Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

WAVE PROPAGATION IN ELECTROSTRICTIVE MATERIALS UNDER BIASED FIELDS

I. INTRODUCTION Di?erent from piezoelectricity which is a linear coupling between mechanical and electric ?elds andcan only exist in anisotropic materials[1], electrostriction refers to the quadratic dependence of strainor stress on electric ?elds[2,3] …     

Effect of a biasing electric field on the propagation of antisymmetric Lamb waves in piezoelectric plates

Based on the theories of anisotropic elasticity, piezoelectricity and elastic waves in solids, the propagation of antisymmetric Lamb waves in a biasing electric field is investigated in this paper. By solving the coupled differential equations of motion under a biasing electric field, the phase velocity equations of antisymmetric Lamb wave modes for electrically open and shorted cases are obtained, respectively. The beating effect arising from the difference between the phase velocity of the zero-order symmetric mode and antisymmetric mode exists in the plate when the plate has a thickness comparable to or slightly larger than the wavelength. The influence of the biasing electric field on the phase velocity, beat wavelength, mechanical displacement and stress fields for the lowest two antisymmetric modes of Lamb waves are discussed in detail. From the calculated results, it is seen that the phase velocity of the fundamental antisymmetric mode is especially sensitive to the applied biasing electric field.     

非线性压电效应下压电层合板的弯曲

考虑非线性压电效应,即电致弹性和电致伸缩效应情况下压电层合板的弯曲。从非线性压电方程和几何方程导出了压电层合板合应力、合力矩与应变之间的广义本构关系,这些关系关于电场是非线性的。利用Ritz法和双傅立叶级数得到四边简支对称压电层合板在高电场作用下的非线性解并进行计算。结果表明,只考虑线性压电效应只能适应于作用电场较低或基础层的刚度比压电层的刚度要大得多的情况。     

Electric field gradient effects in anti-plane problems of polarized ceramics

The equations of elastic dielectrics with electric field gradient effects are specialized to the case of anti-plane motions of polarized ceramics. A general solution is obtained in polar coordinates. Analytical solutions to the static problems of the potential field of a line source, the capacitance of a circular cylindrical ceramic shell, and the dynamic problem of the dispersion relation of plane waves are obtained to examine the electric field gradient effect. Special attention is paid to the case when the shell is thin and the waves are short.     

考虑几何与压电非线性行为压电层合轴对称圆板的精确解

研究可移简支及夹支边界条件下,轴对称压电层合圆板在强电场和机械荷载联合作用下的非线性变形.考虑电致伸缩的非线性压电效应及几何非线性的影响,导出轴对称压电层合圆板的控制方程.通过调整坐标轴的位置对控制方程进行简化,得到关于挠度和径向力的4阶非线性控制方程.再通过简单的积分并引入无量刚变量将控制方程等价地化为2阶非线性耦合微分方程组.利用幂级数法得到可移简支及夹支边界条件下强电场和均布荷载共同作用时的挠度、径向力及径向位移的幂级数精确解.通过对双、单压电晶片执行器的数值计算及分析,得到电场、外载对于位移、径向力的影响关系.     

On solitary waves in plasma

From the equations of hydrodynamics and electrodynamics, a system of coupled nonlinear equations governing the propagation of plane electromagnetic waves in a collisionless electron plasma is obtained. It is shown that solitary wave solutions exists for both the longitudinal and transverse components of the electromagnetic field. It is found that the velocity of the electromagnetic vector solitary wave depends on the amplitudes of all components of the field linearly. The relations among the longitudinal and transverse components that support the solitary waves are determined for different values of plasma temperature. It is shown that while transverse solitary waves cannot exist, except when they are supported by longitudinal waves, the latter can exist by themselves. The interaction of the longitudinal solitary waves with each other is studied and an upper bound on the amplitudes of these waves is obtained. A Lagrangian density function and two conservation laws for the longitudinal wave equation are found. Frequency spectra of the solitary waves are calculated and their low frequency content is emphasized.     

Simple waves and small perturbations in magnetizable or polarizable media

The system of equations describing the behavior of the electrically polarizable media in the case of a sufficiently weak magnetic field and the system of equations for magnetizable media in a weak electric field coincide except for the notations. These equations are used for the investigation of Riemann waves and small perturbations for different given dependences of ε and Μ on ρ and T. The case when the velocity of propagation of simple waves takes complex values is considered. Similar investigation has been carried out in [1, 2] in the case where the dependence of Μ on ρ and T is in the form Μ?1/Μ=cρT (Mossoti formula) and μ=1+4πρk(θ-T)/H. The same problems are investigated here for an infinitely conducting magnetizable gas, whose behavior is described by another system of equations reminiscent of the equations of magnetohydrodynamics.     

Weakly nonlinear long waves in a prestretched Blatz-Ko cylinder: Solitary, kink and periodic waves

This paper studies nonlinear waves in a prestretched cylinder composed of a Blatz-Ko material. Starting from the three-dimensional field equations, two coupled PDEs for modeling weakly nonlinear long waves are derived by using the method of coupled series and asymptotic expansions. Comparing with some other existing models in literature, an important feature of these equations is that they are consistent with traction-free surface conditions asymptotically. Also, the material nonlinearity is kept to the third order. As these two PDEs are quite complicated, the attention is focused on traveling waves, for which a first-order system of ODEs are obtained. We use the technique of dynamical systems to carry out the analysis. It turns out that the system is three parameters (the prestretch, the propagating speed and an integration constant) dependent and there are totally seven types of phase planes which contain trajectories representing bounded traveling waves. The parametric conditions for each phase plane are established. A variety of solitary and periodic waves are found. An important finding is that kink waves can propagate in a Blatz-Ko cylinder. We also find that one type of periodic waves has an interesting feature in the profile slope. Analytical expressions for all bounded traveling waves are obtained.     

偏压电场对压电板中Lamb波相速度的影响

本文研究了偏压电场作用下,Lamb波在压电板中的传播行为，首先给出了偏压电场作用时压电板中的应力场及电位移场，然后通过求解含初应力及初电位移的小幅波动问题的耦合方程，分别给出了Lamb波的对称模态和反对称模态的相速度方程，以典型的PZT－5H压电陶瓷板为例进行了数值计算,并讨论了偏压电场对Lamb波相速度及频散曲线的影响，结果表明，偏压电场可以显著地改变Lamb波的传播速度，借此可使声波器件获得延时效果。     

A hybrid approach for nonlinear computational aeroacoustics predictions

In many aeroacoustics applications involving nonlinear waves and obstructions in the far-field, approaches based on the classical acoustic analogy theory or the linearised Euler equations are unable to fully characterise the acoustic field. Therefore, computational aeroacoustics hybrid methods that incorporate nonlinear wave propagation have to be constructed. In this study, a hybrid approach coupling Navier–Stokes equations in the acoustic source region with nonlinear Euler equations in the acoustic propagation region is introduced and tested. The full Navier–Stokes equations are solved in the source region to identify the acoustic sources. The flow variables of interest are then transferred from the source region to the acoustic propagation region, where the full nonlinear Euler equations with source terms are solved. The transition between the two regions is made through a buffer zone where the flow variables are penalised via a source term added to the Euler equations. Tests were conducted on simple acoustic and vorticity disturbances, two-dimensional jets (Mach 0.9 and 2), and a three-dimensional jet (Mach 1.5), impinging on a wall. The method is proven to be effective and accurate in predicting sound pressure levels associated with the propagation of linear and nonlinear waves in the near- and far-field regions.     

Effect of a biasing electric field on the propagation of symmetric Lamb waves in piezoelectric plates

This paper is concerned with the effect of a biasing electric field on the propagation of Lamb waves in a piezoelectric plate. On the basis of three dimensional linear elastic equations and piezoelectric constitutive relations, the differential equations of motion under a biasing electric field are obtained and solved. Due to the symmetry of the plate, there are symmetric and antisymmetric modes with respect to the median plane of the piezoelectric plate. According to the characteristics of symmetric modes (odd potential state) and antisymmetric modes (even potential state), the phase velocity equations of symmetric and antisymmetric modes of Lamb wave propagation are obtained for both electrically open and shorted cases. The effect of a biasing electric field on the phase velocity, electromechanical coupling coefficient, stress field and mechanical displacement of symmetric and antisymmetric Lamb wave modes are discussed in this paper and an accompanying paper respectively. It is shown that the biasing electric field has significant effect on the phase velocity and electromechanical coupling coefficient, the time delay owning to the velocity change is useful for high voltage measurement and temperature compensation, the increase in the electromechanical coupling coefficient can be used to improve the efficiency of transduction.     

偏压电场作用下磁电弹板中Lamb波传播研究

基于磁电弹性材料的本构关系和运动微分方程,研究了磁电弹板中Lamb波的传播特性.在电磁开路和短路两种边界条件下,导出了偏压电场作用下磁电弹板中Lamb波的对称模态和反对称模态的相速度方程,求解了相应的应力场.通过数值算例分析了偏压电场对相速度和应力场的影响.结果表明,偏压电场对相速度有较大影响,对应力场几乎没有影响.研究成果可为设计磁电换能器提供一定的理论参考.     

Group properties and invariant solutions of equations of an electric field in the case of nonlinear ohm's law

The group properties of one-dimensional nonstationary equations of an electric field in homogeneous isotropic media with nonlinear conductivity are considered. The nonlinear Ohm's laws for which these equations have the broadest symmetry properties are determined. Ordinary differential equations determining invariants solutions are obtained; the order of the equations is lowered or they are integrated to the end.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskaya Fizika, No. 3, pp. 28–36, May–June, 1972.     

WAVES IN PRE-STRETCHED INCOMPRESSIBLE SOFT ELECTROACTIVE CYLINDERS:EXACT SOLUTION

This paper studies wave propagation in a soft electroactive cylinder with an underlying finite deformation in the presence of an electric biasing field.Based on a recently proposed nonlinear framework for electroelasticity and the associated linear incremental theory,the basic equations governing the axisymmetric wave motion in the cylinder,which is subjected to homogeneous pre-stretches and pre-existing axial electric displacement,are presented when the electroactive material is isotropic and incompressible.Exact wave solution is then derived in terms of(modified) Bessel functions.For a prototype model of nonlinear electroactive material,illustrative numerical results are given.It is shown that the effect of pre-stretch and electric biasing field could be significant on the wave propagation characteristics.     

Amplification of acoustic waves in laminated piezoelectric semiconductor plates

Summary Two-dimensional equations for coupled extensional, flexural and thickness-shear motions of laminated plates of piezoelectric semiconductors are obtained systematically from the three-dimensional equations by retaining lower order terms in power series expansions in the plate thickness coordinate. The equations are used to analyze extensional waves in a composite plate of piezoelectric ceramics and semiconductors. Dispersion and dissipation due to semiconduction as well as wave amplification by a dc electric field are discussed.     

Uniaxial wave propagation in a nonlinear thermoviscoelastic medium with temperature dependent properties

The problem of finite wave propagation in a nonlinearly thermoviscoelastic thin rod whose viscoelastic properties are temperature dependent is considered. The rod is subjected to mechanical or thermal time-dependent loading. The coupled equations of motion and heat conduction are based on a constitutive theory of nonisothermal nonlinear viscoelasticity which is described by single-integral terms only. This theory is reformulated here for the uniaxial motion of a compressible rubbery material. The solution of the field equations is obtained by a numerical procedure which is developed for the present case and is able to handle successfully shock waves whenever they built up in the nonlinear material.     

A nonlinear analysis of surface acoustic waves in isotropic elastic solids

With the fast evolution of wireless and networking communication technology, applications of surface acoustic wave (SAW), or Rayleigh wave, resonators are proliferating with fast shrinking sizes and increasing frequencies. It is inevitable that the smaller resonators will be under a strong electric field with induced large deformation, which has to be described in wave propagation equations with the consideration of nonlinearity. In this study, the formal nonlinear equations of motion are constructed by introducing the nonlinear constitutive relation and strain components in a standard procedure, and the equations are simplified by the extended Galerkin method through the elimination of harmonics. The wave velocity of the nonlinear SAW is obtained from approximated nonlinear equations and boundary conditions through a rigorous solution procedure. It is shown that if the amplitude is small enough, the nonlinear results are consistent with the linear results, demonstrating an alternative procedure for nonlinear analysis of SAW devices working in nonlinear state.