On the question of rayleigh waves propagating along the surface of an inhomogeneous elastic body

The expression for the complex intensity of the Rayleigh wave in an inhomogeneous, elastic body is sharpened.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 20–23, 1986.     

Dispersion properties of nonstationary SV Rayleigh waves near the surface of an inhomogeneous elastic body

The physical characteristics of the surface of nonstationary SV Rayleigh waves are studied and formulas are obtained for the phase and group velocities (as functions of time and the observation point on the surface) of the Rayleigh waves. These formulas are in accord with the familiar formulas for the phase and group velocities in the stationary case.Translated from Zapiski Nauchnykh Seminarov Leningradzkogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 173, pp. 172–179, 1988.     

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.     

Love waves of SH type in an inhomogeneous transversely isotropic elastic medium

The asymptotics of high-frequency Love waves, which are analogous to transverse surface SH waves, is considered for a special type of anisotropy (transverse isotropy) of elastic media. The wave field is represented as a sum of the space-time (ST) caustic expansion and two additional ST ray series for faster (relative to the transverse surface wave) body waves, decaying exponentially with depth. Near the surface, the coefficients of the ST caustic and ray series, as well as the eikonals of waves, are determined in the form of expansions in a small parameter, which characterizes the proximity of the caustic of the ray field to the surface. With regard for the specific structure of the elasticity tensor of a transversely isotropic medium, the surface is treated as a plane. Interrelations between the parameters of elasticity, which are consistent with the conditions of the positivity of the elastic deformation energy and provide for the origination of the surface waves considered, are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 243–262.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Propagation of quasielastic Rayleigh waves in an inhomogeneous,anisotropic, elastic medium

The propagation of modulated oscillations along the smooth surface free from normal stresses of an inhomogeneous, anisotropic, elastic body with parameters which vary little over a characteristic wavelength is considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 89, pp. 234–245, 1979.The author thanks the participants of the seminar on the propagation and diffraction of waves of the Leningrad Branch of the Mathematics Institute (LOMI), in particular, V. S. Buldyrev and V. M. Babich, for fruitful discussions and their attention to the work.     

Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

A homogeneous elastic body with stress-free boundary is considered. The boundary of the body, which consists of a smooth cylindrical surface and a half-plane, has a continuous tangent plane, but the curvature of the normal section of the boundary has a discontinuity of the first kind at each point of the junction line. The behavior of two kinds of “whispering gallery” transversal surface waves at transition through the junction line is studied. For waves of the first kind (corresponding to Dirichlet boundary conditions), the displacement vector is normal to the boundary, whereas for waves of the second kind (corresponding to Neumann boundary conditions) the displacement vector is tangent to the boundary and normal to the ray, similarly to the case of Love waves. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 86–102. Translated by N. Ya. Kirpichnikova.     

On quasiphotons of Rayleigh waves (the case of an anisotropic elastic body)

A procedure of constructing analytic formulas for quasiphotons is developed. Quasiphotons are special asymptotic solutions of linear equations, which describe wave processes. These asymptotic solutions correspond to concentrated wave packets propagating along rays on the surface of an elastic body. Bibliography: 20 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 7–18.     

Polarization-spectral anomalies of surface waves in an almost layered elastic half space

The propagation of Love and Rayleigh waves in an isotropic elastic half space whose properties vary little in a horizontal direction is considered. Expressions are obtained for the admixed components of the displacements.Translated from Zapiski Naukchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 102–114.     

A film coating on a rough surface of an elastic body

A solution of the plane problem of the theory of elasticity for a film–substrate composite is solved by a perturbation method for a substrate with a rough surface. An algorithm for calculating any approximation, which ultimately leads to the solution of the same Fredholm equation of the second kind, is given. Formulae for calculating the right-hand side of this equation, which depends on all the preceding approximations, are derived. An exact solution of the integral equation in the form of Fourier series, whose coefficients are expressed in quadratures, is given in the case of a substrate with a periodically curved surface. The stresses on the flat surface of the film and on the interfacial surface are found in a first approximation as functions of the form of bending of the surface, the mean thickness of the film and the ratio of Young's moduli of the film and the substrate. It is shown, in particular, that the greatest stress concentration on the film surface occures on a protrusion of the softer substrate. ©2013     

The diffraction of surface waves by an elastic platform floating on shallow water

The scattering of long gravitational waves by a floating elastic plate is investigated using linear shallow-water theory. For a plate of arbitrary shape, the solution of the problem is reduced to a system of boundary integral equations. Using the example of a rectangular plate, the solution obtained is compared with existing theoretical and experimental results. The behaviour of the buckling of a rectangular plate and of a strip of constant width is compared for oblique incidence of surface waves.     

The effect of surface elasticity and residual surface stress in an elastic body with an elliptic nanohole

The deformation of an elastic plane with an elliptic hole in a uniform stress field is considered, taking into account the surface elasticity and the residual surface tension. The solution of the problem, based on the use of the linearized Gurtin–Murdoch surface elasticity relations and the complex Goursat–Kolosov potentials, is reduced to a singular integrodifferential equation. Using the example of a circular hole, for which an exact solution of the equation is obtained in closed form, the effect of the residual surface tension and the surface elasticity on the stress state close to and on the boundary of a nanohole is analysed for uniaxial tension. It is shown that the effect of the residual surface stress and the surface tension, due to deformation of the body, depends on the elastic properties of the surface, the value of the stretching load and the dimensions of the hole.