Higher-Order Asymptotic Approximations for Surface Love Waves of SH Type in Tranversely Isotropic Elastic Media

A uniform asymptotics of the surface Love modes for a special case of anisotropy (tranverse isotropy) of an elastic media is obtained. In constructing the asymptotics of surface waves, the space-time (ST) ray method is employed. The wave field of each Love mode is represented as the sum of the ST caustic expansion involving the Airy functions with a real eikonal and two correction terms that are ST ray solutions, which in fact are inhomogeneous waves with complex eikonals. The eikonals and coefficients of the caustic and ray series are sought in the form of expansions in powers of two variables. The first variable is the distance from the surface, whereas the other characterizes the proximity of the caustic of a ray field to the boundary surface. Thanks to the specific structure of the elasticity tensor for a transversely isotropic medium, the boundary surface is necessarily a plane. A recursion process of computation of higher terms of the asymptotic expansion allows one to trace the conversion of the formulas obtained to the known ray solutions for isotropic elastic media. Relations between the elasticity parameters of a medium are obtained that ensure the existence of SH Love waves in a transversely isotropic medium and that are consistent with the conditions of the positiveness of the elastic energy of deformation. Bibliography: 6 titles.     

Asymptotic expansion of the solution to the system of elasticity equations with a concentrated pulse force

For the linear system of elasticity equations, we consider the problem of wave excitation by a concentrated pulse force of arbitrary orientation. On assuming that the medium is isotropic and its density and elastic moduli are infinitely differentiable functions constant in some neighborhood of the source point, we write down an asymptotic series for the solution. The coefficients of the series determine the singular component of the solution, as well as the jumps, of the solution and its derivatives, as the characteristic cones are crossed, corresponding to longitudinal and transverse waves.     

Propagation of Rayleigh waves of sv type in transversely isotropic elastic media

The asymptotics of high-frequency surface waves in elastic media is studied for a special case of anisotropy, namely, for transversely isotropic media (where the parameters of elasticity are invariant with respect to rotations about one of the coordinate axes). In the zeroth asymptotic approximation, the slow Rayleigh waves (of SV type) under study are polarized in the plane of the normal section of the surface. The principal term of the asymptotics (which has the form of a space-time (caustic) expansion) is found, and calculations related to the necessity of introducing two additional faster waves with complex eikonals are carried out. The conditions on the elasticity parameters of the medium that insure the origination of the surface waves in question are obtained. Due to the specific structure of the elasticity tensor under consideration, the boundary of the medium is necesarily plane. For appropriate values of elastic parameters, the resulting formulas coincide with the corresponding expressions in the isotropic case. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 278–293. Translated by Z. A. Yanson     

弹性波在平面多连通域中的绕射与动应力集中

本文用复变函数论方法研究了弹性波在平面多连通域中的绕射问题,给出了这一问题解的完备逼近序列及边备条件的一般表示。问题归结为无穷代数方程组的求解,使用电子计算机可直接求得解答。特别是,对弱耦合问题,本文提出了渐近求解方法并且使用这个方法详细地讨论了P波对圆孔群的绕射问题。基于绕射波场的解,文中给出了任意形状空腔动应力集中系数的一般算式。     

Long-wavelength approximation in scattering of elastic waves

We construct the asymptotic form of the solution in the long-wavelength limit for the problem of scattering of plane waves in an elastic medium by a cavity or rigid nonmoving inclusion. The parameters determining the scattered wave at large distances in the first approximation are expressed in terms of the integrated characteristics of the scatterer, such as its volume, the tensor analog of the capacity, the Wiener capacity in the two-dimensional case, and the dipole elastic tensor. Some of these characteristics are new.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 6–19, 1986.     

Approximate Description of the Dynamics of Thin Isotropic Elastic Coatings and Interlayers by Using Asymptotics of High Order of Accuracy

Asymptotically accurate low-frequency models for isotropic elastic coatings and interlayers are developed. The main constraint is the requirement on contact conditions for the layer and the base that at least one of the boundary conditions must include the displacement component in an explicit form. The displacement and stress fields in the three-dimensional elastic system are sought in the form of asymptotic expansion into power series of a small parameter — the ratio between the half-thickness of the layer and the minimum length of the wave in the longitudinal direction. The action of the layer is approximated by impedance boundary conditions, which are transferred to the contact surface with the basic, more thick body. These conditions are obtained with an asymptotic error up to and including the sixth order of magnitude. A numerical testing, which is carried out with the example of partial waves, shows a satisfactory accuracy, comparable with that of high-order theories of plates. The results obtained can be utilized in fast algorithms for calculating spectra of natural waves in half-spaces, thick laminated plates, and shallow shells with coatings and interlayers. The physical limit of applicability of the theory developed is the frequency of the first quasi-resonance in the corresponding deformable system. The number of alternating interlayers is unlimited. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 783–794, November–December, 2005.     

Finite expressions for the characteristic matrices of weakly curved elastic layers

The propagation of two-dimensional waves in cylindrical and spherical, homogeneous, elastic layers is investigated. For these layers finite formulas are found for the characteristic matrices. Comparison of these matrices and use of asymptotic representations for the Hankel functions make it possible to derive expressions in the case of weakly curved elastic layers. The expressions obtained correspond to analogous formulas in the form of matrix series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 156–169, 1981.     

The diffraction of plane elastic unsteady waves by a delaminated inclusion in the case of smooth contact in the delamination region

A solution of the problem of the diffraction of unsteady elastic waves by a thin strip-like delaminated rigid inclusion in an unbounded elastic medium under conditions of planer strain is proposed. We have in mind an inclusion, one side of which is completely bonded with the medium while, the other side is delaminated and conditions of smooth contact are satisfied on it. The method of solution is based on the use of discontinuous solutions of the Lamé equations of motion under conditions of planer strain, which have been constructed earlier in the space of Laplace transforms. As a result, the problem reduces to solving a system of three singular integral equations for the transforms of the unknown discontinuities. The inverse transforms are found by a numerical method, based on the replacement of a Mellin integral by a Fourier series.     

An Inner Source in a Transversely Isotropic Elastic Medium

For a homogeneous, transversely isotropic elastic medium excited by a point force acting on the isotropy plane, an exact displacement field is constructed. This field decomposes into P-SV waves and SH waves. For SH waves satisfying the Lamé equations, boundary conditions are established and rather simple expressions are derived. These expressions are compared with the first terms of the ray series. Bibliography: 9 titles.     

Radiation problem of normal stress loading of bore surface

The radiation of elastic waves at normal harmonic stress loading of a ring strip of the cylindrical bore-surface in the infinite elastic medium is considered. The wave field is represented by contour integrals. By using the saddle-point method stress and displacement of asymptotic expansions in the far field were obtained. It is shown that in the far field P-waves bring radial displacements and stresses, and perpendicular to these, displacements and stresses are caused by S-waves. In the paper orientation diagrams of radial and circular displacements and frequency dependencies of relative distribution of radiation power source on types of waves are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Vibrations of a cylinder in a concentric vessel filled with a two-layer fluid

A hybrid vibrational system containing a solid (a cylinder) with an elastic connection to a coaxial cylindrical cavity, completely filled with a heavy ideal stably stratified two-layer fluid, is considered. The combined self-consistent vibrations of the body and the fluid (of the internal waves) are studied. An explicit solution of the internal boundary value problem of an inhomogeneous liquid in an annular domain for a specified motion of the body is obtained. An integrodifferential equation of the Newton type is constructed on the basis of this. This equation describes the self-consistent oscillations of the cylinder. In the case of weak coupling of the interaction between the solid and the medium, an approximate solution is obtained using asymptotic methods and an analysis is carried out. Qualitative effects of the mutual effect of the motions of the cylinder and the fluid are found.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

Surface waves supported by thin-film/substrate interactions

A systematic approximation to the linear equations for small-amplitudesurface waves in an elastic half space, interacting with a residuallystressed thin film of different material bonded to its planeboundary, is developed in powers of the film thickness, assumingthe latter to be small compared to the wavelength of the disturbance.The theory is illustrated by calculating asymptotic expansionsof the wave speeds for Love and Rayleigh waves valid to secondorder in the dimensionless film thickness for a transverselyisotropic film bonded to an isotropic substrate.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

Convergent Power Series for Fields in Positive or Negative High-Contrast Periodic Media

We obtain convergent power series representations for Bloch waves in periodic high-contrast media. The material coefficient in the inclusions can be positive or negative. The small expansion parameter is the ratio of period cell width to wavelength, and the coefficient functions are solutions of the cell problems arising from formal asymptotic expansion. In the case of positive coefficient, the dispersion relation has an infinite sequence of branches, each represented by a convergent even power series whose leading term is a branch of the dispersion relation for the homogenized medium. In the negative case, there is a single branch.     

Surface Waves on a Cavity in a Semiinfinite Elastic Medium

Biot [5] examined the propagation of waves along the free surface of a cylindrical cavity in an elastic body of infinite extent and obtained a dispersion relation for the velocity of this wave in terms of the ratio of the wavelength to the cavity diameter. This paper contains solutions for waves in a semiinfinite elastic medium with a cylindrical cavity with axially symmetric harmonic loading of the plane surface. The solutions are expressed in terms of Lame potentials which are represented by combinations of integrals containing trigonometric kernels and kernels of Weber transforms. A solution is obtained for volume waves and Biot waves. The relative velocity and relative length of surface waves are studied as functions of the loading frequency.     

Three-scale asymptotics for a diffusion problem coupled with the wave equation

In this article, we present an asymptotic analysis of waves of elastic stress in an infinite solid whose boundary is subject to a rapid thermal load. The problem under consideration couples the wave equation and the heat equation, and the asymptotic approximation of the solution requires three-scaled variables. The asymptotic approximation is supplied with a rigorous remainder estimate and is illustrated numerically.     

Asymptotic study of the elastic seadbed effects in ocean acoustics

A theoretical and asymptotic investigation of the Green' function for the system governing the propagation of time-harmonic acoustic waves in a horizontally stratified ocean with an elastic seabed is presented. Employing the surface Neumann-to-Dirichlet map for the elastic half space, we reduce the problem to an equivalent one in the layer, with a nonlocal boundarycondition at the fluid-bottom interface. The reduced problem is transformedby Hankel transform, to a non-selfadjoint boundary value problem for a second-order ordinary differential equation over the layer depth. The well posedness of this problem is investigated applying analytic Redholm theory for an equivalent Lippmann-Schwinger integral equation. An asymptotic expansionof the transformed nonlocal boundary condition is constructed in the case of a seabed with small shear modulus, and it is used to show that the Green function is a regular perturbation of that one in the case of a fluid bottom.     

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.