Investigation on the dynamic performance of viscoelastic dielectric elastomer oscillators considering nonlinear material viscosity

Abstract

As a typical kind of soft electroactive materials, dielectric elastomers are capable of producing large deformation under external stimuli, which makes them desirable materials for many practical applications in transduction technology, including tunable oscillators and resonators. The dynamic performance of such dielectric elastomer–based vibrational devices is strongly affected by material viscosity as well as electromechanical coupling. Moreover, as suggested by experiments and theoretical studies, dielectric elastomers exhibit deformation-dependent relaxation process, which makes the modeling of the dynamic performance of dielectric elastomer–based devices more challenging. In this work, by adopting the state-of-art modeling framework of finite-deformation viscoelasticity, the effect of the nonlinear material viscosity on the in-plane oscillation and the frequency tuning of dielectric elastomer membrane oscillators is investigated. From the simulation results, it is found that the nonlinear viscosity only affects the transient state of the frequency tuning process. The modeling framework developed in this work is expected to provide useful guidelines for predicting the dynamic performance of dielectric elastomer–based vibrational devices as well as their optimal design.

Keywords

Dielectric elastomer oscillator nonlinear viscosity viscoelasticity

1. Introduction

As soft electroactive polymers, dielectric elastomers (DEs) are capable of producing large deformation under electrical stimuli, which makes them desirable materials for electromechanical transducers (Pelrine et al., 2000). A common design of DE actuator consists of an elastomeric membrane sandwiched by two soft compliant electrodes on both sides. When a voltage is applied on the electrodes, the DE membrane contracts in thickness and expands in area (Pelrine et al., 2000). Due to their unique properties such as large deformation capability, flexibility, and high energy density, DEs have been widely used to design functional actuators with different configurations in practical applications, such as soft robots, adaptive optical elements, programmable haptic surfaces, energy harvesters, oscillators, and resonators (Ahmadi et al., 2013; Anderson et al., 2010; Carpi et al., 2011; Kornbluh et al., 2002; O’Brien et al., 2010; O’Halloran et al., 2008; Pelrine et al., 2000; Xu et al., 2012).

The advantage of DE oscillators and resonators is that the oscillation of the membrane and the resonant frequency can be actively tuned by changing the applied alternating or static voltage (Li et al., 2012). This feature enables DE-based oscillators to be a promising alternative to traditional silicon-based devices. Early studies on analyzing the dynamic performance of DE-based oscillators and resonators mainly focused on their electromechanical response. For example, Feng et al. (2011) demonstrated the dynamic performance of a DE microbeam resonator under an electromechanical load and investigated the oscillation of the device in terms of the quality factor (Q-factor) and the resonant frequency shift ratio. Li et al. (2012) modeled the in-plane deformation and the frequency tuning of a plane membrane resonator using the Gent model (Gent, 1996). Kollosche et al. (2012) demonstrated how pre-stretching of DEs can change their voltage-induced deformation, electromechanical instability, and loss-of-tension. Sheng et al. (2014) proposed a free energy model to investigate the dynamic characteristics of a DE membrane with in-plane deformation. Lv et al. (2018) studied the dynamic performance of a DE balloon with consideration of the stiffening and damping effect. However, DEs are proven to exhibit strong viscoelasticity in nature (Hong, 2011). The material viscoelasticity of DEs exerts a significant effect on their actuation response. Therefore, more efforts have been devoted to studying the viscoelastic effect on the behavior of DEs recently. For example, Yang et al. (2005) developed mechanics models accounting for the viscoelastic effect of DEs under uniaxial and biaxial loading conditions. Plante and Dubowsky (2007) studied the viscoelasticity of DEs with the Ogden (1972) model and experimentally demonstrated the effect of the stretching rate on the performance of DE actuators. Recently, based on the fully coupled field theory for DEs developed by Suo et al. (2008) and the finite-deformation viscoelasticity theory developed by Reese and Govindjee (1998), Hong (2011) has developed a constitutive model that can adopt most hyperelastic relations and evolution laws of viscoelastic solids to capture the viscoelastic response of DEs. Adopting the constitutive model by Hong (2011), Zhang et al. (2015) developed a dynamic model for homogeneously deformed viscoelastic DE actuator under equal-biaxial, uniaxial, and pure shear forces. Based on the same framework, Zhou et al. (2014) demonstrated the effect of the material viscoelasticity on the resonant frequency of DE resonators. Later, Zhou et al. (2016) further investigated the dynamic response of a DE membrane oscillator under a harmonic excitation. The above-mentioned studies assume the linearity of the material viscosity when considering the viscoelastic deformation. In other words, the material viscosity is assumed as a constant in these studies. Nevertheless, according to the theory of polymer dynamics (Doi and Edwards, 1988), the viscosity of polymer chains in elastomers is deformation-dependent, that is, nonlinear, especially when the elastomers undergo large deformation. This argument has also been confirmed by experiments (Hossain et al., 2012; Wang et al., 2016). In fact, much less effort has been devoted to addressing the effect of the nonlinear viscosity of DEs, which may also strongly influence the dynamic performance of DEs. Until recently, Zhou et al. (2018) incorporated the nonlinear material viscosity into the finite-deformation viscoelasticity theory by Reese and Govindjee (1998) to investigate the deformation of elastomers. In this work, the modeling framework by Zhou et al. (2018) will be further employed to study the frequency tuning process and the dynamic response of DE oscillators.

2. Models and formulations

In this work, we will revisit the configuration of a commercial DE oscillator (Biggs and Hitchcock, 2010) developed by Artificial Muscle, Inc., to investigate the dynamic performance of DE oscillators and resonators. As shown in Figure 1, the DE oscillator consists of a DE membrane coated with two compliant electrodes on the top and bottom surfaces. Figure 1(a) shows the DE membrane in the undeformed state with dimensions L₁, L₂, and L₃. In Figure 1(b), the DE is pre-stretched in area and attached to a rigid frame and a rigid bar of mass m with its two edges along 2-direction. The rigid bar is connected to the other side of the frame by a spring and a viscous damper. The stiffness of the spring and damping coefficient of the viscous damper are denoted as k and c, respectively. In the pre-stretched state, the elongation of the spring is denoted as βL₁ and the dimensions of the DE membrane change to l_1p, l_2p, and l_3p with the pre-stretch ratios defined as λ_1p = l_1p/L₁, λ_2p = l_2p/L₂, and λ_3p = l_3p/L₃. Then an electric voltage Φ is applied to the compliant electrodes as shown in Figure 1(c). Under such an electrical load and forces from the spring and the viscous damper, the DE membrane further deforms to the current state with dimensions l₁, l₂, and l₃. Consequently, the stretch ratios of the current state to the undeformed state are defined as λ₁ = l₁/L₁, λ₂ = l₂/L₂, and λ₃ = l₃/L₃, respectively. In the current state, constrained by the rigid frame and the rigid bar, the DE membrane is under tensile forces P₁ and P₂ in 1- and 2-directions. Moreover, the rigid bar is subjected to forces P_s from the spring and P_d from the viscous damper. The absolute position of the rigid bar in 1-direction is defined as x. In Figure 1(d), the oscillator is excited by a displacement field $y (t) = Y \cos (ω_{b} t)$ at the rigid frame.

For such an oscillator configuration according to Figure 1(c), the motion equation of the rigid bar is expressed as

\frac{P_{s}}{G L_{2} L_{3}} + \frac{P_{p}}{G L_{2} L_{3}} - \frac{P_{1}}{G L_{2} L_{3}} + \frac{m}{G L_{2} L_{3}} \frac{d^{2} x}{d t^{2}} = 0

(1)

where the forces from the spring and the damper are $P_{s} = (β - λ_{1} + λ_{1 p}) L_{1} k$ and $P_{p} = c (dx / dt)$ , respectively. It should be mentioned that, in addition to material fracture that could be prevented by limiting the stretches, there are other two possible failure modes of the DE, namely, loss-of-tension and electrical breakdown. To prevent loss-of-tension of the membrane, the in-plane forces in the DE membrane should be ensured as tensile forces, that is, P₁ in 1-direction and P₂ in 2-direction should be greater than 0. In addition, the applied voltage Φ should be maintained below the electrical breakdown voltage of the DE. The breakdown voltage V_EB of the DE is determined by

\frac{V_{EB}}{L_{3}} \sqrt{\frac{ε ε_{0}}{G}} = d λ_{1}^{- 1} λ_{2}^{- 1}

(2)

where $d = E_{EB} \sqrt{ε ε_{0} / G}$ and E_EB is the dielectric strength of the DE (Koh et al., 2011; Zhou et al., 2013).

Since DEs are known to exhibit viscoelastic properties, the deformation gradient F of the DE membrane is commonly decomposed into an elastic component and an inelastic component in a multiplicative form (Hong, 2011), which gives

F = F^{e} F^{i}

(3)

where $F = (\begin{matrix} λ_{1} & 0 & 0 \\ 0 & λ_{2} & 0 \\ 0 & 0 & λ_{3} \end{matrix})$ , $F^{i} = (\begin{matrix} {λ_{1}}^{i} & 0 & 0 \\ 0 & {λ_{2}}^{i} & 0 \\ 0 & 0 & {λ_{3}}^{i} \end{matrix})$ , and $F^{e} = (\begin{matrix} λ_{1}^{e} & 0 & 0 \\ 0 & λ_{2}^{e} & 0 \\ 0 & 0 & λ_{3}^{e} \end{matrix})$ under the current loading condition which assumes uniform deformation of the DE. The superscripts “e” and “i” represent the elastic and inelastic parts, respectively. Assuming material incompressibility of the DE (Hong, 2011; Li et al., 2012; Yong et al., 2011; Zhou et al., 2014), the stretch ratios satisfy $λ_{3} = 1 / λ_{1} λ_{2}$ , $λ_{3}^{e} = 1 / λ_{1}^{e} λ_{2}^{e}$ , and $λ_{3}^{i} = 1 / λ_{1}^{i} λ_{2}^{i}$ . Furthermore, the total Helmholtz free energy $W$ of the deformed DE membrane consists of two parts. One is the equilibrium Helmholtz free energy density W^EQ, which is associated with both the total deformation and the applied electric voltage $Φ$ . The other part is the non-equilibrium Helmholtz free energy density W^NEQ, which is only related to the elastic deformation. Therefore, $W = W^{EQ} (λ_{1}, λ_{2}, V) + W^{NEQ} (λ_{1}^{e}, λ_{2}^{e})$ . It is worth noticing that the electric field is assumed to be in equilibrium, as the electric field always reaches the equilibrium state much faster than the mechanical deformation (Hong, 2011). According to the work of Huang and Suo (2012), the equilibrium Helmholtz free energy density W^EQ takes the form of

W^{EQ} = W_{s} (λ_{1}, λ_{2}) + \frac{ε ε_{0}}{2} {(\frac{Φ}{L_{3}})}^{2} λ_{1}^{2} λ_{2}^{2}

(4)

where $W_{s} (λ_{1}, λ_{2})$ is the strain energy density function of the elastomer, and the second term is the free energy associated with the polarization, ε₀ is the dielectric permittivity of the vacuum, and ε is the relative dielectric constant of the DE. Here, we adopt the Gent model (Gent, 1996) as the strain energy density function for both the equilibrium and non-equilibrium Helmholtz free energy density, that is

W_{s} = - \frac{G^{EQ} J_{\lim}^{EQ}}{2} \ln (1 - \frac{λ_{1}^{2} + λ_{2}^{2} + λ_{1}^{- 2} λ_{2}^{- 2} - 3}{J_{\lim}^{EQ}})

(5)

\begin{matrix} W^{NEQ} = - \frac{G^{NEQ} J_{\lim}^{NEQ}}{2} \\ \ln (1 - \frac{{(λ_{1}^{e})}^{2} + {(λ_{2}^{e})}^{2} + {(λ_{1}^{e} λ_{2}^{e})}^{- 2} - 3}{J_{\lim}^{NEQ}}) \end{matrix}

(6)

where G^EQ and G^NEQ are the equilibrium shear modulus and the non-equilibrium shear modulus, $J_{\lim}^{EQ}$ is a dimensionless parameter related to the stretching limit of the DE, and $J_{\lim}^{NEQ}$ is related to the stretching limit of the viscous polymer subnetwork.

When the DE in the current state is perturbed, the change of the total Helmholtz free energy is equal to the work done by the tensile force P₁ and P₂, the inertia force, and the voltage $Φ$ , resulting in

\begin{matrix} L_{1} L_{2} L_{3} (\frac{\partial W}{\partial λ_{1}} δ λ_{1} + \frac{\partial W}{\partial λ_{2}} δ λ_{2}) \\ = P_{1} L_{1} δ λ_{1} + P_{2} L_{2} δ λ_{2} + Φ δ Q - L_{2} L_{3} \frac{L_{1}^{3}}{3} ρ \frac{d^{2} λ_{1}}{d t^{2}} δ λ_{1} \end{matrix}

(7)

where ρ is the mass density of the DE, and the charge on the DE is expressed as $Q = Φ ε ε_{0} λ_{1}^{2} λ_{2}^{2} L_{1} L_{2} / L_{3}$ . It can be noticed that the work done by the inertia force in the thickness direction is ignored in equation (7) since the oscillator only vibrates in 1-direction and L₃ is much smaller than L₁.

Considering that δλ₁ and δλ₂ are any arbitrary small variations, substituting equations (4) to (6) into equation (7) leads to

\begin{matrix} \frac{P_{1}}{G L_{2} L_{3}} & = \frac{χ J_{\lim}^{EQ} (λ_{1} - λ_{1}^{- 3} λ_{2}^{- 2})}{J_{\lim}^{EQ} - λ_{1}^{2} - λ_{2}^{2} - λ_{1}^{- 2} λ_{2}^{- 2} + 3} \\ + \frac{(1 - χ) J_{\lim}^{NEQ} (λ_{1} {(λ_{1}^{i})}^{- 2} - λ_{1}^{- 3} λ_{2}^{- 2} {(λ_{1}^{i})}^{2} {(λ_{2}^{i})}^{- 2})}{J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1} λ_{2}}{λ_{1}^{i} λ_{2}^{i}})}^{- 2} + 3} \\ + \frac{L_{1}^{2} ρ}{3 G} \frac{d^{2} λ_{1}}{d t^{2}} - \frac{ε ε_{0}}{G} {(\frac{Φ}{L_{3}})}^{2} λ_{1} λ_{2}^{2} \end{matrix}

(8)

\begin{matrix} \frac{P_{2}}{G L_{1} L_{3}} & = \frac{χ J_{\lim}^{EQ} (λ_{2} - λ_{2}^{- 3} λ_{1}^{- 2})}{J_{\lim}^{EQ} - λ_{1}^{2} - λ_{2}^{2} - λ_{1}^{- 2} λ_{2}^{- 2} + 3} \\ + \frac{(1 - χ) J_{\lim}^{NEQ} (λ_{2} {(λ_{2}^{i})}^{- 2} - λ_{2}^{- 3} λ_{1}^{- 2} {(λ_{1}^{i})}^{2} {(λ_{2}^{i})}^{2})}{J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1} λ_{2}}{λ_{1}^{i} λ_{2}^{i}})}^{- 2} + 3} \\ - \frac{ε ε_{0}}{G} {(\frac{Φ}{L_{3}})}^{2} λ_{2} λ_{1}^{2} \end{matrix}

(9)

where G = G^EQ+G^NEQ and χ = G^EQ/G $(0 \leq χ \leq 1)$ . χ represents the fraction of the time-independent polymer networks in the elastomer, which is equal to 1 for a purely elastic medium, while 0 for a viscous fluid.

Furthermore, the inelastic stretch ratio ( $λ_{1}^{i}$ and $λ_{2}^{i}$ ) in equations (8) and (9) must satisfy the thermodynamic evolution law (Reese and Govindjee, 1998), that is

- \frac{1}{2} F \frac{d [{(C^{i})}^{- 1}]}{dt} F^{T} {(b^{e})}^{- 1} = γ^{- 1} : τ_{NEQ}

(10)

where $C^{i} = (F^{i})^{T} F^{i}$ , $(b^{e})^{- 1} = (F^{e})^{T} F^{e}$ , $τ_{NEQ} = 2 F^{e} (\partial W^{NEQ} / \partial C^{e}) (F^{e})^{T}$ , $C^{e} = (F^{e})^{T} F^{e}$ , and $γ^{- 1}$ is an isotropic rank-four tensor. According to the work of Reese and Govindjee (1998), $γ^{- 1}$ takes the form of

γ^{- 1} = \frac{1}{2 η} (I^{4} - \frac{1}{3} I \otimes I)

(11)

where $I^{4} = (1 / 2) (δ_{ik} δ_{jl} + δ_{il} δ_{jk})$ is an isotropic rank-four mobility tensor, I is the second-order identity tensor, and η represents the shear viscosity. According to this thermodynamic evolution law, the time-dependent inelastic stretch ratios are expressed as

\frac{d λ_{1}^{i}}{dt} = \frac{G^{NEQ} J_{\lim}^{NEQ} λ_{1}^{i} [2 {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1}^{- 1} λ_{2}^{- 1}}{{(λ_{1}^{i})}^{- 1} {(λ_{2}^{i})}^{- 1}})}^{2}]}{6 η [J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{- 2} {(\frac{λ_{2}}{λ_{2}^{i}})}^{- 2} + 3]}

(12)

\frac{d λ_{2}^{i}}{dt} = \frac{G^{NEQ} J_{\lim}^{NEQ} λ_{2}^{i} [2 {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{1}^{- 1} λ_{2}^{- 1}}{{(λ_{1}^{i})}^{- 1} {(λ_{2}^{i})}^{- 1}})}^{2}]}{6 η [J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{- 2} {(\frac{λ_{2}}{λ_{2}^{i}})}^{- 2} + 3]}

(13)

It should be noted that η is the viscosity of the material in the current state, which depends on the deformation of the DE (most studies in the literature assume linear viscosity, i.e. η is constant) and should be constitutively prescribed to obtain the inelastic stretch ratios as shown in equations (12) and (13). Here, the theory of polymer dynamics by Doi and Edwards (1988) is used to determine the nonlinear viscosity $η$ .

According to their theory, the viscosity η of elastomers originates from the diffusion of the polymer chains in the material and a polymer chain (A–B) is considered to be confined in a tube-like region due to the topological constraints (see Figure 2(a)). Moreover, the diffusion process of a polymer chain can be classified based on the timescale. In the short timescale, the polymer chain wriggles within the tube, while in the long timescale, the polymer chain reptates along the tube. Moreover, from a mathematical perspective, it is very difficult to describe the wriggling motion of the polymer chains, while the wriggling motion can be represented by the tube diameter a since it always occurs within the tube. Once the tube diameter is obtained, the wriggling motion can be forgotten. According to this theory, the viscosity η can be expressed in terms of the tube diameter as

η = \frac{1}{12} \frac{ζ N^{3} b_{0}^{4} G_{0}}{k_{B} T {〈 a 〉}^{2}}

(14)

where ξ is the monomer friction constant, N is the polymerization degree of chains, b₀ is the effective bond length between monomers, G₀ is the shear relaxation modulus in the undeformed state, T is the temperature, k_B is the Boltzmann constant, and $〈 * 〉$ is the expectation operator of parameter *. Also, the axis of the tube confining chain A–B is defined as the primitive chain of the polymer chain. According to the work of Doi and Edwards (1998), the tube diameter $a$ is a function of the primitive length L of the polymer chain and the end-to-end vector R of the primitive chain, that is, $〈 a 〉 = 〈 R^{2} 〉 / 〈 L 〉$ (see Figure 2(b)). When the deformation of the material is small, $R$ and L can be assumed as unchanged, which leads to a constant tube diameter a and thus a constant viscosity η. However, when the material undergoes large deformation, both L and R should change with the macroscopic deformation (Doi and Edwards, 1988; Zhou et al., 2018), that is, $〈 L 〉 = \int (| F \cdot u | / 4 π) L d^{2} u$ and $〈 R^{2} 〉 = \int {| F \cdot R |}^{2} f_{0} (R) d^{3} R$ , where $u$ is the unit tangent vector at a contour position of the polymer chain in the reference state, $f_{0} (R)$ is the statistical distribution function of the end-to-end vector R in the reference state, and F is the deformation gradient. Therefore, the nonlinear viscosity of the elastomer can be determined as

\frac{η}{η_{0}} = \frac{a_{0}^{2}}{{〈 a 〉}^{2}} = {\frac{{〈 R^{2} 〉}_{0} \frac{| F \cdot u |}{4 π} d^{2} u}{\int {| F \cdot R |}^{2} f_{0} (R) d^{3} R}}^{2}

(15)

where a₀ is the tube diameter and $η_{0} = (1 / 12) (ζ N^{3} b_{0}^{4} G_{0} / k_{b} {Ta}_{0}^{2})$ is the viscosity in the reference state. For the convenience of expression, the viscosity ratio in equation (15) can be expressed as

\frac{η}{η_{0}} = \frac{1}{α {(F)}^{2}}

(16)

with

\frac{1}{α (F)} = \frac{{〈 R^{2} 〉}_{0} \frac{| F \cdot u |}{4 π} d^{2} u}{\int {| F \cdot R |}^{2} f_{0} (R) d^{3} R}

3. Resonant frequency of an oscillator

For the oscillator shown in Figure 1, the deformation of the DE is fixed in 2-direction, that is, $λ_{2} = λ_{2 p}$ and $x = (λ_{1 p} - λ_{1}) L_{1}$ , and equation (1) can be reduced to a second-order differential equation as

\frac{d^{2} λ_{1}}{d t^{2}} + a_{1} \frac{d λ_{1}}{dt} + g (λ_{1}, λ_{1}^{i}, λ_{2}^{i}, V^{*}) = 0

(17)

where

g = a_{2} {\begin{matrix} \frac{χ J_{lim}^{EQ} (λ_{1} - λ_{1}^{- 3} λ_{2}^{- 2})}{J_{lim}^{EQ} - λ_{1}^{2} - λ_{2}^{2} - λ_{1}^{- 2} λ_{2}^{- 2} + 3} \\ + \frac{(1 - χ) J_{lim}^{NEQ} [λ_{1} {(λ_{1}^{i})}^{- 2} - λ_{1}^{- 3} λ_{2}^{- 2} {(λ_{1}^{i})}^{2} {(λ_{2}^{i})}^{2}]}{J_{lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1} λ_{2}}{λ_{1}^{i} λ_{2}^{i}})}^{- 2} + 3} \\ - {(V^{*})}^{2} λ_{1} λ_{2}^{2} - \frac{(β - λ_{1} + λ_{1 p}) L_{1} k}{G L_{2} L_{3}} \end{matrix}}

Here, the dimensionless voltage is defined as $V^{*} = Φ \sqrt{ε ε_{0} / G} / L_{3}$ , $a_{1} = (c L_{1} / G L_{2} L_{3}) / ((m L_{1} / G L_{2} L_{3}) + (L_{1}^{2} / 3 G) ρ)$ , and $a_{2} = 1 / ((m L_{1} / G L_{2} L_{3}) + (L_{1}^{2} / 3 G) ρ)$ . To study the resonant frequency of the oscillator, we adopt the commonly used perturbation method in the literature (Li et al., 2012; Zhou et al., 2014; Zhu et al., 2010). When the membrane is subjected to a small perturbation of amplitude Δ(t) in 1-direction at time t, the total stretch ratio of the membrane in that direction can be expressed as

λ_{1} = λ'_{1} + Δ (t)

(18)

where $λ'_{1}$ is the stretch ratio of the DE in the kinetic equilibrium state in 1-direction before the perturbation. Meanwhile, by combining equations (17) and (18), and expanding the equation into Taylor series, we can rewrite equation (17) as

\frac{d^{2} Δ}{d t^{2}} + a_{1} \frac{d Δ}{dt} + \frac{\partial g ({λ'}_{1}, λ_{1}^{i}, λ_{2}^{i}, V^{*})}{\partial λ_{1}} Δ = 0

(19)

Therefore, the resonant frequency can be determined as

\begin{matrix} ω_{n}^{2} & = \frac{\partial g ({λ'}_{1}, λ_{1}^{i}, λ_{2}^{i}, V^{*})}{\partial λ_{1}} \\ = a_{2} {\frac{χ J_{\lim}^{EQ} (1 + 3 λ_{1}^{- 4} λ_{2}^{- 2})}{J_{\lim}^{EQ} - λ_{1}^{2} - λ_{2}^{2} - λ_{1}^{- 2} λ_{2}^{- 2} + 3} + \frac{2 χ J_{\lim}^{EQ} {(λ_{1} - λ_{1}^{- 3} λ_{2}^{- 2})}^{2}}{{(J_{\lim}^{EQ} - λ_{1}^{2} - λ_{2}^{2} - λ_{1}^{- 2} λ_{2}^{- 2} + 3)}^{2}} - {(V^{*})}^{2} λ_{2}^{2} + \frac{k L_{1}}{G L_{2} L_{3}} \\ + \frac{(1 - χ) J_{\lim}^{NEQ} [{(λ_{1}^{i})}^{- 2} + 3 λ_{1}^{- 4} λ_{2}^{- 2} {(λ_{1}^{i})}^{2} {(λ_{2}^{i})}^{2}]}{J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1} λ_{2}}{λ_{1}^{i} λ_{2}^{i}})}^{- 2} + 3} + \frac{2 (1 - χ) J_{\lim}^{NEQ} {[λ_{1} {(λ_{1}^{i})}^{- 2} - λ_{1}^{- 3} λ_{2}^{- 2} {(λ_{1}^{i})}^{2} {(λ_{2}^{i})}^{2}]}^{2}}{{[J_{\lim}^{NEQ} - {(\frac{λ_{1}}{λ_{1}^{i}})}^{2} - {(\frac{λ_{2}}{λ_{2}^{i}})}^{2} - {(\frac{λ_{1} λ_{2}}{λ_{1}^{i} λ_{2}^{i}})}^{- 2} + 3]}^{2}}} \end{matrix}

(20)

As it can be seen from the expression of the resonant frequency in equation (20), it is a function of the time-dependent inelastic stretch ratios $λ_{1}^{i}$ and $λ_{2}^{i}$ . Therefore, $ω_{n}$ is an instant resonant frequency of the oscillator at time t.

As demonstrated in the literature, the resonant frequency of a DE oscillator can be actively tuned by applying a voltage to the DE membrane (Li et al., 2012; Zhou et al., 2016). Here, we revisit the frequency tuning process of DE oscillators with the new material model accounting for the nonlinear viscosity. Figure 3(a) and (b) shows the change of the stretch ratio during a typical frequency tuning process. Figure 4(a) and (b) plots the variation of the resonant frequency $ω_{n}$ during the same process. The results from both the linear and nonlinear viscosity models are also compared in these figures. The geometric and material parameters take the values from the previous studies (Chiang Foo et al., 2012; Zhou et al., 2016), that is, $α = 4$ , $β = 1$ , $a_{1} = 20$ , $a_{2} = 10^{3}$ , $d = 5$ , $J_{\lim}^{EQ} = 110$ , $J_{\lim}^{NEQ} = 55$ , $χ = 0.3$ , and the relaxation time $τ_{0} = η_{0} / G^{NEQ} = 3$ in the reference state. Moreover, in the simulation, it is ensured that the applied voltage is lower than the electrical breakdown voltage of the DE membrane and the loss-of-tension of the membrane is also avoided. At the very beginning, the oscillator in the pre-stretched state (with $λ_{1 p} = 1.5$ and $λ_{2 p} = 3$ ) is fully relaxed. Therefore, both the stretch ratio λ₁ and the resonant frequency $ω_{n}$ remain unchanged. After sufficient time, for example at t = 10 s, a voltage is applied to the DE membrane at a rate of $d V^{*} / dt = 6.667 / s$ and the voltage is then fixed as $V^{*} = 0.125$ afterward. The response curves for both the linear and nonlinear viscosity models demonstrate the same trend. It is observed that the applied voltage causes a fast rise of the stretch ratio $λ_{1}$ and a sudden drop of the resonant frequency $ω_{n}$ . After the voltage is applied, both the stretch ratio and the resonant frequency fluctuate with a decaying amplitude, which is a typical transient response for a spring–damper system. During this fluctuation, energy of the system dissipates mainly through the damper. After such a period, the stretch ratio starts to increase, while the resonant frequency starts to gradually decrease. This is mainly due to the stress relaxation of the DE membrane. From Figures 3(b) and 4(b), it is observed that after sufficient time, the stretch ratio and the resonant frequency eventually become steady, which indicates that the DE membrane is fully relaxed again and has reached an equilibrium state, and the resonant frequency remains as a constant thereafter. The value of the resonant frequency at this steady state is defined as the tuned frequency by the applied voltage. It is observed that the difference of the response curves between the nonlinear viscosity model and the linear viscosity model is considerable during the transient state, while the DE membrane reaches the same steady state for both the linear and nonlinear models. Moreover, the DE membrane reaches the steady state faster when the nonlinear viscosity is considered. The reason behind this phenomenon is that, as the stretch ratio increases with the applied voltage from the reference state to the current state, $α (F)$ rises rapidly. Therefore, the viscosity η decreases rapidly since it is inversely proportional to the second order of $α (F)$ , as shown in equation (16). In other words, the nonlinear viscosity η decreases from the reference state to the current state. Thus, the DE membrane relaxes faster when the nonlinear viscosity is considered.

The effect of the applied voltage rate on the tuned frequency of the DE oscillator is demonstrated in Figures 5 and 6, with loading rate $d V^{*} / dt = 1 / s$ and $d V^{*} / dt = 0.1667 / s$ , respectively. At a higher rate of the applied voltage, there is more fluctuation on the resonant frequency, which means that the damper has more dominant effect. However, the loading rate has no effect on the tuned frequency, which is only a function of the value of the applied voltage. Furthermore, Figure 7 illustrates a comparison of the tuned frequency between a viscoelastic (χ = 0.5) DE resonator and a purely elastic DE resonator. It can be noticed that the viscoelastic DE membrane resonator has a tuned frequency range about 50% narrower than the purely elastic one. It should be mentioned that the nonlinear material viscosity does not change the tuned frequency of the oscillator.

To further investigate the influence of the nonlinear material viscosity on the dynamic response of DE-based oscillators, an AC voltage $Φ = p \sin (q ω_{n} t)$ is applied to the DE membrane to generate oscillation; p and q are parameters that govern the amplitude and frequency of the alternating voltage. As shown in Figure 8, the difference between the linear viscosity and nonlinear viscosity models is mainly reflected during the transient response of the DE oscillator.

4. Forced response of an oscillator

As introduced in the literature, some DE oscillators for potential sensing applications (e.g. structural health monitoring) are required to be connected to external environment during their operation (Carden and Fanning, 2004). For these DE oscillators, they are usually subjected to external excitation and it is essential to study their forced response. To examine the dynamic response of the DE oscillator under external excitation, the frame of oscillator is excited with a displacement field, $y (t) = Y \cos (ω_{b} t)$ (see Figure 1(d)), where Y is the amplitude of the excitation and $ω_{b}$ is the excitation frequency. Due to the external excitation, the absolute position of the rigid bar is changed to $x = (λ_{1 p} - λ_{1}) L_{1} + y$ . Correspondingly, the equation of motion (17) is rewritten as

\frac{d^{2} λ_{1}}{d t^{2}} + a_{1} \frac{d λ_{1}}{dt} + \frac{Y}{L_{1}} ω_{b}^{2} \cos (ω_{b} t) + g (λ_{1}, λ_{1}^{i}, λ_{2}^{i}, V^{*}) = 0

(21)

Numerical solution of equation (21) gives the response of the oscillator according to the external base excitation. In order to ensure that the loss-of-tension does not occur, the pre-stretched ratio is selected as λ_1p = 4 when high amplitude excitation is implemented. Figure 9 depicts the time response (displacement x/L₁ vs time $t$ ) of the rigid bar to the external excitation for both the linear and nonlinear models under a static voltage, V* = 0.1. For low excitation frequency $ω_{b} = 0.01 ω_{n}$ (Figure 9(a) and (b)), the steady-state response displacement amplitude is approximately the same as the excitation amplitude ((a) Y = 0.01 L₁ and (b) Y = L₁); however, for high excitation frequency $ω_{b} = 1.5 ω_{n}$ (Figure 9(c) and (d)), the response amplitude is attenuated. Furthermore, simulation results from the two viscosity models stay almost the same at both transient and steady states. It is thus concluded that the displacement transmissibility is only frequency dependent.

5. Conclusion

Based on the theory of finite-deformation viscoelasticity and the coupled field theory for DEs, this work investigates the effect of nonlinear viscosity of material on the frequency tuning of a viscoelastic DE membrane oscillator, as well as its oscillation behavior under an AC voltage. Comparison of modeling the oscillation between a nonlinear viscosity model and a linear viscosity model shows that neglecting the deformation-dependent viscosity may lead to error in predicting the transient state response of the oscillator. Moreover, by investigating the time response of an oscillator to external base excitation, it can be concluded that the effect of nonlinear viscosity is negligible. The modeling framework and simulation results are anticipated to provide an increased understanding on the dynamic performance of vibrational DE devices.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

ORCID iD

Liying Jiang

References

Ahmadi

Gooyers

Soleimani

, et al. (2013) Fabrication and electromechanical examination of a spherical dielectric elastomer actuator. Smart Materials and Structures 22: 115004.

Anderson

Hale

Gisby

, et al. (2010) A thin membrane artificial muscle rotary motor. Applied Physics A 98: 75–83.

Biggs

Hitchcock

(2010) Artificial muscle actuators for haptic displays: system design to match the dynamics and tactile sensitivity of the human fingerpad. Proceedings of SPIE—The International Society for Optical Engineering 7642: 76420I.

Carden

Fanning

(2004) Vibration based condition monitoring: a review. Structural Health Monitoring 3: 355–377.

Carpi

De Rossi

Kornbluh

, et al. (2011) Dielectric Elastomers as Electromechanical Transducers: Fundamentals, Materials, Devices, Models and Applications of an Emerging Electroactive Polymer Technology. Amsterdam: Elsevier.

Chiang Foo

Cai

Jin Adrian Koh

, et al. (2012) Model of dissipative dielectric elastomers. Journal of Applied Physics 111: 034102.

Doi

Edwards

(1988) The Theory of Polymer Dynamics. Oxford: Oxford University Press.

Feng

Jiang

Lau

(2011) Dynamic characteristics of a dielectric elastomer-based microbeam resonator with small vibration amplitude. Journal of Micromechanics and Microengineering 21: 095002.

Gent

(1996) A new constitutive relation for rubber. Rubber Chemistry and Technology 69: 59–61.

10.

Hong

(2011) Modeling viscoelastic dielectrics. Journal of the Mechanics and Physics of Solids 59: 637–650.

11.

Hossain

Steinmann

(2012) Experimental study and numerical modelling of VHB 4910 polymer. Computational Materials Science 59: 65–74.

12.

Huang

Suo

(2012) Electromechanical phase transition in dielectric elastomers. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 468: 1014–1040.

13.

Koh

SJA

Zhou

, et al. (2011) Mechanisms of large actuation strain in dielectric elastomers. Journal of Polymer Science Part B 49: 504–515.

14.

Kollosche

Zhu

Suo

, et al. (2012) Complex interplay of nonlinear processes in dielectric elastomers. Physical Review E 85: 051801.

15.

Kornbluh

Pelrine

Pei

, et al. (2002) Electroelastomers: applications of dielectric elastomer transducers for actuation, generation, and smart structures. Proceedings of SPIE—The International Society for Optical Engineering 4698: 254–271.

16.

Yang

(2012) Electromechanical and dynamic analyses of tunable dielectric elastomer resonator. International Journal of Solids and Structures 49: 3754–3761.

17.

Liu

, et al. (2018) Dynamic performance of dielectric elastomer balloon incorporating stiffening and damping effect. Smart Materials and Structures 27: 105036.

18.

O’Brien

Calius

Inamura

, et al. (2010) Dielectric elastomer switches for smart artificial muscles. Applied Physics A 100: 385–389.

19.

Ogden

(1972) Large deformation isotropic elasticity—on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 326: 565–584.

20.

O’Halloran

O’Malley

McHugh

(2008) A review on dielectric elastomer actuators, technology, applications, and challenges. Journal of Applied Physics 104: 071101.

21.

Pelrine

Kornbluh

Pei

, et al. (2000) High-speed electrically actuated elastomers with strain greater than 100%. Science 287: 836–839.

22.

Plante

Dubowsky

(2007) On the performance mechanisms of dielectric elastomer actuators. Sensors and Actuators A: Physical 137: 96–109.

23.

Reese

Govindjee

(1998) A theory of finite viscoelasticity and numerical aspects. International Journal of Solids and Structures 35: 3455–3482.

24.

Sheng

Chen

, et al. (2014) Nonlinear dynamic characteristics of a dielectric elastomer membrane undergoing in-plane deformation. Smart Materials and Structures 23: 045010.

25.

Suo

Zhao

Greene

(2008) A nonlinear field theory of deformable dielectrics. Journal of the Mechanics and Physics of Solids 56: 467–486.

26.

Wang

Decker

Henann

, et al. (2016) Modeling of dielectric viscoelastomers with application to electromechanical instabilities. Journal of the Mechanics and Physics of Solids 95: 213–229.

27.

B-X

Mueller

Theis

, et al. (2012) Dynamic analysis of dielectric elastomer actuators. Applied Physics Letters 100: 112903.

28.

Yang

Frecker

Mockensturm

(2005) Viscoelastic model of dielectric elastomer membranes. Proceedings of SPIE—The International Society for Optical Engineering 5759: 82–93.

29.

Yong

Zhou

(2011) Dynamics of a thick-walled dielectric elastomer spherical shell. International Journal of Engineering Science 49: 792–800.

30.

Zhang

Chen

, et al. (2015) Coupled nonlinear oscillation and stability evolution of viscoelastic dielectric elastomers. Soft Matter 11: 7483–7493.

31.

Zhou

Jiang

Khayat

(2013) Failure analysis of a dielectric elastomer plate actuator considering boundary constraints. Journal of Intelligent Material Systems and Structures 24: 1667–1674.

32.

Zhou

Jiang

Khayat

(2014) Viscoelastic effects on frequency tuning of a dielectric elastomer membrane resonator. Journal of Applied Physics 115: 124106.

33.

Zhou

Jiang

Khayat

(2016) Dynamic analysis of a tunable viscoelastic dielectric elastomer oscillator under external excitation. Smart Materials and Structures 25: 025005.

34.

Zhou

Jiang

Khayat

(2018) A micro–macro constitutive model for finite-deformation viscoelasticity of elastomers with nonlinear viscosity. Journal of the Mechanics and Physics of Solids 110: 137–154.

35.

Zhu

Cai

Suo

(2010) Nonlinear oscillation of a dielectric elastomer balloon. Polymer International 59: 378–383.