Existence of longitudinal waves in anisotropic poroelastic solids

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.     

Longitudinal and transverse waves in anisotropic elastic materials

Summary It is shown that a necessary and sufficient condition for a longitudinal wave to propagate in the direction n in an anisotropic elastic material is that the elastic stiffness C ₁₁ (n) is a stationary value (maximum, minimum or saddle point) at n. Explicit expressions of all n and the corresponding elastic stiffness C ₁₁ (n) for which a longitudinal wave can propagate are presented for orthotropic, tetragonal, trigonal, hexagonal and cubic materials. As to longitudinal waves in triclinic and monoclinic materials, only few explicit expressions are possible. We also present necessary and sufficient conditions for a transverse wave to propagate in the direction n. As an illustration, explicit expressions of all n, the polarization vector a and the wave speed c for which a transverse wave can propagate in cubic and hexagonal materials are given. The search for n in hexagonal materials confirms the known fact that a transverse wave can propagate in any direction. A longitudinal wave is necessarily accompanied by two transverse waves. However, a transverse wave can propagate without being accompanied by a longitudinal wave.     

Pure modes for elastic waves in crystals: mathematical modeling and search

A universal search method of pure longitudinal and pure transverse modes for elastic wave propagation in crystals, in general piezoelectrics, is presented. A mathematical model of pure modes for elastic waves based on adiabatic state equations for an arbitrary anisotropic piezoelectric medium and its equation of motion under elastic deformations in the rotated Cartesian coordinates is constructed. The condition for longitudinal normals is that all non-diagonal matrix elements of the effective elastic stiffness coefficients in the corresponding wave equation are equal to zero. Equating to zero non-diagonal elements only in one row of this matrix, one can obtain the condition for transverse normals. A computer program is prepared which allows to find the pure modes for elastic waves in crystals and to calculate their characteristics when symmetry class, elastic, piezoelectric, dielectric constants, and crystal density are known.     

The coated thermoelastic body within a low-frequency elastodynamic field

A superposition of a longitudinal and a transverse plane elastic wave excites a small body which is embedded in an infinite elastic medium. The interior of the body exhibits thermoelastic behaviour of the Biot type and it contains a core which is also thermoelastic but with different thermal and elastic parameters. Integral representations for the near as well as the far-field are obtained which involve volume integrals over the shell and the core and surface integrals over the surface of the scatterer and the core–shell interface. Complete low-frequency expansions are provided and the scattering problem is reduced to a sequence of transmission problems for the determination of the coefficients of these expansions. It is shown that the thermal character of the interior media is observed in the low-frequency approximations of order higher or equal to three, when we are close to the scatterer and higher or equal to five, when we are far away from it. Furthermore, the thermoelastic behaviour of the scatterer affects only the radial scattering amplitude, which is of the longitudinal type, while the tangential scattering amplitudes, which are of the transverse type, coincide with the corresponding expressions for scattering by an elastic body with a penetrable elastic core.     

Thermoelastic plane waves in a rotating isotropic medium

Summary We discuss the theory of thermoelastic wave propagation in a rotating isotropic material. In any given direction there are four waves. In general, all of these waves are attenuated, and none of them is purely dilatational or transverse. Some earlier published results are found to be false.     

Note A note on finite amplitude transverse waves in a thermoelastic solid

Summary. If thermal and mechanical coupling is neglected, there is a class of isothermal strain energy functions for isotropic compressible hyperelastic solids, that admits the propagation of a finite amplitude transverse wave, without a coupled longitudinal wave, and a subclass that admits the simultaneous propagation of an uncoupled finite amplitude transverse wave and a longitudinal wave [1]. Several examples of this class of strain energy function are discussed. When thermal and mechanical coupling is considered the solid is described as thermoelastic. The purpose of this paper is to investigate the possibility of the propagation of a finite amplitude transverse wave without a coupled longitudinal wave, or the uncoupled simultaneous propagation of a finite amplitude transverse wave and a longitudinal wave, in an isotropic thermoelastic solid that has no underlying deformation. It is shown that an extensive class of isotropic thermoelastic solids does not admit the propagation of an uncoupled finite amplitude transverse wave, with or without an uncoupled longitudinal wave, even if the corresponding hyperelastic solid does.     

Time-resolved line focus acoustic microscopy of layered anisotropic media: application to composites

This paper presents theoretical and experimental studies of the time-domain response of line focus acoustic microscopy from a layered anisotropic medium. A method for elastic constant reconstruction from acoustic microscopy signatures also is presented. The microscopy response is complicated by multiple reflections in the layers and by the anisotropic nature of the material. The model is based on a new, stable recursive stiffness matrix algorithm developed for a multilayered anisotropic medium, which is applied to the interpretation of the time-resolved acoustic microscopy signature. Specific examples are given for unidirectional and multidirectional graphite epoxy composites. It is shown that the fluid load has a significant effect on the leaky surface waves in these composites, increasing surface wave speed above that for the slow transverse wave. This results in its absence from the microscopy signature of the surface wave. The theoretical results are compared with experiments carried out using a line focus PVDF transducer developed at National Institute of Standards and Technology (NIST). Time-resolved acoustic microscopy has been applied to the determination of elastic constants of a unidirectional composite or of one lamina in a cross-ply composite. The lateral waves and multiple reflections of bulk waves appearing in the microscopy signatures are used for the elastic properties reconstruction. The reconstruction results are compared to data obtained by the self-reference double-through-transmission ultrasonic bulk wave method.     

Surfaces of discontinuity for a cubically anisotropic body in the micropolar thermoelasticity theory

We derived an equation for the propagation of thermoelastic waves in cubically anisotropic continuous media with account for the asymmetry of the stress tensor. By means of this equation the existence of longitudinal and transverse waves of displacements and microrotations is established. Translated from Inzhenerno-Fizcheskii Zhurnal, Vol.73, No. 5, pp. 1027–1032, September–October, 2000.     

Propagation of inhomogeneous waves in anisotropic piezo-thermoelastic media

A mathematical model for the propagation of harmonic plane waves in an anisotropic piezo-thermoelastic medium is explained through three relations. Two of them relate the stress-induced harmonic variations in temperature and electric potential to mechanical displacement of material particles. The third is a system that defines modified Christoffel equations for wave propagation in the medium. The solution of this system is ensured by a quartic equation whose complex roots explain the existence and propagation of four attenuating waves in the medium. The effects of piezoelectricity and thermoelasticity on the wave propagation are analyzed in the discussion of special cases. An angle between propagation direction and direction of maximum attenuation defines the attenuated wave as inhomogeneous wave. The complex slowness vector for each of the four attenuated waves in the medium is resolved to calculate the phase velocity and the attenuation factor for its propagation as an inhomogeneous wave along a general direction in three-dimensional space. The variations in phase velocities and attenuation factors with propagation direction are computed, for a realistic numerical model.     

The Character of Deformations at the Interface of Elastic Media under the Slip Condition

Fresnel coefficients obtained when solving the problem of wave propagation through the interface of two elastic media and expressions for components of the elastic-distortion tensor allow one to study the character of dynamic deformations at the interface. Deformation modes different from zero at the interface of the elastic media under the slip-contact condition have been determined. Dependences of deformation amplitudes at the interface on the wave incidence angle and parameters of the adjacent media for incident longitudinal and transverse waves have been constructed and analyzed.     

Energy flux characteristics of inhomogeneous waves in anisotropic thermoviscoelastic media

This study aims to calculate the wave-field characteristics of four attenuating waves in anisotropic thermoviscoelastic medium. An energy balance equation relates the complex-valued energy flux vector to the time-averaged densities of kinetic energy, strain energy and dissipated energy of plane harmonic waves in the medium. A complex slowness vector defines the inhomogeneous propagation of an attenuating wave in the medium. This slowness vector is specified with the phase velocity and the two non-dimensional attenuation parameters of the wave. One of the attenuation parameter defines the inhomogeneity strength of the wave as a measure of its deviation from homogeneous propagation. The phase velocity, attenuation parameters, polarizations of particles, propagation direction are combined to define the group velocity, ray direction and quality factor of attenuation of an inhomogeneous wave in the medium. Numerical examples are considered to study the variations of these characteristics of energy flux with propagation direction and inhomogeneity strength for each of the four attenuating waves in the medium. The effects of anisotropic symmetries are analyzed on the velocities of waves. The decay-rate of energy densities is exhibited with offset in the propagation-attenuation plane.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

S波由饱和土入射于弹性土时在界面上的反射与透射

从地震工程实际出发，借助Biot多孔介质中的波动方程，根据各种界面条件导出了S波从饱和土入射于弹性土时在交界面上反射与透射的一般计算公式。作为算例，数值计算分析了S波从饱和土入射于饱和土与弹性土交界面时，饱和土中P1、P2和S波的反射系数以及弹性土中P波、SV射系数与界面排水条件、入射角以及频率之间的关系。结果表明：各种波的反射、透射系数与入射角、入射频率以及界面排水条件有关系。     

Rayleigh waves in a thermoelastic solid half space using dual-phase-lag model

The present paper deals with the study of Rayleigh waves in a thermoelastic homogeneous isotropic solid half space in the context of dual-phase-lag model. The medium is subjected to stress free, thermally insulated, boundary conditions. The equation for the phase velocity of Rayleigh waves and the analytical expressions for the amplitudes of the displacements, temperature and thermal stresses have been derived. The expressions are obtained for a wave traveling along the free surface. The results discussed numerically and illustrated graphically to show effect of the coupling parameter and phase-lags.     

Internal Reflection in Uniaxial Crystals,III: Transmission and Reflection Coefficients for an Extraordinary Incident Wave

Abstract

The boundary conditions have been solved and the reflection and transmission coefficients calculated for an interface between a uniaxial crystal and an isotropic medium when the incident wave is extraordinary. The existence of a pseudo-Brewster angle for each reflected wave has been verified, and the polarizations of the reflected refracted waves have been determined.     

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.     

一维复合波入射的固-固界面接触特性分析

基于弹性波理论,针对接触面非线性相互作用,建立了一个非线性刚度模型,利用接触应力描述非线性刚度特性。引入包含基波和谐波的一维复合波入射,在二阶微扰近似以及非线性边界条件下,给出了界面的反射波和透射波的表达式,得到反射系数和透射系数。结合界面线性刚度和接触应力的幂律关系,定义了四个非线性系数。通过数值计算分析了固-固界面的非线性特性以及谐波对于接触应力的依赖性。     

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 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 dispersion and attenuation in fiber composites

 This work deals with the dispersion and attenuation of elastic plane waves propagating in a single layer fiber reinforced composite, in a direction which is perpendicular to the fibers. An iterative effective medium model, based on single scattering considerations, for the quantitative estimation of wave dispersion and attenuation is proposed. The single scattering problem is solved numerically by means of a 2-D boundary element methodology. Numerical results concerning the plane velocity and the attenuation coefficient of longitudinal or transverse SH, SV waves propagating in two types of fiber reinforced composite materials, are presented. The obtained results are compared to those taken either experimentaly or numericaly by other investigators.