Excitation of modulated oscillations in inhomogeneous media

The excitation of scalar and elastic waves by stationary sources is considered. The method of the boundary layer is used. Construction of higher-order approximations is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 111–122, 1981.     

Diffusion in inhomogeneous media: Localization and positivity

Summary We study nonnegative solutions of the Emden-Fowler equation on R⁺ in the sub-linear case. The results are used to investigate localization and positivity properties for a class of degenerate parabolic problems.     

Propagation and scattering of kinks in inhomogeneous nonlinear media

The model of kink scattering in inhomogenous nonlinear media is considered. The existence of kink bound states oscillating periodically and chaotically in time is shown. Translated from Teoreticheskaya i Mathematicheskaya Fizika, Vol. 112, No. 3, pp. 384–394, September, 1997.     

Non-stationary problems of generalized elastothermodiffusion for inhomogeneous media

The method of investigation of non-stationary boundary value problems of the theory of thermodiffusion using the Laplace integral transform is described. In the classical theory of elasticity this method was first used by V. Kupradze and the author.     

The scattering of classical waves from inhomogeneous media

By extending Kato's theory of two Hilbert space scattering, we are able to formulate both optical and accoustical scattering from inhomogeneous media as strictly elliptic problems. We use this formulation to present simple proofs of the existence and completeness of scattering states.     

Some problems of the correlation theory of structurally inhomogeneous elastic media

The correlation theory of random functions is used to describe structurally inhomogeneous elastic media with randomly distributed inhomogeneities. Expressions are obtained for the elastic moduli of averaged layered, fibrous and granular media.Mekhanika Polimerov, Vol. 2, No. 3, pp. 365–371, 1966     

Evolution of a liquid-liquid interface in inhomogeneous layers

The basic system of integral and differential equations is obtained that describes the evolution of a liquid-liquid interface in inhomogeneous ground layers such that the square root of their conductivity is described by a metaharmonic function. A discrete scheme for the basic system of equations is constructed using the method of discrete vortex pairs.     

Evolution of perturbations on a weakly inhomogeneous background

The evolution of growing and decaying one-dimensional linear perturbations on a stationary, weakly inhomogeneous background is investigated studied. Attention is focused on the amplification of waves that arise from initial perturbations, localized in regions whose width is small compared with the inhomogeneity scale. A relation between the Hamiltonian formalism (with a complex dispersion equation) and the saddle-point method is established for an asymptotic representation of the integral that expresses perturbations in terms of the initial data. Model examples of the evolution of perturbations are examined.     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.     

The efficient solution of electromagnetic scattering for inhomogeneous media

We consider an electromagnetic scattering problem for inhomogeneous media. In particular, we focus on the numerical computation of the electromagnetic scattered wave generated by the interaction of an electromagnetic plane wave and an inhomogeneity in the corresponding propagation medium. This problem is studied in the VV polarization case, where some special symmetry requirements for the incident wave and for the inhomogeneity are assumed. This problem is reformulated as a Fredholm integral equation of second kind, which is discretized by a linear system having a special form. This allows to compute efficiently an approximate solution of the scattering problem by using iterative techniques for linear systems. Some numerical examples are reported.     

Nonlinear periodic deformation of thixotropic media

The experimental procedure and the processing of results on the nonlinear periodic deformation of polymeric and dispersed system melts are examined. Experimental data are given on the effect of the amplitude of harmonic deformation on nonlinear stress distortions.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 689–696, July–August, 1972.     

Simulation of the deformation processes of nonlinear viscoelastic media

Questions of the similarity and the possibility of simulating the deformation processes of nonlinear viscoelastic medía are investigated. Systems of characteristic parameters are given. It is established that simulation based on the same material is possible only if the characteristic time of the process is the same and the stresses and strains at similar points of the model and the simulated object are equal. The characterstic time for a model made of different material is different from that for the simulated object.Mekhanika Polimerov, Vol. 3, No. 2, pp. 221–226, 1967     

Transient response of inhomogeneous thermoelastic media to a dynamic input

A problem involving a half-space or an infinite space with a cylindrical or spherical cavity subjected to a uniform dynamic input on its surface is studied. The material is elastic and has an inhomogeneity in the direction perpendicular to the boundary surface for the half-space problem and an inhomogeneity in the radial direction for the problem with a cylindrical or spherical cavity. The numerical results, obtained when the dynamic input is a step pressure, indicate that the wave profiles are influenced by two kinds of dispersions: one caused by thermal effects and the other by inhomogeneity. The influence of thermal dispersion is observed more pronouncedly along the initial portion of wave profiles. Compared to the homogeneous case the inhomogeneities in which Lamé's constants increase and decrease with the distance measured from the boundary have respectively amplifying and attenuating effects on both the axial stress and velocity.

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Zusammenfassung Das Problem der gleichmässigen dynamischen Belastung an den Grenzen eines halbunendlichen Körpers oder eines unendlichen Körpers mit einem zylindrischen oder sphärischen Halbraum wurde untersucht.Der Körper ist elastisch und enthält Inhomogenitäten, im halbunendlichen Körper in der Richtung senkrecht zur Grenzebene und im unendlichen Körper mit Hohlräumen in radialer Richtung. Die numerischen Ergebnisse, die für eine Belastung nach einer Schritt-Funktion erzielt wurden, zeigen dass die Profile der Ausbreitungswellen von zwei Streuungen beeinflusst sind: die erste ist eine Folge der thermischen Effekte, die zweite eine solche der Inhomogenitäten. Der Effekt der thermischen Zerstreuung ist im anfänglichen Bereich der Profile stärker. Im Vergleich zum homogenen Fall hat die Zu-beziehungsweise die Abnahme der Lamé'schen Konstanten mit dem Abstand von der Grenze erhöhenden beziehungsweise vermindernden Einfluss sowohl auf die axiale Spannung als auch auf die Geschwindigkeit.

     

Stress distribution near a reinforced elliptical hole in a spherical shell at the elastoplastic stage of deformation

On the basis of Lagrange's variational equation the authors obtained nonlinear resolvent equations and coefficients taking into account the effect of a reinforcing element (a rod) on the state of stress and strain of a spherical shell weakened by a curvilinear (elliptical) hole. The article explains the method of numerical investigation of the inelastic state of the shell based on the application of the variational difference method in combination with the method of elastic solutions. The inelastic state of a shell with a reinforced hole was numerically investigated.Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 80–83, 1990.