A complete direct approach to nonlinear modeling of dielectric elastomer plates

Abstract

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,

Electrostrictive polymer plates as electro-elastic material surfaces: Modeling, analysis, and simulation

“…After reviewing electrostrictive polymers in the context of 3D electro-elasticity in the previous section, we turn our attention to the case we are actually interested in—electrostrictive polymer plates. First, we introduce the governing equations of thin electro-elastic plates modeled as a 2D generalized material surface, see Vetyukov et al (2018), Hansy-Staudigl et al (2018) and Hansy-Staudigl et al (2019). In particular, we consider the plate as a 2D continuum with five mechanical degrees of freedom, three translations and two rotations, resembling the notion of a single rigid director d attached to each particle of the plate, see Naghdi (1972).…”

“…which was used before for piezoelectric plates and shells by Vetyukov et al (2018) and for dielectric elastomer plates by Hansy-Staudigl et al (2018) and Hansy-Staudigl et al (2019). p , m , and σ are purely mechanical external forces and moments and electrical charges per unit area of the reference surface with area A 0 .…”

“…In our previous work on dielectric elastomer plates, see Hansy-Staudigl et al (2019), we have introduced an alternative second strain measure K as…”

“…Concerning the direct approach for elastic plates and shells, we refer to Altenbach and Zhilin (1998), Opoka and Pietraszkiewicz (2009), Eliseev and Vetyukov (2010), and Vetyukov (2014b), and for an extension to nonlinear micropolar shells with surface stresses see Altenbach and Eremeyev (2011). Except for our own works Krommer et al (2016), Vetyukov et al (2018), Hansy-Staudigl et al (2018), and Hansy-Staudigl et al (2019) on piezelastic plates and shells and dielectric elastomer plates no substantial effort toward extending the direct approach for electro-elastic material surfaces is known to the authors. In Hansy-Staudigl et al (2018) the direct approach has been applied to dielectric elastomer plates without constitutive coupling; the formulation was based on a plane stress assumption and numerical integration of the resulting plane stress augmented free energy through the thickness rendered the 2D structural augmented free energy needed for the direct approach.…”

“…In Hansy-Staudigl et al (2018) the direct approach has been applied to dielectric elastomer plates without constitutive coupling; the formulation was based on a plane stress assumption and numerical integration of the resulting plane stress augmented free energy through the thickness rendered the 2D structural augmented free energy needed for the direct approach. More recently, we were able to directly introduce this structural augmented free energy for dielectric elastomer plates in Hansy-Staudigl et al (2019) without the need for a numerical integration through the thickness. Also the concepts of the total stress, the Maxwell stress, the polarization stress and the electrostatic stress were established directly for the electro-elastic material surface.…”