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.     

On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Higher-Order Asymptotic Approximations for Surface Love Waves of SH Type in Tranversely Isotropic Elastic Media

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.     

Light scattering by a homogeneous sphere near a plane boundary

The scattering of electromagnetic waves by a homogeneous sphere near a plane boundary is presented in this paper. The vector wave equations derived from Maxwell’s equations are solved by means of the two orthogonal solutions to the scalar wave equation. Hankel transformation and Erdélyi’s formula are used to satisfy the planar boundary conditions and the determination of the unknown coefficients in the scattered field and internal fields is achieved by matching the electromagnetic boundary conditions on the surface of the sphere. Existence and uniqueness of the solution of the series involving these unknown coefficients are shown.     

Generalization of the method of functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media

A new approach for constructing functionally invariant solutions for dynamic problems of the plane theory of elasticity of anisotropic media is proposed. Solutions of the equations of motion in displacements and potentials, which express plane waves and waves from a point source, and also complex solutions of a general type are obtained and investigated. The problem of the reflection of plane waves from the boundary of a half-space is solved for comparison with earlier results [1]. The solutions obtained agree with the physical meaning of the problems and with the solutions for isotropic media.     

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.     

Two-dimensional Green’s functions for semi-infinite isotropic thermoelastic plane

Based on the two-dimensional steady-state governing equations of isotropic thermoelastic material and the compact general solution expressed in three harmonic functions, the corresponding three harmonic functions contain nine undetermined constants are constructed for a line heat source applied in the interior of a semi-infinite thermoelastic plane. All components of thermoelastic field in the semi-infinite plane can be derived by substituting the harmonic functions into the general solution. And the undetermined constants can be obtained by the compatibility conditions, equilibrium conditions and the different boundary conditions for extended Mindlin problem and extended Lorentz problem. Thus, the Green’s functions in above two cases are obtained, and the numerical results are given graphically by contours.     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity. Received: June 1, 2005     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity.     

Three-dimensional Tollmien-Schlichting waves generated by sound in the boundary layer on an elastic surface at transonic free-stream velocities

Within the framework of the triple-deck theory, the effect of surface elasticity on three-dimensional packets of Tollmien-Schlichting waves generated by acoustic disturbances induced near the boundary layer at transonic free-stream velocities is investigated. It is shown that the elasticity of the surface considerably weakens the most unstable oblique waves but does not change the characteristic horseshoe shape of wave packets with two disturbance peaks propagating at an angle to the incoming flow.     

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].     

具有两个驰豫时间的热弹性立方晶体材料中平面波的传播

研究具有两个驰豫时间的、两个不同弹性和热性质的、广义传热立方晶体固体半空间的有缺陷结合面上,热弹性平面波的反射和折射问题.具有两个驰豫时间的广义热弹性理论,是1972年由Green和Lindsay提出并应用于问题的研究.对有缺陷边界,给出了反射系数和折射系数(即反射波和折射波振幅与入射波振幅之比)的表达式,并推演了法向刚性边界、横向刚性边界、接触传热边界、滑动边界和结合面边界时的表达式.给出了在不同边界条件及出射角时,不同的反射波和折射波的振幅比,在不同的入射波时的变化图.发现反射波和折射波振幅比受到介质刚性和热性质的影响.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. I: General solutions and fundamental solutions

Four steady-state general solutions are derived in this paper for the two-dimensional equation of isotropic thermoelastic materials. Using the differential operator theory, three general solutions can be derived and expressed in terms of one function, which satisfies a six-order partial differential equation. By virtue of the Almansi’s theorem, the three general solutions can be transferred to three general solutions which are expressed in terms of two harmonic functions, respectively. At last, a integrate general solution expressed in three harmonic functions is obtained by superposing the obtained two general solutions. The proposed general solution is very simple in form and can be used easily in certain boundary problems. As two examples, the fundamental solutions for both a line heat source in the interior of infinite plane and a line heat source on the surface of semi-infinite plane are presented by virtue of the obtained general solutions.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson     

表面效应对偏场下介电高弹体表面波传播的影响

采用表面薄层模型考察偏场下介电高弹体的表面效应,针对不同边界情形,建立一阶等效边界条件.基于有限变形电弹性体的线性增量理论,利用Stroh公式和Ting方法,给出等效边界条件的严格推导过程.进一步利用Stroh公式,获得了偏场下具有表面效应的介电高弹体中表面波的频散方程.以可压缩Neo-Hookean介电高弹体为例,分析了表面效应对预变形和电学偏场作用下的介电高弹体表面波传播特性的影响.结果表明,通过施加适当的偏场,可以调控和优化纳米声表器件的性能.     

Boundary value problems in the theory of binary mixtures of thermoelastic solids

In this paper the diffusion model of the linear theory of thermoelasticity of binary mixtures is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established. The interior boundary value problem (BVP) of steady vibration of binary mixtures of thermoelastic solids is formulated. The existence theorems of eigenfrequencies of the BVP of steady vibrations are proved. The connection between plane waves and existence of eigenfrequencies is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-dependent waves of rayleigh type near the surface of an inhomogeneous elastic body

The well studied, high-frequency Rayleigh waves polarized in a plane normal to a cross section of the surface of an inhomogeneous elastic body with phase speed close to the speed of transverse waves are generalized to the case of the time-dependent equations of elasticity. For the wave field uniform asymptotics are obtained in the form of space-time ray expansions of two types: with a real eikonal (for transverse waves diffracted at the surface) and with a complex eikonal (for a longitudinal wave damped away from the surface).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 168–183, 1986.     

Reflection at the free surface of fiber-reinforced thermoelastic rotating medium with two-temperature and phase-lag

The present investigation is concerned with the reflection of plane waves from the free surface of a homogeneous, anisotropic, fiber-reinforced thermoelastic rotating medium under two-temperature and dual-phase-lag model. It has been observed that three coupled plane waves travel in the medium with distinct speeds. Using appropriate boundary conditions, the amplitude and energy ratios for the reflected waves are derived and the numerical computations have been carried out with the help of MATLAB programming. The numerical values of the modulus of reflection coefficients are presented graphically to exhibit the two-temperature, phase lag and rotation parameter effects. The expressions of energy ratios have also been obtained in explicit form and are shown graphically as functions of angle of incidence. It has been verified that during reflection phenomena, the sum of energy ratios is equal to unity at each angle of incidence.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.