Love wave dispersion in pre-stressed homogeneous medium over a porous half-space with irregular boundary surfaces

The paper investigates the existence of Love wave propagation in an initially stressed homogeneous layer over a porous half-space with irregular boundary surfaces. The method of separation of variables has been adopted to get an analytical solution for the dispersion equation and thus dispersion equations have been obtained in several particular cases. Propagation of Love wave is influenced by initial stress parameters, corrugation parameter and porosity of half-space. Velocity of Love waves have been plotted in several figures to study the effect of various parameters and found that the velocity of wave decreases with increases of non-dimensional wave number. It has been observed that the phase velocity decreases with increase of initial stress parameters and porosity of half-space.     

Torsional surface waves in heterogeneous anisotropic half-space under initial stress

The present paper is concerned with the propagation of torsional surface waves in a heterogeneous anisotropic half-space under the initial compressive stress. The heterogeneity in the half-space is caused by the linear variation in rigidity, initial compressive stress and density. The solution part of the problem involves the use of Whittaker function. The dispersion equation has been obtained in a closed form, which shows the variation of phase velocity with corresponding wave number. Effects of anisotropy and initial stress have been shown by the means of graphs for different anisotropic materials. It has found that the phase velocity of torsional waves decreases with increment in initial stress and inhomogeneity. Obtained phase velocity of torsional surface wave is found to be less than the shear wave velocity, which agrees with the standard result.     

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.     

Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress

The present work deals with the possibility of propagation of torsional surface wave in fluid saturated poroelastic layer lying over nonhomogeneous elastic half space. Both the media are assumed to be under compressive initial stress. The half space has two types of inhomogeneity, viz; hyperbolic and quadratic. The dispersion equation for torsional wave in porous layer has been derived and observed that the presence of fluid in pores increases the velocity of the torsional surface wave but the phase velocity diminishes due to the presence of compressive initial stress in the porous layer. It is also observed that the velocity of the torsional surface wave increases due to the increase of initial stress in inhomogeneous half space. The inhomogeneity factor due to quadratic and hyperbolic variations in rigidity, density and initial stress of the medium decreases the phase velocity as it increases.     

Propagation of torsional waves in an inhomogeneous layer over an inhomogeneous half-space

The present paper is concerned with the study of propagation of torsional waves in an inhomogeneous isotropic layer whose material properties vary harmonically with a space variable, lying over a semi-infinite inhomogeneous isotropic half-space. The closed form solutions for the displacement in the layer and half-space are obtained separately. The dimensionless phase velocity has been plotted against dimensionless wave number and scaled wave number for different values of inhomogeneity parameters. The effects of inhomogeneity have been shown in the dispersion curves using 2D and 3D plot.     

Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space

The paper studies the propagation of Love waves in a non-homogeneous substratum over an initially stressed heterogeneous half-space. The dispersion equation of phase velocity is derived. The velocities of Love waves are calculated numerically as a function of kH and presented in a number of graphs, where k is the wave number, and H is the thickness of the layer. The case of Gibson’s half-space is also considered. It is observed that the speed of Love waves is finite in the vicinity of the surface of the half-space and vanishes as the depth increases for a particular wave number. It is also observed that an increase in compressive initial stresses causes decreases of Love waves velocity for the same frequency, and the tensile initial stress of small magnitude in the half-space causes increase of the velocity.     

Dispersion and attenuation of torsional wave in a viscoelastic layer bonded between a layer and a half-space of dry sandy media

Propagation of a torsional wave in a doubly-layered half-space structure of an initially stressed heterogeneous viscoelastic layer sandwiched between a layer and a half-space of heterogeneous dry sandy media is studied. A closed form complex expression for the velocity profile is obtained under effective boundary conditions. The real part of the complex expression provides a dispersion equation, and the imaginary part yields a damping equation. The derived dispersion and damped equations are in well agreement with the classical Love wave condition. In addition, to study the effect of the dissipation factor, the attenuation coefficient, the sandy parameters, the initial stress, the heterogeneity parameters, and the thickness ratio parameter, some noteworthy contemplations are made by numerical calculations and graphical visuals. The results of this paper may present a deeper insight into the behaviour of propagation phenomena in heterogeneous viscoelastic and heterogeneous dry sandy materials that can provide a theoretical guide for the design and optimization in the field of earthquake engineering. The study also reveals that the presence of a damping part due to viscoelasticity affects the torsional wave propagation significantly.     

G-type seismic waves in fibre reinforced media

This paper investigates the possibility of shear wave propagation along the plane surface in the interface of two different types of fibre reinforced media. The upper layer is fibre reinforced and the lower half-space is taken inhomogeneous fibre reinforced. Dispersion equation and condition for maximum energy flow near the surface are obtained in compact form. The dispersion equation coincides with that of Love wave for uniform media. Effect of reinforcement and inhomogeneity on phase and group velocity has been depicted by means of graphs. It is observed that inhomogeneity and reinforcement decreases the phase velocity and presence of reinforcement deviate the group velocity.     

SH-waves in viscoelastic heterogeneous layer over half-space with self-weight

The effect of gravity, heterogeneity and internal friction on propagation of SH-waves (horizontally polarised shear waves) in viscoelastic layer over a half-space has been studied. Using the method of separation of variables, dispersion equation has been obtained and used to recover the damped velocity of SH-waves. Both the real and imaginary parts of dispersion equation are in well agreement with the classical Love wave equation. It has been observed that heterogeneity of the medium affects the velocity profile of SH-wave significantly. Some other peculiarities have been observed and discussed in our study.     

Propagation of SH waves in an irregular monoclinic crustal layer

The present paper discusses the dispersion equation for SH waves in a monoclinic layer over a semi-infinite elastic medium with an irregularity. In the absence of the irregularity, the dispersion equation reduces to standard dispersion equation for SH waves in a monoclinic layer over an isotropic semi-infinite medium. The dispersion curves for different size of the irregularity are computed and compared for the half-space without any irregularity. It can be seen that the phase velocity is strongly influenced by the wave number and the depth of the irregularity.     

各向异性粘弹性多孔截止中平面波的传播

This study discusses wave propagation in perhaps the most general model of a poroelastic medium. The medium is considered as a viscoelastic, anisotropic and porous solid frame such that its pores of anisotropic permeability are filled with a viscous fluid. The anisotropy considered is of general type, and the attenuating waves in the medium are treated as the inhomogeneous waves. The complex slowness vector is resolved to define the phase velocity, homogeneous attenuation, inhomogeneous attenuation, and angle of attenuation for each of the four attenuating waves in the medium. A non-dimensional parameter measures the deviation of an inhomogeneous wave from its homogeneous version. An numerical model of a North-Sea sandstone is used to analyze the effects of the propagation direction, inhomogeneity parameter, frequency regime, anisotropy symmetry, anelasticity of the frame, and viscosity of the pore-fluid on the propagation characteristics of waves in such a medium.     

Effect of loosely bonded undulated boundary surfaces of doubly layered half-space on the propagation of torsional wave

This paper investigates the propagation of torsional wave in an initially stressed poroelastic layer with corrugated as well as loosely bonded boundary surfaces, sandwiched between a corrugated fiber-reinforced layer and a viscoelastic half-space under initial stress. The velocity equation has been obtained in closed form analytically and the substantial effect of affecting parameters on the phase velocity of torsional surface wave has been demonstrated numerically and graphically. Comparative study has been made to observe the effect of flatness parameter, reinforcement, viscoelasticity and porosity on the phase velocity, meticulously. Some particular cases have also been discussed and it is found that velocity equation is in well-agreement to the classical Love wave equation. Moreover, some remarkable observation has been made through numerical computation and graphical demonstration for fiber-reinforced layer of carbon fiber-epoxy resin, poroelastic layer of sandstone and a viscoelastic half-space.     

Love wave propagation in heterogeneous micropolar media

The present paper framed to study the impact of heterogeneity on propagation of Love wave in a heterogeneous micropolar layer over an elastic inhomogeneous stratum, when both rigidity and density are assumed to vary linearly with depth. The equations of motion have been formulated separately for layer and half-space under suitable boundary conditions. Analytical solution for the dispersion equation has been obtained using method of separation of variables by means of the Airy function and Whittaker function. Some particular cases have also been investigated. Further, as a special case the velocity equation for isotropic layer over a homogeneous half-space coincides with the standard result of Love wave. Numerical calculations of frequency relation have been performed and depicted by means of graphs to exhibit the substantial impact of heterogeneity, micropolar parameters and wave number on the phase velocity of Love wave. The wave velocity is strongly influenced by these parameters.     

Influence of linearly varying density and rigidity on torsional surface waves in inhomogeneous crustal layer

The present work deals with the possibility of propagation of torsional surface wave in an inhomogeneous crustal layer over an inhomogeneous half space. The layer has inhomogeneity which varies linearly with depth whereas the inhomogeneous half space exhibits inhomogeneity of three types, namely, exponential, quadratic, and hyperbolic discussed separately. The dispersion equation is deduced for each case in a closed form. For a layer over a homogeneous half space, the dispersion equation agrees with the equation of the classical case. It is observed that the inhomogeneity factor due to linear variation in density in the inhomogeneous crustal layer decreases as the phase velocity increases, while the inhomogeneity factor in rigidity has the reverse effect on the phase velocity.     

梯度介质半空间Rayleigh面波传播性能研究

对剪切弹性模量沿深度以指数函数变化的非均质半空间，本文用摄动法得到了Rayleigh面波的波函数解答及相速度方程。以不同金属与陶瓷复合而成的几种梯度材料为例，用数值方法求解了相速度方程，给出了相应的波的弥散曲线，结果表明，梯度介质半空间自由表面附近的Rayleigh波通常有两种不同的弥散形式，即正常弥散和非正常弥散。     

Dispersion in oceanic crust during earthquake preparation

Dispersion of Stoneley waves is studied in a sedimentary layer of ocean bottom resting over basaltic solid half space. Sedimentary layer is assumed a transversely isotropic poroelastic medium. Lower-most solid half-space is assumed to be embedded with vertically aligned saturated micro-cracks and behaves transversely isotropic to wave propagation.Frequency equation is obtained in the form of determinantal equation. Role of phase angle is eliminated by expressing slowness of waves in terms of phase velocity and elastic constants. Numerical solutions for phase velocity and group velocity are obtained for a particular model. Calculations are made for different depths of ocean and sediments. Effect of thickness and density of cracks on these velocities are observed.Special cases are discussed which represent the absence of ocean and sediments, in the model considered. Changes in dispersion are discussed during the stress accumulation in an earthquake preparation region.     

Propagation behavior of Love waves in a functionally graded half-space with initial stress

The propagation behavior of Love waves in a functionally graded material layered non-piezoelectric half-space with initial stress is taken into account. The Wentzel–Kramers–Brillouin (WKB) technique is adopted for the theoretical derivations. The analytical solutions are obtained for the dispersion relations and the distributions of the mechanical displacement and stress along the thickness direction in the layered structure. First, these solutions are used to study the effects of the initial stress on the dispersion relations and the group and phase velocities, then the influences of the initial stress on the distributions of the mechanical displacement and shear stresses along the thickness direction are discussed in detail. Numerical results obtained indicate that the phase velocity of the Love waves increases with the increase in the magnitude of the initial tensile stress, while decreases with the increase in the magnitude of the initial compression stress. The effects on the dispersion relations of the Love wave propagation are negligible as the magnitudes of the initial stress are less than 100 MPa. Some other results are obtained for the distributions of field quantities along thickness direction. The results obtained are not only meaningful for the design of functionally graded structures with high performance but also effective for the evaluation of residual stress distribution in the layered structures.     

径向非均匀介质中圆形夹杂的动应力分析

基于复变函数理论,研究了径向非均匀弹性介质中均匀圆夹杂对弹性波的散射问题. 介质的非均匀性体现在介质密度沿着径向按幂函数形式变化且剪切模量是常数. 利用坐标变换法将变系数的非均匀波动方程转为标准亥姆霍兹(Helmholtz) 方程. 在复坐标系下求得非均匀基体和均匀夹杂同时存在的位移和应力表达式. 通过具体算例分析了圆夹杂周边的动应力集中系数(DSCF). 结果表明:基体与夹杂的波数比和剪切模量比,基体的参考波数和非均匀参数对动应力集中有较大的影响.      

Inhomogeneous waves at the boundary of a generalized thermoelastic anisotropic medium

The Christoffel equation is derived for the propagation of plane harmonic waves in a generalized thermoelastic anisotropic (GTA) medium. Solving this equation for velocities implies the propagation of four attenuating waves in the medium. The same Christoffel equation is solved into a polynomial equation of degree eight. The roots of this equation define the vertical slownesses of the eight attenuating waves existing at a boundary of the medium. Incidence of inhomogeneous waves is considered at the boundary of the medium. A finite non-dimensional parameter defines the inhomogeneity of incident wave and is used to calculate its (complex) slowness vector. The reflected attenuating waves are identified with the values of vertical slowness. Procedure is explained to calculate the slowness vectors of the waves reflected from the boundary of the medium. The slowness vectors are used, further, to calculate the phase velocities, phase directions, directions and amounts of attenuations of the reflected waves. Numerical examples are considered to analyze the variations of these propagation characteristics with the inhomogeneity and propagation direction of incident wave. Incidence of each of the four types of waves is considered. Numerical example is also considered to study the propagation and attenuation of inhomogeneous waves in the unbounded medium.     

The propagation of initially plane waves in nonhomogeneous viscoelastic media

In this study, the propagation of an initially plane wave in a linear isotropic nonhomogeneous viscoelastic medium, where the nonhomogeneity varies transversely to the direction of propagation, is investigated. For this purpose, first the propagation of waves in a linear isotropic viscoelastic medium of arbitrary inhomogeneity is studied by employing the notion of singular surfaces. The characteristic equation governing wave velocities, and the growth and decay equations describing the change of the strength of the discontinuity as the wave front moves are obtained.In the second part of this work, the propagation of initially plane waves is studied for three types of inhomogeneities by employing the findings established in the first part. The first kind of inhomogeneity considered is of axisymmetrical type where the wave propagation velocity depends on the radial coordinate only, increasing linearly up to a certain radial distance and remaining constant thereafter. The second kind is also axisymmetrical with a wave velocity distribution decreasing linearly till a given value of the radial coordinate. In the third one, the wave velocity is assumed to vary linearly over a given interval along a certain coordinate axis only, which is perpendicular to the direction of propagation, and remain constant outside. The ray and wave front analyses are carried out and the decay or growth of stress and velocity discontinuities are studied for each of the three cases.