Torsional Wave Dispersion Research Articles

Dispersion of torsional surface waves in a threefold concentric compounded cylinder with imperfect interface

The present paper investigates the torsional wave propagation in a threefold concentric pre-stressed compounded cylinder with imperfect contact conditions. The three-dimensional linearized theory of elastic waves and the piecewise homogeneous body model has been employed to formulate the problem. The mathematical modeling has been carried out in two independent cases. In the first case, a solid cylinder encased in a hollow cylinder embedded in an infinite elastic medium has been considered. Whereas the second case comprises a hollow cylinder of finite thickness in place of a solid cylinder. By means of Murnaghan potential, the mechanical characteristics of the three materials have been used. Further, the dispersion relations for both the cases have been obtained in terms of the Bessel and modified Bessel functions. In order to validate the present findings, two particular cases have been derived that matches with the previous works. The first case is obtained by removing the outermost cylinder, while the second case has been derived by removing the imperfection in addition to that. To summarize the computations, a complete numerical simulation has been carried out, and graphical illustrations have been shown to aid the mathematical analyses.

Dispersion of Torsional Waves in a Hollow Bilayered Cylinder with Initial Inhomogeneous Thermal Stresses

The influence of the inhomogeneous initial thermal stresses caused by the heating or cooling of the inner and outer surfaces of a hollow bilayered cylinder on dispersion of torsional waves propagating in this cylinder is studied. The study is performed within the scope of the second version of the 3D linearized theory of elastic waves in initially stressed bodies, according to which the initial inhomogeneous thermal stresses are determined within the scope of the classical uncoupled linear theory of thermoelasticity. The equations related to the propagation of torsional waves are solved by a discrete-analytical method, and the numerical results found are presented and discussed.

Torsional wave dispersion in a bi-layered hollow cylinder with inhomogeneous initial stresses caused by internal and external radial pressures

The present paper studies the influence of the inhomogeneous initial stresses in the bi-layered hollow cylinder and it is assumed that these stresses are caused by the hydrostatic pressures acting on the interior and outer free surfaces of the cylinder. The study is made by utilizing the version of the three-dimensional linearized theory of elastic waves in bodies with initial stresses for which the initial stress-strain state in bodies is determined within the scope of the classical linear theory of elasticity. For the solution to the corresponding eigenvalue problem, the discrete-analytical method is employed. Numerical results are presented and analyzed for concrete selected pairs of materials. According to these results and their analyses, it is established that, unlike homogeneous initial stresses, the influence of the inhomogeneous initial stresses on the torsional wave dispersion has not only quantitative but also qualitative character. For instance, in particular, it is established that as a result of the initial stresses caused by the hydrostatic pressure acting in the interior free surface of the cylinder, the cut-off frequency appears for the fundamental dispersive mode and the values of this frequency increase with the intensity of this pressure.

Dispersion of Torsional Surface Wave in a Pre-Stressed Heterogeneous Layer Sandwiched Between Anisotropic Porous Half-Spaces Under Gravity

The study of surface waves in a layered media has their viable application in geophysical prospecting. This paper presents an analytical study on the dispersion of torsional surface wave in a pre-stressed heterogeneous layer sandwiched between a pre-stressed anisotropic porous semi-infinite medium and gravitating anisotropic porous half-space. The non-homogeneity within the intermediate layer and upper semi-infinite medium is assumed to rise up, because of quadratic variation and exponential variation in directional rigidity, pre-stress, and density respectively. The displacement dispersion equation for the torsional wave velocity has been expressed in the term of Whitaker function and their derivatives. Dispersion relation and the closed-form solutions have been obtained analytically for the displacement in the layer and the half-spaces. It is determined that the existing geometry allows torsional surface waves to propagate and the observe exhibits that the layer width, layer inhomogeneity, frequency of heterogeneity in the heterogeneous medium has a great impact on the propagation of the torsional surface wave. The influence of inhomogeneities on torsional wave velocity is also mentioned graphically by means plotting the dimensionless phase velocity against non-dimensional wave number for distinct values of inhomogeneity parameters.

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.

Dispersion of Torsional Surface Wave in a Pre-stress Anisotropic Porous Medium With Rigid and Traction Free Boundary Surface Over Sandy Half-Space

In this article, an endeavor has been made to infer the dispersion equation of the torsional surface wave in a pre-stressed an anisotropic porous medium over a dry sandy semi-limitless medium. The case insightful investigation of rigid boundary and traction free boundary surface of the upper layer has been examined in points of interest. The elastic equations of motion have been defined independently for different media utilizing Biot’s hypothesis. The dispersion equations of motion for both cases have been acquired under the suitable limit condition in the term of Whittaker function and its derivatives.

Torsional surface wave dispersion in pre-stressed dry sandy layer over a gravitating anisotropic porous half-space

The present paper discusses the dispersion of torsional surface wave in a pre-stressed dry sandy layer over an anisotropic porous half-space under the influence of gravity. The inhomogeneity has been expected as hyperbolic variation in the sandy layer. The mathematical analysis of the problem has been dealt with the asymptotic expansion of Whittaker function and its derivative. The dispersing mathematical statement acquired is in concurrence with the conventional consequence of Love wave in a homogeneous layer over a homogeneous half space when the upper limit plane is thought to be traction free. The velocities of torsional waves are ascertained numerically as a component of kH and introduced in various graphs, where k is the wave number and H is the thickness of the sandy layer. The numerical expression gives the scaffold between displaying results and field application. The impact of non-homogeneity in modulus of rigidity, stress, density, and porosity have been portrayed by the method for numerical data and graphs. The study reveals that the torsional surface wave propagates in an initially stressed dry sandy layer over a gravitating anisotropic porous half-space. It is additionally watched that the vicinity of gravity field builds the speed of the torsional surface wave. This work may be valuable to comprehend the way of seismic wave proliferation amid quake in the considered media.

The torsional surface wave in a prestressed anisotropic intermediate poroelastic layer of varying heterogeneities

The aim of this paper is to study the behavior of the torsional surface wave in a heterogeneous initially stressed vertical fluid-saturated anisotropic layer sandwiched between inhomogeneous and homogeneous porous half-spaces. It has been considered that the mass density and rigidity of the upper half-space and intermediate layer are space dependent. The proposed model is solved to obtain different dispersion relations for the torsional surface wave in a heterogeneous poroelastic medium lying between two half-spaces. The influence of compressive stress and heterogeneity on torsional surface wave dispersion is shown numerically. It has been observed that heterogeneity, porosity, initial stress of the layer and inhomogeneity of the upper and porosity of lower half-spaces affect the torsional wave speed much. The wave analysis further indicates that the torsional surface waves travel faster in elastic half-spaces in comparison than in the fluid-saturated porous layer.

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.

Model of Torsional Waves Based on Transverse and Layered One-Dimensional Component of Infinite Length

The dispersion of torsional wave in transverse and layered one-dimensional component of infinite length is investigated. The study is carried out by treating the one-dimensional component as a cylinder of infinite length firstly, Then put forward a way that the component is divided into a multilayer to analyze torsional wave propagation, the model is constructed in cylindrical coordinates for homogeneous and layered component, the governing equation of torsional wave was derived based on one-dimensional elastic theory and Whittaker’s function, obtain harmonic wave solution of the governing equation by making use of the boundary and continuity conditions of adjacent layers. Finally the result of numerical simulation illustrated the influence of component’s density and shear module on torsional angels and phase velocity. It has been observed the velocity is changing as the change of parameters of shear module and density according to the curve. The conclusion made shown the consistency with the classical theory, and the effective on model such as bearings, engineering mechanics and displacement of particles.

Dispersion and attenuation by transmission, reflection, and mode conversion in welded pipes

Torsional wave dispersion and attenuation in an open empty welded pipe are determined from a multi-receiver position reflection experiment. The fundamental torsional wave is dominantly reflected at the free end and the converted non-axisymmetric flexural modes are naturally attenuated. The resulting phase velocity contours are in agreement with theoretical predictions. The transmission losses are quantified and compared to those reflective elements associated with end and weld reflection. At any reflective node, the incident wave is split between back and forward preserved mode scattering (“reflection/transmission”), conversion to other modes plus energy lost by absorption. The ratios for each element are quantified.

Torsional Wave in a Viscoelastic Layer over a Viscoelastic Substratum of Voigt Types

This article explores the dissipation and dispersion of torsional wave resulting from imperfection in elasticity in terms of internal friction, layer width between dissimilar homogeneous viscoelastic isotropic medium having Voigt-type viscosity. The closed forms solutions for the displacement in the upper layer and lower half space are obtained separately. The generalized torsional wave period equation is obtained and the angular frequency has been plotted against wave number for different values of relevant parametric variation and certain particular cases have been deduced. Dissipation and dispersion are analysed using two and three dimensional plots along with filled contour plots.

Dispersion of torsional surface wave in an intermediate vertical prestressed inhomogeneous layer lying between heterogeneous half spaces

The purpose of this study is to illustrate the propagation of the torsional surface waves in an intermediate inhomogeneous initially stressed vertical elastic layer sandwiched between two heterogeneous half-spaces. It is considered that the mass density and the rigidity of upper and lower half-spaces are space dependent. The proposed model is solved to obtain the different dispersion relations for the torsional surface wave in the elastic medium of different properties. The effects of compressive and tensile stresses along with the heterogeneity on the dispersion of torsional surface wave in the intermediate layer are shown numerically. The wave analysis further indicates that the inhomogeneity, the initial stress of the layer and the heterogeneity of both the half spaces affect the wave velocity remarkably. The results may be useful to understand the nature of seismic wave propagation in geophysical applications and in the field of earthquake engineering.

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.

Existence of torsional surface waves in an earth’s crustal layer lying over a sandy mantle

This paper aims to study the dispersion of torsional surface waves in a crustal layer being sandwiched between a rigid boundary plane and a sandy mantle. In the mantle, rigidity and initial stress vary linearly while density remains constant. Dispersion relation has been deduced in a closed form by means of variable separable method in the form of Whittaker function. The velocity equation for isotropic layer over a homogeneous half-space has been obtained which coincides with the standard result of Love wave under the effect of rigid boundary.

Dispersion of torsional surface waves in anisotropic layer over porous half space under gravity

AbstractIn this paper an attempt has been made to derive the frequency equation of torsional surface waves in initially stressed fiber reinforced layer lying over fluid saturated anisotropic porous half space under gravity. The dispersion equation has been obtained in terms of Whittaker function and its derivatives, which further expanded asymptotically and retain the terms upto second degree. The analysis of frequency equation bear out the pronounced effect of initial stress, transverse and longitudinal rigidity of reinforced material, gravity and porosity of the porous half space on the propagation of torsional surface waves. Moreover, it can be noted that in the absence of initial stress, reinforcement parameters, gravity and porosity, the frequency equation of torsional surface waves reduces to dispersion equation of Love waves in isotropic layer. This ensures the classical result that torsional surface waves do not propagate through isotropic homogeneous medium.

On the influence of initial strains in layers on the propagation of torsional waves in a hollow sandwich cylinder (soft core and stiff face layers)

Within the scope of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in prestrained bodies, the effect of initial strains in layers on the dispersion of torsional waves in a hollow sandwich cylinder is studied. Two cases are considered: in Case 1, it is assumed that initial strains exist in only face layers; in Case 2, initial strains occur in all layers of the cylinder. The elasticity relations for layer materials of the cylinder are given through a harmonic potential. A dispersion equation is derived, and an analytical expression is obtained for the limit values of wave propagation speed for low wave numbers. Numerical results are presented and discussed.

Torsional Wave Propagation in Sandwich Hollow Cylinder with Pre-Strained Face (Core) Layers (Layer)

This paper studies torsional wave dispersion in a three-layered (sandwich) hollow cylinder with pre-strained face (core) layers (layer) made from high elastic materials the mechanical relations of which are described through the harmonic potential. The investigations are carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The analytical expression is obtained for the low wave number limit values of the torsional wave propagation velocity. The numerical results on the influence of the initial strains of the face (core) layers (layer) of the cylinder on the torsional wave propagation velocity are presented and discussed.

Torsional wave dispersion in a finitely pre-strained hollow sandwich circular cylinder

This paper studies torsional wave dispersion in a three-layered (sandwich) hollow cylinder with finite initial strains. The investigations are carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The mechanical relations of the materials of the cylinders are described through their harmonic potential. The analytical expression is obtained for the low wavenumber limit values of the torsional wave propagation velocity. The numerical results on the influence of the initial stretching or compression of the cylinders along the torsional wave propagation direction are presented and discussed.

Non-local continuum modeling of carbon nanotubes: Physical interpretation of non-local kernels using atomistic simulations

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non-local 1D rod theory and axisymmetric shell theory. We found that gradient models incur significant errors in both the phase and group velocity when compared to the atomistic model. Complementing these studies, we have also performed detailed tests on the effect of length of the nanotube on the axial and shear moduli to gain a better physical insight on the nature of the true non-local kernel. We note that unlike the kernel from gradient theory, the numerically fitted kernel becomes negative at larger distances from the reference point. We postulate and confirm that the fitted kernel changes sign close to the inflection point of the interatomic potential. The numerically computed kernels obtained from this study will aid in the development of improved and efficient continuum models for predicting the mechanical response of CNTs.