A mixed finite element formulation for piezoelectric shells

A finite element formulation to analyze piezoelectric shell problems is presented. A reference surface of the shell is modelled with a four node element. Each node possesses six mechanical degrees of freedom, three displacements and three rotations, and one electric degree of freedom, which is the difference of the electric potential in thickness direction. The formulation is based on the mixed field variational principle of Hu-Washizu. The independent fields are displacements u , electric potential φ, strains E , electric field E , stresses S and dielectric displacements D . The mixed formulation allows an interpolation of the strains and the electric field in thickness direction. Accordingly a three-dimensional material law is incorporated in the variational formulation. It is remarked that no simplification regarding the constitutive law is assumed. The formulation allows the consideration of arbitrary constitutive relations. The normal zero stress condition and the normal zero dielectric displacement condition are enforced by the independent stress and dielectric displacement fields. They are defined as zero in thickness direction. The present shell element fulfills the important patch tests: the in-plane, bending and shear test. Some numerical examples demonstrate the applicability of the present piezoelectric shell element. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maxwell's equations in an unbounded structure

This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.     

Towards modelling the curing process in particle-filled electro-active polymers

A large actuation voltage is required to produce a desired mechanical deformation in dielectric elastomers. To reduce the amount of the actuation voltage, several mechanisms can be applied and the inclusion of high dielectric permittivity fillers in the matrix material in the uncured stage is one of them. Moreover, to obtain a maximum advantage from the high dielectric permittivity fillers, an electric field is applied during the curing process which helps the particles to align in a preferred direction. The stiffness gaining during curing is modelled by an appropriate constitutive relation where the temporal evolution of the material parameters is considered. In this paper we present a phenomenologically-inspired large strain framework for simulating the curing process of polymers that can work under the use of an electro-mechanically coupled load. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diffraction by an acoustically transmissive or an electromagnetically dielectric half-plane

This work gives a mathematical model for an acoustically penetrable or electromagnetically dielectric half-plane. An approximate boundary condition is used that depends on the thickness of, and the material constants for, the half-plane. A solution is obtained, by using the approximate boundary condition, for the problem of a line source field diffracted by a penetrable/dielectric half-plane. The asymmetry of the approximate boundary condition results in a matrix Wiener–Hopf problem, which is solved explicitly.     

Investigation of curing an epoxy resin by the polarization current method

Conclusions 1. An investigation of the curing kinetics of an epoxy resin by the polarization current method at constant temperature reveals, on the infralow frequency range f=0.001–1 Hz, the existence of a relaxation process which during polymerization changes shape and is shifted toward longer times.2. In all stages of polymerization the frequency dependences of the complex dielectric constant are satisfactorily described by the Havriliak-Negami dispersion equation.3. Whereas the dielectric characteristics on the audio- and radio-frequency range can be used to explore the initial stage of polymerization, the dielectric characteristics on the infralow frequency range can be used as a source of information on the state of the material in the final stages of cure (the end of the gelling period and the post-gelling phase). The characteristics that provide the most information are the spectrum parameters, which indicate three stages of polymerization.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 140–146, January–February, 1978.     

Harmonic generation in an electromagnetic wave

A plane electromagnetic wave is propagated in a non-linear non-dissipative isotropic dielectric. A method is given for the calculation of the change in the amplitudes and phases of harmonics of all orders, as the wave progresses, for any specified dependence of the dielectric constant on the magnitude of the electric field and for any initial wave form. The calculations are valid provided the distance of travel is sufficiently small that shocks are not formed. The formulae for the amplitudes and phases of the various harmonics take a particularly simple form when the initial wave-form is sinusoidal and the non-linearity of the dielectric is not too great.The results are compared with those obtained by the usual methods of calculation which assume that only harmonics of very low order are generated. The discrepancy in the amplitudes of these low order harmonics, calculated by the two methods is, for most practical purposes, insignificant.

---

Résumé Nous discutons la propagation d'une onde électromagnétique plane dans un milieu diélectrique, non-linéaire, non-dissipatif et isotrope. Nous donnons une méthode pour le calcul du changement des amplitudes et des phases des harmoniques d'ordres variés. Les calculs sont valides sous la condition que la distance parcourue par l'onde n'est pas assez grande pour le développement d'un choc. Nous faisons la comparaison entre nos résultats et ceux qui sont obtenus par les méthodes usuelles dans lesquelles on fait l'hypothèse que seulement les harmoniques de petits ordres sont produites. Les différences entre les amplitudes des harmoniques de petits ordres, calculées par les deux méthodes, ne sont pas importantes du point de vue pratique. D'autre part, notre méthode est valable pour le calcul des amplitudes et des phases des harmoniques de tous les ordres pour une onde d'une forme quelconque.

     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

基于内变量理论的电流变液本构关系

研究了电流变液的微结构本构关系.其理论框架是基于内变量理论和机理的分析.电流变液是由高介电常数的颗粒悬浮在某种液体中组成的.在电场作用下,极化的颗粒将沿着电场方向聚集在一起形成链状结构.颗粒聚集体的大小和方向将随外加电场和应变率的变化进行调整,因而可以通过建立起能量守恒方程和力平衡方程来确定颗粒聚集体的大小和方向的变化.那么,一个三维的、清晰的本构关系可以由相互作用能和系统的耗散能导出.具体考虑和讨论了在简单剪切载荷作用下的系统响应,发现电流变液的切变剪薄粘滞系数同系统Mason数之间近似于幂指数∝(Mn)-082的关系.     

The deformation of a liquid drop by an electric field

Summary The shape and stability of an incompressible dielectric drop which is stressed by a uniform external electric field are re-examined by considering small perturbations from a prolate spheroid. Compared with the shapes predicted by other approximations it is found that, for a given field strength, the drops should be a little longer and consequently a little flatter at the equator in order to satisfy the equilibrium conditions. It is also shown that beyond a certain drop deformation the L. P. E. (Legendre polynomial expansion) method fails because the equilibrium conditions at the surface of the drop are not satisfied.     

High-frequency asymptotic method for investigating the problem of excitation of a thin dielectric coating of an ideally conducting cylinder (H-polarization)

An incomplete Galerkin projection method is constructed and proved as a generalization of the Sommerfeld method for the problem of plane diffraction of the field of a point source on an ideally conducting circular cylinder coated by a thin homogeneous dielectric layer with an arbitrary boundary, in the H-polarization case. An explicit asymptotic solution is obtained in the form of a generalized Watson series. Specific eesonance phenomena are investigated for a boundary layer with corner points. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 132–149.     

Harvesting energy with an out-of-plane dielectric elastomer generator

This note concerns the numerical analysis of the performance of an annular membrane dielectric elastomer generator deforming non-homogeneously out-of-plane. The prestretch of the membrane is crucial for an improved behaviour of this device. We evaluate its positive effects on the performance by employing a hyperelastic formulation. Moreover, fixing the external radius, we compare the behaviour of two generators characterized by different values of the radius ratio R_e/R_i, namely 2 and 4. Numerical results show that the generator with the smaller ratio has a better performance. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence from an electromagnetic fluid system to the full compressible MHD equations

We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.     

One-way propagation in electromagnetic materials

Approaches to the characterization of the direction of wave propagation are examined, in the frequency and in the time domains, for electromagnetic materials with memory. In the frequency domain, the direction is characterized through the sign of the energy flux which is shown to decay as the wave propagates. In the time domain, a review is given of the wave-splitting approach for a dispersive dielectric. The result is a pair of decoupled equations, one being regarded to represent a forward propagating wave and the other a backward propagating wave, both of them involving an appropriate convolution kernel. To establish a connection between the two approaches, the Fourier components associated with the wave-splitting equations are examined. Owing to a thermodynamic restriction on the permittivity, the energy flux criterion, as well as the phase and amplitude criteria, implies that the direction is forward or backward depending on the value of the cosine transform of the convolution kernel and hence on the frequency.     

Insulated antenna in a layered medium

The article presents a generalization of the integral equation of an insulated linear antenna immersed in a cylindrically layered lossy dielectric medium. The insulation is provided by a lossless dielectric layer. The kernel of the integral equation is represented as a superposition of the fundamental solutions of the wave equation with equivalent propagation constants for the given media. A generalization to a plane-layered medium is proposed. The problem of a vertical radiator above a layered half-space is considered. Translated from Chislennye Metody v Matematicheskoi Fiziki, Published by Moscow University, Moscow, 1996, pp. 80–88.     

Direct numerical method for an inverse problem of hyperbolic equations

A coefficient inverse problem of the one-dimensional hyperbolic equation with overspecified boundary conditions is solved by the finite difference method. The computation is carried out in the x direction instead of the usual t direction. The original boundary condition and the overspecified boundary data are used as the new initial conditions, and the original data at t = 0 are used to compute the coefficient directly. The computation time used by this scheme is almost equal to that for solving the hyperbolic equation in the same region once, even though the inverse problem is essentially nonlinear and hence more difficult to solve. An error estimate is obtained that guarantees the stability of the scheme marching in the x direction. Several numerical experiments are carried out to show the convergence and other properties of the scheme. © 1992 John Wiley & Sons, Inc.     

Numerical investigation of trapped modes in an irregular waveguide

It is shown by means of a numerical experiment that an arbitrary asymmetric perturbation of the function describing a dielectric insert in a waveguide, in the general case, results in vanishing of the trapped mode; i.e., it ceases to exist.     

A stability estimate for a solution to an inverse problem of electrodynamics

We consider the integrodifferential system of equations of electrodynamics which corresponds to a dispersive nonmagnetic medium. For this system we study the problem of determining the spatial part of the kernel of the integral term. This corresponds to finding the part of dielectric permittivity depending nonlinearly on the frequency of the electromagnetic wave. We assume that the support of dielectric permittivity lies in some compact domain Ω ⊂ ℝ³. In order to find it inside Ω we start with known data about the solution to the corresponding direct problem for the equations of electrodynamics on the whole boundary of Ω for some finite time interval. On assuming that the time interval is sufficiently large we estimate the conditional stability of the solution to this inverse problem.