Crystals and polycrystals,composites and polymers, grids and multibar systems can be considered as examples of media with microstructure. A characteristic feature of all such models is the existence of scale parameters which are connected with micro geometry or long-range interacting forces. As a result the corresponding theory must essentially be a nonlocal one. The book is devoted to a systematic investigation of effects of microstructure, inner degrees of freedom and nonlocality in elastic media. The propagation of linear and nonlinear waves in dispersive media, static problems, and the theory of defects are considered in detail. Much attention is paid to approximate models and limiting tran sitions to classical elasticity. The book can be considered as a revised and updated edition of the author's book under the same title published in Russian in 1975. The frrst volume presents a self-con tained theory of one-dimensional models. The theory of three-dimensional models will be considered in a forthcoming volume. The author would like to thank H. Lotsch and H. Zorsky who read the manuscript and offered many suggestions.
Table of Contents1 Introduction.- 2 Medium of Simple Structure.- 2.1 Simple Chain.- 2.2 One-Dimensional Quasicontinuum.- 2.3 One-Dimensional Quasicontinuum (continued).- 2.4 Equation of Motion and Elastic Energy Operator.- 2.5 Strain and Stress, Energy Density, and Energy Flux.- 2.6 Boundary Problems.- 2.7 The Dispersion Equation.- 2.8 Kernel of Operator ?? in the Complex Region.- 2.9 Green’s Function and Structure of the General Solution.- 2.10 Approximate Models.- 2.11 Solution of Basic Boundary Problems.- 2.12 Notes.- 3 Medium of Complex Structure.- 3.1 Basic Micromodels.- 3.2 Collective Cell Variables.- 3.3 Phenomenology.- 3.4 Acoustical and Optical Modes of Vibration. General Solution and Green’s Matrix.- 3.5 Long-Wave Approximation and Connection with One-Dimensional Analog of Couple-Stress Theories.- 3.6 Elimination of the Internal Degrees of Freedom in the Acoustic Region.- 3.7 Equivalent Medium of Simple Structure.- 3.8 Diatomic Chain.- 3.9 The Cosserat Model.- 3.10 Notes.- 4 Nonstationary Processes.- 4.1 Green’s Functions of the Generalized Wave Equation.- 4.2 Investigation of the Asymptotics Behavior.- 4.3 Decomposition into Packets and Factorization of Wave Equations.- 4.4 Energy Method and Quantum-Mechanical Formalism.- 4.5 Characteristics of the Evolution of a Packet.- 4.6 Superposition of Packets.- 4.7 Solutions Localized in the Neighborhood of Extrema of the Dispersion Curve.- 4.8 The Case of External Forces.- 4.9 Weakly Inhomogeneous Medium.- 4.10 Local Defects.- 4.11 The Structure of the Green’s Function of an Inhomogeneous Medium.- 4.12 The Scattering Matrix.- 4.13 Connection of the S-Matrix with Green’s Functions.- 4.14 Scattering on Local Defects.- 4.15 Notes.- 5 Nonlinear Waves.- 5.1 Korteweg-de Vries Model.- 5.2 Connection Between the KdV-Model and Nonlinear Wave Equation.- 5.3 Deformed Soliton.- 5.4 The Nonlinear Chain.- 5.5 Conservation Laws.- 5.6 Decay of the Initial Perturbation and the Distribution Function of Solitons.- 5.7 The Soliton Gas.- 5.8 Notes.- 6 Inverse Scattering Method.- 6.1 Basic Idea of the Method.- 6.2 Inverse Scattering Problem for the Operator L = d2/dx2 + u(x).- 6.3 N-Soliton Solutions of the KdV-Equation.- 6.4 Complete Integrability of the KdV-Equation.- 6.5 Shabat’s Method.- 6.6 N-Soliton Solutions for the Equation of Nonlinear String.- 6.7 The Toda Lattice.- 6.8 Fermi-Pasta-Ulam Problem.- 6.9 Perspectives of the Method.- 6.10 Notes.- Appendices.- 1. Summary of Fourier Transforms.- 2. Retarded Functions and Dispersion Relations.- 3. Expansion of Functions, Given at a Finite Number of Points, in Special Bases.- References.