Phase and group velocities of normal waves in an orthotropic layer

The article analyzes the distribution of the phase and group velocities of normal, transversely symmetric waves in an anisotropic layer of the orthrhombic class. The numerical investigation of a complex variance equation made it possible to establish qualitative peculiarities of the behavior of phase and group velocities for stresses different from elastoequivalent ones.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 101–105, 1987.     

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.     

The three-dimensional problem of the axisymmetric deformation of an orthotropic spherical layer

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.     

Rayleigh wave motions in an orthotropic half-space under time-harmonic loadings: A theoretical study

Rayleigh surface wave motion in a homogeneous orthotropic elastic solid half-space under time-harmonic sources is theoretically investigated. In this article, closed-form expressions of the Rayleigh wave field as a result of a time-harmonic concentrated line load are simply determined by using reciprocity theorems. For the verification purpose, analytical predictions are also obtained through Fourier transform technique with the aid of the residue theorem. The solutions found by the two approaches are shown to be mathematically the same. As an example of calculation, surface waves due to a set of buried loadings in the orthotropic half-plane are then examined in detail. A discussion of these results is presented to explore the influence of the loading types on the displacement amplitudes of the generated wave fields.     

Interaction between Griffith cracks in a sandwiched orthotropic layer

A study is made of the interaction between three coplanar Griffith cracks which are located symmetrically in the midplane of an orthotropic layer of finite thickness 2h sandwiched between two identical orthotropic half planes. The Fourier transform technique is used to reduce the elastostatic problem to a set of integral equations which have been solved by using the finite Hilbert transform and Cooke's results. Analytical expressions for the stress intensity factors at the tips of cracks are obtained for large values of h. Numerical results concerning the interaction effects are presented with physical significance. It is shown that interaction effects are either shielding or amplification depending on the location of cracks, spacing of crack-tips, and the thickness of the layer. The stress magnification factors at the crack-tips are also calculated.     

A slow wave in a fluid layer simulating an oil-saturated seam

A simple model of an oil-saturated layer is proposed. A slow wave propagating in a fluid layer sandwiched between two elastic half-spaces is studied. For this wave the phase and group velocities and also the horizontal and vertical components of the displacement are presented. The theoretical results are in good agreement with the experimental data obtained. Bibliography:4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 101–112. Translated by P. V. Krauklis.     

Compression of an orthotropic prism

The formulation and mathematically correct solution of a three-dimensional boundary value problem are given. A numerical calculation is presented for a prism of Canadian spruce and the displacement and stress fields are determined.Presented at a scientific seminar on polymer mechanics held in the Mechanics-Mathematics Department of Moscow State University, February 7, 1968.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 810–815, September–October, 1968.     

A slow wave in a porous layer

It is shown that in a cracked layer sandwiched between two elastic half-spaces, a slow wave propagates. For low frequencies the velocity of this wave is much less than its velocity in an infinite fluid. The dispersion and amplitude characteristics of the slow wave are studied as a function of the porosity and frequency. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1993, pp. 146–153. Translated by P. V. Krauklis.     

Inverse scattering from an orthotropic medium

We consider the inverse scattering problem of determining the shape of an orthotropic homogeneous medium from a knowledge of the far field pattern corresponding to incident time harmonic plane waves. We solve the direct problem by potential theory, show the uniqueness of the inverse problem and then use a ‘simple’ method recently developed by Colton and Kirsch (1996) and Colton and Monk (1997) to solve the inverse problem. Numerical examples are given showing the practicality of our method for solving the inverse problem.     

Calculation of thermal stresses in an orthotropic cylinder

The present article describes the determination of thermal stresses in an orthotropic cylinder with an axially symmetric temperature field taking into account the change in the elasticity constants over the cylinder radius. A closed solution is given for constant elasticity coefficients.P. I. Baranov Central Institute of Aircraft-Engine Building, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 310–314, March–April, 1972.     

Averaged wave operators on singular spectrum

We prove the existence of pairs of unitary (or self-adjoint) operators with singular spectral measure whose difference is a rank-two operator for which the Abel wave operators fail to exist. Also, we discuss the closely related problem of constructing the Hilbert transform with respect to a singular measure on the unit circle.     

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.     

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.     

Investigation of low-frequency normal waves in a Biot layer surrounded by an elastic medium

The porous Biot layer surrounded by two elastic half-plates is considered. For this medium, two dispersion equations are established. These dispersion equations correspond to symmetric and antisymmetric parts of the medium. The investigation of these equations is conducted in the region of low frequencies. The imaginary roots of these equations in this region determine low-frequency normal waves. Bibliography: 6 titles.     

Discreteness of the spectrum in the problem on normal waves in an open inhomogeneous waveguide

We consider the problem on normal waves in an inhomogeneous waveguide structure reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. The inhomogeneity of the dielectric filling and the occurrence of the spectral parameter in the transmission conditions necessitate giving a special definition of what a solution of the problem is. To find the solution, we use the variational statement of the problem. The variational problem is reduced to the study of an operator function. We study the properties of the operator function needed for the analysis of its spectral properties. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.     

The indentation of an orthotropic half plane

Zusammenfassung Der Verfasser findet eine allgemeine Lösung für die Verteilung des Druckes zwischen einem Block und einer orthotropen Halbebene. Es wird gezeigt, dass die Verteilung des Druckes für den flachen Block mit allgemeiner Belastung un-abhängig ist von den elastischen Eigenschaften der Halbebene und auch genau dieselbe, als ob die Halbebene isotrop wäre.     

A hollow ellipsoidal needle in an orthotropic elastic medium

The problem of the stress distribution on the surface of a hollow ellipsoidal needle in an orthotropic elastic medium and a homogeneous external field is solved. Explicit expressions are obtained for the stresses on the needle surface in terms of the elastic constants of the medium and parameters of the ellipsoid in a local system of coordinates connected to the normal to the surface at each point of the needle. The general solution of the problem of the stress concentration on an ellipsoidal inhomogeneity /1/ and the passage to the limit cases of an ellipsoidal cavity based on the presence of small parameters /2/ is used.     

Elastic equilibrium of an inhomogeneous orthotropic parallelepiped

A spatial problem of the elasticity theory of an inhomogeneous body is examined for an orthotropic rectangular domain of finite size with a free lateral surface. The material mechanical characteristics are functions of one of the coordinates. The principle of minimum additional energy and the apparatus of special orthonormal Horvey polynomials are utilized. Different modifications of the boundary conditions at the structure endfaces are analyzed. Numerical examples are presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 9–13, 1987.