A numerical framework for modeling anisotropic dielectric elastomers

Highlights

• Finite element modeling framework for the anisotropic electroactive elastomers.

• A computationally efficient staggered solution algorithm for nonlinear coupled field equations.

• Analytical expressions for the tangent moduli of anisotropic dielectric elastomers.

• Effect of material anisotropy on the thresholds on the stable operation of DEAs.

• Effect of anisotropy on the performance of bending and buckling actuators.

Abstract

This paper reports a finite element based numerical framework for simulating the electromechanical behavior of nonlinear anisotropic dielectric elastomer actuators at finite strains. Based on the existing models for incompressible anisotropic neo-Hookean hyperelastic solids and ideal dielectric elastomers, a theory of anisotropic dielectric elastomers is outlined. The analytical expressions are derived for the tangent moduli of the anisotropic materials. A computationally efficient staggered solution algorithm is presented for solving the coupled nonlinear equations by decoupling displacement and electric potential fields. Selective reduced integration technique is used for alleviating the volumetric locking due to material incompressibility. The model is implemented into an in-house finite element program. We first validate the accuracy of the finite element implementation by considering the cases of homogeneous deformation, with a particular emphasis on the electromechanical instability. Subsequently we demonstrate the utility of the proposed numerical framework by analyzing two representative cases (bending and buckling actuators) involving inhomogeneous deformations. In both the cases, anisotropy in the mechanical properties of the elastomer is found to have a favorable influence on the actuation performance. Finally, for various mesh sizes, a comparison of the computation time required by the monolithic solution approach and the proposed staggered approach is presented to highlight the computational efficacy of the latter.

Introduction

Dielectric elastomers (DEs) are compliant elastic materials and belong to a class of electro-active polymers (EAPs). DEs undergo large deformation when subjected to an electric field [1]. The ability of dielectric elastomers to achieve large deformation with their lightweight, low cost and fast response time [2] have made them attractive for a wide range of applications, such as energy harvesting devices [3], resonators [4], artificial muscles [5], active lenses [6], among many others. However, requirement of high voltage for achieving large strain together with the associated electromechanical instability has limited the widespread use of DE actuators. Several approaches have been proposed so far to circumvent these problems [[7], [8], [9], [10]]. One of such approaches is to impart anisotropy in the mechanical behavior of DE actuators [[11], [12]], which can be realized by various means and at various scales. A significant volume of previous investigations expounds on the electromechanical analysis of electroactive polymer composites, both theoretically [13] as well as experimentally [14]. In this approach, the deformation of the DE actuator is constrained using stiff fibers in the direction that is perpendicular to the desired direction of actuation [[8], [15], [16]]. Using this approach, Lu et al. [14] achieved 28% unidirectional actuation strain in the flat dielectric membranes. The computational modeling these fiber-stiffened DE actuators is performed by employing the simplistic lumped parameter models based on the assumption of homogeneous deformation [17], or using homogenization approaches [[18], [19]] facilitating a numerical solution through finite element method [20]. In another instance, a few recent experiments demonstrate the enhancement in the actuation response of the dielectric elastomers using the mechanical anisotropy of the electrode material [[21], [22], [23]]. With this approach, Cakmak et al. [21] achieved the linear actuation strain of more than 40% at a relatively low electric field (100 V/$\mathrm{\mu }\mathrm{m}$) using highly aligned carbon nanotube (CNT) sheet electrodes on acrylate adhesive films. In addition to fiber-reinforcement and imparting anisotropy in the material used for preparing the electrodes, there has been a continued interest in yet another way of enhancing the actuation by including high dielectric constant fillers into the polymer matrix [[24], [25], [26], [27], [28]]. Because of these inclusions, soft elastomers show anisotropy in both mechanical as well as dielectric behaviors. The direction of anisotropy of such composite material depends on the direction of the electric field applied during the curing process. The computational modeling of this dispersion anisotropy in dielectric elastomer actuators has received an increasing attention in the recent past [[26], [29], [30]].

In connection to the modeling and simulation of dielectric elastomers, the basic theory of non-linear electroelasticity was developed by Toupin [31], for investigating the quasistatic electromechanical behavior of elastic dielectrics. Recently, several researchers reformulated the electroelasticity theory of finite deformations and made significant efforts towards the numerical implementation for modeling the soft dielectrics [[32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]]. The electromechanical behavior and pull-in instability phenomena in finitely deformed isotropic dielectric elastomers, considering the effects of several factors, such as the level of prestress [[44], [45]], time dependent actuation [[46], [47], [48]], and viscoelasticity [[49], [50]], have been reported analytically as well as numerically in several investigations reported previously in the literature. However, limited efforts have been made to model numerically the electromechanical behavior of the DE actuators with dispersion type anisotropy. In this regard, Yong et al. [51] reported a lumped parameter model to investigate the electromechanical instability in transversely isotropic elastomers. Bustamante [52] presented a constitutive model that considers transverse isotropy stemming from the particle reinforced anisotropy. A homogenization approach for electroactive polymers with random particulate microstructure, in the limit of infinitesimal deformations, is presented by Siboni and Castaneda [18]. Recently, Hossain et al. [26] presented a constitutive framework for the modeling of dispersion type anisotropy in electroactive polymers. In view of the growing interest in exploiting anisotropy in the technology of DE actuators, it is imperative to develop a numerical modeling framework to support the design and development of this class of actuators. However, to the best of the authors’ knowledge, a finite element-based numerical framework for modeling the anisotropic dielectric elastomers has not hitherto been discussed in the literature.

To this end, this paper presents a nonlinear, finite deformation, finite element framework for anisotropic dielectric elastomers based on the existing models for incompressible anisotropic neo-Hookean hyperelastic solids [53] and ideal dielectric elastomers [54]. The analytical expressions are derived for the tangent moduli of the anisotropic materials. This material model has been implemented into an in-house finite element code. We propose a computationally efficient staggered algorithm, in which, we decouple the displacement and the electric potential fields and solve for them separately, to avoid the ill-conditioning that arises in the monolithic scheme of solution. A parametric study is performed for investigating the effect of material anisotropy on the pull-in instability of dielectric elastomers with different levels of pre-stretch. Further, we simulate the effect of material anisotropy on the actuation performance of the bending and buckling actuators.

This paper is organized in six sections. In Section 2, we overview the basics of nonlinear continuum electromechanics and the finite element implementation of the coupled electro-elastic problem. A material model for the anisotropic dielectric elastomers and the analytic expressions for tangent moduli are presented in Section 3. A computationally efficient staggered algorithm for the solution of coupled nonlinear equations is proposed in Section 4. In Section 5, we demonstrate the utility of the numerical framework by simulating the effect of material anisotropy on (1) the pull-in instability of the dielectric elastomers, and (2) actuation performance of bimorph bending and buckling actuators. Subsequently, a comparison of the computational efficiency of the proposed staggered algorithm with that of the monolithic approach is presented. The conclusions drawn from the present study are summarized in Section 6.