Suppression of electromechanical instability in fiber-reinforced dielectric elastomers Open Access

Dielectric elastomers (DEs) are soft active materials that can deform in response to electric field.^1,2 The ability of converting the mechanical and electrical energy between each other has made DEs attractive for sensors, actuators and generators.^3–9 However, the electromechanical instability, which is also referred as pull-in instability, has limited the applications of DEs.^10,11

When a voltage is applied on a DE membrane, the thickness of the membrane decreases due to the Maxwell stress (Figure 1). The decrease in the thickness results in a further increase in electric field. This positive feedback could lead DEs to fail at a small deformation due to the pull-in instability.^11,12 To achieve large deformation, several methods have been adopted to suppress the instability.^13–20 The most widely used method is applying a pre-stretch on elastomer using a mechanical load.^13,15,18 or rigid constraint.^1,16 Both experimental results and model predictions have shown that prestretched DEs can either eliminate the pull-in instability¹³ or survive the instability to reach a giant deformation.¹⁵ The physical mechanism behind this method is that elastomers exhibit strain stiffen at high stretches which suppresses the deformation caused by the positive feedback. It is also demonstrated that large deformation can be achieved by spraying charge on elastomers.^21,22 In addition, elastomers with interpenetrating network can also achieve large deformation without mechanical instability.²³ Zhao and Suo²⁴ proposed a theory showing that giant deformation is achievable for DEs with stress-strain curve of a specific form: the mechanical response is compliant at small deformation region and stiffens steeply at modest deformation. Motivated by this observation, we present a theory to investigate the performance of soft fiber-reinforced elastomers under electric field.

Most of biological tissues and some engineering materials such as filled woven fabrics^25–27 can be classified into fiber-reinforced composite. While several works have been carried out to investigate the mechanical performance of these materials,^25,28,29 limited efforts have been made to study the electromechanical response of the materials. In the following, we first present a nonlinear constitutive theory to describe the dielectric and mechanical response of fiber-reinforced DEs.

We simulate a fiber-reinforced dielectric membrane under an electric field in the thickness direction while the fiber is only distributed in the plane. The fiber is assumed to share the same deformation gradient as the elastomer matrix.²⁵ In the above loading condition, we can choose the dominate fiber direction, the perpendicular direction in the plane and the z direction as the principal directions of the strain. The deformation gradient F, which maps the material lines in the reference configuration to the current deformed configuration, can be defined as: F = λ_fe_f ⊗ e_f + λ_pe_p ⊗ e_p + λ_ze_z ⊗ e_z. The composite is assumed to be volumetric incompressible. Thus, the stretch in the z direction can be written as $λ z =1/ λ f λ p$⁠. The free energy Ψ is composed of an isotropic matrix term, an anisotropic fiber term and a polarization term,

where $D ˜$ is the nominal electrical displacement. The nominal electric field can be calculated as: $E ˜ = D ˜ /ϵ$⁠, where ϵ is the permittivity of the dielectric elastomer.

The Arruda-Boyce eight chain model is applied to represent the free energy of stretching polymer chains,^30,31

where $λ eff = 1 3 I 1$⁠, $I 1 = λ f 2 + λ p 2 + 1 λ f 2 λ p 2$⁠, $x= L − 1 λ eff λ L$⁠, $y= L − 1 1 λ L$⁠, μ_N is the stiffness of polymer network and λ_L is the stretch limit of polymer chain. The function $L − 1$ is the inverse Langevin function where $L u =cothu− u − 1$⁠.

The fiber structure is represented by the semicircular von Mises distribution function,

where θ indicates the fiber angle relative to the dominate fiber direction and b is the concentration parameter representing the degree of fiber alignment along the dominate direction.

The square of the fiber stretch can be represented as I_f = a₀ ⋅ Ca₀, where C = F^TF is the right Cauchy strain tensor and $a 0 =cos θ e f +sin θ e p$ is the unit vector of the fiber. Thus, I_f can be represented as $I f = λ f 2 cos 2 θ + λ p 2 sin 2 θ$⁠.The free energy density of stretching fiber is represented as,²⁸ 

where k₁ has unit of stiffness representing the tensile properties of fiber family and k₂ describes the nonlinearity of the fiber, with higher value of k₂ representing the fiber stiffens at lower strain region.

We adopted a linear model for electric field free energy density,³² 

The Cauchy stress can be calculated as $σ f = λ f ∂ Ψ / ∂ λ f$ and $σ p = λ p ∂ Ψ / ∂ λ p$⁠, which can be represented as,

Substituting eq. (2), (4) and (5) into the above equations leads to,

where $μ= μ N 3 λ L λ eff L − 1 λ eff λ L$⁠.

To evaluate the electro-actuated response of the dielectric membrane under the electric field, we obtained λ_f and λ_p as a function of $E ˜$ through numerically solving σ_f = 0 and σ_p = 0. The parameters for elastomers are chosen as μ_N = 10 kPa and ϵ = 4.0 × 10⁻¹¹ F/m to represent the mechanical properties of acrylic dielectric elastomers.¹² The baseline parameters for fiber are chosen as b = 5, k₁ = 10 kPa and k₂ = 1. The values of k₁ and k₂ can be obtained through fitting these parameters to stress response of fiber-reinforced composite under biaxial deformation.^25,28,33 The typical value of k₁ varies from 0-1000 kPa and the typical value of k₂ varies from 0.1-1000.^25,28,33 To investigate the influence of different fiber properties on the electromechanical performance of membrane, a parameter study is performed to vary one of b, k₁ and k₂ while kept the other two the same as the baseline values.

We first investigate the influence of stiffness of fiber on the dielectric response. Due to the anisotropic response in the dominate fiber and perpendicular direction, we plot the nominal electrical displacement as a function of λ_z shown in Figure 2. Without fiber, the dielectric elastomer suffers the pull-in instability when λ_z reaches 0.625. With increasing the stiffness of fiber, the pull-in instability can be suppressed and monotonic dielectric response can be achieved.

Figure 3 compares the response of dielectric composite with different fiber nonlinearity parameters k₂. A larger k₂ represents the fiber stiffen at lower strain which also results a significant increase of electric displacement at a smaller deformation.

The fiber distribution also has a significant influence on the electrical-mechanical response. Figure 4 plots the fiber distribution of different distribution parameter b. When the fiber is mainly distributed in the dominate direction, the electric-induced stretch is mainly distributed in the perpendicular direction. Pull-in instability still occurs as shown in Figure 5. Decreasing the anisotropic distribution of fiber can suppress pull-in instability. However, decreasing b also results that the tremendous increase of electric displacement occurs at a smaller deformation, which may cause electric breakdown. These results suggest there may exist an optimal fiber distribution to achieve maximum deformation. For ideal dielectric elastomer, the nominal breakdown electric displacement can be calculated as: $E ˜ failure = E failure λ z$⁠, where E^failure is the breakdown electric field. Previous experiments have shown that the electric breakdown of dielectric elastomer may depend on the sample thickness and prestretch.^34–36 For simplicity, here we assume that E^failure is constant.²⁴ Specifically, E^failure is chosen as 50 MV/m.^35,36

Figure 5 plots the failure point of dielectric composite. For elastomer with fiber mainly distributed in the dominate direction (b = 50), the materials suffer pull-in instability. The thickness of the specimen dramatically decrease to a lower value. The materials fail before it can reach the stable state. For the case of b = 15, the materials survive the instability and reaches a stable state before electric breakdown. A further decrease b can eliminate the pull-in instability. Thus, the materials fail due to the electric breakdown. With more evenly distributed fiber family, the materials fail at a larger thickness due to the materials stiffen at a smaller deformation. Figure 6 plots the λ_z at the failure point as a function of fiber distribution parameter b. As shown, a transient failure mode from pull-in instability to electric breakdown can be observed when decreasing the fiber distribution parameter b. For both k₁ = 1 KPa and k₁ = 10 KPa, the optimal fiber distribution is b = 15.

In this work, we only simulated the electro-actuated response of the fiber-reinforced membrane without loading in the plane directions. Previous works have shown the electromechanical performance of dielectric elastomer depends on the deformation modes.^37,38 The model developed here can also be applied to describe the electro-actuated performance of fiber-reinforced dielectric elastomers in other deformation modes if the pre-stretch or loading is applied in the dominate fiber direction or perpendicular direction. Otherwise, the membrane will not exhibit homogenous deformation in every location and finite element method needs to be used to obtain the deformation. This is a subject of future work.

In summary, we have developed a nonlinear theory to investigate the electromechanical performance of fiber-reinforced elastomers. The results show that this class of materials can suppress the pull-in instability. It is also demonstrated that the fiber distribution has a strong influence on the performance of the materials. This work provides a promising method to achieve giant deformation of dielectric materials.