On love waves in a stratified hypoelastic solid with material boundary

Transverse surface waves in a half-space covered with a stratum of a different material are investigated in the context of hypoelasticity of grade zero, taking into account the surface stress effects on the boundary. It is found that the wave motion is possible and that unlike in the corresponding problem with classical boundary, the variation of amplitude in the stratum may be either periodic or exponential. When the amplitude in the stratum is periodic, the motion is similar to that in a stratum bounded on both sides by very deep layers of different elastic or hypoelastic materials.     

Gaussian beams of love waves in an anisotropic medium and their summation

Gaussian beams of surface waves are constructed by means of the Kravtsov ansatz. The Gaussian beams are summed which enables us to describe analytically the wave field of a surface Love wave in the vicinity of an arbitrary singularity. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 6–13, 1990. Translated by V. D. Azhotkin.     

On nonstationary love waves near the surface of an anisotropic elastic body

Space-time (ST) ray expansions in the presence of a caustic are used for the construction of an asymptotics of surface waves similar to the SH Love waves. Certain conditions which ensure the existence of surface waves with SH polarization are deduced. Some special cases of anisotropic media are indiceted where these conditions are satisfied. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 166–172. Translated by Z. A. Yason.     

On the relationship between love modes and the set of supercritically reflected waves

Considering the Love problem as an example, we derive relations connecting the following two exact integral representations of its solution: one explicitly involving both damped and undamped modes (residues at the roots of the dispersion equation of the problem) and the other based on expanding the interference field into a series of a geometric progression. In the latter case, to each such summand a generalized ray of a wave of certain multiplicity propagating in the layer can be put into correspondence. By using the methods of contour integrals, a correspondence between the set of multiple waves and interference modes is established. Bibliography: 1 title. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 214, 1994, pp. 200–209. Translated by T. N. Surkova.     

Nonstationary love waves of the SH-type in an anisotropic elastic medium. A kinematic approach

In this paper, the propagation of Love waves in anisotropic elastic media is studied. These waves are a similar to the transverse surface SH waves in the isotropic case. Necessary conditions for the existence of Love waves of this polarization type near the surface Σ of an anisotropic elastic body are deduced. The algorithm developed here makes it possible to find the direction (s) of transverse surface wave propagation (at every point on the surface Σ). The algorithm employed is illustrated by some special anisotropic cases. The space-time method is used to construct the asymptotics of Love waves for those types of anisotropic media the eikonal equation of which is valid on the surface of an elastic body. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 262–276 Translated by Z. A. Yanson     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

Nonstationary love waves of theSH type in an anisotropic elastic medium. A kinematic approach. II

In this paper, the study of the high-frequency Love waves (simular to the well-known transverse waves of theSH type) near the surface of an anisotropic elastic body is continued. The formulation of the boundary-value problem, independent of a specific form of the elasticity tensor, provides the possibility of developing a kinematic approach, which is essential for constructing the asymptotics of these high-frequency waves. To this end, an algorithm is proposed that allows one to relate the transversality of the polarization of surface waves to the directions of their propagation on the surface and to obtain the conditions necessary for the origination of such waves. The algorithm suggested makes it possible to indicate those types of symmetry of media (special cases of anisotropy) for which the directions obtained correspond to the field of rays of Love waves. In these cases, the space-time ray method provides a mathematical tool for constructing uniform asymptotics of the surface waves in question. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 218, 1994, pp. 206–219. This work was supported by the Russian Foundation of Fundamental Research (Grant 93-011-16148). Translated by Z. A. Yanson.     

Asymptotic Stability of Large-amplitude Oscillations in Systems with Hysteresis

We study depending on a parameter periodic systems with the main linear part and a hysteresis nonlinearity; the linearized at infinity system has a one-dimensional subspace of periodic solutions for the critical parameter value. We prove theorems on a number, localization, and asymptotic stability of large-amplitude periodic solutions for the nonlinear system. Received March 1998     

The classical and generalized love problems in a domain of low-frequency interference waves of signal type. I

Low-frequency interference waves propagating through a thin elastic layer, which is in rigid contact with an elastic half-space, are considered. A new method for computing the fields of low-frequency waves suggested by G. I. Petrashen allows us to compute the fields of these waves and to draw some conclusions concerning their behavior. In particular, we may conclude that the thin upper layer significantly distorts a signal sent deep into a medium. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 225, 1996, pp. 5–39. Translated by N. S. Zabavnikova.     

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.     

Friedrich ii and the love of geometry

This is a somewhat expanded version of an article in Architektur Aktuell 169/170 (1994), 88-95, reprinted by permission. English translation by L.L. Schumaker.     

Cardano's waves

The classical formula of Cardano for a cubic equation is deduced as a solution of an initial-value problem for a differential equation. A connection is noted with the wave equation.     

Effects of random noise in a dynamical model of love

This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of “Romeo’s romantic style”, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed.     

Interaction of acoustic waves with shock waves in elastic solids

Summary Transmission and reflexion of plane acoustic waves through a longitudinal shock wave in elastic isotropic solids are investigated. As a result, the amplitude of the transmitted and reflected waves and the jump of the acceleration of the shock are explicitly determined.

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Résumé On envisage la transmission et la réflexion d'une onde acoustique plane par une onde de choc longitudinale dans les solides élastiques isotropes. On détermine explicitement l'amplitude des ondes transmises et réflechies et le saut de l'acceleration du choc.

     

Equilibrium viscoelastic waves

Summary From an asymptotic property of the Fourier integral, it is shown that for any input wave for a one-dimensional viscoelastic medium the waveform will recover its original shape and will propagate with the equilibrium wave speed, provided that the equilibrium modulus exists and the duration of the input wave is long enough.

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Zusammenfassung Wir benutzen eine asymptotische Eigenschaft des Fourierintegrals, um zu zeigen, dass beim ein-dimensionalen viskoelastischen Körper jede Eingangswelle bald in ihrer ursprünglichen Gestalt wiedergewonnen und mit der Geschwindigkeit des Gleichgewichtszustandes propagiert wird, sobald man voraussetzt, dass der Gleichgewichtsmodul existiert und die Eingabezeit hinreichend lang ist.

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