Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

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.     

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.     

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.     

On wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium

The paper examines the problem of wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details.     

Waves in fluid-filled elastic tubes with a stenosis: Variable coefficients KdV equations

In the present work, by treating the arteries as thin-walled prestressed elastic tubes with a stenosis and the blood as an inviscid fluid we have studied the propagation of weakly nonlinear waves in such a medium, in the longwave approximation, by employing the reductive perturbation method. The variable coefficients KdV and modified KdV equations are obtained depending on the balance between the nonlinearity and the dispersion. By seeking a localized progressive wave type of solution to these evolution equations, we observed that the wave speeds takes their maximum values at the center of stenosis and gets smaller and smaller as one goes away from the stenosis. Such a result seems to reasonable from the physical point of view.     

Propagation of weak perturbations in cracked porous media

A three-velocity, three-pressure mathematical model is proposed which enables one to study wave processes in the case of a double porosity, deformable, fluid-saturated medium. This model takes account of the differences in the velocities and pressures in pore systems of different characteristic scales of the pores, fluid exchange between these pore systems and the unsteady forces due to interphase interactions. It is established that a single transverse and three longitudinal waves: one deformation wave and two filtration waves, propagate in such a medium. The existence of two filtration waves is associated with the two different characteristic scales of the pores and the difference in the velocities and pressures of the fluid in these pore systems. The filtration waves decay considerably more rapidly than the deformation and transverse waves. The velocities of the deformation and transverse waves are mainly determined by the elastic moduli of the skeleton. The velocity and decay of the first filtration wave depend strongly on the intensity of the interphase interaction force while the velocity of the second filtration wave depends strongly on the rate of mass exchange between the pores and the cracks. The rate of decay of the second filtration wave is significantly higher than that of the first filtration wave.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Reflection and transmission of couple longitudinal waves at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids

The problem of reflection and transmission of elastic waves due to incident plane couple longitudinal waves at a plane interface between two dissimilar half-spaces of thermo-elastic materials with voids has been investigated. Using the theory of Iesan (1986), the propagation of couple longitudinal waves in the thermo-elastic materials with voids has been explained. The expressions of the reflection and transmission coefficients and energy ratios corresponding to reflected and transmitted waves are obtained. These coefficients and energy ratios of the various reflected and transmitted waves are computed numerically for a specific model.     

Effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids

The effect of micro-inertia in the propagation of waves in micropolar thermoelastic materials with voids has been investigated. Elastic waves are reflected due to incident coupled longitudinal and coupled shear waves from a plane free boundary of micropolar thermoelastic materials with voids. The amplitude ratios corresponding to the reflected coupled longitudinal and coupled shear waves are derived by using appropriate boundary conditions. Energy partition in the free surface has been presented. The amplitude and energy ratios of the reflected waves are also computed numerically for a particular model.     

The effect of stiffness on the transverse and longitudinal motions of a musical string: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The effect of stiffness on the propagation of longitudinal and transverse waves and vibrations in prestretched strings is considered. The contribution of the longitudinal and transverse components to the dynamic load is of the same order. The longitudinal vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena are possible. Low stiffness, which is characteristic for musical strings, leads to a small change in the frequencies of the whole spectrum of transverse and longitudinal vibrations, but to a considerable change in the shape of the string at strike and mounting points and on the transverse wave front.     

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.     

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.     

Stabilization of the uniform Timoshenko beam by one locally distributed feedback

In this article, we study the energy decay rate for an elastic Timoshenko system. This system consists of two coupled wave equations. Only the equation about the rotation angle is damped by one locally distributed feedback at the neighbourhood of the boundary. The equation for the transverse displacement of the beam is only indirectly damped through the coupling. First, we establish an exponential energy decay rate in the case of the same speed of propagation. Next, when the wave speeds are different, a polynomial-type decay rate is obtained. These results are proved by verifying the frequency domain conditions.     

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.     

On acceleration waves in anisotropic thermoelastic media taking account of finiteness of the heat propagation velocity : PMM vol. 38, n 6, 1974, pp. 1098–1104

The propagation of acceleration waves in an anisotropic thermoelastic medium is studied. It is shown that taking account of the finiteness of the heat distribution velocity results in the appearance of four kinds of accelaration waves, whose velocities and damping coefficients depend in an essential way on the direction of wave surface propagation. A comparison between the velocities and damping coefficients of plane acceleration waves in a zinc crystal, obtained with and without the finiteness of the heat propagation velocity taken into account, is presented.The papers [1, 2] are devoted to the influence of the coupling of the strain and temperature fields on the nature of wave propagation in a homogeneous isotropic body in the case of an infinite heat distribution velocity. A number of features due to coupling of the fields is obtained therein, and it is shown in particular that weak and strong discontinuities damp out, and the order of damping is determined by an exponential factor.Taking account of finiteness of the heat distribution velocity results in the appearance of two kinds of longitudinal waves whose propagation velocities depend in an essential manner on the velocity of the heat perturbation [3, 4].     

Eigenwaves in a relativistic heat-conducting viscous fluid according to extended irreversible thermodynamics

We propose an approach to extended irreversible thermodynamics leading to a set of transport equations for the dissipative variables characterizing a relativistic heat-conducting viscous fluid. After this, we study the propagation eigenmodes showing the possible existence of three non-trivial modes: two longitudinal and one transverse. The fast longitudinal wave is a pressure wave associated to a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The general study of the longitudinal case is not possible. Natural special cases are analyzed in detail.     

The Propagation of Finite-Amplitude Waves in a Model Boundary Layer

Finite-amplitude wave propagation is considered in flows of boundary-layer type when the wavelength is long compared to the boundary layer thickness. In this limit, the evolution of the amplitude is governed by the Benjamin-Ono equation and we have computed the coefficients of its nonlinear and dispersive terms for the specific case of Tietjens's model. The propagation of wave packets is also considered, and it is found that for packets centered about an O(1) wavenumber questions again arise relative to long waves, except that now the packet-induced mean flow is the “long wave.” By introducing an appropriate scaling for the far field and employing multiple scales in the direction transverse to the flow, it is shown how the mean-flow distortion can be made to vanish at infinity.