1 Introduction

Dielectric elastomers (DEs) are classified as soft active materials, and they are capable of large deformations. When subject to a voltage across its thickness, a DE membrane sandwiched between two compliant electrodes can induce large deformations by contracting in the thickness and expanding in the area. Pelrine et al.^[1] reported that the voltage-induced strain in DEs can easily exceed 100%. Since then, this unique attribute along with other inherent attractive features, such as light weight, low cost, high efficiency, and silent operation, has attracted intensive interests in recent years^[2-6]. DEs can easily be fabricated into a wide range of shapes and structures to realize electromechanical transduction^[7-10]. Emerging applications include actuators and sensors, energy harvesters, soft robotics, adaptive optics, artificial muscles, and refreshable displays^[11-18]. Suo^[19] has recently reviewed the theory of DEs. Existing models of DEs mostly focus on elastic behaviors^[20-25]. However, most DEs are rubber-like polymers, and they often suffer from time-dependent and dissipative processes, such as conductive relaxation, dielectric relaxation, and viscoelastic relaxation^[26-27]. Experiments^[28-29] have shown that viscoelasticity can significantly affect the electromechanical behaviors of DEs. To further explore the viscoelastic effect on the DE, a lot of theoretical or experimental studies have been reported^[30-34].

In this paper, a viscoelastic model for a circular DE membrane actuator containing a concentric rigid inclusion is established to depict its electro-viscoelastic behaviors. When subject to the pressure and voltage, the membrane undergoes out-of-plane large deformations. The governing equations to characterize large deformations and viscoelasticity of the membrane are derived, respectively. Evolutions of the considered variables and profiles of the deformed shape are obtained and illustrated graphically.

2 overning equations

Figure 1 illustrates the cross section of a circular DE membrane containing a concentric rigid inclusion. In the undeformed state (see Fig. 1(a)), the circular DE membrane is of the thickness H and the radius B and coated on both surfaces with compliant electrodes. A specific particle is at the distance A from the center O, and a general particle is at the distance R from the center. In the deformed state (see Fig. 1(b)), the membrane is centrally attached to a light rigid circular disk of the radius a, and then connected to a fixed ring of the radius b. The particle A moves to the place a, and the particle B moves to the place b. The ratio A/B is defined as the geometric parameter of the membrane, and the ratios a/A and b/B are defined as the pre-stretches of the membrane. These ratios can be used as design parameters to optimize such devices or to enhance the electromechanical performance of such devices. When the membrane is subject to a pressure P and a voltage Φ across its thickness, it inflates into an out-of-plane axisymmetric shape. The thickness of the membrane becomes h, and an amount of charge Q accumulates on both electrodes. Despite the types of the mechanical loadings, such membrane-disk configurations are popular and crucial in the family of DE devices such as universal muscle actuators, marked by Artificial Muscle, Inc., and have attracted much attention in theoretical investigations^{[8, 10, 21]} as actuators or experimental studies on the efficiency as generators^[14]. Such configurations may also be used as balloon catheters, valves, or vibrators. The big differences of our work from the previous ones on inhomogeneous large deformations lie in two aspects. One is our configuration being a membrane-disk combination instead of a membrane only as in Refs. [32] and [35], and the other is our investigation focusing on the viscoelastic behaviors instead of elastic behaviors only as in Refs. [8], [10], and [21].

To characterize the out-of-plane deformation, let z be the coordinate along the symmetric axis of the configuration, r be the coordinate along the radial direction, and the coordinate origin locate at the center of the rigid ring. The DE is viscoelastic in nature so that all the considered variables are time-dependent. In the deformed state, the particle R moves to a place with coordinates (r(R, t), z(R, t)), where t is a time variable. Consider two nearby particles, R and R+dR, in the undeformed state. They move to (r(R, t), z(R, t)) and (r(R, t)+dr, z(R, t)+dz) in the deformed state, respectively. The longitudinal stretch defined by the distance dl between the two particles in the deformed state divided by that in the undeformed state is

(1)

The latitudinal stretch defined by the perimeter of a circle in the deformed state divided by that in the undeformed state is

(2)

Denote θ(R, t) as the slope at the particle (r(R, t), z(R, t)). Then, we get dr=dlcos θ and dz=-dlsin θ, where dr=r(R+dR, t)-r(R, t) and dz=z(R+dR, t)-z(R, t), so that

(3)

(4)

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 by the volume of the element in the undeformed state. To characterize the viscoelasticity of the DE membrane, the free-energy density function W is prescribed as^[27]

(5)

The state of the DE membrane is described by the two independent stretches λ₁ and λ₂ and the nominal electric displacement D together with internal variables (ξ₁, ξ₂, …). When there are small variations (δλ₁, δλ₂, δD, δξ₁, δξ₂, …) of the independent variables, the free-energy density varies by

(6)

The coefficients in front of δλ₁, δλ₂, and δD are defined as

(7a)

(7b)

(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 function 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 requires that an increase in the free energy should not exceed the total work done by the external loads, that is,

(8)

where Q=2π∫DRdR is the total charge accumulated on the electrode, and is the volume enclosed by the membrane and the rigid disk.

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

(9)

(10)

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

(11)

Once the membrane is assumed to be in mechanical and electrostatic equilibrium, the inequality (11) decomposes into

(12)

and

(13)

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

(14)

(15)

(16)

and the boundary conditions,

(17)

(18)

The true quantities relate to the nominal quantities as σ₁ =s₁λ₁, σ₂ =s₂λ₂, E=λ₁λ₂E, and D=D/(λ₁λ₂), where σ₁(R, t), σ₂(R, t), E(R, t), and D(R, t) are the true longitudinal nominal stress, the true latitudinal nominal stress, the true electric field, and the true electric displacement, respectively. In terms of the above relations, Eqs. (14)-(16) can be rewritten as

(19)

(20)

(21)

In the deformed state, the boundary conditions are

(22)

3 Viscoelastic model

Viscoelastic dielectrics have recently been investigated theoretically by using rheological models of springs and dashpots. Here, we adopt a rheological model of two parallel elements^[28]. One element consists of the spring with a shear modulus μ_α, and the other consists of the spring with a shear modulus μ_β connected in series with the dashpot with a viscosity η (see Fig. 2).

In this rheological model, the two stretches λ₁ and λ₂ are the net stretches of both elements. For the top spring, the stretches are λ₁ and λ₂. For the bottom spring, however, the stretches are λ₁^e =λ₁ξ₁^-1 and λ₂^e = λ₂ξ₂^-1, where ξ₁ and ξ₂ are the stretches in the dashpot.

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

(23)

Substituting (23) into (7a)-(7c) and using the relations between the nominal quantities and the true quantities, we obtain the state equations as

(24)

In Eq. (24), ε is the permittivity of the DE membrane.

As proposed by Zhao et al.^[27] and Kollosche et al.^[31], the kinetic model of the membrane can be written as follows by using the free-energy function in Eq. (23):

(25)

In the current model, and denote the rates of deformations in the dashpot, and the dashpot is modeled as the Newtonian fluid. It is noted here that this kinetic model satisfies the thermodynamic inequality when η > 0.

Here, our attention is mainly paid to the viscoelastic relaxation of the DE 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

(26)

Equation (26) implies that, at the initial time t=0, the dashpot still holds, and both springs bear the applied load. The elastic behaviors of the membrane can be described by the two springs. After a quite long time, the spring attached with the dashpot fully relaxes and carries no load. Hence, the elastic behaviors of the membrane can be characterized by a single spring μ_α.

4 Numerical simulation

From Eqs. (19), (20), and (24), we obtain

(27)

A combination of Eq. (18) with the first equation in Eq. (24) gives

(28)

Take μ_α =μ_β =μ/2 and introduce non-dimensional quantities P^* =P/(μH/B), Φ^* = , R^* =R/B, and r^* =r/B. Equation (28) can be rewritten as

(29)

Rewrite Eq. (25) as

(30)

By applying the initial conditions in (26), the values of r(R, t), z(R, t), θ(R, t), and λ₁(R, t) at t=t₀ =0 can be obtained respectively from ordinary differential equations (3), (4), and (27) with the algebraic equation (29) by means of the shooting method. Subsequently, by setting a suitable time interval Δt and using the obtained values of r(R, t), z(R, t), θ(R, t), and λ₁(R, t) at t=t₀, the values of ξ₁ and ξ₂ at the time t₁ =t₀ +Δt can be obtained from Eq. (30) through the improved Euler method. Then, by using the obtained ξ₁ and ξ₂ at the time t₁, the values of r(R, t), z(R, t), θ(R, t), and λ₁(R, t) at t₁ can thus be obtained from Eqs. (3) and (4) with Eqs. (27) and (29) by the shooting method. By repeating the above procedure, all the considered variables can be obtained step by step.

5 Results and discussion

To illustrate the electromechanical viscoelastic behaviors of the membrane, four different cases are considered. In the simulation, we take μ_α =μ_β =μ/2, and the non-dimensional quantities are introduced to normalize the variables, i.e., P^* =P/(μH/B), Φ^* = , σ₁^* =σ₁/μ, σ₂^* =σ₂/μ, E^* = , R^* =R/B, z^* =z/B, r^* =r/B, and t^* =t/t_ν, where t_ν =η/μ_β is called the viscoelastic relaxation time.

(1) Case 1

In Case 1, we investigate how the pressure influences the viscoelastic behaviors of the membrane. The DE membrane is only subject to the pressure, and no voltage is applied. In the calculation, the ratio A/B=0.3, and the pre-stretches are a/A=b/B=1.0. The obtained results are illustrated in Figs. 3-5.

Figure 3 shows evolutions of the longitudinal stretch λ₁, the height z^* , the true longitudinal stress σ₁^*, and the stretch ξ₁ in the dashpot of the particles around the circumference of the disk when the membrane is subject to the pressure. In the calculation, three different pressures P^* =0.7, 0.8, and 0.9 are taken, respectively. At the very beginning of deformations, the dashpot in Fig. 2 does not deform, and the elastic deformation of the membrane is dominated by the two springs. In the process of deformations, the viscoelastic deformation of the membrane is characterized by the two springs and the dashpot. While after a long-term period, because of the full relaxation of the bottom spring, the viscoelastic deformation of the membrane is governed by the top spring and the dashpot. This expected trend is illustrated in Fig. 3. As seen, all the considered variables gradually evolve into a stable state. For example, for the pressure P^* =0.8, the considered variables change little after t^* > 10.0. At 0 < t^* < 5, all the variables increase drastically. With the increase in the pressure, the values of the considered variables increase.

Figure 4 shows the distributions and evolutions of the longitudinal stretch λ₁ and the latitudinal stretch λ₂ in the DE membrane at different time when the membrane is subject to the pressure P^* =0.8. As shown, both λ₁ and λ₂ increase with time and change little after t^* > 8.0. The largest λ₁ appears at the particles around the circumference of the disk, indicating that these particles are more apt to undergo rupture failure when the pressure exceeds some threshold value.

Figure 5 shows evolutions of the profile of the deformed membrane when it is subject to the pressure P^* =0.8. As shown, the profile of the DE membrane changes little after t^* > 8.0, and the membrane eventually becomes stable.

(2) Case 2

In Case 2, we examine how the voltage affects the electro-viscoelastic performances of the membrane when the pressure holds constant. In the simulation, the pressure is fixed at P^* =0.5. The ratio A/B=0.3, and the pre-stretches are a/A=b/B=1.0. The obtained results are shown in Figs. 6-8.

Figure 6 shows evolutions of the longitudinal stretch λ₁, the height z^* , and the longitudinal true stress σ₁^*, the stretch ξ₁ in the dashpot, and the true electric field E^* of the particles around the circumference of the rigid disk. In the calculation, three different voltages Φ^* =0.1, 0.2, and 0.3 are set. As shown, all the considered variables gradually reach a stable state after t^* > 10.0. The steady values of the considered variables increase with the increase in the voltage.

Figure 7 shows evolutions and distributions of the longitudinal stretch λ₁ and the latitudinal stretch λ₂ of the membrane when it is subject to the voltage Φ^* =0.3. As observed, both λ₁ and λ₂ change little after t^* > 10, and the membrane approaches the electromechanical stable state.

Figure 8 shows profiles of the deformed membrane when it is subject to the voltage Φ^* =0.3 while the pressure holds as P^* =0.5. As shown, the profile changes little after t^* > 10, and the membrane tends to be stable.

(3) Case 3

In Case 3, we consider how the ratio A/B influences the viscoelastic behaviors of the membrane. A combination of the pressure P^* =0.5 and the voltage Φ^* =0.2 is set, and the pre-stretches are a/A=b/B=1.0. The obtained results are illustrated in Fig. 9.

Figure 9 shows evolutions of the longitudinal stretch λ₁, the height z^* , the true longitudinal stress σ₁^*, the stretch ξ₁ in the dashpot, and the true electric field E^* of the material particles around the circumference of the disk. In the simulation, three different ratios A/B=0.3, 0.4, and 0.5 are taken. As illustrated, for a short-term time period, the considered variables change dramatically, and then the variables gradually evolve into steady values. It is noticed that the steady values of the variables decrease with the increase in the ratio A/B.

(4) Case 4

In Case 4, we examine how the pre-stretches affect the performance of the membrane. In the simulation, three different pre-stretches a/A=b/B=1.2, 1.3, and 1.4 are taken. A combination of the pressure P^* =0.5 and the voltage Φ^* =0.2 is set. The obtained results are illustrated in Fig. 10.

Figure 10 shows evolutions of the longitudinal stretch λ₁, the height z^* , the true longitudinal stress σ₁^*, the stretch ξ₁ in the dashpot, and the true electric field E^* of the particles around the circumference of the rigid disk. As demonstrated, the steady values of the considered variables increase with the increase in the pre-stretches a/A and b/B.

6 Conclusions

To address the issues of dissipation of the DE membrane actuators, the viscoelastic model is formulated for a circular DE membrane containing a concentric rigid inclusion. When subject to electromechanical loadings, the membrane inflates and deforms into an out-of-plane axisymmetric shape. The electro-viscoelastic model of the actuator is established in the context of the nonlinear theory of the viscoelastic dielectrics and nonequilibrium thermodynamics. In the simulation, emphasis is placed on demonstrating how the electromechanical loadings and the design parameters affect the time-dependent performance of the actuator. From the obtained results, following conclusions can be drawn:

(Ⅰ) Due to the viscoelastic nature of the DE, in a short-term time period, all the considered variables change rapidly. Then, after a long-term period, they reach a stable state, i.e., evolutions of the considered variables are time-dependent.

(Ⅱ) The electromechanical loadings significantly influence the performance of the actuator. In case when the membrane is free from the voltage, and the design parameters hold constant, the steady values of the considered variables increase with the increase in the pressure, while in case when the pressure and the design parameters keep constant, the steady values of the considered variables increase with the increase in the voltage.

(Ⅲ) For particles along the circumference of the rigid disk, they are more apt to undergo failure such as rupture or electric breakdown.

(Ⅳ) The design parameters A/B, a/A, and b/B also play a very important role in the performance of the actuator. In case when the pressure, the voltage, and the pre-stretches a/A and b/B are fixed, the steady values of the variables decrease with the increase in the ratio A/B, while in case when the pressure, the voltage, and the ratio A/B are fixed, the steady values of the variables increase with the increase in the pre-stretches a/A and b/B.

It is hoped that the approach may provide some guidelines in designing and optimizing such DE devices.

Acknowledgements We would like to sincerely thank Huiming WANG, Professor of Zhejiang University, for his great help to the accomplishment of this paper.