures are used in the calcula-

tion: T ¼ 290K for (a), (d), (g), and (j),

T ¼ 300K for (b), (e), (h), and (k), and

T ¼ 310K for (c), (f), (i), and (l).

134101-5 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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Figure 8 shows the dynamic response of the visco-

elastic DE at a relative high frequency of 400Hz around its

natural frequency. The viscoelastic DE system exhibits

beating at this higher frequency, which does not exist at

1 Hz (Fig. 7(a)). At the same time, the amplitude of the total

stretch increases and the value of the viscous stretch in

Fig. 8(b) decreases compared with those at 1 Hz (Fig. 7(b)).

Furthermore, ~E � k plot in Fig. 8(c) and ~E � ~D=e plot in

Fig. 8(d) show hysteresis, indicating that energy is dissi-

pated during the cycle at the higher frequency. The k� _kphase diagram at 400Hz is shown in Fig. 8(e); the corre-

sponding Poincare map forms a closed loop (Fig. 8(f)),

showing that the viscoelastic DE undergoes a nonlinear

quasi-periodic oscillation.

C. Effect of viscoelastic relaxation

In this case, the temperature is held fixed at 300K and

the effect of the viscoelastic relaxation time on the dynamic

response is simulated.

When DE is assumed to be hyperelastic, there will be no

viscoelastic dissipative process, so the governing equation of

the hyperelastic DE specializes to

€k� 3

2

1

kþ c1k4_k2 þ c2

ð1þ c1k3Þ

� YðTÞ3

ðk4 � kÞ þ e ~E2 � k

2e0@er@k

~E2

� �¼ 0: (12)

The effect of the relaxation time on the frequency de-

pendence of the amplitude is shown in Fig. 9. When tak-

ing viscoelastic relaxation into account, the low

relaxation time is seen to have a significant influence on

the amplitude of the DE (Fig. 9(a)). The largest peak is

0.3817 for s ¼ 0:01, which happens at 382 Hz (red curve

in Fig. 9(a)), and the maximal amplitude attains a much

bigger value of 1.339 for the hyperelastic DE which

appears at 375 Hz (black dashed curve in Fig. 9(a)).

However, when the relaxation time is relatively high, for

instance, the relaxation time of VHB 4910 described by

sðTÞ of Eq. (2), the temperature dependence of the visco-

elastic relaxation properties has a relatively minor influ-

ence on the amplitude-frequency response characteristic

in Fig. 9(b). Results show that this viscoelastic relaxation

time has a little effect on the natural frequency of the sys-

tem. The natural frequency of the hyperelastic dielectric

elastomer appears around f ¼ 375 Hz. The peak ampli-

tude changes from 1.330 at 374Hz for the VHB 4910

DE (red curve in Fig. 9(b)) to 1.339 at 375Hz for the

hyperelastic dielectric elastomer (black dashed curve in

Fig. 9(b)), where the changes are very small. That is, for

the VHB 4910 viscoelastic DE system, the viscoelastic

relaxation has a relatively minor influence on the dynamic

performance.

FIG. 7. Time response of the total stretch (a) and the viscous stretch (b), tra-

jectory plots of ~E � k response (c), and ~E � ~D=e response (d). The phase

diagram (e) and the Poincare map are for the excitation frequency f ¼ 1 Hz

at the temperature of 300K.

FIG. 6. Phase diagrams and Poincare

maps of the viscoelastic DE for the fre-

quency of 335Hz with temperature

(a) and (d) 290K, (b) and (e) 300K,

(c) and (f) 310K, respectively.

134101-6 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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129.89.24.43 On: Sat, 31 May 2014 10:52:04

V. CONCLUSIONS

We used the Euler-Lagrange equation to guide the de-

velopment of the dynamic models for viscoelastic DEs. The

approach is illustrated by a specific model to describe the fi-

nite deformation of a viscoelastic DE taking into account the

temperature dependence of the shear moduli, the permittiv-

ity, and the relaxation time.

FIG. 8. Applied frequency of the cycle

voltage is 400Hz. The calculated time

responses are shown for (a) total

stretch, (b) viscous stretch, (c) ~E � kresponse, (d) ~E � ~D=e response,

(e) phase diagram, and (d) the Poincare

map for frequency 400Hz at tempera-

ture 300K.

FIG. 9. Oscillating amplitude of the viscoelastic DE is plotted as a function of frequency at different relaxation times with fixed temperature of 300K.

(a) Comparison of the frequency dependence of the amplitude between the viscoelastic elastomer with a relaxation time of s ¼ 0:01 and a hyperelastic DE.

(b) Comparison of the frequency dependence of the amplitude between the VHB 4910 viscoelastic elastomer with a relaxation time of s ¼ sðT ¼ 300KÞ � 70:25 sand a hyperelastic DE.

134101-7 Sheng et al. J. Appl. Phys. 114, 134101 (2013)

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The results showed that when the temperature increases,

the natural frequency of the viscoelastic DE system decreased

and the maximal amplitude increases. Solving the equation of

motion, the oscillation, hysteresis, phase diagram, and the

Poincare map for the viscoelastic DE system are obtained

under a cycle voltage. The viscoelastic DE underwent a non-

linear quasi-periodic vibration around the natural frequency.

In addition, the magnitude of the total stretch was very small,

and the mean value of total stretch drifted away from the ini-

tial equilibrium state for the low frequency of 1Hz.

Viscoelasticity had a significant influence on the dynamic

response of the system at low frequencies. Moreover, the rel-

ative high relaxation time had little or no effect on the

amplitude-frequency response characteristic and the relative

low relaxation time can have a big influence on the

amplitude-frequency response characteristic of the DE.

ACKNOWLEDGMENTS

This research was supported by the Doctoral Fund of the

Ministry of Education of China (Grant No. 20120201110030)

and the Major Program of National Natural Science

Foundation of China (Grant No. 51290294). The authors

gratefully acknowledge these supports.

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