This book is devoted to selected problems of dynamics of nonhomogeneous linear elasticity. Wave propagation problems in a stratified half-space are considered. By a stratified medium we mean a body whose mechanical characteristics, such as elasticities and mass density, depend on one variable only (the depth of the half-space). A special case of such a body can be a layered medium. Excitations of propagating waves are due to different kinds of sources. In their book the authors mainly consider surface loading (punch problem) surface sources, linear and point sources, vibrating sources and internal sources. They also analyse different types of waves and transport energy.
The book is organized into eleven chapters: 1. Boundary value problems of dynamical elasticity of stratified media. 2. Questions of uniqueness and solvability of dynamical problems for a stratified half-space. 3. Methods of solving integral equations for mixed dynamical problems. 4. Methods of separations of singularities of solutions at corner points. 5. Vibrations of massive punches on an elastic foundation. 6. Analysis of wave fields excited by harmonic surface sources in elastic stratified half-space. 7. Nonstationary waves. 8. Energy of elastic waves excited by surface sources in stratified half-space. 9. Grouping of sources, forming of directional radiation. 10. Internal sources. 11. Dynamical problems of media with relief surfaces.
In the first chapter apart from the necessary preliminaries the construction of Green matrices for stratified half-space is presented in detail. The Green matrix is a crucial notion for the further study in this book. In the second chapter a system of integral equations for contact problem without friction (punch type) is derived. Mixed boundary conditions are taken into account. Moreover, the problem of uniqueness of solution for this system of integral equations is investigated. Chapter 3 contains a discussion on three methods for solving the problem of a vibrating punch on elastic foundation. Namely, factorization method, fictious absorption method and the variational-difference method are proposed to solve the formulated problems. The second method was created and developed by the first author and his coworkers [e.g.: with O. D. Priakhina, Prikl. Mat. Mekh. 45, 725-733 (1981; Zbl 0519.73096)]. All the discussed methods have some advantages and disadvantages which are pointed out in this chapter.
Chapter 4 consists of an analysis of the problem of separation of singularities and applies it to the following: wedge-shaped punch with friction, wedge-shaped punch with adhesive contact and the problem of stress singularities at crack tips. The Mellin transformation is used to solve this kind of problems. Chapter 5 deals with vertical and horizontal vibrations of punches, rectangular in their cross-sections. Numerical calculations and an analysis of the problem are given in detal. Chapter 6 is concerned with solving problems arising from studying wave fields excited by surface loadings. For the neighborhood of the source an integral representation of wave fields is derived and an asymptotic analysis is presented for the points far from the source. Also, harmonic loading moving with a constant velocity along the horizontal axis is studied.
In chapter 7 the problem of propagation of nonstationary impulses in the multilayered half-space is investigated. Asymptotic representation is used to solve the problem. Chapter 8 contains an energetic analysis for the case of an interaction between vibrating sources and an elastic medium. Representation for total power flux through an arbitrary horizontal surface, a lateral cylindrical surface and the lower semisphere with a large radius are discussed. Chapter 9 is devoted to some problems of control of a power flux, problems of an optimal distribution of sources and their interactions. The method of consideration of vertical nonhomogeneities, which is so important from the seismological point of view, is presented. Chapter 10 deals with an analysis of wave fields when sources are placed at some depth of the considered half-space. Sources are modelled by certain distribution of body forces and concentrated body forces. A method of construction of fundamental solution for the space and for the half-space is presented. The last chapter deals with boundary value problems of dynamics for unbounded convex domains with non-plane boundary. A new method of derivation of the boundary integral equation is presented, using the idea of factorization. For the plane boundary the formulated problem is reduced to the integral equation for the elastic half-space.
The book has very few references to western authors, but it is clearly written and may be of interest to seismologists, geophysicists, applied mechanicians and acousticians.