Rayleigh surface waves in the theory of thermoelastic materials with voids

In this paper we analyze the behavior of plane harmonic waves and Rayleigh waves in a linear thermoelastic material with voids. We take into account the damped effects of the thermal field upon the propagation waves. Consequently, the propagation condition is established in the form of an algebraic equation of 9th degree whose coefficients are complex numbers while the eigensolutions of the thermoelastodynamic with voids system are explicitly obtained in terms of the characteristic solutions. We show that the transverse waves are undamped in time and they are not influenced by the thermal and porous effects while the longitudinal waves are all damped in time and they are coupled with the thermal and porous effects. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the eigensolutions in concern. The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic and homogeneous thermoelastic with voids half-space. Furthermore, we use the numerical methods and computations to solve the secular equation for a specific material.     

Rayleigh Surface Waves on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Elasto-thermodiffusive (ETNP) surface waves in semiconductor materials

The present article is devoted to investigate the propagation of elasto-thermodiffusive (ETNP) surface waves in a homogeneous isotropic, thermally conducting semiconductor material of half-space with relaxation of heat and charge carrier fields. The secular equation, a more general functional relation, that governs the propagation of elasto-thermodiffusive (ETNP) surface waves in homogeneous isotropic, thermoelastic semiconductor material halfspace with relaxation of heat and charge carrier fields has been derived by solving a system of coupled partial differential equations. A hybrid numerical technique consisting of Descartes algorithm for solving complex polynomial characteristic equation along with functional iteration scheme has been successfully used to solve the secular equation in order to obtain dispersion curves, attenuation coefficient and specific loss factor of energy dissipation for p-type germanium (Ge) semiconductor. Some particular forms of the general secular equation governing the propagation of elasto-thermodiffusive (ETN/ETP), thermoelastic (ET), elastodiffusive (EP/EN) and thermodiffusive (TP/TN) surface waves have been also deduced and discussed. In order to illustrate the analytical development, the numerical solution of the secular equation and other relevant relations under different situations is also carried out for Ge semiconductor materials to characterize the elasto-thermodiffusive (ETP) and thermodiffusive (TP) surface waves. The computer simulated results have been presented graphically in respect of the dispersion curves, attenuation coefficient and specific loss factor.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

This paper is concerned with the propagation of Rayleigh waves in an incompressible isotropic elastic half-space overlaid with a layer of non-viscous incompressible water under the effect of gravity. The authors have derived the exact secular equation of the wave which did not appear in the literature. Based on it the existence of Rayleigh waves is considered. It is shown that a Rayleigh wave can be possible or not, and when a Rayleigh wave exists it is not necessary unique. From the exact secular equation the authors arrive immediately at the first-order approximate secular equation derived by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. When the layer is assumed to be thin, a fourth-order approximate secular equation is derived and of which the first-order approximate secular equation obtained by Bromwich is a special case. Some approximate formulas for the velocity of Rayleigh waves are established. In particular, when the layer being thin and the effect of gravity being small, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers.     

Free vibration of microstretch thermoelastic plate with one relaxation time

The propagation of waves in 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 Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skew-symmetric wave mode propagation have been obtained. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical computations are found to be in close agreement.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

Propagation of Rayleigh waves on free surface of transversely isotropic generalized thermoelastic diffusion

The present paper is devoted to the study of Rayleigh wave propagation in a homogeneous, transversely isotropic, thermoelastic diffusive half-space, subject to stress free, thermally insulated/isothermal, and chemical potential boundary conditions in the context of the generalized thermoelastic diffusion theory. The Green-Lindsay（GL） theory is used in the study. In this theory, thermodiffusion and thermodiffusion mechanical relaxations are governed by four different time constants. Secular equations for surface wave propagation in the considered media are derived. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient are graphically presented in order to present the analytical results and make comparison. Some special cases of frequency equations are derived from the present investigation.     

Rayleigh Waves on an Exponentially Graded Poroelastic Half Space

In this paper we consider the propagation of seismic waves in isotropic poroelastic half spaces with continuously varying elastic properties, namely with an exponentially decaying depth profile. The present paper shows that the problem leads naturally to a bicubic equation. We obtain explicit inhomogeneous plane wave solutions in an exponential evanescent form with respect to the depth of half space. Further, these solutions are used to solve the boundary value problem of a Rayleigh surface wave and the secular equation is established. The results obtained theoretically are exemplified for numerical data and represented graphically for a representative poroelastic material.     

Rapid Sliding Indentation with Friction of a Pre-Stressed Thermoelastic Material

A rigid indentor travels with a constant speed over the surface of an isotropic thermoelastic half-space. Friction exists between the indentor and half-space, and the latter is initially in equilibrium at a uniform temperature under a uniform normal pre-stress. This pre-stress, below but near yield, is assumed to produce deformations that dominate the additional deformations due to indentation. Thus, the process is treated as one of small deformations superposed upon (relatively) large. The governing equations for the superposed deformation are those of nonisotropic coupled thermoelasticity. A steady-state two-dimensional study uses robust asymptotic analytical solutions to reduce the associated mixed boundary value problem to a classical singular integral equation which can be solved analytically. The solutions show that the pre-stress-induced de facto nonisotropy alters the values of the rotational and dilatational wave and Rayleigh speeds in the half-space and, in the case of a compressive pre-stress, generates a second, lower, critical speed. In addition, pre-stress generates noncritical sliding speeds at which the friction-dependent integral equation eigenvalue changes sign. For purposes of illustration, expressions for the half-space surface temperature change and its average over the contact zone, the equations necessary to determine contact zone size and location, the resultant moment on the indentor, and the maximum compressive stress on the contact zone are presented for a parabolic indentor. This revised version was published online in August 2006 with corrections to the Cover Date.     

Surface waves in a stretched and sheared incompressible elastic material

In this paper, we analyze the effect of a combined pure homogeneous strain and simple shear in a principal plane of the latter on the propagation of surface waves for an incompressible isotropic elastic half-space whose boundary is normal to the glide planes of the shear. This generalizes previous work in which, separately, pure homogeneous strain and simple shear were considered. For a special class of materials, the secular equation is obtained in explicit form and then specialized to recover results obtained previously for the two cases mentioned above. A method for obtaining the secular equation for a general form of strain–energy function is then outlined. In general, this is very lengthy and the result is not listed, but, for the case in which there is no normal stress on the half-space boundary, the result is given, for illustration, in respect of the so-called generalized Varga material. Numerical results are given to show how the surface wave speed depends on both the underlying pure homogeneous strain and the superimposed simple shear. Further numerical results are provided for the Gent model of limiting chain extensibility.     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

Propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space

The purpose of this research is to study the effect of voids on the surface wave propagation in a layer of a transversely isotropic thermoelastic material with voids lying over an isotropic elastic half-space. The frequency equation is derived after developing a mathematical model for welded and smooth contact boundary conditions. The dispersion curves giving the phase velocity and attenuation coefficient via wave number are plotted graphically to depict the effects of voids and anisotropy for welded contact boundary conditions. The specific loss and amplitudes of the volume fraction field, the normal stress, and the temperature change for welded contact are obtained and shown graphically for a particular model to depict the voids and anisotropy effects. Some special cases are also deduced from the present investigation.     

Rayleigh waves in orthotropic fluid-saturated porous media

In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the secular equation of the wave but that secular equation is still in implicit form. The main aim of this paper is to derive explicit secular equation of the wave. By employing the method of polarization vector, the secular equations of Rayleigh waves in explicit form is obtained. This equation recovers the dispersion equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the secular equation obtained is not a complex equation as the one derived by Liu and Liu, it is a really real equation.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Rolling without Slip on a Transversely Isotropic Thermoelastic Half-space

Rolling without slip by a rigid cylinder on a transversely isotropic, coupled thermoelastic half-space at constant subcritical speed is studied. The cylinder is of infinite length, surface heat convection is neglected, and a dynamic steady state of plane strain is treated. The unmixed problem of traction applied to a translating surface strip is addressed first. A robust asymptotic form of the exact transform solution, valid when Fourier heat conduction dominates any thermal relaxation effect, is extracted, and inverted analytically. Use of material characterization and identification of parameters that vanish in the isotropic limit or are invariant under an isothermal–thermoelastic transformation result in compact full-field solutions. These expressions are used to construct analytical solutions that satisfy the mixed boundary value problem and auxiliary conditions of rolling contact. For the hexagonal material zinc, calculations are made for contact zone width and temperature increases near onset of zone yield. Mathematics Subject Classifications (2000) 73B30, 73C25, 73C30, 73C35.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.