In this paper, we present a complete direct approach to nonlinear modeling of thin plates, which are made of incompressible dielectric elastomers. In particular, the dielectric elastomers are assumed to exhibit a neo-Hookean elastic behavior, and the effect of electrostatic forces is incorporated by the purely electrical contribution to the augmented Helmholtz free energy. Our approach does not involve any extraction-type procedure from the three-dimensional energy to derive the plate augmented free energy, but directly postulates the form of this energy for the structural plate problem treated in this paper. Results computed within the framework of this novel approach are compared to results available in the literature as well as to our own three-dimensional finite element solutions. A very good agreement is found. IntroductionThe present paper is dedicated to the memory of Vladimir Vasilyevich Eliseev and his pioneering work on modern versions of the linear and nonlinear theories for thin elastic rods, plates and shells, for which he developed geometrically nonlinear equations in a compact tensorial form based on the principle of virtual work applied to material lines and surfaces. His most essential contributions to the topic of this paper can be found in [1][2][3][4][5][6]. The present paper is based upon Eliseev's work on modeling of thin plates and shells as material surfaces. In particular, we extend the direct approach for elastic shells he presented in [6] and that was further developed by Vetyukov [7] to the case of electro-active plates modeled as electro-elastic material surfaces.The general theory of elastic dielectrics, of which dielectric elastomers are a sub-class, dates back to Toupin [8], and it has been further developed in, e.g., [9][10][11][12]. Elastic dielectrics belong to the class of so-called smart or intelligent materials, with piezoelectric materials and electro-active polymers as prominent examples. Concerning the latter, we refer to, e.g., [13] or [14]. A practically important sub-class of electro-active polymers are dielectric elastomers, which are rubber-type materials that exhibit a polarization when an external electric field is applied at electrodes mounted to its top and bottom surfaces. By this polarization, the electrodes get attracted due to the corresponding electrostatic forces, such that the resulting squeezing yields large in-plane deformations. This property is used for actuation, see, e.g., [15][16][17][18][19] for a survey on soft robotics, as well as for sensing applications. This makes dielectric elastomers a promising technology, posing a soft alternative E. Hansy-Staudigl · M. Krommer (B) TU Wien,
In this article we discuss modeling of electrostrictive polymer plates as electro-elastic material surfaces. A complete direct approach is developed without the need to involve the three-dimensional formulation. Ponderomotive forces and couples as well as constitutive coupling by means of electrostriction are accounted for. We propose a rational formulation for the augmented free energy of electro-elastic material surfaces incorporating electrostriction by a multiplicative decomposition of the surface stretch tensor and an additive decomposition of the surface curvature tensor into elastic and electrical parts. Numerical results computed within the framework of this complete direct approach are compared to results computed with a method that requires the numerical integration of the three-dimensional augmented free energy through the thickness of the plate and to alternative formulations reported in the literature.
Bringing into focus the design aspect of thin film electro-active polymer actuators justifies the deployment of a structural mechanics framework. We propose a physically consistent constitutive model for such actuators, which is valid for plates and shells as material surfaces within a complete direct formulation. To this end, we use the principle of virtual work to deduce the general form of the constitutive law from an augmented Helmholtz free energy, as a function of the structural Green-Lagrange type strain measures and of the material electric field, without the need of a-priori assumptions concerning the state of strain and stress.Mechanical deformations of thin film devices -e.g. made of polyurethane -under the action of an external electric field, are caused by two different sources. On the one hand, the applied electric field causes a dielectric polarization of the polymer matrix, yielding to corresponding attractive electrostatic forces between the electroded surfaces resulting into a squeezing of the film. On the other hand, crystalline graft units with a certain natural, but arbitrarily directed polarization, embedded in between the polymer chains, have to align in the direction of the applied electric field such that a rotation of the whole crystal unit takes place. This rotation results in an additional macroscopic thickness squeeze -known as the electrostrictive effect.We treat both electromechanical coupling phenomena separately, where it turns out, that the electrostatic forces can be accounted for by an electrical contribution to the augmented free energy, whereas, the electrostrictive effect is taken into account in the elastic part of the augmented free energy by virtue of a hybrid multiplicative and additive decomposition of the plate/shell deformation measures.Benefiting from the structural mechanics formulation, we gain a lower -two-dimensional -formulation, which provides a clear physical insight into the nature of the deformation process initiated by the external electric field. E.g. for the linearised problem, a comparison to the literature on thermoelastic plates and shells uncovers the action of the electric field as a combined source of self-stresses. In order to solve particular problems, the constitutive relation of the geometrically and physically nonlinear formulation is implemented into our in-house finite element code. The computed results, which were tested against results from the literature as well as against test problems of our previous work (where numerical integration of the three dimensional plate/shell augmented free energy through the thickness was employed), show a very good agreement.
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