investigation on the viscoelastic behaviors of a circular dielectric elastomer membrane undergoing large deformation

AIP ADVANCES 6, 125127 2016Investigation on the viscoelastic behaviors of a circular dielectric elastomer membrane undergoing large deformation Bing Wang, Zhengang Wang, and Tianhu Hea S

membrane undergoing large deformation

Bing Wang, Zhengang Wang, and Tianhu He

Citation: AIP Advances 6, 125127 (2016); doi: 10.1063/1.4973639

View online: http://dx.doi.org/10.1063/1.4973639

View Table of Contents: http://aip.scitation.org/toc/adv/6/12

Published by the American Institute of Physics

AIP ADVANCES 6, 125127 (2016)

Investigation on the viscoelastic behaviors of a circular dielectric elastomer membrane undergoing large

deformation

Bing Wang, Zhengang Wang, and Tianhu Hea

School of Science, Lanzhou University of Technology, Lanzhou 730050,

People’s Republic of China

(Received 7 November 2016; accepted 22 December 2016; published online 30 December 2016)

To explore the time-dependent dissipative behaviors of a circular dielectric elas-tomer membrane subject to force and voltage, a viscoelastic model is formulated based on the nonlinear theory for dissipative dielectrics The circular membrane

is attached centrally to a light rigid disk and then connected to a fixed rigid ring When subject to force and voltage, the membrane deforms into an out-of plane shape, undergoing large deformation The governing equations to describe the large defor-mation are derived by using energy variational principle while the viscoelasticity

of the membrane is describe by a two-unit spring-dashpot model The evolutions

of the considered variables and the deformed shape are illustrated graphically In calculation, the effects of the voltage and the pre-stretch on the electromechanical behaviors of the membrane are examined and the results show that they significantly influence the electromechanical behaviors of the membrane It is expected that the present model may provide some guidelines in the design and application of such

dielectric elastomer transducers © 2016 Author(s) All article content, except where

otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4973639]

I INTRODUCTION

As a class of electroactive polymers, dielectric elastomers (DEs) possess a unique attribute: large deformation When subject to a voltage across its thickness, a membrane of DEs coated on both surfaces with compliant electrodes reduces in thickness and expands in area As reported by Pelrine

et al in 2000, such voltage-induced strain can readily exceed 100%.1In addition to the capability

of large deformation, DEs have other intrinsic attributes, such as light weight, high efficiency, low cost, noise free etc., which make them attractive for applications as transducers in artificial muscles, actuators and sensors, energy harvesters, soft robotics, adaptive optics etc.2 10

To deform the DE transducers distinctly, usually, the required electric field is huge and the DE transducers commonly consist of thin, flat or cylindrical membranes.1,10,11 As a con-sequence, the DE transducers may suffer from several modes of failure, such as electric break down, electromechanical instability, loss of tension, etc., which would degrade their perfor-mance.12,13 To address such issues and reveal the electromechanical behaviors of DEs, a number

of theoretical or experimental investigations were reported in recent years.14–26 The existing investigations mostly relate to the time-independent elastic behaviors of the DEs However, most DEs are rubber-like materials and they often involve in time-dependent, dissipative pro-cesses, such as conductive relaxation, dielectric relaxation and viscoelastic relaxation.27,28 Exper-iments have proved that viscoelasticity can significantly affect the electromechanical behaviors

of DEs.29 – 32 To further reveal the viscoelastic effect on the DEs, a lot of efforts have been contributed.33 – 37

a Corresponding author E-mail: heth@lut.cn

2158-3226/2016/6(12)/125127/10 6, 125127-1 © Author(s) 2016

In present paper, a viscoelastic model for a circular dielectric elastomer membrane is presented in the context of the nonlinear theory of dissipative dielectrics.38The membrane is centrally attached to

a light rigid disk and then mounted on a rigid ring When subject to force and voltage, the membrane deforms into an out-of plane shape, undergoing large deformation The governing equations for characterizing the large deformation as well as the viscoelasticity of the membrane are formulated respectively The evolutions of the considered variables and the deformed shape of the membrane are obtained numerically and illustrated graphically This work is a further development of that presented

by He et al.,26in which only the time-independent, elastic behaviors were studied

II GOVERNING EQUATIONS FOR DEFORMATION

Fig.1illustrates the schematic cross section of a circular DE membrane In the reference state (Fig.1(a)), the membrane is of thickness H and radius B and coated on both surfaces with compliant

electrodes In the deformed state (Fig.1(b)), the membrane is pre-stretched and fixed to an outer ring

of radius b, and a light rigid circular disk of radius a, is attached to the center of the membrane Particle

A and particle B move to the place a and the place b respectively When subject to a combination of a

constant force F and a voltage Φ, the membrane deforms into an out-of plane axisymmetric shape and the disk moves downward with displacement u Meanwhile, an amount of charge Q is accumulated

on both surfaces

To describe the deformation, let z be the vertical coordinate, r be the horizontal coordinate and

the coordinate origin coincide with the center of the rigid ring The DE membrane is of viscoelastic

in nature so that all the considered physical variables are time-dependent In the deformed state, a

general particle R moves to a place with coordinates (r(R,t),z(R,t)), where t is time variable Consider two nearby particles in the reference state, R and R + dR They move to new places (r(R,t),z(R,t) and (r(R,t) + dr,z(R,t) + dz) respectively in the deformed state Thus, the longitudinal stretch is

λ1=

s

dr dR

!2

+ dz

dR

!2

(1) The latitudinal stretch is

λ2=r

Denote θ as the slope at the particle(r(R,t),z(R,t) Then we get dr = dl cos θ and dz = −dl sin θ, where

dl = p(dr)2+ (dz)2, so that

∂r

∂z

The DE membrane is assumed to be incompressible, so that λ1λ2λ3= 1, where λ3 is the stretch in

the thickness direction Let W be the Helmholtz free energy of an element of the membrane divided

FIG 1 Schematic cross section of a circular DE membrane.

125127-3 Wang, Wang, and He AIP Advances 6, 125127 (2016)

by the volume of the element in the reference state To characterize the viscoelasticity of the DE

membrane, the free-energy density function W is prescribed as38

W = W λ1, λ2, ¯D, ξ1, ξ2,  (5) The state of the DE membrane is described by stretches λ1, λ2 and the nominal electric displace-ment ¯D together with internal variables (ξ1, ξ2, ) When there are small variations with amounts

δλ1, δλ2, δ ¯D, δξ1, δξ2, of the independent variables, the free-energy density varies by

δW =∂λ∂W

1

δλ1+∂λ∂W

2

δλ2+∂W ∂ ¯D δ ¯D +X

i

∂W

The coefficients in front of δλ1, δλ2and δ ¯D are defined as

s1=∂W(λ1, λ2, ¯D, ξ1, ξ2, )

s2=∂W(λ1, λ2, ¯D, ξ1, ξ2, )

¯

E=∂W(λ1, λ2, ¯D, ξ1, ξ2, )

where s1, s2 and ¯E are interpreted as the nominal longitudinal stress, the nominal latitudinal stress

and the nominal electric field respectively Once the expression of the free-energy density

func-tion W  λ1, λ2, ¯D, ξ1, ξ2,  is specified, Eqs (7a)–(7c) constitute the state equations of the DE membrane

For the membrane, non-equilibrium thermodynamics demands that the increase in the free energy should not exceed the total work done by the external loads, that is

2πH

 B

A

where Q = 2π ∫ ¯DRdR is the total charge accumulated on the electrode.

From Eqs (1) and (2), we obtain

δλ1= cos θd (δr)

dR −sin θ

d (δz)

δλ2=δr

Inserting Eqs (6), (7a)–(7c), (9), and (10) into the inequality (8) and integrating by parts, we obtain

[2πRHs1cos θδr] B A + (2πRHs1sin θ − F)δu

+



[2πHd(Rs1sin θ)]δz+



[2πHs2dR − 2πHd(Rs1cos θ)]δr

+



[2πHR ¯ EdR − 2πΦRdR]δ ¯ D + 2πH

 (X

i

∂W

∂ξiδξi )RdR ≤ 0 (11) Once the membrane is assumed to be in mechanical and electrostatic equilibrium, the inequality (11) decomposes into two expressions as

[2πRHs1cos θδr] B A + (2πRHs1sin θ − F)δu

+



[2πHd(Rs1sin θ)]δz+



[2πHs2dR − 2πHd(Rs1cos θ)]δr

+



i

∂W

From Eq (12), we obtain the governing equations as

∂ (Rs1cos θ)

∂ (Rs1sin θ)

and

[s1cos θδr] B A= 0, 2πRHs1sin θ − F= 0 (17) Let σ1(R, t) be the true longitudinal nominal stress, σ2(R, t) be the true latitudinal nominal stress,

E(R,t) be the true electric field and D(R,t) be the true electric displacement respectively The true

quantities relate to the nominal quantities as σ1= s1λ1, σ2= s2λ2, E= λ1λ2E and D¯ = ¯D/λ1λ2 In terms of these relations, Eqs (14)–(16) can be rewritten as

∂

Rσ1

λ 1 cos θ

R∂R −

σ2

∂

Rσ1

λ 1 sin θ

E= λ1λ2Φ

In the deformed state, the membrane deforms into an axisymmetric shape, and the boundary conditions are

r (A, t) = a, r (B, t) = b, z (B, t) = 0 (21)

III VISCOELASTIC MODEL OF THE DIELECTRIC ELASTOMER

Viscoelastic dielectrics have been studied theoretically by using rheological models of springs and dashpots recently Here, we adopt a rheological model including two parallel units (Fig.2).38 One unit consists of a spring with shear modulus µαand the other consists of another spring with shear modulus µβ connected in series with a dashpot with viscosity η In this rheological model, the two stretches of the DE membrane λ1 and λ2 are assumed to be the net stretches of both units For the top spring, the stretches are λ1and λ2 For the bottom spring, however, the stretches can be represented as λe1= λ1ξ−1

1 and λe2= λ2ξ−1

2 , where ξ1 and ξ2 are the stretches due to the dashpot

We adopt the neo-Hookean model to characterize the elasticity of the membrane, and specify the free-energy density function as

W (λ1, λ2, ¯D, ξ1, ξ2)=µα

2 (λ1 + λ1 + λ1−2λ2−2−3) +µβ

2 (ξ1

−2λ1 + ξ2−2λ2 + ξ1 ξ2 λ1−2λ2−2−3)

+D¯2

125127-5 Wang, Wang, and He AIP Advances 6, 125127 (2016)

FIG 2 Viscoelastic model of the dielectric elastomer.

Substituting (22) into (7a)–(7c) and using the relations between the nominal quantities and the true quantities, we obtain the state equations as

σ1= µα λ1 −λ1−2λ2−2 + µβ λ1 ξ1−2−λ1−2λ2−2ξ1 ξ2 

−εE2

σ2= µα λ2 −λ1−2λ2−2 + µβ λ2 ξ2−2−λ1−2λ2−2ξ1 ξ2 

−εE2

D = εE

(23)

In Eq (23), ε is the permittivity of the dielectric elastomer membrane and εE2is the Maxwell stress

As proposed by Zhao et al.,38the kinetic model of the membrane can be written as follows by using the free-energy function in Eq (22)

dξ1

ξ1dt = 1

3η



µβ λ1 ξ1−2−λ1−2λ2−2ξ1 ξ2 

− µβ

2  λ2 ξ2−2−λ1−2λ2−2ξ1 ξ2 

dξ2

ξ2dt = 1

3η



µβ λ2 ξ2−2−λ1−2λ2−2ξ1 ξ2 

− µβ

2  λ1 ξ1−2−λ1−2λ2−2ξ1 ξ2 

(24)

In the current model, ξ1−1dξ1/dt and ξ−1

2 dξ2/dt denote the rate of deformation in the dashpot

and the dashpot is modeled as Newtonian fluid It is noted here that this kinetic model satisfies the thermodynamic inequality when η > 0

Here, we aim to study the viscoelastic relaxation of the dielectric elastomer membrane To simplify the problem, we assume that the relaxation time is much longer than the time scale for the pre-stretch process Upon this assumption, the initial conditions can be determined as

Eq (25) implies that at the initial time t = 0, both springs carry the applied load and the dashpot does

not deform

IV NUMERICAL SIMULATION

A combination of Eqs (18) and (19) along with Eq (23) gives

dθ

dR= − λ1σ2

dλ1

dR = 1

m1

" 1

R

σ2

λ2

cos θ −σ1

λ1

!

−m2

R (λ1cos θ − λ2)+ m3

dξ1

dR + m4

dξ2

dR

#

(27) where

m1 = µα1+ 3λ1−4λ2−2 + µβ ξ1−2+ 3λ1−4λ2−2ξ1 ξ2 

−εE2λ1−2,

m2 = 2f λ1−3λ2−3 µαλ1+ µβλ1ξ1 ξ2 

−εE2λ1−1λ2−1g

,

m3 = 2µβ λ1ξ1−3+ λ1−3λ2−2ξ1ξ2 

,

m4 = 2µβλ1−3λ2−2ξ2ξ1

Rewrite Eq (24) as

dξ1

dt = 1

3η



µβ λ1 ξ1−1−λ1−2λ2−2ξ1 ξ2 

− µβ

2  λ2 ξ1ξ2−2−λ1−2λ2−2ξ1 ξ2 

dξ2

dt = 1

3η



µβ λ2 ξ2−1−λ1−2λ2−2ξ1 ξ2 

− µβ

2  λ1 ξ2ξ1−2−λ1−2λ2−2ξ1 ξ2 

(28)

By means of the initial conditions in (25), the four variables r(R,t), z(R,t), θ(R, t) and λ1(R, t) at

t0= 0 can be obtained numerically from Eqs (26) and (27) and Eqs (3) and (4) by using the shooting

method Subsequently, by setting a suitable time interval ∆t and using the obtained variables at t0= 0,

we can obtain ξ1and ξ2at time t1= t0+ ∆t from Eq (28) through the improved Euler method Then,

by using the obtained ξ1 and ξ2 at t1, we can get r(R,t), z(R,t), θ(R, t) and λ1(R, t) at time t1 from Eqs (26) and (27) and Eqs (3) and (4) by the shooting method By repeating the above procedure, all the considered variables can be obtained step by step

V RESULTS AND DISCUSSIONS

In designing a device using the configuration in Fig.1, three dimensionless parameters can be

var-ied: a/A, b/B and b/a These parameters may be tailored to modify the performance of the transducer.

To illustrate the viscoelastic behaviors of the membrane, two cases are concerned in calculation In simulation, we take µα= µβ= µ/2 To normalize the variables, these dimensionless quantities are

introduced:F/(2π µHa), Φ/(Hpµ/ε), R/a, t/t v Here, tν= η/µβis called as the viscoelastic relaxation time

A Case one

In case one, we examine how the voltage influences the viscoelastic behaviors of the membrane

by fixing a/A = b/B = 1.2,b/a = 4 and F/(2π µHa)= 0.7 The obtained results are illustrated in Figs.3 5

Fig.3shows the evolutions of the longitudinal stretch λ1, the vertical displacement u, the true

longitudinal stress σ1 and the true electric field E of the material particles along along the cir-cumference of the rigid disk In calculation, four different voltages Φ/(Hpµ/ε) = 0.3, 0.35, 0.4 as well as 0.45 are considered As shown in Figs.3(a)–3(d), for small voltages, for example, voltages,

for example, Φ/(Hpµ/ε) = 0.3 and 0.35, all the considered variables change little after a period

FIG 3 The evolutions of (a) the longitudinal stretch λ 1, (b) the vertical displacement u, (c) the true longitudinal stress σ1 ,

and (d) the true electric field E of the particles along the circumference of the rigid disk.

125127-7 Wang, Wang, and He AIP Advances 6, 125127 (2016)

FIG 4 Distributions of the longitudinal stretch λ 1 and the latitudinal stretch λ 2 in the membrane when subject to two different

voltages Φ/(Hp

µ/ε) = 0.35 and 0.4 respectively.

of time and the membrane gradually evolves into stable state, while for larger voltages, for

exam-ple, Φ/(Hpµ/ε) = 0.4 and 0.45, all the considered variables change remarkably and the membrane reaches no stable state

Fig.4plots the distributions of the longitudinal stretch λ1and the latitudinal latitudinal stretch

λ2in the membrane when subject to two different voltages Φ/(Hpµ/ε) = 0.35 and 0.4 respectively

As seen, for small voltage Φ/(Hpµ/ε) = 0.35, both stretches change little after t/t v = 10.0 and

the membrane becomes stable, while, for larger voltage Φ/(Hpµ/ε) = 0.4, both stretches change dramatically and the membrane becomes unstable It also can be observed from Fig.4(a)that the longitudinal stretch λ1in the membrane decreases monotonically from inside to outside In Fig.4(b), the latitudinal stretch λ2keeps constant at both edges, which coincides with the boundary conditions Fig.5shows the profiles of the deformed shape of the membrane when subject to two different

voltages Φ/(Hp

µ/ε) = 0.35 and 0.4 respectively Similarly, for small voltage Φ/(Hpµ/ε) = 0.35, the profile of the deformed shape of the membrane changes little after t/t v= 10 and the membrane

evolves into stable state, while for larger voltage Φ/(Hpµ/ε) = 0.4, the deformed membrane becomes unstable

B Case two

In case two, we examine how the designing parameters, namely, a/A and b/B, affects the elec-tromechanical behaviors of the membrane when b/a = 4 and F/(2π µHa)= 0.7 In calculation, two

FIG 5 The profiles of the deformed shape of the membrane when subject to two different voltages (a) Φ/(Hp

µ/ε) = 0.35

and (b) Φ/(Hp

µ/ε) = 0.4 respectively.

FIG 6 The evolutions of (a) the longitudinal stretch λ 1, (b) the vertical displacement u, (c) the true longitudinal stress σ1 ,

and (d) the true electric field E of the particles along the circumference of the rigid disk when the membrane is subject to voltage Φ/(Hp

µ/ε) = 0.35.

different combinations of a/A and b/B, i.e., a/A = b/B = 1.0 and a/A = b/B = 1.4, are specified, and

the obtained results are illustrated in Figs.6and7

Fig.6shows the evolutions of the longitudinal stretch λ1, the vertical displacement u, the true

longitudinal stress σ1and the true electric field E of the particles along the circumference of the rigid disk when the membrane is subject to voltage Φ/(Hpµ/ε) = 0.35 For a/A = b/B = 1.0, it implies that the membrane has no pre-stretch before deformation, while for a/A = b/B = 1.4, it means that the

FIG 7 The evolutions of (a) the longitudinal stretch λ 1, (b) the vertical displacement u, (c) the true longitudinal stress σ1 ,

and (d) the true electric field E of the particles along the circumference of the rigid disk when the membrane is subject to voltage Φ/(Hp

µ/ε) = 0.4.

125127-9 Wang, Wang, and He AIP Advances 6, 125127 (2016)

membrane undergoes pre-stretch before deformation As observed, the values of these four variables

increase when the membrane is pre-strained Under Φ/(Hpµ/ε) = 0.35, the four variables evolve into stable values after a period of time

Fig.7also shows the evolutions of the longitudinal stretch λ1, the vertical displacement u, the

true longitudinal stress σ1 and the true electric field E of the particles along the circumference of the rigid disk when the membrane is subject to voltage Φ/(Hpµ/ε) = 0.4 By comparing Fig.7to Fig.6, it can be realized that the membrane becomes unstable under voltage Φ/(Hpµ/ε) = 0.4, and

the pre-stretch a/A = b/B = 1.4 acts to trigger the occurrence of the unstable much earlier.

VI CONCLUSIONS

The time-dependent dissipative processes, such as viscoelastic relaxation, conductive relaxation and dielectric relaxation, significantly affect the electromechanical behaviors of dielectric transduc-ers To illustrate such issues, we formulate a viscoelastic model for a dielectric elastomer membrane attached centrally to a rigid disk, deforming into an out-of plane shape when subject to force and voltage The evolutions of the considered variables and the shapes of the deformed membrane are illus-trated graphically In simulation, the effects of the voltage and the pre-stretch on the electromechanical behaviors of the membrane are investigated, and the results show that they significantly influence the electromechanical behaviors of the membrane For small voltage, the membrane reaches stable state after a period of time, while for the voltage beyond a critical value, the membrane becomes unstable For pre-stretch, when the voltage is small, the membrane evolves into stable state, and the considered variables increase with the pre-stretch, while for larger voltage, say, which is beyond a critical value, the membrane becomes unstable and the pre-stretch acts to trigger the occurrence of the unstable much earlier It is expected that the present model may provide some guidelines in the design and application of such dielectric elastomer transducers

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (11372123) The authors gratefully acknowledge the financial support Particularly, we would like to sincerely thank Huiming Wang, Professor of Zhejiang University, for his great help to the accomplishment of this work

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