Nonlinear dynamic analysis of anisotropic fiber-reinforced dielectric elastomers: A mathematical approach

Abstract

In this paper, the primary purpose is to have a complete understanding of the importance of fibers in dielectric elastomers due to the influence of fiber orientation in decreasing the applied voltage to simulate the structure and improving mechanical performance. Fibers will also reduce the chance of failure like Wrinkling instabilities and increase the response rate under electric displacement. Implementing the dielectric elastomers with anisotropic structure, despite their great potential, has not adequately analyzed in previous researches. Therefore, based on nonlinear continuum mechanics and large inelastic deformations, the constitutive relationships and equations governing the behavior of viscoelastic dielectric elastomers under harmonic electrical loading are extracted and analyzed in different states. The numerical results, such as phase and frequency-amplitude diagrams and the oscillation, illustrate the dynamic behavior of an anisotropic dielectric elastomer with different fiber orientations. To describe the viscoelasticity behavior of the material, hyperelastic models and the rheological model, are used in conjunction with electrical coupling. Using the theory of anisotropic dielectric elastomers, a geometrically nonlinear formulation under finite deformations is developed by the Euler-Lagrange equations and solved mathematically.

Keywords

Electroactive polymers synthetic muscles dielectric elastomers anisotropic structure viscoelasticity behavior

1. Introduction

Today, with the advent of science in many fields, smart materials are also rapidly expanding and used in many devices (An et al., 2016; Bar-Cohen, 2004; Bar-Cohen et al., 1999; Su et al., 1999; Thomsen et al., 2001). The term “smart” can be defined as materials that can respond based on understanding the surroundings and conditions (Miriyev et al., 2017). Thus, polymers are no exception and exhibit different reactions to various stimuli such as temperature, electric field, and magnetic field (Bar-Cohen, 2005). Electroactive polymers (EAPs) are smart polymers that deform and resize in response to external stimuli and also generate electricity (Bar-Cohen, 2002, 2007). This extraordinary ability makes these polymers also used as actuators or sensors. EAPs have fast response time, lower density, and higher flexibility compared to shape-memory alloys and also have less density and electrical stimulation than electroactive ceramics (Kornbluh et al., 1998). EAPs are divided into two main groups according to the stimulation mechanism: Electronic EAPs and ionic EAPs (Pelrine et al., 2000). It should also be noted that electric field or coulomb forces stimulate electronic EAPs, whereas, in ionic EAPs, ion movement, and diffusion are the primary triggers (Jo et al., 2011; Shahinpoor and Kim, 2001; Watanabe et al., 2014). Among EAPs, dielectric elastomer (DE) is one of the most widely used and successful types of EAPs that has been much studied and investigated (Brochu and Pei, 2010; He et al., 2009; Lampani, 2010; Oates et al., 2017; O’Halloran et al., 2008; Pelrine et al., 2002; Romasanta et al., 2015; Tavakol et al., 2014).

Dorfmann and Ogden (2005) investigated the nonlinear behavior of dielectric materials. They extracted nonlinear electro-elasticity relationships based on the relationship between the electric field and mechanical deformation and analyzed the boundary conditions under the large deformation theory. The structural equations of the DE were analyzed using free energy function and electric field in both Eulerian and Lagrangian states to investigate the rate of strain. Goulbourne et al. (2005) developed the mechanistic relationships of DEs associated with large-scale deformation, nonlinear behavior, and electrical effects using Maxwell-Faraday and nonlinear elasticity relationships. In this research, an analytical model was used to study the behavior of DE under the influence of applied voltage changes, initial tension, and external pressure. Mockensturm and Goulbourne (2006) modeled and analyzed the behavior of a viscoelastic spherical dielectric thin film with both mechanical and electro-static pressure. Cauchy stresses were calculated based on invariants of the stretch, and the hyperelastic Mooney-Rivlin model was employed to investigate the structure’s behavior. Liu et al. (2009) proposed DE’s behavior using theoretical relations and experimental results. The strain values were measured in different electric fields and investigated by Maxwell stress relation. In the case of large deformations, both the experimental results and the finite element analysis had been compared. It should be noted that the viscoelastic effect had also been neglected. Zhu et al. (2010b) studied the resonant behavior of a thin layer of DE in the form of a circular ring under pressure and applied voltage. Equilibrium equations were derived using nonlinear field theory for analysis and investigation of laboratory conditions. It was observed that as the tension, pressure, or voltage changes, the natural frequencies also change. Applying sinusoidal pressure and voltage also resulted in harmonic and super-harmonic resonance. Chen et al. (2017) extracted the governing equations using the Helmholtz free energy function of DE and the thermal effects. Examining the results revealed that the deformations obtained from the quasi-static model and the dynamic model are the same at the beginning of excitation and then change with increasing stretch rate. When the DE was excited by a low-amplitude sinusoidal voltage, the stretch-induced by the quasi-static and dynamic model were the same. Still, high amplitude, the results were different. Various models were also utilized to simulate the viscoelastic behavior of DE.

To have a better understanding of the nonlinear oscillation of anisotropic dielectric elastomers (ADEs) in different situations, the dynamic behavior of DEs, a vital parameter to use in acoustic actuators, is examined. It has attracted much attention (Sheng et al., 2013; Zhang et al., 2015). Therefore, Sheng et al. (2014) studied the natural frequency and nonlinear dynamic property of a DE considering the damping effect. The proposed free energy model was based on the Gent model, and a perturbation technique was applied to derive the nonlinear governing equation. A mathematical procedure was introduced by Zhu et al. (2010a) to develop the dynamic behavior of a DE balloon under sinusoidal voltage to calculate the nonlinear frequencies. The dynamic response of a DE was studied by Kumar and Sarangi (2018) with analytical modeling. The standard Euler-Lagrange method, as well as the Mooney-Rivlin function, was used to study the dynamic response of DE under harmonic electrical loading.

Due to the widespread use of ADEs in artificial muscles and smart micro-robots, it is essential to have a better understanding of ADEs (Ahmadi and Asgari, 2020; Jiménez and MacMeeking, 2016; Su et al., 2019). Therefore, in recent years, there has been a large increase in research activity on these types of structures, and many studies have been carried out to analyze the properties of ADEs. Yong et al. (2012) developed the electromechanical instability of ADEs with an electric field and axial stress. By applying the Neo-Hookean model, the Cauchy-Green deformation tensor, and the Helmholtz energy, the governing equations for ADE structure had been extracted, and an analytical procedure was used to solve. The results showed that by increasing the anisotropic parameter, the stability of the dielectric elastomer was significantly increased. Subramani et al. (2014) focused on improving the performance of DE with anisotropic properties by improving the dielectric constant. They showed that employing the fibers and considering the elastomers as composite structures lead to anisotropic behavior and improve their efficiency and response rate. In 2018, they also showed that adding the relatively high strength fibers to the matrix of a dielectric elastomer resulted in increasing energy absorption, improving mechanical performance, and increasing the ADE response rate under the electrical field (Subramani et al., 2018). The acrylic elastomer used in this study is VHB4905, which is commonly used. Hossain and Steinmann (2018) introduced a new structural relationship for electroactive polymers with anisotropic properties by fiber diffusion. To formulate the relationships, the laws of thermodynamics were used. To illustrate the performance of cylindrical structures with axial symmetry under electromechanical coupling loading, a non-homogeneous boundary value problem was firstly investigated.

In this paper, the nonlinear mathematical formulation of an ideal ADE is developed to study the electrical response of a viscoelastic ADE membrane. Because of the complexity of governing equations, the effects of viscoelasticity are neglected by most of the researchers. In contrast, most of the DE materials like VHB show strong viscoelasticity through the thickness direction. Hence, the model of viscoelasticity is represented by a rheological model of springs and dashpots associate with both the Neo-Hookean model and the Mooney-Rivlin model for uniaxial tension and pure shear state. Also, most of the researchers have focused on the study of the behavior of isotropic DE employing the finite element analysis. At the same time, an extensive complete theoretical model including the effect of viscoelasticity for both important hyperelastic material models based on virtual work with fiber-reinforced DEs could not be found in literature, to the best of the authors’ knowledge. The kinematic and potential energy based on the Euler-Lagrange equation is used to derive the nonlinear vibration equation for various conditions. Consequently, by solving the governing equations, the dynamic responses of ADE are examined. The phase diagrams and dynamic vibration and oscillation of the system under harmonic and constant electric loading are also studied. Results show that increasing the electric field has a strong effect on the polymer molecules and makes them less closely aligned with each other. Moreover, the jump of oscillating amplitude at specific values of the frequency of excitation is also observed, which shows the hysteresis phenomena.

2. Nonlinear continuum formulation

In this part, an electro-elastic solid continuum is considered occupying a region $ℑ_{0} \subset R^{3}$ in the reference configuration parametrized by material coordinates $X \in ℑ_{0} \cup \partial ℑ_{0}$ at time $t = 0$ , where $\partial ℑ_{0}$ means the boundary of $ℑ_{0}$ . At the time $t > 0$ , $the ℑ$ shows the deformed configuration parameterized by spatial coordinates $x$ in which the particle $X$ has position vector $x$ defined by the nonlinear deformation mapping $x = φ (X, t)$ , as illustrated in Figure 1.

Figure 1.

A continuum motion description of a fiber-reinforced DE.

The deformation gradient tensor is used to map a line element from the reference state to the deformed state and is defined as

F = \nabla_{X} φ, J = det F

(1)

where J is the Jacobian determinant of the deformation gradient taken $J > 0$ and $\nabla_{X}$ states the gradient operator for the material coordinates $X$ in $ℑ_{0}$ . The right Cauchy–Green tensor is introduced by $C = F^{T} F$ , where ^T $represents$ the transpose of a second-order tensor.

The Maxwell’s equations can be represented in the following forms. At the same time, there are no electric charges and magnetic fields (Dorfmann and Ogden, 2006)

curl E = 0, div D = 0

(2)

Here, $E, D$ are the electric field and electric displacement, respectively, while the operators curl and div are shown $in ℑ$ and operators concerning $x$ . It should be noted that there is a relation between the $E, D$ , and $P$ (the polarization density) introduced by the following equation (Dorfmann and Ogden, 2006)

D = ε E + P

(3)

where $ε$ is the electric permittivity of vacuum or free space.

In the case of vacuum or free space, the $P = 0$ and the equation (3) changes to the following form

D = ε E

(4)

The above equations were based on the current state. The presented various electric quantities may transform from the spatial configuration $ℑ to$ the un-deformed configuration $ℑ_{0}$ . The equations would be rewritten in a Lagrangian formulation based on the following integral forms

\int_{ℑ} div D dv = \int_{\partial ℑ} D . n da = 0

(5)

\int_{s} curl E . n da = \int_{\partial s} E . d x = 0

(6)

in which, $s, \partial s, n$ , and da are the surface in the current state, a closed curve bounding, the unit outward normal $to \partial ℑ$ , and current area elements, respectively. Implementing the Nanson formula (Lai et al., 2009) $n da = J F^{- T} N da$ where $N$ is the referential counterpart of $n$ , the following equations may be resulted as

\int_{ℑ_{0}} Div (J F^{- 1} D) dV = \int_{\partial ℑ_{0}} J (F^{- 1} D) . N dA = 0

(7)

\int_{s_{0}} Curl (F^{T} E) . N dA = \int_{\partial s_{0}} (F^{T} E) . d X = 0

(8)

where $s_{0}$ is the surface in the un-deformed state and $\partial s_{0}$ is its boundary. The Curl and Div present the corresponding differential operators to the position vectors $X$ .

According to the equations (7) and (8), the Lagrangian form of an electric field, electric displacement, and the polarization vector is denoted by $E_{l}, D_{l}$ , and $P_{l}$ , respectively, and are shown by

D_{l} = J F^{- 1} D, E_{l} = F^{T} E, P_{l} = J F^{- 1} P

(9)

Therefore, like equation (2), the Maxwell equations will have resulted as

Curl E_{l} = 0, Div D_{l} = 0

(10)

The other form of equation (3) can be obtained using equation (9) as follow

D_{l} = ε J C^{- 1} E_{l} + P_{l}

(11)

For an ADE the Helmholtz free energy function $Ω$ depends on the two fiber directions and the deformation gradient $F$ proposed by the following equation

\begin{matrix} Ω = Ω (F, E_{l}, A_{0}, B_{0}), A_{0} = α_{0} \otimes α_{0}, B_{0} = β_{0} \otimes β_{0}, \\ | α_{0} | = | β_{0} | = 1 \end{matrix}

(12)

Here, $A_{0}, B_{0}$ are the structural tensor to describe the behavior of an anisotropic electro-elastic material and $α_{0}, β_{0}$ are the unit vectors which state the fiber orientations.

The free energy function in equation (12) can be developed as a function of the set of invariants

\begin{matrix} Ω = Ω (I_{1}, I_{2}, I_{3}, I_{4}, I_{5}, I_{6}, I_{7}, I_{8}, I_{9}, I_{10}, I_{11}, I_{12}, I_{13}, \\ I_{14}, I_{15}, I_{16}, I_{17}, I_{18}, I_{19}) \end{matrix}

(13)

These invariants are given as

\begin{matrix} I_{1} = tr C, I_{2} = \frac{1}{2} [(tr C)^{2} - tr (C^{2})], I_{3} = det C = J^{2}, \\ I_{4} = α_{0} . (C α_{0}) \\ I_{5} = α_{0} . (C^{2} α_{0}), I_{6} = β_{0} . (C β_{0}), I_{7} = β_{0} . (C^{2} β_{0}), \\ I_{8} = (α_{0} . β_{0}) α_{0} . (C β_{0}) \\ I_{9} = α_{0} . (C^{2} β_{0}) I_{10} = {| E_{l} |}^{2}, I_{11} = E_{l} . (C^{- 1} E_{l}), \\ I_{12} = E_{l} . (C^{- 2} E_{l}) \\ I_{13} = α_{0} . E_{l}, I_{14} = α_{0} . (C E_{l}) I_{15} = β_{0} . E_{l}, I_{16} = β_{0} . (C E_{l}) \\ I_{17} = (α_{0} \times β_{0}) . E_{l}, I_{18} = (α_{0} \times β_{0}) . C E_{l}, \\ I_{19} = (α_{0} \times β_{0}) . (C^{2} E_{l}) \end{matrix}

(14)

The complete form of the free energy function is presented in equation (13). The principal invariants of the stress are $I_{1}, I_{2}, I_{3}$ which represent the length, area, and volume change of the solid. Since the stress in an anisotropic material is a function of the deformation gradient and, the fiber orientation the set of invariants for fiber-reinforced elastic materials are illustrated as $I_{4}, I_{5}, I_{6}, I_{7}, I_{8}, I_{9}$ . $I_{4}, I_{6}$ are the fiber stretches in the direction of $α_{0}, β_{0}$ , $I_{5}, I_{7}$ are higher-order invariants for $I_{4}, I_{6}$ , and $I_{8}, I_{9}$ are coupled invariants of the directional vectors $α_{0}, β_{0}$ . Also, $I_{4}, I_{5}, I_{6}, I_{7}, I_{8}, I_{9}$ are constant during the deformation. The quadratics of the nominal and true electric field are also presented by $I_{10}, I_{11}, I_{12}$ associated with the electric field. Moreover, $I_{13}, I_{14}, I_{15}, I_{16}, I_{17}, I_{18}, I_{19}$ are coupled invariants between the nominal electric field and the two electroactive directional vectors, where $I_{13}, I_{14}, I_{15}, I_{16}$ represent the coupling between one electro-active directional vector and the nominal electric field and $I_{17}, I_{18}, I_{19}$ represent the coupling between two electro-active vectors and the electric field. Since the electro-active fibers are considered as one-dimensional components and the coupling invariants between the fibers and the nominal electric field are defined as a dot product of two vectors, there is no coupling when the directions of the fibers $α_{0}, β_{0}$ are orthogonal to the nominal electric field. Finally, implementing the above conditions and eliminating the higher-order invariants with $C^{2}$ the simple form of constitutive formulations is derived.

The initial direction of unit vectors $α_{0}$ and $β_{0}$ are written as follows

\begin{matrix} α_{0} = \cos γ \cos α {\hat{e}}_{1} + \cos γ \sin α {\hat{e}}_{2} + \sin γ {\hat{e}}_{3}, \\ β_{0} = \cos γ \cos β {\hat{e}}_{1} + \cos γ \sin β {\hat{e}}_{2} + \sin γ {\hat{e}}_{3} \end{matrix}

(15)

in which $(α_{0}, β_{0}), {{\hat{e}}_{1}, {\hat{e}}_{2}, {\hat{e}}_{3}}, γ$ are the angle between the vector and the $X_{1}$ direction on the $X_{1} - X_{2}$ plane, the angle between the vector and the vector projection on the $X_{1} - X_{2}$ plane, and unit vectors in a Cartesian coordinate system, respectively (Figure 2). The fibers are assumed to be thin, flexible, continuous and lie parallel, and close together on smooth surfaces in the undeformed body. The elastic body is described using a hyperelastic strain energy function and the fibers are considered as structural components. Also, for the structural approach, the fiber families are considered as structural components in a matrix and a formulation can be obtained by describing the geometric relationship of the fibers and the matrix. One of the assumptions for the chosen anisotropic part of the strain energy function is that fibers cannot withstand compression and in that case, should be excluded from the analysis. To find the fibers under compression, one needs to compute $I_{4}$ and $I_{6}$ . If these values are less than unity for a fiber, then the fiber is under compression.

Figure 2.

The orientation of unit vectors $α_{0}, β_{0}$ and electric field $E_{l}$ .

The free energy function for an ADE is composed of the isotropic $Ω_{iso}$ , anisotropic $Ω_{ani}$ , viscoelastic $Ω_{visco}$ , and electrostatic $Ω_{ele}$ parts as follows

Ω = Ω_{iso} + Ω_{ani} + Ω_{visco} + Ω_{ele}

(16)

The proposed model is also considered for modeling the isotropic part as follow

Ω = Ω_{dev} + Ω_{vol}

(17)

in which, $Ω_{dev}$ and $Ω_{vol}$ appear due to the isochoric deformation and volume change, respectively.

The various forms of volumetric functions can be presented as following (Tuan and Marvalova, 2007)

Ω_{vol} = \frac{ϑ_{0}}{2} (J - 1)^{2}

(18)

where $ϑ_{0}$ is the bulk modulus. Moreover, different functions are referred to the reader for more comprehensive details (Doll and Schweizerhof, 2000; Kadapa and Hossain, 2020; Marckmann and Verron, 2006; Steinmann et al., 2012).

The common hyperelastic models can be written as following (Kim et al., 2012)

Neo - Hookean : Ω_{iso} = \frac{μ}{2} ({\bar{I}}_{1} - 3)

(19)

Mooney - Rivlin : Ω_{iso} = c_{1} ({\bar{I}}_{1} - 3) + c_{2} ({\bar{I}}_{2} - 3)

(20)

Ogden : Ω_{iso} = \sum_{i = 1}^{N} \frac{μ_{i}}{ξ_{i}} (λ_{1}^{ξ_{i}} + λ_{2}^{ξ_{i}} + λ_{3}^{ξ_{i}} - 3)

(21)

where $μ$ is the shear modulus and the other coefficients in Mooney-Rivlin and Ogden models are the material constants which could be calculated experimentally. Also, the modified corresponding invariants are mentioned as follow

{\bar{I}}_{b} = J^{- 2 / 3} I_{b} for b = 1, 4, 6, 8

(22)

{\bar{I}}_{c} = J^{- 4 / 3} I_{c} for c = 2, 5, 7

(23)

The anisotropic deviatoric part is developed as polynomial (Nam, 2004) and exponential functions as follow

\begin{matrix} Ω_{ani} = \sum_{i = 2}^{6} m_{i} {({\bar{I}}_{4} - 1)}^{2 i} + \sum_{j = 2}^{6} o_{j} {({\bar{I}}_{5} - 1)}^{2 j} \\ + \sum_{k = 2}^{6} t_{k} {({\bar{I}}_{6} - 1)}^{2 k} + \sum_{l = 2}^{6} u_{l} {({\bar{I}}_{7} - 1)}^{2 l} \\ + \sum_{v = 2}^{6} w_{v} {({\bar{I}}_{8} - ς)}^{2 v} \end{matrix}

(24)

\begin{matrix} Ω_{ani} = \frac{ϑ_{1}}{2 ϑ_{2}} {\exp [ϑ_{2} ({\bar{I}}_{4} - 1)^{2} - 1} \\ + \frac{ϑ_{1}}{2 ϑ_{2}} {\exp [ϑ_{2} ({\bar{I}}_{6} - 1)^{2} - 1} \end{matrix}

(25)

where $m_{i}, o_{j}, t_{k}, u_{l}$ , and $w_{v}$ are the material constants, and $ς = (α_{0} β_{0})^{2}$ . Moreover, $ϑ_{1} > 0$ is a stress-like material parameter and $ϑ_{2} > 0$ is a dimensionless parameter. Both functions are wildly used but in this paper, the second type is employed.

One of the important parameters that can extremely affect the electromechanical behaviors of ADEs and even limit their applications is viscoelasticity. To describe the viscoelastic behavior, the DE membrane is approximately simulated by various models such as the damping model (Zhao et al., 2011), rheological model (Park, 2018), Kelvin–Voigt model (Lei et al., 2013), Maxwell model (Wenchang et al., 2003), and a combination of both models called Kelvin-Voigt-Maxwell model (Zhang et al., 2017). Because of the simplicity and reliability of the rheological model, this model is selected in this study. Based on the rheological model of springs and dashpot in Figure 3, both springs have different shear modulus (the top spring with $κ_{2}$ , the other spring with $κ_{1}$ ) and a dashpot of the viscosity $η$ .

Figure 3.

Schematic representations of a rheological model for a viscoelastic ADE.

Based on multiplication rule (Sheng et al., 2013) as $λ = \bar{λ} ζ$ the $ζ$ is the non-elastic stretch of the dashpot and $\bar{λ}$ is the stretch of both springs. Also, as a Newtonian fluid, the rate of deformation in the dashpot is represented as follow (Hong, 2011)

\frac{\partial ζ}{\partial t} = \frac{- 1}{η} (\frac{\partial Ω}{\partial ζ})

(26)

It should be stated that the relation between the viscosity of the dashpot $η$ and $κ_{2}$ is prescribed as the viscoelastic relaxation time as $σ = η / κ_{2}$ .

In Figure 4 a DE reinforced with fiber is depicted and its dimensions in the un-deformed and uncharged state (original size) are $L_{1} \times L_{2} \times H$ . In a deformed state, the ADE is connected to a power source of voltage V while the dimensions of the membrane are $l_{1} \times l_{2} \times h$ .

Figure 4.

An ADE (a) before applied voltage, (b) after applied voltage, and (c) with the positive and the negative charge on top and below of the surface of the membrane.

2.1. Uniaxial tension

Given that the authors’ approach is to use a semi-analytical method, based on the following assumptions, the scalar equations have appeared.

Based on the homogeneous deformation in each dimension, the stretch ratio can be defined as follows

\begin{matrix} λ_{1} = x_{1} / X_{1} = l_{1} / L_{1} \\ λ_{2} = x_{2} / X_{2} = l_{2} / L_{2} \\ λ_{3} = x_{3} / X_{3} = h / H \end{matrix}

(27)

Applying the incompressibility constraint for an ADE, the time-dependent stretch and the motion of the body change to the following form

\begin{matrix} J = 1, λ_{1} λ_{2} λ_{3} = 1, λ_{2} = λ_{3} \\ λ_{1} = λ, λ_{2} = 1 / \sqrt{λ}, λ_{3} = 1 / \sqrt{λ} \\ x_{1} = λ X_{1}, x_{2} = 1 / \sqrt{λ} X_{2}, x_{3} = 1 / \sqrt{λ} X_{3} \end{matrix}

(28)

Using the above equation, the matrix forms of the deformation gradient and the right Cauchy–Green deformation tensor are calculated as follow

F = [\begin{matrix} λ & 0 & 0 \\ 0 & 1 / \sqrt{λ} & 0 \\ 0 & 0 & 1 / \sqrt{λ} \end{matrix}], C = [\begin{matrix} λ^{2} & 0 & 0 \\ 0 & 1 / λ & 0 \\ 0 & 0 & 1 / λ \end{matrix}]

(29)

The kinetic energy of the DE can also be obtained as follow

\begin{matrix} Γ = \int_{Ξ} χ \sum_{k = 1}^{3} (\frac{\partial x_{k}}{\partial t}) (\frac{\partial x_{k}}{\partial t}) \frac{1}{2} d Ξ \\ = \int_{0}^{H} \int_{0}^{L_{2}} \int_{0}^{L_{1}} χ {(d λ / dt)}^{2} X_{1}^{2} \frac{1}{2} d X_{1} d X_{2} d X_{3} \\ + \int_{0}^{H} \int_{0}^{L_{2}} \int_{0}^{L_{1}} \frac{1}{4} χ {(d λ / dt)}^{2} λ^{- 3} X_{2}^{2} \frac{1}{2} d X_{1} d X_{2} d X_{3} \\ + \int_{0}^{H} \int_{0}^{L_{2}} \int_{0}^{L_{1}} \frac{1}{4} χ {(d λ / dt)}^{2} λ^{- 3} X_{3}^{2} \frac{1}{2} d X_{1} d X_{2} d X_{3} \\ = \frac{1}{6} χ (d λ / dt)^{2} L_{1}^{3} L_{2} H + \frac{1}{24} χ (d λ / dt)^{2} λ^{- 3} L_{2}^{3} L_{1} H \\ + \frac{1}{24} χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \end{matrix}

(30)

where $χ$ and $Ξ$ are the density of the DE and occupied space, respectively. It is assumed that the material is incompressible, therefore the density remains constant through the deformation.

The potential energy of a viscoelastic ADE subjected to electrical force for different models is represented as follow

\begin{matrix} Neo - Hookean : Π = [ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{4} - 1)^{2}) - 1) \\ + κ_{1} (I_{1} - 3) \\ + κ_{2} ({\hat{I}}_{1} - 3) - 1 / 2 ε E^{2} λ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{6} - 1)^{2} - 1)] Ξ \end{matrix}

(31)

\begin{matrix} Mooney - Rivlin : Π = [ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{4} - 1)^{2}) - 1) \\ + c_{1} (I_{1} - 3) \\ + c_{2} ({\hat{I}}_{1} - 3) + c_{3} (I_{2} - 3) + c_{4} ({\hat{I}}_{2} - 3) - 1 / 2 ε E^{2} λ \\ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{6} - 1)^{2} - 1)] Ξ \end{matrix}

(32)

\begin{matrix} Ogden : Π = [ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{4} - 1)^{2}) - 1) \\ + μ_{1} / ξ_{1} (λ_{1}^{ξ_{1}} + λ_{2}^{ξ_{1}} + λ_{3}^{ξ_{1}} - 3) \\ + μ_{2} / ξ_{2} (λ_{1}^{ξ_{2}} + λ_{2}^{ξ_{2}} + λ_{3}^{ξ_{2}} - 3) + c_{5} (λ_{1}^{ξ_{1}} + λ_{2}^{ξ_{1}} + λ_{3}^{ξ_{1}} - 3) \\ + c_{6} (λ_{1}^{ξ_{2}} + λ_{2}^{ξ_{2}} + λ_{3}^{ξ_{2}} - 3) \\ - 1 / 2 ε E^{2} λ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} (I_{6} - 1)^{2} - 1)] Ξ \end{matrix}

(33)

where

\begin{matrix} I_{1} = λ^{2} + 2 λ^{- 1} \\ {\hat{I}}_{1} = 2 λ^{- 1} ζ + λ^{2} ζ^{- 2} \\ I_{4} = λ^{- 1} \sin^{2} θ + λ^{2} \cos^{2} θ \\ I_{6} = λ^{2} \cos^{2} θ + λ^{- 1} \sin^{2} θ \end{matrix}

(34)

The fiber-reinforced material with arbitrary direction for the following analysis is shown in Figure 5.

Figure 5.

The orientation of fiber in an ADE.

To get the nonlinear governing equation of motion for the time-dependent stretch $λ (t)$ , the Euler-Lagrange equation is needed as following (Bustamante, 2009; Hossain, 2020; Kashani et al., 2010; Tomer et al., 2008)

- \frac{d}{dt} (\frac{\partial Λ}{\partial (d λ / dt)}) + \frac{\partial Λ}{\partial λ} = 0, Λ = Γ - Π

(35)

Employing the kinetic energy and potential energy of ADE, the following equations are derived as follows

\begin{matrix} Neo - Hookean : Λ = 1 / 6 χ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2}^{3} H \\ + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \\ - [ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2} \sin^{4} θ + λ^{4} \cos^{4} θ + 1 \\ - 2 λ^{2} \cos^{2} θ - 2 λ^{- 1} \sin^{2} θ \\ + 2 λ \cos^{2} θ \sin^{2} θ)) - 1) + κ_{1} (λ^{2} - 3 + 2 λ^{- 1}) - 1 / 2 ε E^{2} λ \\ + κ_{2} (- 3 + 2 λ^{- 1} ζ + λ^{2} ζ^{- 2})] H L_{1} L_{2} \end{matrix}

(36)

\begin{matrix} Mooney - Rivlin : Λ = 1 / 6 χ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2}^{3} H \\ + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \\ - [ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2} \sin^{4} θ + λ^{4} \cos^{4} θ \\ + 1 - 2 λ^{2} \cos^{2} θ - 2 λ^{- 1} \sin^{2} θ \\ + 2 λ \cos^{2} θ \sin^{2} θ)) - 1) + c_{1} (λ^{2} - 3 + 2 λ^{- 1}) - 1 / 2 ε E^{2} λ \\ + c_{2} (- 3 + 2 λ^{- 1} ζ + λ^{2} ζ^{- 2}) + c_{3} (- 3 + 2 λ + λ^{- 2}) \\ + c_{4} (2 λ ζ^{- 1} + λ^{- 2} ζ^{2} - 3)] H L_{1} L_{2} \end{matrix}

(37)

\begin{matrix} Ogden : Λ = 1 / 6 χ (d λ / dt)^{2} L_{1}^{3} L_{2} H + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2}^{3} H \\ + 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \\ - [ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2} \sin^{4} θ + λ^{4} \cos^{4} θ \\ + 1 - 2 λ^{2} \cos^{2} θ - 2 λ^{- 1} \sin^{2} θ \\ + 2 λ \cos^{2} θ \sin^{2} θ)) - 1) + μ_{1} / ξ_{1} (λ_{1}^{ξ} + 2 λ_{1}^{ξ} - 3) - 1 / 2 ε E^{2} λ \\ + μ_{2} / ξ_{2} (λ^{ξ_{2}} + 2 λ^{ξ_{2}} - 3) + c_{5} (- 3 + 2 λ^{ξ_{2}} ζ^{- ξ_{2}} + λ^{ξ_{2}} ζ^{- ξ_{2}}) \\ + c_{6} (+ 2 λ^{ξ_{1}} ζ^{- ξ_{1}} + λ^{ξ_{1}} ζ^{- ξ_{1}} - 3)] H L_{1} L_{2} \end{matrix}

(38)

where $E = E_{0} \sin (2 π ft)$ .

And the other couple equation based on $ζ$ and $λ$ can be represented as follow

Neo - Hookean : \frac{\partial ζ}{\partial t} = - \frac{1}{η} [- 2 λ^{2} ζ^{- 3} + \frac{2}{λ}] κ_{2}

(39)

\begin{matrix} Mooney - Rivlin : \frac{\partial ζ}{\partial t} = - \frac{1}{η} [c_{2} (2 λ^{- 1} - 2 λ^{2} ζ^{- 3}) \\ + c_{4} (- 2 λ ζ^{- 2} + 2 λ^{2} ζ)] \end{matrix}

(40)

\begin{matrix} Ogden : \frac{\partial ζ}{\partial t} = - \frac{1}{η} [c_{5} (- 2 ξ_{2} λ^{ξ_{2}} ζ^{- ξ_{2} - 1} - ξ_{2} λ^{ξ_{2}} ζ^{- ξ_{2} - 1}) \\ + c_{6} (- 2 ξ_{1} λ^{ξ_{1}} ζ^{- ξ_{1} - 1} - ξ_{1} λ^{ξ_{1}} ζ^{- ξ_{1} - 1})] \end{matrix}

(41)

2.2. Pure shear

The motion of the body for a pure shear state can be proposed as follow

x_{1} = λ X_{1}, x_{2} = X_{2}, x_{3} = 1 / \sqrt{λ} X_{3}

(42)

where

F = [\begin{matrix} λ & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 / \sqrt{λ} \end{matrix}], C = [\begin{matrix} λ^{2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 / λ \end{matrix}]

(43)

Like the previous section, the kinetic energy of the DE can be developed as follow

\begin{matrix} Γ = \int_{Ξ} χ \sum_{k = 1}^{3} (\frac{\partial x_{k}}{\partial t}) (\frac{\partial x_{k}}{\partial t}) \frac{1}{2} d Ξ \\ = \int_{0}^{H} \int_{0}^{L_{2}} \int_{0}^{L_{1}} χ {(d λ / dt)}^{2} X_{1}^{2} \frac{1}{2} d X_{1} d X_{2} d X_{3} \\ + \int_{0}^{H} \int_{0}^{L_{2}} \int_{0}^{L_{1}} \frac{1}{4} χ {(d λ / dt)}^{2} λ^{- 3} X_{3}^{2} \frac{1}{2} d X_{1} d X_{2} d X_{3} \\ = \frac{1}{6} χ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ + \frac{1}{24} χ (d λ / dt)^{2} λ^{- 3} H^{3} L_{1} L_{2} \end{matrix}

(44)

The potential energy of a viscoelastic ADE where $J \neq 1$ is stated as follows

\begin{matrix} Neo - Hookean : Π = [1 / 2 ϑ_{0} (J - 1)^{2} + ϑ_{1} / 2 ϑ_{2} \\ (\exp (ϑ_{2} ({\bar{I}}_{4} - 1)^{2}) - 1) + κ_{1} ({\bar{I}}_{1} - 3) \\ + κ_{2} ({\bar{I}}_{1} - 3) - 1 / 2 ε E^{2} λ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} ({\bar{I}}_{6} - 1)^{2} - 1)] Ξ \end{matrix}

(45)

\begin{matrix} Mooney - Rivlin : Π = [1 / 2 ϑ_{0} (J - 1)^{2} + ϑ_{1} / 2 ϑ_{2} \\ (\exp (ϑ_{2} ({\bar{I}}_{4} - 1)^{2}) - 1) + c_{1} ({\bar{I}}_{1} - 3) \\ + c_{2} ({\bar{I}}_{1} - 3) - 1 / 2 ε E^{2} λ + c_{3} ({\bar{I}}_{2} - 3) \\ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} ({\bar{I}}_{6} - 1)^{2} - 1) + c_{4} ({\bar{I}}_{2} - 3)] Ξ \end{matrix}

(46)

\begin{matrix} Ogden : Π = [1 / 2 ϑ_{0} (J - 1)^{2} + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} ({\bar{I}}_{4} - 1)^{2}) - 1) \\ + μ_{1} / ξ_{1} (λ_{1}^{ξ_{1}} + λ_{2}^{ξ_{1}} + λ_{3}^{ξ_{1}} - 3) \\ + μ_{2} / ξ_{2} (λ_{1}^{ξ_{2}} + λ_{2}^{ξ_{2}} + λ_{3}^{ξ_{2}} - 3) + c_{5} (λ_{1}^{ξ_{1}} + λ_{2}^{ξ_{1}} \\ + λ_{3}^{ξ_{1}} - 3) + c_{6} (λ_{1}^{ξ_{2}} + λ_{2}^{ξ_{2}} + λ_{3}^{ξ_{2}} - 3) \\ - 1 / 2 ε E^{2} λ + ϑ_{1} / 2 ϑ_{2} (\exp (ϑ_{2} ({\bar{I}}_{6} - 1)^{2} - 1)] Ξ \end{matrix}

(47)

where

\begin{matrix} {\bar{I}}_{1} & = λ^{5 / 3} ζ^{- 5 / 3} + λ^{- 1 / 3} ζ^{1 / 3} + λ^{- 4 / 3} ζ^{4 / 3} \\ {\bar{I}}_{1} & = J^{- 2 / 3} I_{1} \\ = λ^{5 / 3} + λ^{- 1 / 3} + λ^{- 4 / 3} \\ {\bar{I}}_{2} {\bar{I}}_{2} & = 2 λ^{1 / 3} ζ^{- 1 / 3} + λ^{- 8 / 3} ζ^{8 / 3} \\ {\bar{I}}_{2} & = J^{- 2 / 3} I_{2} \\ = 2 λ^{1 / 3} + λ^{- 8 / 3} \\ {\bar{I}}_{4} & = J^{- 2 / 3} I_{4} \\ = λ^{5 / 3} \cos^{2} θ + λ^{- 1 / 3} \sin^{2} θ \\ {\bar{I}}_{6} & = J^{- 2 / 3} I_{6} \\ = λ^{- 1 / 3} \sin^{2} θ + λ^{5 / 3} \cos^{2} θ \end{matrix}

(48)

Now the simple form of the Lagrange function can be derived as follow

\begin{matrix} Neo - Hookean : Λ = 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \\ + 1 / 6 χ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ - [1 / 2 ϑ_{0} (1 + λ - 2 λ^{1 / 2}) + ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2 / 3} \sin^{4} θ \\ + λ^{10 / 3} \cos^{4} θ \\ + 1 + 2 λ^{4 / 3} \sin^{2} θ \cos^{2} θ - 2 λ^{5 / 3} \cos^{2} θ - 2 λ^{- 1 / 3} \sin^{2} θ)) - 1) \\ + κ_{1} (- 3 + λ^{5 / 3} + λ^{- 1 / 3} + λ^{- 4 / 3}) - 1 / 2 ε E^{2} λ \\ + κ_{2} (λ^{5 / 3} ζ^{- 5 / 3} + λ^{- 1 / 3} ζ^{1 / 3} + λ^{- 4 / 3} ζ^{4 / 3} - 3)] H L_{1} L_{2} \end{matrix}

(49)

\begin{matrix} Mooney - Rivlin : Λ = 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} \\ + 1 / 6 χ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ - [1 / 2 ϑ_{0} (1 + λ - 2 λ^{1 / 2}) + c_{1} (- 3 + λ^{5 / 3} + λ^{- 1 / 3} + λ^{- 4 / 3}) \\ + ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2 / 3} \sin^{4} θ + λ^{10 / 3} \cos^{4} θ + 1 \\ + 2 λ^{4 / 3} \sin^{2} θ \cos^{2} θ \\ - 2 λ^{5 / 3} \cos^{2} θ - 2 λ^{- 1 / 3} \sin^{2} θ)) - 1) \\ + c_{4} (λ^{- 8 / 3} ζ^{8 / 3} - 3 + 2 λ^{1 / 3} ζ^{- 1 / 3}) \\ + c_{3} (λ^{- 8 / 3} + 2 λ^{1 / 3} - 3) - 1 / 2 ε E^{2} λ + c_{2} (λ^{5 / 3} ζ^{- 5 / 3} \\ + λ^{- 1 / 3} ζ^{1 / 3} + λ^{- 4 / 3} ζ^{4 / 3} - 3)] {HL}_{1} L_{2} \end{matrix}

(50)

\begin{matrix} Ogden : Λ = 1 / 24 χ (d λ / dt)^{2} λ^{- 3} L_{1} L_{2} H^{3} + 1 / 6 χ \\ (d λ / dt)^{2} L_{1}^{3} L_{2} H \\ - [1 / 2 ϑ_{0} (1 + λ - 2 λ^{1 / 2}) + ϑ_{1} / ϑ_{2} (\exp (ϑ_{2} (λ^{- 2 / 3} \sin^{4} θ \\ + λ^{10 / 3} \cos^{4} θ \\ + 1 + 2 λ^{4 / 3} \sin^{2} θ \cos^{2} θ - 2 λ^{5 / 3} \cos^{2} θ - 2 λ^{- 1 / 3} \sin^{2} θ)) - 1) \\ + μ_{1} / ξ_{1} (λ^{ξ_{1}} + 2 λ^{ξ_{1}} - 3) - 1 / 2 ε E^{2} λ \\ + μ_{2} / ξ_{2} (λ^{ξ_{2}} + 2 λ^{ξ_{2}} - 3) \\ + c_{5} (- 3 + 2 λ^{ξ_{2}} ζ^{- ξ_{2}} + λ^{ξ_{2}} ζ^{- ξ_{2}}) + c_{6} (+ 2 λ^{ξ_{1}} ζ^{- ξ_{1}} \\ + λ^{ξ_{1}} ζ^{- ξ_{1}} - 3)] H L_{1} L_{2} \end{matrix}

(51)

Again, to get the time-dependent stretch, another couple equation is needed as follow

\begin{matrix} Neo - Hookean : \frac{\partial ζ}{\partial t} \\ = - \frac{1}{η} [- \frac{5}{3} λ^{\frac{5}{3}} ζ^{- \frac{8}{3}} + \frac{5}{3} λ^{\frac{- 1}{3}} ζ^{\frac{2}{3}} + \frac{4}{3} λ^{\frac{- 4}{3}} ζ^{\frac{4}{3}}] κ_{2} \end{matrix}

(52)

\begin{matrix} Mooney - Rivlin : \frac{\partial ζ}{\partial t} = - \frac{1}{η} [c_{4} (\frac{8}{3} λ^{\frac{8}{3}} ζ^{\frac{5}{3}} - \frac{2}{3} λ^{\frac{1}{3}} ζ^{- \frac{4}{3}}) \\ + c_{2} (- \frac{5}{3} λ^{\frac{5}{3}} ζ^{- \frac{8}{3}} + \frac{1}{3} λ^{- \frac{1}{3}} ζ^{- \frac{2}{3}} + \frac{4}{3} λ^{- \frac{4}{3}} ζ^{\frac{1}{3}})] \end{matrix}

(53)

\begin{matrix} Ogden : \frac{\partial ζ}{\partial t} = - \frac{1}{η} [c_{5} (- 2 ξ_{2} λ^{ξ_{2}} ζ^{- ξ_{2} - 1} - ξ_{2} λ^{ξ_{2}} ζ^{- ξ_{2} - 1}) \\ + c_{6} (- 2 ξ_{1} λ^{ξ_{1}} ζ^{- ξ_{1} - 1} - ξ_{1} λ^{ξ_{1}} ζ^{- ξ_{1} - 1})] \end{matrix}

(54)

3. Results and discussions

Here, the comparison of results has been made to confirm the accuracy of the mentioned method in this paper with Xu et al. (2012) (Figure 6).

Figure 6.

Performance verification of isotropic DE for a different electric field with the present study.

The obtained results are also compared with those experimentally reported by Son (2011) and Schröder et al. (2005). Son introduced the initial dimensions for the simulation as $L_{1} = L_{2} = 200 mm, h_{0} = 0.5 mm$ and the electric field was applied in the $X_{3}$ direction. He proposed a procedure to solve a nonlinear electro-elastic problem in ABAQUS based on the Newton-Raphson method. The electromechanical response of the isotropic DE under the equibiaxial extension concerning the applied voltage is plotted in Figure 7. It can be seen that the numerical results from the analytical solution have good agreement with the numerical data from ABAQUS with the UMAT. The results of the more realistic orthotropic test material are shown in Figure 8(a) to (d) where the Kirchhoff and Second Piola–Kirchhoff stresses are plotted versus the stretch. The fiber orientations are considered as $χ_{0} = (0.8746, 0.4848, 0 {. 0)}^{T}, γ_{0} = (0.8746, - 0.4848, 0 {. 0)}^{T}$ . In each illustration, the presented model is compared to the reference model and we observe qualitatively accurate matchings of both regimes. So it may be concluded that approximating an orthotropic model via a transversely isotropic formulation is the right thing to do.

Figure 7.

The electromechanical response of the pre-stretched isotropic DE based on the applied voltage.

Figure 8.

Comparison of the reference model and the presented model showing the Kirchhoff $σ$ and second Piola–Kirchhoff S stresses in Experiment versus stretch λ₁ with (a, b) $θ = 0 °$ and (c, d) $θ = 29 °$ .

The ADE properties and initial conditions used in this study are listed as follows

\begin{matrix} H = 5 \times 10^{- 3} m, L_{1} = 20 \times 10^{- 3} m, L_{2} = 10 \times 10^{- 3} m \\ κ_{1} = 3.35 \times 10^{4} pa, κ_{2} = 2.1 \times 10^{4} pa, σ = 400 s \\ χ = 1200 kg / m^{3}, ε = 6.198 \times 10^{- 11} F / m \\ E = 12 \times 10^{4} \sin (600 π t), ϑ_{0} = 13.42 \times 10^{4} pa, \\ ϑ_{1} = 10^{6} pa, ϑ_{2} = 100 pa \\ c_{1} = 0.07 \times 10^{6} pa, c_{2} = 0.03 \times 10^{6} pa, c_{3} = 0.035 \\ \times 10^{6} pa, c_{4} = 0.015 \times 10^{6} pa \\ μ_{1} = 50 \times 10^{3} pa, μ_{2} = 2500 pa, ξ_{1} =, 0.8 ξ_{2} = 3.5, \\ c_{5} = 31.25 \times 10^{3} pa, c_{6} = 3.57 \times 10^{2} pa \\ λ (0) = 1, ζ (0) = 1, d λ / dt (0) = 0 \end{matrix}

(55)

Figures 9 to 18 sketch the relation between the time-dependent behavior of the amplitude and the phase diagram for three hyperelastic models with $\hat{θ} = 0 ° - 90 °$ . That is to say, the $λ - d λ / dt$ phase paths are concluded in closed regions. Moreover, based on the diagrams the nonlinear behavior of the structure is quite evident.

Figure 9.

Comparing the phase diagrams and oscillations of various models for uniaxial tension and $\hat{θ} = 0 °$ .

Figure 10.

Comparing the phase diagrams and oscillations of various models for uniaxial tension and $\hat{θ} = 30 °$ .

Figure 11.

Comparing the phase diagrams and oscillations of various models for uniaxial tension and $\hat{θ} = 45 °$ .

Figure 12.

Comparing the phase diagrams and oscillations of various models for uniaxial tension and $\hat{θ} = 60 °$ .

Figure 13.

Comparing the phase diagrams and oscillations of various models for uniaxial tension and $\hat{θ} = 90 °$ .

Figure 14.

Comparing the phase diagrams and oscillations of various models for pure shear and $\hat{θ} = 0 °$ .

Figure 15.

Comparing the phase diagrams and oscillations of various models for pure shear and $\hat{θ} = 30 °$ .

Figure 16.

Comparing the phase diagrams and oscillations of various models for pure shear and $\hat{θ} = 45 °$ .

Figure 17.

Comparing the phase diagrams and oscillations of various models for pure shear and $\hat{θ} = 60 °$ .

Figure 18.

Comparing the phase diagrams and oscillations of various models for pure shear and $\hat{θ} = 90 °$ .

Figures 19 to 28 illustrate the maximum stretch as a function of the frequency of excitation for three hyperelastic models changing with the fiber orientations from $0 ° - 90 °$ . One can find that by increasing the fiber angle the resonance frequency decreases because fibers help the network of the DE not to be further stretched and make the ADE membrane stiffer. As shown in Figures 19 to 28, it can be observed that at the beginning the stretch shows a rapidly increase behavior in a small frequency range for both uniaxial tension mode and pure shear mode with three different hyperelastic models and after that remains constant until the resonance occurs and again remains almost constant.

Figure 19.

The performances of the ADE under excited frequencies with various models for uniaxial tension and $\hat{θ} = 0 °$ .

Figure 20.

The performances of the ADE under excited frequencies with various models for uniaxial tension and $\hat{θ} = 30 °$ .

Figure 21.

The performances of the ADE under excited frequencies with various models for uniaxial tension and $\hat{θ} = 45 °$ .

Figure 22.

The performances of the ADE under excited frequencies with various models for uniaxial tension and $\hat{θ} = 60 °$ .

Figure 23.

The performances of the ADE under excited frequencies with various models for uniaxial tension and $\hat{θ} = 90 °$ .

Figure 24.

The performances of the ADE under excited frequencies with various models for pure shear and $\hat{θ} = 0 °$ .

Figure 25.

The performances of the ADE under excited frequencies with various models for pure shear and $\hat{θ} = 30 °$ .

Figure 26.

The performances of the ADE under excited frequencies with various models for pure shear and $\hat{θ} = 45 °$ .

Figure 27.

The performances of the ADE under excited frequencies with various models for pure shear and $\hat{θ} = 60 °$ .

Figure 28.

The performances of the ADE under excited frequencies with various models for pure shear and $\hat{θ} = 90 °$ .

Based on Figures 29 to 38, rising the electrical field leads to a rise in voltage, therefore a larger equilibrium stretch and a longer deformation time are expected. The reason is that increasing the electric field has a strong effect on the molecules in the polymer and makes them less closely aligned with each other. It should be noted that after a certain amount of applied electric field the stretch ramps up instead of reaching stability. For the smallest electric field, $E_{0} = 12 kV / mm$ the calculated stretch achieves a steady-state of vibration with enough time for the ADE to relax. Also, a higher fiber angle can give rise to a specific larger amplitude of stretch especially for $E_{0} = 30 kV / mm$ . By expanding the ADE, after a certain time for a distinct electric field, the stretch first increases reach a peak, and then decreases. Finally, with the increase in the applied electric field strength the frequency of the vibration decreases.

Figure 29.

The performances of the ADE under electric field with various models for uniaxial tension and $\hat{θ} = 0 °$ .

Figure 30.

The performances of the ADE under electric field with various models for uniaxial tension and $\hat{θ} = 30 °$ .

Figure 31.

The performances of the ADE under electric field with various models for uniaxial tension and $\hat{θ} = 45 °$ .

Figure 32.

The performances of the ADE under electric field with various models for uniaxial tension and $\hat{θ} = 60 °$ .

Figure 33.

The performances of the ADE under electric field with various models for uniaxial tension and $\hat{θ} = 90 °$ .

Figure 34.

The performances of the ADE under electric field with various models for pure shear and $\hat{θ} = 0 °$ .

Figure 35.

The performances of the ADE under electric field with various models for pure shear and $\hat{θ} = 30 °$ .

Figure 36.

The performances of the ADE under electric field with various models for pure shear and $\hat{θ} = 45 °$ .

Figure 37.

The performances of the ADE under electric field with various models for pure shear and $\hat{θ} = 60 °$ .

Figure 38.

The performances of the ADE under electric field with various models for pure shear and $\hat{θ} = 90 °$ .

The detailed responses are plotted in Figures 39 to 41. The actuation frequencies $f = 200, 300, 400 Hz$ are employed. It can be seen that the direction of fibers strongly affects the stretch of ADE. Moreover, another aim of the harmonic electrical loading is to represent a complicated nature due to the presence of nonlinearity in the electro-mechanical system. Based on the Figures 39 to 41 it is found that as the electrical loading goes up, the amplitude of DE’s vibration increases.

Figure 39.

The performances of the ADE under exited frequency $f = 200 Hz$ for various models.

Figure 40.

The performances of the ADE under exited frequency $f = 300 Hz$ with various models for uniaxial tension.

Figure 41.

The performances of the ADE under exited frequency $f = 400 Hz$ with various models for uniaxial tension.

4. Conclusions

The presence of fibers in the elastomer has an extraordinary effect in improving the performances of the DE. The anisotropy is considered because the fiber-reinforced materials have special abilities such as high response and capability of higher force output. Therefore, in this study, the nonlinear field theory of ADEs is utilized to provide explicit expressions for the nonlinear governing equations of motion of an ADE by giving practical hyperelastic theories function. Because of ADEs’ nature, their mechanical properties can extremely effect by the application of an electric field. Therefore, employing the Helmholtz free energy functions considering the dissipation processes the nonlinear Lagrange equations are derived and analyzed with Neo-Hookean, Mooney-Rivlin, and Ogden material properties. The ADE is ideal and the permittivity is not considered anisotropic. Furthermore, the electrodes coated on both side of the ADE have no electrical resistance. Based on numerical results, such as the oscillation and phase diagrams the influence of fibers’ existence is fully presented. Moreover, by comparing the hyperelastic models for uniaxial tension mode and pure shear mode, it can be seen that the fibers orientation and the induced electric field have both special effects on the nonlinear behavior of ADE. Therefore, because of the importance of the quick dynamic response of smart materials with a dispersion-type anisotropy to the external electric field in smart manufacturing technologies, an analytical model is proposed and developed to present the useful and applicable results this field.

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) received no financial support for the research, authorship, and/or publication of this article.

Ethical statement

The article was conducted according to ethical standards.

ORCID iD

Masoud Asgari

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