Propagation Of Torsional Waves Research Articles

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).

Torsional wave in an inhomogeneous prestressed elastic layer overlying an inhomogeneous elastic half-space under the effect of rigid boundary

An investigation has been carried out for the propagation of torsional surface waves in an inhomogeneous prestressed layer over an inhomogeneous half space when the upper boundary plane is assumed to be rigid. The inhomogeneity in density, initial stress (tensile and compressional) and rigidity are taken as an arbitrary function of depth, where as for the elastic half space, the inhomogeneity in density and rigidity is hyperbolic function of depth. In the absence of heterogeneities of medium, the results obtained are in agreement with the same results obtained by other relevant researchers. Numerically, it is observed that the velocity of torsional wave changes remarkably with the presence of inhomogeneity parameter of the layer. Curves are compared with the corresponding curve of standard classical elastic case. The results may be useful to understand the nature of seismic wave propagation in geophysical applications.

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.

Propagation of torsional surface wave in sandy layer sandwiched between a non-homogeneous and a gravitating anisotropic porous semi-infinite media

The present paper attempts to study the propagation of torsional surface waves in a pre-stressed dry sandy layer sandwiched by a pre-stressed non-homogeneous semi-infinite medium and an anisotropic porous half-space under gravity. The non-homogeneity in the non-homogeneous layer is caused by the quadratic variation in rigidity, density and initial stress. The inhomogeneity has been assumed as hyperbolic variation in the dry sandy layer. The dispersion equation of motion has been derived under suitable boundary conditions in a close form, which shows the variation of phase velocity of the corresponding wave number. The velocities of torsional 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 sandy layer. The effect of inhomogeneity in the elastic modulus of rigidity, pre-stress, density, gravity and porosity depicted by means of graph, using MATLAB software. The study reveals that the torsional waves propagate in the pre-stress dry sandy layer sandwiched between the considered media. This work may be useful to understand the nature of seismic wave propagation during earthquakes.

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 Surface Wave in an Anisotropic Porous Medium over a Dry Sandy Half-Space

AbstractThe present paper deals with the propagation of torsional surface waves in an anisotropic porous layer over an inhomogeneous dry sandy half-space. The shear modulus of elasticity and density has been considered as variable in a porous layer, and the modulus of rigidity remains variable in a sandy half-space. The Whittaker function and its derivative give the result of the phase velocity equation using suitable boundary conditions. The effect of a dimensionless wave number on a dispersion curve has been found numerically, and a graphical representation of porosity and sandiness has been given. The derived dispersion equation found that the velocity of a torsional wave is influenced because of the presence of porosity, inhomogeneity, and sandy half-space. The study may be useful to understand the nature of seismic-wave propagation during earthquakes.

Torsional wave propagation in an inhomogeneous crustal layer over an inhomogeneous elastic half space

AbstractThe present study considers the propagation of torsional wave in an inhomogeneous crustal layer over an inhomogeneous half space when the upper boundary plane is assumed to be rigid. The inhomogeneity in density and rigidity in the crustal layer varies linearly with depth, whereas for the elastic half space, the inhomogeneity in density and rigidity is hyperbolic. The dispersion relation for propagation of such waves is derived by using Whittaker’s function. When the upper boundary plane is assumed to be free and in the absence of inhomogeneity of the crustal layer and lower half space, our derived equation is in agreement with the general equation for Love waves. Numerically, it is observed that the velocity of torsional wave changes remarkably with the presence of inhomogeneity parameter of the layer.

Propagation of Torsional Waves in a Fiber Composite Layer Lying over an Initially Stressed Viscoelastic Half-Space

AbstractThe present study investigates the possibility of torsional surface wave propagation in a fiber composite layer lying over an initially stressed viscoelastic half-space. The closed-form expression for the dispersion relation and damping equation has been obtained. Viscoelasticity of the lower half-space, reinforcement, wave number, and horizontal compressive/tensile initial stress acting in an initially stressed lower half-space have substantial effect on the dispersion curve. For the sake of comparative study, numerical computation and graphical demonstration have been carried out by considering some of the special cases of the problem in addition to the problem itself. A remarkable finding is that reinforcement in the superficial layer favors more the phase velocity and damped velocity of a torsional surface wave as compared with the reinforced free superficial layer. As a special case of the problem, it is found that the obtained dispersion relation is well in agreement with the classical Love ...

Propagation of torsional wave in a non-homogeneous crustal layer over a dry sandy mantle

The present paper is to study the propagation of torsional surface wave in a non-homogeneous crustal layer over a pre-stressed dry sandy half-space. In the layer directional rigidities and density are assumed to vary quadratically with depth whereas rigidity, density and pre-stress vary linearly in the lower half-space. The dispersion equation is deducted in a closed form by means of separation of variables method in the form of Whittaker function. When the layer and the half-space both are homogeneous, the dispersion equation agrees with the equation of the classical case. The effects of non-homogeneity, pre-stress and sandy parameter on the phase velocity have also been studied graphically.

On torsional surface wave in dry sandy crust laid over an inhomogeneous half space

An in-depth investigation has been carried out for the propagation of torsional surface waves in a dry sandy layer over an inhomogeneous half space. Both medium are assumed to be under compressive initial stress. The inhomogeneities that arise due to slow variation in rigidity and density in the lower half space is taken as an arbitrary function of depth. Using WKB approximation a closed form dispersion relation of torsional surface wave in the sandy layer has been formulated in terms of the arbitrary bounded functions. This dispersion equation is further employed to study the polynomial and exponential variation in rigidity and density of the half-space. An extensive analysis is carried out through numerical computation to explore the effect of inhomogeneity, initial stress, sandiness and thickness of the layer on the phase velocity of the torsional surface waves. The analysis of dispersion equation manifests the decline of phase velocity in the presence of sandy parameter as well as by increase in thickness of the sandy layer. The results further indicate that the initial stress of both the medium and the inhomogeneity of the half space affect the wave velocity significantly.

Torsional wave in an initially stressed layer lying between two inhomogeneous media

In the present paper, case wise studies have been made to investigate the existence of torsional surface wave in an initially stressed isotropic heterogeneous layer sandwiched between two inhomogeneous semi infinite media. Two cases have been discussed separately for wave propagation in an inhomogeneous layer between homogeneous and non-homogeneous half spaces, respectively. The dispersion relations of torsional surface wave for both the cases have been obtained in closed form. The effect of inhomogeneity parameter and initial stresses on the propagation of torsional surface wave have been discussed and presented graphically by plotting the dispersion curves.

Influence of rigid boundary on the propagation of torsional surface wave in an inhomogeneous layer

The present work illustrates a theoretical study on the effect of rigid boundary for the propagation of torsional surface wave in an inhomogeneous crustal layer over an inhomogeneous half space. It is believed that the inhomogeneity in the half space arises due to hyperbolic variation in shear modulus and density whereas the layer has linear variation in shear modulus and density. The dispersion equation has been obtained in a closed form by using Whittaker’s function, which shows the variation of phase velocity with corresponding wave number. Numerical results show the dispersion equations, which are discussed and presented by means of graphs. Results in some special cases are also compared with existing solutions available from analytical methods, which show a close resemblance. It is also observed that, for a layer over a homogeneous half space, the velocity of torsional waves does not coincide with that of Love waves in the presence of the rigid boundary, whereas it does at the free boundary. Graphical user interface (GUI) software has been developed using MATLAB 7.5 to generalize the effect of various parameter discussed.

ВЛИЯНИЕ УПРУГОЙ НЕЛИНЕЙНОСТИ И ДЕПЛАНАЦИИ НА ПАРАМЕТРЫ КРУТИЛЬНОЙ ВОЛНЫ, РАСПРОСТРАНЯЮЩЕЙСЯ В ТОНКОСТЕННОМ СТЕРЖНЕ

A mathematical model describing the propagation of torsional wave in the rod, including geometric and material elastic nonlinearity and deplanation, i.e. the output of the cross-section, in the deformation process of the rod, from the initial flat. In contrast to most well-known models is the relation between the twist angle of the rod and measure delenatii not postulated, and is in the process of solving the problem. Determined that the presence of delenatii entails the dispersion of the phase velocity of torsional wave. It is shown that the simultaneous presence of nonlinearity and dispersion leads to the formation of non-sinusoidal standing waves in a beam propagating with constant speed without changing shape. Keywords: mathematical model, torsional wave, thin-wall bar, warping, twisting vibrations.

On the torsional wave dispersion in a hollow sandwich circular cylinder made from viscoelastic materials

The dispersion of torsional wave propagation in the sandwich hollow cylinder made of linear viscoelastic materials is investigated. The investigations are carried out within the scope of the piecewise homogeneous body model by utilizing the exact field equations of viscoelasto-dynamics. The mechanical relations of the materials are described through fractional exponential operators. The analytical expression for the dispersion equation is obtained and the algorithm is developed for its numerical solution. The influence of the viscosity of the layers of the sandwich hollow cylinder is studied through the rheological parameters which characterize the characteristic creep time and long-term values of the elastic constants. It is assumed that the materials of the inner and outer cylinders are the same. Numerical results (dispersion curves) are obtained and the influence of the mentioned rheological parameters on these curves is investigated for the following two cases: attenuation of the torsional waves is dispersive (Case 1) and attenuation of the torsional waves is non dispersive (Case 2). The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity which is valid in Case 1. As a result of the numerical investigations, in particular, it is established that in the case where the rheological parameters of the layers are the same, the viscosity of the layers’ materials causes the torsional wave propagation velocity to increase.

Propagation of torsional waves in an inhomogeneous layer sandwiched between inhomogeneous semi-infinite strata

The present work deals with the propagation of torsional waves in an inhomogeneous layer sandwiched between inhomogeneous semi-infinite strata. The superstratum has exponential variation in rigidity and density, whereas inhomogeneities in the layer and the substratum are assumed to arise due to linear and quadratic variations in shear modulus and density. Closed-form solutions for displacements in the layer and the semi-infinite strata are obtained. A generalized implicit dispersion equation is obtained for the phase velocity and expressed in terms of Heun functions and their derivatives. Three possible torsional modes are derived and characterized separately. It is observed that the effects of inhomogeneity parameters are different for different modes of torsional waves and certain torsional modes are found to be independent of certain inhomogeneity parameters. The range of confinement of torsional modes, the combined effect of wave number and inhomogeneity ratio, and the condition of torsional waves becoming nondispersive are derived through numerical simulation and shown using two-dimensional and three-dimensional graphs.

Energy flow models for the out-of-plane vibration of horizontally curved beams

The purpose of this work is to develop energy flow models to predict the out-of-plane vibration of horizontally curved beams in the mid- and high-frequency range. The dispersion relations of waves are approximately separated into relations to the propagation of flexural waves and torsional waves generating the out-of-plane vibration of the horizontally curved beams with Kirchhoff-Love hypotheses. The energy flow models are based on the energy governing equations for the flexural waves and the torsional waves propagating in the curved beams. Those equations are driven to predict the time- and locally space-averaged energy density and intensity in the curved beams. Total values for the energy density and the intensity as well as contributions of each type of waves on those values are predicted. A verification of the energy flow models for the out-of-plane vibration of the horizontally curved beams is performed by comparing the energy flow solutions for the energy density and the intensity with analytical solutions evaluated using the wave propagation approach. The comparison shows that the energy flow models can be effectively used to predict the out-of-plane vibration of the horizontally curved beams in the mid- and high-frequency range.

Existence of torsional surface wave in a porous crustal layer over an initially stressed inhomogeneous half-space

In this short communication, a mathematical model for torsional wave propagation in a porous crustal layer lying over an anisotropic inhomogeneous half-space has been studied. This study reveals that the inhomogeneity in the half-space is assumed to be present in the directional rigidities and density. The anisotropy nature of the half-space is due to the presence of initial stress. Bessels functions are taken to solve the problem and the frequency equations governing the propagation of torsional waves are derived. It has been observed that there are three sets of torsional wave fronts and one shear wave front. Each set of torsional wave front has a pronounced effect on phase velocity, whereas shear wave front remains un-affected. Numerical treatment is applied to seek these effects on phase velocity of the existing waves and presented graphically. Graphical user interface has been developed using MATLAB to generalize the effect of the various parameters discussed.

Magneto-electro-viscoelastic torsional waves in aeolotropic tube under initial compression stress

This study examines the effect of electric and magnetic field on torsional waves in hetrogeneous viscoelastic cylindrically aeolotropic tube subjected to initial compression stresses. A new equation of motion and phase velocity of torsional waves propagating in cylindrically aeolotropic tube subjected to initial compression stresses, nonhomogeneity, electric and magnetic field have been derived. The study reveals that the initial stresses, nonhomogeneity, electric and magnetic field present in the aeolotropic tube of viscoelastic solid have a notable effect on the propagation of torsional waves. The results have been discussed graphically. This investigation is very significant for potential application in various fields of science such as detection of mechanical explosions in the interior of the earth.

Propagation of torsional waves in a thin circular plate of generalized phononic crystals

By introducing periodicity in the direction of variable vectors, the concept of existing phononic crystals (PCs) is expanded to generalized phononic crystals (GPCs), and a one-dimensional thin circular plate of GPCs (GPC plate) consisting of two alternating homogeneous materials is constructed. To show the characteristics of a torsional stress wave propagating along the radial direction, a transfer matrix is derived based on the basic torsional wave equation of the thin circular plate in cylindrical coordinates. A localization factor is introduced to evaluate the average attenuation of torsional stress waves in the structure, thus the band gaps are calculated. The inherent relationships between wave attenuation band, wave front and periodicity are discussed in detail. Finite element method simulations, numerical analyses and the insertion loss method are combined to investigate the effects of the main parameters on the torsional stress wave band in GPC plate. The results show that there are significant band gaps of the torsional stress waves caused by the periodicity of GPC plate, which are similar to those of existing PCs, and the localization factor can be used for the wave analysis of GPCs. In addition, structural and material parameters have a great influence on the wave attenuation band of the GPC plate.

Torsional wave propagation in a self-reinforced medium sandwiched between a rigid layer and a viscoelastic half space under gravity

The present paper constitutes the study of torsional surface wave propagation in a self-reinforced layer resting over a gravitating viscoelastic half space. The layer has an inhomogeneity of linear type associated with the rigidity and density of the medium. Dispersion equation has been obtained in the terms of HypergeometricU and LaguerreL function. The dispersion equation reduces to a classical form when the inhomogeneity of the layer, gravity and viscosity of the half space vanishes. The influence of various parameters has been depicted by means of graphs for both reinforced and reinforcement free medium.