Rayleigh surface waves problem in linear thermoviscoelasticity with voids

In this paper, we study the propagation of the Rayleigh surface waves in a half-space filled by an exponentially functionally graded thermoviscoelastic material with voids. We take into consideration the dissipative character of the porous thermoviscoelastic models upon the propagation waves and study the damped in time wave solutions. The propagation condition is established in the form of an algebraic equation of tenth degree whose coefficients are complex numbers. The eigensolutions of the dynamical system are explicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surface wave problem is expressed as a linear combination of the five analytical solutions, while the secular equation is established in an implicit form. The explicit secular equation is obtained for an isotropic and homogeneous thermoviscoelastic porous half-space, and some numerical simulations are given for a specific material.     

Rayleigh surface wave propagation in an orthotropic rotating magneto-thermoelastic medium subjected to gravity and initial stress

Abstract

The prime objective of the present article is to analyze the effects of rotation and initial stress on the propagation of Rayleigh surface waves in a homogeneous, orthotropic magneto-thermoelastic half space subjected to gravity field. The frequency equations in closed form are derived and the amplitude ratios of surface displacements, temperature change during the Rayleigh wave propagation on the surface of half space have been computed analytically. The highlights of this study are the effects of different parameters (rotation, magnetic field, initial stress, and gravity) on the velocity of Rayleigh waves. Variation in phase velocity of Rayleigh waves against a wave number is shown graphically. Some particular cases have been deduced. Also, the classical Rayleigh wave equation is obtained as a special case of the present study. Numerical example has been carried out and represented by the means of graphs. Impacts of various involved parameters appearing in the solutions are carefully analyzed. In fact, in the absence of various parameters, these equations are in agreement with the results for isotropic medium.     

Rayleigh waves at the boundary surface of modified couple stress generalized thermoelastic with mass diffusion

In this problem, we have studied propagation of Rayleigh waves in an homogeneous isotropic modified couple stress generalized thermoelastic with mass diffusion solid half space in the context of Lord–Shulman (L-S), Green–Lindsay (G-L) theories of thermoelasticity. Secular equations are derived mathematically by using appropriate boundary conditions. The values of determinant of secular equation, Rayleigh wave velocity and attenuation coefficient with respect to angular velocity for different values of wave number and relaxation times in the absence and presence of mass diffusion, are computed numerically. The numerical simulated results are depicted graphically for copper material.     

Rayleigh waves in Cosserat elastic materials

The present paper gives explicit solutions for surface waves propagation in a homogeneous half space filled with an isotropic Cosserat elastic material. Such solutions are important in the study of seismic waves in an earthquake, supposing that the bottom land is modeled as having a microstructure. To construct explicit expressions for the possible surface waves under consideration, we use the Stroh formalism. These solutions are further used to study the Rayleigh waves and to give the explicit equation for the Rayleigh surface wave speed (secular equation). Numerical calculations and graphics corresponding to the analytical solution are given for aluminium-epoxy composite.     

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.     

Behavior of Rayleigh surface waves in an anisotropic media lying over a prestressed orthotropic half-space

The present paper discusses the propagation of Rayleigh waves in an anisotropic layer with finite thickness lying over a prestressed orthotropic half-space. An anisotropic media and orthotropic media are supposed for the upper layer and lower half-space, respectively. Dispersion equation and displacement components are computed in a compact form considering the case that the displacement and stress are continuous at the interface and stress vanishes on a free surface. Graphs are sketched to represent the effect of density, initial stress and height of the layer on wave velocity. The graphs are also configured to exhibit the mode of propagation of Rayleigh waves. This paper is an attempt to explain the nature of Rayleigh waves mathematically.     

Rayleigh waves in magneto-thermoelastic solids with thermal relaxation

Following a linear theory of magneto-thermo-elasticity with thermal relaxation, the propagation of Rayleigh waves in a semi-infinite body permeated by an uniform magnetostatic field parallel to the boundary surface is investigated. It is assumed that the elastic medium under consideration is a homogeneous, isotropic, electrically and thermally conducting one. The roots of the frequency equation are calculated numerically. The approximate solution for small thermoelastic and magnetoelastic coupling is obtained and compared with the exact solution.     

Propagation of rayleigh waves in transversely isotropic generalized thermoelastic diffusion

This paper is devoted to the study of propagation of Rayleigh waves in a homogeneous, transversely isotropic, thermoelastic diffusive half-space that is subjected to stress-free, thermally insulated/isothermal, and chemical potential boundary conditions in the context of the generalized theory of thermoelastic diffusion. The Lord and Shulman theory, where thermal and thermomechanical relaxation as well as diffusion relaxation are governed by two different time constants, is selected. Secular equations for surface wave propagation in the considered media are derived. The amplitudes of surface displacements, temperature change, and concentration are computed. The paths of the surface particles are determined. Transverse isotropy and diffusion effects on the phase velocity, group velocity, and attenuation coefficient are presented graphically.     

Existence of longitudinal and transverse waves in anisotropic thermoelastic media

Four waves propagate in an anisotropic thermoelastic medium. The fastest among them is a quasi-longitudinal wave. The slowest of them is a thermal wave. The remaining two are called quasi-transverse waves. The prefix ‘quasi’ refers to their polarizations being nearly, but not exactly, parallel or perpendicular to the direction of propagation. The polarizations of these four waves are not mutually orthogonal. Hence, unlike anisotropic elastic media, the existence of a longitudinal wave may not imply the existence of a transverse wave, by default. The existence of a purely longitudinal wave in an anisotropic thermoelastic medium is ensured by the stationary characters of three expressions. These expressions involve components of phase direction with elastic (stiffness and coupling) and thermal coefficients of the thermoelastic medium. The existence of a purely transverse wave is ensured by the two equations restricting the choice of thermoelastic (stiffness and coupling) coefficients. The existence of longitudinal and transverse waves along the coordinate axes and in the coordinate planes are discussed for general anisotropy. The discussion is extended to orthotropic materials, and the existence of pure phases is explored along few specific phase directions.     

Analysis of free vibrations for Rayleigh — Lamb waves in a microstretch thermoelastic plate with two relaxation times

The propagation of free vibrations in a microstretch thermoelastic homogeneous isotropic plate subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Green and Lindsay (G—L) theories of thermoelasticity. The secular equations for the microstretch thermoelastic plate in closed form for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. At short wavelength limits, the secular equations for both modes in a stress-free thermally insulated and isothermal homogeneous isotropic microstretch thermoelastic plate reduce to the Rayleigh surface wave frequency equation. The results for symmetric and skew-symmetric wave modes are computed numerically and presented graphically. The theory and numerical computations are found to be in close agreement. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 36–46, January–February, 2009.     

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.     

Viscously damped acoustic waves with the lattice Boltzmann method

Acoustic wave propagation in lattice Boltzmann Bhatnagar-Gross-Krook simulations may be analysed using a linearization method. This method has been used in the past to study the propagation of waves that are viscously damped in time, and is here extended to also study waves that are viscously damped in space. Its validity is verified against simulations, and the results are compared with theoretical expressions. It is found in the infinite resolution limit k→0 that the absorption coefficients and phase differences between density and velocity waves match theoretical expressions for small values of ωτ(ν), the characteristic number for viscous acoustic damping. However, the phase velocities and amplitude ratios between the waves increase incorrectly with (ωτ(ν))(2), and agree with theory only in the inviscid limit k→0, ωτ(ν)→0. The actual behaviour of simulated plane waves in the infinite resolution limit is quantified.     

Coupled Tamm waves guided by an isotropic and homogeneous dielectric layer in a rugate filter

The propagation of Tamm waves (also known as Bloch surface waves) guided by a layer of homogeneous dielectric material in a rugate filter was studied theoretically. A canonical boundary-value problem was set up and a dispersion equation was obtained. The solution of the dispersion equation indicated the existence of coupled modes of Tamm waves that were absent when the homogeneous dielectric material was taken to occupy a half space. The spatial profiles of the electric and magnetic fields of Tamm waves showed that Tamm waves propagate localized to either one or both interfaces. Multiple Tamm waves may lead to multi-analyte chemical sensors and multi-channel communication.     

Effect of surface stresses on surface waves in elastic solids

This paper deals with the propagation of surface waves in homogeneous, elastic solid media whose free surfaces or interfaces of separation are capable of supporting their own stress fields. The general theory for the propagation of surface waves in a medium which supports surface stresses is first deduced, and then this theory is employed to investigate the particular cases of surface waves, viz. (a) Rayleigh waves, (b) Love waves and (c) Stoneley waves. It is seen that the Rayleigh waves become dispersive in nature; and, in case of low frequency with residual surface tension, a critical wavelength exists, below which the propagation of Rayleigh waves is not possible. This critical wave length is directly proportional to the surface tension. Some numerical calculations have been made in the case of Love waves and conclusions have been drawn.     

Singular surfaces in linear thermoelasticity

In this paper we give a detailed account, within the framework of the linear theory of thermoelasticity, of the propagation of surfaces of discontinuity in a homogeneous, isotropic elastic solid which is able to conduct heat. The methods used in the investigation are, in large measure, due to T. Y. Thomas. The early sections of the paper contain a derivation of the principal results of Thomas's theory which enables us to determine, from a consideration of the appropriate Cauchy initial-value problem, the characteristic surfaces of the linear thermoelastic equations. The wavefronts associated with these characteristics are found to propagate with one of the constant speeds

, ₀, E_T, v_T being respectively the density, the isothermal Young's modulus and the isothermal Poisson's ratio of the material in its reference state.

A discontinuity surface of order r in the displacement and temperature fields is referred to as a weak thermoelastic wave if r2 and a strong thermoelastic wave if r=0 or 1. Concerning the properties of these waves our main conclusions are as follows. Weak thermoelastic waves and strong waves of order 1 are characteristic and may be described as dilatational or rotational according as their speed of propagation is v_T or v_S. Dilatational strong waves of order 1 are shock waves and rotational waves of this type are propagating vortex sheets. For all thermoelastic waves of order 1 the strength (defined in a natural way) is completely determined by its distribution on an initial configuration of the wavefront. Irrespective of the shape of this initial configuration, the strength of a dilatational wave decays rapidly as the wave propagates on account of thermoelastic dissipation. For rotational waves, however, the variation of strength during propagation depends solely upon the geometrical form of the initial wavefront. A strong thermoelastic wave of order 0 is an absolute singular surface in the temperature field, discontinuities of displacement being excluded from consideration. A wave of this type may be characteristic, in which case its speed of propagation is v_S; or it may be non-characteristic, in which case it is a dilatational shock wave. In neither case is the strength of the wave completely determined by its distribution on an initial wavefront, a situation which leads us to argue that thermoelastie waves of order 0 cannot in practice be created.

In the final section of the paper the properties of singular surfaces in classical elastokinetics are discussed in the light of the foregoing analysis of discontinuous thermoelastic waves.     

Stress-sensitivity mapping for surface acoustic waves on quartz

A model is presented, relating the velocity shifts of surface acoustic waves (SAW) to the six tensor components of quasistatic stresses. Stress sensitivity is then defined through six independent coefficients, whatever the origin of the stress (direct external forces, thermoelastic stresses) might be. These coefficients, depending on crystal anisotropy, are computed for different cut angles and propagation directions of quartz crystal, and represented as a contour-line mapping. The determination of SAW quartz cuts compensated for both planar isotropic stresses and first-order temperature effects make it possible to define a family of quartz cuts with potentially low stress and temperature sensitivities for oscillator applications.     

Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space

This paper presents a theoretical study on propagation of torsional surface waves in a homogeneous viscoelastic isotropic layer with Voigt type viscosity over an inhomogeneous isotropic infinite half space. The non-homogeneity in half space is assumed to arise due to exponential variation in shear modulus and density. A closed-form solution has been obtained for the displacement in the layer as well as for a infinite half space. The dispersion and absorption relations for an torsional wave under the assumed geometry have been found. Numerical results are presented for propagation characteristics in terms of a number of non-dimensionalized parameters and have been produced graphically. This study investigates the effect of various parameters, namely non-homogeneity parameter, internal friction, the layer width and complex wave number on dissipation function and phase velocity of the torsional wave. Results in some special cases are also compared with existing solutions available from analytical methods, which show a close agreement.     

Damping of loaded Rayleigh waves on lightly loaded interfaces: Application to the Kapitza conductance

We present two methods for solving the Stoneley equation for lightly loaded surfaces, which we apply to interfaces between, on the one hand, copper and quartz, and, on the other hand, liquid and solid ⁴He, solid hydrogen, deuterium, and neon. We look for solutions with velocities near the Rayleigh wave on the free surface of the heavy medium. The methods for solving the equation are, respectively, a first-order expansion near the Rayleigh velocity and a computer iteration. We show that both methods give nearly identical results for the damping, and therefore the first-order approximation gives a useful analytical expression for the damping of loaded Rayleigh waves. The approximate result for the velocity increment is poor. By applying the results to the Kapitza conductance problem we are able to put forward an explanation for some of the experimental features, and to make suggestions for further experimental and theoretical work.     

A unified formalism using effective surface permittivity to study acoustic waves in various anisotropic and piezoelectric multilayers

A unified formalism is presented that uses the effective surface permittivity (ESP) to study surface acoustic waves (SAW) in layered substrates and guided waves in layered plates. Based on known mathematical tools, such as ordinary differential equation and transfer matrix, a generalized surface impedance (GSI) concept is developed and exploited to investigate the acoustic propagation in various anisotropic and piezoelectric layered structures. The ESP function, originally defined for the surface of a homogeneous and semi-infinite piezoelectric substrate, is extended to both the top surface of and an interface in a layered half space, as well as to either surface of a finite-thickness plate. General ESP expressions for all mentioned configurations are derived in terms of an equivalent GSI matrix. It is shown that, when using the appropriate GSI matrices, the same form of the ESP expressions applies no matter whether the structure is a homogeneous half space alone or coated with a layered plate or a layered plate alone. GSI matrices are explicitly given in terms of the bulk partial mode solutions for a substrate and via the transfer matrix for a plate. Modified GSI matrices for structures consisting of both a plate and a substrate are also specified. Analytical development is fully detailed to suit program implementation. To illustrate its versatility, the formalism is also applied to two-substrate configurations, allowing one to analyze guided waves in a plate sandwiched between and interfacial waves existing along the boundary of two different media. Numerical examples are given to illustrate the spectrum features that the ESP shows for various structures. Deduced ESP expressions allow one to locate directly all piezoelectrically active waves in any structure including at least one piezoelectric layer. Acoustic modes that are not piezoelectrically active and those in non-piezoelectric materials can be also obtained by using the intermediate results, such as derived GSI matrices.     

Rayleigh-Lamb Waves in Transversely Isotropic Thermoelastic Diffusive Layer

Propagation of plane harmonic thermoelastic diffusive waves in a homogeneous, transversely isotropic, thin elastic layer of finite width is studied, in the context of the theory of coupled thermoelastic diffusion. According to the characteristic equation, three quasi-longitudinal waves, namely, quasi-elastodiffusive (QED) mode, quasi-mass diffusion (QMD) mode, and quasi-thermodiffusive (QTD) mode can propagate in addition to quasi-transverse waves (QSV) mode and the purely quasi-transverse motion (QSH) mode, which is not affected by thermal and diffusion vibrations, gets decoupled from the rest of the motion of wave propagation. The secular equations corresponding to the symmetric and skew symmetric modes of the layer are derived. The amplitudes of displacements, temperature change, and concentration for symmetric and skew symmetric modes of vibration of the layer are computed numerically. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient, and amplitudes of displacements, temperature change, and concentration are presented graphically in order to illustrate and compare the results analytically. Some special cases of the frequency equation are also deduced and compared with the existing results.