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

Investigation on the viscoelastic behaviors of a circular dielectric elastomer 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 November 2016; accepted 22 December 2016; published online 30 December 2016) To explore the time-dependent dissipative behaviors of a circular dielectric elastomer 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 deformation 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%.1 In 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 consequence, 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 performance.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 processes, such as conductive relaxation, dielectric relaxation and viscoelastic relaxation.27,28 Experiments 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 125127-2 Wang, Wang, and He AIP Advances 6, 125127 (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.38 The 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.,26 in which only the time-independent, elastic behaviors were studied II GOVERNING EQUATIONS FOR DEFORMATION Fig illustrates 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 = dr dR + dz dR (1) The latitudinal stretch is λ2 = r R (2) 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 = (dr)2 + (dz)2 , so that ∂r = λ cosθ ∂R (3) ∂z = −λ sinθ ∂R (4) The DE membrane is assumed to be incompressible, so that λ λ λ = 1, where λ is the stretch in the thickness direction Let W be the Helmholtz free energy of an element of the membrane divided FIG 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 ¯ ξ1 , ξ2 , W = W λ , λ , D, (5) The state of the DE membrane is described by stretches λ , λ and the nominal electric displace¯ together with internal variables (ξ1 , ξ2 , ) When there are small variations with amounts ment D ¯ δξ1 , δξ2 , of the independent variables, the free-energy density varies by δλ , δλ , δD, δW = ∂W ∂W ∂W ¯ δD + δλ + δλ + ¯ ∂λ ∂λ ∂D i ∂W δξi ∂ξi (6) ¯ are defined as The coefficients in front of δλ , δλ and δD s1 = ¯ ξ1 , ξ2 , ) ∂W (λ , λ , D, ∂λ (7a) s2 = ¯ ξ1 , ξ2 , ) ∂W (λ , λ , D, ∂λ (7b) ¯ ξ1 , ξ2 , ) ∂W (λ , λ , D, E¯ = ¯ ∂D (7c) 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¯ ξ1 , ξ2 , is specified, Eqs (7a)–(7c) constitute the state equations of the DE tion W λ , λ , D, 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 B 2πH δWRdR ≤ Fδu + Φ δQ (8) A ¯ where Q = 2π ∫ DRdR is the total charge accumulated on the electrode From Eqs (1) and (2), we obtain δλ = cos θ d (δr) d (δz) − sin θ dR dR (9) δr (10) R Inserting Eqs (6), (7a)–(7c), (9), and (10) into the inequality (8) and integrating by parts, we obtain δλ = [2πRHs1 cos θδr]BA + (2πRHs1 sin θ − F)δu + [2πHd(Rs1 sin θ)]δz + [2πHs2 dR − 2πHd(Rs1 cos θ)]δr + ¯ ¯ + 2πH [2πHREdR − 2πΦRdR]δD ( i ∂W δξi )RdR ≤ ∂ξi (11) Once the membrane is assumed to be in mechanical and electrostatic equilibrium, the inequality (11) decomposes into two expressions as [2πRHs1 cos θδr]BA + (2πRHs1 sin θ − F)δu + [2πHd(Rs1 sin θ)]δz + [2πHs2 dR − 2πHd(Rs1 cos θ)]δr + ¯ ¯ =0 [2πHREdR − 2πΦRdR]δD (12) 125127-4 Wang, Wang, and He AIP Advances 6, 125127 (2016) ∂W δξi ≤ ∂ξi i (13) From Eq (12), we obtain the governing equations as ∂ (Rs1 cos θ) = s2 ∂R (14) ∂ (Rs1 sin θ) =0 ∂R (15) H E¯ = Φ (16) and [s1 cos θδr]BA = 0, 2πRHs1 sin θ − F = (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 ¯ λ In quantities relate to the nominal quantities as σ1 = s1 λ , σ2 = s2 λ , E = λ λ E¯ and D = D/λ terms of these relations, Eqs (14)–(16) can be rewritten as ∂ Rσ λ1 cos θ R∂R − σ2 =0 λ2R ∂ Rσ λ1 sin θ R∂R E = λ1 λ2 =0 Φ H (18) (19) (20) 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) = (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 µα and 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 λ and λ are assumed to be the net stretches of both units For the top spring, the stretches are λ and λ For the bottom spring, however, the stretches can be represented as λ e1 = λ ξ1−1 and λ e2 = λ ξ2−1 , 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 ¯ ξ1 , ξ2 ) = µα (λ + λ + λ −2 λ −2 − 3) W (λ , λ , D, µβ −2 + (ξ1 λ + ξ2 −2 λ 2 + ξ1 ξ2 λ −2 λ −2 − 3) ¯ −2 −2 D + λ1 λ2 (22) 2ε 125127-5 Wang, Wang, and He AIP Advances 6, 125127 (2016) FIG 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 λ − λ −2 λ −2 + µβ λ ξ1 −2 − λ −2 λ −2 ξ1 ξ2 − εE σ2 = µα λ 2 − λ −2 λ −2 + µβ λ 2 ξ2 −2 − λ −2 λ −2 ξ1 ξ2 − εE (23) D = εE In Eq (23), ε is the permittivity of the dielectric elastomer membrane and εE is the Maxwell stress As proposed by Zhao et al.,38 the kinetic model of the membrane can be written as follows by using the free-energy function in Eq (22) µβ dξ1 = µβ λ ξ1−2 − λ −2 λ −2 ξ1 ξ2 − λ 2 ξ2 −2 − λ −2 λ −2 ξ1 ξ2 ξ1 dt 3η µβ dξ2 = µβ λ 2 ξ2 −2 − λ −2 λ −2 ξ1 ξ2 − λ ξ1 −2 − λ −2 λ −2 ξ1 ξ2 ξ2 dt 3η (24) In the current model, ξ1−1 dξ1 /dt and ξ2−1 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 > 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 ξ1 (R, 0) = 1, ξ2 (R, 0) = (25) 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 λ σ2 dθ =− sin θ dR λ σ1 R (26) dλ 1 σ2 σ1 m2 dξ1 dξ2 (λ cos θ − λ ) + m3 = cos θ − − + m4 dR m1 R λ λ1 R dR dR (27) where m1 = µα + 3λ −4 λ −2 + µβ ξ1 −2 + 3λ −4 λ −2 ξ1 ξ2 − εE λ −2 , m2 = λ −3 λ −3 µα λ + µβ λ ξ1 ξ2 − εE λ −1 λ −1 , m3 = 2µβ λ ξ1 −3 + λ −3 λ −2 ξ1 ξ2 , m4 = 2µβ λ −3 λ −2 ξ2 ξ1 125127-6 Wang, Wang, and He AIP Advances 6, 125127 (2016) Rewrite Eq (24) as µβ dξ1 µβ λ ξ1 −1 − λ −2 λ −2 ξ1 ξ2 − = λ 2 ξ1 ξ2 −2 − λ −2 λ −2 ξ1 ξ2 dt 3η µβ dξ2 µβ λ 2 ξ2 −1 − λ −2 λ −2 ξ1 ξ2 − = λ ξ2 ξ1 −2 − λ −2 λ −2 ξ1 ξ2 dt 3η (28) By means of the initial conditions in (25), the four variables r(R,t), z(R,t), θ(R, t) and λ (R, t) at t = 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 t = 0, we can obtain ξ1 and ξ2 at time t1 = t0 + ∆t from Eq (28) through the improved Euler method Then, by using the obtained ξ1 and ξ2 at t , we can get r(R,t), z(R,t), θ(R, t) and λ (R, t) at time t 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 varied: 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/(H µ/ε), R/a, t/t v Here, tν = η/µβ is 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 = and F/(2π µHa) = 0.7 The obtained results are illustrated in Figs 3–5 Fig shows the evolutions of the longitudinal stretch λ , the vertical displacement u, the true longitudinal stress σ1 and the true electric field E of the material particles along along the circumference of the rigid disk In calculation, four different voltages Φ/(H µ/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/(H µ/ε) = 0.3 and 0.35, all the considered variables change little after a period FIG The evolutions of (a) the longitudinal stretch λ1 , (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 Distributions of the longitudinal stretch λ1 and the latitudinal stretch λ2 in the membrane when subject to two different voltages Φ/(H µ/ε) = 0.35 and 0.4 respectively of time and the membrane gradually evolves into stable state, while for larger voltages, for example, Φ/(H µ/ε) = 0.4 and 0.45, all the considered variables change remarkably and the membrane reaches no stable state Fig plots the distributions of the longitudinal stretch λ and the latitudinal latitudinal stretch λ in the membrane when subject to two different voltages Φ/(H µ/ε) = 0.35 and 0.4 respectively As seen, for small voltage Φ/(H µ/ε) = 0.35, both stretches change little after t/t v = 10.0 and the membrane becomes stable, while, for larger voltage Φ/(H µ/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 λ in the membrane decreases monotonically from inside to outside In Fig 4(b), the latitudinal stretch λ keeps constant at both edges, which coincides with the boundary conditions Fig shows the profiles of the deformed shape of the membrane when subject to two different voltages Φ/(H µ/ε) = 0.35 and 0.4 respectively Similarly, for small voltage Φ/(H µ/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/(H µ/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 electromechanical behaviors of the membrane when b/a = and F/(2π µHa) = 0.7 In calculation, two FIG The profiles of the deformed shape of the membrane when subject to two different voltages (a) Φ/(H µ/ε) = 0.35 and (b) Φ/(H µ/ε) = 0.4 respectively 125127-8 Wang, Wang, and He AIP Advances 6, 125127 (2016) FIG The evolutions of (a) the longitudinal stretch λ1 , (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/(H 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 and Fig shows the evolutions of the longitudinal stretch λ , 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/(H µ/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 The evolutions of (a) the longitudinal stretch λ1 , (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/(H 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/(H µ/ε) = 0.35, the four variables evolve into stable values after a period of time Fig also shows the evolutions of the longitudinal stretch λ , 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/(H µ/ε) = 0.4 By comparing Fig to Fig 6, it can be realized that the membrane becomes unstable under voltage Φ/(H µ/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 transducers 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 illustrated 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 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