Dielectric elastomer composites: A general closed-form solution in the small-deformation limit

A solution for the overall electromechanical response of two-phase dielectric elastomer composites with (random or periodic) particulate microstructures is derived in the classical limit of small deformations and moderate electric fields. In this limit, the overall electromechanical response is characterized by three effective tensors: a fourth-order tensor describing the elasticity of the material, a second-order tensor describing its permittivity, and a fourth-order tensor describing its electrostrictive response. Closed-form formulas are derived for these effective tensors directly in terms of the corresponding tensors describing the electromechanical response of the underlying matrix and the particles, and the one- and two-point correlation functions describing the microstructure. This is accomplished by specializing a new iterative homogenization theory in finite electroelastostatics (Lopez-Pamies, 2014) to the case of elastic dielectrics with even coupling between the mechanical and electric fields and, subsequently, carrying out the pertinent asymptotic analysis.Additionally, with the aim of gaining physical insight into the proposed solution and shedding light on recently reported experiments, specific results are examined and compared with an available analytical solution and with new full-field simulations for the special case of dielectric elastomers filled with isotropic distributions of spherical particles with various elastic dielectric properties, including stiff high-permittivity particles, liquid-like high-permittivity particles, and vacuous pores.     

THEORY OF DIELECTRIC ELASTOMERS

In response to a stimulus, a soft material deforms, and the deformation provides a function. We call such a material a soft active material （SAM）. This review focuses on one class of soft active materials： dielectric elastomers. When a membrane of a dielectric elastomer is subject to a voltage through its thickness, the membrane reduces thickness and expands area, possibly straining over 100%. The dielectric elastomers are being developed as transducers for broad applications, including soft robots, adaptive optics, Braille displays, and electric generators. This paper reviews the theory of dielectric elastomers, developed within continuum mechanics and thermodynamics, and motivated by molecular pictures and empirical observations. The theory couples large deformation and electric potential, and describes nonlinear and nonequilibrium behavior, such as electromechanical instability and viscoelasticity. The theory enables the finite element method to simulate transducers of realistic configurations, predicts the efficiency of electromechanical energy conversion, and suggests alternative routes to achieve giant voltage-induced deformation. It is hoped that the theory will aid in the creation of materials and devices.     

Temporal evolution and instability in a viscoelastic dielectric elastomer

Dielectric elastomer transducers are being developed for applications in stretchable electronics, tunable optics, biomedical devices, and soft machines. These transducers exhibit highly nonlinear electromechanical behavior: a dielectric membrane under voltage can form wrinkles, undergo snap-through instability, and suffer electrical breakdown. We investigate temporal evolution and instability by conducting a large set of experiments under various prestretches and loading rates, and by developing a model that allows viscoelastic instability. We use the model to classify types of instability, and map the experimental observations according to prestretches and loading rates. The model describes the entire set of experimental observations. A new type of instability is discovered, which we call wrinkle-to-wrinkle transition. A flat membrane at a critical voltage forms wrinkles and then, at a second critical voltage, snaps into another state of winkles of a shorter wavelength. This study demonstrates that viscoelasticity is essential to the understanding of temporal evolution and instability of dielectric elastomers.     

Effect of matrix viscoelasticity on the electrorheological properties of particle suspensions

We have investigated the electrorheological properties of dispersions of semi-conducting particles in oils and elastomers. We focused on how the dynamic mechanical properties measured under oscillatory shearing change with the viscosity of the oil or the elasticity of the elastomer. The dependence on electric field and strain amplitude were also investigated. We found that the largest increment of the mechanical properties under electric fields was obtained when using oils of low viscosity and elastomers of low elasticity. The strain amplitude which produced the largest variation with electric field was found to be 0.1% for the elastomer systems, but significantly larger (1%) for the oil systems. These results are interpreted in terms of a model based on the competition between the dipole–dipole electrostatic interaction (which acts to maintain neighbouring particles together) and the shearing force due to the deformation of the matrix (which acts to separate the particles). We find that there are parallels between the electrorheological behaviour of particles dispersed in elastomers and the behaviour of particles dispersed in oils. These results should find application in the selection of suitable matrix materials for electrorheological suspensions.     

Temperature effects on the dynamic response of viscoelastic dielectric elastomer

Viscoelasticity and temperature can significantly affect the performance of a dielectric elastomer. In the current study, we use a thermodynamic model to describe the effect of temperature and viscoelasticity on the electromechanical response undergoing a cyclic electric load by taking into account of the temperature dependent dielectric constant. Because of the significant viscoelasticity in the dielectric elastomer, the deformation and the nominal electric displacement can not keep in phase with the electric field at low frequencies. The results show that the magnitude of the cyclic electromechanical actuation strain increases with the decrease of the temperature and decreases with the increasing frequency, and viscoelasticity can result in significant hysteresis for dielectric elastomers under a relative low temperature and a low frequency.     

Electro-viscoelastic behaviors of circular dielectric elastomer membrane actuator containing concentric rigid inclusion

The time-dependent electro-viscoelastic performance of a circular dielectric elastomer (DE) membrane actuator containing an inclusion is investigated in the context of the nonlinear theory for viscoelastic dielectrics. The membrane, a key part of the actuator, is centrally attached to a rigid inclusion of the radius a, and then connected to a fixed rigid ring of the radius b. When subject to a pressure and a voltage, the membrane inflates into an out-of-plane shape and undergoes an inhomogeneous large deformation. The governing equations for the large deformation are derived by means of non-equilibrium thermodynamics, and viscoelasticity of the membrane is characterized by a rheological spring-dashpot model. In the simulation, effects of the pressure, the voltage, and design parameters on the electromechanical viscoelastic behaviors of the membrane are investigated. Evolutions of the considered variables and profiles of the deformed membrane are obtained numerically and illustrated graphically. The results show that electromechanical loadings and design parameters significantly influence the electro-viscoelastic behaviors of the membrane. The design parameters can be tailored to improve the performance of the membrane. The approach may provide guidelines in designing and optimizing such DE devices.     

Modeling of dielectric viscoelastomers with application to electromechanical instabilities

Soft dielectrics are electrically-insulating elastomeric materials, which are capable of large deformation and electrical polarization, and are used as smart transducers for converting between mechanical and electrical energy. While much theoretical and computational modeling effort has gone into describing the ideal, time-independent behavior of these materials, viscoelasticity is a crucial component of the observed mechanical response and hence has a significant effect on electromechanical actuation. In this paper, we report on a constitutive theory and numerical modeling capability for dielectric viscoelastomers, able to describe electromechanical coupling, large-deformations, large-stretch chain-locking, and a time-dependent mechanical response. Our approach is calibrated to the widely-used soft dielectric VHB 4910, and the finite-element implementation of the model is used to study the role of viscoelasticity in instabilities in soft dielectrics, namely (1) the pull-in instability, (2) electrocreasing, (3) electrocavitation, and (4) wrinkling of a pretensioned three-dimensional diaphragm actuator. Our results show that viscoelastic effects delay the onset of instability under monotonic electrical loading and can even suppress instabilities under cyclic loading. Furthermore, quantitative agreement is obtained between experimentally measured and numerically simulated instability thresholds. Our finite-element implementation will be useful as a modeling platform for further study of electromechanical instabilities and for harnessing them in design and is provided as online supplemental material to aid other researchers in the field.     

A mathematical model for cylindrical,fiber reinforced electro-pneumatic actuators

In recent years, dielectric elastomers have received increasing attention due to their unparalleled large strain actuation response (>100%). The force output, however, has remained a major limiting factor for many applications. To address this limitation, a model for a fiber reinforced dielectric elastomer actuator based on the deformation mechanism of McKibben actuators is presented. In this novel configuration, the outer cylindrical surface of a dielectric elastomer is enclosed by a network of helical fibers that are thin, flexible and inextensible. This configuration yields an axially contractile actuator, in contrast to unreinforced actuators which extend. The role of the fiber network is twofold: (i) to serve as reinforcement to improve the load-bearing capability of dielectric elastomers, and (ii) to render the actuator inextensible in the axial direction such that the only free deformation path is simultaneous radial expansion and axial contraction. In this paper, a mathematical model of the electromechanical response of fiber reinforced dielectric elastomers is derived. The model is developed within a continuum mechanics framework for large deformations. The cylindrical electro-pneumatic actuator is modeled by adapting Green and Adkins’ theory of reinforced cylinders to account for the applied electric field. Using this approach, numerical solutions are obtained assuming a Mooney–Rivlin material model. The results indicate that the relationship between the contractile force and axial shortening is bilinear within the voltage range considered. The characteristic response as a function of various system parameters such as the fiber angle, inflation pressure, and the applied voltage are reported. In this paper, the elastic portion of the modeling approach is validated using experimental data for McKibben actuators.     

RATE DEPENDENT STRESS-STRETCH RELATION OF DIELECTRIC ELASTOMERS SUBJECTED TO PURE SHEAR LIKE LOADING AND ELECTRIC FIELD

The performance of dielectric elastomer(DE) transducers is significantly affected by viscoelastic relaxation-induced electromechanical dissipations.This paper presents an experimental study to obtain the rate dependent stress-stretch relation of DE membranes(VHB TM 9473) subjected to pure shear like loading and electric loading simultaneously.Stretching rate dependent behavior is observed.The results also show that the tensile force decreases as the voltage increases.The observations are compared with predictions by a viscoelastic model of DE.This experiment may be used for further studies of dynamic electromechanical coupling properties of DEs.     

Experimental slowing of flexural waves in dielectric elastomer films by voltage

Shape and physical properties of dielectric elastomers are changeable by voltage. Theoretical works show that these changes can be harnessed to tune the propagation of superposed elastic waves. We experimentally demonstrate this concept by manipulating waves in a dielectric elastomer film, focusing on the flexural mode at low frequencies. To this end, we design an experimental apparatus to pre-stretch, actuate, excite waves at low frequencies in a VHB™ 4910 film, and measure the velocity of the fundamental flexural mode. Our results show that the excited wave velocity is slowed down by the applied voltage, and provide experimental proof of concept for the application of deformable dielectrics as tunable waveguides.     

A nonlinear field theory of deformable dielectrics

Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.     

Viscoelastic properties of MR elastomers under harmonic loading

This paper presents both experimental and modeling studies of viscoelastic properties of MR elastomers under harmonic loadings. Magnetorheological elastomer (MRE) samples were fabricated by mixing carbonyl iron power, silicone oil, and silicone rubber and cured under a magnetic field. Its steady-state and dynamic properties were measured by using a parallel-plate rheometer. Various sinusoidal loadings, with different strain amplitude and frequencies, were applied to study the stress responses. The stress–strain results demonstrated that MR elastomers behave as linear visocoelastic properties. Microstructures of MRE samples were observed with a scanning electron microscope. A four-parameter linear viscoelatic model was proposed to predict MRE performances. The four parameters under various working conditions (magnetic field, strain amplitude, and frequency) were identified with the MATLAB optimization algorithm. The comparisons between the experimental results and the model predictions demonstrate that the four-parameter viscoelastic model can predict MRE performances very well. In addition, dynamic properties of MRE performances were alternatively represented with equivalent stiffness and damping coefficients.     

A 3D multi-field element for simulating the electromechanical coupling behavior of dielectric elastomers

We propose a multi-field implicit finite element method for analyzing the electromechanical behavior of dielectric elastomers. This method is based on a four-field variational principle, which includes displacement and electric potential for the electromechanical coupling analysis, and additional independent fields to address the incompressible constraint of the hyperelastic material. Linearization of the variational form and finite element discretization are adopted for the numerical implementation. A general FEM program framework is developed using C ++ based on the open-source finite element library deal.II to implement this proposed algorithm. Numerical examples demonstrate the accuracy, convergence properties, mesh-independence properties, and scalability of this method. We also use the method for eigenvalue analysis of a dielectric elastomer actuator subject to electromechanical loadings. Our finite element implementation is available as an online supplementary material.     

Electromechanical Vibrations and Dissipative Heating of Viscoelastic Thin-Walled Piezoelements

The results of studying the electromechanical response of thin-walled viscoelastic piezoactive elements under harmonic loading are generalized. The nonlinear electrothermoviscoelastic problem for a harmonically deformed body is formulated in a simplified form with regard for the facts that the mechanical, thermal, and electric fields are coupled, the material is physically nonlinear, and its properties depend on temperature. Classical and refined electromechanical models of single-layer and multilayer shells and plates under general and harmonic loading are reviewed. The models consider that the electromechanical characteristics of the material depend on temperature and physical and geometrical nonlinearities. Methods for solving nonlinear coupled electrothermoviscoelastic problems are discussed. Analytical and numerical solutions are given to specific quasistatic and dynamic electrothermoviscoelastic problems for thin-walled elements such as rods, plates, and shells of various shapes under harmonic electric loading. The effect of dissipation, the temperature dependence of the material properties, and physical and geometrical nonlinearities on the harmonic and parametric vibrations and stability of piezoelectric elements is studied     

Electrohydrodynamic instability of two superposed Walters B´ viscoelastic fluids in relative motion through porous medium

Summary  The electrohydrodynamic Kelvin–Helmholtz instability of the interface between two uniform superposed viscoelastic (B′ model) dielectric fluids streaming through a porous medium is investigated. The considered system is influenced by applied electric fields acting normally to the interface between the two media, at which there are no surface charges present. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by either streaming and applied electric fields for the potentially unstable configuration, or streaming only for the potentially stable configuration, as long as perturbations in the direction of streaming are ignored. For perturbations in all other directions, there exists instability for a certain wavenumber range. The instability of this system can be enhanced (increased) by normal electric fields. In the presence of surface tension, it is found also that the normal electric fields have destabilizing effects, and that the surface tension is able to suppress the Kelvin–Helmholtz instability for small wavelength perturbations, and the medium porosity reduces the stability range given in terms of the velocities difference and the electric fields effect. Finally, it is shown that the presence of surface tension enhances the stabilizing effect played by the fluid velocities, and that the kinematic viscoelasticity has a stabilizing as well as a destabilizing effect on the considered system under certain conditions. Graphics have been plotted by giving numerical values to the parameters, to depict the stability characteristics. Received 27 March 2000; accepted for publication 3 May 2001     

Multi-frequency excitation of magnetorheological elastomer-based sandwich beam with conductive skins

The present work deals with the dynamic stability of a symmetric sandwich beam with magnetorheological elastomer (MRE) embedded viscoelastic core and conductive skins subjected to time varying axial force and magnetic field. The conductive skins induce magnetic loads and moments under the application of magnetic field during vibration. The MRE part works in shear mode and hence the dynamic properties of the sandwich beam can be controlled by magnetic fields due to the field dependent shear modulus of MRE material. Considering the core to be incompressible in transverse direction, classical sandwich beam theory has been used along with extended Hamilton's principle and Galarkin's method to derive the governing equation of motion. The resulting equation reduces to that of a multi-frequency parametrically excited system. Second order method of multiple scales has been used to study the stability of the system for simply supported and clamped free sandwich beams. Here the experimentally obtained properties of magnetorheological elastomers based on natural rubber have been considered in the numerical simulation. The results suggest that the stability of the MRE embedded sandwich beam can be improved by using magnetic field.     

温度对介电弹性体材料力电耦合变形的影响（固体力学大会文章）

介电弹性体（Dielectric elastomer,简称DE）材料是一类在电场激励下可以产生大幅度尺寸或形状变化的新型柔性功能材料。DE材料具有非常宽的温度应用范围,这种宽的温度工作范围和快速大变形性能为各种柔性致动器结构提供了良好的基础,但作为一种粘弹性高分子材料,温度对其性能的影响也是非常明显的。然而到目前为止,所有关于DE材料驱动性能的研究仅局限于室温条件下,温度变化对DE材料力电耦合稳定性的影响几乎没有相关报道。基于此,通过实验研究了温度对最常用的DE材料（VHB 4910,3M）力电耦合变形的影响,结果表明：升高温度可以提高DE材料的力电耦合变形;温度越高,DE材料越容易发生力电耦合失效。然后,从热力学和粘弹性力学出发,建立了考虑温度影响后的DE材料的粘弹性力电耦合模型,数值模拟理论结果和实验结果非常吻合。     

Band-gaps in electrostatically controlled dielectric laminates subjected to incremental shear motions

The thickness vibrations of a finitely deformed infinite periodic laminate made out of two layers of dielectric elastomers is studied. The laminate is pre-stretched by inducing a bias electric field perpendicular to the layers. Incremental time-harmonic fields superimposed on the initial finite deformation are considered next. Utilizing the Bloch-Floquet theorem along with the transfer matrix method we determine the dispersion relation which relates the incremental fields frequency and the phase velocity.Ranges of frequencies at which waves cannot propagate are identified whenever the Bloch-parameter is complex. These band-gaps depend on the phases properties, their volume fraction, and most importantly on the electric bias field. Our analysis reveals how these band-gaps can be shifted and their width can be modified by changing the bias electric field. This implies that by controlling the electrostatic bias field desired frequencies can be filtered out. Representative examples of laminates with different combinations of commercially available dielectric elastomers are examined.