Velocity Of Torsional Surface Waves Research Articles

Scale effects on the torsional surface waves propagation in an initially stressed dissipative nanoplate

The effects of small length scale on the torsional surface waves propagating in a pre-stressed dissipative nanoplate were examined in this paper. Biot’s incremental deformation principle as well as nonlocal elasticity theory were implemented to form the governing and frequency equations of torsional surface waves in nanoplate under considered conditions. In this work, we proposed a new approach to determine the phase velocity of torsional disturbances. Different categories are used to investigate the dispersion relations such as: wave number, scale coefficient, initial stress parameter, and dissipation coefficient. The results show that small length scale influences causes torsional dispersion of the wave propagation, which disappear in local elasticity theory. In addition, it is recognized that the torsional surface waves propagating in the pre-stressed dissipative nanoplates can be differentiated in a clear manner from those propagated in the absence of small length scale effects. The initial stress parameter can also be utilized to change the characteristics of the torsional surface wave's phase velocity in nanoplates. It is also observed that, the dissipation has a great effect on the phase velocity of torsional surface waves in propagating in the nanoplate. Therefore, this research is theoretically useful to convey information on nonlocal and dissipation properties of the nanomaterials: for example, through a precise measurement of the phase velocity of torsional surface wave propagates in a nanocomposite material by presence of the medium’s dissipation. This work could be useful in understanding the way of designing the Nano devices and Nano resonators.

Propagation of Torsional Surface Waves in Heterogeneous Half-Space with Irregular Free Surface

This study is an attempt to show the impacts of free surface irregularity on the torsional surface waves propagating in heterogeneous, elastic half-space. The surface irregularity is taken in the parabolic form at the surface of the half-space. The governing equation and corresponding closed form solutions are derived. Then, the phase velocity of torsional surface waves is obtained analytically and the influences of surface irregularity are studied in detail. Numerical results analyzing the torsional surface waves propagation are discussed and presented graphically. The analytical solutions and numerical results reveal that, the surface irregularity and heterogeneity have notable effects on the torsional surface waves propagation in the elastic half-space. Since the Earth crust is heterogeneous medium with irregular surface, thus it is important to consider the effects of heterogeneity and surface irregularity on velocity of torsional surface waves propagating in the Earth medium.

Torsional Waves in a Fiber Composite Medium at a Loosely Bonded Interface Constrained Between Dry Sandy Layer and Gravitating Poroelastic Substrate

The objective of this paper is to study the effect of loosely bonded interface on torsional surface wave propagation in a fiber reinforced composite medium constrained between dry sandy layer and an anisotropic gravitating poroelastic substrate. All the media are assumed to be under initial stress. The dispersion relation on this proposed multilayer ground structure has been derived in closed form under certain boundary conditions, which contain Whittaker function and its derivative, which is further expanded asymptotically, retaining up to only the linear terms. The numerical solution for the limiting case of torsional surface waves is also discussed. As a special case of the problem, when the entire medium is isotropic and one of the upper layer vanishes and removing the initial stress and gravity, the dispersion relation obtained is in agreement with the classical Love type wave equation. The influence of various technical constants, such as sandy parameter, reinforcement parameter, porosity parameter, Biot’s gravity parameter, loosely bonded parameters, initial stress of both the layers and half spaces on the phase velocity of torsional surface wave has been pointed out by means of graphs.

Propagation of Torsional Surface Waves in a Nonhomogeneous Half-Space With Circular Irregularity in Free Surface

The present paper studies the effect of circular regularity on propagation of torsional surface waves in an elastic non-homogeneous half-space. Both rigidity and density of the half-space are assumed to vary inversely linearly with depth. Separation of variable method has been used to get the analytical solutions for the dispersion equation of the torsional surface waves. Also, the effects of non-homogeneity and irregularity on the phase velocity of torsional surface waves have shown graphically.

Analysis of Torsional Waves in a Prestressed Composite Structure with Loosely Bonded and Corrugated Boundaries

A mathematical model is presented to describe the propagation of torsional surface waves in a corrugated loosely bonded orthotropic layer sandwiched between two initially stressed viscoelastic half-spaces. The dispersion relation in a closed form is obtained for the analytical model. It is found that the initial stress, hydrostatic stress, viscoelasticity, and the bonding and flatness parameters have a great effect on the phase velocity of torsional surface waves. The method of separation of variables is employed to obtain an analytical solution in the present study. Some particular cases are discussed, and it is found that the results obtained well agree with the classical Love wave equation. Numerical simulations have also been performed to show results of the present analytical study graphically.

Scattering of torsional surface waves in a three layered model structure

In this article, a comparative study has been made to investigate the scattering behaviour of three layered structure model on torsional surface wave. For such model intermediate layer is taken as fiber reinforced composite, resting over a dry sandy Gibson substratum and underlying by different anelastic media. We consider two distinct mediums for topmost layer. In the first case, topmost layer has been taken as fluid saturated homogeneous porous layer, while in the second case the fluid saturated porous layer has been replaced by a transversely isotropic layer. Simple form expression for the secular equation of torsional surface wave has been worked out in both the cases by executing specific boundary conditions, which comprises Whittaker\'s function and its derivative, for imminent result that have been elaborated asymptotically. Some special cases have been constituted which are in excellent compliance with recorded literatures. For the sake of comparative study, numerical estimation and graphical illustration have been accomplished to identify the effects of the width ratio of the layers, Biot\'s gravity parameter, sandy parameter, porosity parameter and other heterogeneity parameters corresponding to the layers and half spaces, horizontal compressive and tensile initial stress on the phase velocity of torsional surface wave.

Torsional Wave Propagation in a Pre-Stressed Structure with Corrugated and Loosely Bonded Surfaces

AbstractAn analytical model is presented to study the behaviour of propagation of torsional surface waves in initially stressed porous layer, sandwiched between an orthotropic half-space with initial stress and pre-stressed inhomogeneous anisotropic half-space. The boundary surfaces of the layer and halfspaces are taken as corrugated, as well as loosely bonded. The heterogeneity of the lower half-space is due to trigonometric variation in elastic parameters of the pre-stressed inhomogeneous anisotropic medium. Expression for dispersion relation has been obtained in closed form for the present analytical model to observe the effect of undulation parameter, flatness parameter and porosity on the propagation of torsional surface waves. The obtained dispersion relation is found to be in well agreement with classical Love wave equation for a particular case. The cases of ideally smooth interface and welded interface have also been analysed. Numerical example and graphical illustrations are made to demonstrate notable effect of initial stress, wave number, heterogeneity parameter and initial stress on the phase velocity of torsional surface waves.

Mathematical Modeling of Torsional Surface Wave Propagation in a Non-Homogeneous Transverse Isotropic Elastic Solid Semi-Infinite Medium Under a Layer

The present paper deals with the mathematical modeling of the propagation of torsional surface waves in a non-homogeneous transverse isotropic elastic half-space under a rigid layer. Both rigidities and density of the half-space are assumed to vary inversely linearly with depth. Separation of variable method has been used to get the analytical solutions for the dispersion equation of the torsional surface waves. Also, the effects of nonhomogeneities on the phase velocity of torsional surface waves have been shown graphically. Also, dispersion equations have been derived for some particular cases, which are in complete agreement with some classical results.

Influence of initial stress and gravity on torsional surface wave in heterogeneous medium

The present paper investigates the effect of gravity and initial stress on the propagation of torsional surface waves in heterogeneous medium. The dispersion equation has been obtained for rigid and traction free boundaries in terms of Whittaker function. The present study reveals that torsional surface waves can propagate in both the cases. In this paper, we assume the expansion of Whittaker function up to linear term. It is observed that, in presence of initial stress and gravity for both rigid and traction free boundary, the phase velocity of torsional surface waves increases with the growth of rigidity. In both the cases, it has also been noticed that as the gravity increases, the phase velocity of the torsional surface waves decreases in presence of heterogeneity and initial stress. It is concluded that in presence of traction free boundary, the phase velocity of the torsional surface waves is more than the rigid boundary.

Influence of corrugated boundary surface and reinforcement of fibre-reinforced layer on propagation of torsional surface wave

These days fibre-reinforced materials are frequently used in construction sector for example in dams, bridges etc. Also the earth structure and artificial structure made by human may contain irregularity or corrugation, therefore, propagation of waves and vibrations through these structures gets affected by them. Motivated by these facts the present problem aims to study the propagation of torsional surface wave in a fibre-reinforced layer with corrugated boundary surface overlying an initially stressed transversely isotropic half-space. The closed form of the dispersion equation has been deduced and the notable effect of reinforcement, undulatory parameter of corrugated boundary surfaces of the layer, corrugation parameter of upper and lower boundary surfaces of the layer, initial stress acting in half-space and wave number on the phase velocity of torsional surface wave has been exhibited. The numerical computation along with graphical illustration has been carried out for fibre-reinforced layer of carbon fibre-epoxy resin and T300/5208 graphite/epoxy material for the transversely isotropic half-space. As a special case of the problem, deduced dispersion equation is found in well-agreement with the classical Love wave equation. Comparative study for reinforced and reinforced free layer has been performed and also depicted graphically. Moreover some analysis is made to highlight the important peculiarities of the problem.

Propagation of a torsional surface wave in a non-homogeneous anisotropic layer over a heterogeneous half-space

This paper aims to study the dispersion of torsional surface waves in a non-homogeneous anisotropic layer over heterogeneous half-space. We consider the inhomogeneity varies exponentially with depth in the layer and in half-space three types of heterogeneities, namely, quadratic, hyperbolic and exponential are assumed. The dispersion equation has been deducted for each case in a closed form by means of variable separable method. It has been observed that for homogeneous isotropic upper layer over a homogeneous half-space, the velocity of torsional surface waves coincides with that of Love waves. Dispersion curves are plotted for different variation in inhomogeneity parameters. The effects of the medium characteristics on the propagation of torsional surface waves are discussed.

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.

Effect of Rigid Boundary on the Propagation of Torsional Surface Waves in Initially Stressed Gravitating Dry Sandy Medium

The paper aims to study, the effect of rigid boundary on the propagation of torsional surface waves in initially stressed gravitating dry sandy half space. The mathematical analysis of the problem has been dealt with the Whittaker function. Assuming the expansion of the Whittaker function up to linear term, it is concluded that the gravity field will always allow torsional waves to propagate. The expansion of the Whittaker function up to quadratic terms shows two such wave fronts may exist in the medium. Finally, it is concluded that the sandy medium without support of a gravity field cannot allow the propagation of torsional surface waves, where as the presence of gravity field always supports the propagation of torsional surface waves regardless of whether the medium is elastic or dry sandy. Numerical results analyzing the velocity equations are discussed and presented graphically. It has been observed that density, rigidities play an important role for the propagation of torsional surface waves. It has been seen that as the non-homogeneity parameter in the layer increases, the velocity of torsional surface wave also increases. It has also been observed that an increase in compressive initial stresses decreases the velocity of torsional surface wave.

Propagation of torsional surface waves in a double porous layer lying over a Gibson half space

The present study deals with the behavior of torsional surface waves when they propagate through an inhomogeneous fluid saturated porous double layers over a dry sandy inhomogeneous Gibson half space. The inhomogeneities of the porous layers are taken as quadratic and exponential variation with depth in the density, elastic moduli and initial stress. In the half space it varies linearly in the elastic moduli and initial stress. For simplicity of the problem, we have used the separation of variable technique. The dispersion equation has been derived with boundary conditions and solved by an iterative method (Newton Raphson method). We have also converted our dynamical equations into non-dimensional form. It has been observed from the numerical validation of the proposed model that the presence of the initial stress and inhomogeneity of both media affect significantly the phase velocity of torsional surface waves. The effect of initial stress, inhomogeneity parameters, depth ratio, sandy parameter, Biot׳s gravity parameter and porosity of the layer on the dimensionless phase velocity of the torsional surface waves are demonstrated graphically with respect to the non-dimensional wave number kH1 (where k is the wave number and H1 is the thickness of the second porous layer).

Propagation of Torsional surface waves in an inhomogeneous anisotropic fluid saturated porous layered half space under initial stress with varying properties

We study the behavior of torsional surface waves when they propagate through inhomogeneous fluid saturated porous layer over a homogeneous porous half space. The layer has three types of inhomogeneity, viz; linear, quadratic and exponential, varies with depth as rigidity, density and initial stress. We assume both media under compressive initial stresses and the analysis is based on the Biot’s theory. The effect of inhomogeneity of the layer in the propagation of torsional surface waves have been studied. The dispersion equations are derived for each case and solved by an iterative method (Newton–Raphson method). It is observed from the numerical calculation that the presence of initial stress and inhomogeneity of the medium affect significantly to the phase velocity of torsional surface waves. Also, propagation of torsional surface waves depend upon the medium in which they propagate as torsional surface waves propagate fastly in presence of elastic half space in comparison to porous half space.

Torsional Surface Waves in an Inhomogeneous Layer over a Fluid Saturated Porous Half-Space

ABSTRACTIn the present paper the propagation of torsional surface waves is discussed in an inhomogeneous elastic layer lying over a fluid saturated porous half space. The inhomogeneity in rigidity and density in the inhomogeneous layer plays an important role in the propagation of torsional surface waves. The presence of fluid in the pores diminishes the velocity. Further, it is seen that if the layer becomes homogeneous and the porous half space is replaced by a homogeneous half space, the velocity of the torsional surface waves coincides with that of Love wave. The effect of inhomogeneity factors and porosity factor on the phase velocity of torsional surface wave is delimitated by means of graphs.

Propagation of torsional wave in a composite layer overlying an anisotropic heterogeneous half-space with initial stress

In this paper we have discussed the propagation of torsional wave in a composite layer overlying an anisotropic heterogeneous half space with initial stress. The method of separation of variables is applied to find the dispersion equation. Numerical results analyzing the dispersion equation are discussed and presented by means of graphs. The effect of reinforcement, inhomogeneity parameter and the initial stress on the propagation of torsional surface wave is the main objective in our study. It has been observed that the directional rigidities of the half space have a favoring effect on the phase velocity, whereas the density and initial stress parameter in the half-space have the adverse effect on the velocity of torsional surface wave. It is remarkable to quote that reinforcement prevails in the medium, the phase velocity increases significantly.

Torsional surface wave propagation in an initially stressed non-homogeneous layer over a non-homogeneous half-space

It is of great interest to study torsional surface wave propagation in an initially stressed non-homogeneous layer over a non-homogeneous half-space. The method of separation of variables is applied to find the displacement field. It is well known in the literature that the earth medium is not at all initial stress free and homogeneous throughout, but it is initially stressed and non-homogeneous. Keeping these things in mind, we have discussed propagation of torsional surface wave in an initially stressed non-homogeneous layer over a non-homogeneous half-space. It has been observed that the inhomogeneity parameter and the initial stress play an important role for the propagation of torsional surface wave. It has been seen that as the non-homogeneity parameter in the layer increases, the velocity of torsional surface wave also increases. Similarly as the non-homogeneity parameter in the half-space increases, the velocity of torsional surface wave increases. The initial stresses P present in the inhomogeneous layer also have effect in the velocity of propagation. It has been observed that an increase in compressive initial stresses decreases the velocity of torsional surface wave.

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.

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.