The second approximation of the ray method in calculating the intensity of the ray front in an inhomogeneous elastic medium

An algorithm for calculating the second term of the ray series for the vector of longitudinal displacements in an inhomogeneous isotropic elastic medium in the quasi-two-dimensional case is suggested. This algorithm is based on expanding the medium parameters, the eikonal, and the displacement vector into series in powers of the coordinates transversal to the ray.Translated fromZapiski Nauchnykh Semmarov POMI, Vol. 239, 1997, pp. 79–94.This work was supported in part by the State Committee of the Russian Federation for Higher Education under grant No. 95-0-13.1-66.     

On calculating the second term of the series of the ray method for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media

This paper contiues the series of papers devoted to designing algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. The method proposed in this paper essentially differs from the previous ones. It is based on expanding the amplitudes, the eikonal, and the given problem parameters into power series with respect to the coordinates transversal to the direction of propagation of the waves considered. The methods for computing the correction term of the ray expansion for the vector of longitudinal displacements are compared. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 25–43.     

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.     

A procedure for computing successive approximations in the asymptotic space-time (ST) ray method of Gaussian beams. II. (An isotropic inhomogeneous elastic medium)

The method of spatial rays is applied in the construction of the formulas for the asymptotic expansions of the Gaussian beam type in an elastic medium. The coefficients for higher-order approximations of the expansions constructed are subject to a recurrent process. The process is considered for theP (orS) component of the displacement vector in an inhomogeneous elastic medium taken as an example.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 88–100, 1989.     

The ray method in an inhomogeneous chiral medium and admixed components of an electromagnetic field

In this paper, a complete asymptotic expansion of a high-frequency electromagnetic field in chiral media is constructed. Recurrence formulas for the coefficients of the expansion are found. Polarizations of electromagnetic waves are found and explained in the zero approximation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 62–72. Translated by V. S. Buldyrev.     

High-frequency point sources in an inhomogeneous elastic medium

Higher approximations of ray asymptotics are investigated by the boundary-layer method. For sources that are naturally called a center of pressure and a center of rotation, direction diagrams are found for transverse and longitudinal waves, respectively, which are absent in a homogeneous medium.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 28–42, 1980.     

Reconstruction of cracks in an inhomogeneous anisotropic elastic medium

In this paper we give in two and three dimensions a reconstruction formula for determining cracks buried in an inhomogeneous anisotropic elastic body by making elastic displacement and traction measurements at the boundary. The information is encoded in the local Neumann-to-Dirichlet map. With the help of the Runge property, the local Neumann-to-Dirichlet map is connected to the so-called indicator function. This function can be expressed as an energy integral involving some special solutions, called reflected solutions. The heart of our method lies in analyzing the blow-up behavior at the crack of the indicator function, which is by no means an easy task for the inhomogeneous anisotropic elasticity system. To overcome the difficulties, we construct suitable approximations of the reflected solutions that capture their singularities. The indicator function is then analyzed by the Plancherel formula.     

Computation of the second term of the series of the ray method in the dimension two and a half

An algorithm for computing the second term of the series of the ray method suggested in this paper is specified for the case of the dimension two and a half. By this is meant that the elastic paramenters of a medium depend on x, z only and the source and observation points are placed in a plane orthogonal to the y axis. Thus, to compute the wave field we need to take into account the rays which are placed in the plane indicated. This enables us to simplify the algorithm. We discuss the algorithm in detail for the case of a point source. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp 94–108 Translated by N. Ya. Kirpichnikova.     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

The oscillations of a rod in an inhomogeneous elastic medium

The local length-dependence of the natural frequencies and forms of plane transverse oscillations of a thin inhomogeneous rod in an elastic medium with a variable stiffness and arbitrary elastic-fastening boundary conditions is investigated. It is established that the presence of an external elastic medium, described by the Winkler model, can lead to an anomalous effect – an increase in the natural frequencies of lower oscillation modes as the length of the rod increases continuously. The extremely fine properties of this change as a function of the length, the mode number and the method of fastening are revealed. The oscillations in the case of standard methods of fastening are investigated separately. Simple examples, which illustrate the anomalous dependence of the natural oscillation frequencies of the rod in an extremely inhomogeneous elastic medium with different boundary conditions are calculated.     

A solution method for the sub-surface stresses and local deflection of a semi-infinite inhomogeneous elastic medium

This paper proposes analytical Fourier series solutions (based on the Airy stress function) for the local deflection and subsurface stress field of a two-dimensional graded elastic solid loaded by a pre-determined pressure distribution. We present a selection of numerical results for a simple sinusoidal pressure which indicates how the inhomogeneity of the solid affects its behaviour. The model is then adapted and used to derive an iterative algorithm which may be used to solve for the contact half width and pressure induced from contact with a rigid punch. Finally, the contact of a rigid cylindrical stamp is studied and our results compared to those predicted by Hertzian theory. It is found that solids with a slowly varying elastic modulus produce results in good agreement with those of Hertz whilst more quickly varying elastic moduli which correspond to solids that become stiffer below the surface give rise to larger maximum pressures and stresses whilst the contact pressure is found to act over a smaller area.     

A regularization method for stationary maxwell equations in an inhomogeneous conducting medium

We consider the problem of defining a vector potential of a magnetic field with a nonstandard calibration in an inhomogeneous conducting medium. The problem in question is one with constraints on the right-hand side and on the solution itself. A generalized and regularized statement of this problem without constraints is proposed and substantiated. Such a problem statement is equivalent to the original generalized problem with constraints.     

Space-time ray method for waves of small deformation in a nonlinear elastic medium

Space-time ray solutions for longitudinal waves of small deformation in a non-linear elastic medium are constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 61–72, 1984.     

Space-time ray method of calculating waves in a weakly inhomogeneous,layered medium

A space-time ray method is presented for constructing the fields of Love and Rayleigh waves for a half space with parameters that vary slowly in a horizontal direction, while the free surface and boundaries of the layers are weakly curved. A surface wave is interpreted here as a harmonic wave packet with amplitude and frequency modulation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 5–18, 1980.     

Numerical realization of an algorithm for reconstructing an inhomogeneous medium in an X-ray tomography problem

Under study is the X-ray tomography problem that is an inverse problem for the transport differential equation. We take into account the absorption of particles by the medium and their single scattering. The statement of the problem corresponds to multiple probing. The medium is unknown; while the densities of the outcoming flux averaged over energy are given. The object in question is the discontinuity surfaces of the coefficients of the equation. This corresponds to searching for the boundaries between various substances contained in the medium that we probe. The solution is constructive, and a numerical realization of the obtained algorithm is presented.     

Regularization method for solving the quasi-stationary Maxwell equations in an inhomogeneous conducting medium

Nedelec vector finite elements are used for the numerical solution of a regularized version of the quasi-stationary Maxwell equations written in terms of a scalar and a vector magnetic potential with special calibration taking into account the conductivity of the medium. An optimal energy estimate for the error of the approximate solution in Lipschitz polyhedral domains is established. Numerical results are presented that demonstrate the stability of the method.     

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.     

Fundamental solution of displacement equations for a transversely isotropic elastic medium

A fourth-order linear elliptic partial differential equation describing the displacements of a transversely isotropic linear elastic medium is considered. Its symmetries and the symmetries of an inhomogeneous equation with a delta function on the right-hand side are found. The latter symmetries are used to construct an invariant fundamental solution of the original equation in terms of elementary functions.