Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.     

THE EFFECT OF DUAL-PHASE-LAG MODEL ON REFLECTION OF THERMOELASTIC WAVES IN A SOLID HALF SPACE WITH VARIABLE MATERIAL PROPERTIES

The present article represents an analysis of reflection of P-wave and SV-wave on the boundary of an isotropic and homogeneous generalized thermoelastic half-space when the boundary is stress-free as well as isothermal. The modulus of elasticity is taken as a linear function of reference temperature. The basic governing equations are applied under four theories of the generalized thermoelasticity： Lord-Shulman （L-S） theory with one relaxation time, Green-Naghdi （G-N） theory without energy dissipation and Tzou theory with dual-phase-lag （DPL）, as well as the coupled thermoelasticity （CTE） theory. It is shown that there exist three plane waves, namely, a thermal wave, a P-wave and an SV-wave. The reflection from an isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P- and SV-waves. The amplitude ratios variations with the angle of incident are shown graphically. Also the effects of reference temperature of the modulus of elasticity and dual-phase lags on the reflection amplitude ratios are discussed numerically.     

A state space formalism and perturbation method for generalized thermoelasticity of anisotropic bodies

A state space formalism for generalized anisotropic thermoelasticity accounting for thermomechanical coupling and thermal relaxation is developed, which includes the classical thermoelasticity as a special case. By properly grouping the field variables using matrix notations, the basic equations of thermoelasticity are formulated into a state equation and an output equation in terms of the state vector. To obtain the solution for a specific problem it suffices to solve the state equation under the prescribed conditions. For weak thermomechanical coupling an asymptotic solution can be obtained by using the method of perturbation with multiple scales. Propagation of plane harmonic thermoelastic waves in an anisotropic medium is studied within the context.     

Propagation of plane waves in thermoelastic cubic crystal material with two relaxation times

A problem concerned with the reflection and refraction of thermoelastic plane waves at an imperfect interface between two generalized thermally conducting cubic crystal solid half-spaces of different elastic and thermal properties with two relaxation times has been investigated. The generalized thermoelastic theory with two relaxation times developed by Green and Lindsay has been used to study the problem. The expressions for the reflection and refraction coefficients which are the ratios of the amplitudes of reflected and refracted waves to the amplitude of incident waves are obtained for an imperfect boundary and deduced for normal stiffness, transverse stiffness, thermal contact conductance, slip and welded boundaries. Amplitude ratios of different reflected and refracted waves for different boundaries with angle of emergence have been compared graphically for different incident waves. It is observed that the amplitude ratios of reflected and refracted waves are affected by the stiffness and thermal properties of the media.     

On theory of generalized thermoelastic solids with voids and diffusion

Governing equations of thermoelastic diffusion material with voids are modified with the help of Lord and Shulman theory of generalized thermoelasticity. These governing equations are then solved in two-dimension to show the existence of four coupled longitudinal waves and a shear wave. The complex absolute values of the speeds of the coupled longitudinal waves are computed numerically against the frequency for Magnesium material. The reflection of these plane waves from a stress free thermally insulated boundary is also studied, where the dependence of the reflection coefficients on angle of incidence is shown graphically for the incidence of coupled longitudinal wave only. The speeds and reflection coefficients of plane waves are also computed numerically in the absence of voids and diffusion parameters, which are shown graphically to observe the effects of voids and diffusion.     

Analysis of plane waves in anisotropic thermoelastic diffusive medium

This paper concentrates on the study of the propagation of harmonic plane waves in a homogeneous anisotropic thermoelastic diffusive medium in the context of different theories of thermoelastic diffusion. It is found that five types of waves propagate in an anisotropic thermoelastic diffusive medium, namely a quasi-elastodiffusive (QED-mode), two quasi-transverse (QSH-mode and QSV-mode), a quasi-mass diffusive (QMD-mode) and a quasi-thermo diffusive (QTD-mode) wave. The governing equations for homogeneous transversely isotropic diffusive medium in different theories of thermoelastic diffusion are taken as a special case. It is noticed that when plane waves propagate in one of the planes of transversely isotropic thermoelastic diffusive solid, purely quasitransverse wave mode(QSH) decouples from rest of the motion and is not affected by the thermal and diffusion vibrations. On the other hand, when plane waves propagate along the axis of solid, two quasi-transverse wave modes (QSH and QSV) decouple from the rest of the motion and are not affected by the thermal and diffusion vibrations. From the obtained results, the different characteristics of waves like phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically and presented graphically for a single crystal of magnesium. The effects of diffusion and relaxation times on phase velocity, attenuation coefficient, specific loss and penetration depth has been studied. Some particular cases are also discussed.     

Reflection of plane waves at the free surface of a monoclinic thermoelastic solid half-space

Lord–Shulman and Green–Lindsay theories of generalized thermoelasticity are applied to study the reflection from a thermally insulated stress-free thermoelastic solid half-space of monoclinic type. A particular model is chosen for the numerical computations of reflection coefficients. Effects of anisotropy and relaxation times are observed on reflection coefficients.     

Reflection of plane waves from an elastic solid half-space under hydrostatic initial stress without energy dissipation

In this paper, the basic governing equations for isotropic and homogeneous generalized thermoelastic half-space under hydrostatic initial stress are formulated in the context of the Green and Naghdi theory of types II and III. These governing equations are solved analytically to obtain the dimensional velocities in an xy-plane. It is shown that there exist three plane waves, namely a thermal wave, a P-wave and an SV-wave. The reflection from an insulated and isothermal stress-free surface is studied to obtain the reflection amplitude ratios of the reflected waves for the incidence of P- and SV-waves. Numerical computations are carried out and comparisons made with the results predicted in the presence and absence of hydrostatic initial stress. Also the effect of the thermoelastic coupling parameter and the thermal condition on amplitude ratios are shown graphically.     

Plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating half-spaces with two relaxation times

The present paper concentrates on the study of reflection and refraction characteristics of plane waves at an imperfectly bonded interface of two orthotropic generalized thermoelastic rotating half-spaces with different elastic and thermal properties. In this type of problem of orthotropic thermoelastic rotating medium, there are three types of plane waves quasi longitudinal (QL-) wave, quasi thermal (T-mode) wave and quasi transverse (QT-) wave, whose velocities depend on the angle of incidence, imperfection and rotation. The amplitude ratios of reflected waves to that of incident one in each case have been derived. Some special cases of boundaries, i.e. normal stiffness, transverse stiffness, thermal contact conductance, slip boundary and welded contact boundary have been deduced from an imperfect one. Effect of rotation on the amplitude ratios of different reflected and refracted waves with respect to incident QL-wave at different boundaries have been studied graphically. It is observed that thermal properties, imperfect boundary and rotation have significant effect on the propagation of waves.     

Propagation of plane waves at the imperfect boundary of elastic and electro-microstretch generalized thermoelastic solids

The problem of reflection and transmission of plane waves incident on the contact surface of an elastic solid and an electro-microstretch generalized thermoelastic solid is discussed. It is found that there exist five reflected waves, i.e., longitudinal displacement （LD） wave, thermal （T） wave, longitudinal microstretch （LM） wave and two coupled transverse displacement and microrotational （CD（I） and CD（II）） waves in the electro-microstretch generalized thermoelastic solid, and two transmitted waves, i.e., longitudinal （P） and transverse （SV） waves in the elastic solid. The amplitude ratios of different reflected and transmitted waves are obtained for an imperfect boundary and deduced for normal force stiffness, transverse force stiffness, and perfect bonding. The variations of amplitude ratios with incidence angles have been depicted graphically for the LD wave and the CD（I） wave. It is noticed that the amplitude ratios of reflected and transmitted waves are affected by the stiffness, electric field, stretch, and thermal properties of the media. Some particular interest cases have been deduced from the present investigations.     

Reflection and transmission of plane waves at the interface of an elastic half-space and a fractional order thermoelastic half-space

The problem of reflection and transmission due to longitudinal and transverse waves incident obliquely at a plane interface between uniform elastic solid half-space and fractional order thermoelastic solid half-space has been studied. It is found that the amplitude ratios of various reflected and refracted waves are functions of angle of incidence and frequency of incident wave and are influenced by the fractional order thermoelastic properties of media. The expressions of amplitude ratios and energy ratios have been computed numerically for a particular model. The variation of amplitude and energy ratios with angle of incidence is shown graphically. The conservation of energy at the interface is verified.     

Propagation of waves in an electro-microstretch generalized thermoelastic semi-space

The present investigation is concerned with wave propagation in an electro-microstretch generalized thermoelastic solid half space. Two different cases have been discussed： （i） reflection of plane wave at the free surface of an electro-microstretch generalized thermoelastic solid; and （ii） propagation of Rayleigh waves in an electro-microstretch generalized thermoelastic solid half space. In case （i）, the amplitude ratios of the various reflected waves have been computed numerically and depicted graphically against angle of incidence. In case （ii）, the frequency equation is derived and dispersion curves giving phase velocity and attenuation coefficient as a function of wave number, have been plot- ted graphically for a specific model. Some special cases of interest are also deduced, for both the cases.     

Inhomogeneous waves at the boundary of a generalized thermoelastic anisotropic medium

The Christoffel equation is derived for the propagation of plane harmonic waves in a generalized thermoelastic anisotropic (GTA) medium. Solving this equation for velocities implies the propagation of four attenuating waves in the medium. The same Christoffel equation is solved into a polynomial equation of degree eight. The roots of this equation define the vertical slownesses of the eight attenuating waves existing at a boundary of the medium. Incidence of inhomogeneous waves is considered at the boundary of the medium. A finite non-dimensional parameter defines the inhomogeneity of incident wave and is used to calculate its (complex) slowness vector. The reflected attenuating waves are identified with the values of vertical slowness. Procedure is explained to calculate the slowness vectors of the waves reflected from the boundary of the medium. The slowness vectors are used, further, to calculate the phase velocities, phase directions, directions and amounts of attenuations of the reflected waves. Numerical examples are considered to analyze the variations of these propagation characteristics with the inhomogeneity and propagation direction of incident wave. Incidence of each of the four types of waves is considered. Numerical example is also considered to study the propagation and attenuation of inhomogeneous waves in the unbounded medium.     

广义热弹性问题研究进展

本文总结了广义热弹性问题最近10年的研究进展, 包括不同类型广义热弹耦合问题的研究、考虑磁\!--\!电多场耦合的广义电磁热弹耦合问题研究以及计及扩散效应和黏弹性效应的广义热弹性理论的发展、广义热弹性问题基本求解方法等, 通过总结, 使读者对广义热弹性问题的研究现状及发展趋势有较全面的认识, 帮助研究人员进一步开展广义热弹性问题更高层次的研究.      

Wave propagation at interface of heat conducting micropolar solid and fluid media

The present investigation is concerned with the wave propagation at an interface of a micropolar generalized thermoelastic solid half space and a heat conducting micropolar fluid half space. Reflection and transmission phenomena of plane waves are investigated, which impinge obliquely at the plane interface between a micropolar generalized thermoelastic solid half space and a heat conducting micropolar fluid half space.The incident wave is assumed to be striking at the interface after propagating through the micropolar generalized thermoelastic solid. The amplitude ratios of various reflected and transmitted waves are obtained in a closed form. It is found that they are a function of the angle of incidence and frequency and are affected by the elastic properties of the media. Micropolarity and thermal relaxation effects are shown on the amplitude ratios for a specific model. The results of some earlier literatures are also deduced from the present investigation.     

Elastic Waves in Swelling Porous Media

Time harmonic waves in a swelling porous elastic medium of infinite extent and consisting of solid, liquid and gas phases have been studied. Employing Eringen’s theory of swelling porous media, it has been shown that there exist three dilatational and two shear waves propagating with distinct velocities. The velocities of these waves are found to be frequency dependent and complex valued, showing that the waves are attenuating in nature. Here, the appearance of an additional shear wave is new and arises due to swelling phenomena of the medium, which disappears in the absence of swelling. The reflection phenomenon of an incident dilatational wave from a stress-free plane boundary of a porous elastic half-space has been investigated for two types of boundary surfaces: (i) surface having open pores and (ii) surface having sealed pores. Using appropriate boundary conditions for these boundary surfaces, the equations giving the reflection coefficients corresponding to various reflected waves are presented. Numerical computations are performed for a specific model consisting of sandstone, water and carbon dioxide as solid, liquid and gas phases, respectively, of the porous medium. The variations of phase speeds and their corresponding attenuation coefficients are depicted against frequency parameter for all the existing waves. The variations of reflection coefficients and corresponding energy ratios against the angle of incidence are also computed and depicted graphically. It has been shown that in a limiting case, Eringen’s theory of swelling porous media reduces to Tuncay and Corapcioglu theory of porous media containing two immiscible fluids. The various numerical results under these two theories have been compared graphically.     

The reflection phenomena of quasi-vertical transverse waves in piezoelectric medium under initial stresses

The propagation of plane vertical transverse waves at an interface of a semi-infinite piezoelectric elastic medium under the influence of the initial stresses is discussed. The free surface of the piezoelectric elastic medium is considered to be adjacent to vacuum. We assumed that the piezoelectric material is anisotropic of the type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). For an incident of vertical transverse plane wave, four types (two for the displacement and two for the electric potential) of reflected plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves are shown to be exist. The relations governing the reflection coefficients of these reflected waves for various boundary conditions (mixed-free-fixed) are derived. It has been shown analytically that reflected coefficients of (qP) and (qSV) waves depend upon the angle of incidence, the parameters of electric potential, the material constants of the medium as well as the initial stresses presented in the medium. The numerical computations of reflection coefficients for different values of initial stresses have been carried out by computer for aluminum nitride (AlN) as an example and the results are given in the form of graphs. Finally, particular cases are considered in the absence of the initial stresses and the electric potential. Some of earlier studies have been compared to the special cases and shown good agreement with them.     

Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses

This paper is devoted to study a problem of reflection and refraction of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties. One of the two media is aluminum nitride, which is considered the down piezoelectric medium and the above medium is chosen as PZT-5H ceramics. The two piezoelectric media welded are assumed to be anisotropic of a type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). The equations of motion and constitutive relations for the piezoelectric media have been written. Suitable boundary conditions are used to obtain the reflection and refraction coefficients. For an incidence of quasi-longitudinal plane waves, four independent-type amplitude ratios of elastic displacement components for plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves, are shown to exist. Also, it is observed that there exist four dependent amplitude ratios of electric potential, which are proportional to the previous four types. Finally, it is found that the coefficients of reflection and refraction are functions of angle of incidence, elastic constants, piezoelectric potential parameters and the initial stresses. Numerical computations and the results obtained are depicted graphically. In the end, a particular case has been reduced from the present study. This investigation is considered important because the initial stresses in such practical problems are inevitable and may result in frequency shift, a change in the velocity of surface waves and controlling the selectivity of a filter compensation of the devices.     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

Effect of Rotation on Rayleigh—Lamb Waves in an Isotropic Generalized Thermoelastic Diffusive Plate

The present investigation is aimed at studying the effect of rotation on propagation of Rayleigh—Lamb waves in a homogeneous isotropic thermoelastic diffusive plate of finite width in the framework of different theories of thermoelasticity, including the Coriolis and centrifugal forces. The medium is subjected to stress-free, thermally insulated, isothermal, and chemical potential boundary conditions and is rotating about an axis perpendicular to its plane. Secular equations corresponding to the symmetric and skew-symmetric modes of the plate are derived. Phase velocities and attenuation coefficients of various possible modes of wave propagation are computed from the secular equations. Amplitudes of displacements, temperature, and concentration for symmetric and skew-symmetric modes of plate vibrations are computed numerically. The computed results are presented graphically.