Rayleigh-type Wave Propagation Research Articles

Rayleigh-type wave in non-planar pre-stressed nearly incompressible elastic structure

The primary objective of the current work is to establish the impacts of planar boundary and non-planar boundary on the displacement components (DCs) of Rayleigh-type wave propagation in an elastic structure. The elastic structure, precisely, incorporates a pre-stressed nearly incompressible elastic material in the half-space along with the non-planar boundary at the free surface. The methodology applied features variable separable technique and Taylor’s series expansion along with substitution method to establish the expressions for DCs and frequency relation. The frequency relation of Rayleigh-type wave is obtained for the situation considering planar free surface, which is further reduced as a special case of the study and validated with the classical result present in the literature. The DCs of Rayleigh-type wave have been numerically computed, and their reliance on wave number have been exhibited graphically. In addition to this, the influences of affecting parameters viz., hydrostatic stress, principal Cauchy stress and curvature parameter on DCs have been traced out. The computational result also manifests the domain of existence of Rayleigh-type wave propagating in the considered model in the presence of planar free surface.

The Propagation of Rayleigh-Type Waves in Nonlocal Inhomogeneous Transversely Isotropic Half-Space

This paper deals with propagation of Rayleigh-type waves in nonlocal inhomogeneous transversely isotropic half-space. From the characteristic equation of wave for the nonlocal inhomogeneous transversely isotropic medium, the existence of the number of surface waves depends on the inhomogeneity of the medium through the number of solutions satisfying the damping condition of the characteristic equation discussed. It has also been concluded that there exist cut-off frequency and escape frequency for the wave propagating in size-dependent materials based on the nonlocal theory. Dispersion equation for the propagation of Rayleigh-type surface waves at the free surface has been derived. Based on the obtained dispersion equation, the effects of the inhomogeneity of material and nonlocality parameter on the Rayleigh wave propagation are considered through some numerical examples.

A nonlocal strain gradient theory for wave propagation analysis in transversely isotropic elastic medium

This paper has the objective of studying the propagation of surface waves in a transversely isotropic medium based on nonlocal strain gradient theory (NSGT). A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory and classical theory. It has also been concluded that there exist escape frequency and cut-off frequency for the wave propagating in size-dependent materials based on the NSGT. Dispersion equation for the propagation of Rayleigh-type waves at the free surface has been derived. The effect of the nonlocal parameter, the strain gradient parameter on the Rayleigh wave propagation is considered through some numerical examples.

Analysis of higher modes of Rayleigh-type waves in fiber-piezo thermoelastic composite structure

The study presents a theoretical modeling and analysis for the purpose of designing of acoustic devices and sensors with high performance. The propagation of Rayleigh-type waves in a piezoelectric substrate guided with fiber-reinforced layer is studied. Lord-Shulman (L-S) theory is used for thermoelasticity for both layer and substrate. Direct Sturm-Liouville method is employed to obtain the dispersion relation using traction free, thermally insulated, electrically open and short boundary conditions. The effect of different material parameters (e.g., reinforce, thickness ratio, and amplitude of thermal displacement) on the velocity (phase and group) of the considered wave are shown graphically by taking suitable numerical examples. The effect of specific loss factor is presented through graph. The present study may be utilized for better optimization of surface acoustic wave devices and sensors.

Rayleigh type wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer

This paper is concerned with the Rayleigh wave dispersion in an incompressible functionally graded orthotropic half-space loaded by a thin fluid-saturated aeolotropic porous layer under initial stress. Both the layer and half-space have subjected to the incompressible in nature. The particle motion of the Rayleigh type wave is elliptically polarized in the plane, which described by the normal to the surface and the focal point along with wave generation. The dispersion of waves refers typically to frequency dispersion, which means different wavelengths travel at a different velocity of phase. To deal with the analytical solution of displacement components of Rayleigh type waves in a layer over a half-space, we have taken the assistance of different methods like exponential, characteristic polynomial and undetermined coefficients. The dispersion relation has been derived based upon suitable boundary conditions. The finite difference scheme has been introduced to calculate the phase velocity and group velocity of the Rayleigh type waves. We also have derived the stability condition of the finite difference scheme (FDS) for the phase and group velocities. If a wave equation has to travel in the time domain, it is necessary to achieve both accuracy and stability requirements. In such cases, FDS is preferred because of its power, accuracy, reliability, rapidity, and flexibility. The effect of various parameters involved in the model like non-homogeneity, porosity, and internal pre-stress on the propagation of Rayleigh type waves have been studied in detail. Graphical representations for the effects of various parameters on the dispersion equation have been represented. Numerical results demonstrated the accuracy and versatility of the group and phase velocity depending on the stability ratio of the FDS.

Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space

This study is devoted to investigate the propagation of Rayleigh-type waves in transversely isotropic nonlocal piezoelastic half-space. When the stress-free boundary is maintained at charge-free condition, the dispersion equation for the propagation of Rayleigh waves at the free surface of transversely isotropic piezoelastic solids has been obtained. Based on the obtained dispersion equation, the effect of the nonlocality on the speed of Rayleigh wave is numerically considered. The dependence of velocities of plane waves in transversely isotropic nonlocal piezoelastic medium on the direction of propagation as well as non-dimensional frequency parameter has been also illustrated.

Dispersion of Rayleigh-Type Wave in an Exponentially Graded Incompressible Crustal Layer Resting on Yielding Foundation

The objective of this study is to develop a theory to study the propagation of Rayleigh-type waves in an inhomogeneous layer having yielding surface. A detailed study of a Rayleigh-type wave propagating in an exponentially graded incompressible layer resting on yielding surface is considered. The frequency equation being a function of phase velocity, wave number and heterogeneity parameter associated with the yielding parameter and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by the above-stated parameters. In particular cases, the dispersion relation has been discussed for stress-free foundation by taking yielding parameter tending to zero. In numerical and graphical computation, the significant effects of distortional velocity have been carried out. Moreover, the obtained dispersion relation is found in well agreement to the classical case in homogeneous isotropic layer resting on a yielding foundation.

Impact of inhomogeneous fiber-reinforced layer with frictional interface on Rayleigh-type wave propagation

The effect of frictional boundary on the propagation of Rayleigh-type wave in an initially stressed inhomogeneous fiber-reinforced layer overlying an initially stressed homogeneous semi-infinite medium has been analyzed by an approximate analytical method. A realistic model has been considered for sliding boundary friction at the interface. The frequency equation has been obtained in closed form. The substantial effects of various affecting parameters, viz. reinforcement, inhomogeneity, bonding parameter, spectral decay parameter, and horizontal initial stress on phase and damped velocity have been discussed graphically in detail. The remarkable observation has been obtained through the comparative study in the presence and the absence of reinforcement in the layer.

Effect of Loose Bonding and Corrugated Boundary Surface on Propagation of Rayleigh-Type Wave

THE PROBLEMS CONCERNS TO THE PROPAGATION OF SURFACE WAVE PROPA-GATION THROUGH VARIOUS ANISOTROPIC MEDIUMS WITH INITIAL STRESS AND IRREGULAR BOUNDARIES ARE OF GREAT INTEREST TO SEISMOLOGISTS, DUE TO THEIR APPLICATIONS TOWARDS THE STABILITY OF THE MEDIUM. THE PRESENT PAPER DEALS WITH THE PROPAGATION OF RAYLEIGH-TYPE WAVE IN A CORRU-GATED FIBRE-REINFORCED LAYER LYING OVER AN INITIALLY STRESSED ORTHO-TROPIC HALF-SPACE UNDER GRAVITY. THE UPPER FREE SURFACE IS ASSUMED TO BE CORRUGATED; WHILE THE INTERFACE OF THE LAYER AND HALF-SPACE IS CORRUGATED AS WELL AS LOOSELY BONDED. THE FREQUENCY EQUATION IS DEDUCED IN CLOSED FORM. NUMERICAL COMPUTATION HAS BEEN CARRIED OUT WHICH AIDS TO PLOT THE DIMENSIONLESS PHASE VELOCITY AGAINST DIMENSIONLESS WAVE NUMBER FOR SAKE OF GRAPHICAL DEMONSTRATION. NUMERICAL RESULTS ANALYZE THE INFLUENCE OF CORRUGATION, LOOSE BOND-ING, INITIAL STRESS AND GRAVITY ON THE PHASE VELOCITY OF RAYLEIGH-TYPE WAVE. MOREOVER, THE PRESENCE AND ABSENCE OF CORRUGATION, LOOSE BONDING AND INITIAL STRESS IS ALSO DISCUSSED IN COMPARATIVE MANNER.

Rayleigh-type wave propagation in incompressible visco-elastic media under initial stress

Propagation of Rayleigh-type surface waves in an incompressible visco-elastic material over incompressible visco-elastic semi-infinite media under the effect of initial stresses is discussed. The dispersion equation is determined to study the effect of different types of parameters such as inhomogeneity, initial stress, wave number, phase velocity, damping factor, visco-elasticity, and incompressibility on the Rayleigh-type wave propagation. It is found that the affecting parameters have a significant effect on the wave propagation. Cardano’s and Ferrari’s methods are deployed to estimate the roots of differential equations associated with layer and semi-infinite media. The MATHEMATICA software is applied to explicate the effect of these parameters graphically.

Rayleigh-type wave propagation on a transversely isotropic viscoelastic layer with yielding and rigid foundations

ABSTRACTThe present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.

Influence of Rigid, Stress-Free and Yielding Base of a Composite Structure on the Propagation of Rayleigh-Type Wave: A Comparative Approach

AbstractIn this paper, case wise studies have been made to investigate the possibility of propagation of Rayleigh-type wave in a composite structure comprised of two transversely-isotropic material layers with viscoelastic effect. The common interface between the layers is considered to be rigid whereas the base has been considered as rigid, stress-free and yielding in three different cases (Case-I, II and III). Closed-form of frequency equation and damped velocity equation has been established analytically for propagation of Rayleigh-type wave in a composite structure for all three cases. In special cases, frequency equations and damped velocity equations for the case of composite structure with rigid, stress-free and yielding base have been found in well-agreement to the established standard results pre-existing in the literature. Numerical and graphical computation of phase and damped velocity of Rayleigh-type wave propagating in the composite structure comprised of double transversely-isotropic viscoelastic Taylor sandstone material layers (Model-I) and double isotropic viscoelastic material layers (Model-II) have been carried out. Significant effect of anisotropy and width ratio of layers, dilatational and volume viscoelasticity associated with viscoelasticity of layer medium and yielding parameter associated with yielding base of composite structure on phase and damped velocities of Rayleigh-type wave for the considered models have been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the present problem.

Influence of yielding base and rigid base on propagation of Rayleigh-type wave in a viscoelastic layer of Voigt type

The present study aims to study the propagation of Rayleigh-type wave in a layer, composed of isotropic viscoelastic material of Voigt type, with the effect of yielding base and rigid base in two distinct cases. With the aid of an analytical treatment, closed-form expressions of phase velocity and damped velocity for both the cases are deduced. As a special case of the problem it is found that obtained results are in good agreement with the established standard results existing in the literature. It is established through the study that volume-viscoelastic and shear-viscoelastic material parameter and yielding parameter have significant effect on phase and damped velocities of Rayleigh-type wave in both the cases. Numerical calculations and graphical illustration have been carried out for both the considered cases in the presence and the absence of viscoelasticity. A comparative study has been performed to analyse the effect of layer with yielding base, traction-free base and rigid base on the phase and damped velocities of Rayleigh-type wave.

Rayleigh-type wave propagation through liquid layer over corrugated substrate

The propagation of the Rayleigh-type wave in a fluid layer overlying a corrugated substrate is studied. The corrugated substrate is considered as a fluid saturated poroelastic substrate and a quadratically heterogeneous isotropic elastic substrate in Case I and Case II, respectively. Closed form expressions of dispersion relation for Case I and Case II are obtained. The influence of corrugation, porosity, and heterogeneity on the phase velocity of Rayleigh-type wave, for both cases, is highlighted and demonstrated through numerical computation and graphical discussion. Neglecting corrugation at the common interface, expressions of phase velocity of the Rayleigh-type wave for both cases are derived in a closed form as a special case of the problem. Comparison between the presence and the absence of both heterogeneity and poroelasticity in the substrate of the composite structure is a key in the present study.

Propagation of Rayleigh type wave in an initially Stressed Voigt Type Viscoelastic Layer

In this paper the study of propagation of Rayleigh type wave in a homogeneous initially stressed isotropic viscoelastic layer of Voigt-type resting on a rigid base has been dealt. The closed form expression of phase velocity is deduced analytically and obtained result is found to be in well agreement with the pre-established result in a special case. Numerical computation has been carried out for the derived dispersion equation to analyse the effect of viscoelastic parameters and initial stress on phase velocity of Rayleigh type wave. It is observed that viscoelastic parameters and initial stress parameter have significant effect on the phase velocity of Rayleigh-type wave which has been illustrated graphically.

Influence of Heterogeneity and Initial Stress on the Propagation of Rayleigh-type Wave in a Transversely Isotropic Layer

In this paper we study the propagation of Rayleigh-type wave in a heterogeneous transversely isotropic elastic layer with initial stress resting on a rigid foundation. Frequency equation is obtained in closed form. The frequency equation being a function of phase velocity, wave number, initial stress and heterogeneous parameter associated with the rigidity and density of inhomogeneous layer reveals the fact that Rayleigh-type wave propagation is greatly influenced by above stated parameters. In Numerical and graphical computation the significant effects of distortional velocity have been carried out. Moreover, the obtained dispersion relation is found in well–agreement to the classical case in isotropic and transversely isotropic layer resting on a rigid foundation.

Effect of reinforcement, gravity and liquid loading on Rayleigh-type wave propagation

In the present time, self-reinforced materials are the basic requirement for civil engineering construction. These constructions encounter surface wave propagation during earthquakes and similar disturbances. Therefore, the study of surface wave propagation in such material is of great significance. The present paper, in the light of practical situation like dams, canals etc. aims to study the effect of gravity on the propagation of Rayleigh-type surface wave in a self-reinforced semi-infinite medium bounded and loaded by an inviscid liquid layer. Secular equation for the propagation of Rayleigh-type wave has been derived in closed form. The effect of gravity, reinforcement and liquid loading on phase velocity of Rayleigh-type wave has been distinctly observed. Numerical computation has been carried out and graphical illustration is provided for analysis of the presented study. Moreover, the effect of reinforced semi-infinite medium is compared to the effect of reinforced free semi-infinite medium on the phase velocity of Rayleigh-type surface wave to unravel the reinforcement effect. The effect of presence and absence of liquid loading on self-reinforced semi-infinite medium for Rayleigh-type wave propagation is also analysed and depicted by means of graphs.

A simple method for simulating microseism H/V spectral ratio in 3D structure

The understanding of detailed 3D basin structure is very important for seismic hazard analysis because ground shaking can be both amplified and prolonged due to basin effects and causing significant damage. Background seismic noise is ubiquitous and continuous and these characteristics make ambient seismic noise useful to study velocity structure because neither infrequent earthquakes nor expensive explosions are needed. The horizontal to vertical (H/V) spectral ratio method is a common technique to study the structure of the basin using background seismic noise. Many observations of the H/V ratio recorded at seismic stations over basins display systematic decreases in the frequency of the dominant H/V peak with increasing basin thickness. However, some observations cannot be fully explained by theoretical simulations of the H/V ratio based on 1D velocity profiles beneath the stations. In order to study the precise relationship between the H/V ratio and the basin structure, wavefield simulation of background noise for 3D velocity structure can be quite useful. However, this simulation is still far from being routine because our knowledge of the noise source is quite poor. In this paper, we propose a simple technique to directly simulate the H/V ratio instead of the wavefield of background noise. To evaluate the method, we performed numerical experiments for the Santa Clara Valley and the results show that the synthetics can predict the observations well. We also performed several sensitivity tests for the source and the velocity structure of the basin and found that Rayleigh type wave propagation and a minimum velocity contrast between the basin and background media are required to make the H/V ratio sensitive to the basin structure. Although we also found that the H/V ratio technique has a limitation in determining small scale basin structure at deeper depth, this technique can still be used to evaluate pre-existing velocity models and give some constraints on the development of new velocity models for the basins.

Displacement approach for generalized Rayleigh waves in layered solid-fluid media : Bull Seismol Soc AmV79, N4, Aug 1989, P1251–1263

Abstract A finite-element method for the propagation of Rayleigh-type waves in layered solid-fluid media is presented. The method uses displacement as the only parameter to describe the solid and the fluid motions. A penalty function method and a selective integration technique are employed to change the elastic coefficient matrix of fluid from singular to nonsingular. Results obtained from the proposed method agree well with those obtained from available analytical solutions. The method can be used to study the propagation of Rayleigh-type waves in a two-dimensional, irregular solid-fluid system.

Displacement approach for generalized Rayleigh waves in layered solid-fluid media

AbstractA finite-element method for the propagation of Rayleigh-type waves in layered solid-fluid media is presented. The method uses displacement as the only parameter to describe the solid and the fluid motions. A penalty function method and a selective integration technique are employed to change the elastic coefficient matrix of fluid from singular to nonsingular. Results obtained from the proposed method agree well with those obtained from available analytical solutions. The method can be used to study the propagation of Rayleigh-type waves in a two-dimensional, irregular solid-fluid system.