Electroelastic plate instabilities based on the Stroh method in terms of the energy function Ω*(F, DL)
Introduction
In this paper we are concerned with the stability analysis of an electroelastic plate with compliant electrodes coated on its major surfaces, across which an applied potential difference induces deformation of the plate. This problem has been analyzed extensively based on the so-called Hessian approach, as, for example, by Zhao and Suo in [19] and many other papers, including, for example, [3], [15], [17], [18]. In [7] the limitations of the Hessian approach for the analysis of the stability of dielectric elastomer plates were pointed out, and the more general incremental analysis was adopted therein. This followed the incremental theory described in [5] and its application to the quasi-electrostatic theory of acoustic wave propagation in [4]. The theory is based on analyzing incremental deformations and electric fields superimposed on a configuration in which a finite deformation is accompanied by an electric field generated by a potential difference between compliant electrodes on the major surfaces of the plate.
The incremental analysis can be conducted in terms of a constitutive law that involves either the electric field, the electric displacement field [8] or the polarization [16] in addition to a deformation variable, although the formulation in the latter case is less compact than in the other two formulations. In [7] a constitutive law based on the energy function Ω*(F, DL), per unit reference volume, was adopted, where F is the deformation gradient and DL the Lagrangian version of the electric displacement vector field. The latter is the pullback of the true electric displacement field D from the deformed to the undeformed reference configuration and they are related by for an incompressible material, for which and to which attention is confined henceforth in this paper.
The incremental problem can alternatively be formulated in terms of the energy function Ω(F, EL), where EL is the Lagrangian electric field related to the true electric field E in the deformed configuration by the pullback . For full details of the incremental formulations in terms of both the energy function Ω*(F, DL) and Ω(F, EL) we refer to [8]. Indeed, an alternative approach to the incremental analysis of the plate instabilities has been provided in [13] using the Stroh method [11], [12] in terms of the energy function Ω(F, EL), which led to equations that decoupled the symmetric and antisymmetric modes of incremental instabilities. In terms of Ω*(F, DL), this decoupling, which was not noted in [7], was also derived in the review [6], but not by using the Stroh approach.
The incremental method has been used in several other recent papers. For example, Díaz-Calleja et al. [2], following the analysis in [7], used an incremental approach based on Ω*(F, DL), with different forms of the elastic part of the energy function, to investigate the stability of an equibiaxially stretched incompressible isotropic electroelastic plate and the differences in the results obtained by using the Hessian approach. By contrast, using an energy function with the polarization as the independent electric variable, Yang et al. [16] presented an incremental analysis for investigating antisymmetric and symmetric modes of instability of a finite block of a neo-Hookean dielectric under different incremental boundary conditions and compared the results with those for a thin film and a half-space. A detailed analysis of instability modes of a neo-Hookean dielectric plate subject to (inhomogeneous) finite bending deformations have been investigated by Su et al. [14], who formulated the incremental problem using the Stroh approach in terms of the energy function Ω*(F, DL).
In the present paper we revisit the plate stability problem using the Stroh approach in respect of the energy function Ω*(F, DL) and recover in compact form results that have been obtained by other methods or with different forms of the energy function. The paper is organized as follows. In Section 2 we summarize the governing equations and boundary conditions of nonlinear electroelasticity and their incremental counterparts. These are then applied in Section 3 in their two-dimensional specialization in respect of a rectangular plate. The governing incremental equations and boundary conditions are arranged in the Stroh form in Section 4, and in Section 5 the bifurcation equations describing possible symmetric and antisymmetric wrinkling instability modes are derived under incremental voltage control. Finally, in Section 6, the results of the bifurcation analysis are illustrated in respect of a Gent electroelastic material model with two widely different values of the Gent parameter, and Section 7 contains some concluding remarks.
Section snippets
Basic equations
We work in terms of the energy function Ω*(F, DL), from which the constitutive relationsare derived, where T is the total nominal stress tensor and p* is a Lagrange multiplier associated with the incompressibility constraint. The variables T, EL and DL satisfy (in the absence of mechanical body forces) the governing equations where Div and Curl are the divergence and curl operators with respect to the undeformed reference configuration. For the
Application to two-dimensional increments in a plate
We now apply the general equations of the previous section to a rectangular incompressible electroelastic plate. It is subject to homogeneous stretches λ1 and λ3 in the plane of the plate and λ2 through the plate thickness H in the undeformed configuration, which is small compared to its lateral dimensions. The deformed geometry is defined in terms of Cartesian coordinates (x1, x2, x3) according towhere L1 and L3 are the lateral dimensions of the plate in
Stroh formulation
To set up the equations in the Stroh formulation we use the variables with the x1 dependence of the form where the constant k is the wave number of the periodic undulations. To avoid unnecessary factors of k in the equations, we writewhere U1, U2, Φ, Σ21, Σ22, E are functions of kx2. We represent the first three components as the generalized ‘displacement’ and the second three as
Derivation of the bifurcation equations
Now expand the general solution for η in terms of the eigenvectors in the formwith constants .
For ease of representation, we introduce the notations noting that for . Then, the boundary conditions (49) yield
Application to a Gent electroelastic model
To illustrate the solutions of the bifurcation equations (57) and (58) we specialize the constitutive model Ω*(I1, I5) to the formwhere ε is the constant permittivity of the material, so that and W(I1) is given by the Gent form [10]where G, a constant, is the Gent parameter and the constant μ is the shear modulus in the reference configuration. In the following we require the derivativesIn particular, we
Concluding remarks
In this paper we have shown that the formulation of nonlinear electroelasticity in terms of the energy function Ω*(F, DL) and the Stroh method produces the same results as for the energy function Ω(F, EL) described in [13] in respect of the stability analysis of a rectangular plate subject to equibiaxial in-plane deformations and to an electric field generated by compliant electrodes on its major surfaces. This follows the review article [6] in which the same bifurcation equations governing
References (19)
- et al.
Complex bifurcation maps in electroelastic elastomeric plates
Int. J. Solids Struct.
(2017) - et al.
Nonlinear electroelasticity: incremental equations and stability
Int. J. Eng. Sci.
(2010) - et al.
Instabilities of an electroelastic plate
Int. J. Eng. Sci.
(2014) - et al.
Wrinkles in soft dielectric plates
J. Mech. Phys. Solids
(2018) - et al.
Finite bending and pattern evolution of the associated instability for a dielectric elastomer slab
Int. J. Solids Struct.
(2019) Wolfram research
Inc., Mathematica, Version 11, Champaign
(2017)- et al.
On electromechanical stability of dielectric elastomers
Appl. Phys. Lett.
(2008) - et al.
Electroelastic waves in a finitely deformed electroactive material
IMA J. Appl. Math.
(2010) - et al.
Instabilities of soft dielectrics
Phil. Trans. R. Soc. Lond. A, in press
(2019)
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Electromechanical stability of wrinkled dielectric elastomers
2022, International Journal of Solids and StructuresCitation Excerpt :Since electromechanical instability often anticipates dielectric breakdown, it makes sense to propose a method to study the electromechanical stability of a wrinkled membrane. Various approaches to electromechanical instability exist in literature; two main strands are represented by energy methods (Zhao and Suo, 2007; Norris, 2007; Zurlo et al., 2017) and incremental methods (Bertoldi and Gei, 2011; Dorfmann and Ogden, 2014c, 2019a, 2014b; Conroy Broderick et al., 2020; DeTommasi et al., 2013b); connections between the predictions of these methods can be found in Fu et al. (2018a), Dorfmann and Ogden (2014b, 2019b). Wrinkling is a classical topic of structural mechanics and it admits various approaches, depending on the desired level of approximation for the description of wrinkles.
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