Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory

In the present work, the effect of longitudinal magnetic field on wave dispersion characteristics of equivalent continuum structure (ECS) of single-walled carbon nanotubes (SWCNT) embedded in elastic medium is studied. The ECS is modelled as an Euler–Bernoulli beam. The chemical bonds between a SWCNT and the elastic medium are assumed to be formed. The elastic matrix is described by Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation. The governing equations of motion for the ECS of SWCNT under a longitudinal magnetic field are derived by considering the Lorentz magnetic force obtained from Maxwell’s relations within the frame work of nonlocal elasticity theory. The wave propagation analysis is performed using spectral analysis. The results obtained show that the velocity of flexural waves in SWCNTs increases with the increase of longitudinal magnetic field exerted on it in the frequency range; 0–20 THz. The present analysis also shows that the flexural wave dispersion in the ECS of SWCNT obtained by local and nonlocal elasticity theories differ. It is found that the nonlocality reduces the wave velocity irrespective of the presence of the magnetic field and does not influences it in the higher frequency region. Further it is found that the presence of elastic matrix introduces the frequency band gap in flexural wave mode. The band gap in the flexural wave is found to independent of strength of the longitudinal magnetic field.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Propagation of plane waves in microstretch fluid

The propagation of plane harmonic waves are studied in a microstretch fluid medium. It is found that five basic waves can propagate at distinct speeds in an infinite linear homogeneous isotropic microstretch fluid. Out of these five waves, one is a longitudinal micro-rotational wave, two are coupled longitudinal waves and remaining two are coupled transverse waves. The longitudinal micro-rotational wave travels independently and is not influenced by the microstretching of the medium, while the coupled longitudinal waves arise due to the presence of microstretching and coupled transverse waves arise due to the presence of micro-rotation in the medium. Speed of propagation of all the waves are found to be complex valued and dispersive at low frequency, but almost non-dispersive at high frequency. Due to complex valued speeds of propagation, all the waves are attenuating but differently. Coupled sets of longitudinal waves reduce to a longitudinal wave of micropolar fluid in the absence of microstretching. Reflection phenomena of a set of coupled longitudinal waves incident obliquely at the free surface of a microstretch fluid half-space has been investigated. Closed formulae for the reflection coefficients are presented and computed numerically for a particular medium. The real and imaginary parts of the complex speeds of all the waves and their corresponding attenuation coefficients have also been studied numerically and depicted graphically against frequency parameter.     

Nonlinear transparency regimes for three-component acoustic pulses in a system of electron and nuclear spins

We study acoustic solitons consisting of one longitudinal and two transverse components and propagating in the direction perpendicular to an external magnetic field in a crystal containing paramagnetic impurities of electron and nuclear spins. The coupling of the electron spin subsystem to the longitudinal sound allows making the velocity of the latter close to that of the transverse acoustic waves, which provides an effective interaction between all components of the elastic field by means of the nuclear spin subsystem. We derive a three-component system of material and reduced wave equations describing this process and construct its soliton solutions in the form of stationary and breather pulses. Based on them, we study the peculiarities of the dynamics of the elastic field components and reveal the differences from the two-component model. The existence of two families of breathers is an important distinctive feature of the considered case.     

Wave propagation in thermo-chiral elastic medium

The propagation of time harmonic waves through an infinite thermo-chiral elastic material has been investigated. The elastic field of thermo-chiral medium has been described by extending the governing equations and constitutive relations of hemitropic micropolar material to include temperature field. Seven basic waves consisting of three coupled dilatational waves and four coupled shear waves traveling with distinct speeds may exist in the medium. All the waves are found to be dispersive, however the coupled dilatational waves are attenuating and temperature dependent, while the coupled shear waves are independent of temperature field. The phase speeds and corresponding attenuation quality factors of all the coupled dilatational waves have been computed numerically for a specific model. The effect of chirality and temperature field have been shown graphically.     

Interaction of Elastic Waves

We derive and analyze asymptotic equations for the interaction of weakly nonlinear elastic waves. We show that there are resonant triads involving two transverse and one longitudinal wave provided the wave speeds satisfy a certain irrationality condition. We study initial value and signaling problems, and the interaction of sawtooth wave packets.     

Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Reflection of plane waves from the stress-free isothermal and insulated boundaries of a nonlocal thermoelastic solid

The generalized thermoelasticity theory based upon the Green and Naghdi model II of thermoelasticity as well as the Eringen’s nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves which are dispersive in nature and associated with attenuation. In addition to the coupled waves, there also exists one independent vertically shear type wave which is dispersive but without any attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear type wave is found to to be associated with a critical frequency, while the coupled longitudinal waves may have critical frequencies under constraints. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on the phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients as well as the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Two-dimensional wave propagation in a generalized elastic solid

The object of the present study is to investigate the propagation of two-dimensional waves in a weakly nonlinear and weakly dispersive elastic solid. The reductive perturbation method is directly applied to a Lagrangian whose Euler–Lagrange equations give the field equations for a quadratically nonlinear elastic medium with higher order gradients. In the long-wave approximation, it is shown that the long-time behavior of the two transverse waves is governed by the two coupled modified Kadomtsev–Petviashvili (CMKP) equations. Depending on the choice of the direction of perpendicular dynamics, various forms of the CMKP equations are obtained. Some special solutions are also presented for a simplified form of the CMKP equations.     

Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix

This paper reported the result of an investigation into the effect of magnetic field on wave propagation in carbon nanotubes (CNTs) embedded in elastic matrix. Dynamic equations of CNTs under a longitudinal magnetic field are derived by considering the Lorentz magnetic forces. The results obtained show that wave propagation in CNTs embedded in elastic matrix under longitudinal magnetic field appears in critical frequencies at which the velocity of wave propagation drops dramatically. The velocity of wave propagation in CNTs increases with the increase of longitudinal magnetic field exerted on the CNTs in some frequency regions. The critical/cut-off frequency increases with the increase of matrix stiffness, and the influence of matrix on wave velocity is little in some frequency regions. This investigation may give a useful help in applications of nano-oscillators, micro-wave absorbing and nano-electron technology.     

The propagation of elastic waves in composite materials reinforced with piezoelectric fibres

The propagation of longitudinal elastic waves in composite materials, consisting of a polymer matrix reinforced by continuous fibres in one direction, is considered. The reinforcing fibres have piezoelectric properties and have a thin current-conducting coating (“shunted fibres”). The scattering of electric energy in such materials leads to dispersion of the velocity of the elastic waves and to their attenuation. The effective-field method is used to determine the macroscopic electroelastic constants of such composites. These constants enable one to obtain, in explicit form, the frequency dependence of the real and imaginary parts of the wave number of a longitudinal wave, propagating along the reinforcement direction, and also their dependenc on the physical and geometrical characteristics of the components.     

Propagation of nonlinear capillaary waves in an electrically conducting liquid

The article deals with the propagation of periodic capillary waves with finite amplitude on the surface of an electrically conducting liquid subjected to the effect of a magnetic field. It is shown that the evolution of wave packets is described by perturbed nonlinear Schrödinger equations. Its asymptotic solution is obtained, and it is established that the influence of MHD effects manifests itself in reduced frequency and amplitude of the propagating waves.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 97–99, 1990.     

An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

On the first-order speeds in any directions of acceleration waves in prestressed second-order isotropic,compressible, and homogeneous materials

In this paper, using the perturbation method we proposed in [A. Marasco, A. Romano, On the ordinary waves in second-order elastic, isotropic, compressible, and homogeneous materials, Math. Comput. Modelling 49 (7–8) (2009) 1504–1518], the first-order terms of the speeds and the amplitude of the principal waves and of the waves in any propagation direction are determined in second-order elastic, isotropic, compressible, and homogeneous materials. Moreover, for the general waves we determine the relations among the second-order constitutive constants which ensure that the waves are longitudinal or transverse.     

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Multi-displacement continuum modelling of the metamaterial plate with periodical arranged resonators

Multi-displacement continuum modelling of a two-dimensional (2D) elastic metamaterials plate with periodically arranged local resonator over the surface of the plate is studied in this paper. The additional displacement fields are introduced to model the response of the local resonators. The continuous conditions between the adjacent unit-cells are used to reflect the periodicity of the microstructured continuum and resultantly turned into the constraint conditions between the additional displacement field and the other continuous field. The dispersion features of the multiple-displacement coupled wave propagating along the high symmetrical direction and any oblique direction are both studied numerically. It is found that the multi-displacement coupled waves can be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along the highly symmetric direction but cannot be divided into the coupled longitudinal wave and the coupled transversal wave when propagating along any oblique direction. The effects of boundary conditions on the dispersion of acoustic and optical branches of coupled waves are discussed in detail. At last, the influences of the parameters of resonator on the dispersion feature of the multi-displacement coupled waves are investigated numerically.     

Modelling of circumferential waves in cylindrical thermoelastic plates with voids

The propagation of thermoelastic waves along circumferential direction in homogeneous, isotropic, cylindrical curved solid plates with voids has been investigated in the context of linear generalized theory of thermoelasticity. The plate is subjected to stress free or rigidly fixed, thermally insulated or isothermal boundary conditions. Mathematical modeling of the problem for the considered cylindrical curved plate with voids leads to a system of coupled partial differential equations. The model has been simplified by using the Helmholtz decomposition technique and the resulting equations are solved by using the method of separation of variables. The formal solution obtained by using Bessel’s functions with complex arguments is utilized to derive the secular equations which govern the wave motion in the plate with voids. The longitudinal shear motion and axially symmetric shear vibration modes get decoupled from the rest of the motion in contrast to non-axially symmetric plane strain vibrations. These modes remain unaffected due to thermal variations and presence of voids. In order to illustrate theoretical developments, numerical solutions have been carried out for a stress free, thermally insulated or isothermal magnesium plate and are presented graphically. The obtained results are also compared with those available in the literature.     

On the propagation of elastic waves in two-phase media

At the present time a number of papers has been already devoted to the dynamics of two-phase media. One may mention the papers by Frenkel' [1], Rakhmatulin [2], Biot [3,4], Zwikker and Kosten [5], and others. However, the basic problem of the setting up of the equations of motion in two-phase media still cannot be considered solved and requires additional study and experimental verification.

This paper is concerned with the study of the simplest case of motion, which is the propagation of elastic waves in a homogeneous isotropic medium consisting of a solid and a fluid phase. The problems of the reflection of plane waves and surface waves at the free boundary of the half-space are solved. It is shown that the stress-strain relations established by Frenkel' are equivalent to the analogous relations proposed by Biot and that the equations of motion of the latter are more general.