Propagation of Seismic Waves in Block Elastic-Fluid Media. II

A two-dimensional medium consisting of alternating elastic and fluid blocks along the x and z axes is considered. For this block medium, an effective model described by a system of equations is constructed by the method of matrix averaging. An investigation of the equations of this model enables one to separate two body waves from the wave field, to construct their fronts, and to obtain expressions for their velocities along the axes. The effective model is considered in the cases where the block medium is converted to a layered elastic-fluid medium, where all the blocks are of the same size, and where an elastic or a fluid medium occupies the entire volume. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 254–271.     

Propagation of Seismic Waves in Block Elastic-Fluid Media. I

An approach of averaging block elastic-fluid media is proposed, and an effective model for a block medium in which every cell consists of three elastic blocks and one fluid block is constructed. An investigation of the model equations shows that in this model two longitudinal waves and one wave with a concave front set propagate. The limiting cases where the fluid block is narrowed down to a point or where the fluid block occupies the whole cell are considered in the paper. Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 230–253.     

An effective model of a porous-fluid medium

For a medium containing alternating porous Biot layers and fluid layers, an effective model is established by the method of matrix averaging. An investigation of equations of this effective model shows that the wave field consists of a leading front and two triangular fronts. The velocities of these fronts along the axes are determined. If the thicknesses of the fluid layers are very small, then the second triangular front is converted into a back concave front, and a slow wave arises. This slow wave is of interest for seismology. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 190–211.     

Homogenizing the acoustic properties of a porous matrix containing an incompressible inviscid fluid

We undertake a rigorous derivation of the Biot's law for a porous elastic solid containing an inviscid fluid. We consider small displacements of a linear elastic solid being itself a connected periodic skeleton containing a pore structure of the characteristic size ε. It is completely saturated by an incompressible inviscid fluid. The model is described by the equations of the linear elasticity coupled with the linearized incompressible Euler system. We study the homogenization limit when the pore size εtends to zero. The main difficulty is obtaining an a priori estimate for the gradient of the fluid velocity in the pore structure. Under the assumption that the solid part is connected and using results on the first order elliptic systems, we obtain the required estimate. It allows us to apply appropriate results from the 2‐scale convergence. Then it is proved that the microscopic displacements and the fluid pressure converge in 2‐scales towards a linear hyperbolic system for an effective displacement and an effective pressure field. Using correctors, we also give a strong convergence result. The obtained system is then compared with the Biot's law. It is found that there is a constitutive relation linking the effective pressure with the divergences of the effective fluid and solid displacements. Then we prove that the homogenized model coincides with the Biot's equations but with the added mass ρ_a being a matrix, which is calculated through an auxiliary problem in the periodic cell for the tortuosity. Furthermore, we get formulas for the matricial coefficients in the Biot's effective stress–strain relations. Finally, we consider the degenerate case when the fluid part is not connected and obtain Biot's model with the relative fluid displacement equal to zero. Copyright © 2003 John Wiley & Sons, Ltd.     

Investigation of the wave field in an effective model of a layered elastic-fluid medium

For a medium that consists of alternating elastic and fluid layers, an effective model is constructed and investigated. This model is a special case of the Biot medium. The wave field is represented as Fourier and Mellin integrals. In the Mellin integral, the contour of integration is replaced by a stationary contour. In the expressions obtained, the order of integration is changed and the inner integral is calculated. The outer integral is equal to two residues. The corresponding poles are roots of two equations of fourth order. These roots lie on the right half-plane and may be complex or real. The representation obtained for the wave field is in agreement with the expressions derived by the Smirnov-Sobolev method. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 175–192.     

Attenuation of waves propagating in fluid mixtures

Wave propagation in fluid mixtures is investigated on the basis of effective models of block and layered media. These models are anisotropic fluids described by wave equations. In the equations, additional terms describing wave attenuation are introduced. The attenuation is related to a friction force proporitional to the difference of tangent displacements on the boundaries. Owing to attenuation, the total energy of the wave field decreases steadily and the amplitudes of waves are diminished expotentially with time, which is determined by attenuation coefficients. The attenuation coe.cients are found in the cases where two fluids are mixed completely and where the particles of one fluid are inclusions into the other. The approach suggested enables one to consider more complicated fluid mixtures as well. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 148–179.     

Acoustics equations in poroelastic media

We study the acoustics equations in poroelastic mediawhich were obtained by the author previously in result of homogenization of the exact dimensionless equations describing the joint motion of an elastic solid skeleton and a viscous fluid in the pores on the microscopic level. A small parameter in this model is the ratio ɛ of the average size l of the pores to the characteristic size L of the physical region under consideration. The homogenized equations (the limit regimes of the exact model as ɛ tends to zero) depend on the dimensionless parameters of the model, which depend on the small parameter, and are small or large quantities as ɛ tends to zero. On assuming that the solid skeleton is periodic, we analyze the concrete form of acoustics equations for the simplest periodic structures.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Investigation of normal waves in a porous layer surrounded by elastic half-spaces

The wave field and dispersion equations are found for a porous layer surrounded by two elastic half-spaces. The porous layer is described by the effective model of a medium in which elastic and fluid layers alternate. To investigate the normal waves, all real roots of dispersion equations are determined and their movements as the wave number increases are investigated. As a result, the dispersion curves of all normal waves are constructed and the dependence of normal waves on the parameters of the porous layer and elastic half-spaces is analyzed. Bibliography: 6 titles.     

On methods of deriving equations describing effective models of layered media

Methods of deriving equations describing effective models of layered periodic media are presented. Elastic and fluid media, as well as porous Biot media, may be among these media. First, effective models are derived by a rigorous method, and then some operations in the derivation are replaced by simpler ones providing correct results. As a consequence, a comparatively simple and justified method of deriving equations of an effective model is established. In particular, this method allows us to simplify to a degree and justify the derivation of an effective model for media containing Biot layers; this method also produces equations of an effective model of a porous layered medium intersected by fractures with slipping contacts. Bibliography: 15 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998 pp. 219–243. Translated by L. A. Molotkov.     

Equations of an effective model of an anisotropic elastic medium with cracks filled with a fluid

For a periodic layered medium in which every period consists of an elastic anisotropic layer and a fluid homogeneous layer, an effective model is derived by averaging. This model describes wave propagation and has two phases. The equations of this model are deduced in the case of the general anisotropy and in some special cases. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 175–191. Translated by L. A. Molotkov.     

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT

We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Effective Model of a Layered Medium in Which Alternating Porous and Elastic Layers Are in Slide Contact

For a medium in which porous and elastic layers alternate and there is slide contact on the interfaces, an effective model is established. This model is of three phases and includes two elastic phases and one fluid phase. Specific features of this effective model are that two waves with triangular front sets propagate and the second (slow) longitudinal wave is absent in it. In the special case where the thickness of elastic layers is very small but they continue to work as barriers for fluid particles from porous layers, the effective model is of two phases, and one of the triangular front sets disappears. Bibliography: 14 titles.     

Investigation of the wave field in an effective model of layered elastic media with slide contact on the interfaces

An effective model of media consisting of two alternating elastic layers with slide contact on the boundaries is investigated. The wave field in this model is represented as Fourier and Mellin integrals. In the Mellin integrals we replace the contours of integration by stationary contours. In the expressions obtained, we change the order of integration and calculate the inner integral. The outer integral is equal to two residues. The corresponding poles are roots of two equations of the sixth order. These roots may be situated in the right half-plane and may be complex or real. The representation obtained of the wave field corresponds to expressions derived by the Smirnov-Sobolev method. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 187–205.     

The coupling system of Navier–Stokes equations and elastic Navier–Lame equations in a blood vessel

In this paper, the blood flow problem is considered in a blood vessel, and a coupling system of Navier–Stokes equations and linear elastic equations, Navier–Lame equations, in a cylinder with cylindrical elastic shell is given as the governing equations of the problem. We provide two finite element models to simulating the three-dimensional Navier–Stokes equations in the cylinder while the asymptotic expansion method is used to solving the linearly elastic shell equations. Specifically, in order to discrete the Navier–Stokes equations, the dimensional splitting strategy is constructed under the cylinder coordinate system. The spectral method is adopted along the rotation direction while the finite element method is used along the other directions. By using the above strategy, we get a series of two-dimensional-three-components (2D-3C) fluid problems. By introduce the S-coordinate system in E³ and employ the thickness of blood vessel wall as the expanding parameter, the asymptotic expansion method can be established to approximate the solution of the 3D elastic problem. The interface contact conditions can be treated exactly based on the knowledge of tensor analysis. Finally, numerical test shows that our method is reasonable.     

Free and nonsteady-state oscillations of elastic fluid-filled systems situated on an elastic base

We develop a method of computing the nonsteady-state and free oscillations of a framed elastic structure situated on an elastic base and containing an ideal compressible fluid. The solution uses the method of integral transforms in conjunction with the method of orthogonal polynomials. In the transform space the problem reduces to systems of linear algebraic equations. The Fourier transform is applied to return to the original space. Examples of the computation are given.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 69–72.     

Analyzing the harmonic wave propagation in the layered thick tubes

In this study, the propagation of time harmonic waves in prestressed, anisotropic elastic tubes filled with viscous fluid is studied. The fluid is assumed to be incompressible and Newtonian. A two layered hyperelastic anisotropic structural model is used for the compliant arterial wall. The tube is subjected to a static inner pressure P_i and an axial stretch λ. The governing differential equations of tube are obtained in cylindrical coordinates, utilizing the theory of “Superposing small deformations on large initial static deformations”. The analytical solutions of the equations of motion for the fluid have been obtained. Due to variability of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch and material parameters.     

Attenuation of the interference slow wave in a cracked layer

The attenuation of the interference slow wave propagating in a cracked fluid-filled layer sandwiched between two elastic half-spaces is evaluated. It is proposed that the attenuation is induced by the inelasticity of the fluid. It is shown that the attenuation of the wave may be very small if the thickness of the layer is much less than the wavelength. This explains the possibility of observing the slow wave in real field seismic experiments. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 153–160. Translated by P. V. Krauklis.     

The equations of the effective two-phase cracked model with finite length cracks

The two-phase effective model of a cracked medium with cracks filled with liquid is extended to the case where cracks have finite length. On the basis of the equations obtained for the generalized effective model, the expressions of kinetic and potential energies and energy streams along axes are derived. This also shows that attenuation is absent. bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 142–153, 1990. Translated by L. A. Molotkov.