Wave propagation in micropolar mixture of porous media

This paper is concerned with the possible propagation of waves in an infinite porous continuum consisting of a micropolar elastic solid and a micropolar viscous fluid. Micropolar mixture theory of porous media developed by Eringen [A.C. Eringen, Micropolar mixture theory of porous media, J. Appl. Phys. 94 (2003) 4184–4190] is employed. It is found that there exist four coupled longitudinal waves (two coupled longitudinal displacement waves and two coupled longitudinal microrotational waves) and six coupled transverse waves in a continuum of this micropolar mixture. All the waves are found to attenuate and dispersive in nature. A problem of reflection of coupled longitudinal waves from a free boundary surface of a half-space consisting the mixture of a micropolar elastic solid and Newtonian liquid, is investigated. The expressions of various amplitude ratios and surface responses are derived. Numerical computations are performed to find out the phase velocity and attenuation of the waves. The variation of amplitude ratios, energy ratios and surface responses are also computed for a specific model. All the numerical results are depicted graphically. Some limiting cases have also been discussed.     

Wave Propagation in a Green–Naghdi Thermoelastic Solid with Diffusion

The Green and Naghdi theory of thermoelasticity is applied to study plane-wave propagation in an elastic solid with thermo-diffusion. The governing equations of an elastic solid with generalized thermo-diffusion are solved to show the existence of three coupled longitudinal waves and a shear vertical (SV) wave in a two-dimensional model of the solid with thermo-diffusion. The reflection of plane waves from a thermally insulated stress-free surface of an elastic solid with thermo-diffusion is also studied. A non-homogeneous system of four equations in reflection coefficients is obtained. The speeds of the plane waves are computed numerically and plotted against frequency for a particular range. The complex absolute values of the reflection coefficients of all reflected waves are computed numerically and plotted against the angle of incidence of the striking wave at the free surface. The effects of diffusion parameters are shown graphically for speeds and reflection coefficients of plane waves.     

Reflection of Magneto-Thermoviscoelastic Waves Under Generalized Thermoviscoelasticity

Considering each of the thermal fields, viscoelastic and electromagnetic fields contribute to the total deformation of a body and interact with each other. Reflection of magneto-thermoelastic waves under generalized thermoviscoelastic theories is employed to study the reflection of plane harmonic waves from a semi-infinite elastic solid in a vacuum. The expressions for the reflection coefficients, which are the ratios of the amplitudes of the reflected waves to the amplitudes of the incident waves, are obtained, and the reflection coefficient ratios variation with the angle of incidence under different conditions are shown graphically when acrylic plastic materials are considered.     

Elastic wave and energy propagation in angled beams

This investigation comprises an experimental and numerical study of elastic wave propagation in angled beams. Axial impact by two strikers of different lengths was applied to three steel beams, each bent to incorporate a “V” section of different angle in the middle. Finite element simulation using ABAQUS was employed to examine details of the elastic waves generated in the impact tests. The numerical results correlated well with experimental data, and computational simulation was utilized to analyse the propagation of energy associated with the elastic waves. This demonstrated that after several reflections from and transmission across the bends energy is progressively smeared throughout the entire beam and does not concentrate at any particular segment; the bulk of the energy is conveyed via flexural waves. Numerical simulation of wave propagation in a beam with a single angle was also undertaken to study the energy associated with waves reflected from and transmitted across the bend, and how these are affected by the bend angle. The effects of input pulse duration, beam thickness and beam material properties on energy reflection and transmission at a bend are also discussed; this leads to the conclusion that when a longitudinal pulse of a particular frequency impinges on a bend, the ratio between its wavelength and the beam thickness governs the energy reflected from and transmitted across the bend. Moreover, the bend junction geometry (curvature) is found to have a significant influence on the energy reflected and transmitted, especially for obtuse bend angles.     

Reflection of plane micropolar viscoelastic waves at a loosely bonded solid-solid interface

A solution of the field equations governing small motions of a micropolar viscoelastic solid half-space is employed to study the reflection and transmission of plane waves at a loosely bonded interface between two dissimilar micropolar viscoelastic solid half-spaces. The amplitude ratios for various reflected and refracted waves are computed for a particular model for different values of bonding parameter. The variations of these amplitude ratios with the angle of incidence are shown graphically. Effects of bonding parameter and viscosity on the amplitude ratios are shown.     

平面S波在非饱和土自由边界上的反射问题研究

运用非饱和孔隙介质理论阐述了弹性波在非饱和土中的传播特性,分析了平面S波在非饱和土层自由边界上的反射问题。根据边界条件,分别导出了在非饱和土自由边界上的四种反射波：反射P1波、反射P2波、反射P3波及反射S波的振幅反射率及能量反射率的理论表达式,并在此基础上进行了数值计算。算例中讨论了四种反射波的振幅反射率及能量反射率受平面S波入射角度及土层饱和度变化的影响情况。计算结果表明：各反射波的振幅反射率及能量反射率不仅与入射角有关,也受到饱和度变化的影响,这些结论对土动力学的理论研究以及相关工程地震勘探具有一定指导意义。     

A note on energy flux in micropolar elastic waves

The expressions for the time average power per unit area, the kinetic energy per unit volume, strain energy per unit volume and velocity of energy flux are derived for the case of a plane time harmonic micropolar elastic waves and it has been shown that the time-average energy density is equally divided between the time-averages of the kinetic and strain energy densities. The energy ratios of reflected micropolar elastic waves with the angle of emergence of incident longitudinal displacement wave have been shown graphically.     

The effect of rotation on the reflection of magneto-thermoelastic waves under thermoelasticity without energy dissipation

Summary In this paper reflection of magneto-thermoelastic waves is employed to study the effect of rotation and the magnetic field on the plane harmonic waves of a rotating semi-infinite elastic solid nearby a vacuum under Green and Naghdi theory. The expressions for the reflection coefficients, which are the ratios of the amplitudes of the reflected waves to the amplitude of the incident wave, are obtained. The effect of rotation, the magnetic field, coupling parameter and frequency on the reflection ratios are illustrated graphically. Comparisons are made with the results predicted by GN theory of type II in the presence and absence of rotation and magnetic field. A comparison is also made with the results predicted by GN theory of types II and III in the presence of the magnetic field and rotation.     

Reflection of Plane Waves in a Semiconducting Medium under Photothermal Theory

In this paper photothermal theory was used to study the reflection of waves at the surface of a semi-infinite semiconconducting medium. Using the harmonic wave method, the reflection coefficient ratios were obtained analytically under coupled thermoelastic theory and plasma theory for an incident CI wave, which is one coupled thermoelastic plasma wave and an incident rotational wave. The variations of the amplitude of reflection coefficient ratios with the angle of incidence are shown graphically for silicon. Effects of the thermal frequency, the thermoelastic coupling parameter, and the thermoelectric coupling parameter were given by numerical results. Also, the energy ratios for reflected waves were computed to check the numerical results.     

Wave propagation in a cylindrical viscous layer between two elastic shells

Propagation of longitudinal waves in a liquid-filled layer between two thin coaxial shells is investigated. Both liquid viscosity and elasticity of the shells are accounted for. Dynamics of the shells is treated using the Kirchhoff–Love approximation. The elastic deformations of the shells in the sound wave are coupled with the liquid flow in the gap through appropriate dynamic and kinematic boundary conditions. Hydrodynamics of the liquid is described using the quasi-one-dimensional (hydraulic) approach. It is assumed that the external and internal shells are composed of different isotropic elastic materials and have different widths. The dispersion equation for harmonic waves in the system is obtained; it is valid in the low frequency range where the wave length is greater than the external shell radius. In the limiting case for an ideal liquid the dispersion equation yields water hammer speed in the system. The analysis of the dispersion equation has shown strong influence of viscous losses on dispersion and attenuation of pressure signals in the low frequency region. The wave speed and attenuation are highly dependent on the geometrical parameters of the system and elastic properties of the shells.     

Electromagnetic effects on Rayleigh surface wave propagation in a homogeneous isotropic thermo-microstretch elastic half-space

This paper deals with the problem of electromagnetic effect on the propagation of Rayleigh surface waves in a homogeneous, isotropic, thermally-conducting microstretch elastic half-space. In this context, the generalized theory of thermoelasticity is considered. The governing equations for the Rayleigh surface waves in the cases of insulated as well as isothermal boundaries are derived. In the presence of the magnetic effect, the analytical expressions for the displacement, microrotation, microstretch, and temperature changes are obtained. The changes in the phase velocity, microrotation, and path of particles for aluminum epoxy material are presented graphically.     

利用波动法研究曲梁结构中的波形转换和能量传递

基于Flugge理论,建立了薄壁均质常曲率曲梁面内运动的6阶微分控制方程,得到了曲梁的频散特性曲线和6种波的轴向位移和径向位移的比值,推导了位移和内力响应的表达式以及物理域和波数域的变换矩阵。利用波的传递和反射矩阵对曲梁和半无限长直梁耦合时的能量传递系数和反射系数进行了求解分析。对于半无限长直梁中给定的拉伸波或弯曲波入射,得到了和频率,曲率半径和伸展角度相关的各种波传递和反射的能量系数表达式。数值结果表明,纵波和弯曲波在经过曲梁结构之后发生了波形转换,并研究了能量传递和反射系数随频率,伸展角度,曲梁曲率半径和截面尺寸比的变化。结果表明,无限长直梁和曲梁耦合系统中,低频时,经过曲梁反射和传递后的弯曲波和纵波会相互转化;高频时弯曲波和纵波都能够没有散射地通过曲梁而进行传播。为改善高频时曲梁中的能量衰减效果,研究了在曲梁结构中插入单个、多个中间支撑或阻振质量块时的能量传递和反射系数。结果表明,阻振质量块能够很好地阻止高频时曲梁中能量的传递,对于周期分布的多个阻振质量块,能量传递系数随频率的变化存在周期结构的阻带特征。这些研究结果为曲梁结构的设计提供定性的理论基础。     

Reflection of PlaneWaves from Electro-magneto-thermoelastic Half-space with a Dual-Phase-Lag Model

The aim of this paper is to study the reflection of plane harmonic waves from a semi-infinite elastic solid under the effect of magnetic field in a vacuum. The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves, are obtained. Similarly, the reflection coefficient ratio variations with the angle of incident under different conditions are shown graphically. Comparisons are made with the results predicted by the dual-phase-lag model and Lord-Shulman theory in the presence and absence of magnetic field.     

Elastodynamic Behavior of Waves in Thermo-Microstretch Elastic Plate Bordered with Layers of Inviscid Liquid

The elastodynamic behavior of waves in a thermo-microstretch elastic homogeneous isotropic plate bordered with layers of inviscid liquid on both sides subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman and Green and Lindsay theories of thermoelasticity. The mathematical model has been simplified by using the Helmholtz decomposition technique, and the frequency equations for different mechanical situations are obtained and discussed. The special cases such as short wavelength waves and regions of the secular equations are also discussed. Finally, the numerical solution is carried out for a magnesium crystal composite material plate bordered with water. The dispersion curves, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch, and temperature distribution for the symmetric and skew-symmetric wave modes are presented graphically.     

Channelled waves in anisotropic mesh metamaterials

Channelled waves in 2D periodic anisotropic L-C mesh metamaterials have been investigated. Circuit simulation and the newly developed analytical model of a unit cell have demonstrated full qualitative agreement for both lossless and lossy cases. Isofrequencies for a lattice unit cell and the circuit simulations of finite meshes have shown that propagating waves are channelled from a point source as pencil beams which can travel only along specific trajectories. The beam direction varies with frequency, and at resonance frequency omega=1/(LC)^1/2, the phase and group velocities of the transmitted wave are orthogonal. The effect of losses was explored, and it is shown that losses can cause qualitative changes of the channelled wave type. It is shown that the channelled waves do not follow the laws of geometrical optics (Snell's law, specular reflection and so on) at the interfaces of L-C meshes but are governed by the conditions of phase synchronism and impedance matching. Only in the special case of dual L-C and C-L meshes with the interface parallel to the axis of rectangular grid excited at the resonance frequency (X=1) do the channels follow the trajectories of optical rays. A planar mesh test cell has been designed and applied to retrieving the unit cell L-C parameters from the S-parameter measurements     

Acoustic reflection from the boundary of anisotropic thermoviscoelastic solid with fluid

Vertical slownesses of waves at a boundary of an anisotropic thermoviscoelastic medium are calculated as roots of a polynomial equation of degree eight. Out of the corresponding eight waves, the four, which travel towards the boundary are identified as upgoing waves. Remaining four waves travel away from the boundary and are termed as downgoing waves. Reflection and refraction of plane harmonic acoustic waves are studied at a plane boundary between anisotropic thermoviscoelastic solid and a non-viscous fluid. At this fluid-solid interface, an incident acoustic wave through the fluid reflects back as an attenuated acoustic wave and refracts as four attenuating waves into the anisotropic base. Slowness vectors of all the waves in two media differ only in vertical components. Complex values of vertical slowness define inhomogeneous refracted waves with a fixed direction of attenuation, i.e. perpendicular to the interface. Energy partition is calculated at the interface to find energy shares of reflected and refracted waves. A part of incident energy dissipates due to interaction among the attenuated refracted waves. Numerical examples are considered to study the variations in energy shares with the direction of incident wave. For each incidence, the conservation of incident energy is verified in the presence of interaction energy. Energy partition at the interface seems to be changing very slightly with the azimuthal variations of the incident direction. Effects of anisotropy, elastic relaxation and thermal parameters on the variations in energy partition are discussed. The acoustic wave reflected from isothermal interface is much significant for incidence around some critical directions, which are analogous to the critical angles in a non-dissipative medium. The changes in thermal relaxation times and uniform temperature of the thermoviscoelastic medium do not show any significant effect on the reflected energy.     

Interactive mechanisms between the internal blast loading and the dynamic elastic response of spherical containment vessels

The interactive mechanisms between internal blast loading and dynamic elastic response of spherical containment vessels are studied in this paper. The blast loading history in containment vessels can be divided into three periods, i.e. the primary-shock period, the shock-reflection period and the pressure-oscillation period. It is shown that the initial response of the containment vessel depends on both the impulse and the shape of the primary-shock depending on the ratio of the loading period to the breathing mode period. However, during the shock-reflection period, the response of the containment vessel can be coupled with the reflected shock waves in the vessel, especially when the dominant frequency of reflected shock waves is close to the breathing mode frequency of the vessel. During the pressure-oscillation period, the dynamic loading is mainly the oscillation of the internal pressure due to the oscillatory volume change of the vessel, which couples dissipatedly with the vibration of the vessel leading to reduced vibration amplitudes. The effects of the influential non-dimensional parameters on the resonant interaction in shock-reflection period are discussed, based on which guidelines are recommended for avoiding the strain growth in the shock-reflection period in the design of spherical containment vessels.     

Nonlinear wave scattering at the flexible interface of a granular dimer chain

Uncompressed granular dimer chains composed of repetitive pairs of heavy-light spherical, linearly elastic beads exhibit interesting intrinsic responses. The dynamics of these highly discontinuous nonlinear media is governed by the mass ratio scaling the mass disparity of each heavy-light pair of beads. In particular, it has been theoretically and experimentally shown that they support countable infinities of anti-resonances at a discrete set of mass ratios leading to solitary pulses propagating through the dimers with no attenuation or distortion. Conversely, they support countable infinities of resonances at a different discrete set of mass ratios, leading to substantial and rapid attenuation of propagating pulses due to energy scattering from low-to-high frequencies and wavenumbers by means of radiating traveling waves. In this work we computationally study nonlinear scattering of impeding pulses at the interface of an impulsively excited dimer chain with a dispersive elastic boundary, namely, a finite linear string resting on an elastic foundation. We develop a computational algorithm which, through iteration and interpolation at successive time steps, accurately computes (and ensures convergence of) the highly discontinuous contact forces and displacements at the flexible interface of the granular medium. This enables accurate computation of wave transmission, reflection, localization or multi-scale nonlinear scattering at the flexible interface for varying mass ratios of the dimer and the interface parameters. We show that, depending on the mass ratio of the dimer and the stiffness of the elastic foundation, the nonlinear scattering at the flexible interface may lead to significant reduction of the maximum contact force at the interface, and, thus, drastically affect the transmitted and reflected energy at the flexible boundary. In fact, an inverse relation between the stiffness of the elastic foundation and the residual energy transferred from the dimer chain to the flexible boundary is found. Moreover, for sufficiently small mass ratios of the dimer chain transient breathers are realized close to the interface in the form of localized “fast” oscillations of light granules of the dimer that entrap shock energy and then release in a slow time scale back to the chain and the flexible boundary. This work paves the way for studying highly discontinuous and nonlinear scattering phenomena at interfaces of granular media with flexible continua.     

Anomalies in the ultrasound reflection from a solid-dissipative medium interface

The reflection of longitudinal acoustic waves from the interface between a solid and a strongly dissipative medium was theoretically studied. It is shown that the reflection coefficient and the reflected signal phase significantly depend on the coefficient of ultrasound absorption in the dissipative medium. An algorithm for restoring the time variation of the dissipative medium viscosity during solidification is proposed, which is based on the results of measurements of the amplitude reflection coefficient for a pulsed ultrasonic signal. The reflection and transmission coefficients are determined for the ultrasound wave amplitude, velocity, pressure, and intensity. An expression is derived for the acoustic energy absorbed upon reflection from a dissipative medium. A frequency dependence of the amplitude and generalized phase of the signals reflected from and transmitted through the interface are restored from the results of spectral calculations. The theoretical results are confirmed by experimental data on the reflection of acoustic pulses from an organic glass-epoxy compound resin interface measured during solidification of the latter medium. According to these data, the reflection coefficient and the reflected acoustic pulse duration decreased upon solidification of the epoxy compound.     

The transverse elastic impact response of thick hollow cylinders

The two objectives of the studies in this paper were to determine the transient elastic impact response of thick-walled hollow cylindrical structures and to determine how this response was affected by the presence of flaws in the cylinder. These studies were aimed at determining the feasibility of using the impact-echo method for detecting flaws in the cylindrical concrete structures, such as pipes. Three-dimensional finite element models and laboratory specimens containing various types of flaws at known locations were used in the studies. It is shown that, for a hollow cylinder having a length greater than about eight times its outer diameter, the impact response as measured at points close to the impact point is composed of a number of resonant frequencies caused by cross-sectional (flexural) modes and a thickness frequency caused by dilatational waves reflected between the inner and outer wall surfaces. Equations are presented which relate the response of a hollow cylinder to the frequency of the fundamental cross-sectional (flexural) mode of a solid circular bar. It is shown that the location of cracks, areas of reduced wall thickness, and honeycombing as well as the depth of surface-opening cracks can be determined.