Elastic Media with Microstructure I

1 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 andNonlinear 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.