Secular Equation Of Rayleigh Waves Research Articles

Effect of thermal diffusivity on Rayleigh waves in human tissue

This work examines the nature of human tissue under the context of dual phase lag thermoelastic diffusion theory using the principle of propagation of Rayleigh wave along the stress free, thermally insulated/isothermal, and impermeable/isoconcentrated conditions. The governing equations for the tissue are formed by considering the third-order approximation of Taylor’s expansion for the modified Fourier’s law of heat conduction and modified Fick’s law of mass diffusion in terms of phase lag parameters. The irrational secular equations of Rayleigh waves are derived, which are then rationalized to extract their complex roots. The phase speeds and the attenuation coefficients are evaluated from these complex roots. The variation of these phase speeds and attenuation coefficients are plotted graphically against angular frequency for two significant modes. The behavioral patterns of displacement components, temperature, and concentration fields are computed and are expressed graphically. The paths of the particle motion are represented graphically. Some results of published articles are recovered from the present formulation to validate our results.

Rayleigh wave through half space semiconductor solid with temperature dependent properties

The article focuses on Rayleigh wave propagation in a homogeneous isotropic semi-conductor thermoelastic medium rotating with fixed angular frequency with temperature-dependent properties. The elastic constants depend upon the temperature function. The effects of temperature dependency parameter, time derivative, and fractional order are illustrated. By the theory of thermo-elasticity, waves result in the generation of thermal signals that propagate through the medium. A heat conduction model of three-phase lag (3PL) along with fractional order time derivative is used to analyze the thermal signals. The secular equations of Rayleigh waves are derived mathematically at the stress-free, carrier density and thermally insulated boundaries. Some specific properties like velocity, attenuation coefficient, specific heat loss and penetration depth for Rayleigh waves have been evaluated and presented graphically. The secular equations are computed numerically and depicted graphically using Matlab.

An approximate formula for the Rayleigh wave velocity in the machined surface with residual stress

The present paper is concerned with the Rayleigh wave propagation in the machined surface with a thin damaged layer with residual stress. We assume that the layer and the half-space are bonded perfectly to each other. Biot’s theory of small deformations influenced by initial stress forms the basis for this study. With the help of the effective boundary condition method, a third-order approximate secular equation of Rayleigh waves is established for the case that the layer and half-space are both orthotropic. In addition, an explicit third-order approximate formula for the Rayleigh wave velocity is derived from the secular equation. By considering a sample of silicon wafer by fine grinding with fine abrasive grains, the accuracy of the approximate formula has been verified. This study is meant to serve as the mathematical foundation for non-destructive testing of residual stress in the practical applications.

Propagation of two Rayleigh waves in viscoelastic medium: Explicit expressions for complex velocities

AbstractPropagation of Rayleigh waves is considered in an isotropic linear viscoelastic medium. Such propagation is represented through a complex velocity, which lies implicit in an irrational secular equation. A complex analysis technique is used to transform this irrational complex equation into an algebraic equation. A linear equation replaces the secular equation for propagation of the classical Rayleigh wave in viscoelastic medium. A quadratic equation replaces the secular equation for propagation of additional Rayleigh wave, which exists for appropriate value of a complex parameter. Explicit expressions are obtained for the complex velocity, which satisfy the secular equation for Rayleigh waves. Two different analytical expressions for the velocity certify the existence of second Rayleigh wave in some viscoelastic materials with specifically related elastic and dissipative parameters. Numerical examples are solved to analyse the complex velocities and particle motion for the two Rayleigh waves in viscoelastic materials.

Explicit secular equation and H/V ratio formula of Rayleigh waves in orthotropic micropolar elastic half-spaces

The main purpose of this paper is to derive the explicit secular equation and formula of H/V ratio of Rayleigh waves propagating in an orthotropic micropolar elastic half-space. The polarization vector method is used so that the characteristic equation of Rayleigh waves need not to be solved. The obtained secular equation recovers the secular equations of Rayleigh waves propagating in isotropic micropolar elastic and orthotropic (purely) elastic half-spaces. Furthermore, for isotropic micropolar elastic materials of which the micropolar couple modulus is much smaller than the shear elastic modulus, explicit formulas for the velocity and the H/V ratio of Rayleigh waves are derived by utilizing the complex function method. It is shown numerically that the micropolar parameters affect Rayleigh waves weakly in the range of low frequencies (long wavelengths) but considerably in the range of high frequencies. The numerical analysis also indicates that the obtained formulas for the velocity and the H/V ratio are good tools for nondestructively evaluating the micropolar couple modulus using Rayleigh waves of short wavelengths, and if the data of phase velocity and H/V ratio are measured with the same accuracy in practice the H/V ratio formula could be better than phase velocity in inverse problem.

A note on the effect of rigidity and initial stress on the propagation of Rayleigh waves in pre-stressed orthotropic elastic layered medium

This paper contains the secular equation of Rayleigh waves in a prestressed semi-infinite space of orthotropic material which is covered by a prestressed elastic layer of orthotropic material. The semi-infinite space and layer are treated as in spring contact to each other. This paper treats the surface of layer as rigid. The effect of tensile and compressional stress on velocity curve is depicted in form of graphs. Particular cases for perfect and sliding interface is discussed and secular equation is derived. Comparison of velocity curve for welded and spring contacts has been examined graphically. MATLAB programming language is used to plot the graphs. Effect of stiffness constants on velocity of Rayleigh waves has been shown graphically.

The effect of rotation on the propagation of waves in an incompressible transversely isotropic thermoelastic solid

The present paper is concerned with plane and surface wave solutions of the governing equation for an incompressible, homogeneous, transversely isotropic and rotating thermoelastic medium in the context of the Green–Naghdi theory. The velocity equation for homogeneous plane waves is obtained which indicates the existence of two coupled plane shear waves. The secular equation for Rayleigh waves is derived for thermally insulated as well as isothermal boundaries. The non-dimensional speeds of plane waves are plotted against the angle of incidence, non-dimensional anisotropy parameter and rotation/frequency ratio. The non-dimensional speed of Rayleigh surface waves is also computed against the non-dimensional anisotropy parameter and rotation/frequency ratio.

Formulas for the H/V ratio (ellipticity) of Rayleigh waves in orthotropic elastic half-spaces

ABSTRACTThis paper concerns the propagation of Rayleigh waves in compressible and incompressible orthotropic elastic half-spaces. The main aim of the paper is to derive formulas for the H/V ratio (ellipticity) of Rayleigh waves. First, the expression of the H/V ratio in terms of the Rayleigh wave velocity for compressible orthotropic elastic half-spaces is derived using the surface impedance matrix. From this expression and the secular equation of Rayleigh waves, the equation for the H/V ratio is obtained. It is a cubic equation. Employing the theory of cubic equations, the formula for the H/V ratio for compressible orthotropic elastic half-spaces has been derived. The H/V ratio formula for incompressible orthotropic elastic half-spaces has been obtained using the incompressible limit method. Since the H/V ratio of Rayleigh waves is a good tool for nondestructively evaluating mechanical properties of structures before and during loading, the obtained formulas will be very significant in practical applications.

A note on formulas for the Rayleigh wave speed in elastic solids

In the present paper, new analytical, numerical and approximate methods have been presented for the determination of Rayleigh wave speed in isotropic and anisotropic media. The Lagrange’s method is used to provide exact expression for the roots of the secular equation for Rayleigh waves in isotropic media. Then, a simple non-iterative type quadrature method is used to numerically determine the Rayleigh wave speed in isotropic and anisotropic media. Further, an approximate method is presented to determine the velocity of Rayleigh waves. The discrete least square approximation on Chebyshev – Gauss - Lobatto nodes is suggested to transform secular equations to quadratic equations, thereby, providing improved approximations to the Rayleigh wave speed. The analysis is complemented with numerical examples.

Effective boundary condition method and approximate secular equations of Rayleigh waves in orthotropic half-spaces coated by a thin layer

Effective boundary condition method and approximate secular equations of Rayleigh waves in orthotropic half-spaces coated by a thin layer

On a technique for deriving the explicit secular equation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer

The secular equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this secular equation in both papers. This secular equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explicit secular equation of the wave. The technique makes clear the way from the traction-free and continuity conditions to the secular equation and enables us to recognize the misprints appearing in the reported secular equation. The technique can be employed to obtain explicit secular equations of Rayleigh waves for many other cases. Moreover, the paper introduces a transfer matrix in explicit form for an orthotropic layer that is much simpler in form than the one obtained previously.

Exact secular equations of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer

• The propagation of Rayleigh waves in an orthotropic half-space coated by an orthotropic layer is investigated. • The half-space and the layer may be compressible or incompressible and they are in welded contact. • For the compressible/compressible case, the exact secular equation is derived by using the effective boundary condition method. • For three remaining cases, the exact secular equations are obtained by the incompressible limit technique. In this paper, the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary uniform thickness is investigated. The layer and the half-space may be compressible or incompressible and they are in welded contact with each other. The main aim of the paper is to derive explicit exact secular equations of the wave for four possible combinations of a (compressible/incompressible) half-space coated by a (compressible/incompressible) layer. For the compressible/compressible case, the explicit secular equation is derived by using the effective boundary condition method. For three remaining cases, the explicit secular equations are derived from the secular equation for the compressible/compressible case by using the incompressible limit technique along with the expressions of the reduced elastic compliances in terms of the elastic stiffnesses. Based on the obtained secular equations, the effect of incompressibility on the Raleigh wave propagation is considered numerically. It is shown that the incompressibility strongly affects the Raleigh wave velocity and it makes the Raleigh wave velocity increasing.

On a technique for deriving explicit transfer matrices of orthotropic layers

This paper presents a technique by which the transfer matrix in explicit form of an orthotropic layer can be easily obtained. This transfer matrix is applicable for both the wave propagation problem and the reflection/transmission problem. The obtained transfer matrix is then employed to derive the explicit secular equation of Rayleigh waves propagating in an orthotropic half-space coated by an orthotropic layer of arbitrary thickness.

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.

An approximate secular equation of Rayleigh waves in an elastic half-space coated by a thin weakly inhomogeneous elastic layer

In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to depend arbitrarily continuously on the thickness variable. The contact between the layer and the half space is perfectly bonded. The main purpose of the paper is to establish an approximate secular equation of the wave. By applying the effective boundary condition method an approximate secular equation of second order in terms of the dimensionless thickness of the layer is derived. It is shown that the obtained approximate secular equation has good accuracy.

Rayleigh waves with impedance boundary conditions in incompressible anisotropic half-spaces

In this paper, the propagation of Rayleigh waves in an incompressible elastic half-space with impedance boundary conditions is investigated. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane x3=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. For the monoclinic case, the method of polarization vector is used for deriving the secular equation. This is an algebraic equation of eighth-order. When the impedance parameters vanish, the equations obtained coincide with the corresponding secular equations of Rayleigh waves with traction-free boundary conditions.

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. 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.

An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

In this 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 welded. 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 secular equation of fourth order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate secular equation has high accuracy. From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established.

An explicit secular equation of Rayleigh waves propagating along an obliquely cut surface in a directional fiber-reinforced composite

The propagation of Rayleigh waves along an obliquely cut surface in a directional fiber-reinforced composite was studied recently by Ohyoshi [Ohyoshi T. Compos Sci Technol 2000; 60; 2191-6]. The author derived the secular equation of the wave, however, it is still implicit. In this paper, a fully explicit secular equation of the wave is obtained by using the method of first integrals. From it we immediately arrive at the secular equation of the wave for the case when the cut surface is parallel to the fiber direction. This secular equation is much more simple than the ones obtained recently by Cerv [Cerv J. Int Rev Mech Eng (IREME) 2008; 2; 762-72] and Cerv et al. [Cerv J et al. Composite Structures 2010; 92; 568-77]. Based on the obtained secular equations some approximate formulas for the velocity of Rayleigh waves are established and it is shown that they are good approximations. The explicit secular equations and the approximate formulas for the velocity derived in this paper are useful for analyzing the effect of the material properties and the orientation of the fiber direction on the Rayleigh wave velocity, especially they are powerful tools for solving the inverse problem: determining the material parameters from the measured values of the velocity.

An approximate secular equation of Rayleigh waves propagating in an orthotropic elastic half-space coated by a thin orthotropic elastic layer

The present paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space coated with a thin orthotropic elastic layer and 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 secular equation of third-order in terms of the dimensionless thickness of the layer has been derived. From this equation two different third-order approximate secular equations are obtained for the case when the half-space and the layer are both isotropic, one of which recovers the secular equation of second-order derived by Bovik [P. Bovik, A comparison between the Tiersten model and O(H) boundary conditions for elastic surface waves guided by thin layers, J. Appl. Mech. 63 (1996) 162–167]. An explicit second-order approximate formula for the Rayleigh wave velocity has been created based on the obtained approximate secular equation. Since explicit dispersion relations are employed as theoretical bases for extracting the mechanical properties of the thin films from experimental data, the obtained secular equation and formula for the velocity may be useful in practical applications.