Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium

Within the framework of the piecewise homogeneous body model with utilization of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the mathematical modeling of the torsional wave propagation in the initially stressed infinite body containing an initially stressed circular solid cylinder (case 1) and circular hollow cylinder (case 2) are proposed. In these cases, it has been assumed that in the constituents of the considered systems there exist only the normal homogeneous tensional or compressional initial stress acting along the cylinder, i.e. in the direction of wave propagation. In the case where the mentioned initial stresses are not present, the proposed mathematical modeling coincides with that proposed and investigated by other authors within the classical linear theory of elastic waves. The mechanical properties of the cylinder and surrounding infinite medium have been described by the Murnaghan potential. The numerical results related to the torsional wave dispersion and the influence of the mentioned initial stresses on this dispersion are presented and discussed.     

不规则界面对扭转表面波在初应力各向异性多孔弹性层中传播的影响

研究了扭转表面波在一个半无限非均匀半空间中的传播,半空间上覆盖着具有初始应力的各向异性多孔弹性层,弹性层的刚度和密度线性地变化,造成了界面的不规则性．半空间中界面的不规则性,用一个矩形形式表示．可以发现,扭转表面波在这样假定的介质中传播,得到了没有不规则性时的扭转表面波的速度方程．还可以发现,对于均匀半空间覆盖的层状介质,扭转表面波的速度与Love波的速度相一致．     

Possibility of Love wave propagation in a porous layer under the effect of linearly varying directional rigidities

The present paper investigates the Love wave propagation in an anisotropic porous layer under the effect of rigid boundary. Effect of initial stresses on the propagation of Love waves in a fluid saturated, anisotropic, porous layer having linear variation in directional rigidities lying in contact over a pre-stressed, inhomogeneous elastic half-space has also been considered. The dispersion equation of phase velocity has been derived and the influence of medium characteristic such as porosity, rigid boundary, initial stress, anisotropy and inhomogeneity over it has been discussed. The velocities of Love waves have been calculated numerically as a function of KH (where K is the wave number and H is the thickness of the layer) and are presented in a number of graphs.     

Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field

In this paper, propagation of shear waves in a non-homogeneous anisotropic incompressible, gravity field and initially stressed medium is studied. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, gravity field, non-homogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear wave has been obtained. The dispersion equations have been obtained and investigated for different cases. A comparison is made with the results predicted by Abd-Alla et al. [22] in the absence of initial stress and gravity field. The results obtained are discussed and presented graphically.     

Effects of initial stresses on guided wave propagation in multilayered PZT-4/PZT-5A composites: A polynomial expansion approach

Effects of initial stresses on the dispersion curves of Lamb and SH waves in multilayered PZT-4/PZT-5A composites are investigated using the polynomial expansion approach. The piezoelectric layers are considered with arbitrary crystal orientations with a result that only Lamb or SH waves may be transmitted. The problem is solved employing the Legendre polynomial approach that poses the advantages of numerically stability and effectiveness over conventional matrix method. The solution is validated by comparing the wave propagation behavior of piezoelectric materials with those reported in literature, and the convergence properties are examined. Numerical results demonstrate that initial stress has profound influences on the guided wave propagation in multilayered PZT-4/PZT-5A laminates. The phase velocity of Lamb and SH waves increases with initial tensile stresses. In addition, the effects of initial stresses rely on the wave mode and thickness of constituent layers and the stacking sequence of the constituent materials. The results are useful for understanding and optimization of new designs for actuator, electromechanical sensor and acoustic wave devices made of PZT-4/PZT-5A composites.     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

Nonstationary solutions of a generalized Korteweg-de Vries-Burgers equation

Nonstationary solutions of the Cauchy problem are found for a model equation that includes complicated nonlinearity, dispersion, and dissipation terms and can describe the propagation of nonlinear longitudinal waves in rods. Earlier, within this model, complex behavior of traveling waves has been revealed; it can be regarded as discontinuity structures in solutions of the same equation that ignores dissipation and dispersion. As a result, for standard self-similar problems whose solutions are constructed from a sequence of Riemann waves and shock waves with stationary structure, these solutions become multivalued. The interaction of counterpropagating (or copropagating) nonlinear waves is studied in the case when the corresponding self-similar problems on the collision of discontinuities have a nonunique solution. In addition, situations are considered when the interaction of waves for large times gives rise to asymptotics containing discontinuities with nonstationary periodic oscillating structure.     

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.     

Nonlinear transmission eigenvalue problem describing TE wave propagation in two-layered cylindrical dielectric waveguides

The paper is concerned with propagation of surface TE waves in a circular nonhomogeneous two-layered dielectric waveguide filled with a Kerr nonlinear medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of a Green’s function. The existence of propagating TE waves is proved using the contraction mapping method. For the numerical solution of the problem, two methods are proposed: an iterative algorithm (whose convergence is proved) and a method based on solving an auxiliary Cauchy problem (the shooting method). The existence of roots of the dispersion equation (propagation constants of the waveguide) is proved. Conditions under which k waves can propagate are obtained, and regions of localization of the corresponding propagation constants are found.     

Dynamic torsion of viscoelastic cylindrical rods

An exact solution of the problem of the propagation of torsional waves in semiinfinite and finite circular cylinders is obtained within the framework of the linear theory of viscoelasticity. Concrete examples are discussed, and estimates are given for the stresses near the wave front. All the solutions are obtained in series; it is shown that these series converge absolutely for any finite time.     

Torsional vibrations and dispersion of torsional waves in orthotropic composite rods

The Barr’s refined theory of torsional vibrations of isotropic rods of noncircular cross section is generalized for an orthotropic material. An analysis of natural frequencies of torsional vibration of free-free orthotropic prismatic rods of rectangular cross section is carried out with the help of an exact solution of the frequency equation. For orthotropic CFRP and GFRP rods, the improved theory, which takes into account the normal stresses and inertia forces in the axial direction, in some cases, predicts a noticeable raise in the natural frequencies compared with those following from the Saint-Venant classical theory. A good agreement is obtained between the experimental and calculated values of natural frequencies of torsional vibrations of rectangular quartz and fiber glass rods. The dispersion of torsional waves in an orthotropic quasi-homogeneous rod is considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 165–182, March–April, 2008.     

Propagation of SH‐wave in an initially stressed orthotropic medium sandwiched by a homogeneous and an inhomogeneous semi‐infinite media

The paper presents a study of propagation of shear wave (SH‐wave) in an orthotropic elastic medium under initial stress sandwiched by a homogeneous semi‐infinite medium and an inhomogeneous half‐space. The technique of separation of variables has been adopted to get the analytical solutions for the dispersion relation in a closed form. The propagation of SH‐waves is influenced by inhomogeneity parameters and initial stress parameter. Velocities of SH‐waves are calculated numerically for different cases. As a special case when the intermediate layer and half‐space are homogeneous, computed frequency equation coincides with general equation of Love wave. To study the effect of inhomogeneity parameters and initial stress parameter, we have plotted the velocity of SH‐wave in several figures and observed that the velocity of wave decreases with the increases of non‐dimensional wave number. It can be found that the phase velocity decreases with the increase of inhomogeneity parameters. We observed that the velocity of SH‐wave decreases with the increases of initial stress parameter in both homogeneous and inhomogeneous media. GUI has been developed by using MATLAB to generalize the effect of the parameters discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Ray Method Expansions for Surface and Internal Waves in Inhomogeneous Oceans of Variable Depth

A ray method formalism is developed for the analysis of surface and internal waves in an inhomogeneous ocean of variable depth. In this method, we deduce from the governing system of equations a system of first order ordinary differential equations, for the group lines (rays of the ray method) and the propagation of phase and amplitude on them. The dispersion relation for these waves arises as an eigen-condition on an eigen-value problem involving an ordinary differential equation in the depth variable. The deduced equation for amplitude propagation has the interpretation of a statement of conservation of action.     

Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field

In this paper the influence of the gravity field, relaxation times and initial stress on propagation of Rayleigh waves in an orthotropic magneto-thermoelastic solid medium has been investigated. The solution of the more general equations are obtained for thermoelastic coupling by Helmoltz’s theorem. The frequency equation which determines Rayleigh wave velocity have been obtained. Many special cases are investigated from the present problem. Numerical results analyzing the frequency equation are obtained and presented graphically. Relevant results of previous investigations are deduced as special cases from these results. The results indicate that the effect of initial stress, magnetic field and gravity field are very pronounced.     

Stability of solitary waves and weak rotation limit for the Ostrovsky equation

Ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation. In this model dispersion is taken into account while dissipation is neglected. Existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter (which can be either positive or negative). It is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations. The issue of passing to the limit as the rotation parameter tends to zero for solutions of the Cauchy problem is investigated on a bounded time interval.     

Propagation of weakly nonlinear waves in fluid-filled thick viscoelastic tubes

In the present work, we studied the propagation of small-but-finite-amplitude waves in a prestressed thick walled viscoelastic tube filled with an incompressible inviscid fluid. In order to include the dispersion, the wall's inertial and shear effects are taken into account in determining the inner pressure–inner cross-sectional area relation. Using the reductive perturbation method, the propagation of weakly nonlinear waves in the long-wave approximation is investigated. After obtaining the general evolution equation in the long-wave approximation, by a proper scaling, it is shown that this general equation reduces to the well-known evolution equations such as the Burgers, Korteweg-de Vries (KdV), Koteweg-de Vries–Burgers (KdVB) and the generalized Burgers' equations. By proper re-scaling of the perturbation parameter, the modified form of the evolution equations is also obtained. The variations of the travelling wave profile with initial deformation and the viscosity coefficients are numerically evaluated and the results are illustrated in some figures.     

Dispersion of torsional waves in initially stressed multilayered circular cylinders

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the propagation of torsional waves in prestressed multilayered circular cylinders is investigated. The elasticity relations for cylinder components are given through the Murnaghan potential. The influence of variations in the geometric and mechanical parameters of the cylinders on the dispersion curves is analyzed.     

Wave propagation in an initially stressed transversely isotropic thermoelastic solid half-space

The governing equations of thermoelasticity of transversely isotropic solid with initial stresses are formulated at uniform temperature. These equations are solved analytically in two-dimensions to show the existence of three plane quasi waves, namely, Quasi-Longitudinal (QL), Thermal (T-mode) and Quasi-Transverse (QT) waves. Reflection from a thermally insulated stress free surface of an initial stressed transversely isotropic thermoelastic solid half-space is studied. A particular model is chosen for the numerical computations of the propagation speeds, attenuation coefficients and reflection coefficients. Effects of initial stress parameter and thermal disturbances are observed on speeds of propagation, attenuation coefficients and reflection coefficients.     

Propagation of torsional waves in a prestretched compound hollow circular cylinder

The propagation of torsional waves in a prestressed compound (bi-layered) hollow circular cylinder is in vestigated within the frame work of a piecewise homogeneous body model, with the use of a three-dimensional linerized theory of elastic waves in initially stressed bodies. The elasticity relations for components of the compound cylinder are obtained from the Murnaghan potential. Numerical investigations are performed for bronze and steel. According to the results obtained, the effect of variations in the geometric (the ratio between the thickness of the cylinder and its inner radius) and mechanical parameters on the dispersion curves are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 1, pp. 103–116, January–February, 2008.     

Effect of phase-lags on Rayleigh wave propagation in initially stressed magneto-thermoelastic orthotropic medium

The present article deals with Rayleigh surface wave propagation in homogeneous magneto-thermoelastic orthotropic medium. Effect of initial stress and magnetic field on Rayleigh waves is studied in the context of three-phase-lag model of generalized thermoelasticity. The normal mode analysis is used to obtain the exact expressions for the displacement components, stresses and temperature distribution. Various frequency equations are derived and compared with the existing literature. The path of surface particles is elliptical during Rayleigh wave propagation. Effect of phase-lags on Rayleigh wave velocity, attenuation coefficient and specific loss are presented graphically. It is observed from graphical presentation that the effect of magnetic field and initial stress on different wave characteristics is pronounced.